IMPULSIVE CONTROL OF UNDERACTUATED MECHANICAL SYSTEMS
By
Sayyed Rouhollah Jafari Tafti

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
DOCTOR OF PHILOSOPHY
Mechanical Engineering
2012

ABSTRACT
IMPULSIVE CONTROL OF UNDERACTUATED MECHANICAL
SYSTEMS
By
Sayyed Rouhollah Jafari Tafti
Although there has been a significant amount of research in designing and analyzing
impulsive control systems, there are very few applications of impulsive control in mechanical systems. In the first part of this dissertation, we investigate new control strategies for
underactuated mechanical systems based on impulsive inputs. The control problem of underactuated systems is more challenging since such systems have fewer actuators than the
number of their degrees of freedom. We first address the important concern related to application of impulsive control in mechanical systems, namely, implementation of impulse-like
control inputs using standard hardware. This is done through experimental verification of an
impulsive control algorithm for swing-up control of the Pendubot; the control algorithm was
developed earlier in our research group. Showing the effectiveness of the impulsive control
algorithm in experiments, we develop impulsive control algorithms for swing-up control of
the Acrobot; the impulsive control algorithms have distinct advantages over existing algorithms in terms of the time required for swing-up and maximum control torque used by the
continuous controller. Impulsive inputs cause jumps in velocity states of mechanical systems,
and consequently, produce jumps in Lyapunov function candidates used in control design.
This attribute is used to enlarge the region of attraction of equilibria using impulsive inputs
at discrete instants of time. Several case studies of underactuated mechanical systems have
been presented to demonstrate this benefit of using impulsive control. Another advantage of
using impulsive inputs is that such inputs can significantly alter the dynamics of the system

in a very short period of time. This property is used to design a safe fall algorithm for
humanoid robots undergoing a fall, i.e., after the continuous controller has failed to keep
the system trajectories confined to a fixed region around the equilibrium. The algorithm
uses impulsive inputs to change the fall direction of the robot to minimize the damage to
people and objects in the vicinity, as well as to its own self. In many instances, external
disturbances or impact from interaction with the environment can have an adverse effect on
system performance. In the second part of this dissertation, we develop control algorithms
to mitigate these effects in underactuated biped robots. We first develop a disturbance
rejection algorithm for the synthetic-wheel biped to mitigate the effects of external disturbances. A continuous controller is then designed for the synthetic-wheel biped to generate
an impact-free walking gait. This gait consumes zero energy in the ideal case and the necessary conditions to achieve this gait are shown to be general and applicable to a range of
bipedal robotic systems. An underactuated mechanical system can be non-minimum-phase
if its zero dynamics is unstable. We finally investigate the output-tracking problem for linear
non-minimum-phase systems. Intermittent output tracking is achieved under the condition
of finite preview of the reference trajectory; the control algorithm uses switched inputs, which
can be approximated by impulsive inputs in the limit.

This dissertation is lovingly dedicated to my family, especially:
to my wife, Hamideh, without her constant love, endless patience and deep understanding I
could not accomplish this
to my parents, Mahmoud and Sedigheh, whose support and encouragement have sustained
me throughout my life.

iv

ACKNOWLEDGMENTS

I would like to express my sincere gratitude to Dr. Ranjan Mukherjee for all his support,
expertise and dedication throughout my research at the Dynamic Systems and Control Laboratory at Michigan State University. His passion in finding solutions to the challenging
problems, his willingness in pursuing new research ideas and his attitude towards his students will always lead me in achieving my future career goals. I would like to thank Drs.
Hassan Khalil, Steve Shaw and Jongeun Choi for the willingness to be on my Ph.D. committee in spite of their busy schedule and for their insightful suggestions whcih have enhanced
this dissertation. I am also grateful to my colleagues in the Dynamic Systems and Control
laboratory from whom I have benefited over the years; Louis Flynn, Frank Mathis, Dr. Aren
Hellum, Dr. Assaad Awad Abbass, Dr. Nandagopal Methil, Paul Strefling, David Crouse,
Mahdi Jadaliha, Dr. Yunfei Xu, Junho Lee. I would like to sincerely acknowledge all financial supports that I have received from the Department of Mechanical Engineering, the
College of Engineering, the Graduate School at Michigan State University and the National
Science Foundation. I am deeply indebted to my family and above all, my wife, Hamideh,
for their full support, love and encouragement. Without their help, I would not be able to
accomplish my Ph.D. career. Last but not the least, I wish to thank all my friends who have
supported me in many different ways during my stay at Michigan State University. Their
friendship has made it much easier to overcome the difficulties of being far from the family
and the home country.

v

TABLE OF CONTENTS

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

x

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xi

Chapter 1 Introduction . . . . . . . . . . . . . . . .
1.1 Motivation and Objectives . . . . . . . . . . . . .
1.2 Swing-up Control of Acrobot . . . . . . . . . . . .
1.3 Stabilizing Control for Underactuated Systems . .
1.4 Safe Fall Control for Humanoid Robots . . . . . .
1.5 Disturbance Rejection for Bipeds . . . . . . . . .
1.6 Optimizing the Energy Consumption for Bipeds .
1.7 Output Tracking Control for Non-Minimum-Phase
1.8 Scope of Dissertation . . . . . . . . . . . . . . . .

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Chapter 2 Swing-up Control of Pendubot: Experimental Validation
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Background: Swing-up Algorithm for Pendubot . . . . . . . . . . . . .
2.3 Experimental Setup and Results . . . . . . . . . . . . . . . . . . . . . .
2.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Chapter 3 Swing-up Control of Acrobot . . . . . . . . . .
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Acrobot Dynamics and Impulsive Effects . . . . . . . . .
3.2.1 Equations of Motion . . . . . . . . . . . . . . . .
3.2.2 Holding Torque . . . . . . . . . . . . . . . . . . .
3.2.3 Impulsive Torque for Sudden Change in Velocity .
3.2.4 Change in Velocity due to Impulsive Torque . . .
3.2.5 Change in Energy due to Impulsive Torque . . . .
3.3 Swing-Up Control of the Acrobot . . . . . . . . . . . . .
3.3.1 Rest-to-Rest Maneuvers of the Second Link . . .
3.3.2 Acrobot Swing-up Using Rest-to-Rest Maneuvers
3.3.2.1 Swing-up Algorithm . . . . . . . . . . .
3.3.2.2 Numerical Simulation . . . . . . . . . .
3.3.3 Increasing Energy of the Acrobot Using Impulsive
3.3.3.1 Swing-up Algorithm . . . . . . . . . . .
3.3.3.2 Numerical Simulation . . . . . . . . . .
3.4 Modification of an Existing Method . . . . . . . . . . . .

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3.4.2

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Chapter 4 Enlarging Region of Attraction for Underactuated Systems
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.1 Dynamic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.2 Impulsive Inputs . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.3 Effect of Impulsive Inputs on Velocity . . . . . . . . . . . . . . . .
4.3 Stabilization Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4 Case Studies: Linear Stabilizing Controllers . . . . . . . . . . . . . . . .
4.4.1 The Pendubot . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4.1.1 Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4.1.2 Continuous Controller . . . . . . . . . . . . . . . . . . .
4.4.1.3 Numerical Simulations . . . . . . . . . . . . . . . . . . .
4.4.2 The Acrobot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4.2.1 Dynamics and Control . . . . . . . . . . . . . . . . . . .
4.4.2.2 Numerical Simulations . . . . . . . . . . . . . . . . . . .
4.4.3 Rolling Acrobot . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4.3.1 Dynamics and Control . . . . . . . . . . . . . . . . . . .
4.4.3.2 Numerical Simulations . . . . . . . . . . . . . . . . . . .
4.5 Extension for Nonlinear Controllers . . . . . . . . . . . . . . . . . . . . .
4.5.1 Inverted Pendulum on an Inclined Plane . . . . . . . . . . . . . .
4.5.1.1 Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5.1.2 Stabilizing Control Design . . . . . . . . . . . . . . . . .
4.5.1.3 Numerical Simulations . . . . . . . . . . . . . . . . . . .
4.5.2 The Ball and Beam System . . . . . . . . . . . . . . . . . . . . .
4.5.2.1 Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5.2.2 Stabilizing Control Design . . . . . . . . . . . . . . . . .
4.5.2.3 Numerical Simulations . . . . . . . . . . . . . . . . . . .
4.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Chapter 5 Safe Fall Control for Humanoid Robots
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . .
5.2 Dynamic Model . . . . . . . . . . . . . . . . . . . .
5.2.1 Equations of Motion . . . . . . . . . . . . .
5.2.2 Lean Line . . . . . . . . . . . . . . . . . . .
5.3 Impulsive Effects . . . . . . . . . . . . . . . . . . .

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91
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3.5

Existing Control Method . . . . . . . . .
Modified Method Using Impulsive Inputs
3.4.2.1 Impulsive System . . . . . . . .
3.4.2.2 Stability of Impulsive System .
3.4.2.3 Numerical Simulation . . . . .
3.4.3 Alternate Continuous Control Design . .
3.4.3.1 Control Design . . . . . . . . .
3.4.3.2 Numerical Simulation . . . . .
Conclusion . . . . . . . . . . . . . . . . . . . . .

vii

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5.4
5.5
5.6

Safe Fall Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

Chapter 6 Disturbance Rejection for the Synthetic-Wheel Biped
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2.1 Equations of Motion . . . . . . . . . . . . . . . . . . . . . .
6.2.2 Torso Effect on Biped Dynamics . . . . . . . . . . . . . . . .
6.2.3 Interchange of Stance and Swing Legs . . . . . . . . . . . . .
6.2.4 Control Design . . . . . . . . . . . . . . . . . . . . . . . . .
6.3 Impulsive Torques and Effects . . . . . . . . . . . . . . . . . . . . .
6.3.1 Braking Torque for the Swing Leg . . . . . . . . . . . . . .
6.3.2 Braking Torque for the Torso . . . . . . . . . . . . . . . . .
6.3.3 Impulsive Effect on Velocity . . . . . . . . . . . . . . . . . .
6.4 Disturbance Rejection Algorithm . . . . . . . . . . . . . . . . . . .
6.5 Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . . .
6.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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106
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Chapter 7 Energy-Conserving Gaits for Point-Foot Passive-Ankle Bipeds
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2 Energy-Conserving Gaits . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2.1 Assumptions and Definitions . . . . . . . . . . . . . . . . . . . . . . .
7.2.2 Sufficient Conditions for Energy-Conserving Gait . . . . . . . . . . .
7.2.3 Characterization of Trajectories of the Actuated Joints . . . . . . . .
7.3 Three-DOF Bipeds: Two Case Studies . . . . . . . . . . . . . . . . . . . . .
7.3.1 Synthetic-Wheel Biped (SWB) . . . . . . . . . . . . . . . . . . . . .
7.3.1.1 System Description . . . . . . . . . . . . . . . . . . . . . . .
7.3.1.2 Sufficient Conditions for Energy-Conserving Gait . . . . . .
7.3.1.3 Determination of Actuated Joint Trajectories . . . . . . . .
7.3.1.4 Numerical Simulations . . . . . . . . . . . . . . . . . . . . .
7.3.2 Point-Foot Three-Link Biped . . . . . . . . . . . . . . . . . . . . . .
7.4 Five-DOF Point-Foot Biped . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.4.1 System Description . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.4.2 Sufficient Conditions for Energy-Conserving Gait . . . . . . . . . . .
7.4.3 Determination of Actuated Joint Trajectories . . . . . . . . . . . . .
7.4.4 Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . . . .
7.5 Two-Step Periodic Gaits: Five-DOF Bipeds . . . . . . . . . . . . . . . . . .
7.5.1 Generalization to Two-Step Periodic Gait . . . . . . . . . . . . . . .
7.5.2 Characterization of Trajectories . . . . . . . . . . . . . . . . . . . . .
7.5.2.1 Trajectories for the First Step . . . . . . . . . . . . . . . . .
7.5.2.2 Trajectories for the Second Step . . . . . . . . . . . . . . . .
7.5.3 Five-DOF Point-Foot Biped: Similar Leg Links . . . . . . . . . . . .
7.5.4 Five-DOF Point-Foot Biped: Dissimilar Leg Links . . . . . . . . . . .
7.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Chapter 8
8.1
8.2
8.3

8.4

8.5
8.6

Intermittent Output Tracking For Linear Non-Minimum-Phase
Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Discrete Equivalent Systems . . . . . . . . . . . . . . . . . . . . . . . . . . .
Zero Shift for Linear NMP Systems . . . . . . . . . . . . . . . . . . . . . . .
8.3.1 Stable NMP Systems with Distinct Poles - General Case . . . . . . .
8.3.2 Stable NMP Systems with Distinct Poles - Special Case . . . . . . . .
8.3.3 Stable NMP Systems with Repeated Poles . . . . . . . . . . . . . . .
Intermittent Output Tracking Control . . . . . . . . . . . . . . . . . . . . . .
8.4.1 Tracking Control for MP Systems . . . . . . . . . . . . . . . . . . . .
8.4.2 Intermittent Output Tracking for NMP Systems . . . . . . . . . . . .
8.4.3 Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Chapter 9

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Conclusion and Future Work . . . . . . . . . . . . . . . . . . . . . 183

APPENDICES . . . . . . . . . . . . . . .

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Appendix A Procedure for Verifying the Condition in Theorem 3.4.1 . . . . 187
Appendix B Equations of Motion: Synthetic-Wheel Biped . . . . . . . . . . 189
Appendix C Equations of Motion: Five-DOF Biped . . . . . . . . . . . . . . . 191
Bibliography . . . . . . . . . . . . . . .

ix

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LIST OF TABLES

Table 1.1

Mechanical Cost of Transport of Bipeds . . . . . . . . . . . . . . . .

13

Table 5.1

Degrees of freedom for the humanoid robot in Figure 5.1

93

Table 5.2

Kinematic and dynamic parameters for the six-dof robot with cylindrical links. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

Table 6.1

Parameters of the Synthetic-Wheel Biped [1] . . . . . . . . . . . . . 119

Table 7.1

Kinematic and dynamic parameters of Synthetic Wheel Biped

Table 7.2

Kinematic and dynamic parameters of Five-DOF Point-Foot biped
with similar leg links . . . . . . . . . . . . . . . . . . . . . . . . . . 144

Table 7.3

Kinematic and dynamic parameters of Five-DOF Point-Foot biped
with dissimilar leg links . . . . . . . . . . . . . . . . . . . . . . . . . 152

Table 8.1

Linear NMP systems with repeated poles and their corresponding
MP DE systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

x

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LIST OF FIGURES

Figure 2.1

The pendubot is shown in an arbitrary configuration. . . . . . . . .

20

Figure 2.2

Experimental validation of the impulsive control algorithm [2] for
pendubot swing-up. . . . . . . . . . . . . . . . . . . . . . . . . . . .

22

Figure 3.1

The Acrobot in an arbitrary configuration. . . . . . . . . . . . . . .

27

Figure 3.2

˙
Plot showing θ1 -θ2 space where Π is positive and negative for θ1 =
˙
θ2 = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

36

Simulation results for swing-up of the Acrobot using rest-to-rest maneuvers of the second link. . . . . . . . . . . . . . . . . . . . . . . .

38

Configuration of the Acrobot at the time when the impulsive torque
is applied to increase the energy of the system. . . . . . . . . . . . .

41

Simulation results for swing-up of the Acrobot using impulsive inputs
to increase the energy. . . . . . . . . . . . . . . . . . . . . . . . . . .

42

Simulation results for swing-up of the Acrobot: Comparison of the
modified impulsive controller (solid line) and the continuous controller (dashed line) proposed by Xin and Kaneda [3]. . . . . . . . .

52

Simulation results for swing-up of the Acrobot: Comparison of the
controller in Section 3.4.3 (solid line) with the continuous controller
(dashed line) proposed by Xin and Kaneda [3]. . . . . . . . . . . . .

55

ˆ
˙
Plots of the curves V = V and V = 0 and the impulse line correspond
to fixed values of q = (ˆ1 , q2 ) for a two-dof system. The prior-impulse
q ˆ
configuration at A : (ˆ, q − ) is jumped to a configuration at B : (ˆ, q + )
q ˙
q ˙
˙ < 0 and V < V . . . . . . . . . . . . . . . . . . . . . . .
for which V

63

The Pendubot/Acrobot is shown in an arbitrary configuration. For
the Pendubot we have τ2 = 0 and for the Acrobot we have τ1 = 0. .

65

Figure 3.3

Figure 3.4

Figure 3.5

Figure 3.6

Figure 3.7

Figure 4.1

Figure 4.2

xi

Figure 4.3

Simulation results showing joint angles and joint velocities of the
Pendubot and the Lyapunov function, its time derivative, and the
control input for an initial configuration outside RA - First simulation 67

Figure 4.4

˙ ˙
˙
The curves V = V0 , V = 0, and the impulse line are shown in the θ1 -θ2
plane at the initial configuration of the Pendubot for first simulation,
where θ10 = 1.484 and θ20 = 0.087. The impulsive input at t = 0 in
Figure 4.3 results in a jump in the configuration from A : (q0 , q0 ) to
˙−
˙
B : (q0 , q0 ), where V (q0 , q0 ) < 0 and V (q0 , q0 ) < V0 . . . . . . . . .
˙+
˙+
˙+

68

Figure 4.5

Simulation results showing joint angles and joint velocities of the
Pendubot and the Lyapunov function, its time derivative, and the
control input for an initial configuration outside RA - Second simulation 69

Figure 4.6

Simulation results showing joint angles and joint velocities of the Acrobot and the Lyapunov function, its time derivative, and the control
input for an initial configuration outside RA - First Simulation . . .

71

Simulation results showing joint angles and joint velocities of the Acrobot and the Lyapunov function, its time derivative, and the control
input for an initial configuration outside RA - Second Simulation . .

72

A schematic of the rolling acrobot. The link angles θ1 and θ2 are
measured counter-clockwise. . . . . . . . . . . . . . . . . . . . . . .

73

Simulation results including the angular positions, angular velocities,
Lyapunov function and its time derivative and the control input for a
configuration of rolling acrobot with unbounded leg angle originally
outside RA - First simulation . . . . . . . . . . . . . . . . . . . . . .

75

Simulation results including the angular positions, angular velocities,
Lyapunov function and its time derivative and the control input for a
configuration of rolling acrobot with unbounded leg angle originally
outside RA - Second simulation . . . . . . . . . . . . . . . . . . . .

76

Simulation results including the angular positions, angular velocities,
Lyapunov function and its time derivative and the control input for
a configuration of rolling acrobot with bounded leg angle originally
outside RA - First simulation . . . . . . . . . . . . . . . . . . . . . .

77

Figure 4.7

Figure 4.8

Figure 4.9

Figure 4.10

Figure 4.11

xii

Figure 4.12

Simulation results including the angular positions, angular velocities,
Lyapunov function and its time derivative and the control input for
a configuration of rolling acrobot with bounded leg angle originally
outside RA - Second simulation . . . . . . . . . . . . . . . . . . . .

78

Figure 4.13

Cart-pendulum on a slope

. . . . . . . . . . . . . . . . . . . . . . .

81

Figure 4.14

Simulation results including the positions and velocities of the system coordinates, Lyapunov function and its time derivative and the
control input for a configuration of the inverted pendulum on a slope
originally outside the RA - First simulation . . . . . . . . . . . . . .

83

Simulation results including the positions and velocities of the system coordinates, Lyapunov function and its time derivative and the
control input for a configuration of the inverted pendulum on a slope
originally outside the RA - Second simulation . . . . . . . . . . . . .

84

Figure 4.16

Ball and beam system . . . . . . . . . . . . . . . . . . . . . . . . . .

85

Figure 4.17

Simulation results including the positions and velocities of the system
coordinates, Lyapunov function and its time derivative and the control input for a configuration of the ball and beam system originally
outside the RA - First simulation . . . . . . . . . . . . . . . . . . .

88

Simulation results including the positions and velocities of the system
coordinates, Lyapunov function and its time derivative and the control input for a configuration of the ball and beam system originally
outside the RA - Second simulation . . . . . . . . . . . . . . . . . .

89

Figure 5.1

Seven-dof three-dimensional humanoid robot . . . . . . . . . . . . .

93

Figure 5.2

Orientation of the lean line in the inertial frame . . . . . . . . . . .

94

Figure 5.3

A schematic of the safe fall algorithm using two sets of impulsive
inputs. For interpretation of the references to color in this and all
other figures, the reader is referred to the electronic version of this
dissertation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

98

Figure 4.15

Figure 4.18

Figure 5.4

Simulation results showing all the joint angles and angular velocities
for changing the fall direction of the robot . . . . . . . . . . . . . . 104

xiii

Figure 5.5

Simulation results showing the lean line angles for changing the fall
direction of the robot . . . . . . . . . . . . . . . . . . . . . . . . . . 105

Figure 5.6

Simulation results showing the impulsive inputs for changing the fall
direction of the robot . . . . . . . . . . . . . . . . . . . . . . . . . . 105

Figure 6.1

A schematic of the Synthetic-Wheel biped, reproduced from [1]. . . . 107

Figure 6.2

Biped configuration at the time of interchange of stance and swing legs.109

Figure 6.3

Simulation results showing angular positions (rad), angular velocities
(rad/s), and control inputs (N.m) of the SWB in rejecting the impulsive disturbances applied at two instants of time. The dashed lines
˙
correspond to the plots of ψ and ψ. . . . . . . . . . . . . . . . . . . 121

Figure 6.4

Simulation results showing angular positions (rad), angular velocities
(rad/s), and control inputs (N.m) of the SWB in rejecting the impulsive disturbances applied at five instants of time. The dashed lines
˙
correspond to the plots of ψ and ψ. . . . . . . . . . . . . . . . . . . 122

Figure 7.1

A schematic of an n-link point-foot planar biped. . . . . . . . . . . . 126

Figure 7.2

A schematic of the Synthetic-Wheel biped, reproduced from [1]. . . . 132

Figure 7.3

Phase portraits of the Synthetic-Wheel Biped for different amplitudes
of torso oscillation A . . . . . . . . . . . . . . . . . . . . . . . . . . 136

Figure 7.4

Torque plots for an energy-conserving gait for the Synthetic-Wheel
Biped (SWB). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

Figure 7.5

Generalized torque plots for an energy-conserving gait for the SyntheticWheel Biped (SWB). . . . . . . . . . . . . . . . . . . . . . . . . . . 137

Figure 7.6

A schematic of the Point-Foot Three-Link. . . . . . . . . . . . . . . 138

Figure 7.7

A schematic of the five-dof point-foot biped. . . . . . . . . . . . . . 140

Figure 7.8

Torque plots for an energy-conserving gait for the Five-DOF PointFoot Biped with similar leg links. . . . . . . . . . . . . . . . . . . . 144

Figure 7.9

Generalized torque plot for an energy-conserving gait for the FiveDOF Point-Foot Biped with similar leg links. . . . . . . . . . . . . . 145
xiv

Figure 7.10

Configurations of the five-dof point-foot biped with similar leg links
at different instants of time over one step. . . . . . . . . . . . . . . . 146

Figure 7.11

Torque plots for an energy-conserving gait with two-step periodicity
for the Five-DOF Point-Foot Biped with similar leg links. . . . . . . 151

Figure 7.12

Generalized torque plot for an energy-conserving gait with two-step
periodicity for the Five-DOF Point-Foot Biped with similar leg links. 151

Figure 7.13

Configurations of the five-dof point-foot biped with similar leg links
at different instants of time for a two-step periodic gait. The joint
configurations at the beginning of the first step and end of the second
step are the same but they are different from the joint configuration
at the end of the first step. . . . . . . . . . . . . . . . . . . . . . . . 152

Figure 7.14

Torque plots for an energy-conserving gait with two-step periodicity
for the Five-DOF Point-Foot Biped with dissimilar leg links. . . . . 153

Figure 7.15

Generalized torque plot for an energy-conserving gait with two-step
periodicity for the Five-DOF Point-Foot Biped with dissimilar leg links.153

Figure 7.16

Configurations of the five-dof point-foot biped with dissimilar leg links
at different instants of time for a two-step periodic gait. The joint
configurations at the beginning of the first step and end of the second
step are the same but they are different from the joint configuration
at the end of the first step. . . . . . . . . . . . . . . . . . . . . . . . 154

Figure 8.1

The periodic switched input for the system in Eq.(8.1) (in solid line)
and the constant input for DE system in Eq.(8.3) (in dashed line) . 159

Figure 8.2

Simulation results for example 8.2.1 showing state variables and inputs for switched-input system (solid line) and discrete equivalent
system (dashed line). . . . . . . . . . . . . . . . . . . . . . . . . . . 161

Figure 8.3

Simulation results for the first case showing the system output (solid
line) and reference trajectory (dashed line) in the first plot, internal
states in the second plot and control input (Solid line) and tracking
control for MP system (dashed line) in the third plot . . . . . . . . 179

Figure 8.4

Simulation results for the second case showing the system output
(solid line) and reference trajectory (dashed line) in the first plot,
internal states in the second plot and control input (Solid line) and
tracking control for MP system (dashed line) in the third plot . . . . 181
xv

Chapter 1
Introduction

1.1

Motivation and Objectives

Impulsive systems are dynamical systems which can have discontinuous jumps in their state
variables. These jumps are caused by either impulsive inputs which are intentionally applied
to control the system or external disturbances and impacts which are generally undesirable.
The impulsive inputs are theoretically modeled as Dirac-delta functions with unbounded
magnitude and infinitesimal time support. Since the state variables are not always continuous
for impulsive systems, control theories developed for continuous systems need to be revised
for application to impulsive systems. There has been a fair amount of theoretical research
on impulsive control of dynamical systems and credit for some of the early works goes to
Pavlidis [4], Gilbert and Harasty [5], Menaldi [6] and Lakshmikantham [7]. Researchers have
studied the problems of stability, controllability and observability, optimality (see [8, 9, 10],
for example).
Impulsive control has been designed for dynamical systems with different control objectives [11, 12, 13]. However, impulsive control can only be applied to systems for which the
state variables can be changed instantaneously without violating any physical laws. Population growth control systems, financial models or chaotic dynamical systems are some
examples of such systems. In mechanical systems, the discontinuous jumps occur in the
velocity states which are directly changed through the inputs of the system such as motor
1

torques. Therefore, it has to be clarified how to implement ideal delta-function inputs using standard actuators before applying impulsive control algorithms to mechanical systems.
The implementability concern can be the main reason why there are few works in the literature designing impulsive control for mechanical systems [14, 15, 2, 16, 17]. Weibel et al
[14] studied the implementation of impulsive inputs in controlling single-degree-of-freedom
Hamiltonian mechanical systems. They derive the delta-function impulsive inputs to make
desired jumps in the energy levels of the system. The impulsive inputs are approximated by
a series of rectangular pulses and were experimentally verified on a pendulum-cart system.
In this research, we first confirm that impulsive inputs can be implemented in mechanical systems with an experimental validation of swing-up control of the Pendubot [2] using
impulsive inputs. Unlike the approach presented in [14], the impulsive inputs are obtained
from Lagrangian dynamics and are approximated by large-gain continuous inputs applied
during short intervals of time.
After demonstrating the effectiveness of an impulsive control algorithm in experiments,
we develop impulsive control algorithms for underactuated mechanical systems. The control problem of underactuated systems is more challenging since such systems have fewer
actuators than the number of their degrees of freedom. An impulsive algorithm is developed
for swing-up control of the Acrobot, a benchmark problem in underactuated systems, where
impulsive inputs are used to make instantaneous changes in energy level of the acrobot. The
impulsive control algorithm has distinct advantages over existing algorithms in terms of the
time required for swing-up and maximum control torque used by the continuous controller.
Impulsive inputs cause jumps in velocity states of mechanical systems, and consequently, produce jumps in Lyapunov function candidates used in control design. This attribute is used
to enlarge the region of attraction of equilibria using impulsive inputs at discrete instants
2

of time. Several case studies of underactuated mechanical systems have been presented to
demonstrate this benefit of using impulsive control. Another advantage of using impulsive
inputs is that such inputs can significantly alter the dynamics of the system in a very short
period of time. This property is used to design a safe fall algorithm for humanoid robots undergoing a fall, i.e., after the continuous controller has failed to keep the system trajectories
confined to a fixed region around the equilibrium. The algorithm uses impulsive inputs to
change the fall direction of the robot to minimize the damage to people and objects in the
vicinity, as well as to its own self.
In many instances, external disturbances or impact from interaction with the environment
can have an adverse effect on system performance. In the second part of this dissertation, we
develop control algorithms to mitigate these effects in underactuated biped robots. We first
develop a disturbance rejection algorithm for the synthetic-wheel biped [1] to mitigate the
effects of external disturbances. The control algorithm uses impulsive inputs to reject the
effects of the external disturbances as well as cancel the undesirable effects of ground impacts
during leg interchange. Continuous controller is then designed for the synthetic-wheel biped
to generate an impact-free walking gait. This gait consumes zero energy in the ideal case
and the necessary conditions to achieve this gait are shown to be general and applicable to
a range of bipedal robotic systems.
An underactuated mechanical system can be non-minimum-phase if its zero dynamics is
unstable. We finally investigate the output-tracking problem for linear single-input-singleoutput non-minimum-phase systems. Intermittent output tracking is achieved under the
condition of finite preview of the reference trajectory, i.e., the system output tracks a desired
trajectory intermittently in some regular intervals of time rather than continuous tracking
of the trajectory. The control algorithm uses switched inputs, which can be approximated
3

by impulsive inputs in the limit.
In the remainder of this chapter, we give a brief description and review the literature for
each control problem which will be studied in this dissertation.

1.2

Swing-up Control of Acrobot

The Acrobot is a two-link robot in the vertical plane with an actuator at the elbow joint
and a passive shoulder joint. The dynamic model of the Acrobot has four equilibria, all of
which are stabilizable, and the standard control problem refers to swing-up from an arbitrary
initial configuration to the equilibrium configuration with the highest potential energy. The
Acrobot has identical kinematic structure as that of the Pendubot [2], but swing-up of the
Acrobot is more challenging due to the adverse location of its actuated joint. Most of the
results in the literature provide solutions that apply either to the Pendubot or the Acrobot.
The paper by Boone [18], for example, provides a near-optimal solution to the Acrobot
swing-up problem using bang-bang control. More interesting are the papers that can be
applied to the Pendubot and the Acrobot alike. Included in this category are the papers by
Spong [19, 20] based on feedback linearization, and Kolesnichenko and Shiriaev [21] based
on partial stabilization using Lyapunov-like functions.
The feedback linearization method proposed by Spong [19] designs reference trajectories
for the actuated joint that render the zero dynamics unstable and thereby increases the
energy of the system. For the Acrobot dynamic model, this method is highly sensitive
to the control gains. Furthermore, the control design is approximated for implementation
without a rigorous proof. The partial stabilization method proposed by Kolesnichenko and
Shiriaev [21] has been implemented in the Acrobot dynamic model by Xin and Kaneda [3]

4

and Mahindrakar and Banavar [22, 23]. The method uses a Lyapunov-like function and
results in an invariant set that contains points other than the trivial solution. For swing-up
to the desired configuration, constraints have to be imposed on the control gains and certain
initial conditions have to be avoided [3, 22].

1.3

Stabilizing Control for Underactuated Systems

Stabilization of underactuated dynamical systems has been widely studied and several control methods have been proposed. Linear regulation methods such as pole placement and
LQR can be designed based on the linearized system but the Region of Attraction (RA ) of
the desired equilibrium for such controllers is typically small. The main nonlinear control
methods in the literature modify the structure of the original system to yield a closed-loop
system for which the desired equilibrium is asymptotically stable. The Controlled Lagrangian
method, Interconnection and Damping Assignment Passivity-Based Control (IDA-PBC), and
the λ-Method are the three main approaches that have been developed.
The Controlled Lagrangian method, proposed by Bloch et al. [24, 25], is an energy-based
approach that can be applied to underactuated mechanical systems in Euler-Lagrange form.
The kinetic and potential energies of the system are modified to find a desired Lagrangian,
called Controlled Lagrangian. The control gains and the system input are then chosen such
that the closed-loop equations derived from the Controlled Lagrangian match those of the
controlled system. Full phase space stabilization is obtained through kinetic and potential
energy shaping. The control procedure can be complicated for systems with several degrees
of freedom (dof’s).
Passivity-based control can be effectively used to stabilize some underactuated mechanical

5

systems through shaping of the potential energy [26]. However, for most underactuated
systems, the total energy needs to be shaped. For such systems, Ortega et al. [27, 28]
proposed IDA-PBC which considers the Hamiltonian form of the system. The desired energy
function is found by assigning interconnection and damping matrices of the system. Similar
to the Controlled Lagrangian method, where the closed-loop system is maintained in EulerLagrange form, the Hamiltonian form of the closed-loop system is preserved in IDA-PBC.
The key advantage of IDA-PBC is that it provides freedom to change the interconnection
matrix as well as the energy function; the Controlled Lagrangian method can therefore be
viewed as a special case of IDA-PBC. The class of underactuated systems which can be
stabilized by IDA-PBC is limited to those for which two partial differential equations can be
solved. Acosta et al. [29] showed that these equations can be explicitly solved for mechanical
systems with underactuation degree one.
The λ-Method, proposed by Auckly et al. [30, 31], uses differential geometry to assign
a special form for the closed-loop system. This method produces an infinite-dimensional
family of control laws to stabilize the underactuated system. Using a transformation, the
control laws are derived using matching conditions that are recast as linear first-order partial
differential equations. The control laws have been used to stabilize linear systems and the
nonlinear ball and beam system [32].
Among other stabilizing controllers, White et al. [33, 34, 35] proposed a direct Lyapunov
function approach which results in three matching conditions comprised of ordinary and
partial differential equations. The matching conditions can be numerically solved in real
time; this makes it better suited for systems with several dof’s compared to the Lagrangian
and Hamiltonian approaches. Olfati-Saber proposed a method to transform underactuated
mechanical systems into a cascade normal form comprised of linear and nonlinear subsystems
6

[36, 37, 38, 39]. The transformation is chosen such that the control problem is reduced to
control of the reduced-order nonlinear subsystem. The backstepping method is then used to
design a stabilizing controller for the resulting cascade normal form. Sliding mode controllers
have also been proposed to stabilize a class of underactuated systems under uncertainty
conditions [40, 41, 42].

