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Active Synthetic Wheel Prismatic Joint Biped

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ACTIVE SYNTHETIC WHEEL PRISMATIC JOINT BIPED
By
Louis L. Flynn IV

A THESIS

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

MASTER OF SCIENCE
Mechanical Engineering

2009

ABSTRACT
ACTIVE SYNTHETIC WHEEL PRISMATIC JOINT BIPED
By
Louis L. Flynn IV

We have developed a three-link biped with stiff ankles and arc-shaped feet that have
the same radius as the leg length: the MSU Active Synthetic Wheel Biped. This
biped is an underactuated, planar walker with paired legs for lateral stability and
motors between each leg and the torso. The robot is designed to naturally fall in the
direction that the torso is leaning, creating the propulsive torque needed to walk. We
present a dynamic model for the system and its collision dynamics. Because of the
unique shape of the biped, the robot has no‘ vertical component of collision, but ve-
locity mismatches at the switching time results in a horizontal component, which has
been modeled as a disturbance. We then describe two methods of gait creation. The
first method creates a gait by defining predetermined trajectories in the actuated de-
grees of freedom and independently tracking them. The second method creates a gait
by tracking trajectories in the actuated degrees of freedom based on the rolling unac-
tuated degree of freedom at the foot-ground interface through feedback linearization.
We present the results of both kinematics and energy consumption for the two meth-
ods of gait creation. A discussion about the choice of reference trajectories for our
system follows, including methods to create trajectories that minimize the horizontal
impulses at the end of the step and to reduce the total work required to complete a

step.

COPYRIGHT

Copyright by Louis L. Flynn IV, 2009

DEDICATION

To Sandra and my Family

iv

ACKNOWLEDGMENT

I would like to thank my advisor Dr. Ranjan Mukherjee, and my committee members
Dr. Alejandro Diaz and Dr. Brian Feeny.

I would also like to thank the students from the Dynamics and Controls Lab at
MSU. Particularly, Rouhollah J afari for his help in developing the controllers and
debugging the experimental setup and Aren Hellum for his work on the optimization
of trajectories.

Also I would like to thank Sandra Nauwelaerts and the McPhail Equine Perfor-
mance Center at MSU for allowing us to use their motion capture system for measur-
ing kinematics. Ramon Yzmon and Advanced Motion Controls for donating motor
drives for our robot. The MSU HSHSP and David Crouse and for helping set up the
real time operating system. And finally, thank you to Mike Mclean and Roy Bailiff

for their work building the robot.

TABLE OF CONTENTS

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

Nomenclature .................................

Introduction

1.1 Background ................................
1.2 Review of Biped Robots and Biped Controllers .............
1.3 The MSU Active Synthetic Wheel Biped ................

Dynamic Model

2.1 Introduction ................................
2.2 General Dynamic Model .........................
2.3 Impulsive Disturbances due to Swing Leg

Collision ..................................
2.4 Constrained Motion and the Nominal Trajectory ............
2.5 Interchange of Stance and Swing Legs ..................
2.6 Forward and Inverse Relations for Control Transformation for Experi-
ments ...................................

Controller Design

3.1 Introduction ................................

3.2 Decoupled Controller ...........................
3.2.1 Controller for Fixed Torso Angle ................
3.2.2 Controller for Constrained Motion of Swing Leg ........
3.2.3 Simulations of Decoupled Controller ...............

3.3 Coupled Controller ............................
3.3.1 Motivation .............................
3.3.2 Feedback Linearization ......................
3.3.3 Simulations of Coupled Controller ................

Experimental Methods

4.1 Background ................................

4.2 Mechanical Design ............................

4.3 Electrical Design .............................
4.3.1 Version 1 .............................
4.3.2 Version 2 .............................

vi

viii

ix

14
17
21

22

25
25
26
26
27
29
31
31
31
33

35
35

5 Experimental Results 44
5.1 Performance Comparison ......................... 44
5.1.1 Decoupled Controller ....................... 44

5.1.2 Coupled Controller ........................ 47

5.2 Energy Consumption ........................... 51
5.2.1 Decoupled Controller ....................... 52

5.2.2 Coupled Controller ........................ 53

6 Determining Reference Trajectories 54
6.1 Parameterized Gait Functions ...................... 55
6.1.1 Boundary Conditions ....................... 55

6.1.2 Work over One Step ....................... 56

6.1.3 Parameterized Gait Functions .................. 56

6.2 Nominal Trajectory ............................ 58
6.3 Optimized Trajectories .......................... 61
6.3.1 Constraints for Steps with Zero Total Work .......... 62

6.3.2 Equations For Gait Requiring Zero Total Work ........ 62

6.3.3 Term by Term Evaluation of Even / Oddness .......... 64

6.3.4 Proof of Zero Total Work Gait .................. 66

6.3.5 Non-Zero Total Work Gait .................... 71

6.3.6 Discussion of Optimization .................... 74

7 Conclusion and Future Research 76
A Impulse Constant K. 78
Bibliography ..................................................... 79

vii

LIST OF TABLES

4.1 Kinematic and Dynamic Parameters of Synthetic Wheel Biped . . . . 37
4.2 Control System Hardware ........................ 43

5.1 Energy Consumption of Selected Bipeds ................ 53

viii

1.1

1.2

1.3

1.4

1.5

1.6

1.7

2.1

2.2

2.3

2.4

LIST OF FIGURES

Images in this thesis are presented in color

Early Waseda Bipeds ...........................
The RABBIT Planar Biped .......................
The VU Brussels Robot Lucy ......................
Various Successful Passive Dynamic Walkers ..............

Highly Dynamic Robots with Compliant Joints and non-ZMP Based
Controllers .................................

Bipeds with Learning Algorithms ....................

(a) Self-Propelled Wheel, (b) Design Variant Of Self-Propelled Wheel,
(c) Conceptual Design Of Biped With Torso ..............

The Two-dimensional Three Link Model ................
Free-body diagrams of the stance leg, swing leg and torso showing
impulses of all impulsive forces generated as a consequence of swing

leg touchdown ...............................

Illustration of the generalized coordinates 91‘) and 1/2 and their relations
to 0 and a. ................................

Relationships of the generalized coordinates 9, qb, and w to the coordi-
nates 73;, 73w, and a ............................

ix

14

23

3.1

3.2

4.1

4.2

4.3

4.4

4.5

4.6

4.7

5.1

5.2

5.3

Simulation results of the decoupled controller. The motion of the torso
is shown in (a) as a solid line, the desired torso angle, ad is shown as
a dotted line. The leg motion is seen in (b) where the angle of the
stance leg with respect to the vertical is the solid line, and the angle of
the swing leg with respect to the vertical is the dotted line. Graph (c)
shows the two motor torques for Tst (black) and 75w (grey). Compare
this with Figure 3.2. ........................... 30

Simulation results of the coupled controller. The motion of the torso
is shown in (a) as a solid line, the desired torso angle, ad is shown as
a dotted line. The leg motion is seen in (b) where the angle of the
stance leg with respect to the vertical is the solid line, and the angle of
the swing leg with respect to the vertical is the dotted line. Graph (c)
shows the two motor torques for Tst (black) and 73w (grey). Compare

this with Figure 3.1. ........................... 34
MSU Synthetic Wheel Biped With Torso ................ 36
Front View of the Robot .......................... 38
Outer Leg Prismatic Joint ........................ 38
Front View of the Hip Joint ........................ 39
Bottom View of the Hip Joint ....................... 39
Schematic of Version 1 Electrical System ................ 41
Schematic of Version 2 Electrical System ................ 42
Plot of the Torso Angle a with Respect to Time During Walking . . . 45

Plot of the Angles made by the Inner and Outer Legs with respect to
Time during walking. The Outer leg angle with respect to the torso
is the solid line. The Inner leg angle with respect to the torso is the
dotted line. ................................ 46

Plot of control variables V1 and V2 with time. ............. 48

5.4

6.1

6.2

6.3

6.4

6.5

6.6

6.7

6.8

6.9

Plots of angles subtended by the inner and outer legs with the vertical.
The vertical dashed lines indicate the times when stance and swing
legs are interchanged. In the first time interval, the outer leg is the
stance leg (0st) and the inner leg is the swing leg (isw). After the first
interchange, the outer leg becomes the swing leg (osw) and the inner
leg becomes the stance leg (ist). ............ ‘ .........

Simulation of 0, (15 — 7r, and 1,!) over one step starting from rest and torso
angle at 1.5 deg ...............................

The nominal trajectory torques 7'1 and 7'2, corresponding to the torques
on generalized coordinates 45 and t/) respectively. ............

The work and 9f — 60 plotted against initial velocity for a torso angle
of 1.5 deg using the nominal trajectory ..................

Optimized trajectory simulation of 6, ¢—7r, and 1,!) over one step starting
from 0.1 rad/ sec and torso amplitude of -0.06 rad ...........

Zero work trajectory torques 7'1 and 72 corresponding to general coor—
dinates <15 and 1p respectively. Calculations are over one step starting
from 0.1 rad/ sec and torso amplitude of -0.06 rad ...........

9 is shown to be the same at the beginning and end of the step, re-
gardless of the initial velocity, 00. The torso amplitude is constant at
—0.06 rad ..................................

d is shown to be the same at the beginning and end of the step, re—
gardless of the amplitude of the torso angle as long as the robot is able
to complete a step. For small, or positive, torso amplitudes, the robot
is not able to reach the length of a full step and simply wobbles on the
stance leg. .................................

Parameter space in k and d. A rough estimate of the valid space of the
parameters. ................................

Starting initially from rest, relationships between the parameter d and
the final velocity at different torso angle setpoints, ad. ........

