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This is to certify that the
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AN lMPULSIVE-MOMENTUM APPROACH TO CONTROL OF
A CLASS OF UNDERACTUATED MECHANICAL SYSTEMS

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THAMER ALBAHKALI

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AN IMPULSE-MOMENTUM APPROACH TO CONTROL OF A CLASS OF
UNDERACTUATED MECHANICAL SYSTEMS

By

Thamer Albahkali

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Mechanical Engineering

2009

Copyright by
THAMER ALBAHKALI
2009

ABSTRACT

AN IMPULSE-MOMENTUM APPROACH TO CONTROL OF A CLASS OF
UNDERACTUATED MECHANICAL SYSTEMS

By

Thamer Albahkali

Underactuated mechanical systems pose challenging control problems because they
have fewer control inputs than degrees of freedom. The dynamics of these systems are
such that they cannot be always controlled by traditional nonlinear control methods. In
this dissertation we present a new and general methodology to control underactuated
mechanical systems, and we apply this methodology to the control problems of systems
with two and three generalized coordinates with a single underactuated joint. We first
consider the swing-up control problems of the Pendubot and the Acrobot, which are
benchmark problems. A comparison of our methodology with other controllers
previously applied to these problems shows that the systems are stabilized in a settling
time comparable to the best results available in the literature, using the same actuators.
However, a significant advantage of our methodology is that it can be applied to both the
pendubot and the acrobot alike, and only two of the many previously developed control
methodologies share this feature. Our methodology is based on rest-to-rest maneuvers of
the actuated link using impulse-like control inputs. These inputs are designed to make the
energy of the system converge to a level corresponding to that at the desired equilibrium
point while restricting the movement of the actuated link. In order to show the generality
of this methodology we show in this dissertation that it can be used for posture control of
a synthetic-wheel biped robot. The biped robot, which is introduced for the first time in

this dissertation, has two legs and a torso; and we consider stabilization of its standing

configuration after the application of disturbances. Specifically, we consider the case of
small disturbances in which we require the robot to maintain its posture using only the
torso and keeping its feet on the ground. The torso will use impulsive torques to bring the
energy of the system to the desired level. This problem is similar to a person who is
trying to balance and stand without taking any steps. However, not all disturbances can
be handled this way; therefore, we consider the stabilization problem for this biped after
the application of a large disturbance. This time the biped is going to use both legs during
the stabilization process. The biped will walk the necessary steps before stopping and
finally stabilize around the standing configuration. The torso again is responsible for
bringing the energy to the desired level while the legs role are to keep the biped from
falling. The algorithms are provided for solving each control problem mentioned above.

Simulation results are also presented to demonstrate the efficacy of the approaches.

. .

To my dear family

ACKNOWLEDGMENTS

I want to express my deep gratitude to my advisor Dr. Ranjan Mukherjee for his
continuous guidance and support during the years I worked under his supervision. He did

not hesitate to teach me the things I needed in this field and I learned a lot from him.

Moreover, I want to thank all the committee members for their helpful suggestion to
improve my work, especially Dr. Steven Shaw who helped me when my advisor was

gone on sabbatical.

In addition, I want to thank all my friends in the Control Lab for their friendship and
help, especially, Dr. Tuhin Das for teaching me some computers skills that helped me do

my work.

Finally, I want to thank Bob Chalou for helping me create simulation videos of my

work using NX program.

vi

TABLE OF CONTENTS

LIST OF TABLES ................................... ix
LIST OF FIGURES .................................. x
1 Introduction ..................................... 1
1.1 Background ................................... 1

1.2 The Pendubot .................................. 4

1.3 The Acrobot .................................. 6

1.4 An Active Synthetic Wheel Biped Robot with Torso ............. 7

2 The Pendubot .................................... 11
2.1 Background .................................. 11
2.2 Nomenclature ................................. 12
2.3 System Dynamics ............................... 14
2.3.1 Equations of Motion .......................... 14

2.3.2 Force Interaction Between the Two Links ............... 16

2.3.3 Holding Torque ............................ 17

2.3.4 Braking Torque ............................ 17

2.4 Energy Consideration of the Second Link ................... 19
2.4.1 Effect of Sudden Stopping of the First Link .............. 19

2.4.2 Pumping Energy to the Second Link ................. 21

2.4.3 Maintaining the Energy of the Second Link at the Same Level . . . 22

2.4.4 Negative and Positive Work on the Second Link ........... 23

2.4.5 Rest-to-Rest Maneuver of the First Link ............... 24

2.5 Algorithms for Swing-Up Control ...................... 25
2.5.] First Algorithm ............................ 26

2.5.2 Second Algorithm .......................... 32

2.6 Remarks .................................... 4O

3 The Acrobot .................................... 42
3.1 System Dynamics ............................... 42
3.1.1 Equations of Motion ......................... 42

3.1.2 Holding Torque ............................ 43

3.1.3 Braking Torque ............................ 43

3.1.4 Opposite Angular Velocities Torque ................. 44

3.1.5 Impulse Angular Velocity Matching Torque ............. 45

3.1.6 Impulse Constant Angular Velocity Torque ............. 45

3.2 Impulse-Momentum Effect .......................... 46

3.3 Energy Consideration of the System ..................... 50
3.3.1 Energy Change Due to the Impulse-Momentum Approach ...... 50

3.3.2 Rest—to-Rest Maneuver of the Second Link .............. 51

3.4 Algorithms for Swing-Up Control ...................... 53
2.5.1 First Algorithm ............................ 53

2.5.2 Second Algorithm ........................... 60

vii

2.5.3 Third Algorithm ............................ 64

3.5 Remarks .................................... 74

4 Stabilizing an Active Synthetic Wheel Biped due to Small Disturbances . . . 75
4. 1 Introduction .................................. 75

4.2 Nomenclature ................................. 76

4.3 System Dynamics ............................... 78
4.3.1 Equations of Motion .......................... 78

4.3.2 Holding Torque ............................ 80

4.3.3 Braking Torque ............................ 80

4.4 Energy Consideration of the System ..................... 81
4.4.1 Effect of Sudden Stopping of the Torso ............... 81

4.4.2 Rest-to-Rest Maneuver of the Torso .................. 86

4.5 Stabilization Algorithm ............................ 88
4.6 Remarks .................................... 95

S Stabilizing an Active Synthetic Wheel Biped due to Large Disturbances . . . 97
5. 1 Introduction .................................. 97

5.2 Nomenclature ................................. 99

5.3 System Dynamics ............................... 100
5.3.1 Equations of Motion ......................... 100

5.3.2 Torque Expressions of the Actuated Joints .............. 103

5.4 Impulse-Momentum Effect .......................... 106

5.5 Energy Consideration of the System ..................... 111
5.5.1 Energy change when both the second leg and the torso are moving . 111

5.5.2 Energy change due to the impulse-momentum effect ........ 116

5.6 Stabilization Algorithm ............................ 120

5.7 Remarks ................................... 123

6 Conclusions and Future Work .......................... 125
BIBLIOGRAPHY ................................... 130

viii

LIST OF TABLES

2.1 Comparison of simulation results of braking with analytical results .................... 29

ix

2.1

2.2

2.3

2.4

2.5

2.6

2.7

2.8

2.9

3.1

3.2

3.3

LIST OF FIGURES

The pendubot ................................................................................ 12
Forces of interaction between the two links ............................................. 16
Effect of sudden stopping the first link .................................................. l9
Swinging-up control of the first link ..................................................... 25

Plot of joint angles, joint angle velocities, control torque, and energy of the second
link ............................................................................................ 30

Vertically down configurations of the second link during rest-to-rest maneuver... 33

Plots showing regions in the 91-62 space where 1'] > 0 and H < 0 ................... 34
Plot of joint angles, joint angle velocities, control torque, and energy of the second
link ........................................................................................... 35
Plot of joint angles, joint angle velocities, control torque, and energy of the second
link ........................................................................................... 37
2.10 Plot of joint angles, joint angle velocities, control torque, and energy of the second
link ........................................................................................... 40
The acrobot ................................................................................. 43
The impulsive force and moment on the second link and their effect on the first
link ........................................................................................... 47
Sketch of the acrobot during the swing-up process .................................... 56
Solution of the inequality and the selected values of a .............................. 57

3.4

3.5

3.6

3.7

Plot of the first simulation of the first algorithm, which shows the joint angles, joint
angle velocities, control torque, and energy of the system ........................... 59

Plot of the second simulation of the first algorithm, which shows the joint angles,
joint angle velocities, control torque, and energy of the system ..................... 6O

Plot of the simulation of the first algorithm, which shows the joint angles, joint
angle velocities, control torque, and energy of the system ........................... 62

3.8

3.9

3.10

3.11

3.12

3.13

3.14

3.15

4.1
4.2

4.3
4.4

4.5

5.1

5.2

5.3

5.4

5.5

The acrobat at the instant after the application of the impulsive force ............... 66

The change in the acrobot configuration in the moment after the application of the
impulsive force with the change in the negative and positive areas of h1 ........... 67

The change in the acrobot configuration in the moment after the application of the
impulsive force with the change in the negative and positive areas of h2 ........... 68

The change in the acrobot configuration in the moment after the application of the
impulsive force with the change in the negative and positive areas of (111+ 112). .. 69

The angular acceleration of the first link 65 at the moment just after the impulsive

1

force .......................................................................................... 70
The angular acceleration of the second link 652 the moment just after the impulsive
force .......................................................................................... 71

(51 + 52) the moment just after the impulsive force ................................... 72
Plot of third algorithm joint angles, joint angle velocities, control torque, and energy
of the system ................................................................................. 73
An active synthetic wheel biped robot with torso where the legs are aligned ....... 78
Impulsive forces and moments on the biped ............................................. 82
The range selection for the torso angle ((92) .............................................. 92

Plot of joint angles, joint angle velocities, control torque, and energy of the

system ........................................................................................ 93

Plot of joint angles, joint angle velocities, control torque, and energy of the

system ........................................................................................ 95
An active synthetic wheel biped robot with torso ...................................... 98

Impulsive forces and moments on the biped ........................................... 107

(a) The first leg has negative angle value. (b) The first leg has positive angle
value ......................................................................................... 113

The range of rate of change of energy below zero with 63 is fixed to be It rad.. 114

The range of rate of change of energy below zero with 63 is fixed as 2.9 rad. . .. l 15

xi

5.6 (a) The start of event (a). (b) The start of event (b) ................................. 119

5.7 Plot of joint angles, joint angle velocities, control torques, and energy of the
system ..................................................................................... 124

xii

CHAPTER 1

Introduction

1.1 Background
In this dissertation we develop a new and general impulsive control methodology that we
apply to the problem of controlling underactuated mechanical systems. The control of
underactuated mechanical systems have been investigated by many researchers, but our
approach is the first where impulsive forces are used as a part of the control inputs. We
first apply this methodology to the classic benchmark problems for underactuated
systems, the pendubot [6 - 10], [16], [19 - 21], [23 - 24] and [29] and the acrobot [33 -
38], [40 - 45], and [47 - 59]. A comparison of our methodology with other controllers
previously applied to these problems shows that the systems are stabilized in a settling
time comparable to the best results available in the literature, using the same actuators.
However, a significant advantage of our methodology is that it can be applied to both the
pendubot and the acrobot alike, and only two of the many previously developed control
methodologies share this feature. Thus, we have developed a new and competitive control
strategy for these systems, and proven its efficacy on the standard benchmark problems in
the field.

The proposed methodology is quite general and can also be applied to other

underactuated mechanical systems. As an example, we show in this dissertation that it

can be used for the posture control of a synthetic-wheel biped. This class of robots are
currently of great interest since, as compared to wheeled mechanisms, walking machines
have the advantage of better maneuverability over uneven terrain and are better suited to
avoiding moving obstacles, making them ideal for outdoor environments [13], and [60 -
74]. The dynamics of bipeds are, however, strongly non-linear, often inherently unstable,
typically underactuated, involve foot-ground interactions and the attendant abrupt
changes in the number of degrees of freedom, and discrete impact events, all of which
makes analysis and control of these systems quite challenging. We show that the control
methodology developed in the present work can stabilize this synthetic-wheel biped in a
short settling time with reasonable impulsive torques that will not require large motors.
This demonstrates the general applicability of the methodology, and its potential for
application to other underactuated mechanical systems.

Despite the use of impulsive forces, it is noteworthy to mention that it is not
necessary that the system will require the use of large actuators. For example, for
mechanical systems where electric motors are used, our approach will benefit from the
fact that motors generally have a peak torque which is greater than the maximum
continuous torque [28]. We can therefore take advantage of this peak torque, yet still
limit the impulsive torques to be less than the peak torque of the selected motor.

Underactuated mechanical systems are systems that have fewer control inputs than
degrees of freedom [26]. These systems have generated significant interest in the control
community since underactuation reduces cost and weight [81], [82], can help deal with
actuator failure [83], and since many systems are naturally underactuated [84].

Underactuated systems pose challenging problems in control because many of the

methods developed for completely actuated systems ( such as feedback linearization,
Lyapunov theory, passivity, etc.) are not directly applicable to underactuated systems.

The idea of using impulsive forces as control inputs is not new and some of the early
work can be credited to Pavlidis [25], Gilbert and Harasty [17], and Menaldi [22]. In
recent years, researchers have investigated the problems of stability, controllability and
observability, optimality, etc. (see [14], [27], [32] and the references therein). However,
we are the first to use the impulsive forces to control underactuated mechanical systems.
In our approach we impose joint restrictions on the actuated links and make them perform
rest-to-rest maneuvers with small amplitudes around their desired configuration. The rest-
to-rest maneuvers are designed to bring the unactuated joints to their desired
configuration. These joint restrictions imposed on the actuated links help us control both
the passive and active joints using only the active joints. Each of the controlled systems
in this dissertation (the pendubot, the acrobot, and the biped) are locally asymptotically
stabilizable using linear control, and the primary objective of control design is to bring
the configuration of the system within the region of attraction of the equilibrium. This is a
common goal for many researchers working with underactuated systems [6], [7], [21],
[30], [34], [35], [37], [40]. While most of these results are largely applicable to specific
platforms only, our control methodology is general and can be applied to multiple
platforms and has the potential to be applied to underactuated systems with multiple
degrees of underactuation.

This dissertation is organized as follows: in this chapter we present a literature survey
of earlier work done on control problems of the pendubot, acrobot, and biped robots and

summarize our contribution in relation to earlier work. In chapter 2 we deal with the

control problem of the pendubot. The control problem of the acrobat is considered in
chapter 3. Chapter 4 presents the control problem of the biped subject to small
disturbances. The discussion of the control problem of the biped subject to large
disturbances is followed in chapter 5. Finally, in chapter 6 we present our conclusion and
future work.

1.2 The Pendubot

The pendubot (Figure 2.1) is a two-link robot in the vertical plane with an actuator at the
shoulder joint and a passive elbow joint. It is a classical example of an underactuated
system [11], [26], and its control problem has similarities with that of the single and
double inverted pendulums on a cart. The complete control of the pendubot requires
swing-up to the neighborhood of its equilibrium configuration with the highest potential
energy followed by stabilization.

The stabilization problem, also known as the balancing problem, has been addressed
by several researchers. For example, Spong and Block [6] linearized the dynamic
equations and used a linear quadratic regulator, Erdem and Alleyne [15] demonstrated a
large region of attraction using nonlinear control based on state-dependent Riccati
equation, and Zhang and Tarn [20] used hybrid control. We linearize the dynamic
equations and use a linear controller for stabilization, but the contribution of this work
lies in the development of a new methodology for swing-up of the pendubot.

Several methods have been proposed in the literature for swing-up of the pendubot.
Spong and Block [6] proposed a method based on feedback linearization and F antoni, et
al. [1] utilized passivity properties of the pendubot to develop an energy-based controller.

The controller of Fantoni, et al. [1] requires tuning of parameters for an acceptable rate of

convergence and imposes restrictions on the initial conditions to avoid a singularity.
Kolesnichenko and Shiriaev [18] proposed global feedback transformations for passivity
based control and Lai, et al. [19] used a Lyapunov function with a time-varying
parameter to avoid the singularity problem. A different approach to the problem, based
on limit cycle oscillations in zero dynamics of the pendubot, was adopted by Grognard
and Canudas-de-Wit [10] and Orlov, et al. [9]. In contrast to the energy-based method
[1], where the pendubot moves in a homoclinic orbit, these methods achieve orbital
stabilization. Orlov, et al. [9] used sliding mode control to deal with uncertainties and
external disturbances. A sliding mode controller typically provides ultimate boundedness
and this motivated the zeno-mode control design by Orlov, et al. [23]. Zeno mode
controllers require infinite switchings and hardware implementation [24] results in
chattering. The theoretical and experimental results of F reidovich, et al. [16] are similar
to the work of Grognard and Canudas-de-Wit [10] and impose virtual holonomic
constraints to generate periodic motions of the passive link. Other approaches to swing-
up of the pendubot include fuzzy control, [21], for example.

All swing-up methods essentially aim to increase the energy of the pendubot. Our
method is no exception but we focus on the force of interaction between the two links and
the work done by this force on the second link. For swing-up of the pendubot, we
instinctively take the first link to the vertically upright position and conduct a series of
rest-to-rest maneuvers about this configuration that results in swing-up of the second link.
Similar to the work of Fantoni, et al. [1], our approach is based on the energy of the
system, but it does not impose restrictions on the initial conditions or suffer from any

singularity. Furthermore, the rest-to-rest maneuvers allow swing-up in the presence of

joint limit restrictions on the first link. A salient feature of our approach is the use of
impulsive torques for the rest-to-rest maneuvers as mentioned earlier in this text. The use
of impulsive force provides the scope for a large change in velocity over a short time
interval, and this property is exploited in this dissertation for swing-up of the second link
with joint limit restrictions imposed on the first link. The simulation results for the
pendubot show that the approach is very effective, and we will do a comparison with
some other earlier work to show this successfulness.

1.3 The Acrobot

The acrobot (Figure 3.1) is the same physical system as the pendubot, but the elbow joint
is actuated instead of the shoulder joint. The complete control problem of the acrobot
requires swing-up to the neighborhood of its equilibrium configuration with the highest
potential energy followed by stabilization. Several researchers worked on the swing-up
problem and they used different techniques such as partial feedback linearization [34-3 7],
pseudo-linearization [38-39], optimal control [40], intelligent control [41-47], passivity-
based control [48], and energy-based control [49-53]. The last technique is considered the
best between others because it exploits the physical characteristics of the system; and
therefore, one can get a very robust swing-up control law. Banavar and Mahindrakar [52]
proposed an energy-based method to guarantee stability. Xin and Kaneda [53] introduced
a combination technique between partial feedback linearization and robust control. They
also worked on flying acrobot and studied the control its posture with a non-zero initial
angular momentum [54]. Yonemura and Yamakita [55] and Lai et a1. [56] analyzed the
singularity of the problem and introduced algorithms to deal with it. Lai et al. [57]

extended their work and introduced a non-smooth Lyapunov function, which

theoretically guarantees the stability of the acrobot in the whole motion space. Zergeroglu
et al. [58] proposed a Lyapunov based control algorithm, which guarantees that the first
link is asymptotically driven to the desired setpoint provided some conditions are
satisfied. Furthermore, Shiriaev et al. [59] presented motion planning and feedback
stabilization of periodic orbits of the acrobot.

We investigated the applicability of the impulse-momentum approach for swing-up
control of the acrobot. For the acrobot, the elbow joint is actuated instead of the shoulder
joint; and hence, the swing-up control algorithm is based on a series of rest-to-rest
maneuvers of the second link around the zero angle. The rest-to-rest maneuvers are
designed such that each maneuver results in a net gain in energy of the system. Once the
acrobot configuration reaches the region of attraction of the desired equilibrium point, a
linear controller is invoked to stabilize the equilibrium. Although the pendubot and the
acrobot have different system behaviors, we were able to successfully apply the impulse-
momentum approach for swing-up control of the acrobot. However, swing-up control of
the acrobot takes a longer time since the actuated joint has to pump energy into the
system from the lowest potential energy configuration to the highest potential energy
configuration. In the case of the pendubot, the first link is actuated and raised to its
vertical configuration in the first step; this reduces the amount of energy that needs to be
added by the controller. To achieve swing-up control of the acrobot over a shorter
duration of time, a controller can be designed to first increase the overall energy of the
system or an initial condition can be chosen that corresponds to higher potential energy.