1.4

Safe Fall Control for Humanoid Robots

Safety is one of the main concerns behind the independent autonomous existence of humanoid robots in physically interactive human environments. Although the loss of balance
and fall are rare in isolated or controlled environments, the situation would be different
when the physical separation between robots and humans gradually disappear. A fall of a
humanoid robot is a particularly worrisome event; a fall from an upright posture can cause
damage to the robot, to delicate and expensive objects in the surrounding or can inflict
injury to a nearby human being. Regardless of the substantial progress in humanoid robot
balance control strategies, the possibility of fall remains real, even unavoidable. Yet, few
comprehensive studies of humanoid fall encompassing fall avoidance, prediction, control and
recovery exists in the literature.
A humanoid fall may be caused due to external factors such as unexpected or excessive
forces, unusual or unknown floor slipperiness, slope or profile of the ground, causing the
robot to slip, trip or topple. In these cases the disturbances that threaten balance are larger
than what the balance controller can handle. Fall can also result from internal factors such
as a malfunction or a failure of actuator, power or communication system where the balance
controller is partially or fully incapacitated.

7

A controller dealing with an accidental humanoid fall may have two primary, and distinctly different goals: a) self-damage reduction and b) reduction of damage to others. When
a fall occurs in an open space, the self-damage reduction strategy can aim to mitigate the
damaging effects of the ground impact. If, however, a robot falls in close proximity to others,
its primary objective should be to prevent damage to these objects or minimize injury to
persons. Recently, Wilken et al. have reported a third possible goal of a fall controller, that
of a deliberate fall of a humanoid soccer goalie [43]. This is an example of a strategic fall.
A falling humanoid can be easily appreciated as a formidable system to study and control.
During fall, a humanoid behaves as a nonlinear underactuated multi-body system with highDOF, changing morphology and unpredictable contact configurations. Additionally, the
robot is subjected to the relentless gravitational pull, which it cannot fully resist, and which
tries to bring it down to a crash. Consequently, the time interval between the detection of
a fall and the actual event of ground impact becomes a critical parameter for fall control.
Time is at a premium during the occurrence of a fall; a single rigid body model of a full
sized humanoid indicates that a fall from the vertical upright stationary configuration due
to a mild push takes about 800-900 ms [44]. In many situations the time to fall can be
significantly shorter, and there is no opportunity for elaborate planning or time-consuming
control. Yet, through simulation and experiments we are able to demonstrate that meaningful
modification to the default fall behavior can be achieved and damage to the environment
can be avoided.
Some earlier work on humanoid fall direction change has been reported in [45, 46, 47].
A number of recent papers reported on the damage minimization and prediction aspects of
humanoid fall. In their exhaustive work, Fujiwara et al. [48, 49, 50, 51] proposed martial arts
type motion for damage reduction, computed optimal falling motions using minimum impact
8

and angular momentum, and fabricated special hardware for fall damage study. Ogata et
al. proposed two fall prediction methods based on abnormality detection and predicted Zero
Moment Point (ZMP) [52, 53]. The robot improves fall prediction through experimental
learning. Renner and Behnke [54] use model-based approach to detect external forces on
the robot and Karssen and Wisse [55] use principal component analysis to predict fall.
Following human movement based search procedure, Ruiz-del-Solar et al. implemented a
low damage fall sequence for soccer robots [56]. In [57] and [58], fall prediction and control
are treated together using Gaussian mixture models and Hidden Markov model. Ishida et al.
employed servo loop gain shift to reduce shock due to fall [59]. Kanoi et al. worked on fall
detection from walk patterns [60]. Fall damage minimization is obviously of natural interest
in biomechanics [61].

1.5

Disturbance Rejection for Bipeds

A humanoid robot is eventually aimed to operate in real environments with all interactions
and disturbances from humans and surrounding obstacles. Therefore, the control design for
humanoids must be able to reject the external disturbances which can be applied during
stance or walking phases of the robot. The complicated, nonlinear, unstable and hybrid
dynamics of the bipeds make it a challenging problem to find effective balancing strategies
for humanoid robots.
Based on biomechanical studies [62, 63], Hofmann [64] suggested three basic push recovery
strategies for bipeds: Ankle strategy(or CoP balancing) , hip strategy (or CMP balancing)
and stepping. If the external disturbances are small, the ankle joint is used to adjust the
Center of Pressure (CoP) of the robot within the foot support area to maintain the balance.

9

For greater disturbances, other joints such as hip joint and arms are used to create angular
momentum about the Center of Mass (CoM) to counteract the external push. This strategy
leads to an effective CoP outside the support polygon which is called Centroidal Moment
Point (CMP). For large disturbances, the biped has to take steps to maintain balance.
Based on these strategies, different controllers have been proposed to solve push recovery
problem in humanoids. ZMP-based approaches have been widely used to design balance
controllers for humanoid bipeds [65, 66, 67, 68, 69, 70]. Stephens [71] derived decision
surfaces as functions of reference points of a humanoid to determine when each balancing
strategy is needed and to predict whether or not the biped can recover from the push.
He used simplified models for humanoids which result in inaccurate decision surfaces for
actual robots. Yi et al. [72] proposed reinforcement learning to train a high-level controller
which chooses the proper strategy for disturbance rejection. Their method is more accurate
but it requires off-line procedures to train the controller. Linear Inverted Pendulum Model
(LIPM)[73, 74] is used for most of balance controllers using ankle and hip strategies (e.g.
in [75]). Since LIPM assumes zero angular momentum about CoM, other models such
as Linear Inverted Pendulum model plus Flywheel (LIPF) [76] and Angular Momentum
inducing inverted Pendulum Model (AMPM) [77] are proposed to incorporate the significant
role of angular momentum generated by internal joints of the humanoid. Pratt, et al. [76]
introduced the concept of “Capture point”, a point on the ground where the robot can
step to in order to stop and found the analytical expressions for LIPF. Rebula, et al. [78]
presented a learning method to compute the capture points for a general humanoid. The
proposed method uses the estimated capture points predicted by LIPM as the starting points
and finds the desired foot placement to recover from the push. Their method was validated
on a three-dimensional humanoid robot with twelve actuated dof’s. Stephens and Atkeson
10

extended the humanoid balance strategies for compliant robots with force-controlled joints
[79], [80]. Their standing balance controller uses a model-based approach called Dynamic
Balance Force Control (DBFC) to calculate the joint torques based on desired COM motion
[79]. Their stepping controller for larger disturbances was based on feedforward torque
inputs obtained from DBFC to follow desired CoM trajectories generated by a linear model
predictive control [80]. They validated their push recovery strategies through experiments
on Sarcos, a force-controlled humanoid. Komura et al. [81] and Adiwanhono et al. [82]
proposed balance controllers for walking bipeds which uses AMPM and LIPM, respectively.
In [81], the amount of angular momentum needed to counteract the external disturbance in
both sagital and frontal planes is calculated based on a new criteria called the “difference of
inertia”. The push recovery algorithm in [82] employs a different walking gait after the push
which is based on a ZMP-shifting method.

1.6

Optimizing the Energy Consumption for Bipeds

With biped walking machines becoming quite common, robotics engineers are increasingly
using the metric of Cost of Transport (COT) to compare their energy efficiencies. The COT
is a dimensionless measure and is defined as [83]

COT =

energy used
body weight × distance traveled

It has been used to quantify the energetics of locomotion in animals and is now being used
to compare energy efficiencies of walking machines and other mobile robots. For a biped
robot, the COT can be computed based on the total energy used by the robot, or the energy

11

used by the actuators driving the locomotion mechanism. To distinguish the efficiency of the
mechanical design from the efficiency of the complete system, researchers ([84], for example)
have defined the mechanical cost of transport as

Cmt =

energy used by actuators
body weight × distance traveled

Although there is no ambiguity in the definition of Cmt , the “energy used by the actuators”
has been calculated in different ways by different researchers. Hobbelen and Wisse [84]
made the assumption that actuators expend energy irrespective of whether the mechanical
energy is positive or negative. On the other hand, Srinivasan and Ruina [85] assumed that
the energy used by the actuators is positive when the mechanical energy is positive but is
equal to zero when the mechanical energy is negative. Both Hobbelen and Wisse [84] and
Srinivasan and Ruina [85] have attempted to take into account the limitations of technology
in regenerating energy from motion and storing it back in the energy storage unit. Their
assumptions are certainly not consistent with one another and are likely to change further
if actuators other than electric motors are used. To avoid an actuator-specific definition of
Cmt , and in light of the fact that technological advancements (in hybrid cars, for example)
are indeed making it possible to regenerate and store energy, the original definition of Cmt
can be reworded as follows:

Cmt =

mechanical energy required by gait
body weight × distance traveled

where
T

mechanical energy required by gait =
i

12

0

˙
τi θi dt

(1.1)

˙
and where τi and θi are the generalized force and generalized velocity of the i-th actuated
joint, and T is the time period of each step.
Many successful biped designs have been reported in the literature. The Cmt of some
of these designs are compiled in Table 1.1 and compared with that of human walking. All
of these values were derived from experimental data and are therefore based on the energy
used by the actuators. Although different methods may have been used for calculation of
the energy used by the actuators, the Cmt values of these designs span a wide range. At
one end of the spectrum is Dynamite [83], a passive dynamic walker that can only walk
down inclines for certain initial conditions; at the other end of the spectrum is Honda Asimo
[92, 93, 94], an active biped that can walk, run, and climb stairs, and has many other
functionalities apart from locomotion. Dynamite and Asimo are very different platforms
and no meaningful comparisons can be made, but it should be emphasized that Dynamite
relies on natural limb dynamics whereas Asimo uses significant amount of energy to overcome
natural limb dynamics and track prescribed trajectories. The Cornell Ranger [90] has the
Table 1.1: Mechanical Cost of Transport of Bipeds
Bipeds

Robots

Humans

Honda Asimo
Rabbit
Huang et al. - Optimal Trajectory
Twente Dribbel
MSU Synthetic Wheel
Mabel
TU Delft Denise
MIT Spring Flamingo
Cornell Collins 3D
Cornell Ranger
Dynamite
Walking

13

Cmt
1.600
0.380
0.360
0.220
0.140
0.100
0.080
0.070
0.055
0.040
0.040
0.050

Ref.
[86]
[87]
[88]
[89]
[1]
[87]
[86]
[86]
[86]
[90]
[83]
[91]

same Cmt as that of Dynamite but it is an active biped. It has a stable gait on relatively flat
ground and is highly efficient by design; it was recently recorded to walk 65 km on a single
charge. The Cornell Collins 3D [86] is a three-dimensional biped which achieves yaw control
through mechanical coupling between its legs and opposite arms. Like the Ranger, it stores
energy in a spring during the stance phase and releases it during push-off. It is fairly efficient
in terms of energy consumption but it lacks gait stability. The MIT Spring Flamingo is a
seven link planar biped with six actuated degrees-of-freedom. The feet and legs are slender
compared to the upper body and are driven by series elastic actuators. A virtual model
control [95] algorithm was used to make the robot walk over flat terrain and inclines. Denise
[96], developed at TU Delft, has five degrees-of-freedom: one at the hip, two at the knees,
and two at the ankles. Only the hip joint is actuated, it is driven by two antagonistic pairs
of McKibben muscle-like actuators. Denise has arms but they do not add degrees of freedom
since they are mechanically linked to the opposing thighs with belts. Similar to Denise,
Dribble [89] is underactuated and has four degrees-of-freedom with a single active hip joint.
The knee joints are passive but they have active locking mechanisms.
It is important to note that both the gait and the controller used to control the gait have
significant effect on the energy consumption of biped robots. Rabbit [97] and Mabel [87]
are controlled using the Hybrid Zero Dynamics method [98] but they differ widely in their
construction and implementation. Rabbit is a conventional stiff robot with highly-geared
servomotors whereas Mabel is a compliant robot. A significantly lower Cmt for Mabel can
be attributed to its ability to store energy during the gait cycle due to its compliance, and
the ability of the controller to exploit this compliance. Huang et al. [88] presented the
behaviors of a planar three-link biped for two different controllers. The first controller, implemented experimentally, uses an optimization method to determine optimal trajectories
14

for the boundary conditions of a step. The second controller, described as the self-excited
controller, imposes constraints on the degrees-of-freedom of the robot and generates an oscillatory behavior of the system for walking. The self-excited controller results in a lower cost
of transport but its Cmt value is not listed in Table 1.1 since it was not implemented experimentally. The MSU Synthetic-Wheel biped [1] has a moderate Cmt with its symmetric gait.
An optimal gait [99], that eliminates impulsive forces at the time of swing-leg touchdown,
offers to provide a significantly lower Cmt but it has not been implemented in experiments.
Among other methods to improve energy efficiency, Lohmeier [100] proposed to increase limb
stiffness, move the center of mass of the biped higher, and minimize limb inertia; Asano and
Luo [101] and Wisse et al. [102] designed efficient mechanisms such as the bisecting hip
mechanism; and Gomes and Ruina [103] proposed to eliminate energy losses due to impact
through velocity matching.

1.7

Output Tracking Control for Non-Minimum-Phase
Systems

The output tracking problem for linear systems has been studied for more than four decades.
It deals with finding a control input such that the internal dynamics is stable and the system
output asymptotically tracks a desired trajectory. For linear Minimum-Phase (MP) systems,
the tracking problem can be easily solved, however, it is difficult to achieve both asymptotic
tracking and good transient performance in the case that the system is Non-Minimum-Phase
(NMP).
If the reference signals to be tracked are generated by an unstable external autonomous
linear system, i.e., an exosystem, it is referred to as the output regulation problem or the
15

servomechanism problem [104, 105, 106]. The solution to the output regulation problem for
linear systems relies on solving a set of two linear matrix equations which are not solvable
if an eigenvalue of the exosystem is identical to a zero of the linear plant [107, 108, 109].
This makes the class of exosystem-generated reference signals even more limited for NMP
systems with Right-Half-Plane (RHP) zeros.
Inversion-based control techniques have been proposed for output tracking with specified
desired trajectory. For a given desired output, the inversion problem is solved to find the
desired state variables and the desired control input. The tracking control input is comprised of the desired input as a feedforward component and a stabilizing input as a feedback
component. For NMP systems, the inversion techniques in [110] results in causal but unbounded solutions. This problem was resolved using stable inversion [111] for which the
control input is non-causal and requires preactuation. Later, Weng and Chen [112] proposed
a method based on causal stable inversion by requiring the desired trajectory to have a compact support. Stable inversion techniques cannot be applied to systems with non-hyperbolic
internal dynamics, which correspond to imaginary-axis zeros for a linear system. For such
cases, approximate stable inversion [113, 114] was proposed to achieve approximate output
tracking.
For nonlinear systems, Grizzle et al.

[115] proved that asymptotically stable zero-

dynamics is a necessary condition for asymptotic tracking of desired output trajectories.
For NMP systems, only approximate output tracking methods have been developed; in these
methods the output is redefined or the unstable internal dynamics is approximated by stable
dynamics [116, 117, 118]. The unstable zero dynamics of NMP systems also limits the best
achievable transient performance for the output tracking problem. The ideal performance
for the cheap control problem is zero for MP systems; it is limited to the energy needed to
16

stabilize the internal dynamics for NMP systems [119, 120, 121].
Discrete-time systems with unstable zeros have the same control limitations as those of
NMP continuous-time systems [122] and therefore, control methods based on zeroth-order
hold can only achieve approximate output tracking for such systems [123, 124, 125]. By
replacing the traditional zeroth-order hold, multirate control [126, 127] was proposed to
overcome the problem of unstable zeros. Using a multirate feedforward control, Fujimoto et
al. [128] developed a control method for perfect tracking in discrete-time systems.

1.8

Scope of Dissertation

The work in this dissertation is organized as follows: In Chapter 2, we provide the experimental results for the swing-up control of the Pendubot to verify an impulsive control algorithm
developed earlier in [2]. We develop an energy-based algorithm using impulsive inputs for
swing-up control of the Acrobot in Chapter 3. An extension to current swing-up algorithms
to include impulsive inputs is also discussed. In Chapter 4, impulsive inputs are used to
enlarge the region of attraction of a desired equilibrium for underactuated systems. Several
case studies of underactuated mechanical systems are studied to show the effectiveness of
the algorithm. In Chapter 5, we use impulsive inputs to propose a safe fall algorithm for
Humanoid robots. In particular, we design an impulsive control algorithm to change the
fall direction of a six-dof robot to minimize damage to the surrounding objects. In Chapter
6, a disturbance rejection algorithm is developed for the Synthetic-Wheel biped [1]. Impulsive inputs are used to cancel the adverse effects of external impulsive disturbances and
ground impact during the leg interchange. In Chapter 7, Energy-conserving gaits with ideal
zero-energy consumption are generated for a class of underactuated bipeds by avoiding the

17

ground impact at the leg interchange. In Chapter 8, we propose an intermittent output
tracking control method for linear non-minimum-phase systems. The switched inputs used
in the control method can be approximated by impulsive inputs in the limit. Finally, we
provide some concluding remarks and the directions for the future work in Chapter 9.

18

Chapter 2
Swing-up Control of Pendubot:
Experimental Validation

2.1

Introduction

Although impulsive inputs which are ideally modeled as Dirac-delta functions, can not be
implemented, they can be estimated such that the same impulsive effects are resulted in the
state variables. Using numerical simulations, Albahkali et al [2] demonstrated that impulsive
inputs can be estimated by large-gain continuous inputs in their swing-up control algorithm
for the Pendubot. The large-gain inputs are derived from dynamics which result in a fast
convergence of the system velocities to their new values. It is shown that as the gains are
chosen larger, the effects on the system dynamics get closer to the ideal impulsive case. In this
chapter, the impulsive control algorithm in [2] is experimentally verified using a Pendubot
set-up. The experimental results confirms that the impulsive inputs cab be closely estimated
by continuous inputs and therefore, implemented using standard hardware. This chapter is
organized as follows: In Section 2.2, we briefly review the swing-up control algorithm of
the Pendubot in [2]. The experimental set-up and results are presented in Section 2.3 and
concluding remarks are provided in Section 2.4.

19

2.2

Background: Swing-up Algorithm for Pendubot

We briefly summarize the algorithm proposed by Albahkali, et al. [2] for swing-up control
of the pendubot, shown in Figure 2.1:
1. Initialization: Linearize the dynamic equations of the pendubot about the desired
˙
˙
equilibrium (θ1 , θ1 , θ2 , θ2 ) = (π/2, 0, 0, 0) and design a linear controller to render the
equilibrium locally asymptotically stable.
2. First link swing-up: Drive the first link from its initial configuration to any configu˙
ration where θ1 satisfies (π/2 − α) ≤ θ1 ≤ (π/2 + α), θ1 = 0 for some small positive
angle α.
3. Swing-up control of the second link: Conduct rest-to-rest maneuvers of the first link
about the vertically upright configuration satisfying (π/2 − α) ≤ θ1 ≤ (π/2 + α). Use a
continuous torque to drive the first link from rest and stop it with an impulsive torque
when it reaches the boundary of the interval [(π/2 − α), (π/2 + α)]. It was shown [2]
g

θ2

d2

l1

l2

c.m
.

Y

d1

θ1

.
c.m
X

τ1

Figure 2.1: The pendubot is shown in an arbitrary configuration.

20

that this maneuver can increase the energy of the second link. Continue rest-to-rest
maneuvers till the energy of the second link is approximately equal to Edes .
4. Stabilization: When the second link reaches its highest potential energy configuration,
the pendubot configuration will be within the region of attraction of the desired equilibrium. Invoke the linear controller, designed in the first step, to stabilize the desired
equilibrium.

2.3

Experimental Setup and Results

Experiments were done with the P-2 pendubot system, produced by Mechatronic Systems,
Inc. The kinematic and dynamic parameters of the pendubot are as follows:

m1 = 1.0367 kg

m2 = 0.5549 kg

l1 = 0.1508 m

l2 = 0.2667 m

d1 = 0.1206 m

d2 = 0.1135 m

I1 = 0.0031 kgm2

I2 = 0.0035 kgm2

(2.1)

The shoulder joint was driven by a DC motor, powered by an amplifier operating in the
current mode. Both joints have encoders for joint angle sensing. The pendubot was interfaced
with a dSpace DS1104 board and the controller was implemented in the Matlab/Simulink
environment for real-time control. The impulsive torque in the third step of the algorithm
was realized by applying a high-gain braking torque, discussed in [2].
The experiments were performed with α = 0.175 rad ≈ 10◦ ; the results are shown in
Figure 2.2. It can be seen from Figure 2.2 that the second step of the algorithm commences
at around t = 1.1 sec and the first link reaches its upright configuration at around t = 3.0
21

sec. The swing-up of the second link (third step) is initiated at approximately t = 3.2
sec. The pendubot configuration reaches the region of attraction of the desired equilibrium
immediately after t = 7.0 sec; the linear controller is then invoked for stabilization.
The rest-to-rest maneuvers of the first link in the third step of the algorithm can be
seen from the subplot of θ1 in Figure 2.2. The two horizontal dotted lines in this plot
roughly mark the boundary of oscillation of the first joint. The amplitude of this oscillation
is approximately 13.5◦ . This is larger than the value of α ≈ 10◦ and this indicates that
the braking torque was insufficient to stop the joint immediately at the boundary of the
interval [(π/2 − α), (π/2 + α)]. In fact, it can be seen from the plot that θ1 overshot the
boundary significantly during one maneuver. A larger gain for the braking torque could

2 a
0
◦
−2 27
0
−2 b
−6
5
0
−5
10
0
−10
5
0
−5
2
0
−2

θ1 (rad)
θ2 (rad)

c
˙
θ1 (rad/s)
d
˙
θ2 (rad/s)
e
τ1 (N.m)
f
E(J)

0
step 1

2
step 2

4

time (s)
step 3

6

8

10
step 4

Figure 2.2: Experimental validation of the impulsive control algorithm [2] for pendubot
swing-up.

22

not be generated because of the limitation of the power amplifier but this did not pose any
significant problem in swing-up of the second link. Although it is possible to design electrical
circuits that enable the motor to apply torques that closely approximate an impulsive torque,
the experimental results clearly indicate that regular power amplifiers are sufficient for simple
implementations of impulsive control.
The large peaks in the subplot of τ1 (in Figure 2.2) in the third step of the algorithm
correspond to the impulsive braking torques. Each peak is followed by an oscillation in the
torque profile - this is due to amplification of measurement noise. The impulsive braking
˙
torque is computed based on θ1 , which is obtained through differentiation of the measurement
˙
of θ1 . The process of differentiation amplifies the noise, which is visible in the plot of θ1 ,
and results in the torque ripple.
˙
˙
The subplots of θ1 and θ2 indicate that the joint velocities undergo large changes (relatively speaking) at the time of application of the impulsive torques - this is consistent with
the theory of impulsive control. It can be seen from the figure that the energy of the system
also undergoes large changes at the time of application of the impulsive torques. Overall,
the experimental results are consistent with those obtained through numerical simulations
by Albahkali, et al. [2].

2.4

Conclusion

The swing-up algorithm of the Pendubot using impulsive inputs was experimentally validated. Impulsive inputs were approximated by large-gain continuous inputs and are derived
from a fast dynamics which result in desired jumps in velocities. Although the impulsive
inputs were assumed to provide instantaneous braking of the first link, the experimental

23

results indicate that the impulsive torques need not to be Dirac delta functions and the
swing-up algorithm is effective even when the magnitude of the impulsive input is bounded
and its time support is not infinitesimal. The magnitude of the impulsive inputs are however
larger than the continuous torque required by the algorithm. This should not be seen as
a problem since actuators such as motors can apply substantially larger torques than the
maximum continuous torque over small time intervals. This is referred to as the peak torque
[129] and it can be twice to ten times larger than the maximum continuous torque for different motors. The experiment results match the simulation results and this confirms that
impulsive inputs are indeed implementable in mechanical systems using regular actuators
and amplifier circuits.

24

Chapter 3
Swing-up Control of Acrobot

3.1

Introduction

In this chapter, we present an energy-based method for swing-up control of the Acrobot. The
method relies on the use of impulsive inputs together with continuous inputs and was used
earlier to develop an algorithm for swing-up of the Pendubot [2]. The algorithm developed
for the Pendubot can be directly used for swing-up of the Acrobot but it suffers from slow
convergence. A modified algorithm, based on an additional set of impulsive inputs, provides
fast convergence; also, it is not sensitive to the choice of control gains and does not impose
restrictions on the initial conditions. This algorithm is presented in Section 3.3.
The algorithm presented in Section 3.3 is based on the energy of the system and can
therefore be viewed as a modification of Spong’s algorithm [19, 20] through inclusion of
impulsive inputs in the set of admissible controls. This approach to enlarging the set of
admissible controls is general and can be used to modify other algorithms in the literature.
This is demonstrated in Section 3.4 through modification of the algorithms proposed by
Xin and Kaneda [3] and Mahindrakar and Banavar [22], which are based on the work by
Kolesnichenko and Shiriaev [21]. It is shown that impulsive inputs allow us to enlarge the
set of continuous inputs for swing-up, and thereby additionally enlarge the set of admissible
controls. This provides greater flexibility in control design.
This chapter is organized as follows. In Section 3.2 we review the dynamic model of the
25

Acrobot and derive expressions for jump in velocities and change in energy for impulsive
elbow torque. In Section 3.3 we present an algorithm for swing-up that does not suffer
from limitations. In Section 3.4 we demonstrate the flexibility in control design for Acrobot
swing-up by including impulsive inputs in the set of admissible controls. Section 3.5 provides
the concluding remarks.

3.2
3.2.1

Acrobot Dynamics and Impulsive Effects
Equations of Motion

Consider the Acrobot in Figure 3.1. Assuming no friction in the joints, the equation of
motion can be written as [19]

¨
˙ ˙
M(θ)θ + N(θ, θ)θ + G(θ) = T

(3.1)

θ = [θ1 θ2 ]T ,

(3.2)

where
T = [0 τ2 ]T

˙
and M(θ), N(θ, θ), and G(θ), given by the expressions




q2 + q3 C2 
 q1 + q2 + 2q3 C2
M(θ) = 

q2 + q3 C2
q2




˙2
˙1 + θ2 )
˙
−(θ
 −θ

 q4 C1 + q5 C12 
˙
G(θ) = g 
N(θ, θ) = q3 S2 

 ,
˙1
q5 C12
θ
0

(3.3)

are the inertia matrix, matrix containing the Coriolis and centrifugal forces, and vector of

26

g

θ2

d1

θ1

d2

l1

l2

c.m
.

Y

τ2

.
c.m

X

Figure 3.1: The Acrobot in an arbitrary configuration.

gravity forces, respectively. In Eq.(3.3), Sk = sin θk , Ck = cos θk , for k = 1, 2, C12 =
cos(θ1 + θ2 ), g is the acceleration due to gravity and qj , j = 1, 2, · · · , 5 are some constants
with the following expressions

2
q1 =m1 d2 + m2 l1 + I1 ,
1

q3 =m2 l1 d2 ,

q2 = m2 d2 + I2
2

q4 = m1 d1 + m2 l1 ,

q5 = m2 d2

(3.4)

where mk is the mass of the k-th link, dk is the distance between the k-th joint and its
center-of mass, lk is the length of the k-th link and Ik is the mass moment of inertia of the
k-th link, for k = 1, 2.

3.2.2

Holding Torque

˙
We compute the torque required to hold the second link fixed, i.e., maintain θ2 = 0. By
˙
¨
substituting θ2 = θ2 = 0 in Eq.(3.1), we get

27

















 q4 C1 +q5 C12   0 
 q1 +q2 +2q3 C2  ¨  0 

=
 +g 
 θ1 + 

˙2
τ2h
q5 C12
q3 S2 θ1
q2 +q3 C2

(3.5)

¨
Eliminating θ1 from the two equations in Eq.(3.5), the holding torque at the second joint
can be expressed as follows

˙2
τ2h = q3 S2 θ1 +

3.2.3

g
[(q +q C )q C −(q2 +q3 C2 )q4 C1 ]
q1 +q2 +2q3 C2 1 3 2 5 12

(3.6)

Impulsive Torque for Sudden Change in Velocity

˙
Consider the action that results in exponential convergence of the second link velocity, θ2 ,
˙
to a desired value, θ2des , i.e.,

¨
˙
˙
θ2 = −k1 (θ2 − θ2des ),

k1 > 0

(3.7)

˙
where k1 is a constant that determines the rate of convergence of θ2 . To compute the torque
required for this action, we multiply Eq.(3.1) with the inverse of the inertia matrix to get








¨
1
 −(q2 +q3 C2 )τ2 +h1 
 θ1 


=
2 2
q1 q2 −q3 C2
¨
(q1 +q2 +2q3 C2 )τ2 +h2
θ2

(3.8)

where h1 and h2 are given by the expressions

2 ˙2
˙
˙
h1 = q2 q3 (θ1 + θ2 )2 S2 + q3 θ1 S2 C2 + g(q3 q5 C2 C12 − q2 q4 C1 )

28

(3.9)

2
˙
˙
˙2
h2 =−(θ1 + θ2 )2 (q2 q3 +q3 C2 )S2 −(q1 +q3 C2 )q3 θ1 S2

−g {q3 q5 C2 C12 −(q2 +q3 C2 )q4 C1 +q1 q5 C12 }

(3.10)

Substitution of Eq.(3.7) into the second equation of (3.8) gives

τ2 =

−1
˙
˙
k (θ − θ
)(q q −q 2 C 2 )+h2
q1 +q2 +2q3 C2 1 2 2des 1 2 3 2

(3.11)

For a large value of gain k1 , the torque in Eq.(3.11) will be impulsive and converge the
second link velocity to the desired velocity in a short period of time.
As a special case, consider the impulsive torque that results in instantaneous stopping of
the second link. The impulsive braking torque for the second link is obtained by substituting
˙
θ2des = 0 in Eq.(3.11), namely,

τ2b =

−1
2 2
˙
k1 θ2 (q1 q2 − q3 C2 ) + h2
q1 + q2 + 2q3 C2

(3.12)

When the second joint comes to rest, the braking torque is equal to the holding torque. This
can be verified from Eq.(3.6) and Eq.(3.12).

3.2.4

Change in Velocity due to Impulsive Torque

The application of an impulsive torque in the active coordinate θ2 results in velocity jumps
in both active and passive coordinates. The relationship between the jumps in velocities
can be derived from Lagrange’s equations using the approach presented in [1]. Consider
the integral of the equations of motion in Eq.(3.1) over the short interval of time ∆t during

29

which the impulsive torque acts, i.e.
∆t

∆t

¨
˙ ˙
M(θ)θ + N(θ, θ)θ + G(θ) dt =

0

T dt

(3.13)

0

The above equation can be rewritten as:
∆t

˙
˙
M(θ)∆θ+N(θ, θ)∆θ +

∆t

G(θ) dt =
0

∆t

˙
∆θ

¨
θ dt,

T dt,
0

∆t

∆θ

0

˙
θ dt

(3.14)

0

Since ∆t ≈ 0 and the configuration of the Acrobot does not change during this time, i.e.
∆θ = 0, Eq.(3.14) will be simplified to

˙
M(θ)∆θ = Timp

(3.15)

where Timp = [0 τ2i ]T is the vector of applied impulses. From Eq.(3.3), the above equation
can be decomposed into the following equations:

˙ + ˙−
˙ + ˙−
[q1 + q2 + 2q3 C2 ](θ1 − θ1 )+[q2 + q3 C2 ](θ2 − θ2 ) = 0
˙+ ˙−
˙+ ˙−
[q2 + q3 C2 ](θ1 − θ1 )+q2 (θ2 − θ2 ) = τ2i
where

(3.16)
(3.17)

˙−
˙−
θk and θk denote the amgular velocities of the k-link, immidiately before and after

the application of impulse, respectively. Eq.(3.16) can be written as

˙+ ˙ −
˙+ ˙−
[q1 + q2 + 2q3 C2 ](θ1 − θ1 ) = −[q2 + q3 C2 ](θ2 − θ2 )

30

(3.18)

If the velocity of the second link after the impulse is equal to some desired velocity, i.e.,
˙+
˙
θ2 = θ2des , Eq.(3.18) can be used to obtain the velocity of the first link after the impulse:
q2 + q3 C2
˙+
˙−
˙
˙−
θ1 = θ1 −
(θ2des − θ2 )
q1 + q2 + 2q3 C2

3.2.5

(3.19)

Change in Energy due to Impulsive Torque

The configuration, and hence the potential energy, of the Acrobot will not change over the
small period of time that the impulse is applied. Therefore, the change in total energy of
the system is due to the change in the kinetic energy, which can be expressed as

∆E = ∆K1 + ∆K2

(3.20)

where ∆K1 and ∆K2 denote the change in kinetic energy of the first and second link, and
are given by the relations

1
˙+
˙−
∆K1 = (I1 + m1 d2 )[(θ1 )2 − (θ1 )2 )]
1
2

1
1
˙+ ˙+
˙− ˙−
∆K2 = I2 (θ1 + θ2 )2 − I2 (θ1 + θ2 )2
2
2
1
2 ˙+
˙+ ˙+
˙+ ˙+ ˙+
+ m2 l1 (θ1 )2 + d2 (θ1 + θ2 )2 + 2l1 d2 θ1 (θ1 + θ2 )C2
2
2
1
2 ˙−
˙ − ˙−
˙− ˙− ˙−
− m2 l1 (θ1 )2 + d2 (θ1 + θ2 )2 + 2l1 d2 θ1 (θ1 + θ2 )C2
2
2

(3.21)

(3.22)

If we assume the velocity of the second link after the impulse to be equal to the desired
˙+
˙
velocity, i.e., θ2 = θ2des , the change in total energy can be obtained by substituting Eq.(3.19)

31

into Eqs.(3.21) and (3.22) as follows:

∆E =

2 2
q1 q2 − q3 C2
˙
˙−
(θ2des )2 − (θ2 )2
2(q1 + q2 + 2q3 C2 )

(3.23)

Equation Eq.(3.23) implies that the change in total energy of the system depends on the
˙
magnitude of θ2des , i.e.,
˙
˙−
∆E > 0 if |θ2des | > |θ2 |
˙
˙−
∆E = 0 if |θ2des | = |θ2 |

(3.24)

˙
˙−
∆E < 0 if |θ2des | < |θ2 |
˙
If an impulsive braking torque is used, i.e., θ2des = 0, the change in energy is negative
and is given by the relation

∆E = −

3.3
3.3.1

2 2
q1 q2 − q3 C2
˙
(θ − )2
2(q1 + q2 + 2q3 C2 ) 2

(3.25)

Swing-Up Control of the Acrobot
Rest-to-Rest Maneuvers of the Second Link

Consider a maneuver in which the second joint starts from rest under the application of
a continuous torque and is brought back to rest through the application of the impulsive
braking torque in Eq.(3.12), where gain k1 has a large value. Taking into account the loss
of energy due to impulsive braking given by Eq.(3.25), the change in energy during each

32

rest-to-rest maneuver can be computed as follows

∆E =

˙
τ2c θ2 dt −

2 2
q1 q2 − q3 C2
˙
(θ − )2
2(q1 + q2 + 2q3 C2 ) 2

(3.26)

If the torque in Eq.(3.26) is assumed to have the form

¨
τ2c = k2 θ2 ,

k2 > 0

(3.27)

˙−
the work done by the torque to increase the second link velocity from zero to θ2 , which is
the velocity prior to application of the impulsive brake, is expressed as follows

˙
τ2c θ2 dt =

˙ ¨
k2 θ2 θ2 dt =

˙−
θ2
0

1
˙−
˙ ˙
k2 θ2 dθ2 = k2 (θ2 )2
2

(3.28)

The total change in energy due to the rest-to-rest maneuver can now be obtained by substituting Eq.(3.28) into Eq.(3.26)

∆E =

2 2
q1 q2 − q3 C2
1
˙−
k2 −
(θ2 )2
2
(q1 + q2 + 2q3 C2 )

(3.29)

From Eq.(3.29), it can be seen that the gain k2 can be chosen to add or remove energy,
namely
2 2
q1 q2 − q3 C2
(q1 + q2 + 2q3 C2 )
2 2
q1 q2 − q3 C2
k2 <
(q1 + q2 + 2q3 C2 )

k2 >

33

⇒ ∆E > 0
⇒ ∆E < 0

(3.30)

Using Eq.(3.8), the torque expression in Eq.(3.27) can be written as:

τ2c =

2 2
q1 q2 − q3 C2

k2 h2
− k2 (q1 + q2 + 2q3 C2 )

(3.31)

The above expression for the torque will not be singular if k2 is chosen based on Eq.(3.30).