6.10 Starting initially from 1 rad/ sec, relationships between the parameter

(1 and the velocity difference at different torso angle setpoints, ad. . .

xi

49

60

6O

61

68

69

69

70

73

53> Q!

ist

isw

Jst

sz

NOMENCLATURE

distance of the center of mass of the stance leg, (m)

distance of the center of mass of the swing leg, (m)

distance of center of mass of the torso, (m)

vector of gravitational torques (in original equations of motion)
vector of gravitational torques (in constrained equation of motion)
vector of gravitational torques (in transformed equations of motion)
vector of gravitational torques (in reduced-order equations of mo-
tion)

inner leg stance

inner leg swing

acceleration due to gravity, (9.81 ms-2)

mass moment of inertia of the stance leg about its own center of
mass, (kgmz)

mass moment of inertia of the swing leg about its own center of
mass, (kgm2)

mass moment of inertia of the torso about its own center of mass,
(@7712)

length of the stance leg, (m)

length of the swing leg, (m)

length of the torso, (m)

inertia matrix (in original equations of motion)

xii

ost

ad

CIc

inertia matrix (in constrained equation of motion)

inertia matrix (in transformed equations of motion)

inertia matrix (in reduced-order equations of motion)

mass of the stance leg, (kg)

mass of the swing leg, (kg)

mass of the torso, (kg)

matrix of centrifugal terms (in original equations of motion)
matrix of centrifugal terms (in constrained equations of motion)
matrix of centrifugal terms (in transformed equations of motion)
matrix of centrifugal terms (in reduced-order equations of motion)
outer leg stance

outer leg swing

radius of curvature of the feet, (m)

vector of torques (in original equation of motion)

vector of torques (in constrained equation of motion)

generalized coordinate related to the torso angle

generalized coordinate related to the leg angles

angle of the torso with respect to the vertical, measured positive in
the cow direction, (rad)

constant desired value of a, (rad)

% arc angle of the feet, (rad)

vector of generalized coordinates: 6, d) and 2p

vector of (constrained) generalized coordinates: 6, <15

angular displacement of the stance leg with respect to the vertical,
measured positive in the counter-clockwise direction, (rad)
angular displacement of the torso with respect to the stance leg,

measured positive in the counter-clockwise direction, (rad)

xiii

‘80

T2

’l’st

73w

7"st

7'sw

an odd function in theta for the coordinate <15

angular displacement of the swing leg with respect to the stance
leg, measured positive in the counter-clockwise direction, (rad)

an odd function in theta for the coordinate 1p

generalized force corresponding to (b, (N m)

generalized force corresponding to 1p, (N m)

angle between torso and stance leg, (rad)

angle between torso and swing leg, (rad)

generalized force corresponding to first, (N m)

generalized force corresponding to 73w, (N m)

The following nomenclature pertains to instants of time when the swing leg

comes in contact with the ground:

an instant of time when the swing leg comes in contact with the
ground, (3)

horizontal impulse applied to the swing leg by the ground, (N s)
impulse applied to the stance leg by the ground in the a: direction,
(N 8)

impulse applied to the stance leg by the ground in the y direction,
(N 8)

impulse applied to the stance leg at the hip joint in the a: direction,
(N 8)

impulse applied to the stance leg at the hip joint in the y direction,
(N 8)

Iswx

Iswy

9+

q'3_

impulse applied to the swing leg at the hip joint in the x direction,
(N 8)

impulse applied to the swing leg at the hip joint in the y direction,
(N 8)

impulse applied to the torso at the hip joint in the :1: direction, (N s)
impulse applied to the torso at the hip joint in the y direction, (N s)
time derivative of 6, immediately before swing leg makes contact
with the ground, (rad/s)

time derivative of 6, immediately after swing leg makes contact with
the ground, (rad/s)

time derivative of ab, immediately before swing leg makes contact
with the ground, (rad/s)

time derivative of (I), immediately after swing leg makes contact
with the ground, (rad/s)

time derivative of t/J, immediately before swing leg makes contact
with the ground, (rad/s)

time derivative of ’l/), immediately after swing leg makes contact
with the ground, assumed to be zero, (rad/s)

disturbance impulse corresponding to 6, (N ms)

disturbance impulse corresponding to 45, (N ms)

disturbance impulse corresponding to w, (N ms)

vector of generalized coordinates 6, qb and t!) after interchange of
stance and swing legs

coordinate transformation matrix describing stance and swing leg

interchange.

XV

Chapter 1

Introduction

1 .1 Background

While the process of developing a viable general purpose walking biped has been a
long and challenging endeavor, there is still much work that needs to be done in
order to build a useful biped. We currently have machines that can walk and run
in straight lines and in circles, perform many tasks such as climbing stairs, carry
light masses, and perform complicated path planning; each of these accomplishments
stand alone as incredible feats. But current bipeds cannot do many of the things
necessary to really be useful as service robots. Primarily, they still require extremely
controlled and well known environments in which to work and the most capable robots
require a significant amount of energy to complete simple tasks. In comparison to
their human counterparts, robots simply underperform. For example, looking at
energy efficiency, the estimated specific cost of transport (energy used/[weight a:
distance traveled], Get) for human walking is approximately 0.2 and the related
specific mechanical energy efficiency (Cmt) of 0.05. Asimo, the famous Honda robot
which is still one of the most capable today, achieves an estimated 3.2 Get and 1.6 Cmt

[1]. When looking at capability, while these robots can run, they are limited to

extremely flat ground at speeds of 6 — 7 km/hr in short bursts, while the current
fastest marathon runner ran at an average 20.34 lem/hr [2]. There is still much
work to be done regarding stability and efficiency in biped robots and we hope to
understand some of the essential issues in overcoming these problems through the

development of a simple biped.

Bipedal walking is, by nature, a difficult, dynamically challenging controls prob-
lem incorporating underactuation, non-linearities, relatively high numbers of degrees
of freedom, and hybrid system dynamics due to intermittent contact impact from
foot-ground interaction and discrete events such as switching support between legs.
Therefore it is no surprise why, even after 40 years of intense study, the best bipedal
robots fail to provide the kind of stable, robust and efficient locomotion that biological
organisms can achieve [3]. But the advantages of biped design are clear. As compared
to traditional wheeled mobile robots, walking machines have the potential for better
maneuverability over uneven terrain such as outdoor environments with crevasses and
bumps. By virtue of being anthropomorphic, bipeds are well suited to sharing the
workspace with humans, interactng in a manner similar to how another human may
interact. There have been a large number of methods of adding capability to biped
machines including innovative control techniques, drive and actuator designs, and
better combinations of the two which have brought basic ability to these machines.
Here, we have chosen a sample of these developments to illustrate the wide variety of

methods developed that bring unique capabilities and add efficiency to these robots.

1.2 Review of Biped Robots and Biped Controllers

The academic study of robotic bipeds began at Waseda University in Japan with the
WL—l in 1967 [4]. Due to inherent problems with static controllers, such as speed

and stability limitations, researchers at Waseda University developed the first quasi-

dynamic (WL—9DR) and fully dynamic (WL-IORD) bipeds. The completely dynamic
biped, discussed extensively in [5], employs high-gain controllers to track trajectories
satisfying the Zero Moment Point (ZMP) condition, developed by Vukobratovic and
Juricic [6]. The ZMP method and its different interpretations [6—8], including the Foot
Rotation Index [9], have led to the development of the most stable biped platforms to
date [10—14] including the ASIMO line of robots from Honda and similar consumer

and academic robots [15, 16].

 

(a) Waseda WL-l (b) Waseda WL-lORD

Figure 1.1: Early Waseda Bipeds

The designs for these ZMP bipeds began with equipment and control techniques
developed for anchored robotic arms used in the manufacturing industry, and in
general are built in much the same manner: light linkages paired with extremely
stiff joints which help in driving joints along desired trajectories. There have also
been a number of controllers for stiff robots that do not follow the ZMP criterion
for stability. One of these robots is the Rabbit, developed at Centre National de la
Recherche Scientifique in France [17]. Rabbit is controlled using the virtual constraint
and hybrid zero dynamics method [18] which creates a single degree of freedom system
by applying virtual constraints. The ankle joint of Rabbit is passive and all other

joints are controlled to follow trajectories strictly based on the angle of the ankle.

Any locomotion gait involves synchronous movement of relevant degrees of freedom
of the system and in this regard the active synthetic wheel biped, presented in this

thesis, has similarities with Rabbit.

 

Figure 1.2: The RABBIT Planar Biped

In robots built with high mechanical joint stiffness, other techniques such as
impedance control [19—21] and variable compliance controllers [22] have been im-
plemented in high-level controllers to emulate low joint stiffness to eliminate some of
the problems inherent in stiff robots such as collision impulses with the ground. These
techniques generally also incorporate a low-level, high-gain trajectory controller for
fine position control.

This desire for the addition of compliance in joints has led to a different class
of trajectory controlled robots that incorporate variable joint stiffness, not in their
controllers, but in the mechanisms of the robots. Examples such as Lucy [23] and
W'L—14 [24], utilize compliant actuators which ultimately allow some of the natural
system dynamics to drive trajectory creation. In the WL—14, trajectory tracking that
follows the natural dynamics of the system was shown to have an associated reduction
in energy consumption. Compliance controllers have also been applied to these types

of robots. The Lucy robot was able to show this reduced energy consumption in

simulation, but was not able to experimentally show a reduction in energetic cost due

to speed limitations [25].

 

Figure 1.3: The VU Brussels Robot Lucy

Passive Dynamic Walkers (PDW’s), machines that need no actuation to find sta-
ble gaits, epitomize the use of natural dynamics to create joint trajectories. This
type of approach was pioneered by Tad McGeer in the late 1980s when he investi-
gated passive systems using limit cycle analysis together with basin of attraction and
F loquet multipliers as indicators of stability [26]. Inspired by a walking toy from
McMahon’s 1984 paper [27], McGeer developed a kneed walker that can walk with
human-like, periodic, gaits down small slopes with no control, using potential energy
from traversing down a slope to balance the losses associated with the collision of the
foot during heel strike and system frictional losses. McGeer’s methods for analysis
has produced a number of passive dynamic walkers, such as the series built at Nagoya
University [28]. The Nagoya University planar walker took approximately 4000 steps
on a treadmill, illustrating a reasonable level of stability in these walkers without the
need for a controller. The methods for quantification of stability in passive bipeds

has been expanded upon with methods such as the Gait Sensitivity Norm [29]. There

have also been a few successful three—dimensional passive walkers including the Collins
passive dynamic walker developed at Cornell. This walker was able to compensate

for the yaw forces using a pair of arms attached to the opposite leg [30].