1.4 An Active Synthetic Wheel Biped Robot with Torso

For a long time, researchers have tried to develop walking machines and "bipeds in

particular. As we said earlier in this chapter, walking machines are better suited for

outdoor environments, which have uneven terrain and natural obstacles, than wheeled
machines and bipeds have the potential for staying leveled and stable in such
environments. The dynamics of bipeds are, however, non-linear and often unstable,
involving foot-ground interaction and continual change in degrees of freedom and
discrete events that make their analysis and control difficult. It is not surprising that only
recently researchers made a good progress in the design and development of bipeds.
Takanishi, et al. [60] proposed a control method for a dynamic biped walking
under unknown external force. Zheng and Shen [61] achieved control of a static biped to
walk on an unknown sloped terrain using force sensors at the feet and position sensors of
the joints. Although the implementation of the strategy was simple, it was only applied to
static walking. Yamaguchi, et al. [62] introduced an anthropomorphic dynamic biped
which can adapt to structured environments. Chew, et al [63] used simple intuitive
control strategies to compel bipedal robots to walk over sloped terrain. Pratt, et al. [64]
used Virtual Model Control to control a planar biped to walk on flat ground. Stitt and
Zheng [65] proposed a method to generate gaits based on distal supervised learning; this
method incorporates a forward model of the robot dynamics and uses it to convert
stability information into information on how to adjust the robot’s joints so as to regain
stability. Lohmeier, et al. [66] designed a biped robot to achieve a dynamically stable gait
pattern and the robot reached a walking velocity of 2.4 km/hr in experiments. The robot
required accurate sensors and a powerful computer system for performance and stability.
Many of the results in the literature use the “Linear Inverted Pendulum Mode” for control
of bipeds. For example, Kajita and Tani [67] developed a simple control method for

walking on ground constituted of planes and vertical steps, such as stairs. Park and Kim

[68] proposed a gravity-compensated inverted pendulum mode to design reference
motions of biped robots that includes the dynamics of free leg motion. Matsumoto, et al.
[69] designed a combined leg-wheeled biped that has the ability to travel quickly on flat
surfaces using wheels and climb stairs effectively. Ota, et al. [70] proposed similar hybrid
locomotion with twin-frame structure robot.

In this dissertation a new biped robot is introduced. The feet of this robot are part
of a circle, which allows the overall movement of the biped to be similar to a rolling
wheel. We described this robot as “An Active Synthetic Wheel Biped Robot with Torso.”
In order to start walking, the torso will lean to the front to create a moment on the system.
Then, the robot will take a step forward to avoid falling over. Before the first leg tip is
reached, the second leg will take over to complete the circle and the robot will repeat the
process to achieve walking. Our goal in this dissertation is to solve the stabilization
problem of this robot about its standing configuration. We investigate the solution for
small and large disturbances. First, we are going to assume that the system is only
affected by a small disturbance. The control problem then will be similar to a standing
person pushed and tried to maintain the standing position without pulling out his or her
leg. Hence, we have applied our methodology for stabilizing the system where the two
legs are aligned. The objective is to move the torso in a manner that allows the robot to
regain is standing posture after being disturbed. The system is then considered as a two-
link underactuated system depending only on the torso for its movement. Second, the
effect of a large disturbance is considered. Therefore, we have used both the second leg
and the torso this time to regain the standing posture. The second leg role is always going

to be completing the circle of foot (i.e. make the system walk) while the torso is going to

adjust the energy of the system by its movement until the energy level reaches the desired
one. We have obtained very good results and this establishes the generality of our
methodology. This methodology is in fact well suited for the biped model since it
restricts the links from going over certain limits. The biped is different from the pendubot
and the acrobot in that its torso is not allowed to rotate complete 360 degrees for
stabilization. As in the case of the pendubot and acrobot, the control method is based on a
series of rest-to-rest maneuvers of the actuated link, the torso in the case of the biped,
around its vertical configuration. The rest-to-rest maneuvers are designed such that the
energy of the system converges to the desired value. This brings the robot configuration
within the region of attraction of the equilibrium point wherefrom a linear controller can

be invoked to stabilize the equilibrium.

10

CHAPTER 2

The Pendubot

2.1 Background
The pendubot is a two-link planar robot in the vertical plane with an actuator at the
shoulder joint but no actuator at the elbow joint. The control problem of the pendubot
typically refers to the task of stabilizing its equilibrium point with the highest potential
energy, which is unstable. This requires swing-up control of the pendubot from the
lowest potential energy configuration to its equilibrium configuration with the highest
potential energy followed by balance control that stabilizes the equilibrium. The balance
control problem is quite simple; and therefore, we address the swing-up control problem.
Our approach to swing-up control focuses on the force of interaction between the two
links and the change in energy of the second link resulting from it. We will discuss two
algorithms that result in swing-up of the second link as well as the pendubot to the region
of attraction of the desired equilibrium point. At that juncture, a linear controller can be
invoked to stabilize the equilibrium.

This chapter is organized as follows. The next section lists the nomenclature used
throughout this chapter. In section 2.3 we provide the equations of motion of the
pendubot and derive expressions for the force of the interaction between the two links,

the holding torque, and the braking torque. In section 2.4 we discuss several methods for

11

changing the energy of the second link; these methods will be used later in the algorithms
for swing-up control of the pendubot. In section 2.5 we present two algorithms for swing-

up control along with numerical simulations. Concluding remarks are provided in section

2.6.

 

 

 

Figure 2.1. The pendubot

2.2 Nomenclature
The Pendubot is shown in Figure 2.1. For the nomenclature listed below, the subscript
i assumes values of 1 and, 2; and subscript j assumes values of 1, 2, 3, 4, and 5.
l . length of the i-th link, (m)
d. distance between the i-th joint and center of mass of the i-th link, (m)
m. mass of the i-th link, (kg)
1. mass moment of inertia of the i-th link about its center of mass, (kgmz)

t9. angular displacement of the i-th link as shown in Figure 2.1, (rad)

é. angular velocity ofthe i-th link, (rad/s)

é.— angular velocity of the i-th link, immediately before the first link is stopped,

(rad/s)

12

:n .31 .31

'21

angular velocity of the second link, immediately after the first link is stopped,
(rad/s)

velocity of the center of mass of the second link, (m/s)

velocity of the center of mass of the second link, immediately before the first link

is stopped, (m/s)

velocity of the center of mass of the second link, immediately after the first link is F
stopped, (m/s) j
Cartesian reference frame fixed to the second link .:

inertial reference frame with unit vectors 7 and ] along the X and Y axes,

respectively

force acting on the second link at the second joint along the x direction, (N)

force acting on the second link at the second joint along the y direction, (N)
impulsive force acting on the second link at the second joint, (N)

force acting on the second link at the second joint along the direction of motion of

the second joint; it does positive work on the second link (N)

external torque applied on the first link, (Nm)

external torque required to hold the first joint fixed, i.e., maintain 91 = O, (Nm)

external torque required for braking, i.e., causing exponential decay in the
velocity of the first link, (Nm)

control torque applied during rest-to-rest maneuver prior to braking, (Nm)

impulsive moment acting on the second link at its center of mass, (Nm)

total energy of the second link, (J)

13

EZT potential energy of the second link when (6I , (92) = (7r/2, 0), (J)
g acceleration due to gravity, (9.81 m/sz)
q}. constants, who values depend on kinematic and dynamics parameters of the
pendubot
S. sin 9.
I l
C. cos 6.
I I
Sl2 sin (31 + 62)
Cl2 cos (6] + 62)

2.3 System Dynamics

2.3.1 Equations of Motion

Consider the pendubot in Figure 2.1. We assume it to be an ideal system with no friction

in the joints. The kinetic energy of the two links of the pendubot can be written as follows

_1 -2 1 2-2
[(51—5561 +—?:mlai1 91 (2.1)
KE —11(é+9' 2+1 2 22)
2‘22 1 2) zmzvz ('

where

i52=—[léS +d (6' +9 )5 ]i+[le'c +d2(e' +e' )C ]] (2.3)

1 1 1 2 1 2 12 1 1 1 1 2 12
I O 2
v2 4161 +al2 (61+62) +211d261(61+62)C2 (2.4)

The potential energies of the two links are given by the relations

PE] = m1 gdlSl (2.5)

PE2 = ng(lls1 +d2512) (2.6)

The system Lagrangian [4] takes the form

14

L=(KE1 +1<152)—(PE1 +PE2)

and the equations of motion can then be written as

1[24_]_2L_ - .
dt 661 601
agate: : 0
dt 602 662
Putting the results in matrix form, we get

A(e)é+3(9,é)e+o(e)=r
where

A(6)=[q1+q2 +2‘13C2 ‘12 “739]
‘12 +‘13Cz ‘12

13(63):,135 ‘éz “(91 +92)

2
01 0
q4C1+q5C12
G(9)=g[ C
‘15 12

and qi, i = 1, 2, 3, 4, and 5, are constants, given below

q1=md2+m12+1

1 1 21 1
q2 = "1sz + I2
‘13 = ”’2’1d2
q4 = mla’1 + m2]1
%=%%

15

(2.7)

(2.8)

(2.9)

(2.10)

(2.11)

(2.12)

(2.13)

(2.14)

(2.15)

 

2.3.2 Force Interaction Between the Two Links

In this section we derive an expression for the force of interaction between the two links
of the pendubot and the work done by this force in changing the energy of the second
link. By applying the Newton—Euler method [4], the force of interaction between the two

links can be expressed as follows

i 2 .. .
— _ . . _ 2 1 I
Fx—m2 d2(61+62) +11(als2 91C2)+g812:| (2.16)
— '- u .- .. _ .2
Fy—m2_d2(61+62)+11(61C2 6152)+gC12] (2.17)

 

 

 

 

>
X
Figure 2.2. Forces of interaction between the two links

where the directions of F x and FJ) are shown in Figure 2.2. The resultants of F x and F y is

F R’ which can be decomposed into a workless constraint force along the length of the

first link and the component F that does work on the second link. The force F can be

expressed in terms of F x and Fy as follows

F=FxS2+FyC2=m2[llt91+d2(61+62)C2—d2(61+62) S2+gCl] (2.18)

16

The energy of the second link and its derivative can be expressed as follows

. . 2
- _l l 2
E2 _ K52 +1014:2 _ 2 12(61+62) + 2 mzvz +ng(1151 +d2S12) (2.19)
. 1' 1.. .. .. ‘ . . 2
E2 = m2 101 101+d2(61+62)62 —d2 (91 +62) 52 + gC1 (2.20)

It can be verified that E2 = F (1191). This is not surprising since (1191) is the velocity of

the point of application of the force F and have the same direction as that of F.
2.3.3 Holding Torque

During swing-up control, our algorithms will use the technique of holding the first link

fixed at certain times, i.e., maintain (91 = 0. We compute the torque required to hold the

first link fixed by substituting 91 = (91 = O in equation (2.10), as follows

q+qC - é . qC+qC r
i 2 3 ]62—[ 2]q35292+g[ 4 1C5 12 = h (2.21)
(12 0 ‘15 12 0
By eliminating 652 from the two equations in equation (2.21), Th can be expressed as
2' =— 592+ C—33—q—5CC (222)
h ‘13 2 2 g ‘14 (,2 2 12 '

2.3.4 Braking Torque
We consider braking action that results in exponential decay of the motion of the first

joint to zero. To this end, we assume

01 = —k1191, kl > 0 (2.23)

where k1 is a constant that will control the rate of decay of 6’1. To compute the torque

required for braking, we multiply equation (2.10) first with the inverse of the inertia

matrix to obtain 651 and 32 as a function of 61,91,62,92 and r

17

[:1]: [1(6) [3}- B(0,é)[:1]—G(6) (2.24)

2

— — C
A—1(e)=—1—— q2 q2 (13 2 (2.25)
det[A(0)] -q2 - q3C2 q1 + q2 + 2q3C2

Note that: det[A(6)] = qlq2 — q32C§

_ 2 2 2 2 2 2 2
—(11+m1d1 )(12 +m2d2)+m21112 +m211d252 >0

As the determinant is always positive, the angular accelerations of both links will always

remain bounded. Substituting the equations (2.13), (2.14) and (2.25) into (2.24) we get

[.671] 1 qzr + kl (2.26)

z 2 2 _
6)2 qlqz'q3C2 (q2+q3C2)”h2

 

where hl and h2 are given by the expressions
= (9+é)25+2é2CS+( CC — C) (227)
’71‘12‘131 2 2931223‘13‘15212‘12‘141 '

h =-(e‘ +9 )2( + 2C )5 —( + C) 9'25
2 12 ‘12‘13‘13226116132q312
_g [4345C2C12 —(q2 + q3C2)q4C1+q1q5C12j (2.28)

On substituting equation (2.23) in the first equation of equation (2.26), we get
_ _—_1_ ' _ 2 2
rb _ ‘12 [k191(q1q2 (13C2 )+ kl] (2.29)

When the first joint comes to rest, the braking torque becomes equal to the holding

torque. This can be easily verified from equations (2.22) and (2.29).

18

2.4 Energy Consideration of the Second Link

2.4.1 Effect of Sudden Stopping of the First link

A large value of gain k1 in the expression for braking torque in equation (2.29) will result

in sudden stopping of the first link. This action of suddenly stopping the first link has the
effect of application of an impulsive force and an impulsive moment on the second link,
as shown in Figure 2.3. The impulsive force results in a change in the linear momentum
of the second link and the impulsive moment results in a change in its angular
momentum. The change in the linear and angular momentum of the second link can be

expressed as follows

a _ _.+ _ *_
zmpAt — m2 (v2 2 ) (2.30)
—° — _. —o _ -+ — ._ ._
MimpAt — r2 x FAt — 1262 12(01 + 62 ) (2.31)

 

 

 

3r

Figure 2.3. Effect of sudden stopping the first link

19

 

where At is the interval of time over which the impulsive force and impulsive moment

act, {3; and 17; are the velocity vectors of the center of mass of the second link before

and after the action of the impulsive force, respectively, and 6"; and «92— are the angular

velocities of the second link before and after the action of the impulsive moment,

respectively. The angular velocity of the first link, before it is suddenly stopped is l

....

denoted by (9—. The expressions for v+ 17— and F can be written as follows

 

1 2’ 2 2
-.. ... - - a
v2 =d262 (—5121+C12)) (2.32)
v2 =—[119181+d2(61 +92)512]i+[1191C1+d2(91 +92)C12]j (2.33)
r2 = —d2 (C12i +512j) (2.34)

where fand fare unit vectors, as shown in Figure 2.3. By substituting the equations

(2.32), (2.33) and (2.34) into the equation (2.31), we get

mIdC

é+=e‘+ 1+——2—1——2—2— (9‘ (2.35)
2 2 I +m d2 1
2 2 2

Since there is no change in the potential energy of the second link over the At time
interval, the change in the total energy of the second link is due to the change in its

kinetic energy alone, and is equal to

2 2
_ +_ —__1_ 2 -+ _i -— ~—
AEz—KEZ KE2—2(12+m2d2)(62) 212(61H92)

..lm 12(9‘)2+d2(9‘+é‘)2+21dé‘(é"+é')C (236)
2 2 1 1 2 l 2 121 1 2 2 '

By substituting equation (2.35) into (2.36) and expanding the equation we can see that

20

.2 .2 mldC ,2 mIdC ,,
(9+) =(e') + 1+—21——3——22— (91“) +2 1+—ll—2—7§- 61—6,‘ (2.37)
12 + mzd2 12 + m2d2

. 2 . 2 . .
comparing the coefficients of (91—) , (t9; ) and 61—62— we can see

F -1
222.2

.mldc 2mIdC

)1+ 212 2 + 2122 __1_m,2_1md2

2
0 (9—) :l(1 +m d2

l 2 2 2

 

 

(I +m d2

.l

b

22
md(.

—mldC _l1 =lm12__2_;L_1
2122 22 2 211+md2

2 2 2
. (9‘)2-l(1 +m d2)-i1 —lm d2=0
2 '2 2 22 2 2 2 22
ldC
'-'-.i 2 ’"2122 _ _L ( 2 )_
- 6162. 2(12+m2d2) 2 1+———1 m d2 12 2m2 2d2+211d2C2 -0
2 2 2
1 2 ”'2‘122C22 - 2
:>AE =—ml ——-1 (0—) (2.38)

2 22

- 2 2 2 = . - -— _
Since m2d2 C2 < (12 +m2d2 ),AE2 S 0 and AE2 0 1f only 1f61 — 0. Clearly, the total

energy of the second link decreases whenever the first link is suddenly stopped.

2.4.2 Pumping Energy to the Second Link

It was verified in section 2.3.2 that E2 = F (1191). Therefore, if we select 1191 = F and

use a proper gain, we will guarantee the increase of E2. Expanding F and rearranging the

equation we get

491 = k2[1191+d2(61+62)62 -d2(91+62) 52 + gCl] (2.40)

substituting (2.27) into (2.40) and solving for the torque we get

21

 

22 -
qq-qC 6 . .2
r:12 32 —l—gC+d(6+9)S
12 232

 

— 1 1 +d( +h )C] (241)
2[1h1 2’1 2 2 '
1142'd2q3C2

. _ 2_ 2 _ 2 2
Note that. 1qu 212(13C2 —ll(12+m2d2) m211d2C2

2 2
21112 +mzlld2 —m211d2 21112 > 0

Therefore, the control law in (2.41) will have no singularities at any time, which allows

E2 to increase at all times.

2.4.3 Maintaining the Energy of the Second Link at the Same Level
At certain times we need to keep the energy of the second link fixed at a certain level.

Therefore, we want to find an expression for 2' that guarantees this to happen. By looking

to equation (2.21), we can see that E2 = 0 if:

0 (91 = 0 which is predictable since the system is assumed to have no friction

. 1161+d2(l91+62)C2—d2(61+62) 52+gC1=0

substituting (2.27) into the last equation and solving for the torque we get

 

 

2 2
611612-(136‘2 . . 2 1

I", _d C2 'gC1+d2(61+62) 52 "I _d C2l’1h1+d2(h1+h2)C2]
1‘12 2‘13 2 1‘12 2‘13 2

(2.42)

This expression is similar to (2.41) with only dropping the term (9.1/11:2 , which shows

interesting result. When 91 /k2 is positive E2 will increase and vise versa. The first link

angular velocity will go between the positive and negative numbers in order to do the

mission of keeping E2 at the same level.

22

 

2.4.4 Negative and Positive Work on the Second Link
If the second link has some amount of energy and the first link position is at lower
configuration, the second link will spin around itself. In order to move the first link
toward the top, the second link energy will be affected either negatively or positively. We
will discuss in this section two ways to bring the first link to the top configuration while
knowing what is happening to the second link energy.
1. Negative work:
As the second link is having energy that will make it spin, we will take the advantage of
the force interaction between the two links. Once this force is capable of producing a
moment on the first link greater than the moment that is produced from its weight and in
the opposite direction, then the first link is released. This means

I = 0 (2.43)
2. Positive work:
If the interaction force between the two links cannot produce a moment on the first link
greater than the moment that is produced from its weight and we forced the first link to
go in the other direction by a torque, we will produce a positive work on the second link.
Also, we do not want the first link to have a high angular velocity when we stop.

Therefore, we will drive the first link to the top with a constant low angular velocity

 

.. . . .. 6121' + hl
0 2 —k3 (t9 — t9 ) and we know from equation (2.27) that 6 =

l 1 Ides 1 2 .2
‘11"2 “13 C2
_ ;l_ ' _ ' _ 2 2
Z” T ‘ ‘12 [k3 (91 61des)(q1q2 ‘13C2 )+ ’71] (2'44)
where 91 des is the desired first link angular velocity.

23

2.4.5 Rest-to-Rest Maneuver of the First Link

Consider a maneuver in which the first joint starts from rest and is brought back to rest

through the application of a braking torque using a large gain k]. Taking into account the

loss of energy due to sudden stopping, given by equation (2.39), the net work done on the

second link due to the rest-to-rest maneuver can be computed as follows

2 2
md C . 2 . . 2
AE =[Fld6 +im 12 ill-1hr) szledz—im12(e‘) (2.45)
2112211+md21 112211
2 22

 

where F is given by the expression in equation (2.18). If we impose the constraint

d2(61+62)62—d2(61+62) 82+gcl 4411.91, k4>0 (2.46)

we get from (2.19), (2.45) and (2.46)

AE >11F6idt—lm12(e")2=ljm [(1+k )16]édz—lm12(6—)2
2-1 1 2 1 2 4

 

 

21 1 11 1 2 21 1
1+k 2 l+k 2 2
. 4 2 _t 2 -— ___ 4 2 -— _i 2 -—
_[ 2 ]szl [29161dt 2mzl1 (61 ) [ 2 ],,2211 (91 ) 2”“2’1 (91 )
zlk m 12(0‘)2>0 (2 47)
2 4 21 1 -

Using equation (2.11) it can be shown that the constraint in equation (2.46) can be

imposed by applying the torque

 

 

41612-61326}? . .2
r: gC—d (6+6)S
c kl +qu2 1 21 2 2
41% 232
+ 1 {d( +h)C -kl } (248)
k, M, C2 2’1 2 2 41"1 '
41‘12 2‘132

24

 

 

 

Figure 2.4. Swinging-up control of the first link

Clearly, the net energy of the second link will increase if the first joint is driven using the
torque expression in equation (2.48) and then stopped suddenly.
It is important to note that at any time during a rest-to-rest maneuver, while the first

link is still in motion, it is possible to compute:

(a) E2 from the values of 61,61,62, and 62 , and

(b) energy loss that would result from stopping the first link instantaneously, from
equation (2.39).

When the difference of the energy values in (a) and (b) is equal to E2], the motion of the

first link can be quickly stopped to have E2 z E27. .

2.5 Algorithms for Swing-Up Control
In this thesis we will use two successful algorithms for swing-up control of the pendubot
followed by asymptotic stabilization of the desired equilibrium. Both algorithms depend

on the idea of pumping energy to the second link until it reaches EZT while guaranteeing

25

 

that the first link is at the top configuration.

