ACTIVE SLIDING SYNTHETIC WHEEL BIPED:
DYNAMICS, CONTROL, AND VERIFICATION
By
Kyle James Crayne

A THESIS
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
Mechanical Engineering–Master of Science
2014

ABSTRACT
ACTIVE SLIDING SYNTHETIC WHEEL BIPED:
DYNAMICS, CONTROL, AND VERIFICATION
By
Kyle James Crayne
The active sliding synthetic wheel biped is an underactuated, planar, biped robot capable
of sliding while walking. This biped is a modification of the previously developed Michigan
State University (MSU) synthetic wheel biped (SWB). To create the sliding-while-walking
motion, a hybrid controller was created that consisted of a continuous controller over the
course of the step and a discrete controller at the end of the step. Partial feedback linearization was used to control the biped’s motion to a specified gait during the step. By discretely
adjusting gait parameters at the end of each step through chaos control, the biped’s motion
was stabilized over multiple steps. Numerical simulations were used to determine the configurations in which the biped would step and to validate the hybrid control method. New
hardware in the form of belted feet and an electronic controller were added to the previous
MSU SWB to account for the increased complexity of sliding while walking. To implement
the controller on the physical system, the control method was modified from the theoretical design to account for physical realities. Once altered, the biped successfully walked for
multiple steps in experiments by using the hybrid control method.

ACKNOWLEDGEMENTS

I would like to thank my advisor Dr. Ranjan Mukherjee and my committee members Dr.
Steve Shaw and Dr. Hassan Khalil. Their guidance and instruction were fundamental in the
success of this project.
I would also like to thank my colleagues in the Dynamics and Controls lab. Specifically, I
want to thank Frank Mathis for his outstanding support with all aspects of the project and
Connor Boss for his excellent hardware assistance. Without their help, this project may not
have been possible.
The support provided by the National Science Foundation, NSF Grant CMMI 0925055,
is gratefully acknowledged. Advanced Motion Controls donated motor drives and discounted
others in support of this project as well.
Finally, I want to thank my friends and family for their never-ending support throughout
my academic career.

iii

TABLE OF CONTENTS

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

v

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

vi

CHAPTER 1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . .

1

CHAPTER 2 DYNAMICS AND CONTROLS . . . . .
2.1 Mathematical Modeling . . . . . . . . . . . . . . . . .
2.1.1 Biped Dynamics . . . . . . . . . . . . . . . . .
2.1.2 Foot Interchange . . . . . . . . . . . . . . . . .
2.2 Controller Design for Continuous Dynamics . . . . . .
2.2.1 Partial Feedback Linearization . . . . . . . . . .
2.2.2 Desired Gait Design . . . . . . . . . . . . . . .
2.2.3 Simulations of Continuous Dynamics . . . . . .
2.3 Controller Design for Hybrid Dynamics . . . . . . . . .
2.3.1 Periodic Behavior . . . . . . . . . . . . . . . . .
2.3.2 Chaos Control . . . . . . . . . . . . . . . . . . .
2.3.3 Simulations of Hybrid Dynamics . . . . . . . . .

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4
4
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6
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11
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14
15
16

CHAPTER 3 HARDWARE DESIGN . . . . . . . . . . . . . . . . . . . . . . 20
3.1 Sliding Foot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.2 Electronic Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
CHAPTER 4 EXPERIMENTAL VALIDATION . . . .
4.1 Controller Implementation . . . . . . . . . . . . . . . .
4.1.1 Programming and Operation . . . . . . . . . . .
4.1.2 Control Changes . . . . . . . . . . . . . . . . .
4.2 Experimental Results . . . . . . . . . . . . . . . . . . .

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CHAPTER 5 CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . 34
APPENDIX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

iv

LIST OF TABLES

Table 2.1

Physical Parameters for Simulation

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

13

Table 4.1

Experimental Physical Parameters . . . . . . . . . . . . . . . . . . . . . .

30

Table .1

Electrical Hardware on ECU . . . . . . . . . . . . . . . . . . . . . . . . .

43

Table .2

Additional Electrical Hardware . . . . . . . . . . . . . . . . . . . . . . . .

43

v

LIST OF FIGURES

Figure 1.1 Synthetic Wheel Biped (SWB) by Flynn [1] . . . . . . . . . . . . . . . .

3

Figure 2.1 An arbitrary configuration of the active sliding synthetic wheel biped . .

5

Figure 2.2 A schematic of the biped about to step. Stance and swing denoted in
the figure refer to their designations before the step happens. . . . . . . .

7

Figure 2.3 Phase portrait of passive coordinate θ when α = 0. The vertical lines
at θ = −β/2 and θ = β/2 indicate the bounds for which the dynamics
are valid. The dashed line represents the coordinate transformation
q + = T q − at the end of the step. . . . . . . . . . . . . . . . . . . . . . .

14

Figure 2.4 Simulated Angle States. Each vertical dashed line represents a step. . . .

17

Figure 2.5 Plot of error states vs time . . . . . . . . . . . . . . . . . . . . . . . . . .

18

Figure 2.6 Plot of continuous control effort vs time . . . . . . . . . . . . . . . . . .

19

Figure 3.1 Picture of Active Sliding Synthetic Wheel Biped . . . . . . . . . . . . . .

20

Figure 3.2 Comparison of old (top) and new (bottom) feet . . . . . . . . . . . . . .

21

Figure 3.3 ECU block diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

22

Figure 3.4 Picture of ECU mounted on robot . . . . . . . . . . . . . . . . . . . . . .

22

Figure 3.5 Picture of motor amplifiers mounted on robot . . . . . . . . . . . . . . .

23

Figure 4.1 Phase portrait of passive coordinate θ with α = 0 when the inside leg
(left) and the outside leg (right) are the stance leg. The vertical lines
at θ = −β/2 and θ = β/2 indicate the bounds for which the dynamics
are valid. The dashed line represents the coordinate transformation
q + = T q − at the end of the step. . . . . . . . . . . . . . . . . . . . . . .

28

Figure 4.2 Experimental Angle States. Each vertical dashed line represents a step .

30

Figure 4.3 Experimental Angle State Errors . . . . . . . . . . . . . . . . . . . . . .

31

Figure 4.4 Experimental Torques . . . . . . . . . . . . . . . . . . . . . . . . . . . .

32

Figure 4.5 Experimental Chaos Control Error . . . . . . . . . . . . . . . . . . . . .

33

Figure .1

38

Biped Model with forces . . . . . . . . . . . . . . . . . . . . . . . . . . .

vi

Figure .2

ECU Board - Component Layout . . . . . . . . . . . . . . . . . . . . . .

44

Figure .3

ECU Wiring Schematic . . . . . . . . . . . . . . . . . . . . . . . . . . . .