3.3.2

Acrobot Swing-up Using Rest-to-Rest Maneuvers

3.3.2.1

Swing-up Algorithm

A swing-up algorithm for the Acrobot is proposed using rest-to-rest maneuvers of the second
link. This algorithm is similar to the one proposed for the Pendubot [2] in the sense that
energy of the system is increased through rest-to-rest maneuvers, and impulsive control
inputs are used as braking torques. The algorithm is comprised of the following three steps:
1. Initialization:
• Linearize the dynamic equations of the Acrobot in Eq.(3.1) about the equilibrium
˙
˙
configuration (θ1 , θ1 , θ2 , θ2 ) = (π/2, 0, 0, 0).
• Design a linear controller to render the equilibrium locally asymptotically stable.
Let RA define the region of attraction of the equilibrium.
• Choose a small positive constant δ, such that the system configuration lies inside
˙
˙
RA when θ1 ≈ θ2 ≈ 0 and |P − Edes | < δ, where P is the potential energy and
Edes is the total energy of the Acrobot at its upright equilibrium.
2. Increasing the energy of the system:

34

• Conduct rest-to-rest maneuvers to increase E to its desired value, Edes . The restto-rest maneuvers will be implemented with θ2 satisfying −α ≤ θ2 ≤ α, where α is
a small positive angle. Using Eq.(3.30), the control gain for the torque expression
in Eq.(3.31) is chosen such that ∆E > 0, i.e., the energy is increased through the
rest-to-rest maneuvers. In particular, the following procedure will be adopted.
The holding torque in Eq.(3.6) will be applied to hold the second link fixed. To
initiate motion of the second link in the positive (counter-clockwise) direction, the
torque expression in Eq.(3.31) will be used when it is greater than τ2h . To initiate
motion in the negative (clockwise) direction, the torque expression in Eq.(3.31)
will be used when it is less than τ2h . As the second link approaches the boundary
of the interval [−α, α], the braking torque τ2b in Eq.(3.12) will be used with a
large value of gain k1 to quickly stop the motion of the second link.
• Terminate the rest-to-rest maneuvers when the condition |E−Edes | < δ is satisfied
and apply the holding torque in Eq.(3.6) to keep the second link fixed.
3. Stabilization:
˙
With −α ≤ θ2 ≤ α, θ2 = 0, and |E − Edes | < δ, the first link will behave like a
pendulum and reach its configuration with maximum potential energy in finite time.
˙
˙
When the Acrobot has maximum potential energy, i.e. |P −Edes | < δ and θ1 = θ2 = 0,
its configuration will be inside RA . Invoke the linear controller, designed in the first
step of the algorithm, to stabilize the equilibrium.

Remark 3.3.1. For the rest-to-rest maneuvers of the second link in step 2, τ2c in Eq.(3.31)
has to be greater (or less) than τ2h to initiate the motion of the second link in the counter35

˙
clockwise (or clockwise) direction when θ2 = −α (or θ2 = +α) and θ2 = 0. We show that
˙
˙
this condition can be satisfied by choosing proper values for α. For θ1 = θ2 = 0, Figure 3.2
shows the regions in θ1 -θ2 space where Π = (τ2c − τ2h ) is positive and negative. If the second
link is held fixed at θ2 = +α or θ2 = −α, the Acrobot will oscillate like a pendulum and θ1
˙
will oscillate between its maximum and minimum values where θ1 = 0. The amplitude of
oscillation of θ1 will depend on the energy of the Acrobot and the maximum and minimum
values of θ1 will correspond to Acrobot configurations that lie at the end points of the dashed
lines in Figure 3.2. If the end point of the dashed lines are on opposite sides of the curve
Π = 0, as shown in Figure 3.2, it will be possible to initiate the rest-to-rest maneuvers in the
desired direction. To ensure that the end points are on opposite sides of the curve Π = 0,
we will have to properly choose the value of α, i.e., choose a small value when the Acrobot
has low energy.
Remark 3.3.2. It may be desirable to choose a fixed value of α, say α, for all the rest-to-rest
¯
maneuvers during swing-up. It follows from Remark 3.3.1 that it will be possible to conduct
0.8

Π=0

θ2 (rad)

0.4

Π>0

α
α
¯

0.0

η
¯
Π<0

-0.4

-0.8

−3π/2

−α
Π=0
−π/2
−π
θ1 (rad)

Π=0
0

π/2

˙
˙
Figure 3.2: Plot showing θ1 -θ2 space where Π is positive and negative for θ1 = θ2 = 0.
36

¯
the rest-to-rest maneuvers successfully if the initial energy of the Acrobot is greater than P ,
¯
where P is the potential energy of the Acrobot at the configuration (θ1 , θ2 ) = (α, η ) and α
¯ ¯
¯
and η are shown in Figure 3.2.
¯

3.3.2.2

Numerical Simulation

We present numerical simulation results using kinematic and dynamic parameters of the
Acrobot from Xin and Kaneda [3]:

m1 = 1 kg, l1 = 1 m, d1 = 0.5 m, I1 = 0.083 N.m2
m2 = 1 kg, l2 = 2 m, d2 = 1 m, I2 = 0.33 N.m2

(3.32)

˙ ˙
The initial conditions were chosen as (θ1 , θ2 , θ1 , θ2 ) = (−π/4, 0, 0, 0) and the algorithm
parameters were set to be δ = 0.1 J and α = 0.17 rad. For the parameters in Eq.(3.32), Edes
was computed to be 49.05 J. The simulation results are shown in Figure 3.3. By choosing
the gain k2 in Eq.(3.31) appropriately, the total energy of the Acrobot is increased in each
rest-to-rest maneuver. The peaks in the torque plot represent impulsive braking torques
used in the rest-to-rest maneuvers.
The main drawback of the algorithm in Section 3.3.2.1 is that it takes a long time for
swing-up. For the simulation in Figure 3.3, the algorithm is switched to the linear controller
at about t = 115 sec, which is shown by vertical dotted line. This time is much longer than
swing-up times reported in the literature. For example, for the same kinematic and dynamic
parameters and lower initial energy level, the algorithm in [3] achieved swing-up in about 8
sec. The swing-up time required by our algorithm will be longer if the initial energy of the
Acrobot is lower due to the constraint on the choice of α - see Remark 3.3.1. In the next

37

θ1 (rad)
˙
˙
θ2 (rad/s) θ1 (rad/s) θ2 (rad)

0.17
0.0
-0.17
5
0
-5
1
0
-1

τ2 (N.m)

E(J)

3.5
0.0
-3.5

a
b

c

d

50

e

0
10
0
-10

f

0

20

40

60

80

100

120

time (sec)
Figure 3.3: Simulation results for swing-up of the Acrobot using rest-to-rest maneuvers of
the second link.

section, we modify the algorithm to provide faster swing-up.

3.3.3

Increasing Energy of the Acrobot Using Impulsive Inputs

3.3.3.1

Swing-up Algorithm

The algorithm in Section 3.3.2 used impulsive inputs for braking, which reduces the energy
of the system. For faster swing-up, we propose to use impulsive inputs to increase the energy
of the Acrobot. Specifically, we apply the impulsive torque in Eq.(3.11) with large gain k1
˙
and θ2des chosen such that the energy of the system increases based on the condition in

38

Eq.(3.24). If the velocity of the second link after the impulse is chosen as,

˙
˙−
θ2des = −λ θ2 ,

˙−
θ2 = 0,

λ>1

(3.33)

the second link will have a velocity greater in magnitude and opposite in direction to that
prior to the impulse, and the energy of the system will be higher. The change in sign of
the velocity is needed to keep the second link angle bounded within some predefined range
while the energy is raised to the desired level. Most of the energy gained from application
of impulsive torques will be in the form of kinetic energy and needs to be converted into
potential energy for swing-up. To convert kinetic energy into potential energy, the torque
is set to zero and the second joint is made free. This keeps the total energy of the system
constant and allows kinetic energy to be converted into potential energy. The rest-to-rest
maneuvers are used subsequently to regulate the energy of the Acrobot to the desired level
Edes . The modified algorithm is stated next:
1. Initialization: Same as step 1 in Section 3.3.2.1.
2. Increasing the energy of the system:
˙
• At the initial time, if θ2 = θ2 = 0, apply the impulsive torque in Eq.(3.11), where
˙
θ2des is chosen to be nonzero. From Eq.(3.24), we know that this will increase the
energy of the Acrobot.
• Using Eq.(3.33), apply the impulsive torque in Eq.(3.11) when the second link
approaches a bound of the interval [−γ, γ] from inside, where γ is some positive
angle - see Figure 3.4. Note that γ can always be chosen such that the second link
approaches the bound of the interval [−γ, γ] for any initial conditions or initial
39

˙
velocity of the second link, θ2des . Applying these impulsive torques will increase
the energy of the system, mainly in the form of kinetic energy. Continue this
process till E ≥ Edes .
• Release the system by setting τ2 = 0. This will cause the links to swing freely
resulting in exchange between kinetic and potential energies while the total energy
remains constant.
• Stop the second link using the impulsive braking torque in Eq.(3.12) when the
˙
¨
potential energy of the Acrobot reaches its local maxima, i.e, P = 0, P < 0. The
braking action will reduce the total energy of the system according to Eq.(3.25)
but the potential energy will remain at its local maxima.
3. Regulation of energy to the desired level:
• Same as step 2 in Section 3.3.2.1: Conduct rest-to-rest maneuvers with θ2 ∈
[−α, α]. If the second link is initially out of this bound, it will be brought inside
by the first rest-to-rest maneuver. Depending on the current level of energy, the
gain k2 is chosen from Eq.(3.30) to regulate (increase or decrease) E to Edes .
• Terminate the rest-to-rest maneuvers when the condition |E−Edes | < δ is satisfied
and apply the holding torque in Eq.(3.6) to keep the second link fixed.
4. Stabilization: Same as step 3 in Section 3.3.2.1.
Remark 3.3.3. The above algorithm is not sensitive to the choice of control gains and does
not impose restrictions on the initial conditions. It also provides fast swing-up of the Acrobot
- this is demonstrated next through simulations.
40

Y

X

τ2
˙+
˙−
θ2 = −λ θ2

+γ

−γ

Figure 3.4: Configuration of the Acrobot at the time when the impulsive torque is applied
to increase the energy of the system.
3.3.3.2

Numerical Simulation

We present numerical simulation results using kinematic and dynamic parameters of the
Acrobot from Xin and Kaneda [3], given in Eq.(3.32). The parameters of our algorithm were
chosen as:
λ = 2.5,

δ = 0.10 J,

γ = 60◦ ,

α = 5◦

(3.34)

˙ ˙
The initial conditions were chosen at (θ1 , θ2 , θ1 , θ2 ) = (−π/2, 0, 0, 0)1 and the initial im˙
pulsive torque was chosen to result in θ2des = 3.0 rad/s. For the parameters in Eq.(3.32),
Edes was computed as 49.05 J. The simulation results are shown in Figure 3.5. The total
energy (solid line) is discretely increased through the application of impulsive torques. After five discrete applications of impulsive torque, the condition E ≥ Edes is satisfied and
the second joint is released. The potential energy (dashed line) subsequently increases, and
1 This

initial configuration corresponds to zero kinetic energy and lowest potential energy and
cannot be handled by the algorithms in Xin and Kaneda [3] and Mahindrakar and Banavar [22].

41

0
−4
8
4
0
10
0
−10

a
θ1 (rad)
b

θ2 (rad)

c
˙
θ1 (rad/s)
˙
θ2 (rad/s)

20 d
0
−20
40
0
10
0
−10

e
E, P (J)
f
τ2c (N.m)

100 g
0
−100
0

2.42

3.86

2
step 2

4

time (sec)

step 3

τ2i (N.m)

5.35

6

8

step 4

Figure 3.5: Simulation results for swing-up of the Acrobot using impulsive inputs to increase
the energy.

when it reaches its local maxima, the second link is stopped using an impulsive brake. This
event, which occurs at approximately t = 2.42 sec, results in a drop in the total energy of
the system. After application of the impulsive brake, the condition |E − Edes | < δ is not
satisfied and therefore one rest-to-rest maneuver is used to regulate the energy to Edes . The
rest-to-rest maneuver is completed at approximately t = 3.86 sec and the linear controller
is invoked at t = 5.35 sec. In comparison to the algorithm proposed in [3], our algorithm
requires shorter time for swing-up: 5.35 sec compared to 7.33 sec, and lower maximum continuous torque: 12.5 N.m compared to 20.0 N.m, despite starting from a lower initial energy
configuration.

42

Numerical simulations results of the algorithm using Acrobots parameters in Mahindrakar
and Banavar [22] can be found in [130].

3.4
3.4.1

Modification of an Existing Method
Existing Control Method

Kolesnichenko and Shiriaev [21] developed a general methodology for control of underactuated systems; this method was used for Acrobot swing-up by Xin and Kaneda [3] and
Mahindrakar and Banavar [22]. We first review this method and then modify it to include
impulsive control inputs.
Consider the dynamics of the Acrobot in the following form:

x = f (x, τ2c )
˙

(3.35)

˙ ˙
where x = (θ1 , θ2 , θ1 , θ2 )T and f (x, τ2c ) can be determined using Eq.(3.8). A continuous
controller is designed to achieve the following objectives:

lim θ2 = 0,

t→∞

˙
lim θ2 = 0,

t→∞

lim E = Edes

t→∞

(3.36)

˙
With θ2 = 0, θ2 = 0, and E = Edes , the Acrobot behaves like a pendulum and its trajectory
converges to a heteroclinic orbit that passes through the desired upright equilibrium. To
achieve the control objective, the following “positive storage function” is used:

1
2
˙
˙
kp θ2 + kd (θ2 )2 + ke (E − Edes )2
V (θ, θ) =
2

43

(3.37)

where kp , kd , ke > 0. The following control input is chosen:

τ2c = −

˙
˙
kp θ2 + kd a(θ, θ) + φ(θ2 )
kd b(θ2 ) + ke (E − Edes )

(3.38)

˙
where a(θ, θ) and b(θ2 ) are given by the expressions

˙
a(θ, θ) =
b(θ2 ) =

1
h
2 2 2
q1 q2 − q3 C2

1
(q + q2 + 2q3 C2 )
2 2 1
q1 q1 − q3 C2

(3.39)

and the condition
kd min{b(θ2 )} > ke Edes
θ2

(3.40)

is satisfied to avoid singularity in the control torque in Eq.(3.38). Using Eq.(3.38), the
derivative of the storage function is shown to be:

˙
˙
˙
˙
V (θ, θ) = −θ2 φ(θ2 )

(3.41)

˙
˙
where φ(θ2 ) is chosen such that V ≤ 0, and therefore the system is stable. In [3] and [22],
˙
the function φ(θ2 ) is chosen as:

˙
˙
φ(θ2 ) = kc θ2 ,

kc > 0

(3.42)

The invariant set for the closed-loop system is given by

M = V0 ∪ Ω

44

(3.43)

where V0 corresponds to the desired equilibrium and Ω is the finite set of undesirable equilibria:

V0 = {x ∈ R4 | V (x) = 0}
˙
Ω = {x ∈ R4 | V (x) > 0 , V ≡ 0}

(3.44)

In [3] and [22], the gains kp , kd , and ke are chosen such that the equilibria in Ω are hyperbolic
and V0 is attractive. This choice of gains ensures that the the control objective is achieved
for all initial conditions except a set from which system trajectories do not converge to the
desired equilibrium.

3.4.2

Modified Method Using Impulsive Inputs

3.4.2.1

Impulsive System

We modify the control method in [3] and [22] to include impulsive inputs. Consider the
following impulsive dynamical system:

x(t) = f (x, τ2c ), t ∈ η(x)
˙
/
∆x(t) = H(x, τ2i ), t ∈ η(x)

(3.45)

where x, f (x, τ2c ) and τ2c are given in Eqs.(3.35) and (3.38), and correspond to the closedloop system dynamics of the Acrobot designed by Xin and Kaneda [3] and Mahindrakar and
Banavar [22]. H(x, τ2i ) represents a jump in the states of the Acrobot due to the application

45

of the impulse τ2i . Using Eqs.(3.16) and (3.17), it can be shown

H(x, τ2i ) =

(x+

− x− )

q +q +2q C
q +q C
= 0 0 − 2 32 2 2 1 2 2 3 2 2
q1 q2 −q3 C2
q1 q2 −q3 C2

T

τ2i

(3.46)

In (3.46), x+ and x− represent the right and left limits of x at the instant of the impulse,
respectively. By choosing the impulse τ2i to be

τ2i = −

2 2
q1 q2 − q3 C2
˙
θ−
q1 + q2 + 2q3 C2 2

(3.47)

˙+
we get impulsive braking which results in θ2 = 0. This can be verified from H(x, τ2i ), which
can be rewritten as:

H(x, τ2i ) = 0 0

T
q2 + q3 C2
˙−
˙−
θ2 − θ2
q1 + q2 + 2q3 C2

(3.48)

We choose the set η(x) in Eq.(3.45) to be:

η(x) = {t ∈ [t0 , ∞) | x ∈ ρ(x)}

(3.49)

˙ ¨
ρ(x) = {x ∈ R4 | θ2 θ2 < 0}

(3.50)

˙ ¨
ρ⊥ (x) = {x ∈ R4 | θ2 θ2 ≥ 0}

(3.51)

where

We define the set ρ⊥ (x) as

46

such that
ρ(x) ∩ ρ⊥ (x) = ∅

(3.52)

˙+
Since θ2 = 0 from our choice of τ2i in Eq.(3.47), x+ lies on the boundary of ρ⊥ (x). By
˙ ¨
taking the derivative of θ2 θ2 with respect to time immediately following the impulse, we get

˙ ¨
¨+
˙+ + (3) = (θ+ )2 ≥ 0
¨
D(θ2 θ2 ) = (θ2 )2 + θ2 θ2
2

(3.53)

˙ ¨
which means that θ2 θ2 is nondecreasing after the impulse. It follows from continuity that
x will be in the interior of ρ⊥ (x) after an impulse and there exists a nonzero time interval
ǫ > 0 before x can enter the set ρ(x). Therefore there is no instant with multiple impulses.

3.4.2.2

Stability of Impulsive System

The jump in the positive storage function in Eq.(3.37) due to the impulse in Eq.(3.47) can
be computed as follows

∆V = V (x+ ) − V (x− )
1
1
˙−
= − kd (θ2 )2 + ke (E + −Edes )2 −(E − −Edes )2
2
2
1
1
˙−
= − kd (θ2 )2 + ke (E + +E − −2Edes ) ∆E
2
2
− )2
˙
˙−
(θ2
ke (θ2 )2
=−
kd b(θ2 )+
+ke (E + −Edes )
2b(θ2 )
4b(θ2 )
˙−
1 kd b(θ2 )+ke(E + −Edes ) ˙− 2 ke (θ2 )4
(θ2 ) + 2
=−
2
b(θ2 )
4b (θ2 )

47

(3.54)

where Eqs.(3.25) and (3.39) were used. From the condition in Eq.(3.40) we have
kd b(θ2 ) + ke (E + − Edes )
>0
b(θ2 )

⇒

∆V ≤ 0

(3.55)

The continuous control input given by Eqs.(3.38) and (3.42) and the impulsive input in
Eq.(3.47) results in

D+V ≤ 0

∀t ∈ η(x)
/

∆V ≤ 0

∀t ∈ η(x)

(3.56)

where D + V denotes the upper right Dini derivative of V with respect to time. Using the
stability theorems in [9] and [131], we can claim stability of the impulsive dynamical system
in (3.45).
To prove the asymptotic convergence to the desired equilibrium, we use the Invariance
Principle for impulsive dynamical systems in [132] and [133]. To find the largest invariant
set, we define the set

˙
Z ={x ∈ R4 | V = 0, x ∈ ρ⊥ (x)} ∪ {x ∈ R4 | ∆V = 0, x ∈ ρ(x)}

(3.57)

Using Eqs.(3.41), (3.50) and (3.54), it can be shown

˙
Z = {x ∈ R4 | θ2 = 0, x ∈ ρ⊥ (x)} ∪ ∅

48

(3.58)

Therefore, the largest invariant set M is given by

˙
M = {x ∈ R4 | θ2 ≡ 0, x ∈ ρ⊥ (x)}

(3.59)

which is the same set as in Eq.(3.43) obtained in [3] and [22]. The invariant set is not simply
V0 ; it includes Ω which contains a finite set of equilibrium points. In [3] and [22], a set
of conditions were imposed such that all continuous trajectories that do not start in Ω are
attracted to V0 . For an impulsive dynamical system we have both continuous and discrete
trajectories, and therefore, we must extend the results in [3] and [22] to include discrete
trajectories.
We need to find conditions such that discrete trajectories do not end in Ω. To this end,
we define the set
˙ ¨
S(x) = {x ∈ R4 | θ2 θ2 = 0}

(3.60)

which is the boundary between ρ(x) and ρ⊥ (x). Let S1 (x) be the subset of S(x) from where
trajectories enter ρ(x) from ρ⊥ (x), i.e.,

S1 (x) = {x ∈ S(x) |

d ˙ ¨
(θ θ ) < 0}
dt 2 2

(3.61)

Each element of S1 is arbitrarily close to a point in ρ(x) from where a discrete trajectory
originates and satisfies

d ˙ ¨
˙ (3)
¨
˙ (3)
(θ2 θ2 ) = θ2 θ2 + (θ2 )2 < 0 ⇒ θ2 θ2 < 0
dt

49

(3.62)

In addition, define the set S2 as

S2 (x) = {x ∈ S(x) | x + H(x, τ2i ) ∈ Ω}

(3.63)

Each element of S2 is arbitrarily close to a point in ρ(x) from where a discrete trajectory ends
in Ω. The set S1 ∩ S2 contains all points from where trajectories of the closed-loop system
end in Ω. We are now ready to state the additional condition that needs to be satisfied for
asymptotic convergence of trajectories to the desired equilibrium.
Theorem 3.4.1. Consider the impulsive dynamical system in Eq.(3.45) with continuous
control given by Eqs.(3.38) and (3.42); and impulsive control given by Eq.(3.47) and applied
at t ∈ η(x), defined in Eq.(3.49). If the positive gains kp , kd , ke , and kc are chosen such
that
˙ (3)
θ2 θ2 ≥ 0 ∀ x ∈ S2

(3.64)

then no discrete trajectories will end in the set Ω of the invariant set M.
Proof. From the definition of S1 in (3.61) it can be shown that the condition in (3.64) implies
S1 ∩ S2 = ∅.
Remark 3.4.1. The algorithms in [3] and [22] impose constraints on kp , kd and ke values
to guarantee convergence of Acrobot trajectories to the upright equilibrium provided that
certain initial conditions are avoided. The constraint in (3.64) has to be additionally satisfied
for implementation of the modified approach that includes impulsive inputs, presented above.
Although the modified approach imposes an additional constraint on the gains, the use of
impulsive inputs provides us with greater flexibility in choosing the continuous control inputs.
This will be illustrated in Section 3.4.3.
50

3.4.2.3

Numerical Simulation

We provide simulation results for the modified impulsive controller presented above and
compare them with the results of the continuous controller in [3]. The kinematic and dynamic
parameters of the Acrobot are taken from [3] and are given here in Eq.(3.32). The initial
conditions and gains were chosen as follows:

˙ ˙
(θ1 , θ2 , θ1 , θ2 ) = (−1.4, 0, 0, 0)
100
kp = 61.2, kd = 35.8, ke = 1, kc =
1+0.14 V (x)

(3.65)

The initial conditions and the gains kp , kd and ke in Eq.(3.65) are the same as those in [3]
and they guarantee convergence of the Acrobot trajectory to the upright equilibrium under
continuous control. To establish convergence using the impulsive control approach, we used
the procedure in the Appendix A to determine the elements of S2 , defined in (3.63). For
the initial conditions and gain values in Eq.(3.65), the positive storage function V (x) for
all x ∈ S2 , was found to be larger than the value of V at the initial time. This implies
that S2 = ∅ for positive-time solution trajectories and therefore the condition in Eq.(3.64)
is trivially satisfied.
The simulation results are shown in Figure 3.6. For the parameters in Eq.(3.32), Edes
was computed to be 49.05 J. The solid and dashed lines in the plots for energy E, positive
storage function V , and control input τ2 correspond to the modified impulsive controller
and the controller in [3], respectively. Note that the torque plot for the modified controller
includes both continuous and impulsive torques, i.e., τ2 = τ2c + τ2i . Both controllers achieve
swing-up in approximately 8 sec, after which a linear controller is invoked for stabilization.

51

10
0
-10
5
0
-5
5
0
-5
10
0
-10
40
0

a

θ1 (rad)

b

θ2 (rad)

c

˙
θ1 (rad/s)

d

˙
θ2 (rad/s)

e

E (J)

1000
0

V

f

0
g
-200
0

τ2 (N.m)

2

4

time (sec)

6

8

10

Figure 3.6: Simulation results for swing-up of the Acrobot: Comparison of the modified
impulsive controller (solid line) and the continuous controller (dashed line) proposed by Xin
and Kaneda [3].
Although the time required for swing-up is similar, the impulsive controller requires larger
continuous torques and multiple impulses within a short interval of time.

3.4.3

Alternate Continuous Control Design

3.4.3.1

Control Design

Consider the impulsive dynamical system in Eq.(3.45) with positive storage function given
by Eq.(3.37). We choose to apply the same impulsive control input in Eq.(3.47) at time

52

instants given by Eq.(3.49) but modify the continuous control input in Eq.(3.38) as follows:

τ2c = −

¨
˙
kp θ2 + kd a(θ, θ) + kc θ2
,
kd b(θ2 ) + ke (E − Edes )

kc > 0

(3.66)

This results in
˙ ¨
D + V = −kc θ2 θ2

(3.67)

From the definitions for η(x) and ρ(x) in Eq.(3.49) and Eq.(3.50), we have

˙ ¨
D + V = −kc θ2 θ2 ≤ 0

∀t ∈ η(x)
/

(3.68)

¨
Using Eqs.(3.8) and (3.39), the expression for θ2 can be found and substituted into Eq.(3.66)
to get
τ2c = −

˙
kp θ2 + (kd + kc ) a(θ, θ)
(kd + kc ) b(θ2 ) + ke (E − Edes )

(3.69)

Since b(θ2 ) > 0 and kc , kd , ke > 0, the singularity in the torque expression in Eq.(3.69) can
be avoided by simply choosing the gains as follows

kd min{b(θ)} > ke Edes
θ

(3.70)

The jump in V due to the impulse is the same as in Eq.(3.54) and therefore the stability of
the system is guaranteed - this follows from our discussion in Section 3.4.2.2.
The invariant set M is found to be the same as the set in Eq.(3.59), which is identical
that derived in [3] and [22]. To prove the convergence to the desired upright configuration,
we follow the same procedure as in Section 3.4.2.2. The gain values kp , kd , ke and kc need to
be chosen such that the condition in Theorem 3.4.1 is satisfied in addition to the conditions
53

in [3] and [22].

3.4.3.2

Numerical Simulation

We compare the performance of the controller presented above with that of the continuous
controller in [3]. The kinematic and dynamic parameters of the Acrobot are taken from [3]
and are given in Eq.(3.32). The initial conditions and gains were chosen as:

˙ ˙
(θ1 , θ2 , θ1 , θ2 ) = (−1.4, 0, 0, 0)
kp = 61.2, kd = 35.8, ke = 1, kc = 75.6

(3.71)

The initial conditions and the gains kp , kd and ke are the same as those in [3] and they
guarantee convergence of the Acrobot trajectory to the upright equilibrium under continuous control. To establish convergence using the impulsive control approach, we used the
procedure in Appendix A to determine the elements of S2 , defined in Eq.(3.63). For the initial conditions and gain values in Eq.(3.71), the set S2 was found to be empty and therefore
the condition in Eq.(3.64) is trivially satisfied.
The simulation results are shown in Figure 3.7. As in the last simulation, the solid and
dashed lines in the plots for energy E, positive storage function V , and control input τ
correspond to our controller and the controller in [3], respectively. Our controller achieves
swing up in approximately 7.33 sec. This is slightly faster than the controller in [3], which
requires 8 sec. For both controllers, a linear controller was invoked for stabilization after
swing-up. It can be noted that the number of impulses used by our controller are fewer
and farther apart than those in Figure 3.6. Furthermore, the maximum continuous torque
is approximately 15 N.m, which is lower than 20 N.m required by the controller in [3].

54

0

θ1 (rad)

-5 a
0

θ2 (rad)

-4 b
0
-4 c
5
0
-5 d
40 e
20
0
1000
500
0 f
80 g
40
0
0

˙
θ1 (rad/s)
˙
θ2 (rad/s)
E (J)
V
τ (N.m)
2

4

time (sec)

6

8

10

Figure 3.7: Simulation results for swing-up of the Acrobot: Comparison of the controller in
Section 3.4.3 (solid line) with the continuous controller (dashed line) proposed by Xin and
Kaneda [3].

The impulsive control approach allows us to replace the continuous controller in Eq.(3.38)
with the controller in Eq.(3.69) and comparison of simulation results in Figures 3.6 and 3.7
indicate improvement in performance.
Numerical simulations results of the algorithm using Acrobots parameters in Mahindrakar
and Banavar [22] can be found in [134].

3.5

Conclusion

We proposed two swing-up algorithms for the Acrobot based on continuous and impulsive
control inputs. The first algorithm is energy-based and is a modification of the algorithm

55

developed earlier for the Pendubot [2]. The algorithm for the Pendubot can be directly used
for swing-up of the Acrobot but provides slow convergence; a modification of the algorithm,
based on a set of impulsive inputs that increase the energy of the system, provides rapid
swing-up. As compared to other general algorithms in the literature, our algorithm is not
sensitive to control gains and does not impose constraints on the control gains or initial
conditions. Numerical simulations were presented to show swing-up from an initial condition
that corresponds to the equilibrium configuration with the lowest potential energy. This
initial condition cannot be handled by existing algorithms in the literature.
The idea of enlarging the set of admissible controls to include impulsive input is quite
general and can be used to modify existing algorithms in the literature. This is demonstrated
by the second swing-up algorithm presented in this chapter, which is a modification of
the algorithms in [3] and [22]. The modification based on impulsive inputs imposes an
additional constraint on the control gains but provides greater flexibility in choosing the
continuous control inputs. This results in improved performance, which is demonstrated
through numerical simulations. The numerical simulations also indicate that the additional
constraint is trivially satisfied.

56

Chapter 4
Enlarging Region of Attraction for
Underactuated Systems

4.1

Introduction

The main algorithms for stabilization of underactuated systems were discussed in Section
1.3. Regardless of the algorithm, the system trajectories will not approach the desired
equilibrium if the configuration of the system lies outside the region of attraction, RA . In this
chapter, we propose a control algorithm which can enlarge the RA of a desired equilibrium
for underactuated systems. This implies that it may be possible to stabilize an equilibrium
from a configuration which lies outside the RA for an existing controller. Our algorithm uses
impulsive inputs in conjunction with the continuous control inputs and its success depends
on two sufficient conditions that need to be satisfied by the impulsive inputs.
This chapter is organized as follows: In Section 4.2, we model the effect of impulsive
inputs on the dynamics of underactuated systems. An algorithm to enlarge the RA of an
equilibrium of the system is proposed in section 4.3. In Section 4.4, the algorithm is applied to
three underactuated systems, namely, the Pendubot, the Acrobot, and the Rolling Acrobot,
for all of which linear stabilizing controllers were designed. In Section 4.5, the algorithm is
applied to the inverted pendulum on slope and the ball-beam system for which nonlinear
stabilizing controllers were designed. Concluding remarks are provided in section 4.6.
57

4.2
4.2.1

Background
Dynamic Model

For an underactuated dynamical system, Lagrange’s equations have the form:

M(q)¨ + N(q, q)q + G(q) = u
q
˙ ˙

(4.1)

where q ∈ Rn represents the vector of generalized coordinates, and M(q) ∈ Rn×n , N(q, q) ∈
˙
Rn×n and G(q) ∈ Rn denote the positive definite inertia matrix, the matrix of centrifugal
and Coriolis forces, and the vector of gravitational forces, respectively. u ∈ Rn is the vector
of generalized forces and has the following form:

u = [0, · · · , 0, u1, · · · , um ]T ,

m<n

(4.2)

The degree of underactuation of the system is equal to n − m.