 

(a) McGeer’s Dyna— (b) Nagoya Passive Dy- (c) Cornell 3D PDW
mite namic Walker

Figure 1.4: Various Successful Passive Dynamic Walkers

There are quite a few other robots that are not built to follow limit cycles like
the passive walkers, but are built with the intention of exploiting natural dynamics
during some portion of the gait cycle and use different controllers to remain stable
with a high level of mechanical compliance throughout the robot [1, 31—35]. Unlike
the pure passive dynamic walkers, they are able to have flat feet, which when ac-
tuated either passively or actively are able to increase the control authority of the
ankle when the foot is on the ground. Also, they champion the use of series elastic
actuators in order to reduce the impedance of the joints, which allows fast adaptation
for unknown disturbances without necessarily knowing them in advance [32]. The
virtual model control approach developed at MIT is a way to model the robot and
its behaviors as virtual objects. In this approach, the joint torques can be calculated
to mimic the virtual components of the model without the need for generating joint

trajectories [36—38]. Other controllers such as learning algorithms and neural os-

 

(a) MIT Planar Biped (b) MIT Spring (c) TU Delft
Flamingo Flame

Figure 1.5: Highly Dynamic Robots with Compliant Joints and non-ZMP Based
Controllers

cillators have also been implemented successfully in a number of different types or
robots. Neural oscillators or central pattern generators have been shown to exist in
biological systems and have provided insight into how biological systems might control
their limbs while performing tasks such as walking [39]. This motivated the approach
based on local neural networks [40] that react to inputs from local sensors. Machine
learning algorithms employ a learning controller to analyze its steps after training.
The Toddler [41], developed at MIT, can learn quickly and is efficient because of its

extensive use of natural dynamics.

This thesis describes the design for a new biped robot that we believe is a solid
platform to study the wide range of problems that bipedal robots encounter. We
wanted a design that would remain simple to model, namely having a low number
of degrees of freedom, but also included the issues seen in most bipeds such as un—
deractuation, intermittent contact, and instability. The design is an extension on
the concept of a self-propelled wheel [42], and it promises to provide impact-free and

stable motion during walking.

      

(a) RunBot with Learn- (b) MIT Toddler
ing Neural Oscillator Con-
troller

Figure 1.6: Bipeds with Learning Algorithms
1.3 The MSU Active Synthetic Wheel Biped

As an extension of previous work from Das and Mukherjee [42], we decided to un-
dertake the development of a new dynamic biped that was based on the concept
of a self—propelled wheel. This design began as a rolling wheel, as shown in Figure
1.7 (a), where three eccentric masses are constrained and driven along radial spokes
120° apart to create a mass unbalance used to drive the wheel. Similarly, a mass
unbalance can also be created by changing the relative angular position of the spokes
while keeping the radial distance of the masses fixed. This design variant is shown in
Figure 1.7 (b). We can also restrict the range of motion of these spokes within the
wheel, such that one spoke is always near the top of the disk, as a torso would be in
an upright walking biped, and the other two oscillate around the bottom of the disk,
as two legs would. From this variant, we can then cut the outer rim of the wheel and
create feet-like sections. In our design, shown in Figure 1.7 (c), these sections of the
rim are used as arc-shaped feet that “synthesize” the rim of a wheel as the legs move
along.

Although our design came from completely different work, a similar type of robot

 

 

m2

I
I
I

 

 

 

 

(b) (c)

Figure 1.7: (a) Self-Propelled Wheel, (b) Design Variant Of Self-Propelled Wheel, (c)
Conceptual Design Of Biped With Torso

had been imagined previously by Tad McGeer, though his emphasis was on using
the natural dynamics of the system to create the gait cycle [26]. This is where
the nomenclature of “synthetic wheel” originated as the wheel would be synthesized
naturally as the walker rolled along. Our robot differs in that we use active control of
the hip joints to drive the swing and stance angles along desired trajectories. Ideally
if we used the same analysis as McGeer, we could determine passive trajectories for
our particular robot, but run into limitations of our hardware, particularly foot size,
required to fully implement this at reasonable velocities. Future work may include a
limit cycle analysis of this particular biped, with hopes of obtaining a more efficient

gait, as discussed in Chapter 7.

Because the synthetic wheel concept provides a large range of stable gaits and has
a low number of degrees of freedom, we believe this is an excellent platform to study

the challenges of biped locomotion. Although our biped is normally unstable due to

the center of mass lying above the center of rotation, due to its geometry, it is able
to remain standing using a relatively simple criterion: one foot always needs to be
below the hip joint at all times. Of course, the complexity comes in how we choose
to satisfy this criterion while remaining stable and preferably with little work. This
design also allows us to avoid collisions. Vertical components of collision are avoided
through the nature of the synthetic wheel design, and the horizontal component of
the collision during foot interchange can be avoided by exchanging when the velocity
of the stance and swing legs are equal. Avoiding the horizontal collisions simplifies
analysis, but imposes constraints on our final gait.

Experimentally this resulted in a planar biped, with paired legs for lateral stability.
The outer pair of legs had a pair of motors that acted in unison to control the leg
length to avoid the swing leg from scuffing on the ground as it swung forward. This
thesis describes the process that we have taken to develop a simple 3-link biped. We
developed two different controllers that were both successfully implemented on our
experimental robot. These models and controllers are discussed in Chapters 2 and 3.
We created two different versions of the biped. The first version did not successfully
accomplish our goals and required a revision which enabled us to easily implement
complex controllers. These mechanical and electrical designs are discussed in Chapter
4. We then discuss the experiments we ran and results in Chapter 5. Optimization
of the swing leg and torso trajectories is discussed in Chapter 6, describing a method
of obtaining an efficient gait and determining velocity changes over a step, as well as

discussion on how we might find further improvements.

10

Chapter 2

Dynamic Model

2.1 Introduction

Here we present the necessary models to analyze the general motion of our three-link
biped. First, we derive the general equations of motion for the biped system and
also derive the impulse disturbances for the system when the swing leg contacts the
ground with a velocity that is not the same as the stance leg velocity. We then impose
constraints to define a gait and derive the equations of motion of the constrained
system. Finally, we show the transformation from our mathematical models to the
motor torques on our experimental platform in order to physically implement the

controllers.

A schematic of the biped is shown in Figure 2.1. It has three degrees of freedom
and can be described by the generalized coordinates: 6, (15, and t/i. These angles
denote the angle of the stance leg with the vertical and the angles of the swing leg
and torso with respect to the stance leg, respectively. The system has two actuators
corresponding to the generalized coordinates c3 and 1p and is an underactuated system.
The system also contains two prismatic joints in the outer legs to avoid foot scuffing

at center stance. The prismatic joints are assumed to be dynamically insignificant

11

 

 

 

 

 

 

 

 

 

\\\%’\\\\\\\
Figure 2.1: The Two-dimensional Three Link Model

and are therefore not included in this analysis.

2.2 General Dynamic Model

Using Lagrange’s equations, the dynamics of the system shown in Figure 2.1 can be
written in the form
M(Q)ri+N(q.ti)<i+G(q) = T (2-1)
where q and T are given by the relations
T

q=[e¢¢]T, T=[oT,.2] (2.2)

and q is the vector of generalized coordinates; T is the vector of the generalized forces

with 1'1 and T2 denoting the torques of the motors that control the relative angles

12

¢ and tb, respectively; M (q) is the symmetric inertia matrix; N (q,q) is the matrix
containing centrifugal and Coriolis terms; and G (q) is the vector of torques produced
by the gravitational force. In this model, 6 is the passive degree of freedom and
therefore its corresponding generalized force is zero. The terms M (q), N (q, (j), and

G (q) are defined as follows

M = [Mijl3x3’N = [Nijl3x3’G =[Gz‘13x1 (23)

where

Mu = J, + Jst + sz + rm [(1% + 212(12 + (d, — R) cos(6 + a) — (10]
+m3t [613, + 2R(R + (as, — R) cos(6) — c130]
+m3w [dgw + 2R(R + (dsw -— R) cos(6 + t/J) — dsw)]

M12 = M21 = Jt + mt(dt - R)(dt — R + Rcos(6 + (15))

M13 = M31 = sz + msw(d3w — R)(dsw — R + Rcos(6 + #0)

M22 = Jt + mt(dt - Rlz

M23 = M32 = 0

M33 = sz + m3w(dsw — a)2

N11 = mtR(R — dt) 311109 + ¢)(9 + (13) + mstR(R — dst) Sinwé
+m3wR(R — dsw) sin(6 + We + 6)

N12 = mtR(R — dt) Sin(9 + (”(9. + <23)

N13 = mszm — d3...) sinw + W + zi)

N21 = N22 = N23 = 0

N31 = N32 = N33 =0

13

G1 = ["3th — dt) $111“) + <15) + mst(R - dst) 513(9)
+msw(R — dew) sin(9 + ¢)lg
G2 = "MR - dt) Si11(6 + 45)!)

03 = msw(R — dsw) sin(6 + tb)g (2.4)

and mt is the mass of the torso and rust and mm are the masses of the stance leg
and swing leg, respectively; Jt, Jst and sz are the mass moments of inertia of the
torso, stance and swing legs about its center of gravity; R is the radius of curvature
of the feet; dst, dsw and dt are the distances of the center of gravity of the legs and

torso; and g is acceleration due to gravity, as shown in Figure 2.1.