45

vii

CHAPTER 1
INTRODUCTION

Bipedal robots and walking machines have superior mobility over non-uniform terrain when
compared to wheeled mobile robots. However, wheeled robots are significantly more efficient
on flat terrain when compared to walking machines. Because human-friendly environments
incorporate large stretches of flat terrain combined with non-uniform features such as stairs,
it is desirable to have machines capable of traversing both terrains in the most efficient
manner possible. A wheeled bipedal robot that can combine the functionalities of a wheeled
robot and a traditional biped could provide such mobility.
Active wheeled bipeds designed by Matsumoto et. al. [2] and Hashimoto et. al. [3] are
capable of rolling on their wheels like a car and stepping like a biped with locked wheels.
Itabashi et. al. [4] studied bipeds with passive wheels and demonstrated skating motion.
These passive skating bipeds have the ability to roll on wheels, but they cannot control their
wheels directly. Neither the passive skating bipeds by Itabashi [4] nor the active wheeled
bipeds of Hashimoto [3] or Matsumoto [2] exploit the full potential of the hybrid platform
since they do not consider sliding while walking.
The main advantage of the sliding-while-walking motion is increased versatility in terrain
navigation. While Matsumoto [2] can roll on flat ground and step over obstacles, it needs
an explicit transition point between rolling and walking. If the sliding and walking motions
are combined, it would be possible to transition between different terrain without changing
control strategies. Such control strategies, by design, guarantee stability during the transition
phase. If separate controllers are used for rolling and sliding motions, one would have to
additionally ensure stability of the switched system during transition. The sliding-whilewalking motion can enable transport over discontinuous terrain such as a step between
a side-walk and a road without needing a complete stop. Sliding-while-walking motion

1

is a combination of the efficiency of rolling and the versatility of walking. While active
wheeled bipeds will require more sophisticated control strategies when compared to those
used by Matsumoto [2], Hashimoto [3], and Itabashi [4], the added complexity could enable
slip rejection in wheeled bipedal robots. A simple, active biped design based on McGeer’s
Synthetic Wheel [5] was used for the development of the control methods for this hybrid
motion.
The “Synthetic Wheel” by McGeer [5] refers to bipeds with circular arc feet that, when
walking, create a continuous rolling surface like a wheel. Asano et. al. [6] showed that
the speed of bipeds with this arc foot design can be increased by decreasing the energy lost
due to foot-ground impact. These biped designs are primarily passive walkers, but can be
modified with active joints. Different active designs such as MABEL [7] have more mobility
than their passive counter-parts but use significant energy to maintain stability and track
trajectories. In an effort to strike a balance between the mobility of active bipeds and the
efficiency of passive bipeds, Flynn et. al. developed the active synthetic wheel biped with
torso (SWB) [8].
Starting from Flynn’s synthetic wheel design shown in Figure 1.1, we have added sliding
capabilities that can be realized through the use of an active belt on the feet. While this
work explicitly analyzes sliding-while-walking motion, it may also be beneficial to those who
study motion in changing inertial frames, such as stepping onto a conveyor belt or escalator.
This hardware also has applications to slip rejection control, such as remaining stable when
slipping due to spilled water on a tiled floor. Instead of building precise low-friction surfaces
to test slip rejection methods, this hardware could be used to simulate slip at any time
through the use of the active belt on the feet. While not all of these applications will
be addressed in this paper, the hardware and overall control design being used would be
applicable to these important problems. This paper investigates the control of sliding-whilewalking locomotion, and the effectiveness of the control is analyzed through simulations.
In Chapter 2 we discuss the dynamic model and theoretical control development for

2

Figure 1.1: Synthetic Wheel Biped (SWB) by Flynn [1]

the biped. In Section 1 we present the mathematical model of the biped platform, which
includes the dynamics of stepping and the foot-ground interaction. Section 2 covers the
controller design for trajectory tracking during each step. This includes designing the desired
trajectories and verifying that the biped will step. Section 3 presents the discrete chaos
control approach used to guarantee the stability over multiple steps. Numerical simulations
are presented to demonstrate the sliding-while-walking hybrid gait.
Chapter 3 explains the changes in hardware to the SWB to accommodate the desired
motion. Section 1 presents the design of the sliding foot, and Section 2 covers the design of
the electronic control unit (ECU).
The experimental results are discussed in Chapter 4. Section 1 explains the many changes
to the theoretical controller that are required for the biped to operate. Section 2 presents
experimental data of the biped while it is walking while sliding.

3

CHAPTER 2
DYNAMICS AND CONTROLS

2.1
2.1.1

Mathematical Modeling
Biped Dynamics

The active sliding synthetic wheel biped is shown in Figure 2.1, which is a slight modification
of Flynn’s SWB [8]. It has four generalized coordinates: θ, φ, ψ, and x. The biped stands
on its stance leg, while the swing leg and torso are connected to the stance leg at the hip.
The angular displacement of the stance leg, measured counter-clockwise from the downward
vertical axis, is defined as θ. The angular displacement of the torso with respect to the
stance leg is φ and the angular displacement of the swing leg with respect to the stance leg
is ψ. The linear displacement of the point of contact of the stance leg is defined as x. The
length of the torso is denoted as lt and the center of mass of the torso along the axis of the
torso is measured as dt from the end of the torso. Similarly, R is the length of each leg and
dsw and dst are the locations of center of mass for the swing leg and stance leg respectively.
The arc angle of the foot is defined as β and a measurement of the torso with respect to the
upward vertical axis is given as α.
The vector of generalized coordinates is given by
T

q=

θ φ ψ x

(2.1)

The equations of motion (EOM) for the biped when the stance leg is in contact with the
ground can be determined using Lagrange equations as

M(q)¨
q + N(q, q)
˙ =Q

4

(2.2)

dt

φ
α

lt

β

j
R

i
dsw

ψ

g

R
θ

x

dst

Figure 2.1: An arbitrary configuration of the active sliding synthetic wheel biped
where M is the mass and inertia matrix while N is the vector of Coriolis and gravitational
terms as shown in the Appendix (1). The vector of generalized forces, Q is given by
T

Q=

RF τ1 τ2 F

(2.3)

where τ1 and τ2 are the generalized forces acting on the φ and ψ coordinates, and F is is the
horizontal force produced corresponding to the generalized coordinate, x. This force creates
a moment on the stance foot, hence its term RF in the θ equation. A full derivation of the
generalized forces can be seen in the Appendix.
Walking will be achieved by interchanging the swing and stance legs at each step. It is
assumed that the generalized forces acting on the systems are non-impulsive; and, therefore,
there are no jumps in velocities caused by the foot-ground interaction at the time of foot
interchange. This assumption will be justified in later sections.

5

2.1.2

Foot Interchange

A discrete coordinate transformation occurs during the foot interchange due to the switching
of the stance and swing leg. The old swing leg is now defined as the new stance leg, and
vice-versa. This coordinate transformation is given by
q+ = T q−

(2.4)

q˙+ = T q˙−

(2.5)



T =













1 0



1 0 

0 1 −1 0

0 0 −1 0
0 0

0 1












(2.6)

where q − and q + are the generalized coordinates shown in equation (2.1) before and after the
foot interchange respectively. Figure 2.2 shows the biped about to step and the coordinates
before and after the step.
Normally a horizontal impulse from foot-ground interaction would affect the velocities
of the coordinates, as shown by Flynn et. al. [8], which would change the coordinate
transformation. However, we are modeling the foot-ground interaction force, F , to be nonimpulsive and controllable. Physically, this can be realized by a belt that interacts between
the foot and the ground but is frictionless between the belt and the connecting foot. At the
time of contact with the ground, the belt will grip the ground but will allow the foot to slide
without impulse on the belt. This frictionless belt will ideally eliminate all of the horizontal
impulsive forces and the wheel-like design of the SWB will ideally eliminate vertical impulsive
forces [8]. Additionally, it is possible to minimize horizontal impulsive forces through careful
gait design as shown in future sections.

6

torso

φ+
φ−

θ−
stance
leg heel

θ+
ψ+

swing
leg toe

ψ−
swing
leg heel

stance
leg toe

Figure 2.2: A schematic of the biped about to step. Stance and swing denoted in the figure
refer to their designations before the step happens.