2.5.1 First Algorithm

A five-step algorithm is proposed as follows

1. Initialization:

Linearize the dynamic equations of the pendubot in equation (2.11) about the

desired equilibrium (6161,6262) = (7r/ 2 ,0, 0,0). The linear system will be
f(x,u) = 5c = Ax + Bu (2.49)

where x is the state space representation defined as

x =6 x =6 x =6 x =6 r=u (2.50)

Therefore, the matrices A and B can be found at the top position as

 

' 0 1 0 0‘ — o -
6x 6x ——
A: 1 3 ,3: a“ (2.51)
o 0 0 1 0
?f_4 0 9:4 0 _‘9f_4
6x1 6x3 _au]

 

 

 

Using the model of the linearized system, design a linear controller to render the
desired equilibrium of the pendubot (nonlinear system) locally asymptotically

stable. The system is controllable as the controllability matrix C is full rank
C = [B AB A28 A313] (2.52)

Since this equilibrium point is not stable, the matrix A is expected to have

eigenvalues in the right hand side (RHS) plane. Thus, to stabilize the system

26

using a linear controller, new eigenvalues are assigned to the controlled system

(A-BKI) in the left hand side (LHS) plane. Then, the desired feedback gain matrix

K is calculated to find the control law
1 = —K1(x—xeqj (2.53)

This control law is applied to the nonlinear system and stabilizes it at top position.
2. Pumping energy into the second link:
Drive the first link from its initial configuration using the control law in equation (2.41)

until the energy E2 reaches the level E27.

3. Maintain the energy of the second link:

When E2 reaches EZT in the last step, the value of 61 could be high. Therefore, if we

suddenly stop the first link using the braking torque in equation (2.30), E2 will decrease

very much as we can see in equation (2.39). Hence, in this step we will use the control

law in equation (2.42) to maintain E2 at the level of E2]. while waiting 61 to reach a

reasonably small value and then brake. The first link angular velocity will definitely go

through zero as the second link is spinning and the first link is trying to maintain E2 on its

level by going back and forth.
4. Swing-up control of the first link:

In this step we will bring the first link to the top configuration and main E2 at the level of
E2]. The problem with the expression in equation (2.42) is that it does not specify a clear

path for the first link. Therefore, we cannot predict how the first link can reach the top

27

position. Moreover, even if the first link reaches the desired position, it usually reaches
there with an angular velocity that makes the second link loose a high amount of energy
if we suddenly stop the first link. Therefore, in this step we will use a series of positive
and negative work on the second link to bring the first link to the top configuration and

make E2 fluctuate around E27. While the first link is in its way to the top, the level of E2
is tested regularly. Hence, if E2 is at a level higher than E2T and the second link can push

the first link to the top by itself, the control law will use the negative work approach
described in equation (2.43). Otherwise, the control law will stay using the holding torque

described in equation (2.23) waiting to the previous condition to be met. Likewise if E2 is

at a level lower than EZT and the second link cannot push the first link to the top by itself,

the control law will use the positive work approach described in equation (2.44).
Otherwise, the control law will stay using the holding torque described in equation (2.23)
waiting to the previous condition to be met. However, the control law can switch to
equation (2.30) when the conditions of the equations (2.43) and (2.44) are not met during

this process. For example, if the control law in equation (2.43) is operating the system

and the first link angular velocity reached the maximum value allowed (i.e.61des ), then

the control law will switch to the one in equation (2.30) in order to avoid losing too much

energy in E2. Finally, when the first link reaches within a small angle near the top

configuration, the braking torque in equation (2.30) is applied. This will stop the first link
near the desired position and make the second link swing back and forth like a pendulum

near to the upright configuration.

28

5. Stabilization:

By the end of the last step, the first link will be fixed within a small angle of the upright

configuration and the second link energy is almost equal to E27. This will make the

second link behave like a pendulum that swing and can reach the upright configuration in

a finite time. Once the pendubot configuration enters the region of attraction of the

desired equilibrium point (61,61,6262 ) = (7r/ 2,0, 0,0) , the linear controller in equation

(2.53) will be invoked to stabilize the equilibrium point.
Numerical Simulations of the First Algorithm

The kinematic and dynamic parameters of the pendubot were assumed to be

l.=1.0m, d.=0.5 m, m.=1.0kg, l.=im.l.2=0.0833 kgmz, i=l,2
z I I 1 12 l t
For this choice of parameters, EZT was evaluated to be 14.715 J.

Table 2.1 Comparison of simulation results of braking with analytical results

 

 

 

 

 

 

 

 

 

 

 

kl At (sec) A61 (rad) 6; (rad/s) difference (%) AE2 (J) difference (%)
10 0.800 0.200 5.890 26.38 0.724 44.76
100 0.080 0.020 7.330 8.37 0.478 4.30
1000 0.008 0.002 7.899 1.26 0.495 0.92

 

 

Impulse-momentum model of braking: In section 2.3.1 we modeled the sudden
stopping of the first link by the action of an impulsive force and an impulsive moment on
the second link. Here we show that this modeling assumption is accurate for large values
of gain k1 in the expression for the braking torque in equation (2.30), which we know will

cause a sudden stopping of the first link. We consider the pendubot configuration

(6161,6262)=(0.0,2.0,0.0,3.0), where the units are rad and rad/s. Ifthe first joint is

29

stopped instantaneously, the velocity of the second joint and change in energy of the

second link can be computed using equations (2.36) and (2.39), respectively. Specifically,

using 61— = 2.0 rad/s and 62— = 3.0 rad/s, these values can be computed as

a"; = 8.0 rad/s, AE2 = —0.5 J (2.54)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

‘ l i E/ 1

0 5 10 15 20 25 30 35
time (see)

Figure 2.5. Plot of joint angles, joint angle velocities, control torque, and energy of the

second link

./ . .

 

The values of 63' and AEZ’ obtained from simulations, are tabulated above in Table 2.1

for different values of gain k1 used in the expression for the braking torque in equation

30

(2.30). It is clear that the differences of these values from those in equation (2.54) are

negligible for large values of gain kl' Also, as expected, large values of kl require less

time for the first link to come to rest and small angle of travel of the first link before it
comes to rest. In our simulation of swing-up control, presented next, we used a moderate

value ofk1 = 100.

Swing-up control and stabilization: For this simulation we started with initial
conditions which represents the lowest potential energy, i.e. the initial conditions were

chosen to be

(6161,6262 ) = (—72'/2,0.0, 0.0, 0.0) (2.50)

where the units are rad and rad/s. It can be seen from Figure 2.5 that E2 reaches the value

E att=tS

2T = 5.65 sec. Therefore, the control law switched from equation (2.41) to

l

(2.42), which happened in this simulation to be for a very small time. At that time, since

the second link has energy equal to EZT and the first link at a position far from the top,

the second link started to spin fast around itself. This was expected since the potential
energy of the second link is low and most of the link energy is expressed by the kinetic
energy. Then, the control law used the procedure explained in the forth step of the
algorithm (i.e. a series of positive and negative work on the second link) until the first

link is brought near to the upright position. Finally, the control law switched to the linear

controller at t = t5 = 27.4 sec when the pendubot configuration reaches the region of

2

attraction of this controller at the top equilibrium point. This approach showed very good

results. However, the swing-up time is relatively long because of the way we use to bring

31

the first link to the top. Thus, in the next approach we will bring the first link to the top

first and then pump energy to the second link.

2.5.2 Second Algorithm
A four-step algorithm is proposed as follows
1. Initialization:
o The same model of the linear controller used in the first approach will be used

again in the second approach.

0 Choose a small angle a, a > 0, such that (6 ,6 ,6 ,6 )=(6 ,0, 0,0) lies in the
l 1 2 2 1
1’ 2’ 2

all values of 61 satisfying (Jr/2 —a) S 61 S (7r/2 + a).

region of attraction of the desired equilibrium (61,6 6 6 )=(7r/ 2,0, 0,0) for

2. Swing-up control of the first link:

Drive the first link from its initial configuration to any configuration that satisfies

(7r/2—Ct)S61 S(7r/2+a),61 =0, as shown in Figure 2.4.

3. Swing-up control of the second link:

If the configuration of the first link satisfies (7r/2—a) S 61 S (Jr/2+ar),6l = 0, the

second link will automatically swing up to the configuration (62,62)=(0,0) if

E2 = E2]. . To increase the energy of the second link to E2], we will use a series of rest-
to-rest maneuvers of the first link, described above. Additionally, to ensure that 6. will

always lie within the region(7r/2—a) S 6] S (7r/2+a) , the following procedures will be

used:

To hold the first link fixed, the holding torque T h in equation (2.23) will be applied. To

move 61 in the positive (counter-clockwise) direction from rest, the torque expression in

32

equation (2.48) will be used provided it is greater than T h at the initial time. To move 61

in the negative (clockwise) direction from rest, the torque expression in equation (2.48)

will be used provided it is less than Th at the initial time. As 61 approaches the boundary

of the interval [(7t/ 2 — a),(7r/ 2 + 05)], the braking torque T b in equation (2.30) will be
used; a large value of k1 will be used to quickly stop the motion of the first link. Conduct

the rest—to-rest maneuvers until E2 = E27,. This can be accomplished by monitoring the

states of the pendubot as we discussed earlier in section 2.4.5.

In order to swing-up the second link, TC has to be greater than 2' h for some values of

62,62, and lesser than Th for other values of 6 ,62, when (7r/2—a)S6 S(7r/2+a)

l

and 61 = 0. We now show that these conditions will indeed be satisfied. For 61 = 0 , the

difference in the torques can be shown to be equal to

 

 

 

 

 

 

 

 

. _ _ _ _ _ .2
H(61,62,62)—(rc Th)9'1=o—g [C1 46C2C12l d25292 (2'51)
where
m d2
= 2 2 1
‘16 " 2 <
mzd2 +12
(a) ' ' (b)
Figure 2.6. Vertically down configurations of the second link during rest-to-rest
maneuver

33

Now consider the joint configuration (61,62) = (7r / 2 — 6, —7r + 6), 6 e (0,a] , shown in

Figure 2.6(a), where the second link is vertically down. During swing-up, the second link

has to pass through this configuration. Since 52 = —sin 6 < 0, we have from Eq.(2.51)

n 2 glcl "1662C12l
A plot of the right-hand side of the above equation, shown in Figure 2.7, indicates II > 0

for 6 ,6 = 72' / 2— 6, -7r +6 ,6 6 0,6: for feasible values of a and q . This implies
1 2 6

that the control torque Tc can move the first link in the counter-clockwise direction. A

similar analysis indicates II < 0 for the configuration (61,62)=(7r/2-6,—7r+6),

6 e (0,a] , shown in Figure 2.6(b). This configuration is therefore conducive for the first

link to move in the clockwise direction.

 

 

 

 

-160 1 1 1 1 [1 1
- *, _86=0.2‘/ I _-
. q6=0,4 /
-17()_ II;>O- . "$540.6
0
'3-180— —
N
a -— —t
q6=0.7 1
-190/ .. II<O . .3
_. ,,‘16=,0~5. , .., .
/ . (16 "-" 0.3 (16": 0.1
_200 1 1 / 1 1 1 1 1
70 80 90 100 110

91 (deg)

Figure 2.7. Plots showing regions in the 61-62 space where II > 0 and II < 0

34

 

4. Stabilization:

With (7r/2—a) S 61 S (7r/2+a),61 = 0, and E2 = E2]. , the second link will behave like

a pendulum and will reach the vertically upright configuration in finite time.
Concurrently, the configuration of the system will reach the region of attraction of the

desired equilibrium point. Invoke the linear controller, as it was done in the first

approach, to stabilize the desired equilibrium when (6 6 )z (0, 0).

 

 

 

 

 

 

 

 

 

 

 

 

     

 

 

 

 

 

350

 

2’ 2
A 120 'l 1 1 1I | 11 ' I
00 ‘l 1 1 1 I 11 '
8 1 1 1 1 11 '
v 90W
H I |
" 1 1 '
a 60 :1 : E 51 E i: : .
" 1 1 1 1 11 i
a 130 ;; : : 1' : II 7 ‘
go 0 _/.,:: : fl : 11.4
A w
3 M : : .: 1
.5 130 4': 1 1 11 1 11 i 1
' 1 rest-to-rest i I I i ll
: maneuvers 1 1 1 1 _.1 1.__ hqld first
'HJT—Hn [S 1 1 1 1 jomt fixed
a. 0 1? : : " : :1 : '
d.) 1 1 1 1 11 :
3 -200 : : : . :: 1 -
N 1 | 1 1 1 '
Q A/ r- 1- + 1- —1
.400 I I 1
IT I 1 1
i I 1
l l
l |
I l I
| l I

 

 

 

62 (deg/s)
O

 

-350 “ ' l 1 . . .
i swing-u "—— stabilizatlon —»——
fi contro

 

 

 

 

 

 

 

T (Nm)
p—s
\O
O O
l

 

 

 

 

 

 

'190 l 1m 9 I | 1
18 i. 1 . .1 1 11 I I
612E211==::aas
6 1 1 m1 1 14 L 1
0 2 time(sec) 4 6

Figure 2.8. Plot of joint angles, joint angle velocities, control torque, and energy of the
second link

35

Numerical Simulation of the Second Algorithm

As part of the initialization (the first step), a linear controller is designed to stabilize the
desired equilibrium. Through repeated simulation of the closed-loop system behavior, a

is estimated to be 10 deg. The initial configuration of the pendubot is chosen as

(191,611,19262) = (90.0,0.0,—135.0,0.0) (2.52)

where the units are deg and deg/s. This choice of the initial configuration eliminates the
need for the first step of the algorithm, which is trivial. The simulation results for the
second and third steps of the algorithm are shown in Figure 2.6; the plots show the two

joint angles, their velocities, the control torque, and the energy of the second link.
It can be seen from Figure 2.6 that E2 reaches the value of EZT at t = ts = 3.15 sec.
Since the second link is close to its vertically upright configuration at this time, the linear

controller is invoked immediately for stabilization. The swing—up control of the second

link is achieved over the interval te [0,3.15]sec through a series of rest-to-rest

maneuvers separated by periods of time over which the first joint is held fixed. It can be

seen from Figure 2.6 that E2 increases for each rest-to-rest maneuver but remains
constant during times when the first link is held fixed. The increase of Ezduring each

rest-to-rest maneuver is achieved through positive work done by the first link followed by
energy loss due to braking. During braking, the control torque peaks. The large

magnitude of these peaks, which can be attributed to the large value of gain k1, is not of

concern since they are applied intermittently and for short time intervals.

36

 

110 I: E : I :I I I

 

61 (deg)

 

 

 

Y L
‘ 1
1 1 1
1
00 H , , .. ,, T .
I 1 1 :
I l '
1 1 1

 

 

 

61 (deg/s)

 

-200 1 E E i 1 l1 1 1
-—>5 114—5— first joint held fixed
——>§ i4— rest-to—rest maneuver

O '1 I I I .I I I
1 1 I 1
1 I 1 1
I 1 1

 

 

92 (deg)

 

 

 

-400
650

 

 

 

 

 

 

62 (deg/s)

_650 1 l l l 1 :1 1 1
swing-u >;<——— stabilization

Icolntr'o 1/t=ts

 

 

--II---
II

 

    

 

13 (Nm)

 

 

.1.0 1: E E 1 l1 4 1
(13 i i i

 

 

132 (J)

 

 

0.1

 

 

4 6

time (see)

Figure 2.9. Plot of joint angles, joint angle velocities, control torque, and energy of the
second link

Comparison with experimental results: It behooves us to point out that the maximum
torque required by our algorithm, by virtue of being impulsive in nature, will be limited
by the peak torque of the motor and not the maximum continuous torque. The peak
torque of a motor is greater than the maximum continuous torque by a factor that varies
from motor to motor. A search of the literature published by motor manufacturers

indicates that this factor can vary in the range of 2-10 and is equal to 4 for a specific

37

example worked out in the Handbook of Electric Motors [28]. To check the feasibility of
hardware implementation of our algorithm, we now compare our simulation results with
experimental results published in the literature. We present simulation results for two sets
of kinematic and dynamic parameters of the pendubot. The first set of parameters, taken
from Orlov, et al. [9], are presented below:

m] = 0.132 kg, ’1 = 0.203 m, d1 = 0.1574 m, II = 0.00362 11ng

m2 = 0.088 kg, 12 = 0.254 m, d2 = 0.1109 m, I2 = 0.00114 kgm:2

For this choice of parameters, EZT was evaluated to be 0.2710 J. For initial configuration

of the pendubot given by equation (2.52), the simulation results are shown in Figure 2.9.

It can be seen from this Figure that E2 reaches the value of E2 T at t = tS = 2.97 sec. At this

time the second link is close to its vertically upright configuration and the linear
controller is invoked for stabilization. The swing-up control of the second link is
achieved over the interval ta [0, 2.97] sec through a series of rest-to-rest maneuvers
separated by periods of time over which the first joint is held fixed. It can be seen from

Figure 2.9 that E2 increases for each rest-to-rest maneuver but remains constant during
times when the first link is held fixed. The increase of E2 during each rest-to-rest

maneuver is achieved through positive work done by the first link followed by energy
loss due to braking. During braking, the control torques peak, but the peak torques act
over short intervals of time. This is expected since the braking torques are impulsive in

nature due to the choice of a large value of gain k1. Despite its impulsive nature, the

maximum value of the torque required by our algorithm is 1.0 Nm - this is less than the

maximum torque of 1.7 Nm required by the algorithm proposed by Orlov, et al. [9].

38

For the second set of parameters we choose the following kinematic and dynamic
parameters of the pendubot:
m1 = 1.0367 kg, 11 = 0.1508 m, d1 = 0.1206 m, 11 = 0.0031 kgm2
m2 = 0.5549 kg, 12 = 0.2667 m, d2 = 0.1135 m, 2 = 0.0035 kgm2
These parameters result in the following values of q}, j e { 1, 5}:
q1 = 0.0308 kgmz, q2 = 0.0106 kgmz, q3 = 0.0095 kgmz,

q4 = 0.2087 kgm, q5 = 0.0629 kgm

which are almost identical to the values of 6j( j = qj), j 6 {1,5} , in the paper by Orlov,

et al. [24] with experimental results for the control torque.

For these parameters E2]. was computed as 1.438 J. For the initial configuration of

the pendubot given by equation (2.52), the simulation results are shown in Figure 2.10. It
can be seen from this figure that swing up and stabilization is achieved in less than 4 sec.
The maximum torque required by our algorithm is approximately 15 Nm. In the
experimental work by Orlov, et al. [24], the maximum torque required was 7 Nm for the
nominal model and 30 Nm for the model with disturbances. It suffices to say that the
motor used in the experimental work [24] can be used for implementation of our
algorithm. A comparison of our results with the results for the nominal model alone
indicates that our maximum torque is greater than the maximum torque in [24] by a factor
of 2.1. This is acceptable since the torque required by our algorithm is intermittent and its
maximum value is limited by the peak torque of the motor, whereas the maximum torque
in [24] is limited by the maximum continuous torque (which is much less than the peak

torque, [28]) due to its high-frequency components.

39

 

115 :lz': l :l l I

 

91 (deg)

 

 

 

320

 

 

 

 

6 1 (deg/S)

-320 i i 1 1 ¢ 1
——>§ <§—— first joint held fixed
—+—>§ E4—— rest-to-rest maneuver

 

 

92 (deg)

 

 

-400
900

 

 

 

 

 

02 (deg/s)

 

 

 

-900 4 1 ' ' l
swing-u >4—————— stabilization Ar
15 rigoyuo . Pr’tzts

 

 

 

I T

     

 

 

 

13 (Nm)

 

-15 . é: , .
1.5 . 1.

 

 

 

52 (J)

 

 

0.3

 

 

time (sec) 6

Figure 2.10. Plot of joint angles, joint angle velocities, co'ntrol torque, and energy of the
second link

2.6 Remarks

In this chapter we showed two new successful algorithms to swing-up the pendubot. Both
algorithms benefit from using the force of interaction between the two links. However,
the second one had much better settling time and used more convenient ways in
swinging-up the pendubot because it was based on taking the first link to its vertically

upright position and executing a series of rest-to-rest maneuvers about this position with

40

a small amplitude of oscillation. Using the principles of work-energy and impulse and
momentum, the rest-to-rest maneuvers were designed to increase the energy of the
second link. The rest-to-rest maneuvers were carried out until the energy of the second
link equals its maximum potential energy. This results in the second link swinging up to
its vertically upright position and the pendubot reaching a configuration from which the
desired equilibrium can be stabilized using a linear controller. Simulation results were
presented to demonstrate the feasibility of both proposed algorithms. In the next chapters
we extend our work in the second algorithm to swing-up control of the acrobot and

control of other under-actuated robotic systems.

41

CHAPTER 3

The Acrobot

3.1 System Dynamics

In this section we will discuss the equations of motion of the acrobot, which are not far
from that in the pendubot. In addition, we will derive the holding and breaking torque,
which will be applied on the second link in a way analogous to that done for the pendubot
in chapter 2. Moreover, we will derive an expression for the torque, which makes the
system have opposite angular velocities. Then, we are going to derive the impulsive
torque that can match the angular velocity of the two links in magnitude but opposite in
direction and show how it can match the last torque at a certain time. Finally, an
expression for torque, which makes the second link move in a constant angular velocity,

is going to be derived.