2.3 Impulsive Disturbances due to Swing Leg

Collision

 

Figure 2.2: Free-body diagrams of the stance leg, swing leg and torso showing impulses
of all impulsive forces generated as a consequence of swing leg touchdown

14

The prismatic joints on the biped prevent collision between the swing leg and the
ground in the swing phase. Additionally, because the feet are of the same radius as
the leg length, they ensure that there are no vertical impulsive forces on the swing leg
at the time of touchdown. The swing leg may, however, be subjected to horizontal
impulsive forces since we may have 16— 7e 0. Assuming that the impulsive force on
the swing leg results in 62+ = 0, we investigate its effect of the motion of the stance
leg, swing leg and torso using the free-body diagrams in Figure 2.2. A summation
of linear impulse in the a: and y directions and angular impulse in the z direction
about the center of mass of these links provide the following nine (three sets of three)
equations:

Equations for stance leg:

[stat + 101; = mst [R — (R - dst) COS 9] (6+ — 9—)
13,, + 10, = — m3t(R — d3.) sin6 (9+ —- 9-)
J3t(6+ — 6‘) = [13153; cos6 + Isty sin 6] (R — dst)
— 169: [R — (R — dst) cos 6]

+ Icy [R — (R — dst) sin 6] (2.5)
Equations for swing leg:

1wa + I = msw [R — (R — dsw) cos(6 + m] (9+ — (9‘ — 6‘)
1w, = — m3w(R — as...) sin(6 + ¢)(é+ — 9'- — 215—)
sz(é+ — 9- — 213-) = [1m cos(6 + 1p) + 13,, sin(6 + in] (R - List)

— I [R — (R — dsw) cos(6 + 111)] (2.6)

15

Equations for torso:

In 2 mt [R — (R — dt) cos(6 + ¢)] (6+ — 6_ + d)+ -- (15—)
1.3, = — mic-2 — d.) sin(6 + ¢><é+ — 6' + <i+ — if)

Jt(6+ — é“ + 43+ — (if) = [In cos(6 + a) + 1., sin(6 + ¢)] (R — at) (2.7)

The above nine equations, along with the two equations for force balance at the hip

joint:

Its: 'l' Istx + Iswa: = O

Ity + Isty + Iswy = 0 (2-8)

allow us to solve for the eleven unknowns, namely: 13”, I stya 13m, Iswy, In, Ity,

IGI, 10y, I, 6+ and 65"". A part of the solution is as follows:

Its: = 0
9+ = (9‘ + mi;- (2.9)

where K. is a constant whose value depends on the kinematic and dynamic parameters
of the biped and is defined in Appendix A. Substituting Equation 2.9 into Equations
2.5, 2.6 and 2.7, the angular impulses on the stance leg, swing leg and torso are found

to be
0 1 J3t(0.+ — 9—) = Jst Rd.)—

6 + «p : J.w(é+ — é— — r): J... (n — mi- (210)
0+¢ : Jt(é+—é-+qi+—gi-) =0

The angular impulse expressions in Equation 2.10 correspond to the generalized co-

16

ordinates 6, 6 + 1P and 6 + 45, respectively. The angular impulses corresponding to the

generalized coordinates 6, (b and ’t/) can therefore be shown to be

0 I d9: [Jstrt+sz(K»-1)]1/1_
¢ : 61¢ = 0 (2.11)
a : d.) = J... (i. —1)¢-

An accurate dynamic model of the biped, that takes into account the impulsive dis-

turbances due to swing leg touchdown, can now be written as follows
M(a)éi+N(q,ci)é+G(q)=T+D5(t-tc) (2-12)

where D is the vector of angular impulses, defined as

T
D=|:dg dd) (11),]

and d9, d¢, and d1!) are defined in Equation 2.11. It is clear from the expression for d9
in Equation 2.11 that swing leg touchdown results in an impulsive disturbance in the

passive degree of freedom 6. This is also obvious from the last relation in Equation

2.9.

2.4 Constrained Motion and the Nominal Trajec-

tory

Due to our synthetic wheel design, we must only guarantee the stance foot is always
placed with a portion directly below the hip to remain stable as the robot walks.
Therefore we are free to arbitrarily choose torso and swing leg trajectories that satisfy

this condition.

17

 

 

 

\\\\\\\\\\\

Figure 2.3: Illustration of the generalized coordinates (15 and t!) and their relations to
6 and a.

We have chosen a couple of general parameterized gait functions which we can
modify to create different types of constraint trajectories. We chose parameterized
gait functions with 5 parameters to minimize the number of parameters needed to be
determined while simplifying the trajectory to be tracked. These parameterized gait
functions are:

¢=(7r+ad—6)+dsin (%+f) (2-13)

for the angle between the stance leg and the torso and

111 = k6 + asin (g) + csin (359) (2.14)

for the angle between the stance leg and the swing leg. Here, ad is the mean value
of the desired torso angle offset with respect to the vertical, ,8 is half of the foot arc

angle, and the parameters It, a, c, d, and f are available to be determined.

18

We will later discuss different modifications to these general parameterized gait

functions to allow us to satisfy different gait boundary conditions in Section 6.

The first control trajectory that we investigated was what we call our “Nominal

'Itajectory”. We begin by imposing the following two constraints:

C : w = —26
1 (2.15)
C2 a = ad
where a, depicted in Figure 2.3, is defined by the relation
a=d>+6—7r (2.16)

The constraint C1 ensures that the swing leg is symmetric with respect to the stance
leg about the vertical at all times. The constraint C2 ensures that the torso maintains

angle ad with respect to the vertical.

To satisfy these constraints, we require our parameterized gait functions of Equa-
tions 2.13 and 2.14 to have parameter k = —2 and all the other parameters equal to

zero. The equations of the constrained motion then simplify to

¢=7T+Ozd—9 (2.17)

and

w = ——29 (2.18)

The torques (7'1 and 7'2) required to impose these constraints can be easily com-
puted from Equation 2.1. The constrained system has one passive degree of freedom

with the following dynamics which is derived from Equation 2.1.

Mc(6) a + New, 9') + CC = 0 (2.19)

19

where

MC = 1,, — 1..., + mstdgt — mswdgw
+(mt + 2m3t)R2 — 2R(mstd3t — mswdsw)
+mtR(R — dt) cos ad — 2m3tR(R — dst) cos6
NC = R [R(m3t — msw) — mstdst + mswdsw] sin662
CC = [R(m3t — msw) — mstdst + mswdsw] g

—mt(R - dt) sin ad 9 (2.20)

When the stance and swing legs are identical, i.8., mst = msw, dst = dsw, and

I st = 13w, the expressions in Equation 2.20 simplify to the form

MC = (mt + 2msj)R2 + mtR(R — dt) cos ad — 2m3tR(R — 61306086
NC: 0

CC = —mt(R — dt) sin ad g (2.21)

For any set of reasonable parameter values, it can be verified that 6 will be positive for
positive angle ad and vice versa. This will theoretically result in unbounded growth
of 6 but in reality 6 will saturate at some maximum value due to friction and damping
effects that have not been modeled.

These constraints for the swing leg and torso were chosen due to an intuitive
physical interpretation of the system; if the legs have the same mass and moment of
inertia, the total motive torque on the system is directly proportional to the angle
of the torso. This means that the robot should constantly accelerate at a rate that
is entirely determined by the angle of the torso. As the robot leans forward, it
accelerates the robot forward (or slows it down if it traveling backward at the time)

just as leaning backward will do the opposite. This is not at all an optimal trajectory,

20

and necessitates an impulse at foot switching because it necessitates that 2/2— 75 0 if
6 75 0, as discussed in Section 2.3. This causes a number of problems for analysis
of stability of the system but, as shown in our experiments, it does not cause the
experimental system to destabilize. A more detailed discussion about the impulse
disturbance due to leg switching at unmatched velocities is covered in Section 2.3.

Better choices for trajectories are discussed later in Chapter 6.

2.5 Interchange of Stance and Swing Legs

If we consider the constrained motion of the biped for some positive angle ad, it
results in positive acceleration 6. This will result in a positive velocity 6 from zero
initial velocity. When the biped has a positive velocity, it will roll on its stance leg
and the point of contact with the ground will move from the heel to the toe. The
stance and swing legs can be interchanged at any time, but to have the maximum
step size the swing leg should touch down when the heel of the swing leg is right in
front of the toe of the stance leg. As the swing leg becomes the new stance leg and
vice-versa, the definitions of the generalized coordinates change. We can determine a
transformation matrix from the coordinate set during the step to the coordinates of
the new step.

Without assuming maximum step size, an interchange in the stance and swing
legs will result in the following transformation of the generalized coordinates and

their velocities, irrespective of the direction of motion of the biped

 

 

l 0 1 ]
Qnew =P qold: P: O 1 —I (222)
0 0 —-1

where gold is the set of generalized coordinates during the step and qnew is the new

21

set of generalized coordinates after the step has been made. Therefore,

 

 

 

 

anew 901d + ibozd
¢new = ¢old " lbold (223)
_ ibnew _ _ -¢ozd .
and _ _ _ -
612.22» 901d + tiara
43mm 2 450m — lbold (2-24)
_ a... _ . to... ,

 

 

 

 

The elements of M, N and G matrices in Equation 2.1 are functions of mst, msw, dst,
dsw, J at and J3“), and consequently their values need to be updated in the dynamic

model after every interchange of stance and swing legs.

2.6 Forward and Inverse Relations for Control Trans-
formation for Experiments

In our biped platform, two actuators control the relative angles between the torso and
the stance leg and the torso and the swing leg, respectively. We denote the torques
of these motors as Tst and 73w and their corresponding generalized coordinates as cyst
and 73w, respectively. In order to apply our torques generated from our generalized
model, we must transform the torques T1 and 72 into the torques of Tst and Tsw. The
angles a, Vst and 7310 are related to the generalized coordinates 6, ()5, and 11), as shown

in Figure 2.4, by the forward and inverse relations

a = 6+¢—7r 6 = a+73t
'l’st = 7r — Cb => (b = 7r — 'Yst (2.25)
7310 = 7r - (15+ 2P 16 = 73w _ 'Yst

22

 

 

 

 

 

K\\<’\\\\\\\

Figure 2.4: Relationships of the generalized coordinates 6, (25, and ’t/) to the coordinates
"Yst, 73w: and 0-

According to the virtual work principle [43],the total virtual work done by the

actuators in each coordinate is the same, z'.e.,

Tst6'73t ‘l‘ 7.8106732!) = 7—1695 + 7261/} (2°26)

On the other hand, from Equation 2.25, we can Show

565 = ‘6’7’313

51/1 = 6’73w"6’73t (2-27)

23

Substitution of Equation 2.27 into Equation 2.26 gives

Tst = ”Tl-T2

73w = 7'2 (2.28)

which are the actual torques applied by the motors.

24

Chapter 3

Controller Design

3.1 Introduction

Two different robot controllers have been implemented successfully on our biped. We
call the first our “Decoupled” controller, because the stance and swing leg torques
were determined independently of each other and without considering the coupled
dynamics of the system. The more complete “Coupled” controller contains the full
dynamics of the robot and utilizes feedback linearization for the swing leg and stance
leg torque calculations. Both of these controllers were designed using our “Nominal
Trajectory”, as described in Section 2.4 and use the interchange transformation dis—
cussed in Section 2.5. The model is a two dimensional model, and assumes lateral
stability in the experimental setup for all controllers. Using an optimized trajectory,
such as those described in Chapter 6, would necessitate a change in the control objec-
tives, but analysis could be conducted in a manner similar to that described in this

section.