2.2

Controller Design for Continuous Dynamics

Planar biped walkers have been explored in great detail by Westervelt et. al. [9]. The hybrid
nature of these robots was ascribed to the continuous motion over the step and the discrete
events that occur when switching legs. In this paper we control the hybrid dynamics through
the use of a continuous controller over the step and a discrete controller at the discrete foot
interchange. This is similar to the work by Mathis et. al. [10], [11] which controls the hybrid
dynamics of a hopping robot. Grizzle et. al. [12] employed an alternate control method
which used a finite time controller to ensure convergence by the end of a step. The combined
continuous and discrete controller was used instead of the finite time control method due
to the challenges of implementing finite time controllers in practice, such as discontinuous
control torques and chattering.

7

2.2.1

Partial Feedback Linearization

The biped shown in Figure 2.1 is underactuated; and, therefore, it cannot independently
actuate all degrees of freedom (DOF). Partial feedback linearization was chosen to control
the motion of the biped. We define the controllable states as φ, ψ, and x where θ is uncontrollable. To this end, we partition the coordinates in equation (2.1) as follows:




q=




q1 
q2






=















θ 

φ

ψ
x












(2.7)

Similarly, the matrices from equation (2.2) can be partitioned as




M1 M2 

M(q)−1 = 



M3 M4



N(q) = 




N1 

N2


Q=








Q1 
Q2







=













(2.8)

(2.9)




RF 



τ1 



τ2 




(2.10)

F

This partitioning results in the matrices M1 , M2 , M3 , and M4 being 1×1, 1×3, 3×1, and
3×3 sized matrices respectively. N is partitioned identically to q.
From equations (2.2) and (2.8)-(2.10), q¨2 is given by
q¨2 = M3 [Q1 − N1 ] + M4 [Q2 − N2 ]

(2.11)

Through equation (2.10), Q1 can be rewritten as
Q1 = 0 0 R Q2
8

(2.12)

Substituting equation (2.12) into equation (2.11) gives
q¨2 = C2 Q2 − C1

(2.13)

C1 = M3 N1 + M4 N2

(2.14)

where

C2 = M4 + M3 0 0 R

(2.15)

¨ ψ,
¨ x¨) are in terms of the inputs (Q2 ), we
Now that the controllable accelerations (φ,
define the desired dynamics of the controllable system as
¨ = −KP E − KD E˙
E

(2.16)

where E, KP , and KD are given by


E=











φ − φd 



ψ − ψd 



x − xd


KP =




2
ωn 




(2.17)



100 
010
000








KD = 2ζωn I

(2.18)

(2.19)

Note that ωn and ζ are positive constants and I is the identity matrix. The last diagonal
term of KP is 0 while the last diagonal term of KD is 1, allowing us to define a desired
velocity for our biped without defining a desired displacement.
Substituting equation (2.17) into (2.16), the desired dynamics can be written as


q¨2 = K




+




9



φ¨d 



ψ¨d 



x¨d

(2.20)

where K is the desired linear controller, given as


K=




−KP 










 φ˙ − φ˙ d 



φ − φd 




ψ − ψd 
 − KD  ψ˙ − ψ˙ d 






x˙ − x˙ d
x − xd






(2.21)

With the desired controller defined, the control inputs (Q2 ) can be written as




Q2 =



−1 
C2 
 C1



+K




+






φ¨d 




ψ¨d 



x¨d

(2.22)

For this derivation, the desired accelerations (φ¨d ,ψ¨d ,¨
xd ) are allowed to be arbitrary functions
¨ A parametrization of this can be written as
of the states and θ.











φ¨d 



ψ¨d 



x¨d



˜ q)
= K(q,
˙ 





θ¨ 

1

(2.23)




˜ q)
where K(q,
˙ is a 3×2 matrix that is defined by the desired accelerations of our controllable
states. The constant 1 is used to allow the desired accelerations to have components that
˜ used in this controller
are only functions of the measured states. The explicit terms of K
will be defined after the definition of the desired gait.
To write our desired accelerations as a function of only the state variables, the equation
of motion for θ is used. Using the partitions from equations (2.8)-(2.10) and equation (2.2),
the desired accelerations can be written as




¨ 
 φ
d









ψ¨d 



x¨d





 −M1 N1 − M2 N2 + M1 0 0 R Q2 

˜ q)
= K(q,
˙ 


1




(2.24)

Rewriting equation (2.24) gives




¨d 
 φ










ψ¨d 
 = C3 Q2 + C4

x¨d




10

(2.25)

where



 M1

˜ q)
C3 = K(q,
˙ 






0 0 R + M2 
000

 −M1 N1

˜ q)
C4 = K(q,
˙ 







− M2 N2 

1




(2.26)

(2.27)

Substituting equations (2.25) and (2.20) into equation (2.13) gives the control inputs (Q2 )
as



Q2 =











τ1 


τ2 




= [C2 − C3 ]−1 [C1 + C4 + K]

(2.28)

F

It is assumed that [C2 − C3 ] is invertible. If it is not, a pseudo inverse can be used in place
of the inverse to compute the minimum norm solution.

2.2.2

Desired Gait Design

Flynn et. al. developed the “Impact-Free” gait for the synthetic wheel biped with torso [8].
This gait ideally removed impulses due to impact by making the velocity of the swing leg
zero at the end of the step. We propose a modified version of this gait as follows:
φd = π + α − θ
ψd = −β sin(πθ/β)

(2.29)

x˙ d = −Rθ˙ + Va
where β is the arc angle of the foot; α is a constant that defines the desired torso angle
counterclockwise from the upright-vertical axis as shown in Figure 2.1; and Va is a constant
that determines the added velocity due to sliding. The angle at which the biped switches
feet is θ = −β/2. Note that we choose to not define a desired trajectory for the coordinate
x but we do define a desired trajectory for x.
˙ This is reflected in our choice of gains in
˜ the second derivatives of the gait described in
equations (2.18) and (2.19). To obtain K,

11

equation (2.29) are taken, resulting in


˜ =
K









−1

0

−π cos(πθ/β) (π 2 /β)θ˙ 2 sin(πθ/β)
−R

0










(2.30)

Note that the first column is multiplied by θ¨ and the second column is just a constant.
With the gait defined, we need to guarantee that the biped will step. To this end, we
first show that if E ≡ 0 at the end of each step, then E ≡ 0 at the start of the next step.
This can be shown by setting the coordinates and their derivatives equal to their desired
trajectories at the end of the step. Note that we do not know the value of θ˙ at the end of
the step and x is arbitrarily set to zero. At the end of a step we have from equation (2.29)
θ− = −β/2
φ− = π + α + β/2

φ˙ − = −θ˙−

ψ − = −β sin(−π/2) = β ψ˙ − = −π θ˙− cos(−π/2) = 0

(2.31)

x˙ − = −Rθ˙− + Va

x− = 0

Substituting equation (2.31) into the coordinate transformations in equations (2.4) and (2.5),
we find the values at the beginning of the next step as follows:
θ˙+ = θ˙−

θ+ = β/2

φ+ = π + α − β/2 φ˙ + = −θ˙+
ψ+

ψ˙ + = 0

= −β

(2.32)

x˙ + = −Rθ˙+ + Va

x+ = 0

Knowing that θ = β/2 at the beginning of a step when E ≡ 0, we can find the values
of the desired trajectories at the beginning of the step. Note that θ˙− is unknown and x is
arbitrarily set to zero as stated above. These desired values are then given as
θd+ = β/2
φ+
d = π + α − β/2

˙+
φ˙ +
d = −θ

ψd+ = −β sin(π/2) = −β ψ˙ d+ = −π θ˙+ cos(π/2) = 0
x+
d =0

˙+
x˙ +
d = −Rθ + Va
12

(2.33)

A comparison of the results in equations (2.32) and (2.33) shows that E ≡ 0 before the step
results in E ≡ 0 after the step.
If we assume “perfect control”, that is E ≡ 0, the dynamics of θ, φ, and x are described by
equation (2.29). These can be substituted into equation (2.2), which then reduces to a single
DOF system in terms of the passive dynamics of θ as shown in the Appendix, equation (25).
While the passive dynamics are too complicated to solve analytically, we can numerically
investigate the dynamic behavior of θ when E ≡ 0 to determine if the biped will step.