3.1.1 Equations of Motion
Consider the acrobot in Figure 3.1. Assuming an ideal system with no friction in the
joints, the equations of motion is going to be similar to that in the pendubot with the

exception of changing the torque equation (2.12) to be

T-T (31)
_0 ,

42

 

 

 

Figure 3.1. The acrobot

3.1.2 Holding Torque

We compute the torque required to hold the second link fixed, i.e., maintain (92 = 0. By

substituting (92 = 512 = 0 in the modified equation (2.11), we get

q1+q2+2q3C2 .. 0 . €14C’l+615C12 _ 0
+ C 61- 6., q35261+g C - r (3.2)
q2 (13 2 1 ‘15 12 h

By eliminating é] from the two equations in equation (3.2), 2' h can be expressed as

 

.2 g 1 W
2' =qS¢9 + [(q +qC )qC —(q +qC )q :l (3.3)
h 3 21 1 3 2 512 2 3 2 41
q1+qz+2q3C2

The holding torque in equation (3.3) will be used in one of the algorithms for swing-up

control.

3.1.3 Braking Torque

We consider braking action that results in exponential decay of 92 to zero. Therefore,

we assume

62 = —k56’2, k5 > 0 (3.4)

43

where k5 is a constant that will control the rate of decay of 612. To compute the torque

required for braking, we use the same procedure done in chapter 2. Therefore, 51 and 52

will become as a function of 6’ 61 6 0 and r

1’ 1’ 2’ 2
g] _ 1 _(q2+q3C2)T+h1
é2

— 2 2 (3.5)
‘1qu ‘613 C2 (ql + q2 + 2q3C2)r + h2

 

 

where h1 and h2 are the same expressions given in chapter 2. On substituting equation

(3.4) in the second equation of equation (3.5), we get

,_ —1

2 2 -
_ — C k6 +h (3.6)
b {(4142 ‘13 215 2 21
q1+q2+2q3C2

When the second joint comes to rest, the braking torque becomes equal to the holding

torque. This can be verified from equations (3.3) and (3.6).
3.1.4 Opposite Angular Velocities Torque

We compute the torque required to make the angular velocity of the second link stay at

the same magnitude of the angular velocity of the first link but in the opposite direction

(i.e. 611 = —612 , 511 = —é2 ).By substituting these conditions in the modified equation (2.11),

we get

q +q C .. 6’ . q C +q C 0
[ 1 (:1 161+1911435291+gl 4 1C5 12]=[ 1 (3.7)
‘13 2 1 ‘15 12 7

By eliminating 9.1 from the two equations in equation (3.7), 2' can be expressed as

_ 1 -2 ~ _
’- q1+q3C2 [q1q35291 +gl‘11‘15C12 q3q4C1C211 (3'8)

The torque in equation (3.8) will be used in one of the algorithms for swing-up control.

44

3.1.5 Impulsive Angular Velocity Matching Torque

We consider the action that result in exponential matching of 92 to (—611). Therefore, we

assume

91+92=—k6(191+62), k6>0 (3.9)

where k6 is a constant that will control the rate of matching of 62' To compute the torque

required for this action, we have from equation (3.5) 9.1 and 92 as a function of

61,91,492,612 and 1. Therefore, on substituting equation (3.9) in both equations of
equation (3.5), we get

_ —1 - - _ 2 2
t—ql+q3C2{k6(61+62)(qlq2 q3C2)+hl+h2} (3.10)

When the second link angular velocity is equal to (491), the torque in equation (3.10)

becomes equal to the torque in equation (3.8). This can be verified by substituting

((91 = —92) in equation (3.10) and comparing the result with equation (3.8).

3.1.6 Impulsive Constant Angular Velocity Torque

We consider the action that result in exponential matching of 612 to (612 des ). Therefore,

we assume

62=—k7(6?2—62des), k7>0 (3.11)

where k7 is a constant that will control the rate of matching of (92. To compute the torque

required for this action, we have again from equation (3.5) 51 and 62 as a function of

61,é1,62,6’2 and 1. Therefore, on substituting equation (3.9) in the second equation of

equation (3.5), we get

_ _1 - _- _ 2 2
T ‘ q] ”,2 +2q3C2 {1‘7 (‘92 62des)(qlq2 ‘13C2 1+h21 (3'12)

 

45

When the second link angular velocity reaches the desired one (62de51’ it will stay at the

same magnitude using the same torque expression.

3.2 Impulse-Momentum Effect

In this section the effect of the sudden impulsive force and moment on the angular
velocities of the Acrobot’s links is going to be studied. By looking to the free body
diagram of the second link in Figure 3.2, the impulsive force and moment on the center of

mass can be written as

- _ *+_g_
inmpAt—m2(v2 v2) (3.13)

- _ .+ .+_ ._ ._

MzimpA"12(61 +62) 12(61 +62) (3.14)

F . . . a . . . a
~+__ + + + - + + + -
v2 _ _119151+d2(61 +62)S12]z+[1161C1+d2(61 +92)C12]) (3.15)

 

v2 9.1191 S1 +d2(01 +62)S12]i+[1161C1+d2(61 +192 )C12]j (3.16)
In order to study the effect of the impulsive force and moment on the first link
angular velocity, we will discuss the effect on the second joint first. The impulsive force

will stay with the same magnitude and direction. However, there will be impulsive

moment added due to the distance

’2 :‘lzlclzi +5121.1 (3'17)

Therefore, the addition on the impulsive moment will be

_. —o — 2 .+ 0+ — o_ -_ .+ .—
r2xF2impAt—m2 {d2 [(61 +92) (91 +92)]+d211C2(91 +01 )} (3.18)

Now, if we look to the free body diagram of the first link in Figure 3.2, the impulsive

force and the impulsive moment on the center of mass can be written as

46

- __ 3 _- = ,+_,_
FimpAt—( limp inmij’ m1(vl v1) (3.19)
M. At=1(6l+-6l") (3.20)
Imp
where

{31+ = c1161+ (—Sli'+clj'), 171— : dlél- (—51t+C1]) (3.21)

and Flimp is the impulsive reaction force at the first joint which can be found from

equation (3.19) to be

- __ .+_._ .+ .+ _ ._ ._
F11mpA" {”7211151191 611+d25121191 +621 (61 +6211]

.+_._ 7 .+_._ .+ .+_ ._ ._
+m1d1(6’l 191 )Sl}z+{m2[llCl(61 61 )+d2C12{(01 +62) (6] +62 111

+m d (191+-t9'1—)C 1] (3.22)

11 1

 

 

Figure 3.2. The impulsive force and moment on the second link and their effect on the
first link

Now, we want to calculate the impulsive moment on the first link center of mass due to

the impulsive forces applied to its joints. The second joint has an impulsive force with the

47

same magnitude and the opposite direction from that in the second joint in the second
link free body diagram. This impulsive force will create an impulsive moment at the first

link center of mass due to the distance

73 ‘1’1‘d111C1;+51]1 (3'23)

Therefore, the addition on the impulsive moment from this impulsive force will be

73 X 1": 21mm) = ”121—111—6111111611+ “91—1-11““llldizc’lelé1++ 6; Hal— +6511}
(3.24)

In addition, the impulsive reaction force will add another impulsive moment at the first

link center of mass due to the distance

71 = —d1 (C1? + 51]) (3.25)

Hence, the impulsive moment added due to this reaction force is

.. -° __ 2 -+_°— _ '+_'— "1' '1' — .— .—
’1"F1impA" “1‘11 (61 ‘91) ’"2‘111'11‘91 91 1+d2C21191 +62) 16‘ +6211}

(3.26)

Therefore, the impulsive moments at the first link center of mass due to the impulsive

forces can be written as

MlimpAt = _MZimpAt — r2 x F2impAt + r3 x (-F2impAt) + r x F . At (3.27)
equating equation (3.27) to equation (3.20) and substituting the equations (3.14), (3.18),
(3.24) and (3.26) we can get a formula that can specify the relation between the angular

velocities of the links before and after the impulsive force and moment applied to the

second link. This can be shown as the following

2 2 2 '+_ '— _
[11+m1d1 +12+m2(d2 +11 +2d211C2)](61 191 )—

[12 + m2 (.122 +d211C2 ”(19'2— 43“) (3.28)

48

Now, we will consider three cases in which we specify some of the angular velocities and
solve for the angular velocity of the first link after the application of the impulsive force
and moment.
1. First case:

The two links are having arbitrary angular velocities and then the impulsive force and

moment will suddenly stop the second link (i.e. 65" = 0 ). Therefore, if we substitute this

condition into the equation (3.28) and solve for the new angular velocity of the first link

we will get

2
I +m (d +dIC) _
2 2 2 212 9— (3.29)

2 2 2 2
11+m1d1 +12+m2(d2 +1l +2d211C2)

.+_._
191—61+

 

2. Second case:
The first link has an arbitrary angular velocity while the second link is held fixed. The
impulsive force and moment then will make the angular velocity of the second link match

the magnitude of the angular velocity of the first link but in the opposite direction (i.e.

05' = 0 , 6; = —t91+). Hence, if we substitute that in equation (3.28) and solve for the new

angular velocity of the first link we will get
2 2 2
11+ mldl + 12 + m2 (d2 +l1 + 2d211C2) 9‘
1

2 2
11 +m1d1 +m2 (11 +d211C2)

61+:

1 (3.30)

 

3. Third case:

Both links are at a stationary configuration and then the impulsive force and moment are

applied to achieve a desired second link angular velocity (i.e. 611- = (9; = 0). Thus, if we

49

substitute that in equation (3.28) and solve for the new angular velocity of the first link

we will get

1 +m (d2+dlC) .
2 2 2 212 0+ (331)

2 2 2 2
Il+mldl +12+m2(d2 +11 +2d211C2)

 

.+__
61—

3.3 Energy Consideration of the System

3.3.1 Energy Change Due to the Impulse-Momentum Approach

Since there is no change in the potential energy of the system over the At time interval,
the change in the total energy of the system is due to the change in its kinetic energy

alone, and is equal to

AE = AE1+ AE2 = AKE1+ AKE2 (3.32)
2 2
0K -1 2 '+ _ '-
AKE =11 (é++é+)2+lm 12(é+)2+d2(19++é+)2+21d9+(0'++19'+)C
222122211212121122

-.1—1 (6-+a-)2+lm 12(6-)2+22(9-+9-)2+21d6-(6-+6-)c (334)
2 2 1 2 2 2 1 1 2 1 2 121 l 2 2 '

Now, we want to find the energy change in each case we introduce in the last section.

1. First case:

If we induce the condition 19; = O and substitute equation (3.29) into equations (3.33)

and (3.34) and solve the equation (3.32) we will get
2 2 2 2 2
AE = _(11 + mldl )(12 + m2d2 )+ m2]1 (12 + ”7261252)

2 2 2
2(11 +m1dl + 12 +m2d2 +m211 + 2m211d2C2)

 

(9;)2 (3.35)

which is always negative and indicates an energy loss.

50

2. Second case:

If we induce the conditionség =0, 6; 2491+ and substitute equation (3.30) into

equations (3.33) and (3.34) and solve equation (3.32), we will get

2 2 2
I+md +1 +m (d +1 +21dC ))
1 11 2 2 2 1 12 2 [(1 +md2)(1 +m d2)

2 2 2 l l 1 2 2 2
2(Il+m1dl +m211 +m211d2C2)

,Ej

 

2
2 2 2 -—
“"211 (12+m2d252)](61 ) (3.36)

which is always positive and indicates an energy increase.

3. Third case:

If we induce the conditions (91— : 612— : 0 and substitute equation (3.31) into equations

(3.33) and (3.34) and solve equation (3.32) we will get

2 2 2 22
(I+md)(1+md)+ml(1+mdS). 2
215:1 11 2 22 21 2 222(6+) (3.37)

2 2 2 2
2(11 + mldl + 12 + m2d2 + mzll + 2m211d2C2)

 

which is always positive and indicates an energy increase.
3.3.2 Rest-to-Rest Maneuver of the Second Link

Consider a maneuver in which the second joint starts from rest and is brought back to rest

through the application of a braking torque using a large gain k5. Taking into account the

loss of energy due to sudden stopping, given by equation (3.35), the net work done on the

second link due to the rest-to-rest maneuver can be computed as follows

' (11+m1d12)(12 +m261§)+m2112 (12 +m2d2283) ' 2
AE = jEdt— )(63) (3.38)

2 2 2
2(11 + mldl + 12 + "12.12 + mzll + 2m211d2C2

 

51

From the energy equations and the equations of motion the rate change of the energy of

the system is equal to

E1+ E2 = E = r62 (3.39)
Therefore, if we choose I = k236i2 such that
. .. . . k8 ._ 2
jEdt .—. ”892612.11: [k862d62 = 7(62 ) (3.40)

Thus, substituting equation (3.40) into equation (3.38) and selecting k8/2 as the upper

bound of the coefficient appeared in the second term of equation (3.38), the total energy
change of the system will be positive for each rest-to-rest maneuver. Therefore, the

constant k8 will be selected as the following:

2 2 2
k > (12+m2d2 )(11+m1d1 +m211)

8‘ 2 2 2 )
11+12+m1dl +m2(l1 +d2 +211d2

 

(3.41)

From the second equation of equation (3.5) and the condition 1' = A7892 the torque

expression can be found as

k h
r = 3 2 (3.42)

_ 2 2-
‘11"2 q3C k81q1+q2+2q3C21

 

In order not to have singularity in equation (3.42)

2 2 2 2 2 2 2)
¢ q1q2_q3C2 _(11+m1d1)(12+m2d2)+m211 (12+m2d282

8 C ’ 2 2 2
q +9 +29 ( )
1 2 3 2 Il+mld1 +12+m2d2 +m211 +2m211d2C2

(3.43)

The upper bound of equation (3.43) is the same fraction as in (3.41). Therefore, k8 should

be strictly greater than that fraction to guarantee the increase of the system energy and

not having any singularity at any time. Thus,

52

(I +m d2)(11+md2+m12)

 

k8> 2 222 211 2 21 (3.44)
11+12+mld1 +m2(ll +d2 +211d2)

Clearly, the net energy of the second link will increase if the second joint is driven using
the torque expression in equation (3.42) and then stopped suddenly.

3.4 Algorithms for Swing-Up Control

In this thesis three successful algorithms is proposed for swing-up control of the acrobot
followed by asymptotic stabilization of the desired equilibrium. Each algorithm will use a

series of a different impulsive-like torque to add energy to the system until it reaches E T
while ensuring that the second link is restricted to be within small angle a from 62 = 0.

3.4.1 First algorithm
A three-step algorithm is proposed as follows
1. Initialization:

o Linearize the dynamic equations of the acrobot about the desired equilibrium

(191,611,62,612 ) = (at/2,0, 0,0). The linear system will be

f(x,u)= x = Ax + Bu (3.45)

where x is the state space representation defined as

x1=¢91, x2=61, x3=92, x4=62, r=u (3.46)

Therefore, the matrices A and B can be found at the top position as

”0100‘

 

 

F o '
6x1 5"3 3—
A = , B = u (3.47)
0 0 0 1 0
.654 5% 24.
O 0
6x1 6x3 .. all 5

 

 

53

It should be noticed that there will be some difference between the linearized
equations of the pendubot and that for the acrobot due to the change in the torque
postions. Therefore, if we compare equation (2.51) to equation (3.47) we will see
that the A matrix stays the same, however, the vector B changes.

0 Using the model of the linearized system, design a linear controller to render the
desired equilibrium of the acrobot (nonlinear system) locally asymptotically

stable. The system is controllable as the controllability matrix C is full rank
C = [B AB A28 A38] (3.48)

Since this equilibrium point is not stable, the matrix A is expected to have
eigenvalues in the right hand side (RHS) plane. Thus, to stabilize the system
using a linear controller, new eigenvalues are assigned to the controlled system

(A-BKZ) in the left hand side (LHS) plane. Then, the desired feedback gain matrix

K2 is calculated to find the control law

I =—K2(x—xeq) (3.49)

This control law is applied to the nonlinear system and stabilizes it at top position.
0 Choose a small angle a, a > 0, such that (62,612): (6’2 ,0) lies in the region of
attraction of the desired equilibrium (6,61,62,61,): (it/200,0) for all values of
(92 satisfying — a S (92 S a.
2. Swing-up control of the first link:

If the configuration of the second link satisfies —a $62 361,92 = 0, the first link will

automatically swing up to the configuration (61,91) z (0,0) if E z ET' To increase the

54

energy of the first link to make the energy of the system reach Er we will use a series of
rest-to-rest maneuvers of the second link, described in section 3.3.2. Additionally, to

ensure that (92 will always lie within the region —a S 62 S a , the following procedures

will be followed:

To hold the second link fixed, the holding torque T h in equation (3.3) will be applied.

To move 62 in the positive (counter-clockwise) direction from rest, the torque expression

in equation (3.42) will be used, provided it is greater than Th at the initial time. To move

92 in the negative (clockwise) direction from rest, the torque expression in equation
(3.42) will be used provided, it is less than T h at the initial time. As (92 approaches the

boundary of the interval [— ma], the braking torque 2' b in equation (3.6) will be used; a
large value of k5 will be used to quickly stop the motion of the second link. Conduct the
rest-to-rest maneuvers until E z ET'

3. Stabilization:

By the end of the second step the second link position will be within a small angle a from

192 = 0 and the system energy is almost equal to ET This will make the acrobot behave

like a pendulum that swings and can reach the upright configuration in a finite time. Once

the acrobot configuration enters the region of attraction of the desired equilibrium point

(191,191,62,92)= (7r/ 2,0, 0,0), the linear controller in equation (3.49) will be invoked to

stabilize the equilibrium point.
Numerical Simulations of the First Algorithm
The kinematic and dynamic parameters of the acrobot were assumed to be similar to that

used in the pendubot, which are

55

.=1.0m, d.=0.5m, m.=l.0kg, 1.:im.t.2=0.0833kgm2,i=1,2
1 I I t 12 It

For this choice of parameters, ET was evaluated to be 19.62 J.

Y? lg

 

><V

(91— rc/2)

 

 

Figure 3.3. Sketch of the acrobot during the swing-up process
As part of the initialization (the first step), a linear controller is designed to stabilize

the desired equilibrium. Through repeated simulation of the closed-loop system behavior,
a value is estimated to reach a maximum value of around 0.5 rad. Therefore, it is
expected to select this value for a and then make the second link do rest-to-rest
maneuvers to pump energy to the system just as we did in the pendubot case. However,
this turns out not to be the case. The torque expression derived in equation (3.42) makes
another constraint on the value of a, especially if the system started with initial
conditions corresponding to the lowest potential energy. To find the values for a
throughout the swing-up process one can look to the example illustrated in Figure 3.3. In

this example, we are trying to find the maximum value allowed for 62 so following
argument is always true. In this figure, the second link is supposed to move to the

56

negative (clock-wise) direction, which means that the torque expression in equation

(3.42) should be less than the holding torque T h in equation (3.3).

Therefore, if substitute the expressions for the torques we will end with the relation
_—_1 2 2 2 ( ) W211 )
U {11"qu ‘13 C2161 ‘12 + q3C2 k862q1 +"2 + 2‘13C2 ‘1352

2 2 _ .
+glqlqz q3C2111q1+q3C21q5C12 lq2+q3C21q4C11l<OO°SO>

where
_ 2 2
U ‘1‘]1‘12 ‘1362 "”81 q1+q2 +2‘13612111‘11Jr‘72 +2‘13C‘21 (3 ’5 1)
9
1 1.00A 2
I
i 0.75 ~-
1 + +
' 0501
l / /
l / /
I -_ {0‘

l

 

 

 

— 311/2 — 551/4 - n/2 Jar/4 0
(1.

mes/Oi ’ ’ ~— — 0.25

acctcdia’ I

S , , ,
’ 4 — 0.50
+

4 — 0.75
1* — 1.00

 

Figure 3.4. Solution of the inequality and the selected values of a
Because of the constraint in equation (3.44) we know that U in equation (3.51) is going to

be always negative at all times. Moreover, the second link is at stationary state as it is

57

hold by the holding torque (i.e. 612 = 0) . In addition, to explore the boundaries we will

solve the equation for (611 = 0). Therefore, we will end up with the terms

1 _ 2 2 _
U1q1‘12 q3C2111q1+q3C21q5C12 l‘lz‘Lq3C’21q4Cl1<0 (3°52)

the term (-g/ U) is always positive and also the term (qlq2 —q§C§ ). Therefore, in order

to val1date this inequallty we need the term [(q1 +q3C2 )qSC12 —(q2 + q3C2)q 4C1] to

be negative. Using the kinematic and dynamic parameters above and for different values

for 61 we can solve for 62. The solution is shown as the curves in Figure 3.4. However,

selecting a to be in these boundaries is not a wise decision because both links will soon

have some angular velocities, which make the constraint on a get back to the equation
(3.50). This will make the inequality vanish directly after the release of the second link.