4.2.2

Impulsive Inputs

Since ideal impulses can not be implemented in practice, we approximate impulsive inputs
by continuous control inputs with very large gains. A set of impulsive inputs corresponding
to the actuated coordinates can be obtained from the dynamics:

qi = −ki (qi − qi ) ,
¨
˙
˙des

i = 1, · · · , m

(4.3)

˙
where ki is a large positive gain and qi is the desired velocity for qi immediately after the
˙des

58

application of the impulse. Using Eq.(4.1), Eq.(4.3) can be written as:

Ji M −1 (q) [u − N(q, q)q − G(q)] = −ki (qi − qi ) ,
˙ ˙
˙
˙des

i = 1, · · · , m

(4.4)

where Ji ∈ R1×n is a row vector with all zero elements except (n − m + i)-th element which
is unity. Equation (4.4) can be rewritten as follows:

Ji M −1 (q)[N(q, q)q + G(q)] − ki (qi − qi ) = Ji M −1 (q)u ,
˙ ˙
˙
˙des

i = 1, · · · , m

(4.5)

The set of m equations in Eq.(4.5) can be solved to obtain the impulsive inputs uimp =
imp

[u1

imp

, · · · , um ]T . It was shown in [2] that the dynamics in Eq.(4.3) closely approximates

an impulsive input for large gains ki .

4.2.3

Effect of Impulsive Inputs on Velocity

Impulsive inputs cause velocity jumps in both active and passive coordinates of the system.
The relationship for the velocity jumps can be derived from Lagrange’s equations. The
integral of the equations of motion in Eq.(4.1) over the short interval of time ∆t, during
which the impulsive inputs act, gives:
∆t

∆t

[M(q)¨ + N(q, q)q + G(q)] dt =
q
˙ ˙
0

u dt
0

The above equation can be rewritten as:
∆t

M(q)∆q + N(q, q)∆q +
˙
˙

∆t

G(q) dt =
0

59

u dt,
0

(4.6)

∆t

∆q
˙

∆t

q dt,
¨

∆q

0

q dt
˙

(4.7)

0

Since the configuration of the system does not change over ∆t ≈ 0, i.e. ∆q = 0, Eq.(4.7) is
simplified to the form:
M(q)∆q = I
˙

(4.8)

where I = [0, · · · , 0, I1 , · · · , Im ]T is the vector of applied impulses. The above equation can
be decomposed into the following equations:

M1 (q + − q − ) = 0
˙
˙

(4.9)

ˆ
M2 (q + − q − ) = I
˙
˙

(4.10)

ˆ
where M1 ∈ Rn−m×n , M2 ∈ Rm×n and I ∈ Rm are defined as:




 M1 
M =
 ,
M2

ˆ
I = [I1 , · · · , Im ]T

(4.11)

and q − and q + denote the vector of velocities immediately before and after the application
˙
˙
of impulse, respectively. Knowing the impulses applied on the system, Eq.(4.8) can be solved
to determine the velocity jumps in the active and passive coordinates.
Since ideal impulses I can not be implemented in practice, we use the inputs uimp obtained in section 4.2.2 for our algorithm, discussed next.

60

4.3

Stabilization Algorithm

Consider the underactuated system in Eq.(4.1) and assume that a controller has been designed to stabilize an equilibrium of the system. Also assume that a Lyapunov function
V (q, q) can be found to establish asymptotic stability of the equilibrium, and let RA be the
˙
region of attraction of this equilibrium. We propose a control algorithm which uses impulsive
inputs that are applied in conjunction with the stabilizing controller. The algorithm may be
able to stabilize the equilibrium from a set of configurations which lie outside the RA . The
algorithm is presented below:
1. Initialization: Find the Lyapunov function V (q, q) for the underactuated system in
˙
Eq.(4.1) with stabilizing control inputs u = [u1 , · · · , um ]T .
ˆ
imp
imp
˙
2. Impulsive Inputs: If V ≥ 0, apply impulsive inputs uimp = [u1 , · · · , um ]T such

that:
V (q, q + ) < V (q, q − ) ,
˙
˙

˙
V (q, q + ) < 0
˙

(4.12)

Note that the continuous inputs u provided by the stabilizing controller is applied at all
ˆ
˙
times, i.e., the impulsive inputs are applied on top of the continuous inputs when V ≤ 0.
˙
To prove stability of the impulsive system, we need to show that V < 0 during the
continuous phase, and V has negative jumps due to application of the impulsive inputs
[131, 9]. These conditions are satisfied by the proposed algorithm since impulsive inputs
which satisfy the conditions in Eq.(4.12) will force the system to be inside the region where
˙
V < 0 and ensure a negative jump in the value of V .
We illustrate the algorithm with the help of a two-dof system with underactuation degree
one. Let q1 and q2 be the generalized coordinates of the system. Since the configuration of
61

the system does not change at the time of application of the impulsive inputs, the effect of
impulsive inputs can be shown in the q1 -q2 plane - see Figure 4.1. In this figure, the curves
˙ ˙
˙
V = V and V = 0 correspond to fixed values of q = q
ˆ

(ˆ1 , q2 ). The point A : (ˆ, q − )
q ˆ
q ˙

˙
denotes a configuration which lies outside the region where V < 0 prior to the application of
the impulse. Since the impulsive input must satisfy the conditions in Eq.(4.12), the system
˙
configuration jumps from A : (ˆ, q − ) to B : (¯, q + ); B is a point inside the dashed region in
q ˙
q ¯
˙
Figure 4.1 where V < 0 and V < V . This jump has to be along the impulse line, which can
be obtained from Eq.(4.9):
(q2 − q2 ) = −
˙
˙−

M11
(q − q1 )
˙
˙−
M12 1

(4.13)

In Eq.(4.13), M11 and M12 are the elements of the matrix M1 (¯) = [M11 , M12 ] in Eq.(4.11).
q
From Eq.(4.13), the slope of the impulse line depends on the elements of the mass matrix
evaluated at q = q, and this slope determines if the conditions in Eq.(4.12) can be satisfied.
ˆ
An impulse line which does not enter the dashed region in Figure 4.1 corresponds to a
configuration of the system for which the required impulsive inputs can not be found. For
˙
˙
the case of V = 0, i.e., A : (ˆ, q − ) lies on the curve V = 0, the conditions in Eq.(4.12) can
q ˙
˙
not be satisfied if the impulse line is tangent to either V = V or V = 0 curves.
The magnitude of the impulsive input is determined based on its impulsive effects on the
system velocities, i.e., we first find q + which corresponds to a configuration in the dashed
˙
region and then find the impulsive input as discussed in Section 4.2.2.
Remark 4.3.1. There is no guarantee that impulsive inputs can be found that satisfy the
˙
conditions in Eq.(4.12) at a configuration of the system where V ≥ 0. In such a situation,
the closed-loop system is allowed to evolve in time and the search for impulsive inputs that
satisfy Eq.(4.12) should be continued. With change in configuration of the system, the
62

˙
V =0
V
=

q2
˙

V

˙
V >0
B : (ˆ, q + )
q ˙

A : (ˆ, q − )
q ˙

se
ul
p
Im ine
L

˙
V <0
V <V

q1
˙

ˆ
˙
Figure 4.1: Plots of the curves V = V and V = 0 and the impulse line correspond to fixed
values of q = (ˆ1 , q2 ) for a two-dof system. The prior-impulse configuration at A : (ˆ, q − ) is
q ˆ
q ˙
+ ) for which V < 0 and V < V .
˙
jumped to a configuration at B : (ˆ, q
q ˙
conditions in Eq.(4.12) may be satisfied at a later point in time.
˙
Remark 4.3.2. Although V ≥ 0 does not imply that the system configuration is outside
RA , a configuration outside RA will result in a trajectory which does not approach the
equilibrium. For the latter case, the impulsive inputs provide the possibility of bringing the
system trajectories back into the RA .
Remark 4.3.3. For systems with bounded states, impulsive inputs can be additionally applied
to ensure that the states of the system do not exceed their limits. In such cases, the step 2
in the stabilization algorithm is modified as below:
imp
˙
Step 2. If V ≥ 0 or hi (q) > 0 for i = 1, 2, · · · , w, apply impulsive inputs uimp = [u1 , · · ·
imp

, um ]T such that:
V (q, q + ) < V (q, q − ) ,
˙
˙

˙
V (q, q + ) < 0
˙

(4.14)

where hi (q) denotes the i-th constraint function imposed on the states of the system.

63

The effectiveness of the stabilization algorithm presented above will be illustrated through
several numerical simulations in the following sections.

4.4

Case Studies: Linear Stabilizing Controllers

4.4.1

The Pendubot

4.4.1.1

Dynamics

The Pendubot is shown in Figure 4.2. Assuming no friction in the joints, the equations of
motion of the Pendubot are in the form of Eq.(4.1), where

q = [θ2

θ1 ]T ,

τ1 ]T

u = [0

(4.15)

and M(q), N(q, q) and G(q), given by the expressions
˙



M(q) = 

α2

α2 + α3 C 2

α2 + α3 C 2





,

α1 + α2 + 2 α3 C 2



˙
α5 C12
0
θ1 



, G(q) = g 
N(q, q) = α3 S2 
˙


˙
˙
˙
α4 C1 + α5 C12
−(θ1 + θ2 )
−θ2

(4.16)

In Eq.(4.16), Cj = cos θj , Sj = sin θj for j = 1, 2, C12 = cos(θ1 + θ2 ), and the positive
constants αj , j = 1, 2, · · · , 5, have the following expressions
2
α1 = m1 d2 + m2 l1 + I1 ,
1

α3 = m2 l1 d2 ,

α2 = m2 d2 + I2
2

α4 = m1 d1 + m2 l1 ,

64

α5 = m2 d2

(4.17)

g

θ2

d1

θ1

d2

l1

l2

c.m
.

Y

τ2

.
c.m

X

τ1

Figure 4.2: The Pendubot/Acrobot is shown in an arbitrary configuration. For the Pendubot
we have τ2 = 0 and for the Acrobot we have τ1 = 0.

where, for k = 1, 2, mk and Ik are the mass and moment of inertia for the k-th link, and lk
and dk are link lengths that are shown in Figure 4.2.
The jump in the velocity of the passive coordinate θ2 due to the application of impulse in
the θ1 coordinate can be obtained from Eqs.(4.9) and (4.16) as follows:

˙ + ˙−
(θ2 − θ2 ) = −
4.4.1.2

α2 + α3 C 2 ˙ + ˙ −
(θ1 − θ1 )
α2

(4.18)

Continuous Controller

˙
˙
To stabilize the system about its upright equilibrium (θ1 , θ1 , θ2 , θ2 ) = (π/2, 0, 0, 0), we use
the pole-placement method for the linearized system. This controller renders the equilibrium
locally asymptotically stable, and this can be established using the Lyapunov function:

V = XT P X

65

(4.19)

˙
˙
In Eq.(4.19), X = (θ1 , θ1 , θ2 , θ2 )T and P is the solution of the Lyapunov equation:

(A − BK)T P + P (A − BK) + Q = 0

(4.20)

where A and B describe the linearized system in state-space form, K is the vector of control
gains, and Q is the identity matrix.
In the next section, we use the numerical simulations to show that our algorithm can enlarge
the RA of the upright equilibrium.
4.4.1.3

Numerical Simulations

Consider the Pendubot with the following parameters:

m1 = 1, l1 = 2, d1 = 1, I1 = 0.333
m2 = 1, l2 = 2, d2 = 1, I2 = 0.333

(4.21)

The poles of the linearized system were placed at λdes = (−0.5, −1, −1.2, −1.5).
˙
˙
For the first simulation, the initial condition was assumed to be (θ10 , θ10 , θ20 , θ20 ) =
(1.484, −0.45, 0.087, 0.15), where the units are in rad and rad/s. This configuration, which
˙
corresponds to V0 = 15.037 and V0 = 2.7746, lies outside RA of the upright equilibrium.
This can be verified through numerical simulations over a short interval of time for which the
trajectories go unbounded. Figure 4.3 shows simulation results obtained using our algorithm.
Impulsive inputs are applied at seven different time instants: at t = 0, and six more times
in the interval t ∈ [0.1, 2.25]. It can be observed that the impulsive inputs result in jumps
˙
˙
in the joint velocities, and jumps in V to maintain V < 0. Since the initial impulse is
66

80
60
50

a
b

0
0
-1

θ2 (deg)

c
˙
θ1 (rad/s)
˙
θ2 (rad/s)

1
0

d
e

20
10
0
5
0
-5
0
-50

θ1 (deg)

V

f

˙
V

g
0

0.5

1

1.5

2
2.5
time (sec)

3

3.5

τ1 (N.m)
4
4.5

5

Figure 4.3: Simulation results showing joint angles and joint velocities of the Pendubot and
the Lyapunov function, its time derivative, and the control input for an initial configuration
outside RA - First simulation

large in magnitude compared to other impulsive inputs, the negative jump in V can be
observed for the first impulse only. The stabilization of the upright equilibrium establishes
the effectiveness of our algorithm.
The effect of the impulsive input applied at t = 0 is illustrated with the help of Figure
˙ ˙
˙
4.4. This figure plots the curves V = V0 , V = 0, and the impulse line in θ1 -θ2 plane at the
initial configuration. The impulsive input satisfies the conditions in Eq.(4.12) and causes
the configuration to jump from A : (q0 , q0 ) to B : (q0 , q0 ) along the impulse line. At B,
˙−
˙+
˙
V (q0 , q0 ) < 0 and V (q0 , q0 ) < V0 . A numerical search was used to find the impulsive input
˙+
˙+
˙
which results in a maximum decrease in V while ensuring V < 0.
˙
˙
For the second simulation, the initial condition was assumed to be (θ10 , θ10 , θ20 , θ20 )

67

6
4

B : (q0 , q0 )
˙+
˙
V <0

˙
θ2 (rad/s)

2

˙
V =0

˙
V >0

0

A : (q0 , q0 )
˙−

-2

V

=

V

lse
pu
Im

0

-4

ne

Li

-6

-3

-2

-1

0
˙
θ1 (rad/s)

1

2

3

˙ ˙
˙
Figure 4.4: The curves V = V0 , V = 0, and the impulse line are shown in the θ1 -θ2
plane at the initial configuration of the Pendubot for first simulation, where θ10 = 1.484 and
θ20 = 0.087. The impulsive input at t = 0 in Figure 4.3 results in a jump in the configuration
˙
from A : (q0 , q0 ) to B : (q0 , q0 ), where V (q0 , q0 ) < 0 and V (q0 , q0 ) < V0 .
˙−
˙+
˙+
˙+
= (1.745, 0, 0.174, 0), where the units are in rad and rad/s. This configuration, which cor˙
responds to V0 = 32.489 and V0 = 4.433, lies outside RA of the upright equilibrium. This
can be verified through numerical simulations over a short interval of time for which the trajectories go unbounded. Figure 4.5 shows simulation results obtained using our algorithm.
Impulsive inputs are applied at four different time instants: at t = 0, and three more times
in the interval t ∈ [1.6, 2.25]. It can be observed that the impulsive inputs result in jumps
˙
˙
in the joint velocities, and jumps in V to maintain V < 0.

68

a

0
-40
2
1
0

θ1 (deg)

b

120
100
80

θ2 (deg)
˙
θ1 (rad/s)

c

0
-2

d

40
20
0

e

0
-10
60

f

˙
θ2 (rad/s)
V

˙
V
g

τ1 (N.m)

0
0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

time (sec)
Figure 4.5: Simulation results showing joint angles and joint velocities of the Pendubot and
the Lyapunov function, its time derivative, and the control input for an initial configuration
outside RA - Second simulation

4.4.2

The Acrobot

4.4.2.1

Dynamics and Control

The Acrobot is shown in Figure 4.2. Assuming no friction in the joints, the equations of
motion has the form of Eq.(4.1), where

q = [θ1

θ2 ]T ,

u = [0 τ2 ]T

and M(q), N(q, q), and G(q), given by the expressions
˙

69

(4.22)





 α1 + α2 + 2 α3 C 2 α2 + α3 C 2 
M(q) = 
,
α2 + α3 C 2
α2




˙2 −(θ1 + θ2 )
˙
˙
 −θ

 α4 C1 + α5 C12 
G(q) = g 
N(q, q) = α3 S2 
˙
,

˙1
θ
0
α5 C12

(4.23)

The parameters αj , j = 1, 2, · · · , 5 in Eq.(4.23), were defined in Eq.(4.17).
The jump in the velocity of the passive coordinate θ1 due to the application of impulse
in the θ2 coordinate can be obtained from Eq.(4.9) and Eq.(4.23) as follows:

˙+ ˙ −
(θ1 − θ1 ) = −

α2 + α3 C 2
˙ + ˙−
(θ2 − θ2 )
α1 + α2 + 2 α3 C 2

(4.24)

Using the pole-placement method, a stabilizing controller was designed to render the
upright equilibrium locally asymptotically stable. The procedure followed is similar to that
used for the Pendubot in Section 4.4.1.2.

4.4.2.2

Numerical Simulations

Consider the following parameters for the Acrobot:

m1 = 1, l1 = 1, d1 = 0.5, I1 = 0.083
m2 = 1, l2 = 2, d2 = 1, I2 = 0.333

(4.25)

The poles of the linearized system were placed at λdes = (−2, −3, −4, −5).
˙
˙
For the first simulation, the initial conditions are assumed to be (θ10 , θ10 , θ20 , θ20 ) =
70

100

θ1 (deg)

a

50

100 b
50
0
0 c
-4
5
0
-5 d

θ2 (deg)

˙
θ1 (rad/s)
˙
θ2 (rad/s)

10
0 e
0

V

f

-40
50
0

˙
V

g
0

τ2 (N.m)
0.5

1

1.5
time (sec)

2

2.5

3

Figure 4.6: Simulation results showing joint angles and joint velocities of the Acrobot and
the Lyapunov function, its time derivative, and the control input for an initial configuration
outside RA - First Simulation
˙
(1.658, 0, 0.087, 0). This configuration corresponds to V0 = 24.56 and V0 = 1.04. Using
numerical simulations, it was ascertained to be outside the RA of the upright equilibrium.
The simulation results obtained using our algorithm is shown in Figure 4.6. Four impulsive
inputs were applied which includes one impulse at t = 0 and three other impulses in the
˙
˙
interval t ∈ [0.4, 0.9]. The jumps in the joint velocities, and jumps in V to maintain V < 0
can be observed in the figure. Also, the first two negative jumps in V which are large in
magnitude can be seen from the plot. The stabilization of the equilibrium from outside the
RA establishes the efficacy of our algorithm.
˙
˙
For the second simulation, the initial conditions were assumed to be at (θ10 , θ10 , θ20 , θ20 ) =
˙
(1.658, 0.08, −0.087, 0.1). This configuration corresponds to V0 = 17.7 and V0 = 0.82 which
is outside the RA of the upright equilibrium as well. The simulation results are shown in

71

100

a

0
2
0
-4
5
0

θ1 (deg)

b

50
150

θ2 (deg)

c
˙
θ1 (rad/s)
˙
θ2 (rad/s)
d

20
10
e
0
0 f
-40
50
0
-50

V
˙
V

g
0

τ2 (N.m)
0.5

1

1.5
time (sec)

2

2.5

3

Figure 4.7: Simulation results showing joint angles and joint velocities of the Acrobot and
the Lyapunov function, its time derivative, and the control input for an initial configuration
outside RA - Second Simulation

Figure 4.7. Four impulsive inputs including the initial impulse were required for this case
and the system is ultimately stabilized using the control algorithm.

4.4.3

Rolling Acrobot

4.4.3.1

Dynamics and Control

A rolling acrobot, a modified version of the Acrobot, is a two-link planar robot with a passive
rolling shoulder joint and an active elbow joint. Figure 4.8 shows a schematic of the robot.
Assuming no friction in the joints, the equations of motion take the form of Eq.(4.1) where

q = [θ1

θ2 ]T ,

u = [0 τ2 ]T

72

(4.26)

d2

l2

X
Y

τ2

θ2

g
l1
θ1
d1
Figure 4.8: A schematic of the rolling acrobot. The link angles θ1 and θ2 are measured
counter-clockwise.
and M(q), N(q, q), and G(q), given by the expressions
˙




 γ1 + γ2 − 2γ3 C1 − 2γ4 C12 γ2 − γ4 C12 
M(q) = 
,
γ2 − γ4 C12
γ2


˙1 + 2γ4 S12 θ2 γ4 S12 θ2
˙
˙

 (γ3 S1 + γ4 S12 )θ
N(q, q) = 
˙
,
0
0


 γ5 S1 + γ6 S12 
G(θ) = g 

γ6 S12

(4.27)

In Eq.(4.27), the positive constants γi , i = 1, 2, · · · , 6 have the following expressions
2
γ1 =I1 + m1 d2 − 2m1 l1 d1 + (2m1 + m2 )l1 ,
1

γ3 =m1 l1 (l1 − d1 ),

γ4 = m2 l1 (l2 − d2 ),

γ2 = I2 + m2 (l2 − d2 )2
γ5 = m1 (l1 − d1 ),

73

γ6 = m2 (l2 − d2 ) (4.28)

The velocity jump in θ1 coordinate due to the application of impulse in θ2 coordinate
can be obtained from Eq.(4.9) and Eq.(4.27), namely,

˙ + ˙−
(θ1 − θ1 ) = −

γ2 − γ4 C12
˙ + ˙−
(θ2 − θ2 )
γ1 + γ2 − 2γ3 C1 − 2γ4 C12

(4.29)

The same pole placement controller as in Section 4.4.1.2 is designed for the rolling acrobot
˙
˙
to render the upright equilibrium (θ1 , θ1 , θ2 , θ2 ) = (0, 0, π, 0) locally asymptotically stable.

4.4.3.2

Numerical Simulations

The kinematic and dynamic parameters of the rolling acrobot is chosen as:

m1 = 1, l1 = 1, d1 = 0.5, I1 = 0.083
m2 = 2, l2 = 1, d2 = .05, I2 = 0.167

(4.30)

For simulation results, we consider two cases of the rolling acrobot; In the first case, we
assume there is no limit for θ1 and the system can roll on its arc foot without any constraint.
In the second case, the arc foot is limited and the angle θ1 is constrained to be inside a
prescribed range during the stabilization process.
First Case - Unbounded Foot Arc Angle
The poles of the linearized system are placed at λdes = (−0.5, −1, −1.5, −2) using pole
˙
˙
placement control. For the first simulation, the initial conditions are set to (θ10 , θ10 , θ20 , θ20 ) =
˙
(0.436, 0, 3.491, 0). This configuration corresponds to V0 = 12.5 and V0 = 5.33 and is outside
the RA of the upright equilibrium which can be verified through a numerical simulation. The
simulation results are shown in Figure 4.9. The applied impulsive inputs can be recognized
74

40
20
0 a
200

θ1 (deg)

150 b
1
0
c
-1
0 d
-2
-4

θ2 (deg)

5
0
5
0
-5

˙
θ1 (rad/s)
˙
θ2 (rad/s)

e

V

f

0
-40 g
0

˙
V

1

2

3

4
time (sec)

5

6

τ2 (N.m)
7

8

Figure 4.9: Simulation results including the angular positions, angular velocities, Lyapunov
function and its time derivative and the control input for a configuration of rolling acrobot
with unbounded leg angle originally outside RA - First simulation

˙
by the picks in the control input subplot. All impulsive inputs result in a negative V and a
drop in V as can be seen in the figure. Using these impulsive inputs, the upright equilibrium
is stabilized and the configuration which was originally outside the RA , is now attracted to
the desired equilibrium.
For the second simulation, we choose the initial conditions with initial velocities to be at
˙
˙
(θ10 , θ10 , θ20 , θ20 ) = (0.087, 0.5, 3.229, 1). This configuration corresponds to V0 = 23.36 and
˙
V0 = −1.195 which is outside the RA of the upright equilibrium as well. The simulation
results are shown in Figure 4.10. Three impulsive inputs including the initial impulse were
required for this case and the system is ultimately stabilized using the control algorithm.

75

60
20
-20 a
200
150
100 b
2
0 c

θ1 (deg)

θ2 (deg)
˙
θ1 (rad/s)

0 d

˙
θ2 (rad/s)

-4
10 e
0

V

f

0
-20
20
0
-20

˙
V

g
0

1

2

3

4
time (sec)

5

6

τ2 (N.m)
7

8

Figure 4.10: Simulation results including the angular positions, angular velocities, Lyapunov
function and its time derivative and the control input for a configuration of rolling acrobot
with unbounded leg angle originally outside RA - Second simulation

Second Case - Bounded Foot Arc Angle
The leg angle θ1 is constrained as follows:

h1 (q) = |θ1 | − η1 ≤ 0

(4.31)

where η1 = 15◦ . The poles of the closed-loop linearized system are placed at λdes =
(−1, −1.5, −2, −2.5). The modified algorithm mentioned in Remark 4.3.3 is followed to
stabilize the upright equilibrium while ensuring the constraint in Eq.(4.31) is satisfied.
˙
˙
For the first simulation, the initial conditions are assumed to be at (θ10 , θ10 , θ20 , θ20 ) =
˙
(0.087, 0, 3.229, 0). This configuration corresponds to V0 = 1.218 and V0 = 0.366. Using

76

20
a
10
0
200
180
160 b
0.4
0 c
0
-1
-2

θ1 (deg)
θ2 (deg)
˙
θ1 (rad/s)

d

˙
θ2 (rad/s)

1 e
0
0
-2

V

f

0
-5
-10

˙
V
τ2 (N.m)

g
0

0.5

1

1.5

2
2.5
time (sec)

3

3.5

4

4.5

5

Figure 4.11: Simulation results including the angular positions, angular velocities, Lyapunov
function and its time derivative and the control input for a configuration of rolling acrobot
with bounded leg angle originally outside RA - First simulation

the pole placement controller, it can be verified through a numerical simulation that the
constraint in Eq.(4.31) is not satisfied. Figure 4.11 shows the simulation results for our
algorithm where impulsive inputs were used to stabilize the upright equilibrium while satisfying the state constraints. Two impulsive inputs were applied for this simulation; the first
˙
impulse, applied at the initial time, is due to V0 > 0 while the second impulsive input is
applied when θ1 = η1 to keep θ1 inside its bound. Both impulsive inputs result in negative
˙
V as well as a negative jump in V . Using these impulsive inputs, the upright equilibrium is
stabilized without exceeding the bound on θ1 .
For the second simulation, we choose the initial conditions with initial velocities to be
˙
˙
at (θ10 , θ10 , θ20 , θ20 ) = (0, −0.1, 2.967, −0.25). This configuration corresponds to V0 = 1.298
77

0
-10
-20
200
180
160
0
-0.5

θ1 (deg)

a
b

θ2 (deg)

c
˙
θ1 (rad/s)
˙
θ2 (rad/s)

1
0

d
e

1
0
0

V

f
˙
V

-4
10
5
0

g
0

τ2 (N.m)
0.5

1

1.5

2

2.5
time (sec)

3

3.5

4

4.5

5

Figure 4.12: Simulation results including the angular positions, angular velocities, Lyapunov
function and its time derivative and the control input for a configuration of rolling acrobot
with bounded leg angle originally outside RA - Second simulation
˙
and V0 = 0.094 and it is verified that the bound on θ1 will be exceeded using the pole
placement controller. The simulation results for our algorithm are shown in Figure 4.12.
˙
As the first simulation, the initial impulsive input is applied since V0 > 0 and the second
impulsive input is intended to keep θ1 inside its limit.

4.5

Extension for Nonlinear Controllers

The case studies studied in Section 4.4 consider the systems with linear stabilizing controllers
which is only valid in a small neighborhood of the desired equilibrium. For such systems,
˙
the derivative of the corresponding Lyapunov function V is more likely to get positive when
the system configuration lies outside the RA since the controller is designed based on the
78

˙
linearized system. For system with nonlinear stabilizing controllers, however, the V is designed to be less than or equal to zero1 for the nonlinear system and therefore, the condition
˙
to apply the impulsive inputs which is V ≥ 0 as discussed in Section 4.3, might never be
satisfied. In order to use the benefits of impulsive inputs in enlarging the RA for systems
with nonlinear controllers, we modify the stabilization algorithm proposed in Section 4.3 as
follows:
1. Initialization: Find the Lyapunov function V (q, q) for the underactuated system in
˙
Eq.(4.1) with stabilizing nonlinear controller u = [u1 , · · · , um ]T .
ˆ
˙
2. Impulsive Inputs: If V ≥ 0 or the system configuration lies outside the set Ω, apply
imp
imp
impulsive inputs uimp = [u1 , · · · , um ]T such that:

V (q, q + ) < V (q, q − ) ,
˙
˙

˙
V (q, q + ) < 0
˙

(4.32)

where Ω is an estimate of the RA calculated based on the Lyapunov function of the
closed-loop system.
Remark 4.5.1. Compared to the algorithm in Section 4.3, we need an off-line calculation of
the set Ω for the algorithm above. Although it might be hard to find a set which estimates the
RA of the desired equilibrium closely, any conservative estimate of the RA will be adequate
for the proposed algorithm to be applied.
Remark 4.5.2. When there exist constraints on the states of the system, following the discussion in Remark 4.3.3, the additional impulsive inputs are applied to ensure the constraints
˙
that ensuring V ≤ 0 does not necessarily mean that the desired equilibrium is stable.
This is because the states of the system might get unbounded using the designed controller or the
system approaches another equilibrium of the system.
1 Note

79

are satisfied.
In the rest of this section, we consider two well-known underactuated systems in the literature with nonlinear stabilizing controllers and apply the stabilization algorithm proposed
above to enlarge the RA of the desired equilibrium.

4.5.1

Inverted Pendulum on an Inclined Plane

4.5.1.1

Dynamics

Consider the inverted pendulum system on an inclined plane shown in Figure 4.13. Assuming
no friction in the joints, the equations of motion has the form of Eq.(4.1) where

q = [φ s]T ,

u = [0 F ]T

(4.33)

and M(q), N(q, q), and G(q), given by the expressions [25]
˙




β cos(φ − ψ) 
,
β cos(φ − ψ)
γ




0
0 
 −β sin φ 

N(q, q) = 
˙

 , G(q) = g 
˙ 0
−γ sin ψ
−β sin(φ − ψ)φ


M(q) = 

α

(4.34)

where
γ = M + m , α = ml2 , β = ml

(4.35)

and φ and s are the generalized coordinates as shown in Figure 4.13, m and l are the point
mass and the length of the pendulum, M is the mass of the cart and ψ is the slope angle.