2.2.3

Simulations of Continuous Dynamics

Simulations were used to explore the behavior of the system assuming E ≡ 0. The system
parameters are given in Table 2.1. Note that Jt , Jst , Jsw , and Mt , Mst , Msw denote the
moment of inertia about the center of mass and the mass of the torso, stance leg, and swing
leg respectively. All physical values are given in SI units and all angles were measured in
radians. Note from the values that we assume both legs are identical.
Table 2.1: Physical Parameters for Simulation
Parameter
Jt
Jst
Jsw
dt
dst
dsw
Mt
Mst
Msw
R
lt
β
g

Value
0.2376
0.0547
0.0547
0.2415
0.3175
0.3175
12.22
1.628
1.628
0.635
0.483
0.3927
9.810

Units
kgm2
kgm2
kgm2
m
m
m
kg
kg
kg
m
m
rad
m

The phase portrait of the passive system is shown in Fig 2.3 when α = 0. The shaded
region denotes the space where θ will reach the foot switching angle, −β/2 (≈-0.2 rad). From

13

equation (2.32), we know that θ after the step will be β/2 and θ˙ does not change during the
step. This means that once the stepping angle is reached, the beginning of the next step will
be inside the shaded region. Therefore, if the biped steps, it will continue stepping when
E ≡ 0 and α = 0. If α = 0, Flynn et. al. [8] showed that the magnitude of θ˙ will continually
increase for the ideal case but will remain bounded due to friction and other unmodeled
dynamics. Note that this phase portrait is independent of Va because Va effectively changes
the inertial frame without affecting any internal dynamics.
β/2

−β/2
0.2

˙
θ[rad/s]

0.1

0

- 0.1

- 0.2

- 0.3

- 0.4
- 0.3

- 0.2

- 0.1

0

0.1

0.2

0.3

θ[rad]
Figure 2.3: Phase portrait of passive coordinate θ when α = 0. The vertical lines at
θ = −β/2 and θ = β/2 indicate the bounds for which the dynamics are valid. The dashed
line represents the coordinate transformation q + = T q − at the end of the step.

2.3
2.3.1

Controller Design for Hybrid Dynamics
Periodic Behavior

We have shown that the “perfectly controlled” biped (where E ≡ 0) will keep stepping for
certain initial conditions. In general, E will not be identically zero; and, therefore, the biped
will not perfectly track its desired gait. The coordinate transformation at the time of foot
interchange could magnify this error. It is therefore necessary to control the behavior of the

14

biped across multiple steps and ensure that the biped asymptotically approaches a periodic
configuration, which will induce stepping.
Assume the initial conditions at the end of the kth step to be
p(k) =

˙
˙
˙
φ(k) ψ(k) θ(k)
φ(k)
ψ(k)
x(k)
˙

T

(2.34)

Note that these are our generalized coordinates and their derivatives excluding θ and x. We
do not need θ because it is constrained at the end of the step to be −β/2. We also do not
need x because it is allowed to be arbitrary.
For the initial conditions given, the Poincar´e map P (·), that describes the motion from
one step to the next, is given as
p(k + 1) = P (p(k))

(2.35)

The initial conditions in (2.34) can be defined as a period-1 periodic point [13] if
P (p∗ ) − p∗ = 0

(2.36)

The periodic points of (2.36) were found numerically in a defined space of p. Multiple
periodic points were found and we chose p∗ that resulted in behavior that mimics the phase
portrait in Figure 2.3, i.e. E ≡ 0 and α = 0. Note that we had to fix values for ωn , ζ,
α, and Va to simulate the dynamics and find the periodic points. Knowing that there are
multiple periodic points in our dynamic space, we want to force our desired periodic point
to be asymptotically stable.

2.3.2

Chaos Control

With our desired periodic point found, we now want to control the biped to this periodic
point, i.e. asymptotically stabilize the periodic point. To do this, we will use the OGY
method of Chaos Control [14]. We start by defining error states for our initial conditions as
x˜(k) = p(k) − p∗
15

(2.37)

where p∗ is the desired periodic point. We then define the input at each step k as
T

u(k) =

(2.38)

ωn (k) ζ(k) α(k) Va (k)

and the error input as
u˜(k) = u(k) − u∗

(2.39)

where u∗ is the desired set of control values used to find the periodic point in the previous
section. We can now write a discrete linear controller as
x˜(k + 1) ≈ A˜
x(k) + B u˜(k)

(2.40)

Note that A is the Jacobian of the Poincar´e map, i.e. A = δP/δ˜
x|x˜=0 , and that A and B
were computed numerically. The discrete control u˜(k) is defined as follows:
u˜(k) = −K x˜(k)

(2.41)

where K can be found by using the discrete LQR method [15]. This would use ωn (k), ζ(k),
α(k), and Va (k) to control p(k) to p∗ when we start from initial conditions arbitrarily close
to the periodic point.

2.3.3

Simulations of Hybrid Dynamics

Simulations were used to validate the hybrid controller described above. The physical parameters from Table 2.1 were used for these simulations. The desired periodic initial conditions
at the end of the step were assumed to be
p∗ =

T

3.338 0.393 −0.664 0.664 0.000 0.722

(2.42)

and the desired control values are
u∗ =

T

10 0.7 0 0.3

(2.43)

The starting configuration was assumed to be
T

q0 =

0 π 0 0
T

q˙0 =

0 0 0 0
16

(2.44)

Note that this configuration is one where the biped is at rest with both legs straight down
and the torso straight up. The gain used in the chaos control feedback is


K=













0.02

0.02 −0.09 −0.08

0.04



0.23 

0.02 −0.00 −0.09 −0.08

0.05

0.24

0.07 −0.00

0.05

0.04

0.00 −0.11

0.05 −0.01

0.08

0.07 −0.01 −0.17












(2.45)

To move from rest, α(0) or Va (0) will have to be non-zero. To this end, we choose u(0)
to be
T

u(0) =

(2.46)

10 0.7 −0.08 0

Figure 2.4 shows the simulated angle states. The vertical dashed lines represent a step,
with the step number listed next to them. The initial step is not shown so the consistency
of the steps can be seen. Transient effects are seen in the first few steps after which periodic
behavior can be observed.
0.4

ψ
0.3

2

3

4

5

6

7

8

θ

0.2

[rad]

0.1
0
−0.1

φ−π

−0.2
−0.3
−0.4

2

3

4

5

6

7

time [sec]

Figure 2.4: Simulated Angle States. Each vertical dashed line represents a step.