Therefore, it is important to choose a in more conservative way. In our simulation we

choose a to be as the dotted line shown in Figure 3.4.

As it can be seen from the plot in Figure 3.5, the initial conditions were selected to be
(61,91,62,92)= (7r/ 2,0, 0,0). In addition, one can notice that the energy of the system
was increasing gradually until it reaches the desired value E 7. Furthermore, by looking to
the plot of the second link angle, we can see that the second link did a series of rest-to-
rest maneuvers around 62 = 0. As the energy of the system increases, the first link started

to swing-up. This can be seen in the first link plot. Finally, the control law switched to the

linear controller in equation (3.49) at t = tS = 243.2 see, which guaranteed the local

58

stability of the system at the top equilibrium point. Although this approach is successful,

it takes a long time to pump energy to the system when the initial configuration of the

system is (61, 61, 62,62) = (7r/ 2,0, 0,0). However, when the system has more energy at

the initial stage, the swing-up time will be cut dramatically. For example, if we start with

zero initial conditions the swing-up time will be dramatically as shown in Figure 3.6.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

A 2 I l I I ;
“U l
g _ 2 WWWVWMANWVWWVWMNWV‘AAN E q
5 I
.. 6 1 1 1 1 :
A 6 l I u I 2
€ :
s 0 °
05 _ 6 1 1 1 1 E
A 0. 5 ' ' T ' .
“O I
E o J’ ,
N 1
¢ 1
_. 0_ 5 1 1 1 1 I
g 4 . . , .
e 0 ‘‘‘‘‘‘‘‘‘‘‘‘‘‘‘‘‘‘‘‘‘‘‘‘‘‘‘‘‘‘‘‘‘‘‘‘‘‘‘‘‘‘‘‘‘‘‘‘ 1 1‘1‘1‘1‘1'1‘1‘1‘1‘1‘1‘1‘1‘1‘1‘1‘1‘1‘1‘1‘1‘1‘1‘1‘1‘1‘l‘l‘l‘l‘l‘l‘l‘l‘I‘l‘l‘l‘I‘I‘I‘III‘I‘I‘I‘“I "I “I ‘1”! 111‘“ I i i i i i H H J 1;; _i'nv
..g "HHHIHW :
_ 4 1 1 1 1
A 2 5 I I f I :r
E. s
a 0 .
1» E
'- 25 1 l l 1 1:
20 . . . . g
c: E
a: 0' ts
_ 20 ‘ g 1 1 i
0 5 0 100 l 5 0 200 2 5 0
time (see)

Figure 3.5. Plot of the first simulation of the first algorithm, which shows the joint
angles, joint angle velocities, control torque, and energy of the system

59

 

 

 

 

 

   

 

 

——-——-1__--— -----1-..4...1___-..

 

1—1
0
£11

 

---.

 

 

92 (rad/s)
c
S
l

_10 1 1

 

 

 

 

 

 

 

 

 

 

 

 

0 1 1 N J
0 4 8 12 l 6
time (sec)

 

Figure 3.6. Plot of the second simulation of the first algorithm, which shows the joint
angles, joint angle velocities, control torque, and energy of the system

The linear controller was invoked at t = ts = 11.41 sec after the second link did a

couple of rest-to rest maneuvers. Therefore, the effectiveness of this algorithm depends

on the initial conditions of the system.

3.4.2 Second algorithm
A four-step algorithm is proposed as follows
1. Initialization:

The same linear controller proposed in the first algorithm is used here.

60

2. Matching the angular velocities in magnitude but different in direction:
In a case in which the second link is held fixed by the holding torque in equation (3.3) at
an arbitrary angle and the first link is free to move since it is not actuated, we will see the

system swing back and forth around 61 = -m’2. When the first link is slowing down and

most of the kinetic energy of the system transferred to potential energy, we will use the
torque expression in equation (3.10) to make the second link angular velocity match the
first link angular velocity in the magnitude but in the opposite direction. For example, if

62 is positive, like it can be seen in Figure 3.3, the torque will be imposed when the first

link is going in the positive (counter-clock-wise) direction and started to slow down. This
will make the first link angular velocity increase suddenly as it can be calculated in
equation (3.30) and consequently the energy of the system will increase as it was
discussed in equation (3.36).

3. Brake when the second link become to rest:

With k6 selected to be a large constant, the torque in equation (3.10) will match the

magnitude of the angular velocity of the two links with opposite signs in a short time and
keep this relation true at all times. Therefore, we will see at the beginning of applying this
torque that the magnitude of the angular velocity of both links increases suddenly and
then starts to slow down again. Then, we will wait until the second link angular velocity
reaches almost zero and apply a sudden braking torque as in equation (3.6). In this
process the system will not loss energy as it can be seen in equation (3.35). Hence, the

combination of the second and third steps will create rest-to-rest maneuvers analogous to

61

the first algorithm. Moreover, the energy of the system will increase after each maneuver;
and after some time, the energy will reach the desired one.
4. Stabilization:

As was done in the first algorithm, by the end of the third step the system energy is
almost equal to ET When both links enter the region of attraction of the desired

equilibrium point 9 ,9 ,9 ,9 = 7: 2,0, 0,0 , the linear controller in equation (3.49)
1 1 2 2

will be invoked to stabilize the equilibrium point.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2 1 I 1 T
=3 5
35-2 : .
5 5
—6 1 1 1 t
1.? 6 ' ' E
a E
3 o -
.¢_6 1 1 1 :
Ao.7 . . . '
“O
3 o
N
CD
-017 J l 1 :fi
7’? 6 I I I r
E
g o
8'
. .1-6 1 1 1 :1
A250 1 1 1 '
E. I
E 0 .
-250 1 1 1 4
21 I I I .
5 . 3 1
Lu 3 ts ;
__15 1 1 1 \—
0 25 55 85 110

time (sec)

Figure 3.7. Plot of the simulation of the second algorithm joint angles, joint angle
velocities, control torque, and energy of the system

62

Numerical Simulations of the Second Algorithm

Using the same kinematic and dynamic parameters used in the first algorithm and the

initial conditions (91,91,92,92)=(—7r/4,0,—7r/2 0,0) it can be seen from the plot in

Figure 3.7 that the energy increased with each rest-to-rest maneuver done by the second
link. Furthermore, the first link started to swing-up more and more until it reached the

upright position. Finally, the control law switched to the linear controller in equation

(3.49) at t = t5 = 99.67 sec, which guaranteed the local stability of the system at the top

equilibrium point. When applying the impulsive torque in equation (3.10), the control
torque peaks. The large magnitude of these peaks, which can be attributed to the large

value of gain k , is not of concern since they are applied intermittently and for short time

intervals.

When the system gain energy close to the desired one (i.e. ET) during the swing-up
process, the travel of the second link was restricted to a reasonable angle from 92 = 0.

This was done to ensure that the system can be controlled by a linear controller when its
configuration enters the region of attraction of the desired equilibrium point. This means
that we needed to stop the second link after applying the impulsive force and moment and
before its angular velocity goes to zero. However, this should not be a concern since we
know that the angular velocity of the second link at the time of braking is less than what
it was after the impulsive force and moment. Therefore, the net energy change of the rest-
to-rest maneuver will be positive. This can be verified by substituting equation (3.30) into

equation (3.36), we will get

63

 

 

2 2 2 2 2
AE=(Il+mld] )(12+m2d2)+m211 (12 261.111211252)“+ +2) (3 53)
2 2 2
2(Il+mld1 +12+m2al2 +m21l +2m211612C2)l
and we know from the conditions discussed in this case that 9;=—91+, which means
that
( 2)( 2) 2( 2 2)
I+md I+md +mI I+mdS 2
AE= l 11 2 22 21 2 22 2 (92) (3.54)

2 2 2
2(11 + mldl + 12 + 111de + 11121] + 2112111129)

The expression in equation (3.54) is similar to equation (3.35) but with a positive gain in
energy. It means that if we brake suddenly after the impulsive force and moment, the
system will lose the same amount of energy added. But since we will brake after some
time, we know that the second link angular velocity will be less than that after the
impulsive force and moment. Hence, the net energy change in this process will be

positive.

3.4.3 Third algorithm

A five-step algorithm is proposed as follows

1. Initialization:

The same linear controller proposed in the first algorithm is again used here.

2. Matching the second link angular velocity to a desired value:

In a case in which the second link is held fixed by the holding torque in equation (3.3) at
an arbitrary angle and the first link is free to move since it is not actuated. We will use the

torque expression in equation (3.12) to make the second link angular velocity match a

specified value éZdes when the first link is slowed down completely (i.e. 91 z 0) and all

of the kinetic energy of the system transferred to potential energy. This torque will be

64

applied only for a short period of time as an impulsive force and moment applied to the

second link. Moreover, since the system is considered to be a non-minimum phase,

éZdes is selected to be negative when the first link is slowing down from positive angular

velocity values and vice versa. Hence, looking again to the case in Figure 3.3, the second
link is held fixed at a positive angle and the first link is going to the positive (counter-
clockwise) direction. Once the first link slows down completely we will apply the

impulsive torque that will make the second link angular velocity reach a negative value of

éZdes in a short time. This will make the first link gain some angular velocity in the

positive direction as it can be calculated from equation (3.31) and consequently gain
some energy as it can be calculated from equation (3.37).

3. Make the second joint free:

After the second link angular velocity reaches the desired value, the control law will be

switched to make the second joint free or

r = 0 (3.55)
Therefore, we will see at the beginning of applying the torque in equation (3.12) that the

magnitude of the angular velocity of both links increases suddenly and then after
applying the torque in equation (3.55), they start to slow down again as we are going to
discuss later. The energy of the system will not change since the torque is zero.

4. Brake when the second link become to rest:

After the torque in equation (3.55) is applied, the second link angular velocity will slow
down eventually. Therefore, we will wait until the second link angular velocity reaches
almost zero and apply a sudden braking torque as in equation (3.6). In this process the

system will not loss energy as it can be seen in equation (3.35). Hence, the combination

65

of the second, third and, fourth steps will create rest-to-rest maneuvers analogous to the
first algorithm. Moreover, the energy of the system will increase after each maneuver and
after some time the energy will reach the desired one.

5. Stabilization:

In a way similar to the first two algorithms, by the end of the forth step the system energy

is almost equal to ET When the acrobot configuration enters the region of attraction of

the desired equilibrium point (91,91,92,92)=(7r/2,0,0,0), the linear controller in

equation (3.49) will be invoked to stabilize the equilibrium point.

YA lg

 

><V

(91- n/2)

 

 

Figure 3.8. The acrobat at the instant after the application of the impulsive force
To prove that the second link will always slow down in the third step one can look to

equation (3.5) and apply equation (3.55) to get
61 _ l h]
.. — 2 2 h (356)
92 ‘1qu — q3 C2 2

66

 

  

 

1.00 A c 1'6 3%?
H v 9..
CS. 5 H
0.75 ‘— II a '+~
— 0 a
+ Q egg (.5?
0.50 -- v <1?
N
<1:
(a) 6" "
0.25 ~~ (b) (C) 1
\2
1 1 1 1 e
- n/2 — 11/4 0 n/4 11/2 91

 

Figure 3.9. The change in the acrobot configuration in the moment after the application
of the impulsive force with the change in the negative and positive areas of h1

we want to show that the second link will slow down afier the application of the

impulsive force. Hence, if we look to the example in Figure 3.8 and equations (3.56),

(2.27) and (2.28), we can select the value of 9; so that both links will slow down after

the application of the impulsive force (i.e. 91 < 0,92 > O in this example). This can be

done because we know that 91+ is equal to

1 +m2(d2+d 1C)

 

 

. 2 2 21 2 . q +q C .
91+ =_ 2 2 2 a§=—[q +31 +32qzc i6; (3'57)
11+m1dl +12+m2 (d2 +11 +2d211C2) 1 2 3 2

Substituting equation (3.57) into (2.27) and (2.28) we get

2 2
q352[q2(q1+q3C2) +‘13C2l‘12‘L‘I3C2) ] .+ 2
"1 = ‘92) +gl‘13‘15C2C12”‘12‘14C1)

 

2
(‘11 +‘12 +ch3C2)

(3.58)

67

q H] C S 2 . 2
h2=‘( (+1 322:2)2[(qzqfqiczilql+q3C2)+q3lq2+q3C ) Mai)
‘11 ‘12 ‘13 2

 

“g [(41 + q3C2)q5C12 ’(qz +‘13C2)"4C1](3'59)
2C2

Since the term (qlq2 —q3 2 ) is always positive, we should select 9; value so that hl

be negative and h2 be positive in order for both links to slow down. However, it is

important to notice that the moment after the application of the impulsive force equation

(3.57) is no longer valid. This means that I11 and h2 is then only defined by equations

(2.27) and (2.28), respectively.

 

 

92
1.00 A .5
II
. N
0.75 -— 71’
_ <13~
0.50 ~—
0.25 -- (a) 1,) +
g“)
—« I; 1 1 1+
—1t/2 —Tl:/4 0 “It/4 n/2 91

 

Figure 3.10. The change in the acrobot configuration in the moment after the application
of the impulsive force with the change in the negative and positive areas of h2

In order to understand the argument that both links will continue slowing down after
the application of the impulsive force, one can look to the example at Figure 3.8 and the
corresponding Figures 3.9, 3.10 and 3.11 to realize the following

0 After the application of the impulsive force it is important to know that

68

> 9+ is always true and this can be figured out from equation (3.57).

'+
'62 1

 

 

 

After the application of the impulsive force the system configuration is located at

point 1 (i.e. the negative area of hl and the positive area of I22 and (h]+ 112)),

which means that both links will slow down in the moment after the application of

the impulsive force. Moreover, the sum of the angular velocities of both links

(19141192), which 1s dominated by 92, Wlll also slow down (1e. (01+192)> 0 in

this example).

After the application of the impulsive force the border line between the positive
and negative areas can be described by the curve (c) in Figures 3.9, 3.10 and 3.11.
If both links are stationary, the border line between the positive and negative areas
can be described by the curve (a) in Figures 3.9, 3.10 and 3.11. This means that
the first two terms in equations (2.27) and (2.28) are responsible of changing the

border line from curve (a) to curve (c).

 

92
1.00 -‘-
0.75 1
0.50 ._ _
0.25 :—
-1r/2 —1'1/4

 

 

Figure 3.11. The change in the acrobot configuration in the moment after the application
of the impulsive force with the change in the negative and positive areas of (111+ h2)

69

0 In the next step after the application of the impulsive force we know that the first
link will move in the counter-clockwise direction and the second link will move
in the clockwise direction. This means the system configuration will move from
point 1 to 2. However, we know also that both links will slow down and the

magnitude of their angular velocities is going to be less than that in the last step.

In addition, we know that the sum of the angular velocities of both links (91 + 92)

will also slow down. This mean that the border line between the negative and
positive areas will move from curve (c) to curve (b), which helps the links to

continue in slowing down.

 

Figure 3.12. The angular acceleration of the first link 91 at the moment just after the

impulsive force

70

 

Figure 3.13. The angular acceleration of the second link 92 the moment just after the

impulsive force

For the kinematic parameters we used for our simulation, it can be shown -with the
help of Mathematica- that the regions of the desired solution are in Figures 3.12, 3.13 and
3.14. In order to make the understanding of the plots easier on the reader we only
consider the changes of the angle of the first link to be in the region [-1t/2, 112/2] rad
because in the other region we will have a similar solution with an opposite sign. In
addition, the region of the second link angle is [-0.5, 0] rad because the positive region of
this angle means the centrifugal forces will join the gravitational forces in decelerating
the links. Finally, we selected 92+ values to be negative only [0, -10] rad/s because we
want the link to go to the clock-wise direction for this example and we want to see how

far we can go in selecting this value.

71

     
 
 

:11-

\\\\\

 
    
  

 

\\".___;J
\\\“

   
 

\

 

\\\
, “\“kk‘

< \\\ " 1'
\Ҥ\\\\\

Figure 3.14. (91 + 92) the moment just after the impulsive force

Numerical Simulations of the Third Algorithm

Using the same kinematic and dynamic parameters used in the first two algorithm and the

initial conditions (91,91,92,92)=(—rr/4,0,0,0) it can be seen from the plot in Figure
3.15 that the energy of the system increased with each rest-to-rest maneuver done by the
second link. In addition, the effect can be noticed on the first link as it started to swing-up

until it reached the upright position. Finally, the control law switched to the linear
controller in equation (3.49) at t = t8 = 23.76 sec, which guaranteed the local stability of

the system at the top equilibrium point. When applying the impulsive torque in equation
(3.12), the control torque peaks. The large magnitude of these peaks, which can be

attributed to the large value of gain k , is not of concern since they are applied

72

intermittently and for short time intervals.

 

 

 

 

 

 

 

 

A 6 1 1

S

33 0 .

"D—6 1 1
1.8 1 1 ;

 

 

 

I 92 (rad)
Ox 60 c

—-—1_— -_

‘?

 

 

 

1'12 (rad/s)
O

=
z

—6

h
P

 

 

 

-— -fi——-——

 

mm
c 3

 

 

 

 

 

 

 

- 14o - 1

21 l n rL TL T

a: E

m 3 ts

- 15 1 1 \1
o 10 time (sec) 20 30

Figure 3.15. Plot of third algorithm joint angles, joint angle velocities, control torque,

and energy of the system

When the system gain energy close to the desired one (i.e. ET) during the swing-up

process, the travel of the second link was restricted to a reasonable angle from 92 = O.

This was done to ensure that the system can be controlled by a linear controller when its

configuration enters the region of attraction of the desired equilibrium point. This means

that we needed to stop the second link after we made the second joint free and before its

angular velocity goes to zero. However, this should not be a concern since we know that

73

 

the angular velocity of the second link at the time of braking is less than what it was after
we made the second joint free. Therefore, the net energy change of the rest-to-rest
maneuver will be positive. This can be verified by comparing equation (3.35) and
equation (3.37). The two expressions look similar to each other; however, there is a sign
difference. This means that if we brake immediately after we made the second joint free,
the system will loss the same amount of energy added. But since we will brake after some
time, we know that the second link angular velocity will be less than that after we made

the second joint free. Hence, the net energy change in this process will be positive.
3.5 Remarks

In this chapter we presented three new successful algorithms to swing-up the acrobot. All
these algorithms used an impulsive-like torque to pump energy to the system using rest-

to-rest maneuvers of the second link provided that it is within some angle a from 92 = 0.

However, there are some differences between these algorithms, which make some of
them superior to the others. The third algorithm gave the best results because we can
specify the amount of impulsive torque done on the system. Therefore, the initial
conditions do not affect the time of the swing-up process. On the other hand, in the first
and second algorithms the torque is decided based on the status of the system. Hence, if
we started from a configuration corresponding to the lowest potential energy for example;
it will take the system longer time to swing-up. Simulations for all algorithms were
provided and showed these remarks. In the next chapter we will discuss the stabilization
of an under actuated biped robot with torso and use some of the techniques discussed in

this chapter.

74

 

CHAPTER 4

Stabilizing an Active Synthetic Wheel Biped due to Small Disturbances

4.1 Introduction

In this dissertation a new biped robot is introduced. The feet of this robot are part of a
circle, which allows the overall movement of the biped to be similar to a rolling wheel.
We described this robot as “An Active Synthetic Wheel Biped Robot with Torso.” In
order to start walking, the torso will lean to the front to create a moment on the system.
Then, the robot will take a step forward to avoid falling over. Before the first leg tip
reaches the ground, the second leg will take over to complete the circle and the robot will
repeat the process to achieve walking. Our goal in this dissertation is to solve the
stabilization problem of this robot about its standing configuration. In this chapter we
discuss the control problem due to small disturbances. The objective is to move the torso
in a manner that allows the robot to regain its standing posture after being disturbed.
Using the impulse-momentum approach introduced in the previous chapters, the control
law is going to stabilize the system about its standing configuration. The two legs of the
biped are going to be aligned; and therefore, the system will have two degrees of
freedom. Hence, the control problem will have some similarities to that for the acrobat.
We are going to design the controller to do rest-to-rest maneuvers such that the energy of

the system converges to the desired value. This brings the robot configuration within the

75

region of attraction of the equilibrium point wherefrom a linear controller can be invoked
to stabilize the equilibrium.

This chapter is organized as follows. The next section lists the nomenclature used
throughout this chapter. In section 4.3 we provide the equations of motion of the biped
and derive expressions for the holding torque and the braking torque. In section 4.4 we
discuss the effect of sudden stopping of the torso on the energy of the system, and then
we design the rest-to-rest maneuver for the torso to alter the system energy. In section 4.5
we present the algorithm for stabilizing the biped about its standing configuration along
with numerical simulations. Concluding remarks are provided in section 4.6.

4.2 Nomenclature

The biped is shown in Figure 4.1. For the nomenclature listed below, the subscripts i and

j assume values land, 2.