80

r

φ
l

g

F
ψ
s
Figure 4.13: Cart-pendulum on a slope
The jump in the velocity of the passive coordinate φ due to the application of impulsive
force in active coordinate s, can be obtained from Eqs.(4.9) and (4.34), namely,

˙
˙
(φ + − φ − ) = −
4.5.1.2

β cos(φ − ψ)
( s+ − s− )
˙
˙
α

(4.36)

Stabilizing Control Design

˙
The nonlinear controller to stabilize the zero equilibrium at (φ, φ, s, s) = (0, 0, 0, 0) is ob˙
tained from [25] using the Lagrangian Controlled method:

F = −γg sin ψ +

˙
˙
κβ[α sin(φ − ψ)φ2 + D cos(φ − ψ) sin φ] − B ∂H + B c γ y
∂s
α−

β2
2
γ (κ + 1) cos (φ − ψ)

(4.37)

where
αγ − β 2 cos2 (φ − ψ)
, D = −mgl
ργ
ρ−1
β
ǫDγ 2 2
)[sin(φ − ψ) + sin ψ]
y , y = s + (κ +
H = mgl cos(φ) +
2
γ
ρ
2β
κ = 1/σ,

B=

(4.38)

and σ, ρ < 0, ǫ > 0, and c > 0 are some scalar constants. The asymptotic stability of the
81

equilibrium is established using the following Lyapunov function which is obtained from the
controlled energy of the system [25]:

κ
1
˙ ˙
˙
V =mgl − αφ2 − β cos(φ − ψ)[s + β cos(φ − ψ)φ]φ
2
γ
1
κ
κ
˙
˙
− γ[s + β cos(φ − ψ)φ]2 + β 2 cos2 (φ − ψ)φ2
˙
2
γ
2γ
1
β
˙
− (ρ − 1)γ[s + (κ + 1) cos(φ − ψ)φ]2 − H
˙
2
γ

(4.39)

An estimate of the RA is obtained as:

˙
Ωc = z = (φ, φ, s, s) ∈ T Q | V (z) ≤ c
˙
¯
¯

(4.40)

where T Q is a bounded set, and c is a constant scalar.
¯

4.5.1.3

Numerical Simulations

The parameters for the inverted pendulum system shown in Figure 4.13 are chosen as:

m = 0.14 Kg,

M = 0.44 Kg,

L = 0.215 m,

ψ = π/9 rad

(4.41)

and the control parameters are chosen as:

κ = 20,

ρ = −0.02,

ǫ = 0.00001,

c = 0.015

(4.42)

For the parameters in Eqs.(4.41) and (4.42), the estimate of the RA in Eq.(4.40) is determined
by c = 0.26. We also assume that the pendulum angle is constrained to be inside a pre¯

82

60
0
-60
4
0

a

φ (deg)

b

˙
φ (rad/s)

5
0 c
4
0 d
0.5 e
0
0
-2

s (m)
s (m/s)
˙
V
˙
V

f
g

60

F (N)

0
0

5

10

15

20

25

30

time (sec)
Figure 4.14: Simulation results including the positions and velocities of the system coordinates, Lyapunov function and its time derivative and the control input for a configuration
of the inverted pendulum on a slope originally outside the RA - First simulation
defined range, i.e., h1 (q) = |φ| − φmax ≤ 0 where φmax = 1.222 rad = 70 deg.
˙
For the first simulation, we choose the initial conditions of the system as (φ0 , φ0 , s0 , s0 ) =
˙
˙
(π/6, 4, 3, 3). This configuration corresponds to V0 = 0.564 and V0 = −2.669 and lies outside
the set Ωc in Eq.(4.40). From numerical simulations for a short period of time, it can be
¯
confirmed that the sates of the system get unbounded using the stabilizing controller in
Eq.(4.37) and therefore, this configuration is outside the RA of the desired equilibrium.
Figure 4.14 shows the simulation results for the proposed stabilization algorithm in Section
4.5. The peak in the control input subplot shows the initial impulsive input applied at t = 0
sec due to the fact that the initial configuration is outside the set Ωc . This impulsive input
¯
˙
causes a large negative jump in V with V < 0 as required in Eq.(4.32). Using the proposed
algorithm, the desired equilibrium at the origin is stabilized from a configuration which was

83

80
40
0
4
0

a
b

˙
φ (rad/s)

c

s (m)

d
e

0
-2
4
0
-4
0.3

φ (deg)

s (m/s)
˙
V

0
0

˙
V

f
g

-0.6
40

F (N)

0
0

5

time (sec)

10

15

Figure 4.15: Simulation results including the positions and velocities of the system coordinates, Lyapunov function and its time derivative and the control input for a configuration
of the inverted pendulum on a slope originally outside the RA - Second simulation

originally outside the RA .
For the second simulation, the pendulum is assumed to be farther from upright equilib˙
rium position compared to the first simulation by choosing (φ0 , φ0 , s0 , s0 ) = (π/3, 4, −2, −3.5).
˙
The corresponding values of the Lyapunov function and its time derivative are calculated
˙
as V0 = 0.251 and V0 = −0.528. Although this configuration lies inside Ωc , the numerical
¯
simulation using the stabilizing controller in Eq.(4.37) shows that the angular position of the
pendulum exceeds its limit. Simulation results for our algorithm is shown in Figure 4.15.
The only impulsive input is applied a little bit after the starting time when the pendulum
angle reaches its limit, φ = 70 deg. The impulsive input kicks back the pendulum inside
the allowed region and cause a small drop in V which is enough to stabilize the desired
equilibrium without exceeding the limit for φ.
84

4.5.2

The Ball and Beam System

4.5.2.1

Dynamics

The ball and beam system is shown in Figure 4.16. The equations of motion is in the form
of Eq.(4.1) where
q = [r

θ]T ,

u = [0 τ ]T

(4.43)

and M(q), N(q, q), and G(q), given by the expressions
˙



M(q) = 



N(q, q) = 
˙

Jb
R2

+M

0
Mr 2 + J + Jb

0



˙
−Mr θ 
,
˙
2Mr θ
0
0




,





 M sin θ 
G(q) = g 

Mr cos θ

(4.44)

In Eq.(4.44), r and θ are the generalized coordinates as shown in Figure 4.16, M, Jb , and R
are the mass, mass moment of inertial and radius of the ball, respectively, and J is the mass
moment of inertia of the beam. The length of the beam is assumed to be equal to 2L.
Using Eqs.(4.9) and (4.44), it can be shown that application of an impulsive input in

τ
r
θ
Figure 4.16: Ball and beam system

85

active coordinate θ does not cause any velocity jump in the passive coordinate r, i.e.,

r+ = r−
˙
˙

4.5.2.2

(4.45)

Stabilizing Control Design

A nonlinear stabilizing controller for the ball and beam system is given using the IDA-PBC
method in [28] as follows:

τ =√

r
2(L2

+ r2)

−p2
1

+ gr cos θ − kp

−g

L2 + r 2 +

√

2p1 p2 + √

1
L2

+ r2

p2
2

L2 + r 2
1
r
θ − √ arcsinh( )
2
L
2

kv
p1 − p2
2(L2 + r 2 ) sin θ + 2
L + r2

L2

2
+ r2

(4.46)

where
p1 = r ,
˙

˙
p2 = (L2 + r 2 )θ

(4.47)

and kp and kv are the constant control gains. The control input in Eq.(4.46) was obtained
assuming that
M =1,

J = L2 ,

Jb = 0

(4.48)

˙
Asymptotic stability of the desired equilibrium at (r, r, θ, θ) = (0, 0, 0, 0) is established
˙
using the following Lyapunov function which is the Hamiltonian of the desired closed-loop

86

system [28]:

1
−1
V = P T Md P + Vd
2

(4.49)

where




 p1 
P =
,
p2




Md = 


√
√ 2
L2 +r2

1

1
2(L2 + r 2 )

kp
1
r 2
θ − √ arcsinh( )
Vd = g(1 − cosθ) +
2
L
2





(4.50)

The RA of the equilibrium such that the ball remains on the beam for all the time, i.e.,
h1 (q) = |r(t)| − L ≤ 0 for all t ≥ 0, is estimated as [28]:
¯
Ω=

(q, p) |

kp
1
1 kp g
r
1 T −1
θ − √ arcsinh( ) <
P Md P + g(1 − cosθ) +
2
2
L
8 2kp + g
2

(4.51)

which gives a conservative estimate based on the desired Hamiltonian of the system.
Remark 4.5.3. Although the coordinate r is constrained for the ball and beam system,
application of an impulsive input is not helpful in kicking back the constrained state inside
the allowed region when it gets to its limit as proposed in Remark 4.3.3. This is because the
impulsive input does not cause any velocity jump in r as derived in Eq.(4.45). The only way
˙
to ensure a configuration of the system approaches the origin without exceeding the limit on
¯
r is to check whether the configuration lies inside the set Ω in Eq.(4.51).

87

4.5.2.3

Numerical Simulations

For simulations, the following parameters for the system and the controller are chosen:

L = 1,

kp = 1,

kv = 2

(4.52)

˙
For the first simulation, we choose the initial conditions as (r0 , r0 , θ0 , θ0 ) = (0.8, 0, 0.175, −1).
˙
˙
¯
This configuration corresponds to V0 = 1.113 and V0 = −2.439 and lies outside the set Ω in
Eq.(4.51). Using the stabilizing controller in Eq.(4.46), the ball will exceed its limit and gets
off the beam and therefore, this configuration is outside the RA of the desired equilibrium.
Figure 4.17 shows the simulation results for the proposed stabilization algorithm. The initial
impulsive input shown by the peak in the control input subplot, is due to the fact that the

1
0.5
0 a
0 b
-0.4
10
0
-10 c

r (m/s)
˙

0 d
-0.4

˙
θ (rad/s)

r (m)

θ (deg)

1 e
0
0
-2 f

V
˙
V

50 g
0

0

τ (N.m)

5

time (sec)

10

15

Figure 4.17: Simulation results including the positions and velocities of the system coordinates, Lyapunov function and its time derivative and the control input for a configuration
of the ball and beam system originally outside the RA - First simulation

88

¯
initial configuration is outside the set Ω. This impulsive input causes a large negative jump
˙
in V with V < 0 as required in Eq.(4.32). Using the proposed algorithm, the desired
equilibrium at the origin is stabilized without exceeding the limit for r.
˙
For the second simulation, the initial conditions were chosen as (r0 , r0 , θ0 , θ0 ) = (−0.6,
˙
−0.5, 0, 0.5). The corresponding values of the Lyapunov function and its time derivative are
˙
¯
calculated as V0 = 0.689 and V0 = −1.98. This configuration is also outside the set Ω and
it can be shown with the help of numerical simulations that the ball exceeds its limit using
the controller in Eq.(4.46). Simulation results for our algorithm is shown in Figure 4.18.
The only impulsive input is applied at the initial time causing a negative jump in V . The
continuous controller is applied thereafter. The initial impulse in enough to enlarge the RA
of the desired equilibrium to include the initial configuration of the system.

0
-0.5
-1
0.5
0
-0.5
10
0
-10
0.5
0
-0.5
0.5
0
0

a

r (m)
r (m/s)
˙

b
c

θ (deg)
d

˙
θ (rad/s)

e

V
˙
V

-2 f
0
-40 g
0

τ (N.m)

4

time (sec)

8

12

Figure 4.18: Simulation results including the positions and velocities of the system coordinates, Lyapunov function and its time derivative and the control input for a configuration
of the ball and beam system originally outside the RA - Second simulation

89

4.6

Conclusion

An impulsive control algorithm is proposed to enlarge the region of attraction of stabilized
equilibria for underactuated systems. The algorithm requires impulsive inputs to be applied
in conjunction with the stabilizing controller. The impulsive inputs are applied when the
derivative of the Lyapunov function is non-negative and two conditions are satisfied. These
conditions require that the impulsive inputs result in a negative jump in the Lyapunov
function and a jump to a configuration where the derivative of the Lyapunov function is
strictly negative. Although there is no guarantee that these conditions can be satisfied at any
given configuration, numerical simulations show that it is possible to stabilize an equilibrium
point from certain configurations that lie outside its region of attraction. The effectiveness of
the algorithm is demonstrated using several examples of underactuated mechanical systems
with linear and nonlinear continuous controllers.

90

Chapter 5
Safe Fall Control for Humanoid
Robots

5.1

Introduction

Impulsive control inputs are a natural candidate for emergency control when the available
time to influence the dynamic system is limited. Furthermore, due to a very short “time
of application” of the control, the system configuration does not evolve much during the
application of the control. A direct outcome of this property is that the control can be
applied even when the system is close to the boundary of its configuration space. For
example, a robot can accept control command even when it is close to its joint limits. In this
chapter, we propose a control strategy for changing the default fall direction of the robot so
that it avoids contact with surrounding objects or people as a mean of reducing damage to
others. A six-dof robot with three actuated joints is considered here to better understand the
impulsive algorithm before applying to higher dof humanoids. Note that we consider only
those situations in which a fall is caused by an external factor; in particular the sensors and
the motor power must remain intact such that the robot can execute a prescribed control
strategy.
We should clarify that a fall controller is not a balance controller. A fall controller
complements, and does not replace, a balance controller. Further, a fall controller is not a
91

push-recovery controller. A push-recovery controller is essentially a balance controller, which
specifically deals with external disturbances of larger magnitude when the robot must take a
step in order to regain balance [135, 136, 137]. The fall controller is activated only when the
default balance controller or the push recovery controller has failed to stabilize the robot.
The rest of this chapter is organized as follows: In Section 5.2, dynamic model of the
six-dof robot is described. we derive the impulsive effects on the system velocities in Section
5.3. The safe fall algorithm is presented in Section 5.4 which is followed by the numerical
simulations in Section 5.5. Concluding remarks are provided in Section 5.6.

5.2
5.2.1

Dynamic Model
Equations of Motion

We considered a three-dimensional seven-dof robot as shown in Figure 5.1. The degrees of
freedom of the robot are listed in Table 5.1. Assuming that the knee joint is locked, i.e.,
θ1 ≡ 0, we have a six-dof robot with three active joints. Using Lagrange’s formulas, the
equations of motion can be derived as follows:

M(q)¨ + N(q, q)q + G(q) = T
q
˙ ˙

(5.1)

q = [α, β, γ, θ2 , θ3 , θ4 ]T , T =[0, 0, 0, τ2 , τ3 , τ4 ]T

(5.2)

where

92

z4
y4
x4

z3
y3

x3 z
2
Z

y2

x2
z1

X

Inertial
Frame

Y

y1

Front View

Side View

x1

Figure 5.1: Seven-dof three-dimensional humanoid robot
and M(q) ∈ R6×6 , N(q, q) ∈ R6×6 and G(q) ∈ R6 represent the mass matrix, the matrix
˙
of centrifugal and Coriolis forces and the vector of forces due to gravity, respectively. Full
expressions of the matrices M(q), N(q, q) and G(q) are lengthy and are not provided for the
˙
sake simplicity.
Table 5.1: Degrees of freedom for the humanoid robot in Figure 5.1
Link No. of dof’s
Coordinates
Actuation
1
3
α, β, γ: z-x-z Euler angles
passive
2
1
θ1 : Rotation about x2
active
θ2 : Rotation about x2
3
2
active
θ3 : Rotation about z3
4
1
θ4 : Rotation about y4
active

93

5.2.2

Lean Line

To better understand the orientation of the robot during fall, we consider the orientation of
the line connecting the foot contact point with the ground to the center of mass of the whole
system which we call the Lean Line. The vector representing the lean line can be calculated
from:
4
n=1 mn rn
4
n=1 mn

rcm =

(5.3)

where mn is the mass of the nth link and rn is the position vector of the center of mass of
the nth link in the inertial frame. The orientation of the lean line in the inertial frame can
be determined through the angles φ and ψ shown in Figure 5.2 where:

rcm = |rcm | [sin(φ) cos(ψ) i + sin(φ) sin(ψ) j + cos(φ) k]

Z

φ
rcm
Y

ψ
X

Figure 5.2: Orientation of the lean line in the inertial frame

94

(5.4)

5.3

Impulsive Effects

Consider a set of impulsive inputs inputs applied in active coordinates θ2 , θ3 and θ4 at time
t = ti . We integrate the equations of motion in Eq.(5.1) over the short period of time, ∆t
when the impulses are applied:
ti +∆t

M(q)¨ dt +
q

ti

ti +∆t

N(q, q) q dt +
˙
˙

ti

ti +∆t

G(q) dt =

ti

ti +∆t

T dt

(5.5)

ti

Since ∆t is very small and the configuration of the robot does not change over this short
interval of time, Eq.(5.5) can be simplified as:

M(q)(q + − q − ) = I
˙
˙

(5.6)

where q − and q + are the vectors of angular velocities immediately before and after the
˙
˙
application of impulses, respectively, and I represents the vector of impulses, i.e.:

I = [0, 0, 0, I2 , I3 , I4 ]T

(5.7)

The first three equations in Eq.(5.6) which correspond to the passive coordinates of the first
link are written as:
M (q)(q + − q − ) = 0
˙
˙

(5.8)

where M (q) ∈ R3×6 is obtained from partitioning the mass matrix M(q). Eq.(5.8) can be

95

rewritten as:





+ − θ−
+ − α−
˙
˙
˙
˙ 
θ2
α
2




 ˙+ ˙− 
 ˙+

−  + M2 (q) θ − θ  = 0
˙
M1 (q) β − β
3
 3






˙+ ˙−
θ4 − θ4
γ+ − γ−
˙
˙

(5.9)

where M1 (q) ∈ R3×3 and M2 (q) ∈ R3×3 are obtained from:
M (q) = M1 (q) M2 (q)

(5.10)

Equation (5.9) can be used to find the relationships between the system velocities immediately before and after the application of impulse. If values of three velocities after the
application of impulsive inputs are known, we can solve for the other three velocities after
the application of impulsive inputs from Eq.(5.9).
As the first case, assume that the desired values for the active coordinate velocities after the
˙+
˙
˙+
˙
˙+
˙
application of impulsive inputs are specified as θ2 = θ2des , θ3 = θ3des and θ4 = θ4des , then
from Eq.(5.9) we have:


   
−
−
+
˙
˙
˙
˙
θ2des − θ2 
α  α 


   
˙
 ˙ +  ˙ −
−1
˙− 
β  = β  − M1 (q) M2 (q) θ3des − θ 
3

   


   
−
+
˙
˙−
θ4des − θ4
γ
˙
γ
˙

(5.11)

where M1 is invertible since it is a diagonal block of the positive definite matrix M(q).
As the second case, assume that the desired values of the passive coordinate velocities after
˙
˙
the application of impulsive inputs are specified as α+ = αdes , β + = βdes and γ + = γdes .
˙
˙
˙
˙

96

Then from Eq.(5.9) we can write:


   
−
˙−
˙+
˙
˙
αdes − α 
θ2  θ2 


   


 +  −
#
˙
˙
˙
˙
θ  = θ  − M2 (q) M1 (q)  βdes − β − 


 3  3


   
−
˙−
˙+
θ4
γdes − γ
˙
˙
θ4

(5.12)

#

where M2 (q) denotes the Pseudo-inverse of M2 and it is used since the matrix M2 might
be singular.
˙+ ˙+ ˙+
Remark 5.3.1. If the matrix M2 is singular, the velocities [θ2 , θ3 , θ4 ] obtained from Eq.(5.12)
˙
do not exactly correspond to the desired values for the passive coordinates [αdes , βdes , γdes ].
˙
˙
The resulted velocities, however, will give the closest solution in terms of the least Euclidean
˙
norm to the active coordinate velocities which correspond to [αdes , βdes , γdes ]. The veloci˙
˙
ties for the active coordinate obtained from Eq.(5.12) can be substituted into Eq.(5.11) to
yield the corresponding velocities of the passive coordinates.
To find the applied impulses which result in some desired velocity jumps, we consider the
last three equations in Eq.(5.6), namely,

¯
¯
M (q)(q + − q − ) = I
˙
˙

(5.13)

¯
¯
where M (q) ∈ R3×6 and I = [I2 , I3 , I4 ]T . Having the velocities after the application of
the impulsive inputs from Eqs.(5.11) and (5.12), the vector of the applied impulses can be
calculated using Eq.(5.13).
Remark 5.3.2. The ideal impulses obtained from Eq.(5.13) can not be implemented in the
standard actuators. However, we can use the high-gain continuous control inputs to estimate

97

these impulses as discussed earlier in Chapter 2.

5.4

Safe Fall Algorithm

We propose a safe fall algorithm for the system in Eq.(5.1) using two sets of impulsive inputs.
The first set of impulsive inputs is applied in order to reach some desired velocities for the
passive coordinates and the second set is intended to brake the active coordinates which
are held fixed thereafter. The robot undergoes a free fall for the time period between the
impulsive sets. A schematic of the proposed algorithm is shown in Figure 5.3 and can be
described as follows:
¯
The fall is realized at A where the lean line angle in Figure 5.2 reaches φ = φA . The first set
of impulsive inputs are then applied to reach some desired velocities for passive coordinates
which takes the robot configuration to B. After the application of the first set of impulsive
inputs, the joint torques are set to zero such that the robot undergoes a free fall till it reaches
Free fall with no torque inputs

First Impulse to make
desired velocity jumps
in passive coordinates

C : (qC , qC )
˙

B : (qB , qB )
˙
A : (qA , qA )
˙
Fall is recognized at A

Second Impulse
to brake active
coordinates

D : (qD , qD )
˙
Free fall with all active
joints locked
E : (qE , qE )
˙
Desired Configuration
on floor

Start point:
(Robot at upright configuration)

Figure 5.3: A schematic of the safe fall algorithm using two sets of impulsive inputs. For
interpretation of the references to color in this and all other figures, the reader is referred to
the electronic version of this dissertation.

98

¯
the configuration at C which is determined by φ = φC . The second set of impulsive torques
are applied then to brake the active coordinates. This corresponds to a jump in the robot
configuration form C to D. The active coordinates are held locked thereafter till the robot
¯
hits the floor at φ = φE . In particular, the following steps are adopted to design the safe
fall algorithm:
A. Off-line calculations to find a set of solutions at B:
1. Knowing the desired configuration on the floor and having all active coordinates
locked, integrate the equations of motion backward in time to find the configuration at D : (qD , qD ). This configuration is realized when the lean line angle φ,
˙
¯
shown in Figure 5.2, reaches φ = φC . Since the control objective is to change
the fall direction which is mainly described by the angle ψ - see Figure 5.2, we
have some freedom in choosing the angular positions of the active coordinates and
angular velocities of the passive coordinates at the desired configuration on the
floor. Each of these configurations will result in a different solution for D.
2. For each of the solutions at D obtained in step A.1, use the impulsive relations
to find the robot configurations at C : (qC , qC ). In particular, from Eq.(5.11) we
˙
have:



   
˙
˙
˙
θ2C 
αC  αD 


   


   
−1
˙
˙
˙ C  =  βD  − M1 (qD ) M2 (qD ) θ3C 
β


   


   
˙4C
θ
γD
˙
γC
˙

(5.14)

˙
˙
˙
where θ2D = θ3D = θ4D = 0. Many solutions can be found for C by choosing a
˙
˙
˙
range of velocities for the active joint coordinats, θ2C , θ3C , and θ4C . Also, having
different solutions for D obtained in step A.1 results in additional solutions for C.
99

Note that the the angular positions of the active and passive coordinates remain
constant during the jump from D to C.
3. For each of the solutions at C, integrate the equations of motion backward in time
¯
to reach the configuration at B : (qB , qB ) which is realized when φ = φA . This is
˙
done assuming a free fall for the robot, i.e., control inputs are zero.
Remark 5.4.1. After this step, we have a set of solutions for B which are resulted from
different choices of desired configuration at E and different choices of velocities at C.
Each of these solutions will indeed result in the desired change in fall direction of the
robot, i.e., desired ψ angle when the robot hits the floor.
B. On-line calculations to change the fall direction:
¯
1. Once the fall is realized at φ = φA , among the set of solutions for B obtained
˙
in step A, find the closest solution QB = [qB , q B ] to the current configuration
QA = [qA , qA ] where the distance si from each solution QBi = [qBi , qBi ] is defined
˙
˙
as:
si = ||QA − QBi ||2

(5.15)

2. Apply the impulsive inputs to reach the desired velocities for the passive coordi˙ ˙
˙
nates, (αB , βB , γB ) corresponding to QB . The impulsive relations in Eq.(5.12)
ˆ ˆ ˆ
can be used to find the velocity jumps in the active coordinates as follows:



 

˙
˙
˙
ˆ
˙
αB − αA 
θ2B  θ2A 



 

˙


 

ˆ
˙
˙
˙3B  = θ3A  − M2# (qA ) M1 (qA )  βB − βA 
θ



 




 

˙ B − γA
˙4A
˙4B
γ
ˆ
˙
θ
θ
100

(5.16)

3. Let the robot have a free fall with applying no torque in the active joints till it
gets to C.
¯
4. When φ = φC , apply the impulsive torques to brake the active coordinates and
¯
hold them fixed till the humanoid hit the floor at φ = φE .
Remark 5.4.2. The application of the first set of impulsive inputs in step B.2 will not
exactly result in the desired solution QB . This is because: first, the application of
impulsive inputs does not change the angular positions and this yields a deviation
from QB if qA = qB and second, the angular velocities resulted from the application
˙
of impulsive inputs will not match the angular velocities q B if qA = qB or the matrix
M2 (qA ) is singular.
Remark 5.4.3. Following the algorithm in step B, there is no guarantee that we reach the
exact desired configuration on the ground. This is mainly because we can not exactly reach
a desired solution for the configuration at B through applying the first set of impulsive
inputs as discussed in Remark 5.4.2. However, we can get close to the desired change in the
fall direction of the robot as we get closer to a desired solution at B, obtained in off-line
calculations in step A. Our chance of reaching the desired change in the fall direction will
be higher for a larger set of of-line solutions for B as is discussed in Remark 5.4.1.

5.5

Numerical Simulations

The parameters for the robot in Figure 5.1 are listed in Table 5.2. The control parameters
used in the safe fall algorithm are chosen as:

¯
φA = 30 deg,

¯
φC = 70 deg,
101

¯
φE = 90 deg

(5.17)

A wide range of velocities are chosen for passive coordinates in Step A.1 and the active
coordinates in step A.2 which result in a large set of solutions for B. The desired configuration
of the robot on the floor which is used in off-line calculations in Step A, is chosen such that
the desired change in the fall direction is 20 deg. The initial configuration of the robot are
assumed to be:

q0 =[α0 , β0 , γ0 , θ20 , θ30 , θ40 ]T = [90, 10, 0, −10, 0, 0]T
˙ ˙
˙
˙
˙
q0 =[α0 , β0 , γ0 , θ20 , θ30 , θ40 ]T = [0, 5, 0, 0, 0, 0]T
˙
˙

(5.18)

where the units are in deg and deg/s. The simulation results are shown in Figures 5.4, 5.5
and 5.6. The joint positions and velocities of all the coordinates are shown in Figure 5.4
where the impulse moments can be seen from the jumps in the velocity subplots. The lean
line angles φ and ψ are shown in Figure 5.5 and Figure 5.6 shows the impulsive inputs for
the active coordinates. Starting from the initial configuration, the robot goes under a free
fall with zero torques till the fall is realized by the algorithm at φ = 30 deg. The first
impulsive inputs are then applied in accordance with Step B.2. The joint torques are set
to zero thereafter till the robot reaches the second impulsive instant at φ = 70 deg when
the active joints are brought to full stop. The active coordinates will remain locked using a
locking mechanism till the robot hit the floor at φ = 90 deg.
Table 5.2: Kinematic and dynamic parameters for the six-dof robot with cylindrical links.

First link
Second link
Third link
Fourth link

Mass (Kg) Length (m)
1
0.5
1
0.5
4
0.8
2
0.3

Radius (m) Distance from C.M. (m)
0.1
0.25
0.1
0.25
0.25
0.4
0.05
0

102

The change in the fall direction can be seen from the ψ plot in Figure 5.5 where it is shown
the change in the fall direction angle is close to the desired change angle, ψdes = 20 deg.
Note that the dashed lines in Figures 5.4 and 5.5 show the simulation results when a single
set of impulsive inputs are used to brake the active joints at φ = 30 deg.
Remark 5.5.1. Although the fall direction has been changed using the proposed algorithm,
the impulsive inputs result in very large velocity jumps in active coordinates which may
cause the robot joints to exceed their limits. This is mainly because it is physically hard to
change the fall direction of the robot with current active coordinates and very large impulsive
inputs for active coordinates are needed to reach desired changes in the passive coordinates.

5.6

Conclusion

A safe fall algorithm was proposed to change the default fall direction of a six-dof robot
with three active joints. This is done to avoid hitting the surrounding objects to minimize
damage to them. The algorithm take over when the regular control has failed to keep the
robot balanced and a fall is unavoidable. Since there is a short period of time before the
robot hit the floor, only two sets of impulsive inputs are applied at specific configurations of
the robot. The first set of impulsive inputs is applied to give desired velocity jumps to the
passive coordinates while the second impulsive set is intended to stop the active coordinates.
Simulation results show the effectiveness of the algorithm in changing the fall direction of the
robot. For the robot considered here, large impulsive inputs are needed to have a significant
change in the fall direction which may cause large velocity jumps and exceeding joint limits
for the active coordinates. This issue can be addressed for higher-dof robots with more active
joints which make it physically easier to change the fall direction.
103

140
120
100
100

150

α (deg)

α (deg/s)
˙

a
0
200

β (deg)

50

0
400

γ (deg)

c

θ2 (deg)

i

j

0

θ3 (deg)
θ4 (deg)

-100
0

γ (deg/s)
˙

˙
0 θ2 (deg/s)
-200

d

0
-600
100

h

0

0
0
-50

˙
β (deg/s)

b

80
40

g

0.3
0.6
time (sec)

e

-2000

˙
θ3 (deg/s)

k

2000

f

0

0.9

˙
-2000 θ4 (deg/s)
0
0.3
0.6
time (sec)

l
0.9

Figure 5.4: Simulation results showing all the joint angles and angular velocities for changing
the fall direction of the robot

104

100
φ (deg)

80

a

60
40
20
0
20

ψ (deg)

b

10

0
0

0.3

time (sec)

0.6

0.9

Figure 5.5: Simulation results showing the lean line angles for changing the fall direction of
the robot

τ2 (N.m)

150

a

0
τ3 (N.m)

60 b

τ4 (N.m)

-60
5 c

-10
0

0.3

time (sec)

0.6

0.9

Figure 5.6: Simulation results showing the impulsive inputs for changing the fall direction
of the robot

105

Chapter 6
Disturbance Rejection for the
Synthetic-Wheel Biped

6.1

Introduction

In this chapter, we propose a push recovery strategy for the Synthetic-Wheel Biped (SWB)
[1]. Stepping strategy is used to maintain balance when the biped is pushed due to some
external disturbances. The external disturbances are impulsive in nature, i.e., cause some
sudden jumps in the system velocities. To reject these disturbances, our control algorithm
uses impulsive inputs. The push recovery algorithm is pretty robust and capable of rejecting
any pushes during stance and walking phases. Impulsive inputs are also used to impose some
virtual constraints on the motion of the legs and the torso. This work is mainly different
from the approaches in the literature because first, the SWB is an underactuated biped with
curved feet and no ankle joints and second, the set of admissible controls are enlarged to
include impulsive inputs. This chapter is organized as follows: The dynamics and control of
the Synthetic-Wheel biped is reviewed in Section 6.2. We derive the expressions for impulsive
inputs and their effects on system velocities in Section 6.3. Disturbance rejection algorithm
is discussed in Section 6.4 and two numerical examples are presented in Section 6.5. We
provide the concluding remarks in Section 6.6.

106

6.2
6.2.1

Background
Equations of Motion

The SWB [1] is shown in Figure 6.1. It has a torso and two legs with arc-shaped feet. The
arc-shaped feet have the same radius as the leg length and therefore the functionality of a
wheel is maintained by proper placement of the swing leg. The biped has a mechanism that
enables the swing leg to be slightly shorter in length than the stance leg and thereby avoid
scuffing with the ground. The dynamics of this mechanism is insignificant compared to the
overall dynamics of the system. The SWB is characterized by three generalized coordinates:
θ, φ, and ψ. The coordinates φ and ψ are active and are controlled by two actuators; the
coordinate θ is passive. The equations of motion are obtained as follows,

M(q)¨ + B(q, q)q + G(q) = T
q
˙ ˙

(6.1)

dt

lt
torso

R
x

β
y

dsw

φ
ψ

g

R

R

swing foot

θ
dst

stance foot

Figure 6.1: A schematic of the Synthetic-Wheel biped, reproduced from [1].
107

where
q = [θ φ ψ]T ,

T = [0 τ2 τ3 ]T

(6.2)

The matrices M(q), N(q, q) and G(q) are represented as:
˙

M(q) = Mij 3×3 , N(q, q) = Nij 3×3 , G(q) = [Gi ]3×1
˙

(6.3)

and are defined in Appendix B.

6.2.2

Torso Effect on Biped Dynamics

The specific design of the SWB provides the ability to effectively change the direction of the
walking through changing the torso angle. To demonstrate this, consider a “symmetric gait”
of the biped which is generated by imposing the following constraints on the motion of the
torso and the swing leg [1]:
C1 : α = αdes

(6.4)

C2 : ψ = −2θ
where α, depicted in Figure 6.2, is the angle of torso with respect to the vertical axis and is
defined as
α= θ+φ−π

(6.5)

The constraint C1 ensures that the torso maintains a desired angle αdes with respect to the
vertical axis while constraint C2 ensures that the swing leg is symmetric with respect to the
stance leg about the vertical axis at all times. The constrained system has one passive dof

108

with the following dynamics, which is derived by substituting Eq.(6.4) into Eq.(6.1):

¨
˙ ˙
Mc (θ) θ + Nc (θ, θ) θ + Gc = 0

(6.6)

˙
where the expressions for Mc (θ), Nc (θ, θ) and Gc can be found in [1]. For any set of reasonable
¨
parameter values, it can be verified that θ will be positive for positive angle αdes and vice
˙
versa. This implies that we can slow down the biped velocity θ by applying the constraints
˙
in Eq.(6.4) and choosing αdes to have the opposite sign of θ.

6.2.3

Interchange of Stance and Swing Legs

Assuming that the biped has a positive velocity, it will roll on its stance leg and the point
of contact with the ground will move from the heel to the toe. The stance and swing legs
can be interchanged at any time; but to have the maximum step size, the swing leg should

torso

α
φ = −θ+π+α

X
Y

β
swing
leg toe

θ
−ψ=2θ

swing
leg heel

stance
leg heel

stance
leg toe

Figure 6.2: Biped configuration at the time of interchange of stance and swing legs.

109

touch down when the heel of the swing leg is right in front of the toe of the stance leg, as
seen in Figure 6.2.
Without assuming maximum step size, an interchange of stance and swing legs will result
in a transformation of the generalized coordinates and their velocities given by the following
relations [1]:
qnew = P qold

(6.7)

qnew = P qold
˙
˙
where qold and qnew denote the values of the vector q before and after the interchange of
stance and swing legs, respectively, and the entries of the transformation matrix P are




1 
 1 0




P =  0 1 −1 




0 0 −1

(6.8)

which can be obtained from Figure 6.2. The dynamic model in Eq.(6.1) involves kinematic
and dynamic parameters of the stance and swing legs and their values need to be interchanged
during interchange of stance and swing legs as well.

6.2.4

Control Design

We develop the controller for imposing the constraints described in section 6.2.2. For the
symmetric gait with constraints in Eq.(6.4), the new set of generalized coordinates is defined
as
T

q=
¯

θ v1 v2

110

(6.9)

where

v1 = α − αdes
v2 = ψ + 2 θ

(6.10)

The original generalized coordinates φ and ψ are related to the new coordinates v1 and v2
according to the following relations, which can be derived from Eqs.(6.5) and (6.10):

φ =v1 − θ + π + αd
ψ =v2 − 2 θ

(6.11)

Substituting Eq.(6.11) into Eq.(6.1), we obtain the dynamics of the system in terms of the
new generalized coordinates as follows:

¯ q ¨ ¯ q ¯ ¯ ¯ q
˙ ˙
M (¯) q + N (¯, q ) q + G(¯) = T
¯

(6.12)

¯ ¯
¯
where M , N , and G have the same dimensions as M, N, and G respectively. The generalized
¨
force corresponding to θ in Eq.(6.12) is zero and this allows us to eliminate θ from the
two equations corresponding to the generalized coordinates v1 and v2 . The reduced-order
equations have the form
ˆ q ¨ ˆ q ¯ ˆ ˆ q
ˆ
˙ ˙
M (¯) q + N (¯, q ) q + G(¯) = T
ˆ

(6.13)

ˆ
ˆ
ˆ
where M ∈ R2×2 , N ∈ R2×2 and G ∈ R2×1 , and
q = [v1 v2 ]T ,
ˆ

ˆ
T = [τ2 τ3 ]T

111

(6.14)

Equation (6.13) represents a completely actuated system and we use feedback linearization
to design our controller as follows:

˙
ˆ
ˆ q ¯ˆ ˆ q
ˆ q
˙ ˙
T = N (¯, q )q + G(¯) − M (¯)(Kd q + Kp q )
ˆ
ˆ

(6.15)

where Kd and Kp are diagonal, positive-definite matrices of dimension two. Indeed, substitution of Eq.(6.15) into Eq.(6.13) yields

¨
˙
q + K d q + Kp q = 0
ˆ
ˆ
ˆ

(6.16)

which implies q → 0 as t → ∞. This simply follows that v1 , v2 → 0 as t → ∞, i.e., the
ˆ
constraints in Eq.(6.4) are satisfied.
The controller in Eq.(6.15) has been implemented and experimentally verified in our
laboratory biped [1]. The experimental results show that v1 and v2 do not converge to
zero but oscillate around zero due to impulsive disturbances from the ground at the time of
swing-leg touchdown. The algorithm presented in this chapter provides a way to avoid these
impulsive disturbances from the ground while taking steps due to an external disturbance.