Figure 2.5 shows the components of x˜(k) vs. time for the same simulation as Figure 2.4.1
1 A video of the simulation results can be found at www.egr.msu.edu/

~ mukherji/Active_

Sliding_Synthetic_Wheel_Biped_Simulation.mp4
17

[rad]

0.5

x
˜2
0

x
˜1
−0.5

0

1

[rad/s]

1

2

3

x
˜5

4

5

6

7

8

9

10 11 12 13 14 15

4

5

6

7

8

9

10 11 12 13 14 15

4

5

6

9

10 11 12 13 14 15

x
˜3

0

x
˜4

[m/s]

−1
0
0

1

2

x
˜6

−0.5
−1
0

3

1

2

3

7 8
kth step

Figure 2.5: Plot of error states vs time
Note that the error of the chaos controller has converged to zero within 15 steps. Figure 2.6
shows the generalized forces and how they decreased in magnitude over time as the biped
approached its periodic gait. The jumps in magnitude are due to jumps in the non-linear
terms of the input calculations, which can be attributed to the coordinate transformation
at the end of the step. Note that these torque discontinuities are an idealization and it will
not be possible to reproduce them exactly in experiments.
From Figure 2.3, it is clear that the sign of θ changes during the step, but θ˙ remains the
same. This is because θ is an odd function, while θ˙ is an even function. This implies that θ¨
is an odd function. Since θ and θ¨ are odd functions, their discrete jumps at the end of each
step cause the discontinuities in the inputs. Even and odd function characterization of the
passive dynamics of the SWB were explored by Flynn [1].

18

4

τ1

[Nm]

2
0
−2

τ2

−4
0

2

4

6

8

10

12

2

4

6
time [sec]

8

10

12

15

[N]

10
5
0
−5
0

F

Figure 2.6: Plot of continuous control effort vs time

19

CHAPTER 3
HARDWARE DESIGN

To experimentally validate the control algorithm developed, the Synthetic Wheel Biped
(SWB) created by Flynn [1] was re-purposed to allow for additional sliding motion at the
foot. The original SWB as seen in Figure 1.1 was a planar biped with two hip motors and
solid arc shaped feet. The new biped as seen in Figure 3.1 maintains the majority of the
hardware from the previous version but has new feet and new electronics. To create the
sliding motion, a mechatronic foot was designed to replace the previous solid foot. The
Electronic Control Unit (ECU) on the SWB did not have the capacity for the extra motors
and encoders needed to operate the feet, so a complete redesign of the robot’s electronics
was done.

Figure 3.1: Picture of Active Sliding Synthetic Wheel Biped

20

3.1

Sliding Foot

Figure 3.2 shows a comparison of the old foot (top) and new foot with one side panel removed
(bottom). On the new foot, a belt is driven by a small motor to allow for slip between the
ground and the foot. The torque on the motor creates a linear force (F ) at the ground due
to the traction provided by the belt. An oil-impregnated plastic was used for the contact
surface of the belt and Delrin rollers were used to reduce friction while snaking through the
foot. The motors used for the feet were Faulhauber’s 2342-024CR with a 14:1 gearhead and
Faulhaber IE2-512 encoders. The new sliding foot has the same curvature as the previous
static foot created by Flynn [1] to maintain the synthetic wheel design. The two bolts in the
center of each foot are used to mount each foot to the biped’s legs.

Figure 3.2: Comparison of old (top) and new (bottom) feet

21

3.2

Electronic Controller

With 8 motors and 8 encoders needed to run the newest iteration of the Synthetic Wheel
Biped, a new ECU was designed. Figure 3.3 shows the major components and logic flow to
operate the robot while Figure 3.4 shows the ECU when mounted on the robot. Figures depicting the ECU schematic and physical board layout are shown in the Hardware Appendix.
To accommodate all of the different voltages required for operation, a 30V Nickle-Cadmium
(NiCd) battery was used with several different DC/DC voltage regulators to power every
component on the ECU. A separate 24V NiCd battery was used to power the motors and
motor amplifiers.
Wireless Comm.

Encoders
Microcontroller

Motor Controllers

IMU

Data Storage

Figure 3.3: ECU block diagram

Figure 3.4: Picture of ECU mounted on robot

22

An STM32F4-Discovery board was used as the microcontroller for the ECU. This was
chosen for its large number of serial communication buses and high clock speed. A CH
Robotics UM6 Inertial Measurement Unit (IMU) was used to measure the biped’s angle
relative to the earth, and encoders on each motor allowed for angular measurements of each
joint. A Digi Xbee S1 wireless module was used to communicate with the microcontroller
during operation by sending commands from a computer wirelessly. Data was stored using
a microSD card reader.
The motor controllers were also changed in this version of the Synthetic Wheel Biped
to account for the added motors. AMC motor amplifiers Z6A6 and Z12A8 operating in
torque control mode were used because it was possible to mount 8 of them on the biped
torso by using AMC two-axis MC2XZQD mounting cards as shown in Figure 3.5. Six Z6A6
controllers were used to control the feet and linear joints, while two Z12A8 controllers were
used for the hips due to their higher amperage capacity. All of these amplifiers use a ±10
V analog input to control the output torque. This voltage was not readily available from
the microcontroller, so an amplifier circuit was designed to convert a 0-5V output from the
Digital-to-Analog Converter (DAC) to the required ±10 V input of the motor amplifiers. A
diagram of this system can be seen in the ECU schematic in the Hardware Appendix.

Figure 3.5: Picture of motor amplifiers mounted on robot

23

CHAPTER 4
EXPERIMENTAL VALIDATION

4.1
4.1.1

Controller Implementation
Programming and Operation

The STM32F4-Discovery breakout board uses an ARM Cortex-M4 processor for its computations. To program this chip, code was written in C and compiled using proprietary
software to run on the chip. Several Integrated Development Environments (IDEs) allow for
this compilation, and the Embedded Workbench from IAR Systems was used.
The program used software interrupts for timers and serial data input. This allowed
the system to have a precise, fixed-time loop and read all of the information sent from the
IMU and Xbee. The data coming from the IMU contain checksum information to verify the
data being received, while the Xbee does not. This is acceptable because no data is being
streamed over the wireless connection, only commands from a computer to enable/disable
operation and data logging.
The software has two main phases, being a setup phase and a fixed loop phase. In
the setup phase, all of the communication buses are initialized before any data is being
manipulated. Once setup is complete, the system enters an infinite, fixed-time loop phase
where the main body of the program runs. This phase reads all of the information from the
sensors, computes the desired torques, and sends the information to the motor amplifiers and
data storage if enabled. The motors and data storage are initially disabled until a command
is sent to the Xbee. To start the robot, the encoder values and angle from the IMU are
recorded to initialize the system in a set configuration. Once initialized, a command is sent
to the Xbee to start the data collection and robot walking.