I . length of the i—th link, (m)

d distance between the end point of the first link and its center of mass, (m)
d2 distance between the second joint and center of mass of the second link, (m)
R the radius of the feet, (m)

m. mass of the i-th link, (kg)

1. mass moment of inertia of the i-th link about its center of mass, (kgmz)

9. angular displacement of the i-th link as shown in Figure 4.1, (rad)

9. angular velocity ofthe 1111 link, (rad/s)

9.— angular velocity of the i-th link, immediately before the first link (legs) is stopped,

(rad/s)
9+ angular velocity of the second link (torso), immediately after the first link is

stopped, (racfls)

76

V2

~11

l imp

"1‘1

2imp

..5‘1

mp

.3

WP

velocity of the center of mass of the second link, (m/s)

velocity of the center of mass of the second link, immediately before the first link
is stopped, (m/s)

velocity of the center of mass of the second link, immediately after the first link is
stopped, (m/s)

inertial reference frame with unit vectors 1" and ] along the X and Y axes,

respectively

impulsive reaction force acting on the feet, (N)

impulsive force acting on the second link at center of mass, (N)
impulsive force acting on the first link at center of mass, (N)

impulsive moment acting on the j-th link at its center of mass, (Nm)

external torque applied on the torso, (Nm)

external torque required to hold the torso fixed, i.e., maintain 92 = 0, (Nm)

external torque required for braking, i.e., causing exponential decay in the
velocity of the torso, (Nm)
total energy of the system, (J)
potential energy of the system when (91, 92) = ( 0 , 7:), (J)
acceleration due to gravity, (9.81 m/sz)
sin 9.
1
cos 9.
1
sin (91 + 92)

cos (91 + 92)

77

Xj firfio
Y ‘1‘
41,91

18 I“,
Q

R , Q \1
11
d1

 

 

\\\\5\\\\\\\

Figure 4.1. An active synthetic wheel biped robot with torso where the legs are aligned

4.3 System Dynamics
4.3.1 Equations of Motion

Consider the pendubot in Figure 4.1. Assuming an ideal system with no friction in the

joints, the kinetic energies of the two links can be written as follows

-1 '2 l '2
KE1 -21191 +2m1r1 (4-1)
KE :11 (19' +19 )2+-1-m 12 (42)
2 2 2 l 2 2 22 '
where
r1=[R91—(R—d1)Sl]i+[—R+(R-dl)C-l]j (4-3)
r2 =[R91-(R—11)Sl—d2S12:li+[—R+(R-11)C1+d2612]j (4.4)
andtherefore
2 .
.2_ 2 _ _ _ 2
,1 _[R +(R d1) 2R(R d1)C1:l91 (4.5)

78

122 = 112912 +(R—11)2 9'12 +d§(a'1 +92)2 ”(R—10.1291 ((11 +112)c2
—2R912(R—11)

C1—2R91(91+ 192 )d2C12 (4.6)
The potential energies of the two links are as follows

PE1=—m1g[—R+(R—d )Cl]

1 (4.7)
PE2 =m2g[—R+(R—11)Cl +d2C12] (4.8)
Therefore, we can formulate the Lagrangian formulation as follows
L=(KE1+KE2)—(PE1 +1352) (4.9)
—d- 6_L ——(?—L— = O (4.10)
dt 69 69
1 l
’ 2 2
putting the results in matrix form we get
A(9)9+B(9,9)9+G(9)=T (4.15)

where A(9) , B(9,9) , G(9) , 9and Tare given by the expressions

a a _ b b g 9 0
A(9)=[ 11 12],B(9,9)=[b11 £2],G(9)=[ 11],9=[61],T=[ ] (4.16)
“12 “22 21 g21 2 T

and the terms in the inertia matrix A(9) are defined as

all=Ilp+12+m2[R2+(R—ll)2+d22+2(R—ll)d2C —2R(R—l1)C —2Rd C ]

1 2 12
(4.17)
a :1 +m [d2+(R——l)d C —Rd C ] (4.18)
12 2 2 2 1 2 2 2 12
_ 2
a22 — I2 +m2d2 (4.19)

79

where

11p =11+m1[R2+(R—dl)2—2R(R—d1)Cl] (4.20)

and the terms in the matrices B(9,9) , 6(9) are

1111 = m1R(R—d1)9lSl +1112 [R91 (18-1051 4111191112512] (4.21)
1112 = m2 [—(R—11)d2(20'1 +92)52 +Rd2(291+92)S12] (4.22)
b21=m2(R”1)d29152 (4'23)

g11 =m1g(R—dl)Sl +1112,c,r[(11—11)s1 +d2812] (4.24)
1:21=ngdzs12 (4.25)

4.3.2 Holding Torque

During the stabilization process, our algorithms will use the technique of holding the

torso fixed at certain times, i.e., maintain 92 = 0. We compute the torque required to hold

the torso fixed by substituting 92 = 92 = 0 in equation (4.15), as follows

a .. b . g 0
[11]61_[b11]61+[ 11]=[1] (4.26)
“12 21 g21 h

By eliminating 91 from the two equations in equation (3.2), Th can be expressed as

—a
._ 12 ' '
’h‘ “11 [1161+g11l+b2191+g21 (4'27)
4.3.3 Braking Torque

We consider braking action that results in exponential decay of the motion of the torso to

zero. Therefore, as it was in pervious chapters, we assume

80

92 = 419192, 19 > o (4.28)

where k9 is a constant that will control the rate of decay of 92. To compute the torque

required for braking, we multiply equation (4.15) first with the inverse of the inertia

matrix to obtain 91 and 92 as a funct1on of 91,91,92,92 and 2'

91 _ 1 —a121'+h3
é _ 2 h (4.29)
2 ”11"22"’12 all“ 4

where h3 and h; are given by the expressions

 

h3 = ‘a22(b1161+b1292)+“12b21 1‘0225’11 +"125121 (4'30)
'74 = “12(b11‘91+b1262)""11b2161+“12811"“11g21 (4'31)

Note that: 11111122 —a122 = 12 {11p +1112 [R2 +(R —11)2 —2R(R 41M“
411112115 {111) + m2 [st122 +(R —11)2 522 — 2R(R «11)(Cl —C2C12)]} > 0

On substituting equation (4.28) in the second equation of equation (4.29), we get
1 =;L(a a -112 )k9 +1 (432)
b a“ 11221292 4 '

When the torso comes to rest, the braking torque becomes equal to the holding torque.
This can be easily verified from equations (4.27) and (4.32).

4.4 Energy Consideration of the System

4.4.1 Effect of Sudden Stopping of the Torso

A large value of gain k9 in the expression for braking torque in equation (4.28) will result

in the sudden stopping of the torso. This action of suddenly stopping the torso has the

81

effect of applying an impulsive force and an impulsive moment on the leg, as shown in

Figure 4.2. As we did in the first case of the acrobot analysis, we are going to start

looking to the free body diagram of the torso in Figure 4.2. The impulsive force and

moment on the center of mass can be written as

Y ' \‘x F2imp
2 ; 'O
l g inmp ‘1 12 MZimp
\\ lVl2imp+ I'2 X F2imp

rl

l:‘imp

 

 

\ \
Flimp

I-O

<\\\\\

Figure 4.2. Impulsive forces and moments on the biped

inmpm = m2 (17; —-v2—) (4.33)
M21111)“: 1291+ — 12 (6'11" + 6'12") (4.34)
where
a; =[R91+—91+(R—11)C1—91+d2C12]I—:91+(R—11)S1+9ll'd2S12]] (4.35)
17— =[R91‘ —91—(R—11)C1-(91— +9£)d2C12:I°—[91‘(R—11)Sl +(19'l— +e'2-)d2 12]]

 

82

(4.36)

In order to study the effect of the impulsive force and moment on the angular velocity of
the leg, we will discuss the effect on the second joint first. The impulsive force will stay
with the same magnitude and direction. However, there will be impulsive moment added

due to the distance

r2 =d2(—Slzi +C12j) (4.37)

Therefore, the addition on the impulsive moment will be

2 1 _ _ _ .+ ._ _ ._
rszZimpAt_m2d2{[(R 11)C2+d2 RC12](61 +91) 11292} (4.38)

Now, if we look to the free body diagram of the leg in Figure 4.2, the impulsive force and

the impulsive moment on the center of mass can be written as

a _ _. _- = ,+__,_
Fimpm_(r‘lim FZimijt 1111 (v1 v1) (4.39)
_. _ .+_._
MlimpAt—Il (191 91) (4.40)
where
1+ =[R—(R—d )C ]19’+2’—(R—d )5 9+] (4.41)
1 1 1 1 1 11
v1 =[R—(R—dl)C1]9lz—(R—dl)Sl9l j (4.42)

and Flimp is the impulsive reaction force at the first joint which can be found from

equation (4.39) to be as
FlimpAt = {[1111 (R—(R—d1)C1)+m2 (R—(R—11)C1 —d2C12)](91+ —19'1“)
+m2d29gC12}7+{[—m1(R-d1)Sl+m2(—(R-11)Sl—d2S12)](91+—91—)

+m2d292 S12ij (4.43)

83

 

Now, we want to calculate the impulsive moment on the leg center of mass due to the
impulsive forces applied to its joints. The second joint has an impulsive force with the
same magnitude and the opposite direction from that in the second joint in the torso’s free
body diagram. This impulsive force will create an impulsive moment at the leg center of

mass due to the distance

’3 "‘(11 ‘d1)(51i ‘CIJ ) (4'44)
Therefore, the addition on the impulsive moment from this impulsive force will be
_. _‘° = _ _ , _ '+ _ '_ _ '_
r3 x( F2impAt) 1112(11 dl){[ RC1+(R ll)+d2C2](91 91 ) 92 81262} (4.45)
In addition, the impulsive reaction force will add another impulsive moment at the leg

center of mass due to the distance
r1=(R—d1)Slz +[R—(R—dl)cl]j (4.46)

Hence, the impulsive moment added due to this reaction force is

—

F XFlimpAt =—{m1 [R2 +(R‘d1iz ’2lR‘d1lRC1]+m2(R‘dliilR‘I1)+d2C2]

’mzRiR"lR"1)C1'lR‘d1)C1 “d2C12]i(‘91+‘91_)

411112112 [(11 — d1)C2 — RC12]92 (4.47)
Therefore, the impulsive moments at the first link center of mass due to the impulsive

forces can be written as

—o

MlimpAt — _'M2impAt — r2 x F2impAt + r3 x (_FZimpAt) + r1 x FlimpAt (4.48)
equating equation (4.48) to equation (4.40) and substituting the equations (4.34), (4.38),

(4.45) and (4.47) we can get the new angular velocity of the leg after the impulsive force

84

and moment is applied to the torso in terms of the old angular velocities of both links

before the application of the impulsive force and moment. This can be shown as the

following

. . I+m[d2+R—ldC—RdC]_
9+=6_+ 2 2 2 l 1) 2 2 2 12 a“ (4.49)
1 1 “11 2

Now to find the energy lose from the system due to the sudden stopping of the leg, we

calculate the difference in the kinetic energy

 

AE = A131 + AE2 = AKEI + AKE2 (4.50)
KE 11 9+2 61—2 451
A1_21p(1)-(1) (')

AKE =%{1 1.111.2[112 +(R—11)2 +1122 +2(R—l1)d2C2 —2R(R—11)C1

—2Rd2C12]}[(91+)2 +9.1— )2]_%l12 +m2d%}(6;)2

— 2 — — ._._
{12+m2[d2 +(R ll)d2C2 Rd2C12]}91192 (4.52)
:>AE=——-1—I 1 +m R2+(R-l)2—2R(R—I)C

2a“ 2 1p 2 1 1 1

2 2
2 2 2 _ 2 _ _ _ '-
+m2d2 {11p +1112 [R 512 +(R 11) 52 2R(R 11)(C1 C2C12)m(62 ) (4.53)
Since an and all the results of the terms between parentheses are greater than zero; then

AE S 0 , and AE2 = 0 if only if 9; = 0. Clearly, the total energy of the system decreases

whenever the torso is suddenly stopped.

85

4.4.2 Rest-to-Rest Maneuver of the Torso

In order to stabilize the biped at the standing position after the application of small
disturbances, we only going to make the legs behave like one leg only. Therefore, the
biped degrees of freedom will be reduced to two instead of three. The control problem
then is going to be analogues to that for the acrobot. However, in the biped case the total
energy of the system at the beginning of the control problem might be higher or lower
than the desired level. Hence, we are going to discuss two ways to do rest-to-rest
maneuvers of the torso. The first one can reduce energy from the system, and the other
one will add energy to the system. Both ways depend on the fact that the rate change of

energy of the system can be put as

E1+ E2 = E = 2'92 (4.54)

Therefore, with a positive gain k10 choosing the torque expression to be
1 = —k {9' (4.55)
will reduce energy from the system. Similarly, with a positive gain kl] choosing the
torque expression to be
1 = k 61' (4.56)
will add energy to the system.

1. Reducing Energy

The system will start from rest using the torque expression in equation (4.55) and then
stop using the torque expression in equation (4.32). We know that the braking action will
make the system lose energy as it can be seen in equation (4.53). Hence, we are only

going to verify that the moving action will not make the system gain energy. Therefore, if

86

 

we substitute the equation (4.55) into (4.54) and integrate for the interval of starting from

rest to the time just before braking we will get

1 — — 1. 1 _ — 1 1 = _l .2
jEdt _ 1 k109292dt _ 1111 O92d92 2 11092 (4.57)

This means that using this torque in equation (4.55) the energy of the system will be
decreasing at all times. From the second equation of equation (4.29), and equation (4.55)
the torque expression can be found in terms of the system angles and angular velocities as

—k h
2': 10 4 (4.58)

2
“11022 ”“12 + k10"11

 

The dominator of equation (4.58) is always positive; therefore, there is no risk of having

singularities in this expression.

2. Pumping Energy
Consider a maneuver in which the torso starts from rest using the torque expression in
equation (4.56) and is brought back to rest through the application of a braking torque in

equation (4.32) using a large gain k , the work done on the system due to the moving

action can be found by substituting the equation (4.56) into (4.54) and integrating for the

interval of starting from rest to the time just before braking as follows

. _ n 1 _ 1 1 -l .2
jEdt —jk119292dt —jk1192d92 — 2 k1192 (4.59)
This means that using this torque in equation (4.55), the energy of the system will be
increasing at all times. However, since we are going to make the torso brake at some

point, and we know from equation (4.53) that the system will lose energy, it is necessary

for us to show that the net work done on the system due to the rest-to-rest maneuver is

87

 

always positive. Hence, comparing the terms in equations (4.53) and (4.59) we can find

that k11 should meet the following condition in order to make the net work done positive

k1] >—1—{12 {11p +m2[R2 +(R—11)2 —2R(R—11)C1]}

“11
2
2 2 2 _ 2 _ _ 1 _ . .
+m2d2 {11p +m2[R 512 +(R 11) 52 2R(R [1)(C1 czclzflflmfim
From the second equation of equation (4.29), and equation (4.56) the torque expression

can be found in terms of the system angles and angular velocities as

k h
I: 11 4 . (4.61)

2
“11"22 "“12 ’k1 1"11

In order not to have any singularities in equation (4.61) it is required to have

 

k $011022”
11
“11

12

 

, which is the same fraction as in equation (4.60) after expanding

terms. Therefore, selecting kll to be greater than the upper bound of the fraction at

equation (4.60) will guarantee the increase of energy of the system and avoiding

singularities.

4.5 Stabilization Algorithm
In this section we discuss a three-step algorithm for stabilizing the biped about its
standing configuration. The basic idea of our algorithm is to make the energy of the

system reaches the desired level ET and the links of the system to be located inside the

region of attraction of a linear controller, which guarantee asymptotic stabilization of the

desired equilibrium.

88

 

l. Initialization:

Linearize the dynamic equations of the biped in equation (4.15) about the desired

equilibrium 6 ,9 ,6 ,6 = O,0,7r,0 .The linear system will be
1 1 2 2

f(x,u) = x = Ax + Bu (4.62)
where x is the state space representation defined as

x1=6ll, x :6? x =6 x =9 r=u (4.63)

Therefore, the matrices A and B can be found at the top position as

 

(0 1 o 0‘ —O- I
_af_2 0 .65; 0 6f2
6x1 6x3 73—
A: ,8: u (4.64)
0 0 o 1 0
6L4 0 .24 0 514
6x1 6x3 _aur

 

 

 

 

Using the model of the linearized system, design a linear controller to render the
desired equilibrium of the biped (nonlinear system) locally asymptotically stable.

The system is controllable as the controllability matrix C is full rank
C = [B AB A28 A33] (4.65)

Since this equilibrium point is not stable, the matrix A is expected to have
eigenvalues in the right hand side (RHS) plane. Thus, to stabilize the system
using a linear controller, new eigenvalues are assigned to the controlled system

(A-BK3) in the left hand side (LHS) plane. Then, the desired feedback gain matrix

K3 is calculated to find the control law

89

r=—K3 (x—xeq) - (4.66)
This control law is applied to the nonlinear system and stabilizes it at standing
position.

2. Bringing the energy of the system to E T:

Since the torso is actuated, we are going to apply on it the rest-to-rest maneuver
technique as it was mentioned in the last section. The torso will be restricted to travel
within small angle a from the top position. If the torso happens to start from an angle
outside this region, the algorithm will make sure that the torso will be brought to this
region. However, the algorithm should choose the torque expression to do all that based

on the level of energy of the system. If the system is having energy higher than ET then

the torque expression in equation (4.5 8) is going to be chosen. On the other hand, if the

system is having energy less than E7, then the torque expression in equation (4.61) is

going to be chosen. For both torques to start from rest moving to the positive (counter-
clock-wise) direction they need be higher than the holding torque in equation (4.27).
Similarly, to start from rest moving to the negative (clock-wise) direction the torque
expression need to be lower than the holding torque. Once the torso reaches the border of
the designed region for movement, the braking torque in equation (4.32) is applied.

In order to verify that the torque selected can be greater or less than the holding
torque at the time we apply it, let us follow this example. The torso is needed to go to the
negative (clock-wise) direction; therefore, the torque selected should be less than the
holding torque at the time we start switching. This can be put as

r-rh <0 (4.67)

90

Assuming that we want to reduce the energy of the system and knowing that the torso

will start from rest (i.e. 92 =0), and since the legs angular velocity is not very high

because of the small disturbances and is going to be used to the power of two in the

torque equation, its effect will be minimal, which make it possible to examine the

solution with 91 =0. Therefore, using equations (4.27) and (4.58) the equation (4.67)

will be reduced to the following condition

 

‘k10("12311'“11821)+“12

—a2 +k a

g —g <0 (4.68)
a a a 11 21
11 22 12 10 11 11

with simplifying and rearranging, equation (4.68) can be reduced to the following

“12

Similarly, if want the torso to go the negative (clock-wise) direction but with the
intention of adding energy to the system, the same conditions will hold. Therefore, using

equations (4.27) and (4.61) the equation (4.67) will be reduced to the following condition

k a g —a g a

11(1211 1121)+12g _g <0 (470)
a a -a2 —k a a 11 21

1122 12 1111 11

with simplifying and rearranging, equation (4.70) can be reduced to the following

 

fllg —g >0 (471)
11 21 '

which is the same condition as in equation (4.69) but multiplied by a negative sign. Both
conditions depend on the parameters of the system. Hence, using the parameters used in
the numerical simulations at the end of this chapter the solution can be seen in Figure 4.3.

The figure shows the results in degrees for simplicity. The angle of the legs (61) can not

91

exceed i11.25 deg in order for the biped not to fall down. The selection of the maximum

angle of the torso (62) depends however on the conditions (4.69) and (4.71) and on the

fact that it should be small enough to be captured by linear controller.

92 (deg)

A

195
190-

185

.—

12 9:(deg)

4:1
00

—12 —8 —4

1751

170

l

 

1656

Figure 4.3. The range selection for the torso angle ((92)

3. Stabilization:

By the end of the second step the torso position will be within a small angle a from the

top position, i.e. 62 = 7t and the system energy is almost equal to ET This will make the

biped swing back and forth around the desired configuration(61,91,62,92 ) = (0,0,7r,0).

The linear controller in equation (4.66) will be invoked to stabilize the equilibrium point
once the biped configuration enters the region of attraction of this equilibrium point.
Numerical Simulations

The robot kinematic and dynamic parameters are assumed to be

92

 

R =11: 1.0 m, m] = 2.8 kg, d] = 0.32 m, 11 = 0.32 kgmz,
12 = 1.0 m m2 = 1.0 kg, d2 = 0.5 m, 12: 0.0833 kgmz

For this choice of parameters, E T was evaluated to be 23.5048 J.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

N
DJ
b

 

time (sec)

Figure 4.4. Plot of joint angles, joint angle velocities, control torque, and energy of the
system

There are two examples shown in this section. The first one is shown in Figure 4.4,
which represents the case of a disturbance force applied on the robot while it was initially

standing. This case was assumed by having initial conditions as follows:

93

(2,91,92,92)=(0,0,n,1.4)

where the units are rad and rad/s for the joint angles and joint velocities, respectively. It
can be seen from the plots that the system at the beginning of the simulation had more
energy than the desired one. This was expected as the system initially had the desired
potential energy and we added a force that created a kinetic energy. Hence, the system
started to do a series of rest-to-rest maneuvers in order to reduce this energy and bring the

robot configuration back to the desired configuration. Then, the control law switched at t

= ts = 3.76 sec to a linear controller that guaranteed the local stability of system at the

desired configuration.
The second simulation is shown in Figure 4.5, which represents the case of the second
link having some angle from the vertical posture. This case was assumed by having initial

conditions as follows:

(61,91,62,92)=(0,O,2.4,O)
It can be seen from the plots that the system at the beginning of the simulation had less
energy than the desired one. This was expected because the torso initially was not at the
top position, which means that it does not have the highest potential energy.
Furthermore, the control law as it can be seen from the plots started to raise the
energy of the system by bringing back the torso to the top. The system did a one time

rest-to-rest maneuver, which was enough in this case to bring the energy to the desired

level. Then, the control law waited until the robot configuration entered the region of

attraction of the linear controller. Finally, the control law switched at t = ts = 1.71 sec to

94

the linear controller that guaranteed the local stability of system at the desired

configuration.