25

3.2 Decoupled Controller

3.2.1 Controller for Fixed Torso Angle

The control objective is to reach a fixed desired angle for the torso, i.e., a -—> ad as

t ——> 00. From Equation 2.25, we can write
6 = (a + 73!.) (3-1)

If ad is the desired angle of alpha and where 7% is the desired angle of the stance

leg, then

9 = (ad + iii) (3.2)

If we define the error

61 = (ad — a) (33)

then Equation 3.1 gives an alternate expression for el, namely
«91 = (’i’st — 73‘.) (3.4)
To achieve the first control objective, we propose the following error dynamics
61 + kdlél + kplel = 0, kp1,kd1 > 0 (3.5)

where km and kdl are positive constant gains. Since ad is constant, we can use

Equations 3.3 and 3.4 to rewrite Equation 3.5 as follows

73. ~73. = —kd,a+kp1<ad —a) (3.6)

26

Since ‘jfit is the desired acceleration, the motor control torque Tst will be proportional
to (”7% — fist); 01'

Tst = C163. - ’Y'ét) (3-7)

where c1 is some positive constant whose value depends on the kinematic and dynamic
parameters of the stance leg and its drive. Using Equations 3.6 and 3.7, the first torque

controller is expressed in the form

Tst = 01 [-kdld + kpl (01d -- (1)] (3.8)

In the above equation, cl, km and kdl can be lumped into two independent parame-
ters c1 kpl and cl kdl. The values of these parameters were found during experiments
through tuning. We also implemented a method to choose independent values for
these gains for each outer and inner legs while in stance. This allowed us to better

correct for differences in inertia, mass and friction between the legs.

3.2.2 Controller for Constrained Motion of Swing Leg

The second control objective is to ensure that the swing leg and the stance leg move
symmetrically with respect to the vertical, i.e., satisfy the first constraint in Equation

2.15. Using Equation 2.25, this constraint can be written as follows
i/J + 29 = (73w + ’73t + 201) = 0 (3-9)
The desired value of the swing leg is therefore

if... = ‘(73t + 2a) (3-10)

27

We define the error

62 = (7?... — ”73112) (3.11)

and impose the following error dynamics
éiz + kd2é2 + [€1,262 = 0, kPQ’ kd2 > 0 (3.12)

such that 62 —> 0 as t —) 00. Using Equations 3.10 and 3.11, we can write

62 = "(73w + 7323 + 20‘)
(.32 = —("st + ”m + 2(5!) (3-13)
at = i3... — is...

Substitution of Equation 3.12 into Equation 3.13 yields

yg’w - as.” = 19,12 (73w + ‘yst + 2a) + kpghsw + 73t + 20:) (3-14)

Since rig,” is the desired acceleration, the motor control torque Tsw will be proportional
to (fl-[gm _ flaw): 01'

T... = C263... — a...) (13-15)

where c2 is some positive constant whose value depends on the kinematic and dynamic
parameters of the swing leg and its drive. Using Equations 3.14 and 3.15, the second

torque controller is expressed in the form

7'sw = C2 [kdz (7310 ‘l‘ ’l’st 'l' 2d) + kpg (7310 + Vst + 20)] (3-16)

In the above equation, c2, [C192 and kdg can be lumped into two independent param-
eters 62er2 and c2kd2. Once again the appropriate values of these parameters were

found experimentally and their values were dependent on the current swing or stance

28

leg.

3.2.3 Simulations of Decoupled Controller

A numerical simulation of the system and controller was created and the outputs can
be found in Figure 3.1. This simulation uses the mass and inertial properties of our
experimental platform, provided in Table 4.1 with initial conditions of 6 = —11.25 deg,
(25 = 91.25 deg, and tb = 22.5 deg. The robot starts from rest and the motion of the
torso and legs with respect to the vertical are plotted. The joint torques for the stance

and swing legs are also plotted. The desired torso angle, ad, was set to 2 deg.

The controller gains were set to kp = 400 and kd = 20. The controller gains
were chosen to keep the maximum motor torques near the 15 N m limit of our system
within the arbitrary 5 sec of the simulation. Because the robot has a forward—leaning
torso, it will continue to accelerate due to the lack of dissipative elements in the
simulation. This means that the joint torques will continue to increase in magnitude
as the robot continues to accelerate. In the actual robot, the robot would at some
point stop accelerating when the losses match the increase in energy caused by the

torso angle, and therefore reach a steady state velocity.

The decoupled controller has difficulty tracking the desired torso and leg angles.
As the robot’s velocity increases, the tracking error becomes worse as the dynamic
effects of the uncompensated parts of the system become larger. This is particularly
apparent in the torso angle. The torso oscillates with each step, reaching approxi-
mately 3 degrees of error in within the 5 sec. The next section discusses our coupled

controller, which dramatically reduces these errors.

29

 

 

a (deg)

 

 

Leg Angle (deg)

 

 

 

Torque (Nm)

 

 

 

 

Time (s)

Figure 3.1: Simulation results of the decoupled controller. The motion of the torso
is shown in (a) as a solid line, the desired torso angle, ad is shown as a dotted line.
The leg motion is seen in (b) where the angle of the stance leg with respect to the
vertical is the solid line, and the angle of the swing leg with respect to the vertical
is the dotted line. Graph (0) shows the two motor torques for T3t (black) and Tsw
(grey). Compare this with Figure 3.2.

30

3.3 Coupled Controller

3.3.1 Motivation

In this section, we discuss the design of torque controllers that will enable the biped
walk forward or backward along a straight line using a full dynamic model, as opposed
to the decoupled model discussed earlier. It was determined that our decoupled
controller could not track the desired trajectories sufficiently, and a possible reason
for this is that the swing leg controller has no information about the torso angle
or the torso dynamics. In the decoupled control method, one controller could not
compensate for the dynamics of the other directly, possibly leading to relatively large

oscillations in the torso angle as the robot walked along.

3.3.2 Feedback Linearization

To design a controller that achieves the control objectives stated in Section 2.4, we

consider a new set of generalized coordinates for the system

 

 

6
a = v1 (3.17)
V2
where
V1 = a — ad
V2 = 26 + "(,6 (3.18)

These new variables are related to the original generalized coordinates 45 and 2/1 by

31

the following relations,

()5 = V1—6+7r+ad

a = V2—26 (3.19)

Substituting Equation 3.19 into Equation 2.1, we obtain the dynamics of the system

in terms of the new generalized coordinates as follows

M (a) «'7' + N(q, ci) 6+ 6(6) = T (3.20)

where [[4, N and C have the same dimensions of M, N and C, respectively. The
generalized force corresponding to 6 in Equation 3.20 is zero and this allows us to
eliminate 6 from the two equations corresponding to the generalized coordinates V1

and V2. The reduced-order equations have the form

A U A

M (2) (i + M2. «7) 2 + 6(6) = T (321)

where M E RQX2, N E R2X2 and G E R2X1 are functions of all three generalized

coordinates 6, V1 and V2, and

'01

Q)
H
H)
II

(3.22)
'02 7'2

Equation 3.21 represents a completely actuated system and we use feedback lineariza-

tion to design our controller as follows

A A

T = N (r. ci) (i + 6(a) - M(c‘r‘Xchi + Kp 2) (3.23)

where Kd and Kp are diagonal positive-definite matrices of dimension two. Indeed,

32

substitution of Equation 3.23 into Equation 3.21 results in
éj+Kdé+Kpcj=0 (3.24)

which implies (j -—) 0 as t —> 00. From the relations in Equations 3.22 and 3.19 it

simply follows that
”01,112 —) 0 :5 a —> ad, 1,1) —) —26 (3.25)

as t ——) 00, therefore the constraints in Equation 2.15 are satisfied.

3.3.3 Simulations of Coupled Controller

A numerical simulation of the system and coupled controller was created and the
outputs can be found in Figure 3.2. This simulation uses the mass and inertial prop-
erties of of our experimental platform, provided in Table 4.1 with initial conditions
of 6 = —11.25 deg, ()5 = 91.25 deg, and it = 22.5 deg. The robot starts from rest and
the motion of the torso and legs with respect to the vertical are plotted. The joint
torques for the stance and swing legs are also plotted. The desired torso angle, ad,
was set to 2 deg.

The controller gains were set to kp = 400 and kd = 41. The controller gains
were chosen to keep the maximum motor torques near the 15 N m limit of our system
within the arbitrary 5 sec of the simulation. Because the robot has a forward-leaning
torso, it will continue to accelerate due to the lack of dissipative elements in the
simulation. This means that the joint torques will continue to increase in magnitude
as the robot continues to accelerate. In the actual robot, the robot would at some
point stop accelerating when the losses match the increase in energy caused by the

torso angle, and therefore reach a steady state velocity.

33

The coupled controller exhibits much better tracking of the desired torso and leg
angles. This better tracking also results in decreased torque values. The torso error
has been reduced to a fraction of a degree for all velocities. The better tracking of the
torso has led to a reduced acceleration over the simulation time because the average

position of the torso is closer to the desired torso angle.

 

(a)

 

(1 (deg)

 

 

Leg Angle (deg)

 

 

 

Torque (Nm)

 

 

 

 

Time (s)

Figure 3.2: Simulation results of the coupled controller. The motion of the torso is
shown in (a) as a solid line, the desired torso angle, ad is shown as a dotted line.
The leg motion is seen in (b) where the angle of the stance leg with respect to the
vertical is the solid line, and the angle of the swing leg with respect to the vertical
is the dotted line. Graph (c) shows the two motor torques for Tst (black) and Tsw
(grey). Compare this with Figure 3.1.

34

Chapter 4

Experimental Methods

4.1 Background

During the period of our research two major versions of this robot were built. Al-
though the mechanical system has remained largely unchanged, the architecture of
the electronics and the controllers changed significantly. Originally we wanted to use
a simple central computer together with integrated position controllers of the motor
amplifiers to drive the hip positions to the desired locations. This proved to not work
very well as discussed later. The main impetus for changing our computing system
was the desire for better inertial measurements. Our original inertial measurement
unit was a crude device made with simple accelerometers which were then filtered to
approximately determine the direction of gravity. This technique was not accurate
enough for a controlled gait and it was decided that we should switch to a more
powerful device. This change to a commercial inertial measurement unit required an
upgrade to our central computer and therefore it seemed prudent to change our focus

from light computing to ease of programming and communications.