24

4.1.2

Control Changes

To account for differences between the theoretically modelled control system and the implementable system, several changes were made to the controller as described in the previous
section. In the dynamic derivation, it was assumed that the biped could slide with perfect
slip to uncouple the coordinates x and θ. In reality though, the designed foot is not frictionless enough for this assumption to hold true. To compensate, the control algorithm was
reworked as a 3 DOF system where the biped will roll with a set slip speed, meaning
x˙ = −Rθ˙ + Va

(4.1)

Instead of the system having 4 DOF where sliding is caused by controlling the force at
the ground, the biped’s controller uses the 3 DOF model described by Flynn [8] and has a
velocity controller on the motors of the foot. The added velocity Va just changes the inertial
frame of the biped and none of the dynamics include this term.
Instead of re-deriving the entirety of the dynamics and control as done in the previous
sections, the general form of the Lagrange equations as shown in equation (2.2) can have the
terms manipulated to convert the 4 DOF system into a 3 DOF system as follows:

ˆ q )¨qˆ + N
ˆ (ˆ
ˆ
M(ˆ
q, qˆ˙ ) = Q

(4.2)

where
T

qˆ =
ˆ=
Q

(4.3)

θ φ ψ
T

0 τ1 τ2

(4.4)

ˆ is now a 3×3 matrix and N
ˆ is a 3 element column vector. The values of M and N from
M
ˆ and N
ˆ . We can couple the x dynamics
the Appendix (1), can be manipulated to find M
into the other terms by equating F in Q and using equation (4.1). Note that no terms in the
original M and N contained x or x,
˙ therefore the Va term does not appear in the dynamics

25

as stated earlier. The conversion from 4 DOF to 3 DOF can be seen as
ˆ 11 = M11 − 2RM14 + R2 M44
M
ˆ 12 = M12 − RM24
M
ˆ 13 = M13 − RM34
M
ˆ 21 = M21 − RM24
M
ˆ 22 = M22
M
ˆ 23 = M23
M

(4.5)

ˆ 31 = M31 − RM34
M
ˆ 32 = M32
M
ˆ 33 = M33
M
ˆ1 = N1 − RN4
N
ˆ2 = N2
N
ˆ3 = N3
N

With these new equations, the partial feedback linearization derivation follows the same
steps as equations (2.8) to (2.22), resulting in




 τ1 




τ2

= [C2 ]−1 [C1 + K]

(4.6)

where
C1 = M3 N1 + M4 N2

(4.7)

C2 = M4


K = −ωn2 







φ − φdes 

ψ − ψdes


 − 2ωn ζ 



(4.8)


φ˙ − φ˙ des 

ψ˙ − ψ˙ des




 − 2ωn ζ 
d




φ˙ 

ψ˙




(4.9)

remembering that M1 , M2 , M3 , M4 are the partitioned matrices from the inversion of the
mass matrix. The full derivation can be found in the Appendix, equations (8)-(24). In
this derivation, the desired accelerations were assumed to be zero. If the accelerations are
included, numerical inaccuracies can accumulate from velocity squared terms and matrix
26

inversions which can cause instability. It is also important to note that K now includes
an additional viscous damping term with a small positive constant ζd based on the angular
speeds compared to the damping term based on the error of angular speeds. This serves to
filter the torques applied to the biped based on the speed of the robot. Instead of filtering
the torques using a generic low-pass filter or first-order filter, the viscous damping term was
added. This results in slower, smoother motion of biped.
Now that the torques (τ1 and τ2 ) have been found, the speed of the belt still needs to be
controlled. This was done by using a simple, linear, proportional-integral (PI) controller on
the speed of the motors. The desired speed was set to Va , and an individual PI controller
was used for each foot to account for the differences in friction of each foot. To keep the
biped walking straight, the speed controllers had additional terms from each foot that were
coupled to the other foot. This controlled the speed between feet to each other as well as
the desired speed, resulting in straight walking.
Besides the change in DOF, the theoretical model also assumed that both legs had equal
physical parameters while the physical system has legs with different masses and moments
of inertia. To account for this, the robot had to keep track of which leg was in contact with
the ground and apply the physical parameters appropriately in the control algorithm. As
expected, these changes in physical parameters lead to differences in the gait of the robot
depending on which foot is in contact with the ground. Similar to Figure 2.3, which showed
the desired region of operation for the theoretical biped during perfect control, Figure 4.1
shows the phase portrait of the passive coordinate θ when the stance leg is on the inside
(left) and outside (right) leg respectively. As shown, the desired path of θ differs per step,
but the symmetry about θ = 0 allows for a single desired period-1 periodic point instead of
a period-2 point. This allows the chaos controller to still update every step instead of once
every two steps, making the convergence twice as quick.
Besides the physical parameters, there are measurements needed by the controller that are
˙ φ,
˙ ψ,
˙ and x.
not explicitly available with the sensors on the biped, being θ,
˙ While encoders

27

β/2

−β/2

β/2

−β/2
0.2

˙
θ[rad/
s]

0.1
0
− 0.1
− 0.2
− 0.3
− 0.4
− 0.3

− 0.2

− 0.1

0

0.1

0.2

0.3

− 0.3

− 0.2

− 0.1

0

0.1

0.2

0.3

θ[rad]
Figure 4.1: Phase portrait of passive coordinate θ with α = 0 when the inside leg (left) and
the outside leg (right) are the stance leg. The vertical lines at θ = −β/2 and θ = β/2
indicate the bounds for which the dynamics are valid. The dashed line represents the
coordinate transformation q + = T q − at the end of the step.
allow for measurement of angular rotation, they do not measure angular rate. However,
the velocity measurements can be obtained by numerically differentiating the encoder signals and passing the output through a low-pass filter. The filter is required to avoid large
discontinuities in the velocity measurements, which could induce chattering in the system.
In the theoretical model, it is assumed that the center of mass (COM) of the torso is
along the axis of the torso. In the physical system, however, this is not the case. It was
emprically determined that the COM of the torso is actually 2 degrees counter-clockwise of
the torso’s axis as shown in Figure 2.1. To compensate, all terms with φ in the EOM have an
offset of 2 degrees added. This ensures the gravity compensation in the non-linear controller
is as accurate as possible.
The final change is in the design of the chaos controller. In the theoretical model, all
states at the periodic point are used to update gains in the controller (ωn , ζ) and desired
values (α, Va ). In the physical system, it was decided that the chaos controller should only
update the desired values (α, Va ). It is undesirable to update the controller gains because
it can cause instability and chattering in the system after usable gains have been found via

28

experimentation. From Figure 4.1, we see that the dynamics of the system collapse to the
2D manifold of θ × θ˙ when the continuous control is perfect (E ≡ 0). Because we fix θ at
the end of the step to define our periodic point, the periodic point can be uniquely described
by the value of θ˙ at the end of the step. Therefore, instead of 6 values, only the value of
θ˙ will be used in the chaos control as the control objective. In the actual implementation,
the measurement of θ˙ is averaged over several time steps to reduce errors caused by noise in
the velocity data. Therefore, ¯θ˙ will be the implemented control objective. With all of this
information, the chaos controller used is as follows:





4.2



α(k + 1) 

Va (k + 1)








 αdes 
¯˙

= −K θ(k)
+


Vades

(4.10)

Experimental Results

The physical parameters of the biped are given in Table 4.1. Note that these are different
values from those in the theoretical simulations. As physical changes to the biped were
made, the parameters were updated to make the controller as accurate as possible. Feedback
linearization is sensitive to differences between the model and the physical device, so these
changes were crucial in obtaining the desired performance.
Figure 4.2 shows the angle states of the biped during an experimental trial. Each vertical
dashed line represents a step, with a step number next to it. The biped was started at step
0. The data before step 1 was not shown so that the details of the following steps would be
visible. Due to the way the biped is started, the first step takes a long time because the biped
does not immediately start walking. Note that the experimental data is qualitatively similar
to that of the simulations in Figure 2.4. There are two major distinctions, being the lack of
similarity between steps in the experimental data and the small upswing in θ on some steps
of the experimental data. The lack of similarity between steps of the experimental data could
be due to external stimuli in the form of impulses from the ground or ground irregularity.
29