112

6.3

Impulsive Torques and Effects

6.3.1

Braking Torque for the Swing Leg

˙
Consider the action that results in exponential convergence of the swing leg velocity ψ to
˙
zero while keeping θ unchanged. To this end, the following dynamics are assumed,

¨
θ =0
˙
¨
ψ = − k1 ψ

(6.17)

˙
where k1 is a positive constant determining the rate of convergence of ψ to zero. To compute
the torque required for this action, we multiply Eq.(6.1) with the inverse of the inertia matrix
to obtain



¨1
 θ 


1
 ¨ 
 θ2 =

 A33 (A11 A22 − A2 ) − A22 A2
12
13


¨
θ3





 h1 − A12 A33 τ2 − A13 A22 τ3 




 h2 +(A11 A33 −A2 )τ2 +A12 A13 τ3 
13




h3 +A12 A13 τ2 +(A11A22 −A2 )τ3
12

(6.18)

where h1 , h2 and h3 are given by the following expressions,

˙
˙
˙
h1 = − A22 A33 (B11 θ + B12 φ + B13 ψ) − A22 A33 G1 + A12 A33 G2 + A22 A13 G3
˙
˙
˙
h2 =A12 A13 (B11 θ+B12 φ+B13 ψ) +A12 A33 G1 +(A2 −A11 A33 )G2 −A12 A13 G3
13
˙
˙
˙
h3 =A22 A13 (B11 θ + B12 φ + B13 ψ) +A13 A22 G1 −A12 A13 G2 +(A2 −A11 A22 )G3
12
Substituting Eq.(6.17) into the first and third equations in Eq.(6.18) results in

113

(6.19)

τ2 =

1 A2 h1 −A22 (A11 h1 +A13 h3 )
12
˙
+k1A13 A22 ψ
A12 A2 A22 +(A2 −A11 A22 )A33
13
12

A13 h1 + A33 h3
˙
τ3 = 2
− k1 A33 ψ
A13 A22 + (A2 − A11 A22 )A33
12

(6.20)

If the constant k1 has a very large value, the torque expressions in Eq.(6.20) will be impulsive
˙
in nature and will stop the swing leg relative velocity, ψ, in a very short period of time without
changing the stance leg velocity.

6.3.2

Braking Torque for the Torso

Consider an action which results in exponential convergence of the torso absolute velocity,
α, to zero while maintaining equal and opposite velocities for the stance and swing legs, i.e.
˙
˙
˙
ψ = −2 θ, in conformity with the symmetric gait. Using Eq.(6.5), we consider the following
dynamics to achieve the goal,

¨ ¨
˙ ˙
θ + φ = − k2 (θ + φ)
¨
¨
˙
˙
θ + 2ψ = − k3 (θ + 2ψ)

(6.21)

where k2 and k3 are some positive constants determining the rate of convergence of the
velocities to their desired values. The control inputs that result in the dynamics in Eq.(6.21)
can be obtained by substituting Eq.(6.18) into Eq.(6.21) and solving for τ2 and τ3 . The
massive torque expressions are not explicitly mentioned here for the sake of simplicity. If we
choose very large values for k2 and k3 in Eq.(6.21), the resulted torque expressions will be
impulsive in nature and stop the motion of the torso in a very short period of time without
114

affecting the symmetric velocity condition associated with the symmetric gait.

6.3.3

Impulsive Effect on Velocity

The application of impulsive torques for φ and ψ generalized coordinates results in velocity
jumps in all three coordinates θ, φ and ψ. The relationship between the jumps in velocities
can be derived from Lagrange’s equations. Consider the integral of the equations of motion
in Eq.(6.1) over the short interval of time ∆t during which the impulsive forces and moments
act, i.e.
∆t

∆t

[M(q)¨ + N(q, q)q + G(q)] dt =
q
˙ ˙
0

T dt

(6.22)

0

The above equation can be rewritten as:
∆t

M(q)∆q + N(q, q)∆q +
˙
˙

∆t

G(q) dt =
0

T dt

(6.23)

0

Since ∆t is very short time interval and the configuration of the biped does not change during
this time, i.e. ∆q = 0, Eq.(6.23) will be simplified to

M(q)∆q = Tim
˙

(6.24)

where Tim = [0 I2 I3 ]T represents the vector of impulses applied to the biped. The above
equation can be decomposed into the following three equations:

˙
˙
˙
˙
˙
˙
M11 (θ+ − θ− )+M12(φ+ − φ− )+M13 (ψ + − ψ − ) = 0

(6.25)

˙
˙
˙
˙
˙
˙
M21 (θ+ − θ− )+M22(φ+ − φ− )+M23 (ψ + − ψ − ) = I2

(6.26)

115

˙
˙
˙
˙
˙
˙
M31 (θ+ − θ− )+M32(φ+ − φ− )+M33 (ψ + − ψ − ) = I3

(6.27)

where the superscripts “−” and “+” denote the values immediately before and after the
application of the impulsive inputs, respectively.
As the first special case, consider the impulsive action described in Section 6.3.1 in which
˙
˙
˙
ψ + = 0 and θ+ = θ− . Using Eq.(6.25), the relative velocity of torso after the impulsive
action can be obtained as
M
˙
˙
˙
φ+ = φ− + 13 ψ −
M12

(6.28)

As the second case, consider the impulsive action in section 6.3.2 which results in α+ = 0
˙
˙
˙
and ψ + = −2 θ+ . Then the velocity of the swing leg after the impulse can be obtained from
Eq.(6.25) as follows:
˙
˙
˙
M θ− + M12 φ− + M13 ψ −
˙
θ+ = 11
M11 − M12 − 2M13

(6.29)

It should be noted that the impulses I2 and I3 can be computed from Eqs.(6.26) and (6.27).
These impulses can also be approximated by the time integral of the torque expressions
obtained in sections 6.3.1 for large gain k1 and the torque expressions in section 6.3.2 for
large gains k2 and k3 .

6.4

Disturbance Rejection Algorithm

In this section, we develop an algorithm to reject the external disturbances applied to the
Synthetic-Wheel Biped. The external disturbances are assumed to be of the form of impulsive
forces and their effects are modeled by jumps in the generalized velocities. The disturbances
are assumed to be large enough that precludes the possibility of stabilizing the upright

116

posture without taking a step. The problem at hand is therefore to reject the disturbances
and stabilize the upright posture by taking a few steps in the forward or backward direction.
The disturbance rejection algorithm is based on slowing down the biped using the symmetric
gait while imposing constraints on the legs and torso. These constraints are described as
follows:

C3 : − β/2 ≤ θ1 ≤ β/2

(6.30)

C4 : − γ ≤ α ≤ γ

(6.31)

where β is the foot arc angle (see Figure 6.2) and γ is a small positive angle. We propose
the following three-step algorithm to reject external impulsive disturbances and stabilize the
upright posture of the Synthetic-Wheel Biped:
1. Initialization:
(a) Linearize the dynamic equations in Eq.(6.1) about the desired equilibrium point
˙
˙
˙
(θ, θ, φ, φ, ψ, ψ) = (0, 0, π, 0, 0, 0).
(b) Design a LQR controller for the linearized system to render the desired equilibrium
locally asymptotically stable 1 . Let RA be the region of attraction of the desired
equilibrium.
(c) Define a quadratic Lyapunov function for the linearized system as V = X T SX
˙
˙
˙
where X = [θ, θ, φ − π, φ, ψ, ψ]T and S is the solution of the Riccati equation
associated to the LQR design.
T
(d) Define a set point for the Lyapunov function as V0 = X0 SX0 where X0 =
1 This

is always possible because the linearized system is controllable

117

[η, 0, αdes − η, 0, −2η, 0]T and η < β/2 is a positive angle. η and αdes values are
chosen sufficiently small such that if V ≤ V0 , the biped configuration will lie inside
RA and will approach the upright equilibrium without exceeding the bound on θ
in Eq.(6.30).
2. Disturbance Rejection:
(a) Following the external disturbance, apply the torque expression in Eq.(6.15) to
implement the symmetric gait. The torque expression in Eq.(6.15) depends on
the value of the torso desired angle, αdes , which is chosen to have the opposite
˙
sign of the stance leg velocity, θ, so the biped slows down. This follows from our
discussion in Section 6.2.2.
(b) If the stance leg angle θ reaches the boundary of the interval [−β/2, β/2], ap˙
ply the torque expressions in Eq.(6.20) to quickly enforce ψ = 0 while keeping
the stance leg velocity unchanged. The stance leg and swing leg are immediately
interchanged using Eq.(6.7) and the control torques in step 2(a) are applied there˙
after. By enforcing ψ = 0, we avoid impulsive disturbance from the ground to the
biped (swing leg) at the time of leg interchange. It can be seen from Eq.(6.7) that
˙
this also eliminates jumps in θ at the time of leg interchange, which is necessary
for smooth walking.
(c) If the torso angle α reaches the boundary of the interval [−γ, γ], apply the torque
expressions obtained in section 6.3.2 to quickly stop the torso while keeping the
velocity of the legs symmetric. Then continue to apply the control torques in step
2(a).
˙
(d) If V ≤ V0 , terminate this step and go to step 3. The stance leg velocity, θ, will
118

˙
go to zero since αdes was chosen to have a sign opposite to that of θ. Imposing
˙
˙
the symmetric gait constraints in Eq.(6.4), φ and ψ will approach zero as well.
Therefore, By choosing small enough value for αdes , the inequality V ≤ V0 is
eventually satisfied.
3. Stabilization:
With V ≤ V0 and the biped satisfying the constraints in Eq.(6.30) and Eq.(6.31),
the biped configuration will be inside the region of attraction RA. Invoke the linear
controller designed in step 1 to stabilize the desired equilibrium.

6.5

Numerical Simulations

We present two simulations to demonstrate the effectiveness of the proposed algorithm in
rejecting the impulsive disturbances on the Synthetic-Wheel biped. For both simulations,
the kinematic and dynamic parameters in Table 6.1 are used.
For the first simulation, we assume two impulsive disturbances are applied on the biped
Table 6.1: Parameters of the Synthetic-Wheel Biped [1]

Inner leg
Outer leg
Torso

Inner leg
Outer leg
Torso

Kinematic parameters
Length (m) Foot radius, R (m) Foot arc, β (deg)
0.635
0.635
22.5
0.635
0.635
22.5
0.457
Dynamic parameters
Mass (kg)
Inertia (kgm2 )
d in Fig.2 (m)
1.64
0.094
0.285
3.64
0.128
0.355
11.87
0.198
0.307

119

at two instants of time which can be modeled by the following values of the angular velocities:

˙ + ˙+ ˙ +
(θ1 , θ2 , θ3 ) = (0, −5, −1) , at t = 0 sec
˙ + ˙+ ˙ +
(θ1 , θ2 , θ3 ) = (−0.1, 3, −1) , at t = 2 sec

(6.32)

The initial angular positions and the parameters needed in the algorithm are chosen as:

(θ10 , θ20 , θ30 ) = (5◦ , 180◦ , −8◦ )
˙
η = β/3, αdes = −sgn(θ) 5◦ , γ = 10◦

(6.33)

Figure 6.3 shows the simulation results including the plots of the joint angles and angular
˙
velocities and the control inputs. The angular position ψ and the angular velocity ψ of the
swing leg are shown by dashed lines. The horizontal lines in the angular position subplots
represent the bounds on the stance leg and the torso. Due to the initial impulsive disturbance,
the torso gets to its lower bound and the impulsive inputs are applied to keep them inside
˙
the allowed region. The biped starts walking in the negative direction with θ < 0 and
the continuous controller will slow down the biped motion by imposing the symmetric gait
constraints. when the stance leg gets to its bound, the impulsive inputs are applied to make
˙
ψ = 0 right before the interchange of the legs. When the second impulsive disturbance is
applied at t = 2 sec., the biped changes its walking direction to positive and consequently,
the continuous controller switches the desired angle of torso to negative to slow down the
biped using the symmetric gait constraints. Torso reaches its upper bound after the second
disturbance as well and is kicked back inside the region using impulsive inputs. The peaks
in the torque plots correspond to impulsive torques which are applied when the stance leg

120

and the torso reach their bounds. As can be seen from Figure 6.3, the biped is able to reject
the impulsive disturbance after taking few steps in the negative and positive directions. The
linear controller is invoked at t = 6.04 sec. (shown by vertical dashed line) to stabilize the
upright configuration of the biped.
For the second simulation, we applied five identical impulsive disturbances to the biped
at time instants t = 0, 2, 4, 6, 8 sec. which are modeled by the following changes in the
velocities:

˙ ˙
˙
τ3 (N.m) τ2 (N.m) α(rad/s) θ, ψ(rad/s) α(rad) θ, ψ(rad)

˙+ ˙ + ˙ +
(θ1 , θ2 , θ3 ) = (−0.5, 6, −2)

(6.34)

0.4
0
-0.4
0.17
0
-0.17

a

b

c

2
0
-2
2
0
-4
45
0
-40
120
0
-200
0

d

e

f

1

2

3

4

time (sec)

5

6

7

8

Figure 6.3: Simulation results showing angular positions (rad), angular velocities (rad/s),
and control inputs (N.m) of the SWB in rejecting the impulsive disturbances applied at two
˙
instants of time. The dashed lines correspond to the plots of ψ and ψ.

121

the initial conditions for joint angles and the other algorithm parameters were chosen as:

(θ10 , θ20 , θ30 ) = (0, π, 0)
˙
η = β/3, αdes = −sgn(θ) 5◦ , γ = 20◦

(6.35)

The simulation results are shown in Figure 6.4. As in the first simulation, dashed lines represent the swing-leg angle and angular velocity and the horizontal lines in angular positions
subplots show the bounds on the stance leg and torso motion. The disturbances make the
biped to walk in the positive direction with symmetric gait to recover its upright configuration. In this case, the limit on the torso motion is not reached and all the impulsive

θ, ψ(rad)

0.35
0
-0.35

˙ ˙
τ3 (N.m) τ2 (N.m) α(rad/s) θ, ψ(rad/s) α(rad)
˙

inputs shown by peaks in the torque plots, are applied when the stance leg gets to its bound

0.35
0.17
0
2
0
-2
-4
6
2
-2
20
10
0
-10
100
50
0
0

a

b

c
d

e

f

5

time (sec)

10

15

Figure 6.4: Simulation results showing angular positions (rad), angular velocities (rad/s),
and control inputs (N.m) of the SWB in rejecting the impulsive disturbances applied at five
˙
instants of time. The dashed lines correspond to the plots of ψ and ψ.
122

at θ = β/2. Using the proposed algorithm, the biped is able to reject the disturbances.
The vertical dashed line in Figure (6.4) represents the time t = 12.3 sec. when the linear
controller is invoked to stabilize the upright equilibrium of the biped.

6.6

Conclusion

We proposed an algorithm to reject the effects of impulsive disturbances applied to the
Synthetic-Wheel Biped in its stance and walking configurations. The impulsive disturbance
is modeled by jumps in system velocities. The algorithm uses a combination of continuous
and impulsive control inputs to generate a symmetric gait for the SWB while imposing
some constraints on the system states. Using the symmetric gait, the biped takes a few
steps in order to reject the effects of the external disturbances and recover its balance. The
constraints on the system are imposed to avoid additional impulsive disturbances from the
ground at the interchange of legs. Simulation results show the effectiveness of the algorithm
in rejecting the impulsive disturbances applied at different instants of time in both stance
and walking phases of the biped.

123

Chapter 7
Energy-Conserving Gaits for
Point-Foot Passive-Ankle Bipeds

7.1

Introduction

In the previous chapter, we studied a case study where impulsive inputs in the form of external disturbance are undesirable and the control objective is to cancel their effects on the
system. In this chapter, we consider another undesirable form of impulsive inputs which
exists in walking bipeds: the interaction impact from ground applied during the legs interchange. In addition to the velocity jumps for the biped links, energy is lost during the
impact. To make a bipedal walking gait more energy efficient, controllers need to be designed
to avoid these impacts.
In this chapter, we present a general method for designing energy-conserving gaits for
point-foot passive-ankle planar bipeds. Assuming no energy dissipation due to friction, an
energy-conserving gait is one that has a zero Cmt based on the definition in Eq.(1.1). The
sufficient conditions for such a gait are equal potential and kinetic energies at the beginning
and end of each step and no energy loss due to impact at the time of swing-leg touchdown.
These conditions are presented in Section 7.2. The conditions are applied to two three-dof
bipeds in Section 7.3; the results indicate that energy-conserving gaits can be designed for
the Synthetic-Wheel biped [1] but not for the biped with point feet. Energy-conserving gaits
124

are designed for five-dof bipeds in Sections 7.4 and 7.5. One-step periodic gaits are designed
in Section 7.4 for a biped with similar leg links and more general two-step periodic gaits are
designed for bipeds with similar and dissimilar leg links in Section 7.5. Concluding remarks
are provided in Section 7.6.

7.2

Energy-Conserving Gaits

7.2.1

Assumptions and Definitions

Consider the general n-link planar biped comprised of a torso and two legs, shown in Figure
7.1. The equations of motion of the biped are assumed to be of the form

M(q) q + N(q, q) q + G(q) = T
¨
˙ ˙

(7.1)

where q = (θ1 , θ2 , · · · , θn )T and T = (u1 , u2, · · · , un )T are the vectors of generalized coordinates and generalized forces1 , M(q) ∈ Rn×n is the symmetric mass matrix, N(q, q) ∈ Rn×n
˙
is the matrix of centrifugal and Coriolis terms, and G(q) ∈ Rn is the vector of gravitational
forces. We assume that
A1. Only one leg of the biped is in contact with the ground at any given time, i.e., there
is no double-support phase.
The leg in contact with the ground is commonly referred to as the stance leg; the other leg
is referred to as the swing leg. All joints are measured counter-clockwise with respect to
the vertical. They are numbered arbitrarily except that θ1 denotes the angle of the stance
1 Since

the generalized coordinates are absolute joint angles, the generalized forces are the net
external moments acting on each link.

125

torso

θj

hip joint

θn
θ2
y

θ1

swing foot

stance foot
x
Figure 7.1: A schematic of an n-link point-foot planar biped.

leg link in contact with the ground, and θn denotes the angle of the swing leg link that will
come in contact with the ground. For any biped gait, we define the term “step” as follows:
Definition 7.2.1. A step is the complete motion of the biped on the stance leg which is
followed by interchange of the stance and swing legs.
We make the following additional assumptions:
A2. The stance and swing legs of the biped have point feet and the stance foot does not
slide or leave the ground during a step.
A3. All joints of the biped are actuated except the ankle joint of the stance leg, which is
passive.
A4. The biped walks on level ground and there is no energy dissipation due to friction.

126

At the beginning of each step, the joint angles and their velocities are denoted by

i i
i
q i = (θ1 , θ2 , · · · , θn )T ,

˙i ˙i
˙i
q i = (θ1 , θ2 , · · · , θn )T
˙

Similarly, the joint angles and their velocities at the end of each step are denoted by

f

˙f
˙f ˙ f
q f = (θ1 , θ2 , · · · , θn )T
˙

f

f

q f = (θ1 , θ2 , · · · , θn )T ,
We also assume that:

A5. The joint angles of the stance leg ankle at the beginning and end of each step are
f

i
symmetric with respect to the vertical, i.e., θ1 = −θ1 .

The velocity of the swing foot at the time of contact with the ground (at the end of each
step) is denoted by
f
f
f
Vn = Vn,x ˆ + Vn,y ˆ
i
j
f
f
where ˆ and ˆ are unit vectors along the inertial x and y axes, and Vn,x and Vn,y are the
i
j

corresponding velocity components.

7.2.2

Sufficient Conditions for Energy-Conserving Gait

For a biped, the mechanical energy required by its gait will be zero over each step if the
following conditions are satisfied:
1. The potential energies at the beginning and end of each step are identical. This is
achieved if
f

i
θj = −θj ,

127

j = 1, 2, · · · , n

(7.2)

2. The kinetic energies at the beginning and end of each step are identical. This is
achieved if
˙i
˙f
θj = θj ,

j = 1, 2, · · · , n

(7.3)

Additionally, during interchange of the stance and swing legs, the energy of the biped will
remain conserved if:
3. Swing foot touchdown does not cause loss of energy. This is achieved if there is no
impact due to foot-ground interaction, i.e.

f

f

Vn,x = Vn,y = 0

(7.4)

The conditions in Eqs.(7.2) and (7.3) can be easily satisfied for θj , j = 2, 3, · · · , n, since
all joints of the biped except the stance leg ankle are actuated. The condition in Eq.(7.2) is
satisfied for θ1 as well - this is a direct consequence of assumption A5. Since the stance leg
˙
ankle is passive, the main challenge is to satisfy the condition in Eq.(7.3) for θ1 .

7.2.3

Characterization of Trajectories of the Actuated Joints

˙
The condition in Eq.(7.3) can be satisfied by the ankle joint velocity θ1 if it is an even
function2 of θ1 , i.e.,
e
˙
θ1 = g1 (θ1 )

(7.5)

This follows from assumption A5. We now state necessary conditions for the actuated joint
trajectories with the help of the following Theorem.
2 We

use superscripts “e” and “o” to denote even and odd functions, respectively.

128

Theorem 7.2.1. The conditions in Eqs.(7.2), (7.3) and (7.5) will be satisfied only if the
velocities of the actuated joints are even functions of θ1 , i.e.

e
˙
θj = gj (θ1 ),

j = 2, 3, · · · , n

(7.6)

˙
Proof. To guarantee θ1 in Eq.(7.5) is a function of θ1 alone, all actuated joints must be
functions of θ1 , i.e.
θj = fj (θ1 ),

j = 2, 3, · · · , n

(7.7)

∂fj
,
∂θ1

(7.8)

Taking the derivative of Eq.(7.7), we get

˙
˙
θj = hj (θ1 ) θ1 ,

hj

j = 2, 3, · · · , n

The hj ’s can be written as a sum of even and odd functions as follows

˙
˙
θj = he (θ1 ) + ho (θ1 ) θ1 ,
j
j

j = 2, 3, · · · , n

(7.9)

From Eq.(7.3), we have for j = 2, 3, · · · , n
f
f
i
i
˙i
˙f
he (θ1 ) + ho (θ1 ) θ1 = he (θ1 ) + ho (θ1 ) θ1
j
j
j
j

⇒

f

f

i
i
he (θ1 ) + ho (θ1 ) = he (θ1 ) + ho (θ1 )
j
j
j
j

129

(7.10)

Rearranging terms and using Eq.(7.2) and the property of even and odd functions, we get

f

f

i
i
ho (θ1 ) − ho (θ1 ) = he (θ1 ) − he (θ1 )
j
j
j
j

⇒

i
i
i
i
ho (−θ1 ) − ho (θ1 ) = he (θ1 ) − he (−θ1 )
j
j
j
j

⇒
⇒

i
i
ho (−θ1 ) = ho (θ1 )
j
j

ho (θ1 ) = 0
j

(7.11)

It simply follows that hj (θ1 ) = he (θ1 ), j = 2, 3, · · · , N. Using Eqs.(7.5) and (7.8) we can
j
now write
e
˙
˙
θj = he (θ1 ) θ1 = he (θ1 ) g1 (θ1 )
j
j

e
gj (θ1 )

(7.12)

˙
which implies that θj , j = 2, 3, · · · , n is even function of θ1 . This completes the proof.
Since the derivative of an odd function is always an even function, the theorem above
implies that the trajectories of the actuated joints consist of an odd function and a constant.
We choose the following general form for the trajectories

o
θj (θ1 ) = fj (θ1 ) + aj θ1 + bj ,

j = 2, 3, · · · , n

(7.13)

where aj and bj are constants, and the term aj θ1 is the linear part of the odd function.
The constants bj and the coefficients of the odd functions in Eq.(7.13) will be chosen such
that the boundary conditions in Eqs.(7.2), (7.3) and (7.4) are satisfied. Of these boundary
˙
conditions, the condition on θ1 in Eq.(7.3) is satisfied indirectly by choosing the constants and
¨
the coefficients such that θ1 is an odd function. This requires the substitution of Eq.(7.13) in
¨
the expression for θ1 , which is obtained from the equation of motion for θ1 . Although there

130

is no guarantee that trajectories of the form given by Eq.(7.13) can be found for energyconserving gaits for arbitrary bipeds, we show that such trajectories do exist for several
planar bipeds.

7.3

Three-DOF Bipeds: Two Case Studies

In this section, we design energy-conserving gaits for two 3-dof biped platforms: the SyntheticWheel biped3 [1] and a point-foot three-link biped with two legs and a torso.

7.3.1

Synthetic-Wheel Biped (SWB)

7.3.1.1

System Description

The SWB [1], which was described in Section 6.2.1, is shown in Figure 7.2. The equations
of motion of the SWB have the same form as Eq.(7.1), with q and T defined as follows:

q = [θ φ ψ]T ,

T = [0 τ2 τ3 ]T

(7.14)

where τ2 and τ3 are motor torques corresponding to the relative coordinates φ and ψ. The
equations of motion of the SWB are provided in Appendix B.
Remark 7.3.1. The generalized coordinates in Eq.(7.14) are different from those chosen for
the general case in the previous section. This is because the dynamics of the SWB, already
presented in [1], is based on relative generalized coordinates shown in Figure 7.2. This
difference does not change the sufficient conditions for an energy-conserving gait, namely,
3 The

Synthetic-Wheel biped has arc-shaped feet but is investigated here along with point-foot
bipeds because of its point contact with the ground.

131

dt
lt

torso
R
x

β
y

dsw

φ
ψ

g

swing foot

R

R
θ
dst

stance foot

Figure 7.2: A schematic of the Synthetic-Wheel biped, reproduced from [1].
the kinetic and potential energies at the beginning and end of each step are the same, and
there is no loss of energy at the time of swing-leg touch down.

7.3.1.2

Sufficient Conditions for Energy-Conserving Gait

The boundary conditions for the gait are determined as follows:
1. The potential energies at the beginning and end of each step are identical. This is
achieved by
θf = −θi ,

φf = −φi ,

ψ f = −ψ i

(7.15)

2. The kinetic energies at the beginning and end of each step are identical. This is
achieved by
˙
˙
θf = θi ,

˙
˙
φf = φi ,

˙
˙
ψf = ψi

(7.16)

3. The synthetic-wheel design ensures that there are no vertical impulsive forces, i.e.,
132

f

Vn,y = 0. At the time of swing foot touchdown, there will be no horizontal impulsive
forces if
˙
ψf = 0

7.3.1.3

(7.17)

Determination of Actuated Joint Trajectories

Using Eq.(7.13), we consider the following functional dependence for φ and ψ

o
φ(θ) = (a1 θ + b1 ) + f1 (θ)

(7.18)

o
ψ(θ) = (a2 θ + b2 ) + f2 (θ)

(7.19)

˙
˙
¨
The condition θi = θf in Eq.(7.16) can be satisfied if θ is an odd function of θ. To this end,
¨
we derive the expression for θ as follows

¨ N
θ=
D

(7.20)

where N = N(q, q, q ) and D = D(q) are defined as follows
˙ ¨

¨
N(q, q, q ) =− φ {It +mt (dt −lt )(dt −lt +R cos [θ + φ])}
˙ ¨
¨
− ψ {Isw +msw (dsw −R)(dsw −R+R cos [θ+ψ])}
˙
˙ ˙
+ mst (dst − R)(g + Rθ2 ) sin θ + mt (dt − lt ) g + R(θ + φ)2 sin [θ + φ]
˙ ˙
+msw (dsw −R) g+R(θ+ ψ)2 sin [θ+ψ]

133

D(q) = It + Ist + Isw + mst (dst − R)2 + R2 − 2R(−dst + R) cos θ
2
+ mt d2 +lt +R2 +2R(dt −lt ) cos(θ + φ)−2dt lt
t

+msw (dsw −R)2 +R2 −2R(−dsw +R) cos [θ + ψ]
¨
Using Eqs.(7.18) and (7.19), a term-by-term analysis of N and D shows that θ is an odd
function of θ for any values of a1 and a2 provided that b1 and b2 are chosen as [138]

b1 = mπ,

m, n = 0, ±1, ±2, · · ·

b2 = nπ,

(7.21)

For the torso to be upright, the choice of m for b1 = mπ should be limited to m =
±1, ±3, ±5, · · · . Similarly, for the swing-leg to point downwards, the choice of n for b2 = nπ
should be limited to n = 0, ±2, ±4, · · · . By simply setting b1 = π and b2 = 0 and assuming
o
o
f1 and f2 in Eqs.(7.18) and (7.19) to be sinusoidal functions, we have

φ(θ) = (a1 θ + π) + A sin (Bθ)

(7.22)

ψ(θ) = a2 θ + C sin (Dθ)

(7.23)

If the SWB moves in the positive x direction with maximum step length, we have

θf = −θi =

β
2

ψ f = −ψ i = −β

(7.24)
(7.25)

where β is the arc angle of the feet - see Figure 7.2. Using Eq.(7.24) and the boundary
conditions corresponding to θ and φ in Eqs.(7.15) and (7.16), the coefficients a1 and B can
134

be determined. Using Eqs.(7.24) and (7.25) and the boundary conditions corresponding to
θ and ψ in Eqs.(7.15), (7.16) and (7.17), the coefficients a2 , C, and D can be determined.
The energy-conserving trajectories of the actuated joints finally take the form

φ(θ) = π − θ + A sin(2πθ/β)

(7.26)

ψ(θ) = −β sin(πθ/β)

(7.27)

The choice of the constant A is discussed in the next section.
7.3.1.4

Numerical Simulations

The kinematic and dynamic parameters of the SWB are listed in Table 7.1. The constant
A corresponds to the amplitude of oscillation of the torso and determines if the biped will
succeed or fail to take a step. To illustrate this point, we plot phase portraits of the motion of
the SWB in Figure 7.3 for different values of A with initial conditions θi = −β/2 = −11.25
˙
˙
deg and θi = 45 deg/s. For A equal to 0.10 and 0.08, the SWB fails to take a step because θ
changes sign before the step is completed. As A is reduced, for A equal to 0.06 and smaller,
Table 7.1: Kinematic and dynamic parameters of Synthetic Wheel Biped

Leg 1
Leg 2
Torso

Leg 1
Leg 2
Torso

Kinematic parameters
lst , lsw , lt
R
0.635 m
0.635 m
0.635 m
0.635 m
0.457 m
Dynamic parameters
mst , msw , mt Ist , Isw , It
2.640 kg
0.111 kg m2
2.640 kg
0.111 kg m2
11.87 kg
0.198 kg m2

135

β
22.5 deg
22.5 deg
dst , dsw , dt
0.320 m
0.320 m
0.307 m

100

θi = −11.5
˙
θi = 45

-0.40
-0.20
0.0

0

˙
θ (deg/s)

50

0.06
0.10

0.02
0.04

0.08
-50
−15

−5

5

15

θ (deg)
Figure 7.3: Phase portraits of the Synthetic-Wheel Biped for different amplitudes of torso
oscillation A

the biped rolls on its stance foot from heel to toe and is able to take a step. The values of
A for which the SWB is able to take a step depends on its inertial properties and its initial
velocity.
For the particular case of A = 0.0, we plot the torques τ2 and τ3 over one step in Figure
7.4; these torques are responsible for generating the trajectories of φ and ψ in Eqs.(7.26)
and (7.27). The mechanical work done at these joints over one step can be expressed as an

τ2 , τ3 (Nm)

5.0
θi = −11.25

0.0

τ3
τ2

τ2
τ3

-5.0
-15

θf = 11.25

-5

θ (deg)

5

15

Figure 7.4: Torque plots for an energy-conserving gait for the Synthetic-Wheel Biped (SWB).

136

θ θ
τ2 , τ3 (Nm)

5.0
θf = 11.25

θ
τ3

θ
τ2

0.0
θi = −11.25

-5.0
-15

-5

5

15

θ (deg)
Figure 7.5: Generalized torque plots for an energy-conserving gait for the Synthetic-Wheel
Biped (SWB).
integral over θ, as follows

dφ
) dθ
dθ
t
θ
dψ
˙
W3 = τ3 ψ dt = τ3 ( ) dθ
dθ
t
θ

W2 =

˙
τ2 φ dt =

τ2 (

θ
θ

θ
τ2 dθ

(7.28)

θ
τ3 dθ

(7.29)

θ
θ
The plots of τ2 and τ3 are shown in Figure 7.5. The area under these curves are equal to

zero and this confirms that the gait described by Eqs.(7.26) and (7.27) conserves mechanical
energy.