35

4.2 Mechanical Design

We designed and fabricated our synthetic wheel biped based on the schematic in
Fig.2.1 but used paired legs to avoid lateral instability, a common method in 2D

bipeds ( e. g. [26,35,44]). A picture of the robot can be seen in Figure 4.2.

    

 

(a) CAD of MSU Biped

(b) Picture of MSU Biped

Figure 4.1: MSU Synthetic Wheel Biped With Torso

The inner pair of legs is solid, and therefore cannot adjust for leg swing clearance.
We compensate for this by adding a pair of prismatic outer legs that were able to
lengthen during the stance phase and shorten during the swing phase, but still remain
dynamically similar throughout the leg’s range of motion, as in Figure 4.2. A drawing
of the prismatic joint can be seen in Figure 4.3. These joints were only used in the
outer legs to reduce the complexity and number of required motors. Note that at this
time we do not use the prismatic legs for propulsion, but rather only as a means for
providing foot clearance. Because the prismatic joints are in the outer legs only, it
necessitates that when the outer legs are the stance legs, the prismatic joints must

lift the entire mass of the robot, and therefore the motors must be of relatively high

36

power. The robot has 3 internal degrees of freedom (as defined in Fig. 2.1) and stands
1.09 m tall. The main structural material is 6061 Aluminium. Dimensions and mass

distribution of the robot are described in Table 4.1.

Table 4.1: Kinematic and Dynamic Parameters of Synthetic Wheel Biped

 

Kinematic Parameters
Length (m) Foot radius, R (m) 5400215212, 6 (deg)

 

 

 

 

Inner leg 0.635 0.635 11.25
Outer leg 0.635 0.635 11.25
Torso 0.457 — -

 

 

 

 

 

Dynamic Parameters .
Mass (kg) Inertia (kgmz) d in Fig. 2.1 (m)

 

 

 

 

Inner leg 1.64 0.094 0.285
Outer leg 3.64 0.128 0.355
Torso 11.87 0.198 0.307

 

 

 

 

 

 

The robot has four 24 Volt DC servo motors: two of them (Maxon RE40, 150W)
are located at the hip and control the angle of the stance leg and swing leg with
respect to the torso; the other two (Faulhaber 3243 CR, 26.3W) provide simultane-
ous extension and contraction of the outer paired legs. Each hip drive is connected
between the torso and a leg through a 43:1 planetary gearhead (Maxon 42C) and a
final 2:1 helical gear drive. The leg extension motors have a 3.71:1 planetary gear-
head (Faulhaber 38/1) and are connected to the bottom part of the leg through a
4.72 turn/ cm ACME screw drive. The leg extensions ride on linear bearings for a
smooth action. The hip joints are made of two concentric tubes, the inner tube
is connected to both the inner legs while the outer tube is connected to both of the
outer legs. Each tube is also attached solidly to a driven gear, as in Figures 4.4 and
4.5. The motors drive the drive gear which in turn rotates the driven gear, and there-
fore changes the angle of the leg with respect to the torso. The motors are mounted
slightly offset fore-aft with respect to each other, but retain a symmetry with respect

to the midline.

37

   
    
 
 
 

  

Drive
Motors

Inner Tube Outer Tube

 

 

 

 

Motor With Gearhead

Helical Coupler

ACME Screw and Nut “.t

Linear Bearing \\ . /:\/

Figure 4.3: Outer Leg Prismatic Joint

38

Driv Gear Bellows Outer Leg Motor

Coupler w/ gearhead

 

  
    

Inner Leg

Outer Leg //

 

I riven Gear '

Figure 4.4: Front View of the Hip Joint.

Outer Leg Bellows Outer Leg Motor Inner Leg
Drivel Gear youpler w/ ge/arhead Driven Gear

 

Inner Leg Motor

Outer Leg W/ gearhead Inner Leg
Driven Gear Drive Gear

Figure 4.5: Bottom View of the Hip Joint.

39

4.3 Electrical Design

4.3.1 Version 1

The original computing system for the robot was a AVR ATMega128 chip on a
STK300 development board from Kanda (Embedded Results Ltd., PO. Box 200
Aberysth, SY23 2WD, UK). This board was used to compute position commands
that would be sent to the DZRALTE drives from AMC through a step and direction
method of communication. The motor amplifiers would then calculate a current to
output to the motor based on an internal PID controller that could be independently
programmed.

This simplified the design because complex encoder velocity calculations could be
performed by the motor amplifier rather than having the main computer count the
actual values of the incremental encoder. For this method to work, the main computer
has to assume that when a joint is commanded to a specific position, that the joint
arrives at that commanded position within a reasonable time. This generally requires
large gains for fast response.

The largest problem with this method came from our inertial measurement unit
(IMU). It was decided early on that the measurement of the direction of gravity could
have a relatively large error because as long as the robot has the leg near the correct
position (i.e. on the correct side of the body) at the switching time, it would remain
stable due to the synthetic wheel design. To reduce the expense of the robot, a simple
accelerometer based device was built and it was assumed that if the robot was slowly
accelerating horizontally or was traveling at a constant velocity, the measurement of
gravity would be relatively accurate. This, along with heavy filtering, was thought to
provide a reasonable measurement, even though the signal would be subject to lag.
The system provided some decent results when the device was slowly rotated through

its range of motion, but was found to be severely deficient during actual walking

40

exercises, encouraging us to abandon this method and either buy a professional IMU
or to spend considerable resources designing and developing a better IMU containing
rate gyros and accelerometers with appropriate Kalman filters. We happened to have
a commercial inertial sensor available to us. The use of this type of off the shelf
IMU required an upgrade of our computing resources, and necessitated the Version 2
redesign.

DZRALTE
IMU (PID Controller)

 

 
 
 
 
   
 
 
 

Current 533:”

Position Faulhaber 3243

jib
3 r 53:?"

Maxon RE40

fl

 

Torso
Attitude

 

 

 

Kanda AVR

 

 

 

—* Analog
—’ HV Current

—> Inc. Encoder (Pulses)
—> Digital l/O

 

 

 

 

 

Figure 4.6: Schematic of Version 1 Electrical System

4.3.2 Version 2

Version 2 saw a number of changes in the way sensor data was collected, what infor-
mation was available to the main controller and a significant increase in computing
speed. The system redesign included a new embedded computer, new digital and
analog acquisition board, new encoder counters, and a new power supply.

The largest change in the system was the main computer. The robot was con-
trolled using a relatively low power (12W) embedded computer (WinSystems EBC-
855) running a C program on a Linux operating system. The operating system was
later changed to incorporate the Xenomai real-time framework with the Linux ker-
nel, which allowed for more accurate time measurement by measuring time from the

computer system, rather than having to poll an external timer.

41

DZRALTE

(PID Controller) Current

          
 

InertiaCube

WinSystems

‘ t ~ ~ ~ (Torque Controller)

‘~
\ ‘~

 

 

—> - -> Analog (+/- 10V) 3555‘, aMxon RE40
e x

—> R8232 Serial
PC104 Bus
—> HV Current
—9 Inc. Encoder (Pulses)
—> Digital I/O

 

 

 

 

 

Figure 4.7: Schematic of Version 2 Electrical System

Data acquisition was provided through a PC104 format data acquisition board also
by WinSystems (PCM-MIO). In combination, these boards were capable of reading
all of the sensors that we needed while providing plenty of computational overhead,
802.11g wireless and Ethernet.

To allow our computer to read the hip positions, we added encoder counter chips
(Avago HCTL—2017) to the hip encoders. These allow our robot to easily determine
the absolute position of the legs through the use of parallel digital inputs on our data
acquisition board.

The walker relies on accurate absolute angle measurement of the torso to operate.
To measure this, we used one axis of an InterSense Inc. Inertiacube2+ 3D inertial
sensor. This provides torso angle data with an angular resolution of 0.01 degrees at

a rate of 180Hz.

The hip joint drives were programmed to provide current control, requiring a +/-

42

10VDC analog control signal that was proportional to current. The leg extension

motors were run in a position control mode, requiring a +/- 10VDC analog control

signal that was proportional to position. A comparison of both versions of control

system hardware system components can be found in Table 4.2.

 

 

 

[ [ Version 1 Version 2 ]
Computer AVR ATmega128 (Kanda) WinSystems BBC-855
I / O Card On board I/ O WinSystems PCM—MIO

 

Inertial Sensor

Self-Built, accelerometer

 

Inertiacube2+

 

Motor Encoders

Agilent HEDS—5500 x4

 

 

 

Amplifiers AMC DZRALTE x4
Battery 20 cell - 3800 mAh SubC NiMH
Hip Motors Maxon RE40, 150W x2

 

 

 

Leg Extension Motors

Faulhaber 3243 CR, 26.3W x2

 

 

Table 4.2: Control System Hardware

43

Chapter 5

Experimental Results

5.1 Performance Comparison

We measured the kinematics of the robot walking on flat ground while using both our
controllers. Both of our controllers qualitatively worked and were able to complete
many successful steps down a hall, but the coupled controller, discussed in Chapter
3.3, showed better trajectory tracking and better disturbance rejection of step to step

impulses.

5.1.1 Decoupled Controller

The kinematics of the biped were measured using an eight camera, 3-D, passive marker
motion capture system, a product of Motion Analysis Inc., at 120Hz. Reflective
markers were attached to the robot to allow external measurement of the torso and
both legs with respect to gravity as it walked down a runway. The experimental
results are shown in Figures 5.1 and 5.2.