Table 4.1: Experimental Physical Parameters
Parameter
Jt
Jout
Jin
dt
dout
din
Mt
Mout
Min
R
lt
β
g

Value
0.192
0.128
0.094
0.3429
0.3096
0.2048
11.744
3.956
2.132
0.638
0.467
0.35368
9.81

Units
kgm2
kgm2
kgm2
m
m
m
kg
kg
kg
m
m
rad
m

In simulation, all of the states smoothly traversed from their starting position to their final
position. In some portions of the experimental data (namely steps 2, 3, 7, & 8), it seems
that the biped hesitated and even moved backwards slightly at certain spots. This indicates
that the passive dynamics are underdamped based on the controller used, resulting in an
overshoot and oscillation about the desired path. Despite the differences from the simulated
results, the biped still continued to step in experiments.
0.4

2

3

4

5

6

7

8

0.3
0.2

[rad]

0.1

φ−π

0
−0.1
−0.2
−0.3
−0.4

ψ
10

θ
11

12

13

14

15

time [sec]

Figure 4.2: Experimental Angle States. Each vertical dashed line represents a step

Given E1 = φ − φdes and E2 = ψ − ψdes from equation (2.17), these errors and their
30

derivatives are shown in Figure 4.3 for the experimental trial. Note at the beginning of each
step, the velocity error E2 corresponding to the swing leg ψ has a discrete jump. This error
then moves towards zero, while the position error grows. The discrete jumps in velocity
error are due to impulses from collision with the ground at each step. If the velocity of
the swing leg is not exactly zero as the gait specifies, these impulses noticeably affect the
motion of the biped. From Figure 4.3, it is clear that the position error of the biped is
not constantly decreasing. However, when the position error is increasing, the velocity error
is converging towards zero. The relationship between which errors converge at what rates
could be manipulated by adjusting the gains (ωn , ζ, ζd ). If the steps took more time, it may
be possible for the errors and their derivatives to both converge to zero before the next step
occurs.
0.1

2

E1

3

4

5

6

7

8

[rad]

0
−0.1
−0.2

E2

−0.3
10

11

E˙ 2 2

2

12

3

13

4

5

14

6

7

15

8

[rad/s]

1
0
−1
−2

E˙ 1
10

11

12

13

14

15

time [sec]

Figure 4.3: Experimental Angle State Errors

The torques for the experimental trial are shown in Figure 4.4. Note that these do not
match well with the torques shown in the simulated results in Figure 2.6. This could be
due to many factors including the difference in models, physical parameters, external noise,
impulses, and measurement filtering. It is important to note that the torques on the swing
leg (τ2 ) are large at the start of the step. This is most likely due to τ2 trying to counter the

31

effects of the impulse from the contact with the ground. As shown, the experimental torques
are still bounded and the biped did successfully walk. These differences just show the lack of
correlation between simulated and experimental control input to result in walking motion.
2

12
10

3

4

5

6

7

8

τ2

8
6
[Nm]

4
2
0
−2
−4

τ1

−6
−8

10

11

12

13

14

15

time [sec]

Figure 4.4: Experimental Torques

Figure 4.5 shows the experimental chaos controller data over multiple steps. Note that
the value of ¯θ˙ does not completely converge compared to the theoretical estimations. This
could be due to external stimuli such as impulses from impact pushing the biped’s motion
away from the periodic point. There is a large jump in magnitude of all values at step 8. It
is unclear what caused the jump, but it was possibly in response to ground irregularities or
some other stimulus.

32

[rad/s]

0.6

¯˙
θ

0.4
0.2
0
−0.2
0

1

2

3

4

5

6

7

8

9

2

3

4

5

6

7

8

9

3

4
5
kth step

6

7

8

9

[deg]

1
0
−1

α
−2

0

1

0.05
[m/s]

0
−0.05

Va

−0.1
−0.15

0

1

2

Figure 4.5: Experimental Chaos Control Error

33

CHAPTER 5
CONCLUSIONS

In this work, the previous MSU SWB was modified to allow for sliding-while-walking motion. The first model of the active sliding synthetic wheel biped had the ability to control
the horizontal ground interaction force, thus allowing the biped to slide. A partial feedback
linearization controller was designed to track desired trajectories during each step. It was
shown that once the biped steps, it will continue stepping when the error of the continuous
controller is zero. However, this will not be always true. Due to the asymptotic convergence
of the tracking controller, finite time of each step, and any impulses from impact, the continuous controller will likely not be converged by the end of the step. A chaos controller was
therefore designed to stabilize the biped’s motion to a desired periodic gait in the presence
of these issues. This hybrid controller was shown to effectively control the motion of the
biped during each step and over multiple steps through simulation.
To implement this controller, the synthetic wheel biped built by Flynn [1] was modified to
allow sliding-while-walking motion. The feet were changed to incorporate motors and a new
ECU was designed to handle the increased complexity. It was clear in early experimentation
that the physical feet were not capable of near-frictionless motion, so the theoretical biped
system was re-modeled and the control was changed to account for these differences. After
changes were made, the biped exhibited sliding-while-walking motion in an experimental
trial. While the gait did not converge as expected, the chaos controller did add a level of
stability in that it can account for external stimuli that change between steps and the robot
was able to walk for multiple steps.
In the future, it would be beneficial to investigate a new design of the foot that would allow
frictionless motion at the ground. This could be accomplished by redesigning the current
foot to remove sources of friction. Another possibility would be to eliminate the synthetic

34

wheel design and substitute wheels for arc feet. While the latter option completely changes
the model, it can still create the desired sliding-while-walking motion. This biped would
need to be designed with ”point feet” instead of the synthetic wheel, but the capabilities
would remain essentially the same. This wheeled biped could also include knee joints, which
would allow for other motions like climbing ledges or stairs to greatly extend the biped’s
mobility.

35

APPENDIX

36

Equations
Lagrange Equations - 4 DOF
Terms of the 4 DOF Lagrange equations. Note that the mass matrix, M, is symmetric.
M11 = Jst + Jsw + Jt + mst (dst − R)2 + msw (dsw − R)2 + mt (dt − lt )2
M12 = Jt + mt (dt − lt )2
M13 = Jsw + msw (dsw − R)2
M14 = msw (R − dsw ) cos(ψ + θ) + mt (lt − dt ) cos(φ + θ) + mst (R − dst ) cos(θ)
M22 = Jt + mt (lt − dt )2
M23 = 0
M24 = mt (lt − dt ) cos(φ + θ)
M33 = Jsw + msw (R − dsw )2

(1)

M34 = msw (R − dsw ) cos(ψ + θ)
M44 = Mst + msw + mt
N1 = [mt (lt − dt ) sin(θ + φ) + mst (R − dst ) sin(θ) + msw (R − dsw ) sin(θ + ψ)]g
N2 = mt (lt − dt ) sin(θ + φ)g
N3 = msw (R − dsw ) sin(θ + ψ)g
˙ 2
N4 = msw (dsw − R) sin(θ + ψ)(θ˙ + ψ)
˙ 2
+mt (dt − lt ) sin(θ + φ)(θ˙ + φ)
+mst (dst − R) sin(θ)θ˙2

37

Generalized Force Derivation
The generalized forces in equation (2.3) are derived here from Figure .1 and the analysis
below.

dt
lt

R

j
i

φ, τ1

dsw

ψ, τ2

g

R

z
x

θ
dst
C

F

Figure .1: Biped Model with forces

The vector C points from the center of mass of the stance leg to the contact point on the
ground.