7.3.2

Point-Foot Three-Link Biped

The point-foot three-link biped is obtained by replacing the arc-shaped feet on the SWB
with point feet. A schematic of this biped is shown in Figure 7.6. Using the same set of
generalized coordinates and generalized forces as the SWB, we first look at the sufficient
condition for energy-conserving gait that corresponds to foot-ground interaction without

137

dt

lt
torso

lsw
x
y

swing foot dsw

φ
ψ

g

lst
θ
stance foot

dst
Figure 7.6: A schematic of the Point-Foot Three-Link.

impact. This condition, given by Eq.(7.4), can be written as

˙
˙
˙
lst cos θf θf −lsw cos (θf +ψ f )θf −lsw cos (θf +ψ f )ψ f = 0
˙
˙
˙
lst sin θf θf −lsw sin (θf +ψ f ) θf −lsw sin (θf +ψ f ) ψ f = 0

(7.30)

The above equation was obtained by taking the derivative of the swing foot position coordinates relative to the stance foot - see Figure 7.6. Applying condition Eq.(7.2) next and
assuming that the stance leg angle and swing leg angle satisfy (similar to Eqs.(7.24) and
(7.25) for the SWB)
θf = β/2,

θ + ψ = −β/2

138

for some angle β, we get from Eq.(7.30)

˙
˙
˙
cos(β/2) lst θf −lsw θf −lsw ψ f = 0
˙
˙
˙
sin(β/2) lst θf +lsw θf +lsw ψ f = 0

(7.31)

˙
˙
Since β = 0, Eq.(7.31) implies θf = 0, which implies θi = 0 from Eq.(7.3). At the beginning
and end of each step, the stance and swing legs will be symmetric with respect to the vertical
and have zero velocity; and therefore, the biped will roll on its stance foot only if the torso
is inclined with respect to the vertical. The sufficient condition in Eq.(7.2) dictates that the
torso angle with respect to the vertical at the end of the step be equal and opposite to that
at the beginning of the step. This implies that the biped will only roll back and forth on its
stance leg. Clearly, it is not possible to design an energy-conserving gait for the point-foot
three-link biped using the sufficient conditions in Eqs.(7.2), (7.3) and (7.4).

7.4
7.4.1

Five-DOF Point-Foot Biped
System Description

Consider the general five-dof point-foot biped shown in Figure 7.7. The generalized coordinates and generalized forces of the biped are as follows
T

q=

θ1 θ2 θ3 θ4 θ5
T

T =

u1 u2 u3 u4 u5

(7.32)

Assuming that the j-th link is driven by a motor mounted on the (j −1)-th link, j = 2, 3, 4, 5,
139

l3

θ3

torso

l4
θ4

l2

θ5
l5

θ2
θ1
y

l1

swing foot
stance foot

x
Figure 7.7: A schematic of the five-dof point-foot biped.

the generalized force vector can be rewritten in the form
T

T =

−τ2 (τ3 − τ2 ) (τ4 − τ3 ) (τ5 − τ4 ) τ5

(7.33)

where τj , j = 2, 3, 4, 5, are the corresponding motor torques. Note that τ1 = 0 since the
ankle joint of the biped is passive. The equations of motion of the biped and description of
its parameters are provided in Appendix C.

7.4.2

Sufficient Conditions for Energy-Conserving Gait

The boundary conditions for the gait are determined as follows:
1. The potential energies at the beginning and end of each step are identical. This is
achieved by
f

i
θj = −θj ,

140

j = 1, · · · , 5

(7.34)

2. The kinetic energies at the beginning and end of each step are identical. This is
achieved by
˙i
˙f
θj = θj ,

j = 1, · · · , 5

(7.35)

3. At the time of swing foot touchdown, there will be no impact due to impulsive forces
f

f

if Vn,x = Vn,y = 0, i.e.

f ˙f
f ˙f
f ˙f
f ˙f
l1 cos θ1 θ1 +l2 cos θ2 θ2 +l4 cos θ4 θ4 +l5 cos θ5 θ5 = 0
f ˙f
f ˙f
f ˙f
f ˙f
l1 sin θ1 θ1 +l2 sin θ2 θ2 + l4 sin θ4 θ4 +l5 sin θ5 θ5 = 0

(7.36)

The above equation was obtained by taking the derivative of the swing foot position coordinates relative to the stance foot - see Figure 7.7.

7.4.3

Determination of Actuated Joint Trajectories

Using Eq.(7.13), we consider the following functional dependence for θj

o
θj (θ1 ) = (aj θ1 + bj ) + fj (θ1 ),

j = 2, 3, 4, 5

(7.37)

˙i
˙f
¨
The condition θ1 = θ1 in Eq.(7.35) can be satisfied if θ1 is an odd function of θ1 . To this
¨
end, we derive the expression for θ1 as follows
N
¨
θ1 =
D

141

(7.38)

where N = N(q, q, q ) and D = D(q) are defined as follows (see Appendix C):
˙ ¨
5

N(q, q, q ) = −
˙ ¨

5

j=2 i=1
5

5

¨
M(j, i)θj −

5

j=1 i=1

5

˙
N(i, j)θj −

G(j)
j=1

M(j, 1)

D(q) =
j=1

¨
Using Eq.(7.37), a term-by-term analysis of N and D shows that θ1 is an odd function of θ1
for any values of aj ’s provided that bj ’s are chosen as follows:

bj = kj π,

kj = 0, ±1, ±2, · · ·

(7.39)

o
For a realistic walking configuration, we set b2 = b3 = 0 and b4 = b5 = π. Assuming fj in

Eq.(7.37) to be sinusoidal functions, we have

θ2 (θ1 ) = a2 θ1 + A sin (Bθ1 )
θ3 (θ1 ) = a3 θ1 + C sin (Dθ1 )
θ4 (θ1 ) = a4 θ1 + π + E sin (F θ1 )
θ5 (θ1 ) = a5 θ1 + π + G sin (Hθ1 )

(7.40)

f

i
We assume θ1 = −θ1 = β/2 for some angle β and choose B = D = F = 2π/β; this choice

ensures that actuated joint trajectories of θ2 , θ3 and θ4 include one period of sinusoidal
oscillation in each step. For a reasonable clearance between the swing foot and the ground,
we choose H differently, equal to 2.5π/β.
To have identical configuration of the legs at the beginning and end of each step, links
1 and 5 and links 2 and 4 should have the same length and be symmetrical with respect to
142

the vertical at the beginning of each step. From the definition of the joint angles in Figure
7.7, this implies that

i
i
θ5 = −θ1 + π

(7.41)

i
i
θ4 = −θ2 + π

(7.42)

Furthermore, the biped should have the same speed before and after interchange of the stance
and swing legs. This implies
˙f
˙i
θ5 = θ1

(7.43)

We use Eq.(7.42) to obtain a4 = −a2 and use Eqs.(7.41) and (7.43) to solve for a5 and G in
terms of β. The actuated joint trajectories in Eq.(7.40) can now be described with the help
of five parameters, namely, a2 , a3 , A, C and E, as shown below:
θ2 (θ1 ) = a2 θ1 + A sin(2πθ1 /β)
θ3 (θ1 ) = a3 θ1 + C sin(2πθ1 /β)
θ4 (θ1 ) = −a2 θ1 + π + E sin(2πθ1 /β)
θ5 (θ1 ) = a5 θ1 + π + G sin(2.5πθ1 /β)

(7.44)

Of the five parameters, a2 , A and E are chosen such that the two boundary conditions in
Eq.(7.36) are satisfied. This provides some flexibility in choice of these parameters. The
remaining two parameters, a3 and C, are related to the motion of the torso and therefore
do not show up in the no-impact boundary conditions in Eq.(7.36); they can be chosen
arbitrarily and provide additional flexibility in design of energy-conserving gaits.

143

Table 7.2: Kinematic and dynamic parameters of Five-DOF Point-Foot biped with similar
leg links
Kinematic and dynamic parameters
j
lj (m) dj (m) mj (kg) Ij (kg.m2 )
1, 5 0.600 0.300
2.000
0.060
2, 4 0.400 0.200
2.000
0.027
3
0.600 0.300
4.000
0.120

7.4.4

Numerical Simulations

The kinematic and dynamic parameters of the biped with similar leg links (l1 = l5 , l2 =
l4 ) are listed in Table 7.2. Although there is significant flexibility in choosing the gait
parameters, many choices will result in gaits that are not feasible. For example, certain
choices of parameters will result in the swing leg passing through the ground. A set of
parameters that result in a feasible gait can be obtained by trial and error; one such set of
f

θ1 = −22.5

τ2 , τ3 , τ4 , τ5 (Nm)

40.0
20.0

τ3
τ2

τ4

0.0

τ5

-20.0

τ2

τ5
τ4
i
θ1 = 22.5

τ3

-40.0
-20

-10

0

10

20

θ1 (deg)
Figure 7.8: Torque plots for an energy-conserving gait for the Five-DOF Point-Foot Biped
with similar leg links.

144

parameters for a step angle of β = 45.0 deg is given below:

a5 = −1.683 G = −0.739
(7.45)

a2 = 1.200,

a3 = 0.000

A = 0.308,

C = −0.550, E = 0.008

˙i
For an initial velocity of θ1 = −100 deg/s, the torques τj , j = 2, 3, 4, 5, are shown in Figure
7.8. The total mechanical work done by the actuators over one step can be expressed as an
integral over θ1 , as follows
5

W =
j=2 t

˙
τj θj dt =

f
θ1
i
θ1




5
j=2


dθj
τj (
) dθ1
dθ1

f
θ1
i
θ1

τ θ dθ1

(7.46)

The plot of τ θ is shown in Figure 7.9. The area under this curve is equal to zero and this
confirms that the gait described by Eqs.(7.44) and (7.45) conserves mechanical energy. The
configurations of the biped at different instants of time over a single step, moving from left to
f

θ1 = −22.5

τ θ (Nm)

100.0

0.0

-100.0

i
θ1 = 22.5

-20

-10

0

10

20

θ1 (deg)
Figure 7.9: Generalized torque plot for an energy-conserving gait for the Five-DOF PointFoot Biped with similar leg links.

145

Figure 7.10: Configurations of the five-dof point-foot biped with similar leg links at different
instants of time over one step.
right, are shown in Figure 7.10. It was verified that the swing leg does not scuff the ground.
Since actuated joint trajectories of the biped, θj (t), j = 2, 3, 4, 5, are odd functions of the
passive coordinate θ1 , a right-to-left step of the biped will have the same set of configurations
as in Figure 7.10.

7.5

Two-Step Periodic Gaits: Five-DOF Bipeds

7.5.1

Generalization to Two-Step Periodic Gait

The energy-conserving gait designed in the last section is one-step periodic. This is because
of Eqs.(7.41), (7.42) and (7.43), which impose additional conditions that make the configurations4 of the five-dof point-foot biped identical at the beginning and end of each step. It
is not necessary to have identical configurations at the beginning and end of each step to
have an energy-conserving gait, and therefore, energy-conserving gaits need not be one-step
periodic. Energy-conserving gaits designed using the sufficient conditions in section 7.2.2
will however be two-step periodic, i.e., result in identical configurations of the biped after
two steps. This is because two times application of the boundary conditions in Eqs.(7.2) and
(7.3) will revert each link back to its original configuration.
4A

biped configuration is described by the position and velocity of its links.

146

7.5.2

Characterization of Trajectories

For a two-step periodic gait, the configurations of the biped will be different at the beginning
and the end of the first step. Consequently, we will have a different initial configuration for
the second step after the interchange of the stance and swing legs. This will result in a new
set of parameters for the actuated joint trajectories for the second step. In this section, we
modify some conditions in Section 7.4.3 to obtain the trajectories of the actuated joints for
each step. In spite of different walking gaits for the first and second steps, we still ensure
that the energy is conserved for each step by satisfying the sufficient conditions in Eqs.(7.34),
(7.35) and (7.36).

7.5.2.1

Trajectories for the First Step

Consider the trajectories in Eq.(7.40) with the parameters B, D, F and H chosen the same
as those in Section 7.4.3, i.e.,

θ2 (θ1 ) = a2 θ1 + A sin(2πθ1 /β)
θ3 (θ1 ) = a3 θ1 + C sin(2πθ1 /β)
θ4 (θ1 ) = a4 θ1 + π + E sin(2πθ1 /β)
θ5 (θ1 ) = a5 θ1 + π + G sin(2.5πθ1 /β)

(7.47)

Since the condition in Eq.(7.42) is not necessary for a two-step periodic gait, the parameters
a2 and a4 are now chosen independently. The condition in Eq.(7.41) is replaced with the
following condition which ensures that the swing foot is on the ground at the beginning of
the step:
i
i
i
i
l1 cos θ1 + l2 cos θ2 + l4 cos θ4 + l5 cos θ5 = 0

147

(7.48)

In addition, for a two-step periodic gait, we can have different initial velocities of the stance
leg in the first and second steps. Therefore, the condition in Eq.(7.43) is modified to:

˙f
˙i
θ5 = cv θ1 ,

cv > 0

(7.49)

f

i
Knowing that θ1 = −θ1 = β/2, Eqs.(7.48) and (7.49) can be solved for a5 and G in terms of

β. The parameters A and E are chosen such that the two boundary conditions in Eq.(7.36)
are satisfied. The remaining two parameters, a3 and C, which are related to the motion of the
torso, can be chosen arbitrarily and provide some flexibility in design of energy-conserving
gaits.

7.5.2.2

Trajectories for the Second Step

The actuated joint trajectories for the second step should be chosen such that the sufficient
boundary conditions for energy conservation are satisfied, and the configuration of the biped
at the beginning of the second step (after interchange of the legs) matches the configuration
at the end of the first step (before interchange of the legs). This is a direct consequence of
having no impact at the instant of swing-foot touchdown at the end of the first step. To this
end, the actuated joint trajectories in Eq.(7.47) are modified as follows:

ˆ
ˆ
θ2 (θ1 ) = a2 θ1 + A sin(2πθ1 /β)
ˆ
ˆ
ˆ
θ3 (θ1 ) = a3 θ1 + C sin(2πθ1 /β)
ˆ
ˆ
ˆ
θ4 (θ1 ) = a4 θ1 + π + E sin(2πθ1 /β)
ˆ
ˆ
ˆ
θ5 (θ1 ) = a5 θ1 + π + G sin(2.5πθ1 /β)
ˆ

148

(7.50)

ˆ
ˆ
ˆ ˆ ˆ ˆ
where aj ’s, A, C, E, G and β are new parameters to be determined. The parameter β
ˆ
represents the total angle traveled by the new stance leg and is obtained as:

i(2)
f (1)
ˆ
β = 2θ1 = 2θ5
= −2[a5 (β/2) + G sin(2.5π/2)]

(7.51)

where the superscripts (1) and (2) denote the corresponding variables for the first and second
step, respectively. To ensure that the position and velocity of the torso is the same before
and after the interchange of the legs, we have:

i(2)

f (1)

⇒

ˆ
a3 = −(β/β) a3
ˆ

˙f (1)
˙i(2)
θ3 = θ3

⇒

ˆ
β
1
2π
ˆ
C=
[ˆ3 − (a3 −
a
C)]
2π
cv
β

θ3

= θ3

(7.52)

˙i(1)
˙i(2)
ˆ
where we used the relation θ1 = cv θ1 from Eq.(7.49) in deriving C. To ensure that the
positions of the upper links of the stance and swing legs are the same before and after the
interchange, we have:

i(2)

θ2

i(2)

θ4

f (1)

⇒

ˆ
a2 = −(β/β) a4
ˆ

f (1)

⇒

ˆ
a4 = −(β/β) a2
ˆ

= θ4

= θ2

(7.53)

To ensure continuity in the position and velocity of the lower link of the swing leg, we impose
the conditions:

i(2)

θ5

f (1)

= θ1

˙i(2)
˙f (1)
θ5 = θ1

149

(7.54)

ˆ
The above equations can be solved for the parameters a5 and G. The remaining two paramˆ
ˆ
ˆ
eters A and E in Eq.(7.50) can be obtained by solving the no-impact boundary conditions
ˆ
ˆ
in Eq.(7.36) for the second step. Note that these choices of A and E will guarantee identical
velocities for the links 2 and 4 before and after the interchange.
Remark 7.5.1. For the two-step periodic gait, the actuated joint trajectories for the first step
in Eq.(7.47) and the second step in Eq.(7.50) will be sequentially repeated.

7.5.3

Five-DOF Point-Foot Biped: Similar Leg Links

Consider the five-dof point-foot biped with similar leg links, presented in section 7.4 with the
same kinematic and dynamic parameters listed in Table 7.2. To design an energy-conserving
gait with two-step periodicity, we remove the conditions in Eq.(7.41) and Eq.(7.42) to make
the initial configuration of the legs asymmetric with respect to the vertical. The procedure
in Section 7.5.2 is followed to determine gait parameters for the first and second steps. A
feasible gait is obtained by using the following parameters for the first step, described by
Eq.(7.47):

β = π/4

a3 = −0.300 C = −0.500

a2 = 1.200,

a4 = −1.500 a5 = −1.267

G = −0.321, A = 0.261,

150

E = −0.059

(7.55)

θ1 (deg)
first step

τ2 , τ3 , τ4 , τ5 (Nm)

80.0

f
θ1

-10

0

f
θ1

= −22.5

10

= −15.5

40.0
τ4

0.0

τ5

τ5
τ2

-40.0
-80.0

τ3

τ3
i
θ1 = 22.5

-20 -10

0

10

τ2

τ4

i
θ1 = 15.5

20

second step

θ1 (deg)
Figure 7.11: Torque plots for an energy-conserving gait with two-step periodicity for the
Five-DOF Point-Foot Biped with similar leg links.
and the following parameters for the second step, described by Eq.(7.50):

ˆ
β = 0.172π

a3 = 0.435
ˆ

a2 = 2.176,
ˆ

a4 = −1.741 a5 = −2.288
ˆ
ˆ

ˆ
C = −0.293

ˆ
ˆ
G = −0.321, A = 0.276,

(7.56)

ˆ
E = −0.073
θ1 (deg)

τ θ (Nm)

150.0

first step

-10

0

10

0.0

-150.0

i
θ1 = 22.5

-20 -10

0

10

i
θ1 = 15.5

20

second step

θ1 (deg)
Figure 7.12: Generalized torque plot for an energy-conserving gait with two-step periodicity
for the Five-DOF Point-Foot Biped with similar leg links.
151

first step

second step

Figure 7.13: Configurations of the five-dof point-foot biped with similar leg links at different
instants of time for a two-step periodic gait. The joint configurations at the beginning of
the first step and end of the second step are the same but they are different from the joint
configuration at the end of the first step.

ˆ
The value of cv in Eq.(7.49) was set to unity. Note that β < β, which implies that the second
˙ i(2)
˙i(1)
step is shorter than the first step. For initial velocities of θ1 = θ1 = −100 deg/s, the
torques τj , j = 2, 3, 4, 5, are shown for the first and second steps in Figure 7.11. The total
mechanical work done by the actuators over each step can be expressed by Eq.(7.46). The
plot of τ θ for both steps is shown in Figure 7.12. The area under this curve for each step is
equal to zero and this confirms that the gait conserves mechanical energy in each step. The
configurations of the biped at different instants of time over two steps, moving from left to
right, is shown in Figure 7.13. It was verified that the swing leg does not scuff the ground.

7.5.4

Five-DOF Point-Foot Biped: Dissimilar Leg Links

Consider the biped in Figure 7.7 and assume the upper link of each leg to have the same
length as the lower link of the other leg, i.e. l1 = l4 and l2 = l5 . The kinematic and dynamic
Table 7.3: Kinematic and dynamic parameters of Five-DOF Point-Foot biped with dissimilar
leg links
j (1st step)
1, 4
2, 5
3

Kinematic and dynamic parameters
j (2nd step) lj (m) dj (m) mj (kg) Ij (kg.m2 )
2, 5
0.600 0.300
2.000
0.060
1, 4
0.400 0.200
2.000
0.027
3
0.600 0.300
4.000
0.120
152

θ1 (deg)

first step
τ2 , τ3 , τ4 , τ5 (Nm)

80.0

0

20

f

f

θ1 = −31.77

τ5

τ5 τ4

θ1 = −22.5

40.0
0.0
-40.0

-20

τ4

τ2

-80.0
-20

τ2
τ3

τ3
i
θ1 = 31.77

i
θ1 = 22.5

0

20

θ1 (deg)

second step

Figure 7.14: Torque plots for an energy-conserving gait with two-step periodicity for the
Five-DOF Point-Foot Biped with dissimilar leg links.
parameters of the biped are listed in Table 7.3. Since the leg links are not similar, the
kinematic and dynamic parameters of the links are interchanged for the first and second steps
- see Table 7.3. To generate a two-step periodic gait, we choose an asymmetric configuration
of the legs at the beginning of the first step. Following the procedure in Section 7.5.2, a
feasible gait is obtained by using the following parameters for the first step, described by
θ1 (deg)

first step

200.0

-20

τ θ (Nm)

20

f

θ1 = −31.77

f

100.0

0

θ1 = −22.5

0.0
-100.0
-200.0

i
θ1 = 22.5

-20

0

20

θ1 (deg)

i
θ1 = 31.77

second step

Figure 7.15: Generalized torque plot for an energy-conserving gait with two-step periodicity
for the Five-DOF Point-Foot Biped with dissimilar leg links.

153

first step

second step

Figure 7.16: Configurations of the five-dof point-foot biped with dissimilar leg links at different instants of time for a two-step periodic gait. The joint configurations at the beginning
of the first step and end of the second step are the same but they are different from the joint
configuration at the end of the first step.

Eq.(7.47):
β = π/4

a3 = −0.300 C = −0.550

a2 = 1.200,

a4 = −0.800 a5 = −2.236

G = −0.458, A = 0.364,

(7.57)

E = −0.013

and the following parameters for the second step, described by Eq.(7.50):

ˆ
β = 0.353π

a3 = 0.213
ˆ

a2 = 0.567,
ˆ

a4 = −0.850 a5 = −1.292
ˆ
ˆ

ˆ
ˆ
G = −0.458, A = 0.223,

ˆ
C = −0.671
(7.58)

ˆ
E = 0.152

ˆ
The value of cv in Eq.(7.49) was set to unity. Note that β > β, which implies that the second
˙i(1)
˙i(2)
step is longer than the first step. For an initial velocity of θ1 = θ1 = −130 deg/s, the
torques τj , j = 2, 3, 4, 5, are shown for the first and second steps in Figure 7.14. The total
mechanical work done by the actuators over one step can be expressed by Eq.(7.46). The
plot of τ θ for both steps is shown in Figure 7.15. The area under this curve for each step is
equal to zero and this confirms that the gait conserves mechanical energy in each step. The
configurations of the biped at different instants of time over two steps, moving from left to
right, are shown in Figure 7.16. It was verified that the swing leg does not scuff the ground.

154

7.6

Conclusion

We proposed a general method for designing gaits for point-foot passive-ankle planar bipeds
that require zero mechanical energy over each individual step. The proposed method provides
trajectories for all actuated dof of the biped which can be tracked using classical control
techniques. These trajectories ensure that the biped has identical potential energy and
kinetic energy at the beginning and end of each step and there is no loss of energy due to
impact at the time of swing-foot touchdown. To show its effectiveness, the proposed method
has been applied to several three-dof and five-dof bipeds; the designed gaits are one-step
periodic in some cases and two-step periodic in others. The method can be easily applied to
bipeds with greater number of dof’s.
The proposed method provides significant flexibility in choosing joint trajectories for the
actuated dof of the biped. This flexibility can be used to design a wide range of gaits that
have different step lengths, walking speeds, and range of joint torques, for example, and can
therefore be used for optimizing the energy-conserving gaits. For implementation, controllers
will have to be designed to track the desired trajectories accurately. Any deviation from the
desired trajectories will result in a gait that is not energy-conserving but continuous dependence of the mechanical energy on solution trajectories dictates that small deviations in the
trajectories will only require a small amount of mechanical energy. A small deviation in the
desired trajectories can be due to inaccurate tracking or may be introduced purposely to
provide uniformly larger ground clearance while walking on rough terrain, for example. An
energy-conserving gait requires zero mechanical energy but will have zero energy consumption if the negative work done by the actuators can be recaptured and stored. This will
require particular robot capabilities, such as backdriveable actuators and electrical circuits

155

capable of energy regeneration and storage.
The boundary conditions presented here provide a simple framework for designing energyconserving gaits but since these conditions are sufficient there may exist other gaits that
require zero mechanical energy. Of the three conditions presented, the no-impact boundary
condition is necessary to avoid energy dissipation but the other two conditions relating to
kinetic and potential energies of the links can be relaxed. In particular, it may be possible
to find energy-conserving gaits by requiring the total kinetic energy and potential energy of
the biped to be the same at the beginning and end of each step or even multiple steps.

156

Chapter 8
Intermittent Output Tracking For
Linear Non-Minimum-Phase Systems

8.1

Introduction

Underactuated systems with unstable zero dynamics are Non-Minimum-Phase (NMP) systems. The limitations in output tracking for NMP continuous-time systems motivated us
to look for a controller that ensures output matching with the reference trajectory at regular intervals of time. The control objective is similar to the output tracking controllers
for discrete-time systems. Although perfect output tracking can be achieved for discretetime systems, we propose a continuous-time controller to intermittently match the system
output with the reference trajectory for two reasons: First, in the continuous-time control
design, we do not need to incorporate the dynamics of holders and samplers associated with
discrete-time sampled systems. Second, the discretized systems require strong conditions to
be satisfied to preserve the stability of the sampling zeros [139, 140, 141]. For SISO linear
systems with relative degree greater than two, Astrom et al. [142] showed that at least one
of the the limiting zeros (the sampling zeros for the limiting case of zero sampling period)
is outside the unit circle, irrespective of their location for the original continuous system.
This implies that even for a MP continuous-time system, we might face control limitations
induced by unstable zeros for the discretized system.
157

In this chapter, we propose an output tracking controller for linear Single-Input SingleOutput (SISO) NMP systems. A continuous-time controller is designed to take the output
tracking error to zero at regular intervals of time. The controller for the NMP plant is based
on switched inputs, which are derived from the continuous control inputs of a “Discrete
Equivalent” (DE) [143] MP system. The control design ensures zero output tracking error
at regular intervals of time and guarantees the stability of the internal dynamics. A finite
preview of the desired trajectory is required for this method.

8.2

Discrete Equivalent Systems

Consider a linear SISO system represented in the following state-space form:

x(t) = Ax(t) + Bu(t) , x(t0 ) = x0
˙
y(t) = Cx(t)

(8.1)

Let the system input u, shown in Figure 8.1, be defined as:

u(t) =



M

0

if (j −1)δ < t − t0 < (j −1+p)δ
if

j = 1, 2, ...

(8.2)

(j −1+p)δ < t − t0 < jδ

The switching parameter δ > 0 represents the period of the switched input and p ∈ (0, 1)
defines the fraction of the time interval when the input has a nonzero constant value M.
Definition 8.2.1. [143] Consider the SISO linear time-invariant (LTI) system with constant
input u = M:
¯

158

PSfrag

u, u
¯

δ

δ

M
pδ

z

t0

pδ

t1

t2

t

Figure 8.1: The periodic switched input for the system in Eq.(8.1) (in solid line) and the
constant input for DE system in Eq.(8.3) (in dashed line)

¯x
¯¯ ¯
˙
x(t) = A¯(t) + B u , x(t0 ) = x0
¯
¯¯
y (t) = C x(t)
¯

(8.3)

This system is a Discrete Equivalent (DE) of the system in Eq.(8.1) with switched input in
Eq.(8.2) if the state variables and the output of two systems match exactly at t = tj , where
tj = t0 + jδ and j = 1, 2, · · · , after starting from the same initial conditions at t = t0 .
To obtain the relations between switched-input system and its DE, we first find the state
variables of the two systems at times t = t0 + jδ, j = 1, 2, · · · . For the switched-input system
in Eq.(8.1), we can write:

x(t1 ) = eA(t1 −t0 ) x0 +

t1
t0

eA(t1 −τ ) Bu(τ ) dτ = eApδ x0 + A−1 (eApδ − I)BM

x(t2 ) = eA(t2 −t1 ) x(t1 ) = eAδ x0 + A−1 (eAδ − eA(1−p)δ )BM

159

(8.4)

where I represents the identity matrix. For the DE system in Eq.(8.3), we have:

¯
x(t2 ) = eA(t2 −t0 ) x0 +
¯

t2
t0

¯
¯
¯
¯
¯
¯
eA(t2 −τ ) BM dτ = eAδ x0 + A−1 (eAδ − I)BM

(8.5)

From Definition 8.2.1:
x(t2 ) = x(t2 ), y(t2 ) = y (t2 )
¯
¯

(8.6)

Using Eqs.(8.4) and (8.5), we get:

¯
A =A
¯
B =(eAδ − I)−1 (eAδ − eA(1−p)δ )B
¯
C =C

(8.7)

We demonstrate the concept of DE systems with the example below.
Example 8.2.1. Consider a linear system with the following transfer function:

G(s) =

(s + 1)(s − 2)
(s + 1)(s + 2)(s + 3)

(8.8)

It has the following state-space representation:






0 
 −1 0
 1






A =  1 −5 −2.45  , B =  0






0 2.45
0
0

160






 , C = [1 − 6 − 3.27]



(8.9)

If we assume:

M = 2 , δ = 0.8 , p = 0.2

(8.10)

the DE system will have the following state-space representation:






0 
 −1 0
 0.14




¯ 
¯ 
A =  1 −5 −2.45  , B =  0.028






0 2.45
0
0.015






¯
 , C = [1 − 6 − 3.27]



(8.11)

The transfer function of the DE system can be shown to be:

(s + 1)(s + 7.137)
¯
¯
¯
¯
G(s) = C(sI − A)−1 B =
(s + 1)(s + 2)(s + 3)
a

x1 , x1
¯

0.5

x2 , x2
¯

0
0.04

x3 , x3
¯

(8.12)

b

0

0.1

c
0
2

u, u
¯

d
0
0

0.8

1.6

2.4

3.2

4

time (sec)
Figure 8.2: Simulation results for example 8.2.1 showing state variables and inputs for
switched-input system (solid line) and discrete equivalent system (dashed line).

161

A simulation of the response of the switched-input system and its DE is shown in Figure 8.2.
It can be seen that all state variables of the two systems match exactly at regular intervals
of time, δ = 0.8 for u = 2.

8.3

Zero Shift for Linear NMP Systems

The poles of the DE system are identical to the poles of the switched-input system since
¯
A = A in Eq.(8.7). The location of the zeros of the systems can however be different as
shown in Example 8.2.1. In this example, the original system has a RHP zero at s = 2. This
zero is moved to the LHP at s = −7.137 for the DE system. This indicates that a NMP
system can have a DE system which is MP.

8.3.1

Stable NMP Systems with Distinct Poles - General Case

Consider the stable linear system with distinct poles λi , i = 1, · · · , n:
b sn−1 + b2 sn−2 + · · · + bn−1 s + bn
Y (s)
= 1
U(s)
(s − λ1 )(s − λ2 ) · · · (s − λn )
=

c1
c2
cn
+
+···+
s − λ1 s − λ2
s − λn

(8.13)

where bi ’s are the constant coefficients of the zero polynomial and ci ’s are constant residues
of partial fraction expansion of the system transfer function. Assume that the system has at
least one zero in RHP, and therefore it is NMP. The transfer function in Eq.(8.13) can be

162

represented in Diagonal Canonical form [144] as:

x =Ad x + Bd u
˙
y =Cd x

(8.14)

where:
T

Ad =diag λ1 λ2 · · · λn

, Bd = 1 1 · · · 1

C d = c1 c2 · · · cn

(8.15)

For the corresponding DE system, we have from Eq.(8.7):

¯
¯
Ad =Ad , Cd = Cd
¯
Bd =(eAd δ − I)−1 (eAd δ − eAd (1−p)δ )Bd = h1 h2 · · · hn

T

(8.16)

where hi ’s are some positive constants obtained using the properties of diagonal matrix Ad :

hi =

eλi δ − eλi (1−p)δ
eλi δ − 1

,

i = 1, . . . , n

(8.17)

The transfer function of the DE system can be evaluated as:

c h
cn hn
c h
¯
¯
¯
¯
G(s) =Cd (sI − Ad )−1 Bd = 1 1 + 2 2 + · · · +
s − λ1 s − λ2
s − λn

163

(8.18)

The zero polynomial of the DE system (with distinct poles) is obtained as:

¯
Z(s) = c1 h1 (s − λ2 )(s − λ3 ) · · · (s − λn )
+ c2 h2 (s − λ1 )(s − λ3 ) · · · (s − λn )
+···
+ cn hn (s − λ1 )(s − λ2 ) · · · (s − λn−1 )

(8.19)

which can be expanded into the following form:

¯
Z(s) = ¯1 sn−1 + ¯2 sn−2 + ¯3 sn−3 + · · · + ¯n−1 s + ¯n
b
b
b
b
b

(8.20)

where
n

¯1 =
b

ci hi
i=1

¯m =(−1)m−1
b

n

n

λj1 λj2 · · · λjm−1 )ci hi ,

(
i=1

m = 2, · · · , n

(8.21)

j1 , j2 , · · · , jm−1 = i
j1 < j2 < · · · < jm−1
Remark 8.3.1. For the NMP system in Eq.(8.13) and switching parameters δ and p, the DE
system will be MP if the zero polynomial in Eq.(8.20) does not have any root in the closed
RHP. This can be verified by applying Routh’s stability criterion [144] without actually
solving for the zero locations.
Remark 8.3.2. The order of the zero polynomial of the original NMP system and its DE may
be different. This implies that the number of zeros for the DE system may be different from

164

that of the original system. The number of zeros for both systems is however less than n.