The angle of the torso with respect to the vertical, or, is shown in Figure 5.1. The
desired angle, ad, was set to 2.5 deg and the results indicate that a did not stabilize

to this value. It had a steady oscillation and this in part due to the fact that the

44

torque controller for stabilization of a was designed assuming that the torso dynamics
is decoupled from the dynamics of the legs. Another reason for this oscillation in the
torso angle is from the impulse at switching due to the fact that the swing leg has
a different velocity from the ground. This impulse causes a jump in the velocity of
the torso to which the controller cannot instantly correct. The results shown below
indicate that the oscillation in 0 did not adversely impact the gait of the biped.
The angles of the inner and outer legs are plotted in Figure 5.2. This figure shows a
symmetric motion of the legs which implies that the torque controller for constrained
leg motion performed well. The average amplitude of the legs were 23 deg, which is
close to the ideal case of 22.5 deg. Ideally, the legs must change their direction of
rotation at 3:11.25 deg but switching occurs at slightly different angles because of

error in initialization as well as lag in the controller.

 

(ID
_[

 

 

 

 

 

 

00 2 4 6 8 10

Time (sec)
Figure 5.1: Plot of the Torso Angle a with Respect to Time During Walking

45

_L
01

 

      
  

 

 

I T
D ' _[
810r ,'. ". i'. I".
(D f ,1 " H II
3 n ,‘ 1‘ '1 11 '1
U) A " ,' 1' :1 11 '1
CD 5- r“ " ,' 1' 1' 1. '1 _
9 r -- :: -: -: .1 -- :-
h "\ 1| ' | .‘ 1| '1 1 1 '1
9 | I l I '1 I. l
3 I 1 '\ ' ‘ I ‘ I ‘ | I I I I
O 0-: . 1 I : : 1 : 1 : : I I: :1 g
1
g ‘1 I '1 I. 'l f '1 f ' : l f' :I
(U 1 | 1 '
l
h ' l
“=’ -5‘- : : : . ~
5 1 j 1 | 1
(I) 1 ‘| : | : 'l l I 1' 1 1 'l 1 1 l 1
5 : | l I l I I. |. I l I l I l I
“—_ __ II E. ‘ I I II |' I I I _
O 10' I | ‘l I [I I .[ |l
’ 1' 1 ' |' I. 1! .' 1 I I'
a) 1 1 1 . 1 1 1
E) p ., I" ‘1 1, .1 '1 1| 1'
. 1
c 1' 1} ‘1 ‘g' I". 1' '1: ‘1'
<—15L V l' ' |' n V—
1' -
_2G I l l L

 

 

O

N
(I)
—L
0

Time (see)

Figure 5.2: Plot of the Angles made by the Inner and Outer Legs with respect to
Time during walking. The Outer leg angle with respect to the torso is the solid line.
The Inner leg angle with respect to the torso is the dotted line.

The distance traveled by the biped can be estimated from the following equation,
D = nsRA (5.1)

where as is the number of steps taken by the robot, R is the leg length, and A is the
angular amplitude of rotation of each leg during a single step. From Figure 5.2, 17.3
and A are found to be 16 steps and 23 deg (0.4 rad), respectively. This results in an
estimated distance of 4.07 m, which closely approximates the actual traveled distance
of 3.9 m.The distance was covered in roughly 10 sec and therefore the average walking

speed of the biped was 0.4 m/ sec.

46

5.1.2 Coupled Controller

The synthetic wheel biped in Figure 2.1 has no preferential direction of motion. With
reference to Figure 1.7, it can walk in the direction of the positive :1: axis as well as
negative .7: axis. To make the biped walk in the direction of the negative x axis, the
desired angle of the torso with respect to the vertical was set to a negative value,
equal to ad = —1.5 deg. Since the torso has a large mass (as 70% of total mass -
see Table 4.1), a small torso angle is sufficient to cause the biped to walk. The biped
was programmed to walk with the maximum step size. Since the biped was walking
in the direction of the negative 2: axis, the stance and swing legs were interchanged
when the point of contact of the stance leg with the ground reached the heel, i.e.,

when 6 = —fi = —-11.25 deg (see Table 4.1).

The experimental results are shown in Figures 5.3 and 5.4 for some arbitrary
initial conditions. Figure 5.3 shows the variation of V1 and V2 with time. Both these
variables show oscillatory behavior although they were expected to converge to zero
with the controller designed in Section 3.3. The oscillating behavior can be attributed
to the periodic impulsive disturbance acting on the biped, not taken into consideration
in the control design. The amplitudes of oscillation are small, approximately 2 deg
for V1 and 5 deg for V2, respectively, and as such do not pose any problem with the
gait. The mean value of V1 (—0.278 deg, likely due to the large error at the initial
time) is almost zero and this indicates that the controller compensates the effect of
the disturbance. The same cannot be said in regards to control variable V2. The mean
value of V2 is —3.68 deg (see Figure 5.3) and this negative offset can be explained

with the help of Figure 5.4.

Figure 5.4 plots the angles of the inner and outer legs with the vertical and the
vertical dashed lines in the figure indicate the times at which the stance and swing

legs are interchanged. In the first time interval, the outer leg is the stance leg (ost)

47

V1(deg)

V2 (deg)

 

 

 

 

 

 

 

 

4.0
mean value = -0.278
I I I I I I I I I
r I I I I I I I I I
' ‘ /l ' /) ‘ ' ‘ /\ /\
01)" j ’i l i“ l 4“ Z/i\ /f\ar i [k
V. IVIVIV .Vl .Vl
I l I l | I I I I
- I I I I l I | I I
I I I l I I |
'4.0 I I I l I
4.0
ost I 05w [ ost [ ,,,,, [ e= error at the time of interchange
l I I I of stance and swing legs
I | I | I I l | |
0.0 I}, I I {\II I /\l I /\I I
e e I e I I I I
I | I I I
' I l I |
' I l I I I I . .
'4-0 * i i i i i i i i i
‘-. I | I I I | | I l
_ “(I | I | I I I I I
1| | I l I | | I I
mean value I l | | | I I
-8.0 — =-3.68 I l | | | I I
. I . I . | I | | | I |
lsW| lst | lSW | ..... | | | | | |
0
time (s)

Figure 5.3: Plot of control variables V1 and V2 with time.

48

 

 

angle of inner leg with the vertical
— — — angle of outer leg with the vertical

 

 

 

11.25

\
10.0 I

 

angle (deg)
o
o

 

 

 

 

 

time (s)

Figure 5.4: Plots of angles subtended by the inner and outer legs with the vertical.
The vertical dashed lines indicate the times when stance and swing legs are inter—
changed. In the first time interval, the outer leg is the stance leg (ost) and the inner
leg is the swing leg (isw). After the first interchange, the outer leg becomes the swing
leg (osw) and the inner leg becomes the stance leg (ist).

49

and the inner leg is the swing leg (isw). After the first interchange, the outer leg
becomes the swing leg (osw) and the inner leg becomes the stance leg (ist). The
plots of inner and outer leg angles with the vertical are both comprised of segments
with positive and negative slopes, with change in slope occurring approximately at
the time of interchange of stance and swing legs. The segments with negative slopes
represent the stance leg angle with the vertical (dashed line in the first time interval,
solid line in the second time interval) and are therefore plots of 6’ versus time. The
segments with positive slopes represent the swing leg angle with the vertical (solid
line in the first time interval, dashed line in the second time interval) and are plots
of (0 + 1,0) versus time. The legs are interchanged based on stance leg angle, and
specifically when 0 = —fl = —11.25 deg, i.e., when the segments with negative slope

reaches the lower horizontal line.

The control objective of V2 = 0 translates to (0 + 1,11) = —6, i.e., maintaining the
swing leg symmetric with respect to the stance leg about the vertical. This symmetry
is not achieved at the time of interchange of stance and swing legs since the swing
leg angle with the vertical fails to reach the desired value of +1125 deg. This error,
denoted by e in both Figures 5.3 and 5.4, can be reduced by increasing the control
gains but this results in a stiffer biped that is prone to instability due to the impulsive

disturbances.

To maintain a symmetric gait, the swing leg angle with the vertical should also
have a positive slope immediately after interchange of stance and swing legs. This
is not the case, as evident from the undershoots below the lower horizontal line in
Figure 5.4. Each leg, for a brief period of time continues to move in the same direction
in which it was moving in the stance phase. This effectively results in large negative
values of V2 (see Figure 5.3) immediately after interchange of the legs as well as
a negative mean value of V2. It is possible to improve overall gait symmetry by

increasing the control gains. This is, however, avoided since higher gains leads to

50

instability as mentioned earlier.

The inner and outer legs of the biped have different inertia properties and the
impulsive disturbance has distinct effects on the gait depending on which leg is the
stance leg. This is evident from the nature of the oscillations of V2 in Figure 5.3. It
may be useful to note that It in Equation 2.9, for the data provided in Table 4.1, has
a value of 0.412 when the outer leg is the stance leg and a value of 0.832 when the

inner leg is the stance leg.

5.2 Energy Consumption

Energy consumption was not a major goal of this particular project, but it should
be noted that simple changes in the designs of bipeds can have a significant effect
on energy consumption. Here, we have shown that even though the power values
for our robot were determined using a non-optimized leg trajectory, we still have
relatively efficient and stable locomotion. The robot is able to perform a number of
steps every time it is started. It is expected that simply with trajectory optimization,
our robot could achieve much lower energy consumption values while retaining the
same amount of capability. Additional savings could be found through slight changes
in the geometry and inertial parameters of the system.

Although a fair comparison is not possible, Table 5.1 is offered as a reference for
how efficient our robot is compared to the some of the efficient and capable robots
today. Our robot does not have nearly the capabilities of Asimo, but it is stable and
has a robust gait on reasonably flat ground as well as on slightly sloped ground. The
most efficient robots, such as the fully passive Dynamite, are extremely sensitive to
initial conditions and take some operator skill to achieve a stable gait. They are also

unable to climb slopes, as our robot is capable of doing.

We can calculate a measure called the total specific cost of transport which is

51

energy used/ [weight * distance traveled] and abbreviated as Get for our robot and
a number of other robots as a comparison for overall efficiency. Also related to Get
measure is the mechanical energy efficiency, or Cmt. This is calculated similarly, but
only accounts for the mechanical work, or energy the motors consume to walk, rather
than the total energy. This is used to determine the efficiency of the gait itself, in an
attempt to separate computing and sensor energy from actual motor work.