 

0.2

q
d

 

0

61 (rad)

/'

- 0.2
1.2

 

 

d
-

 

 

--—-— -—-— p—-

 

' 01 (rad/s)
S o

 

 

q
q

 

l

 

 

 

 

_-_—- pan-q --——1—-—-—_—q-—-
.

 

 

02 (rad/s)
c #-

 

 

 

 

 

 

 

   

 

 

 

 

—-1———-—p—-p-—q-—--— —-—¢.-—-p-————_

E (J)
13

.(

 

 

22

 

time (sec)

Figure 4.5. Plot of joint angles, joint angle velocities, control torque, and energy of the
system

4.6 Remarks
In this chapter we discussed the stabilization of an under actuated biped robot with torso
using the impulsive-momentum approach introduced in previous chapters. We have

obtained very good results which establish the generality of this approach. The impulse-

95

momentum approach is in fact well suited for the biped model because it restricts the
links from going over certain limits. The biped is different from the pendubot and the
acrobot in that its torso is not allowed to rotate complete 360 degrees for stabilization.
The control approach was very effective and managed to stabilize the biped in a short
time. In the next chapter we discuss the stabilization problem due to large disturbances.
The biped is expected to use both legs this time to avoid falling down and the system will

have three degree of freedom.

96

CHAPTER 5

Stabilizing the Active Synthetic Wheel Biped due to Large Disturbances

5.1 Introduction

In this chapter we discuss the control problem of the biped due to large disturbances. The
objective is to move the legs in a manner that allows the robot to walk the necessary steps
until the system regains its standing posture. Using the impulse-momentum approach
introduced in the previous chapters in slightly different way, the control law is going to
stabilize the system about its standing configuration. Once the system is disturbed, it is
going to move in the direction of the disturbance. The first leg is the only link touching
the ground, and the shape of the feet will make the movement looks like a part of a circle.
Meanwhile, the second leg is going to move faster than the first leg in the opposite
direction in away that make the two legs look symmetric around the vertical. Once the
first leg tip reaches the ground, the second leg will take over and the movements of the
two legs are going to be switched. At the same time, the torso is going to reduce the
energy of the system, and therefore the velocity of the two legs, by going to the opposite
direction of the first leg’s movement and countering by its weight the movement of the
two legs. The torso will also try to stay within a small angular range from the vertical in
order to be within the region of attraction of the linear controller. This linear controller is

invoked once the energy of the system is reduced to the desired level and the two legs

97

enter the region of attraction, too. Finally, the equilibrium will be stabilized after the

application of this linear controller.

 

 

 

 

 

0l

d1

 

a} ‘.
\\\‘P\\\\\\\

Figure 5.1. An active synthetic wheel biped robot with torso

This chapter is organized as follows. The next section lists the nomenclature used
throughout this chapter. In section 5.3 we provide the equations of motion of the biped
and derive expressions for the torques of the actuated joints for different cases. In section
5.4 we discuss impulse-momentum effect of sudden movement of the links on the angular
velocities of the system. In section 5.5 we discuss the energy change of the system due to
certain movements. In section 5.6 we present the algorithm for stabilizing the biped
about its standing configuration along with numerical simulations. Concluding remarks

are provided in section 5.7.

98

5.2

Nomenclature

The biped is shown in Figure 5.1. For the nomenclature listed below, the subscript i

assume values 1, 2 and 3.

l3

~9~

"In

1 imp

"11

2imp

'11

3imp

NE

mp

length of the torso, (m)

distance between the free end point of i-th link and its center of mass, (m)
the radius of the feet, (m)

mass of the i-th link, (kg)

mass moment of inertia of the i-th link about its center of mass, (kgmz)

angular displacement of the i-th link as shown in Figure 5.1, (raa’)

angular velocity of the i-th link, (rad/s)
angular velocity of the i-th link, immediately before the impulsive effect, (rad/s)

angular velocity of the i-th link, immediately after the impulsive effect, (rad/s)

velocity of the center of mass of the i-th link, (m/s)

velocity of the center of mass of the i-th link immediately before the impulsive

effect, (m/s)

velocity of the center of mass of the i-th link immediately after the impulsive
effect, (m/s)

inertial reference frame with unit vectors 1" and ] along the X and Y axes,

respectively

impulsive reaction force acting on the first leg’s feet, (N)

impulsive force acting on the second link (second leg) at center of mass, (N)
impulsive force acting on the third link (torso) at center of mass, (N)

impulsive force acting on the first link (first leg) at center of mass, (N)

99

Mi-imp impulsive moment acting on the i-th link at its center of mass, (Nm)

T2 external torque applied on the second leg, (Nm)

T3 external torque applied on the torso, (Nm)
E total energy of the system, (J)

E potential energy of the system when (191 , 92, 193) = ( 0 , 0, 7r ), (J)

g acceleration due to gravity, (9.81 m/sz)
Si sin 6?].

Ci cos 61.

$12 sin (6] + 62)

C12 cos (61 + 62)

S13 sin (19] + 613)

C13 cos (61 + 63)

5.3 System Dynamics
5.3.1 Equations of Motion

Consider the pendubot in Figure 5.1. Assuming an ideal system with no friction in the

joints, the kinetic energies of the links can be written as follows

__1 -2 1 .2

KEY-51161 +§m1r1 (5.1)
KE —11 (19 +9 )2+l '2 (52)

2‘22 1 2 2”“2’2 '
KE —11(6'1 +19 )2+—1— r'2 (5 3)

3‘23 1 3 2’"33 °

where

r1=[R191—(R—d1)51]i+[—R+(R—d1)C1]j (5.4)

100

 

 

 

 

;2 = "R91_(R_d2)812-2‘+:—R+(R—d2)C12jf
F3 =FR91 ‘(13 ‘d315131’7+__R+(13 _d3)C13-]

and therefore

112 =[R2 +(R—d1)2 —2R(R—d )C

p.111:
[___—__l

’22 = R2912 +(6'1 +6"2)2(R’d2)2 _2Ré1(é1 +é2)(R_d2)C12

12 =R2192+(é +9 )2(13—d3)2—2R191(él+613)(l3—d3)C13

The potential energies of the links are as follows

101v:1 =—mlgl:—R+(R—dl)C1]

PE2 =—m2g[—R+(R—d2)C12]

PE3 = —m3gI:—R +(l3 —d3)C13]
Therefore, we can formulate the Lagrangian formulation as follows

L=(KE +KE +KE

1 2 3)-(PE +PE +PE)

1 2 3

putting the results in matrix form we get

A(6)9’+B(0,6‘)6’+G(6)=T

101

(5.5)

(5.6)

(5.7)

(5.8)

(5.9)

(5.10)

(5.11)

(5.12)

(5.13)

(5.14)

(5.15)

(5.16)

(5.17)

where A(B), B(6,9), C(19) , Hand T are given by the expressions

1- - F' -1

 

 

 

 

 

 

 

 

 

 

“11 “12 “13 . 1’11 [’12 [’13 g11
A(6)= a12 2:22 0 , B(6,6)= 0 0 0 ,G(61)= g2],
. 0 0 0
_“13 0 “33, — . _331-
1 0
19: 192 , T: 2'2 (5.18)
193- 3'3-
and the terms in the inertia matrix A(0) are defined as
2
_ 2 _ _ _
all—11+m1[R +(R d1) 2R(R d1)C1]
1 R2 R d 2 2R R d C
+2"”‘2 +( ‘2)- ( '2) 12
1 R21d22RldC 519
+3””"3 +(3'3l' (3‘3) 13 (')
2
2:12:12”:2 (R—dz) —R(R—d2)C12 (5.20)
2 521
“13=I3+’"3 (’3‘d3) 'Rll3‘d3lci3 (' )
2 22
a22=12+m2(R—d2) (5. )
1 d 2 523
“33=I3+’"3(3‘ 3) (' )

and the terms in the matrices B(6,é) , 6(6) are

’1 1 = ’"1R(R —d1)élsl +”’2R(R ’d2)(‘91 +26.2)512 +m3Rll3 “’3 )(61 +29'3)513

(5.24)

612 =m2R(R—d2)192S12 (5.25)

102

b =m R(l -—d )6’ s (5.26)

13 3 3 3 313
811=m1g(R-d1)Sl+m2g(R—d2)812+m3g(l3—d3)S13 (5.27)
g21=m2g(R‘d21512 (5-23)
5’31 = ”135’ (13 “131513 (5.29)

5.3.2 Torque Expressions of the Actuated Joints

In this chapter we will obtain the torque expressions of the two joints using a slightly
different way from previous chapters. This is because the system has two motors which
create some coupling when solving for the torque expression. We are going to set

conditions for both links to achieve and then solve for the torque expressions by

eliminatingé ’s from the equations of motion. There are four cases that can happen and
they can be shown as follows
1. Both the second leg and the torso are moving

In our algorithm we impose a condition to make both legs look symmetric at all times

(i.e. 192 = -2611). In order to do that we consider, as we did in previous chapters, the

action that results in exponential decay of the motion of the second leg to the negative

double of the first one. Therefore, condition can be written as follows

(261+612)=—k12(219'1+02) (5.30)

Moreover, we will make the torso move with a constant selected angular velocity which

is going to be discussed later in this chapter. Hence, condition can be written as follows

(5.31)

93 = "k13 (‘93 ‘ 93d“)

103

In order to solve for the torque expressions we look into the equations of motion and

rearrange equation (5.17) as follows

 

 

 

 

 

 

 

 

 

 

6'=A—1(6)[T—B(6,6)—G(6)] (5.32)
where
—a_1 a_1 a—r Pa 0 a a a a -
11 12 13 22 33 12 33 13 22
-1 _ —1 —1 -1_ 1 __
A 09)" “12 “22 “23 ‘det(,4(g)) “12“33 “11“33 “13 “12“13
—1 —1 —1 2
,“13 “23 “33, _“13“22 “12“13 “11“22‘“12,
(5.33)
equation (5.32) can be expand as follows
""7 ra_lr +a—lr H — “
6’1 12 2 13 3 115
.. _ _1 _1
6.2 — a2212+a23r3 + h6 (5.34)
6 -1 -1 h
— 3- _“23’2+“33’3_ - 7—
where
h5 1
116 =—A (6)[B(6,6)+G(6)] (5.35)
_“7_

 

 

substituting the second and the third equations of equation (5.34) into equation (5.30) and

(5.31) and solving for the torques we get

_1 - - —1 —l ' '
T =“33["12(2“1+“2)+2“5+“6l‘(2“13 +a23)[k13(63_63des)+h7]

 

2 1 1 1 1 1 1 “ (5'36)
[“23 (2“13 +“23 )‘“33 (2“12 “22”
z' =—:1——[a_lr +k (61 —6’ )+h :l (5 37)
3 a—l 23 2 13 3 3des 7 '
33

The dominator in equations (5.36) and (5.37) is always negative, and in equation (5.36) if

expanded it is equal to the determinant of the matrix (A).

104

2. The second leg is moving and the torso is braking
In our algorithm we will face a situation that needs to make the torso brake while the
second leg is still moving. Therefore, the condition for the second leg will stay the same

(i.e. equation (5.30)) while the condition for the torso will be as follows

193 = —k1463 (5-33)

substituting the second and the third equations of equation (5.34) into equation (5.30) and

(5.38) and solving for the torques we get

 

_1 - - —l —1 '
= 2:33 [k12(291+62)+2h5 +h6]—(2a13 +2123 )[kl4193 +117] (5 39)
2 0-1(2a-1+a_1)—a—1(2a_1+a—1) . l
23 13 23 33 12 22
— ’1 ‘1 +k 6 +11 540
’3‘? “2372 143 7 (' )
“33

3.The second leg is braking and the torso is moving
This case is the opposite of the previous one; as a result, the condition for the torso in

equation (5.31) will hold while the condition for the second leg will be as follows

62 = -k1562 (5.41)

substituting the second and the third equations of equation (5.34) into equation (5.31) and

(5.41) and solving for the torques we get

_1 . . _1 .
T = “23 [1‘13 (“3 ”“3des)+“7l‘“33 (“15“2 “76)

 

2 _1 _1 _1 2 (5.42)
“22“33 "(“23)
z' =—‘1 a_lr +k (6 —6 )+h] (543)
3 —l 23 2 l3 3 3des 7 '

“33

The dominator of equation (5.42) is always positive.

105

4.Both the second leg and the torso are braking

The last case we want to find torque expressions when both the second leg and the torso
are braking; hence, the conditions in equations (5.3 8) and (5.41) are applied. Substituting
the second and the third equations of equation (5.34) into these equations and solving for

the torques we get

 

_1 . _1 .
T _“23 (k14“3+“7)‘“33 (k1562+h6) (5 44)
2 — 1 1 1 2 '
“22“33 ‘(“23)
— ’1 '1 +k 6 +h 545
’3‘? “2372 14 3 7 (' )
“33

5.4 Impulse-Momentum Effect

In this section the effect of the sudden impulsive force and moment on the angular
velocities of Robot’s links is going to be studied. By looking to the free body diagram of
the second leg in Figure 5.2, the impulsive force and moment on the center of mass can

be written as

- _ ~+_~—
F2impAt—m2 (v2 v2) (5.46)
- _ .+ .+_ ._ ._
11421.],11)211_12(61 +192) 12(191 +192) (5.47)
where
~+_r '+_ '+ '+ _ 2-, '+ '+ __ *-
v2 —LR61 (61+62)(R d2)C1241 _(61 +192)(R d2)512(1 (5.48)

‘72— = FR“1_ "(“1— +“2)(R‘“2)C12

~.1
l

 

 

 

(191 +62 )(R—d2)812]j (5.49)
In order to study the effect of the impulsive force and moment on the angular velocity of

the first leg, we will discuss the effect on the second joint first. The impulsive force will

106

stay with the same magnitude and direction. However, there will be impulsive moment

added due to the distance

’2 =(Ii—‘llzll’slz’?+c12j) (5'50)

 

 

 

 

Y
( g
F .
F3imp 21mp
M21mp + M31mp gijL
+ r2 X FZImp ‘ M2imp
+ ['3 X F3imp

 

 

E 1"
\ Flimp
Figure 5.2. Impulsive forces and moments on the biped

Therefore, the addition on the impulsive moment will be

“2"F21mpA’=m2(R'“2){klR'dzl'Rclzlléf+gf)+(R_d2)(駗6}2_)} (5'51)

Similarly, looking to the free body diagram of the torso, the impulsive force and moment

on the center of mass can be written as

- _ ~+_~—
F3impAt—m3(v3 v3) (5.52)

107

- _ .+ .+ _ ._ ._
1143impm.13(61 +63) 13(61 +193) (5.53)

where

N

P . . . Ta P . . _.
~+_ +_ + + _ -_ + + _ -
v __R61 (61 +63 )(13 d3)C13J b(61+193)(l3 d3)sl3() (5.54)
“3 =_R“1 ’(“1 +“3 )(13‘“3)C13_

N.‘
l

 

 

 

r ._ ._ 2.
L(61 +63 )(13 —d3)sl3() (5.55)

Again to study the effect of the impulsive force and moment on the angular velocity of
the first leg, we will discuss the effect on the second joint first. The impulsive force will

stay with the same magnitude and direction. However, there will be impulsive moment

added due to the distance

“3 =(l3 _d3)(_Sl3F+C13]) (5'56)

Therefore, the addition on the impulsive moment will be

_. a _ _ _ _ .+ ._ _ .+ _ ._
r3 xFBimpAt _m3 (13 d3)([(l3 d3) RC13](191 +61 )+(13 d3)(193 63 )( (5.57)
Now, if we look to the free body diagram of the first leg in Figure 5.2, the impulsive

force and the impulsive moment on the center of mass can be written as

- __ - _- _- : fl+_,_
Fimply—”(blimp F2imp F3impjm m1 (v1 v1) (5°58)
- _ .+_ ._
1141imm_11(6l 61) (5.59)
where
9+ = _R—(R—d )C _6+7—(R—d )8 6+] (5.60)
1 L 1 1_ 1 1 1 1
—.— 7 - '—‘.' '—‘:
v1 =_R—(R—d1)C1d1911—(R—d1)81611 (5.61)

 

 

and Flimp is the impulsive reaction force at the first joint which can be found from

equation (5.58) to be

" _ ~+_~— ~+_~— ~+_—-—
FlimpAt—m1(v1 v1 )+m2 (v2 v2 )+m3 (v3 v3 ) (5.62)

108

Now, we want to calculate the impulsive moment on the first leg center of mass due to
the impulsive forces applied to its joints. The second joint has an impulsive force with the
same magnitude and the opposite direction from that in the second joint in the second
leg’s and the torso’s free body diagrams. This impulsive force will create an impulsive

moment at the leg center of mass due to the distance

F4 = (R—d1)(sli—C1j) (5.63)

Therefore, the addition on the impulsive moment from this impulsive force will be
_. —o — — _ _ — _ 0+ — -_ — 1+ — .-
r4XFSum ‘m2(R “1)(_(R “2)“2 RC1_(“1 “1 )+(R “2)C2(“2 “2 )}

(1 —d )C —RC (6+ —6‘)+(1 —d )C (6+— —6 3)((5. 64)

+”73(R‘“1)(_3 3 3 1, 1 1 3 3 3 3

 

 

where

FSum = -(FZimp + F3imp ) N

In addition, the impulsive reaction force will add another impulsive moment at the first

(5.65)

leg center of mass due to the distance
r1=(R—d1)Slz +[R—(R—d1)C1]j (5.66)

Hence, the impulsive moment added due to this reaction force is

“1X“11mpA’P(’" 1R2( “(:( “12)2 W1)RC1(+’" +(‘R (“1)(R’“2)C2
‘R(R’“1)C1“2)12)m3(R2+(R “1)(l3‘ “3)C3 “(R “1)“1
‘R(’3‘“3)C13()(;" “1—"”’) 2(R “2)((R “12)“ ‘CR C12)(“2‘“2)
—m3(l3— d3)[(R—) d )3C R—C3193+]( —63‘) (5.67)

Therefore, the impulsive moments at the first link center of mass due to the impulsive

forces can be written as

109

—. —.

At - r2 x FZimpAt

M. zit—~11? . At—M.

Izmp 21mp 3zmp
-r3 x F3impAt + r4 x FSum + rl x FlimpAt (5.68)

equating equation (5.68) to equation (5.59) and substituting the equations (5.47), (5.51),

(5.53), (5.57), (5.64) and (5.67) we can get a formula that can specify the relation

between the angular velocities of the links before and after the impulsive force and

moment applied to the second leg and torso. This can be shown as the following

.+_._: .__.+ .__.+
2111(61 6l ) 2112(62 192)+a]3(63 63 ) (5.69)

where a a and al are defined before in equations (5.19), (5.20) and (5.21),

11’ 12 3

respectively. In our algorithm we are going to see two cases in which we specify some of
the angular velocities and solve for the angular velocity of the first leg after the
application of the impulsive force and moment.

M

In our algorithm the torso will be either moving with a constant specified angular velocity

or held fixed for a certain time. In both ways, if we suddenly stopped the second leg (i.e.

193' = 0) while the torso continuing on its condition, the new angular velocity of the first

leg can be found from equation (5.69) to be

.+ _ ._ ._
191 — 61 + a1262 (5.70)
and with the condition 612— = —2611— is always maintained this expression can be put as
_ a — 2a _
6+ = (L—J—Z—(K (5.71)
1 a1 1 1

Similarly, if we suddenly brought the second leg from zero angular velocity to

19; = —2191+ then, the new angular velocity of the first leg can be found as

110

. a .
191+ =(——l—21——(61‘ (5.72)
“11 ‘ “12

Case 2

In this case the second leg is always maintaining the condition 192 = —2191 before and after

the impulsive force and moment, and the torso will be stopped suddenly (i.e.6’; = 0 ).