8.3.2

Stable NMP Systems with Distinct Poles - Special Case

In this section, we investigate a special case of the system in Eq.(8.13) with n = 2 and stable
real modes, i.e.,

b1 s + b2
c1
c2
Y (s)
=
=
+
U(s)
(s − λ1 )(s − λ2 )
(s − λ1 ) (s − λ2 )

(8.22)

Using Eqs.(8.20) and (8.21), the zero polynomial for the DE system is obtained as follows:

¯ b
Z = ¯1 s + ¯2
b

(8.23)

where

¯1 = c1 h1 + c2 h2 ,
b

¯2 = −(c1 h1 λ2 + c2 h2 λ1 )
b

(8.24)

and h1 and h2 can be obtained from Eq.(8.17)
Lemma 8.3.1. If λ2 < λ1 < 0, then h2 < h1 .
λδ λ(1−p)δ
is strictly increasing, i.e., dh > 0 in
Proof. We prove by showing that h(λ) = e −e
dλ
eλδ −1

λ ∈ (−∞, 0). To this end, we define a new variable q:
q = eλδ − 1
From λδ < 0, we know that −1 < q < 0. The function h can be rewritten in terms of the
165

new variable q as:
h(q) =

q + 1 − (q + 1)(1−n)
q

Therefore,
∂h ∂q
1 + pq − (q + 1)p
dh
) (δeλδ )
=( )( )=(
dλ
∂q ∂λ
q 2 (q + 1)p
Since δeλδ > 0 and q 2 (q + 1)p > 0, the sign of (dh/dλ) is identical to the sign of (1 + pq −
(q + 1)p ). Since |q| < 1, (q + 1)p can be expanded using the binomial series [145]. Therefore,

1+pq−(1+q)p =1 + pq −

∞





p(p−1) 2 p(p−1)(p−2) 3
 p  j
q +
q + ···)
  q = −(
2
3!
j=0
j

which is always positive for 0 < p < 1 and q < 0. This completes the proof.
The following theorem states the sufficient conditions for the DE system of Eq.(8.22) to
be MP.
Theorem 8.3.1. Consider the second-order linear system in Eq.(8.22), which has two negative real and distinct poles λ1 and λ2 and a RHP zero. Without loss of generality, let
λ1 > λ2 . The DE system will be MP if any of the following conditions is satisfied:
(a) c1 c2 > 0
(b) c1 c2 < 0 and |c1 | ≥ |c2 |
(c) c1 c2 < 0, |c1 | ≤ |c2 | and c1¯1 > 0
b
Proof. Since h1 , h2 > 0 and λ1 , λ2 < 0, condition (a) implies that ¯1 and ¯2 in Eq.(8.24) will
b
b
have the same sign, which implies that the DE system is MP.

166

For condition (b), first consider the case that c1 > 0 and c1 ≥ −c2 . Since λ1 > λ2 , Lemma
8.3.1 implies:

h1 > h2 ⇒ c1 h1 > −c2 h2 ⇒ ¯1 > 0 ,
b
c1 h1 > −c2 h2 ⇒ c1 h1 λ2 < −c2 h2 λ2 < −c2 h2 λ1 ⇒ ¯2 > 0
b
If c1 < 0 and c1 ≤ −c2 , it can be shown that ¯1 , ¯2 < 0. This proves that (b) is a sufficient
b b
condition for the DE system to be MP.
For condition (c), first consider the case that c1 < 0, c2 > −c1 and ¯1 < 0. Since λ1 > λ2 ,
b
Lemma 8.3.1 gives:

h1 > h2 ⇒ λ1 h2 > λ1 h1 > λ2 h1 ⇒ λ1 h2 c2 > λ2 h1 c2 > −λ1 h1 c1 ⇒ ¯2 < 0
b
For the case that c1 > 0, c1 < −c2 and ¯1 > 0, we can show that ¯2 > 0. Therefore,
b
b
condition (c) will ensure that the DE system is MP.

8.3.3

Stable NMP Systems with Repeated Poles

For a linear system with repeated poles, the state-space matrices can be written in the
Jordan Canonical Form [144]. Deriving a general expression for the zero polynomial of the
DE system is not as simple as the case of distinct poles. This is because of the complexity
¯
in calculating B in Eq.(8.7) due to non-diagonal matrix A. The DE system can nevertheless
be designed to be MP even when the original system has RHP zeros - this is illustrated with
the help of numerical examples in Table 8.1. Using a numerical search for proper switching
parameters δ and p, it is shown that the DE system can be MP when the original NMP

167

Table 8.1: Linear NMP systems with repeated poles and their corresponding MP DE systems
Original Zeros
Poles
δ
p
DE zeros
5
[-2,-2,-5,-5] 0.95 0.15 [-33.61,-6.72±1.84i]
[-1,5]
[-2,-2,-2]
0.85 0.3
[-5.86,-1]
[3,4]
[-4,-4,-5,10]
1
0.1 [-13.49,-9.46±1.45i ]
[-1,3±i]
[-3,-3,-5±i] 0.5 0.75 [-0.22,-0.59,-29.39]

system has repeated poles. It should be noted that Table 8.1 includes an example of a
system with relative degree three. For such a system, sampling will necessarily result in a
NMP discrete-time system [142].
Remark 8.3.3. In this section, we only considered stable systems. If the system is unstable,
it can be first stabilized using a state-feedback controller provided that the system is stabilizable. The switching parameters can then be found for the stabilized system. This will be
discussed further in the next section.

8.4

Intermittent Output Tracking Control

It is not possible to design controllers for asymptotic output tracking of arbitrary reference
trajectories for NMP systems. However, it may be possible to find a Discrete Equivalent
Minimum-Phase (DEMP) system for a NMP system through proper choice of switching
parameters δ and p. From Definition 8.2.1, the state variables and the output of the NMP
system will match with those of DEMP system at regular intervals of time. Therefore, if the
output tracking control problem is solved for the DEMP system, the output of the NMP
system will also match with the reference trajectory at regular intervals of time δ. We define
Intermittent Output Tracking next:
Definition 8.4.1. For a linear SISO NMP system, Intermittent Output Tracking (IOT)
168

problem refers to the task of finding a control input u(t) such that for any desired reference
trajectory r(t) and any initial condition x0 :
a) The system is internally stable, i.e, all state variables are bounded for any bounded
r(t).
b) The output y(t) matches with r(t) at regular intervals of time δ, i.e., y(t0 + jδ) →
r(t0 + jδ) as j → ∞ for j = 1, 2, · · · .
To solve the IOT problem, we first review the tracking control design for MP systems.

8.4.1

Tracking Control for MP Systems

Consider a MP system which is represented by the following transfer function:
m
m−1 + · · · + β
m−1 s + βm
¯ N(s) = β1 s + β2 s
G=
n + a sn−1 + · · · + a
D(s)
s
1
n−1 s + an

(8.25)

where m < n and β1 = 0. To obtain a state-space representation of the system in the
normal, we follow the procedure outlined in [146]. First, the denominator polynomial D(s)
is written as:
D(s) = Q(s)N(s) + R(s)

(8.26)

1 ρ
s + q1 sρ−1 + · · · + qρ−1 s + qρ
β1

(8.27)

where
Q(s) =

is the quotient polynomial of order ρ = (n − m), where ρ is the relative degree of the system,
and R(s) is the remainder polynomial. In normal form, the state-space representation is

169

found to be:

¯
˙
η =A0 η + B0 Cc ζ
¯
¯
˙
¯
¯
¯
¯
¯
ζ =Ac ζ + Bc (γ T ζ − β1 C0 η + β1 u)
¯
y =Cc ζ
¯

(8.28)

¯
˙
where [¯, ζ]T is the vector of state variables with η ∈ Rn−ρ , and ζ = [¯, y , · · · , y (ρ−1)]T ∈ Rρ
η ¯
¯
y ¯
¯
and


0 1


0 0


.
Ac =  .
.


0 · · ·


0 ···

 



· · · 0
0
 

 

0

1 · · · 0
 
 

. , B = . , C =
..
.
.
.
c
c
.
1 0 ··· 0 0
.
 

 

0
0 1
 

 

0 0
1
0

γ T = − β1 qρ qρ−1 · · · q2 q1

(8.29)

and (A0 , B0 , C0 ) is a minimal realization of transfer function R(s)/N(s). Note that the
eigenvalues of A0 are identical to the zeros of the MP system in Eq.(8.25) and therefore A0
is Hurwitz.
We assume that the reference trajectory r(t) and its derivatives up to r (ρ) (t) are available
on-line, and r (ρ) (t) is piecewise continuous. The tracking error is defined as:

e = ζ − Rd = y − r
¯ ¯
¯

T

˙
y−r
¯ ˙

···

y (ρ−1) − r (ρ−1)
¯

(8.30)

Using the change of variables e = ζ − Rd , the state equations in Eq.(8.28) can be rewritten
¯ ¯
170

as:

˙
η =A0 η + B0 Cc (¯ + Rd )
¯
¯
e
˙
e =Ac e + Bc [γ T (¯ + Rd ) − β1 C0 η + β1 u − r (ρ) ]
¯
¯
e
¯
¯

(8.31)

The following state-feedback controller,

u=
¯

1
¯¯
[−γ T (¯ + Rd ) + β1 C0 η + r (ρ) − K e]
e
¯
β1

(8.32)

¯
¯
with control gain K chosen to make (Ac − Bc K) Hurwitz, yields the closed-loop system:
˙
η =A0 η + B0 Cc (¯ + Rd )
¯
¯
e
¯ e
˙
e =(Ac − Bc K)¯
¯

(8.33)

¯
Since (Ac − Bc K) is Hurwitz, e → 0 as t → ∞. The internal stability of the closed-loop
¯
system is also ensured since A0 is Hurwitz.

8.4.2

Intermittent Output Tracking for NMP Systems

Consider the normal form of a MP system in Eq.(8.28) which can be rewritten as:

¯x
¯¯
˙
x(t) = A¯(t) + B u(t) , x(t0 ) = x0
¯
¯
¯¯
y (t) = C x(t)
¯

(8.34)

171

where x = [¯, ζ]T and
¯
η ¯








0 
A0
B0 Cc 
¯
¯ 
¯ 
A =
 , C = 0 Cc
, B = 
β1 Bc
−β1 Bc C0 Ac + Bc γ T

(8.35)

¯ ¯ ¯
Now, assume that there are switching parameters δ and p such that (A, B, C) in Eq.(8.35)
represents a DE system for the following NMP system:

x(t) =Ax(t) + Bu(t) , x(t0 ) = x0
˙
y(t) =Cx(t)

(8.36)

where the state-space matrices are obtained using Eq.(8.7) as:

¯
¯
A =A , C = C
¯
B =(eAδ − eA(1−p)δ )−1 (eAδ − I)B

(8.37)

From Definition 8.2.1, state variables and outputs of the systems in Eqs.(8.34) and (8.36)
will be discretely identical provided that over each interval δ, u(t) is constant and u(t) is
¯
switched based on Eq.(8.2). The tracking controller in Eq.(8.32) will however not result in an
input that is constant over every time interval δ. To overcome this problem, we will replace
u in Eq.(8.32) by its averaged value over every time interval δ. The error due to averaging
¯
is given by the following lemma:

172

Lemma 8.4.1. Consider the system in Eq.(8.34) with the input u replaced by ua :
¯
¯

¯x
¯¯
˙
xa (t) = A¯a (t) + B ua , xa (t0 ) = x0
¯
¯
¯¯
ya (t) = C xa (t)
¯

(8.38)

where ua is the average of the Lipschitz control input u(t) over the time interval [t0 , t0 + δ].
¯
¯
Define exa (t) = x(t) − xa (t) and eya (t) = y (t) − ya (t), then:
¯
¯
¯
¯
¯
¯
exa (t0 +δ) ≤
¯
|¯ya (t0 +δ)| ≤
e

¯
¯
eA(t0 +δ−τ ) B L δ 2

sup
τ ∈[t0 ,t0 +δ]

sup
τ ∈[t0 ,t0 +δ]

¯ ¯
¯
|CeA(t0 +δ−τ ) B| L δ 2

(8.39)

where L is the Lipschitz constant for u(t) in time interval [t0 , t0 + δ].
¯
Proof. For the linear systems in Eqs.(8.34) and (8.38), we can write:

exa (t0 + δ) =
¯

t0 +δ
t0

¯

¯u
eA(t0 +δ−τ ) B[¯(τ ) − ua ] dτ
¯

(8.40)

Form real analysis [147], for a bounded linear operator T (ˆ) such as the above integral, we
x
have:
T (ˆ) ≤ T
x

x
ˆ

(8.41)

where x ∈ A and
ˆ ˆ
T = sup{ T (ˆ) : x ∈ A, x ≤ 1}
x
ˆ ˆ ˆ

173

(8.42)

Applying Eq.(8.41) to Eq.(8.40), we have:
t0 +δ

exa (t0 + δ) ≤
≤
≤

dτ

sup

t0

τ ∈[t0 ,t0 +δ]

sup
τ ∈[t0 ,t0 +δ]

¯

¯u
eA(t0 +δ−τ ) B[¯(τ ) − ua ]
¯

¯

¯
{ eA(t0 +δ−τ ) B

sup

u(τ ) − ua } δ
¯
¯

¯
¯
eA(t0 +δ−τ ) B L δ 2

(8.43)

τ ∈[t0 ,t0 +δ]

where we used u(τ ) − ua ≤ Lδ for the Lipschitz function u(t) in Eq.(8.32). The second
¯
¯
¯
¯ ¯
inequality in Eq.(8.39) is similarly proved from eya = C exa .
¯
Let eo (t) = y(t) − r(t) and eo (t) = y (t) − r(t) be output tracking errors for the NMP and
¯
¯
MP systems in Eqs.(8.36) and (8.34), respectively. The following theorem states a relation
between these errors:
Theorem 8.4.1. Consider the NMP system in Eq.(8.36) and MP system in Eq.(8.34) for
t ∈ [t0 , t0 + δ] with the same initial conditions x(t0 ) = x(t0 ) = x0 . Assume that both systems
¯
are stable. For any desired reference trajectory r(t):
(a) State variables x(t) are bounded for any bounded r(t)
(b) eo (t0 + δ) = eo (t0 + δ) + eya (t0 + δ)
¯
¯
if the system input is chosen as

u(t) =



 ua if
¯

0

t0 ≤ t ≤ t0 + pδ

(8.44)

if t0 + pδ < t < t0 + δ

where
1 t0 +δ
u(t) dt
¯
ua =
¯
δ t0
174

(8.45)

and u(t) defined in Eq.(8.32) is the tracking control input designed for the MP system over
¯
the time interval [t0 , t0 + δ].
Proof. Eq.(8.38) is a DEMP system of the NMP system in Eq.(8.36) with the control input
in Eq.(8.44). Therefore, their state variables will match at t = t0 + δ. Furthermore, the
states of the NMP system will be bounded during the entire interval since linear systems
do not have finite escape time. From Lemma 8.4.1, the difference between state variables of
DEMP system in Eq.(8.38) and MP system in Eq.(8.34) at t = t0 + δ is a bounded value.
Since the tracking control u(t) ensures the internal stability for the MP system in Eq.(8.34),
¯
we conclude that all state variables of the NMP system will be bounded for any bounded
r(t). This completes the proof of part (a). The proof of part (b) is very similar and is not
provided here.
Corollary 8.4.1. For any time domain [ti , tf ], the results in Theorem 8.4.1 is valid for
each time interval [ti + (j − 1)δ, ti + jδ] where j = 1, 2, · · · , if the initial time t0 and initial
conditions x0 in Theorem 8.4.1 are redefined as below for each interval:

t0 = ti + (j − 1)δ
x0 = x(ti + (j − 1)δ)

(8.46)

This implies that the tracking control input u(t) for the MP system is designed with new
¯
initial conditions x(t0 ) = x0 for each time interval.
¯
Remark 8.4.1. The conditions for the IOT problem in Definition 8.4.1 can be satisfied for the
NMP system in Eq.(8.36) using the results in Theorem 8.4.1 and Corollary 8.4.1 provided
that:
175

(a) eya (t0 + δ) ≈ 0, which is achieved by choosing a sufficiently small switching interval δ.
¯
¯
(b) eo (t0 + δ) → 0 as t → δ, which is achieved by choosing a sufficiently large gain K in
¯
Eq.(8.32).

8.4.3

Main Result

We can summarize the proposed controller for intermittent output tracking in the following
theorem:
Theorem 8.4.2. Consider the linear SISO NMP system in Eq.(8.36). Assume that it is
stable, without loss of generality. Let δ and p be the switching parameters, as shown in
Figure 8.1, such that Eq.(8.34) is a DEMP system for Eq.(8.36). For sufficiently small δ
and sufficiently large control gain, intermittent output tracking in the interval [ti , ∞], as
defined in Definition 8.4.1, is achieved if the control input is chosen as:

u(t) =



 uaj if (j − 1)δ ≤ t − ti ≤ (j − 1 + p)δ
¯

(8.47)

1 ti +jδ
u (t) dt
¯
δ ti +(j−1)δ j

(8.48)


0

if

for j = 1, 2, · · · where
uaj =
¯

(j − 1 + p)δ < t − ti < jδ

and uj (t) in Eq.(8.32) is the tracking control input designed for the MP system for each time
¯
interval [ti + (j − 1)δ, ti + jδ] with initial conditions given in Eq.(8.46).
Proof. The proof is followed from earlier discussions in sections 8.4.1 and 8.4.2.
Remark 8.4.2. If the NMP system is not stable, a state-feedback controller can be used first
to stabilize the system provided (A, B) are stabilizable. Then, the discussion to design the
176

IOT controller holds for the stabilized system. In this case, the control input in Eq.(8.44) is
changed to:
u(t) = −Kx(t) +



 ua if
¯

0

t0 ≤ t ≤ t0 + pδ

(8.49)

if t0 + pδ < t < t0 + δ

where K is the state-feedback gain chosen such that A − BK is Hurwitz. It should be noted
that the switching parameters δ and p are found for the stabilized system (A − BK, B, C).
For the case that the NMP system is stable and (A, B) is controllable, the state-feedback
control in Eq.(8.49) can be used to move the poles of the system. A relocation of the poles
can be helpful in finding appropriate switching parameters for the NMP system.
Remark 8.4.3. The first necessary step in solving the IOT problem for a NMP system is
to find a DEMP system. This can be done following the discussion in Section 8.3 through
proper choice of parameters δ and p. Although there is no guarantee that a DEMP system
exists for the general case, a numerical search for a range of δ values can be carried out. If
there are multiple choices for the pair (δ, p) for DEMP systems, a trade off between IOT
error, eo (t0 + δ), and the input switching frequency can be used to choose the appropriate
DEMP system. A smaller value of δ implies less IOT error but higher switching frequency.

8.5

Numerical Simulations

In this section, we present two numerical simulations for solving the IOT problem for NMP
systems. The system parameters and reference trajectories are chosen in a way that standard
continuous-time output regulation methods are inapplicable. We summarize the steps to
design the IOT controller as follows:
1. If the NMP system is unstable and stabilizable, design a state-feedback controller to
177

stabilize the system.
2. Find switching parameters δ and p such that all RHP zeros of the stable NMP system
are moved to open LHP. This can be done following the discussion in Section 8.3.
3. For the resulting MP system in step 2, design the tracking control in Eq.(8.32) for each
time interval [ti + (j − 1)δ, ti + jδ] where j = 1, 2, · · · , with initial conditions given in
Eq.(8.46).
4. Apply the control input in Eq.(8.44) for each time interval. In the case of unstable
NMP system, the control input is given by Eq.(8.49).
For the first simulation, consider the following stable NMP system:

G(s) = 10

s(s − 1)
(s + 1)2 (s + 10)

(8.50)

and let the reference trajectory be defined by:

r(t) =



 exp(t) − 1 ,


 exp(3) − 1 ,

t≤3

(8.51)

t>3

This reference trajectory can be generated by the following exosystem:



1 0
w = Sw , S = 
˙

0 0

(8.52)

Since the unstable eigenvalue of the exosystem is identical to the RHP zero of the system,
this tracking problem can not be solved by output regulation techniques [108]. The choice

178

y, r(t)

20 a
10

η1 , η2 (t)

0
0.4 b
0.2
0

-0.2
u, u(t)
¯

0

c

-20
-40
0

0.75

1.5

3

2.25

3.75

4.5

5.25

6

t(sec)
Figure 8.3: Simulation results for the first case showing the system output (solid line) and
reference trajectory (dashed line) in the first plot, internal states in the second plot and
control input (Solid line) and tracking control for MP system (dashed line) in the third plot

of switching parameters δ = 0.75 and p = 0.75 yields the following DEMP system:

(s + 110.22)(s + 0.28)
¯
G(s) = −0.17
(s + 1)2 (s + 10)

(8.53)

Figure 8.3 shows the simulation results for the system in Eq.(8.50). The system output
(solid line) and the reference trajectory (dashed line) are presented in the first plot in Figure
8.3. The second plot shows the system internal states, η1 (t) and η2 (t), which represent the
internal dynamics of the system in the normal form. The third plot shows the control input
for NMP system (solid line) and the tracking control for MP system which are related by
Eq.(8.45) for each time interval δ. These time intervals are illustrated by vertical dashed
lines in Figure 8.3. Simulation results ensure that the system output meets the reference
trajectory with a sufficiently small error while the internal states of the NMP are bounded.

179

For the second simulation, consider the following unstable NMP system:

G(s) =

(s2 + 1)(s − 15)
(s − 1)(s − 2)(s2 + 2s + 5)

(8.54)

with complex zeros and poles. The reference trajectory is defined by:

r(t) = 5 arctan(t)

(8.55)

This trajectory can not be generated by an exosystem so the output regulation methods
can not be applied to this example. Since the NMP system is controllable, a state-feedback
control is first designed to stabilize the system by moving the unstable poles s = 1 and s = 2
to s = −1 and s = −2, respectively. Through a numerical search, the switching parameters
are chosen as δ = 0.5 and p = 0.7 for the stabilized system and result in the following DEMP
system:
2

(s + 25.02)(s + 0.08s + 1)
¯
G(s) = −0.45
(s + 1)(s + 2)(s2 + 2s + 5)

(8.56)

The simulation results are shown in Figure 8.4. It includes the system output (solid line)
and reference trajectory (dashed line) in the first plot, the internal states in the normal
form of the system in the second plot and the control input in the third plot. The control
input is comprised of a continuous state-feedback control input and a switched control input
using (8.49). As can be seen from the figure, the system output is approximately matched
with the reference trajectory at regular intervals of time δ shown by vertical dashed lines.
Furthermore, the internal states remain bounded for the NMP system.
Remark 8.5.1. The deviation of the system output from the desired trajectory during the

180

y, r(t)

8 a
4
0
0.5
η1 , η2 , η3 (t)

0

u(t)

b

0

-0.5
10 c

-10
0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

t(sec)
Figure 8.4: Simulation results for the second case showing the system output (solid line)
and reference trajectory (dashed line) in the first plot, internal states in the second plot and
control input (Solid line) and tracking control for MP system (dashed line) in the third plot
time interval t ∈ (t0 , t0 + δ) depends on the system parameters as well as the switching
parameters δ and p. In the case that there exist multiple choices for switching parameters,
another criterion can be developed to minimize this deviation. This can be investigated more
as a future work.

8.6

Conclusion

A continuous-time control method for intermittent output tracking in linear SISO NMP
systems was proposed. The concept of discrete equivalent system have been used to move
the unstable zeros to the LHP. The switched control input is obtained from a tracking control
input for the discrete equivalent minimum-phase system. The proposed method ensures that
the system output matches with the reference trajectory at regular intervals of time while

181

keeping the internal dynamics stable. Although the controller does not guarantee exact
output matching with the reference trajectory, the matching error can be made sufficiently
small by adjusting the control gains and switching parameters of the input. The reference
trajectories to be tracked are not constrained to any specific set, however, a finite preview
of the trajectory is required. The switched inputs used in the control can be estimated by
impulsive inputs as their magnitudes get larger and their time supports get smaller.

182

Chapter 9
Conclusion and Future Work
In this research, we investigated different aspects of impulsive control in underactuated
mechanical systems. As the first step, we ensured that impulsive inputs are indeed implementable in mechanical systems using standard hardware. This was proven through an
experimental verification of a swing-up algorithm for the Pendubot. Having addressed the
implementability concern, we used the properties of impulsive inputs to develop several algorithms for underactuated mechanical systems with different control objectives. Impulsive
inputs are capable of causing instantaneous jumps in the energy of a mechanical system. This
can be used to develop new energy-based control algorithms or improve current algorithms in
the literature. To illustrate this, we considered the swing-up control of the Acrobot, which
is a benchmark problem in underactuated mechanical systems and compared our results
with those in the literature. To show another aspect of impulsive inputs, we looked at the
stabilization problem for underactuated systems. Impulsive inputs can also be useful in this
category of controllers as they can result in instantaneous jumps in the Lyapunov function
corresponding to a stabilizing controller. We used this property to develop a stabilization
algorithm which results in enlarging the region of attraction of equilibria for underactuated systems. Several examples with linear and nonlinear controllers were studied and it
was shown that adding impulsive inputs at some specific instants of time to the continuous
control inputs can enlarge the region of attraction for a desired equilibrium. As another
property of impulsive inputs, they can affect the dynamics of a system significantly and in a
183

very short period of time. This is beneficial especially when there is not much time available
for continuous inputs to control the system behavior. We show this advantage with a control
problem of changing the fall direction for humanoid robots in order to avoid damaging the
surrounding objects. A six-dof robot was considered and two sets of impulsive inputs were
applied to change the direction of the fall to a desired angle. Impulsive inputs were also
shown to be useful in rejecting external disturbances with the help of balance maintenance
control for the Synthetic-Wheel biped. The last two pieces of work are not using impulsive
inputs but still related to the concept of impulsive control for underactuated systems. The
first one dealt with designing zero-energy walking gaits for underactuated bipeds by avoiding
undesirable impacts from the ground at the leg interchange. The second work proposed an
intermittent output tracking control for non-minimum-phase systems. The switched inputs
can be approximated by impulsive inputs in the limit.
We suggest the following to continue this field of research further in the future. Although
the experiments on the Pendubot showed satisfactory results, a new amplifier circuit may
be designed to get closer to the ideal impulses in experiments. The new circuit will have
a large capacitor which enables the amplifier to provide enough current for the motor at
the impulsive times. The experimental verification of all the algorithms developed in this
dissertation would add a lot to prove that impulsive control is a reliable method for underactuated mechanical systems. For the work presented in Chapter 4, more analysis is needed
to predict if the impulsive algorithm is capable of stabilizing the desired equilibrium from
the given configuration. The safe fall algorithm in Chapter 5 is a preliminary work to prove
the concept of using impulsive inputs for such platforms. A more realistic configuration of a
humanoid with higher dof’s and two legs should be studied. There is some flexibility in the
algorithm which may be used to optimize the safe fall algorithm for a specific humanoid. For
184

example, the impulsive times which are based on specific configurations of the robot may
be changed to get better results. Energy-conserving gaits which are presented in Chapter 7
are not practical yet and the limitations of the generating such gaits in actual bipeds need
to be considered. For example, generated gaits for five-dof bipeds require the knee bent
backward or the swing-leg touches the floor in the middle of its swing phase which are not
practical assumptions. In addition, finding proper electrical circuits which are capable of
regenerating and storing energy from negative work during each step is essential. For the algorithm in Chapter 8, future work may focus on extending the current results to multi-input
multi-output linear systems. Also, the effect of switching parameters on output deviations
from desired trajectory can be investigated to find the best control performance in between
intermittent tracking times.

185

APPENDICES

186

Appendix A
Procedure for Verifying the Condition
in Theorem 3.4.1
Assume that the gains kp , kd , and ke , and the initial condition x0 are chosen based on the
conditions stated in [3], [22]. This guarantees asymptotic convergence of the acrobot to
its upright configuration using the continuous controller designed in those works. We now
outline the procedure to check the condition in Eq.(3.64) in Theorem 3.4.1:
¯ ¯
1. Find all points x = (θ1 , θ2 , 0, 0) in the set Ω defined in Eq.(3.44) using the relations in
¯
¯
[3]. To this end, we first solve the following equation to find θ2 :
kp ¯
θ2 =
g 2 q4 q5
g

¯
sin(θ2 )
2
2
¯
q4 +q5 +2q4 q5 cos(θ2 )

[g

2
2
¯
q4 +q5 +2q4 q5 cos(θ2 )+0.5Edes ]

(A.1)

¯
Then find θ1 from the following equation:

kp θ2 + (E − Edes )τh = 0

(A.2)

¯ ¯ ˙− ˙ −
2. Find the points (θ1 , θ2 , θ1 , θ2 ) in the set S2 (x) defined in Eq.(3.63). Using Eq.(3.18)
˙+
˙+
with θ1 = θ2 = 0, we have:

˙−
θ1 = −

q2 + q3 C2
˙
θ−
q1 + q2 + 2q3 C2 2
187

(A.3)

and from the definition of S2 (x) we know:

¨
˙
θ2 = a(θ, θ− ) + b(θ2 )τc = 0

(A.4)

Using the expression for τ2c in Eqs.(3.38) or (3.69), as appropriate, Eqs.(A.3) and (A.4)
˙−
˙−
can be solved to find the values of θ1 and θ2 .
3. For the points in S2 (x) which satisfy V (x) ≤ V (x0 ), check the condition in Eq.(3.64)
˙ (3) (3)
¨
by computing θ2 θ2 . θ2 is computed by taking the derivative of θ2 given in Eq.(3.8).

188

Appendix B
Equations of Motion:
Synthetic-Wheel Biped
The symmetric matrix M, matrix N, and vector G in Eq.(7.1) have the following entries:

2
M(1, 1) = It + Ist + Isw +mt d2 +lt +R2 +2R(dt −lt ) cos(θ + φ)−2dt lt
t

+ mst d2 + 2R(R + (dst − R) cos(θ) − dst )
st
+msw d2 +2R(R+(dsw −R) cos(θ+ψ)−dsw )
sw
M(1, 2) = It + mt (dt − lt )(dt − lt + R cos(θ + φ))
M(1, 3) = Isw + msw (dsw − R)(dsw − R + R cos(θ + ψ))
M(2, 2) = It + mt (dt − lt )2
M(2, 3) = 0
M(3, 3) = Isw + msw (dsw − R)2
˙ ˙
˙
N(1, 1) = mt R(lt − dt ) sin(θ + φ)(θ + φ) + mst R(R − dst ) sin(θ)θ
˙
˙
+ msw R(R − dsw ) sin(θ + ψ)(θ + ψ)
˙ ˙
N(1, 2) = mt R(lt − dt ) sin(θ + φ)(θ + φ)
˙
˙
N(1, 3) = msw R(R − dsw ) sin(θ + ψ)(θ + ψ)

189

N(2, 1) = N(2, 2) = N(2, 3) = 0
N(3, 1) = N(3, 2) = N(3, 3) = 0
G(1) = [mt (lt − dt ) sin(θ + φ) + mst (R − dst ) sin(θ) + msw (R − dsw ) sin(θ + ψ)] g
G(2) = mt (lt − dt ) sin(θ + φ) g
G(3) = msw (R − dsw ) sin(θ + ψ) g
where mt , mst and msw are the masses of the torso, stance leg and swing leg; It , Ist and Isw
are the mass moments of inertia of the torso, stance leg and swing leg about their respective
center of mass; R is the radius of curvature of the feet; lt is the length of the torso; dt , dst
and dsw are the distances shown in Figure 7.2, and g is the acceleration due to gravity.

190

Appendix C
Equations of Motion: Five-DOF
Biped
The symmetric matrix M, matrix N, and vector G in Eq.(7.1) have the following entries:

2
M(1, 1) = I1 + d2 m1 + l1 (m2 + m3 + m4 + m5 )
1

M(1, 2) = l1 [d2 m2 + l2 (m3 + m4 + m5 )] cos(θ1 − θ2 )
M(1, 3) = d3 l1 m3 cos(θ1 − θ3 )
M(1, 4) = l1 (d4 m4 + l4 m5 ) cos(θ1 − θ4 )
M(1, 5) = d5 l1 m5 cos (θ1 − θ5 )
2
M(2, 2) = I2 + d2 m2 + l2 (m3 + m4 + m5 )
2

M(2, 3) = d3 l2 m3 cos(θ2 − θ3 )
M(2, 4) = l2 (d4 m4 + l4 m5 ) cos(θ2 − θ4 )
M(2, 5) = d5 l2 m5 cos(θ2 − θ5 )
M(3, 3) = I3 d2 m3
3
M(3, 4) = M(3, 5) = 0
2
M(4, 4) = I4 + d2 m4 l4 m5
4

M(4, 5) = d5 l4 m5 cos(θ4 − θ5 )

191

M(5, 5) = I5 + d2 m5
5
N(1, 1) = N(2, 2) = N(3.3) = N(4, 4) = N(5, 5) = 0
˙
N(1, 2) = l1 [d2 m2 + l2 (m3 + m4 + m5 )] sin(θ1 − θ2 )θ2
˙
N(1, 3) = d3 l1 m3 sin(θ1 − θ3 )θ3
˙
N(1, 4) = l1 (d4 m4 + l4 m5 ) sin(θ1 − θ4 )θ4
˙
N(1, 5) = d5 l1 m5 sin(θ1 − θ5 )θ5
˙
N(2, 1) = − l1 [d2 m2 +l2 (m3 +m4 +m5 )] sin(θ1 −θ2 )θ1
˙
N(2, 3) = d3 l2 m3 sin(θ2 − θ3 )θ3
˙
N(2, 4) = l2 (d4 m4 + l4 m5 ) sin(θ2 − θ4 )θ4
˙
N(2, 5) = d5 l1 m5 sin(θ1 − θ5 )θ5
˙
N(3, 1) = − d3 l1 m3 sin(θ1 − θ3 )θ1
˙
N(3, 2) = − d3 l2 m3 sin(θ2 − θ3 )θ2
N(3, 4) = N(3, 5) = N(4, 3) = N(5, 3) = 0
˙
N(4, 1) = − l1 (d4 m4 + l4 m5 ) sin(θ1 − θ4 )θ1
˙
N(4, 2) = − l2 (d4 m4 + l4 m5 ) sin(θ2 − θ4 )θ2
˙
N(4, 5) = d5 l4 m5 sin(θ4 − θ5 )θ5
˙
N(5, 1) = − d5 l1 m5 sin(θ1 − θ5 )θ1
˙
N(5, 2) = − d5 l2 m5 sin(θ2 − θ5 )θ2
˙
N(5, 4) = − d5 l4 m5 sin (θ4 − θ5 )θ4
G(1) = − g[d1m1 + l1 (m2 + m3 + m4 + m5 )] sin θ1
G(2) = − g[d2m2 + l2 (m3 + m4 + m5 )] sin θ2
192

G(3) = − gd3m3 sin θ3
G(4) = − g(d4m4 + l4 m5 ) sin θ4
G(5) = − gd5m5 sin θ5
where mi , Ii , and li , i = 1, · · · , 5, are the mass, mass moment of inertia, and length of the
i-th link; di is the distance of the center of the mass of the i-th link from the i-th joint; and
g is the acceleration due to gravity.

193

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