C = −(R − dst ) sin(θ)i − [R − (R − dst ) cos(θ)]j

(2)

Therefore, the moment produced by F on the stance leg about its center of mass is given by
[R − (R − dst ) cos(θ)]F

38

(3)

The net moments operating on each link about their respective centers of mass are given as:
Mst : [R − (R − dst ) cos(θ)]F − τ1 − τ2
(4)

Msw : τ2
Mt : τ1
The virtual work done on the system can now be expressed as
δW = ([R − (R − dst ) cos(θ)]F − τ1 − τ2 )δθ + τ1 (δθ + δφ) + τ2 (δθ + δψ) + F δz

(5)

z = x + (R − dst ) sin(θ)

(6)

where

Upon simplification, the virtual work is given as
δW = RF δθ + τ1 δφ + τ2 δψ + F δx

(7)

Thus, the generalized forces corresponding to θ, φ, ψ, and x are RF , τ1 , τ2 , and F respectively.

39

Partial Feedback Linearization - 3 DOF Derivation
The following equations derive the feedback linearized controller in 3 DOF. We start by
defining the state and input vectors of the generalized Lagrange equations.
T

q=

θ φ ψ

Q=

0 τ1 τ2

(8)

T

(9)

M(q)¨
q + N(q, q)
˙ =Q

(10)

With the equations of motion defined, the vectors and matrices are partitioned to separate
the controllable (φ, ψ) and uncontrollable (θ) coordinates.

q=







 q1 



=




q2





N(q) = 



 Q1 

Q2

ψ

M3 M4


Q=

φ








(11)

M1 M2 






θ 



M(q)−1 = 






















N1 
N2



=









(12)

(13)





0 



τ1 



τ2

(14)

With the partitioned matrices, we can solve for q¨2 as
q¨2 = −M3 N1 + M4 [Q2 − N2 ]

40

(15)

which simplifies to
q¨2 = C2 Q2 − C1

(16)

C1 = M3 N1 + M4 N2

(17)

C2 = M4

(18)

where

¨ ψ)
¨ defined, the desired controller is defined as
With the controllable accelerations (φ,
¨ = −ω 2 E − 2ζωnE˙
E
n
where

(19)





φ − φd 



E=


ψ − ψd

(20)




Note that ωn and ζ are positive constants. Rewriting this in terms of the desired accelerations
results in





¨ 
 φ
 d 

q¨2 = K + 

ψ¨d

(21)



where K is the desired linear controller, given as


K = −ωn2 







φ − φd 


 − 2ζωn 



ψ − ψd



φ˙ − φ˙ d 

ψ˙ − ψ˙ d

(22)




For the actual controller, an additional viscous damping term based on angular speeds instead
of angular error speeds is added, resulting in


K = −ωn2 







φ − φd 

ψ − ψd


 − 2ζωn 





φ˙ − φ˙ d 

ψ˙ − ψ˙ d




 − 2ζ ωn 
d





φ˙ 

ψ˙




(23)

where ζd is a small positive constant.
The desired accelerations φ¨d , ψ¨d are assumed to be zero in the implemented controller to
avoid numerical inaccuracies caused by velocity squared terms and matrix inversions in the
desired accelerations.

41

The final control torques can then be written as



Q2 = 




τ1 

τ2



= [C2 ]−1 [C1 + K]

(24)

Passive Dynamics of θ
If E ≡ 0, the passive dynamics of θ can be written as
θ¨ = (Msw g sin(θ)(dsw − R) − R(πMsw θ˙ cos(πθ/β) sin(θ − β sin(πθ/β))
˙ sw θ˙ sin(θ)(dsw − R)
(θ˙ − π θ˙ cos(πθ/β))(dsw − R) − θ(M
+Msw sin(θ − β sin(πθ/β))(θ˙ − π θ˙ cos(πθ/β))(dsw − R))
+(Msw π 2 θ˙2 cos(θ − β sin(πθ/β)) sin(πθ/β)(dsw − R))/β)
+Mt g sin(α)(dt − lt ) + Msw g sin(θ − β sin(πθ/β))(dsw − R)
−(π 2 θ˙2 sin(πθ/β)(Msw dsw2 − 2Msw dsw R + Msw R2 + Jsw ))/β)
/(2Jsw + R(Msw cos(θ − β sin(πθ/β))(dsw − R)
+Msw cos(θ)(dsw − R) − Mt cos(α)(dt − lt ))
+R(R(2Msw + Mt ) + Msw cos(θ − β sin(πθ/β))(dsw − R) + Msw cos(θ)(dsw − R)
−πMsw cos(πθ/β) cos(θ − β sin(πθ/β))(dsw − R)) − d2t Mt + 2Msw (dsw − R)2 − Mt lt2
+Mt (dt − lt )2 + 2dt Mt lt − π cos((πθ)/β)(Msw d2sw − 2Msw dsw R + Msw R2 + Jsw ))
(25)

42

Hardware
Table .1: Electrical Hardware on ECU
Component Name
Microcontroller
DAC
Encoder Counters
Wireless UART
SD Card Reader
Voltage Regulator (12V)
Voltage Regulator (24V)
Voltage Regulator (5V)
Voltage Regulator (3.3V)
Header (Encoders)
Header (Amps)
Header (IMU)
Header (Battery)
Header (STM Power)

Part Number
STM32F4-Discovery
MAX510
LSI LS7366R
Xbee S1
uDRIVE uSD-G1
CUI V7812-1000
CUI V7812-1000
CUI V7805-1500
CUI V7803-1500
–
–
–
–
–

Schematic Name
STM
DAC1-DAC2
EC1-EC8
XBee
SD
VR+12V
VR+24V
VR+5V
VR+3.3V
EnH1-EnH8
AMPH1-AMPH2
IMUH
Batt
STM Pow

Table .2: Additional Electrical Hardware
Component Name
IMU
Motor amplifier (hips)
Motor amplifier (feet)
Motors (feet)
Motors (hips)
Motors (linear)
Encoders (feet)
Encoders (hips)
Encoders (linear)
Battery (motor)
Battery (ECU)

Part Number
CH Robotics UM6
AMC Z12A8
AMC Z6A6
Faulhauber 2342-024CR
Maxon RE40
Faulhauber 3243 CR
Faulhauber IE2-512
Agilent HEDS-5500
Agilent HEDS-5500
24V NiCd
30V NiCd

43

Figure .2: ECU Board - Component Layout

44

Figure .3: ECU Wiring Schematic

45

BIBLIOGRAPHY

46

BIBLIOGRAPHY

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University, 2010.
[2] O. Matsumoto, S. Kajita, M. Saigo, and K. Tani. Biped-type leg-wheeled robot. Advanced Robotics, 13(3):235–236, 1998.
[3] K. Hashimoto, T. Hosobata, Y. Sugahara, Y. Mikuriya, H. Sunazuka, M. Kawase,
H. Lim, and A. Takanishi. Realization by biped leg-wheeled robot of biped walking
and wheel-driven locomotion. In Robotics and Automation, 2005. ICRA 2005. Proceedings of the 2005 IEEE International Conference on, pages 2970–2975. IEEE, 2005.
[4] Ken Itabashi and Masaaki Kumagai. Development of a human type legged robot with
roller skates. In System Integration (SII), 2010 IEEE/SICE International Symposium
on, pages 120–125. IEEE, 2010.
[5] T. McGeer. Passive dynamic walking. The International Journal of Robotics Research,
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robot. IEEE International Conference on Robotics and Automation, 2013.
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47

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48