Therefore, from equation (5 .69) and the condition specified above we can find the new

angular velocity of the first leg to be as

. . a .
61+ = 61‘ +[———1—:-—(19; (5.73)
“11 ‘ “12
Similarly, if the torso suddenly brought to a certain angular velocity from zero the new

angular velocity of the first leg can be found as
. . a .
191+ = 191-—|:———1%—(193— (5.74)
“11 ‘ “12
5.5 Energy Consideration of the System

5.5.1 Energy change when both the second leg and the torso are moving

In our system we consider two symmetric legs that always keep a symmetrical distance

between them. This means that both legs have the same kinematic parameter and the

conditions (62 = —261 , and 192 = —2191) are hold, which reduces equations (5.19), (5.20)

and (5.22) to be the following

all =2(11 4.1721(R2 +(R—d1)2 —2R(R—d1)C1(}

+13 +m3 (R2 +(13 —d3)2 —2R(13 _d3)C13( (5.75)

111

a12=11+ml[(R—d1)2—R(R—d1)Cl] (5.76)

2
a22 =11+m1(R—d1) (5.77)

The rate change of the energy of the system is known as
E = 72192 + r319 (5.78)

and since both the second leg and the torso are moving, the torque equations are defined

in equations (5.36) and (5.37). In addition, the angular velocity of the torso is going to be

selected as é3des and with the condition 92 = —291 is taking in place and using equation

(5.37), we can show that equation (5.78) becomes as

_1 h
'— ' “236 7 6 579
E’ ““1171 3des ’27:? 3des ( ' )
33 33

Moreover, equation (5.35) can be redefined based on the above changes as

r
r - —1 —1 —1
h5 “11N1+“12N2+“13N3

_ —1 —1 —1
h6 _ a12N1+622N2+aZ3N3 (5.80)

h -1 —1 —1
— 7~ _“13N1+“23N2 +“33N3_

 

 

 

 

where
P . . 2 1
N1 —m3 ([3 'd3)S13 [(61 + 63.1.1.1) + g]
N, = m,g(R—ar,)sl (5.81)
N3 —m3g(l3 —d3)S13
_ .1

 

 

The torque expression in equation (5.36) can then be rewritten as

2' = a33l(2a1-1]+a1_2l)_al_3l(201-31+a2-31) N _ N (5 82)
2 a; (2013' +a;_.‘)—a;.‘ (2a.? +4.1) ' 2 '

112

This will make equation (5.79) to be a function of 61,61,6363 and the kinematic
parameters of the system. Therefore, depending on the kinematic parameters of the

system we can select values for 613 that will reduce the energy of the system.

§‘\.'; “'1/">§

Torso motion

  
   

Torso motion

Biped motion

Biped motion

 

 

\\\\\\\\\\\ \\\\\\\\\\\
(a) (b)

Figure 5.3. (a) The first leg has negative angle value. (b) The first leg has positive angle
value.

In order to reduce the energy of the system, the torso should always be located on the

opposite side of the vertical to counter the motion of the biped by its weight as shown in

Figure 5.3. Therefore, by selecting the value of 63 the range of values63 can be

predicted. For instance, one can see from Figure 5.3 (a) that when the first leg start from
the maximum negative angle value 611, the torso angle can be more than 180 degrees and

stays in the desired region. This true until the first leg angle reaches zero, then the

113

situation will be reversed as seen in Figure 5.3 (b). Hence, from this fact we can study the

range that can be selected for 93 assuming the range for 611,191 .

 

Figure 5.4. The range of rate of change of energy below zero with 193 is fixed to be 72' rad

Using the kinematical parameters discussed in the numerical simulation at the next
section and with the help of Mathematica we can plot the solution of equation (5.79) that
will show the energy decrease of the system. We will do two plots that illustrate the cases
(a) and (b) discussed above. For case (a), 63 is selected to be (It) in order to make the
torso in opposite side of the motion of the biped for all values of 611 which is [-0.2, 0]
rad. Then, the ranges of values for 91,193 are selected to be reasonable. Since the motion

of the biped is to the positive direction we will select the range of 191 to be [0, 8] rad/s.

The torso range is selected to be [20, 5] rad/s to show the effect of the positive and

114

negative values on the solution. The plot of this case is illustrated in Figure 5.4.

Similarly, for case (b) we select 03 to 2.9 rad in order to make the torso stay on the

opposite side of the motion for all values of 61. The range of 61 now is [0, 0.2] rad, and
for the angular velocities we select the same ranges as in case (a). The plot of this case is
illustrated in Figure 5.4. Looking into these plots, one can deduce that in order to keep
the torso on the opposite side of the motion and therefore decreasing the energy of the

system, the angular velocity 613 should be higher in magnitude and opposite in signs with

the angular velocity of the leg 91. This fact is an intuitively obvious and it was expected

to get.

 

 

 

Ox \\

(_1
)—a
\ 1
\\\ \n \\ x \‘
\}\\ \\\\\\ \

\} _\.
5‘
\\\

 

 

 

 

 

 

 

Figure 5.5. The range of rate of change of energy below zero with 613 is fixed as 2.9 rad

115

5.5.2 Energy change due to the impulse-momentum effect
In section 5.4 we discussed two cases in which the impulse-momentum effect can occur.
In this section we want to discuss how this might affect the energy of the system.
9&4.
The biped is walking with only the first leg on the ground. In addition, the second leg is
moving forward faster that the first leg in order to take over its role of walking once the
first leg tip reaches the ground. The way to represent this in our algorithm is to make the
second leg move twice the distance of the first leg in the opposite direction. Then, before
the second leg angle reaches the maximum value, it will suddenly stop in order to stay on
the first leg velocity only. This was represented in equation (5 .71) and we will consider
this as event (a) as the following
14>.=(f‘e:—2“1—211a1
11 a
Immediately after that, the two legs are going to switch the roles and therefore, the
coordinates are redefined. The second leg is going to move with the first leg angular
velocity (which was already happening after the sudden stop), and the first leg will be
impacted with a sudden torque that make its angular velocity double the angular velocity
of the second leg and opposite in direction. This was represented in equation (5.72) and
we will consider this as event (b) as the following
(“1). (Elli—((91),. (5'84)
11 12
It should be noted that the difference in time between the two events are very short.

Therefore, the difference in the constants a and a12 are minimal or unnoticed. This will

11

116

make the final change of the first leg angular velocity (611+ )b depend only on its status

before the application of event (b) (i.e. (191— )b ). In the short time between event (a) and

(b) the energy of the system is going down due to the movement of the torso as discussed

above. Hence, we can deduce from that

(61")b 3(61+)a (5.85)
Substituting (5.83) and (5.85) into (5.84) we get
(61+)b s (61")a (5.86)

which means that the system slows down or at least stays at the same velocity during this
events. This result is desired because it means that the energy did not increase by these
events and we can do the switching without the fear of disturbing the slowing down
process of the system due to the impulsive forces.

QM

During the process of walking and before the two legs do the switching discussed in the
first case; the torso might reaches the border of the angle designed not to exceed. Hence,
it will be stopped suddenly as described in equation (5.73) and we consider this event as

event (a) again as the following

(61+)a = (61’ L (“(8143—2113), (63")a (5.87)

Once the two legs switch the roles, an impulsive force will bring the torso’s angular
velocity to the same value as before event (a) or less depending on the value of the first
leg’s angular velocity before event (b) is applied. Event (b) can be deduced from equation

(5.74) as follows

117

o c a o I
141 =14) (+1161
b b “11 ‘ “12 b ’7
During the time between events (a) and (b), the first leg’s angular velocity will slow
down because the torso is producing a moment by its weight in the opposite direction of

the biped motion. However, for the sake of argument will assume that this angular

velocity stays the same or

(6‘) = (6+) (5.89)
1 b 1 a
Thus, if we substitute equation (5.87) into equation (5.89) and used the result in equation

(5.88) we will get

a - a o a 1
(61+) =14) (—-—( (a) (———] (63*) <53»
b a “11" “12 a a “11‘ “12 b b
Because of the assumption in equation (5.89) the new desired angular velocity of the

torso will stay the same or
6“) =(6‘) (5.91)
( 3 b 3 a
Therefore, equation (5.90) can be simplified to be

. . a a _
141 1411—13—1 (+1 141
b a “11' “12 a “11’ “12 b a

This means that the change in the constants a1 1, 012 and “13 is the primary effect of the

result. By looking to Figure 5.6 we can get the following

(“13), ‘(“13)b = ""3R(’3 ‘“3)((C13)a ‘(C13)b) < 0 (5'93)

(“11 ‘2“12)a _(“11‘2“12)b = ‘2’"1R(R"“1 )((C1 )1 _(Cl )1.)

118

‘2’"3R(’3 "“3 )((C13 )a "(C13 )bj < O (5'94)

  

  

Biped motion Biped motion

 

 

  

o

\\\\\()\\\\\\ \\\\\(b)\\\\\\
a

Figure 5.6. (a) The start of event (a). (b) The start of event (b).

 
 

From equations (5.93) and (5.94) we find that

a a
(+( >(——l%——( (5.95)
“11' “12 a “11' “12 1,
From this relation and knowing the fact that (6; ) is always has the opposite sign of
a

(61—) , one can deduce from equation (5.92) that the angular velocity of the first leg will
a

slow down after performing events (a) and (b). This is a very comforting result because it
means that the energy did not increase (in fact over all decreased) by these events and we
can perform impulsive-like forces on the torso without the fear of disturbing the slowing

down process of the system.

119

5.6 Stabilization Algorithm

In this section we discuss a three-step algorithm for stabilizing the biped about its

standing configuration. The basic idea of our algorithm is to make the energy of the

system reaches the desired level ET and the links of the system to be located inside the

region of attraction of a linear controller, which guarantees asymptotic stabilization of the

desired equilibrium.

1. Initialization:

o Linearize the dynamic equations of the biped in equation (5.17) about the desired

equilibrium point (

linear system will be

61,61,62,62,63,63

f(x,u)=x=Ax+Bu

where x is the state space representation defined as

x=6 x=6

11’2

l,

x=6

3

2’

’2

=u

)= (1.0.05, 0.0, :0. 1, 0.0, II :c 0.05, 0.0). The

(5.96)

2 , r3 = u3 (5.97)

Therefore, the matrices A and B can be found at the top position as

—

 

h

0
6f2

“x1

0
6f4

“xi

0
6f 6

“x1

1

0

0
6f2

“x3

0
«27,,

“x3

0
6f 6

“x3

0 0
af
0 a.“
5
1 0
5f
0 31
x5
0 0
6f
0 a“
5

0

0

0

—

 

.11

 

b

0
6f2

6112

0
6f4

6u2

0
6f6

6u2

0 "1
6f2

6u3

0
6f4

““3

0
“L6
au

 

3.

(5.98)

0 Using the model of the linearized system, design a linear controller to render the

desired equilibrium of the biped (nonlinear system) locally asymptotically stable.

120

The system is controllable as the controllability matrix C is full rank
C = (13 AB A23 A313) (5.99)

Since this equilibrium point is not stable, the matrix A is expected to have
eigenvalues in the right hand side (RHS) plane. Thus, to stabilize the system
using a linear controller, new eigenvalues are assigned to the controlled system

(A -BK4) in the left hand side (LHS) plane. Then, the desired feedback gain matrix

K 4 is calculated to find the control law

’2
T =‘K4(x‘xeq) (5.100)

3
This control law is applied to the nonlinear system and stabilizes it at standing

position.
2. Bringing the energy of the system to E T:

After the system is disturbed with a high disturbance, the energy of the system will
increase. The biped then is going to lean to the direction of the disturbance standing only
on the first leg. In addition, the second leg will move with double the angular velocity of
the first leg in order to get ready to take over the role of walking from the first leg. At the
same time, the torso will move in the opposite direction of the motion of the biped to
slow it down and consequently reducing the energy of the system. The torso will move
with angular velocity higher in magnitude than the first leg’s angular velocity. Once the
torso exceeded the vertical line going to the opposite side of the biped motion, the energy
will start to go down as discussed in section 5.5.1. During this process the biped links

will be facing impulse-like forces in order to switch the roles of the two legs and to keep

121

the torso inside a selected small angular region from the vertical. We already showed in
details in section 5.5.2 that these impulsive forces overall will not add energy to the
system. In fact, the energy of the system can go down after the completion of the
switching between the two legs or after starting moving the torso again.

3. Stabilization:

By the end of the second step the energy of the system is near the desired one (i.e. ET)“

Then, the torso is going to be hold at its position, which is within a small angle a from

the top position, i.e. 63 = 7:. The biped will continue the motion slowly and once the
second leg angle reaches a position within a small angle ,6 from the desired one, i.e. 62 =
-0.1 rad if the biped. is moving forward and 62 = 0.1 rad if the biped is moving backward.

This will make the biped swing back and forth around the desired configuration

(61,61,62,62,63,63 ) = (:0.05,0.0,;L'0.1,0,7r$0.010). The linear controller in equation

(5.100) will be invoked to stabilize the equilibrium point once the biped configuration
enters the region of attraction of this equilibrium point.
Numerical Simulations
The kinematic and dynamic parameters of the biped were assumed to be
R :10 m, m1 = m2 =1.4 kg, d1=d2= 0.32 m, 11: 12 = 0.16 kgmz,
13: 1.0 m, m3 = 1.0 kg, d3 = 0.5 m, 13 = 0.08333 kgm2
For this choice of parameters, ET was evaluated to be 23.5312 J.
In Figure 5.7 we show the simulation for our algorithm using these parameters. It can

be seen from this figure that the system managed to stand in a very short time after

walking two steps. The initial conditions we imposed were as follows

122

(61,61,62,62,63,63)=(O,O,O,O,7r,2)

The dashed lines illustrated on the figure between arrows indicate the period in which
events (a) and (b) of cases 1 and 2 happened. The first set of arrows happened between
the time period (t = 0.15 — 0.24 sec). It started with event (a) of case 1 and then, at the
time t = 0.19 sec event (a) of case 2 started. Finally, at the end of this period, event (b) of
cases 1 and 2 happens simultaneously. The overall energy of the system decreased at that
time. Similarly, the second set of arrows happened between the time period (t = 0.41 —
0.74 sec). It started with event (a) of case 2 and then, at the time t = 0.66 sec event (a) of
case I started. Finally, at the end of this period, event (b) of cases 1 and 2 happens
simultaneously. Again the overall energy of the system decreased at that time. After the
energy of the system reaches the desired value, the control law switched to the linear

controller at the time t = 1.21 sec which stabilized the equilibrium.
5.7 Remarks

In this chapter we discussed the stabilization of an under actuated biped robot with torso
after the application of large disturbances using the impulsive-momentum approach
introduced in previous chapters. The biped used both legs to avoid falling down and
walked few steps before stabilization. We have obtained very good results which
establish the generality of the impulse-momentum approach. This approach is in fact, as
we mentioned in chapter 4, well suited for the biped model since it restricts the links from
going over certain limits. Hence, the idea of using this approach on different systems is

very promising and it will Open the door for a new way of research.

123

 

 

 

 

 

 

 

3

l
I
l
l
I
I
l
U
L
l
l
I
l
I
I

 

 

 

 

time (sec)

 

 

 

l

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2 0

0.
see a

 

0

2. . 3
3

M 5. 0 5. 4. . . 0
I. .I. o
_ Amhoflv S _ 3a.: 3 Amhufiv Nc 3a.: 3 2 Ago—2v mm 336 N».

5.
2
_

Figure 5.7. Plot of joint angles, joint angle velocities, control torques, and energy of the

system

124

CHAPTER 6

Conclusions and Future Work

In this dissertation we have developed a new control methodology for underactuated
mechanical systems with a single degree of underactuation. From a classical mechanics
point of view, underactuated mechanical systems are those that have fewer generalized
forces than their degrees of freedom. Such systems are quite common in the literature and
some of the systems that have been recently investigated include walking robots, mobile
robots, space and undersea robots, ships, helicopters hovercrafts, and satellites.
Underactuated systems have generated significant interest in the control community since
underactuation reduces cost and weight, can help deal with actuator failure, and since
many systems are naturally underactuated.

Underactuated systems pose challenging problems in control because many of the
methods developed for completely actuated systems, such as feedback linearization,
Lyapunov theory, passivity, etc. are not directly applicable to underactuated systems. The
dynamics of underactuated systems may contain non-minimum phase zero dynamics,
nonholonomic constraints, impulsive disturbances and many other characteristics that
make their control problems difficult, and hard to be generalized. A majority of the
research on underactuated systems has focused on benchmark problems such as the

inverted pendulum, the rotational inverted pendulum, the pendubot, and the acrobot; but

125

control methods developed for these systems, in most cases, are not general where they
can be applied to many systems. For example, the pendubot and the acrobot are similar in
kinematic structure, and although many control algorithms have been proposed, only two
methods are applicable to both systems.

In this dissertation we have developed an impulsive control methodology for swing-
up control of the pendubot and the acrobot. The method is general and can be additionally
used for balance control of biped robots. Each of these systems, the pendubot, the
acrobot, and the biped, are locally asymptotically stabilizable using linear control and the
primary objective of control design is to bring the configuration of the system within the
region of attraction of the equilibrium. This is a common goal for many researchers
working with underactuated systems. While most of these results are largely applicable to
specific platforms only, our control methodology is general and can be applied to
multiple platforms and has the potential to be applied to underactuated systems with
multiple degrees of underactuation.

Underactuated mechanical systems have been investigated by many researchers but
our approach is the first where impulsive forces are used as control inputs for these
systems. Although there has been prior work on impulsive control of dynamical systems,
control of underactuated systems has not been explored. In our approach we impose joint
restrictions on the actuated links and make them perform rest-to-rest maneuvers with
small amplitudes around their desired configuration. The rest-to-rest maneuvers are
designed to bring the unactuated joints to their desired configuration. These joint
restrictions imposed on the actuated links help us control both the passive and active

joints using only the active joints.

126

 

In spite of the copious work on impulsive control, there has been no attempt to look at
the challenges of practical implementation. Most of the work in the literature has
modeled the effect of the impulse as a discontinuous jump in the states and the problem is
dealt mathematically. In our approach, we have investigated the challenges of practical
implementation. The impulsive forces are calculated based on change in energy of the
system and we include the analysis of how this change in energy is affected when the
torques are not perfect impulses. Despite the use of impulsive forces, it is important to
mention that it is not necessary that the system will require the use of large actuators. For
example, for mechanical systems where electric motors are used, our approach benefited
from the fact that motors in general have a peak torque which is greater than the
maximum continuous torque. The maximum torque required by our algorithm has been
limited by the peak torque.

In this dissertation we apply our impulsive control methodology to the control
problems of the pendubot, the acrobot which are benchmark problems. We compared our
results to the best results available in the literature and prove that our results are
comparable and very effective. In addition, we applied our methodology to another
underactuated system the synthetic-wheel biped to show the generality of the approach.
For each of the systems mentioned above, control is achieved by making the energy of
the system converge to the level that the system would have when its configuration is at
the desired equilibrium. In the case of the pendubot, impulsive braking torques are used
and this results in energy loss. For the acrobot, impulsive braking torques can be used,
but a method based on impulsive torques that add to the system energy is more efficient.

In the case of the biped, impulsive torques are used to increase and decrease the energy of

127

the system depending on the location of the system in the configuration space. Clearly,
impulsive forces can be used to increase and decrease the energy of a system; and
therefore, our approach has the potential to be generalized to other systems. In fact, our
approach can be applied to a wide range of dynamic systems, not only underactuated
systems.

The jump in generalized velocities and energetics of impulsive forces were derived
from force and moment relations of the underactuated link for the pendubot, acrobot, and
synthetic—wheel biped. This process becomes cumbersome when the degrees of freedom
and/or underactuation increases. To overcome this difficulty for the general case, we

consider Lagrange's equation for an n-dof system with m actuators:
A(6)6+B(6,6)6+G(6)=T, 6=(6,,6,,...,6,,) (6.1)
where 6,, i = l, 2, ..., m and i = m + 1, m + 2, ..., n, are the generalized coordinates

corresponding to the actuated and passive joints, respectively. The generalized forces

corresponding to the passive joints are zero and hence T =(r,,...,z'm,0,0,...,0). If we

denote the lower blocks of A, B, and G matrices with (n - m) rows by A2, 82, and G2, then,
integration of the equations of motion of the passive joints over the time interval of action

of the impulsive forces yields

( A, (6)6At + (B, (6,6)6At + (G, (6)At = 0 (6.2)
In the above equation, (G, (6) At #0 since At is infinitesimal and (B,(6,6)6At=

IB,(6,6)d6 z 0 since d6 z 0 over the duration of the impulse. Thus, equation (6.2)
gives

(A,(6)6'Az=0:> (A,(6)d6=0:>A,(6)[6+—6“]=0 (6.3)

128

The above equation provides (n - m) constraints involving 11 unknowns, namely, 6: , i = 1,

2, , 11. To solve these constraint equations, an additional m equations are required in
terms of the unknowns. The constraints are dependent on the entries of A2(6), a sub-
matrix of the mass matrix; and hence, on the configuration of the system at the time of
application of the impulse. The pendubot, acrobot, and synthetic-wheel biped with locked
feet are systems with n = 2 and m = l and we have verified that the equations, which
relate the jump in the angular velocities for these systems, can be derived from equation
(6.3). Equation (6.3) will be invaluable in the analysis of general underactuated systems.
In this dissertation we have derived the relation in equation (6.3) for each system, the
pendubot, the acrobot, and the biped using free body diagram analysis which verified the
analysis discussed above. This equation is the core of our analysis when dealing with any
new system in the future. Our future work will investigate the effectiveness of this
methodology to control multi-link systems with multi-degree of underactuation and also

we will look to tracking control problem of bipeds.

129

[1]

[21
[3]
[41

[5]
[61

I71

[8]

[9]

[10]

[11]

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