We walked the robot down a hall and measured the distance and time during
travel. Total power was determined by measuring the battery voltage and amperage
using a 0.2 ohm resistor and external data acquisition system at 500 Hz. By sub-
tracting the energy required to run our computers and controllers, we determined
a mechanical energy efficiency. Finally by determining and subtracting the power
required to run the prismatic joints, we could determine a cost of transport using just
the hip joints. This was done because the prismatic joints are high power joints that
are being driven with high gains to precise positions. The design of these joints is
also inefficient due to the use of inefficient high-friction parts such as ACME screws
as opposed to a more expensive ball screw system. The results of five trials were
averaged and recorded in Table 5.1.

The velocity that the robot has during walking can greatly influence the measure-
ment of Cat, especially in robots such as ours where there is a large constant cost due
to our computing and sensor systems. Because of the nature of our two controllers,
the robot had a different preferred speed while making these measurements. There—
fore the efficiencies of the two different controllers Show a large difference in Get, but

similar Cmt values.

5.2.1 Decoupled Controller

The total power required to walk at 0.37 m/s averaged 46 W, corresponding to a

total specific cost of transport of 0.98. The power required to run the motors while

52

walking averaged 20 W, corresponding to a mechanical energy efficiency of 0.33. The
power consumption without the leg extension motors averaged 8.7 W, corresponding
to a cost of transport of 0.14. A comparison of the robot’s energy consumption to a

number of other robots as well as a couple of human activities is shown in Table 5.1.

5.2.2 Coupled Controller

The total power required to walk at 0.45 m/s averaged 55.4 W, corresponding to
a total specific cost of transport of 0.73. The power required to run the motors
while walking averaged 26.6 W, corresponding to a mechanical energy efficiency of
0.35. The power consumption without the leg extension motors averaged 13.4 W,
corresponding to a cost of transport of 0.18. A comparison of the robot’s energy
consumption to a number of other robots as well as a human walking is shown in

Table 5.1.

Table 5.1: Energy Consumption of Selected Bipeds

 

 

 

 

Cet Cmt
Robots1 MSU Synthetic Wheel Biped

Decoupled Controller 0.98 0.33/0.14
Coupled Controller 0.73 0.35/0.18

Asimo 3.2 1.6

TU Delft Denise 5.3 0.08

MIT Spring Flamingo 2.8 0.07

Cornell Collins 3D 0.20 0.055

McGeer Dynamite - 0.04

Humans Walking 0.2 0.05

 

 

 

 

1 Values found in [30] and [1]

53

Chapter 6

Determining Reference

Trajectories

Due to its synthetic wheel geometry, our robot has a wide variety of possible gaits
with which the robot will remain walking. Therefore, we should be able to find swing
leg and torso trajectories that can reduce energy consumption as well as control the

impact at the switching times.

This can be done though creating a trajectory for the swing leg that is a function
of the stance angle, 6, that also has a zero velocity in «p at the end of the swing phase.
Doing this completely eliminates the impact during switching discussed in Section
2.3. In a similar manner, we can create a trajectory for the torso angle as a function
of the stance angle, so that the combination of the two trajectories minimizes the

work to take a step.

Note that for this system, because our model is completely lossless, any changes in
energy of the system results in an increase of kinetic energy of the system if the robot
ends its step in the same position as it started, but with legs in reversed positions.
Therefore, for all of the reference trajectory calculations, it is beneficial to assume

the legs have the same mass and moment of inertia to increase the symmetry in the

‘54

system. This could be achieved in our experimental robot by adding mass to our

inner leg.

6.1 Parameterized Gait Functions

6.1.1 Boundary Conditions

To create functions to relate the swing leg angle and torso angle to the uncontrollable
stance state, we can start by imposing the boundary conditions that guarantee that
we satisfy the zero velocity condition in It at the end of each step. For example if
we choose to use the entire foot during the stance phase (an arbitrary condition that
makes each step the same length), then the stance angle 0 at the start and end of the
step is

42(0) = 0(T) = [3 (6.1)

if we choose to move in the positive X direction as defined in Figure 2.3. T is the
time at the end of the step, 5 is defined as half the foot are as in Table 4.1, and 0 is

as defined in Figure 2.1. By extension, this also imposes the condition

em = —20 (5.2)

where 21) is the angle from the stance to the swing leg, because the swing leg must be
in the correct spot to become the new stance leg. To match our velocities of swing

and stance legs at transition, we can impose the condition

122(0) = 2])(T) = 0 (53)

Note here that ,8 could in reality be any step size by changing the arc length of

the foot or simply using a portion of the foot instead of the full length.

55

6.1.2 Work over One Step

We would also like the trajectory functions to minimize the work required to make a

step. This necessitates the minimization of the work equation

T. .
W: [0 [an+¢¢2]dt. (6.4)

Another way to look a this is through total energy of the system. Because the
system is lossless, any energy that is put in or taken out through the motors is found
as a reduction or increase of the total energy of the system. This implies that if the
robot has the same position at the beginning and end of the step, the only change will
be in the kinetic energy of the system. Therefore we can also calculate the change in
kinetic energy over one step to derive the amount of work done by, or on, the motors
during the step.

For most analyses, we choose to model negative work as perfectly regenerated
energy. At best, our motors can recapture around 80% of the negative work, and
if this energy needs to be stored in the battery, estimations for recapture fall to at
least less than 50% of the total work (coulometric charging efficiency of nickel metal
hydride batteries is typically 66% for average charges. For faster charges this rate
will decrease even lower). Theoretically, we can change the amount of work needed
by the motors by adding passive springs to store energy during one part of the gait
cycle and returning it at another time instead of using motor work to provide the

necessary torques.

6.1.3 Parameterized Gait Functions

We now return to our generalized parameterized gait functions. We originally chose
these gait functions because they can satisfy the boundary conditions defined in the

previous section, and used 5 parameters to minimize the number of parameters needed

56

to satisfy our desired trajectories as well as simplify the trajectory to be tracked.

These parameterized gait functions are once again:

¢=(7T+ad-0)+dSIIl(ZTI'B-0-+f) (6'5)

for the angle between the stance leg and the torso and

2/2 = k6 + asin ($2) + csin (7:361) (6.6)

for the angle between the stance leg and the swing leg. Here, ad is the mean value
of the desired torso angle offset with respect to the vertical, ,8 is half of the foot arc
angle, and the parameters It, a, c, d, and f are available to be determined. However,

due to the boundary condition in 6.2, we have the relationship
a = —z3(2 + k) (67)
and due to the second boundary condition 6.3
c = — (6.8)

We are now ready to determine the remaining parameters, k, d, and f, by solving
the differential equation in I9 and calculating the work required to take one step.
First, we must derive the differential equation in 0 from the equations of motion of
the system. We begin with the equations of motion for the general system (Equation

2.1) and solve the first equation
M119. + M1203 + M1320 + N119 + N120 + N130 + 01 = 0 (6.9)

while substituting our parameterized gait functions for ab and 1,0. This reduces the

57

equation to a differential equation in 0 of the form

N15+N0+C¥=0 (6.10)

We can numerically solve this equation for 6 and 6 in time, and at each time step
determine the corresponding values of 45, «p and their derivatives. These solutions
can then be substituted into the two remaining equations of the general equations of

motion,

M210 + A1220 + lI/[23'Ill + N219 + N220 + N230 + G2

7'1

M315 + Iv132<i5+ M3312} + N310 + N32<I> + N331!) + 03 = 72 (6.11)

to solve for the torques T1 and 72 in time. With the numerical values of the torques
calculated, we can solve for the work done over a single step for given initial conditions

and a desired torso setpoint ad.

6.2 Nominal Trajectory

As a baseline measurement for energy, we can determine the work over one step using
the trajectories previously described in Section 2.4. Substituting the nominal trajec-
tories of Equations 2.15 and 2.16 into the parameterized gait functions of Equations
6.5 and 6.6 would require parameter values of k = —2 and all the other parameters
equal to zero, as shown earlier. The equations of the constrained motion then simplify
to

¢=7r+ozd—6 (6.12)

and

a = —26 (6.13)

58

Therefore, the velocities of the states are
<15 = —6 (6.14)

and

II = —29 (6-15)

both of which can never be zero if 0 74 0

For a torso angle of 1.5 deg, and an initial velocity of zero, we can determine the
work for the step as well as time traces for the states 0, qb and 11), as shown in Figure

6.1 (Note for ease of viewing, (15 is plotted as qt -— 7r).

Total work over the step, assuming a 100% regeneration rate, is .258 J with a
step time of 2.47 3. Note that at the end of the step 1b is not zero, implying that
there must be an impulse if the legs were switched at this time. This impulse is not
included in the analysis of the trajectory. The positive work value also implies that
there must be a positive change in the kinetic energy at the end of the step because
the change in potential energy must be zero. The torque curves in Figure 6.2 show
that the torque needs to be discontinuous at the end of each step, jumping from one
value to the other as it switches legs.

This also illustrates another issue with this method of optimization. Because all
of the parameters are used to control the trajectories with the boundary conditions
we have selected, we have no ability to separately control the velocity in the uncon-
trollable state, 0. This means that the end velocity can and will be different from
the initial velocity of the system without carefully choosing our parameterized gait
functions. In this case, 9 starts initially at zero and progresses to 0.319 rad / sec. This
is intuitively so because the torso is leaning forward and therefore, while using the
nominal trajectory, it is constantly accelerating forward. As we increase the initial

velocity of the robot, the difference in velocity at the beginning and end of the step

59

 

0.6

Radians

 

 

 

 

 

 

 

 

 

Figure 6.1: Simulation of 0, (15 — 7r, and ’t/J over one step starting from rest and torso

angle at 1.5 deg.

I

1.0
Time (s)

1.5

2.0

 

2.0

1.04

Torque (Nm)
0
O

 

 

 

 

 

 

 

0.0 0.5

T

1.0

Time (s)

1.5

2.0

2.5

2.5

Figure 6.2: The nominal trajectory torques T1 and r2, corresponding to the torques
on generalized coordinates <15 and 1/) respectively.

60

grows smaller, as the system has less time between the beginning and end of the step

but is still accelerating at the same rate. This is illustrated in Figure 6.3.

 

l l l I

Work (J/step)

 

 

 

0.2 I w .
0.0 0.5 . 1.0 1.5 2.0

 

0.35
030 - . . . . .
0.25 . FInal Velocny lnItIal Velocuy
0.20 -
0.15 ~
0.10 I
0.05 .
0.00 . a .
0.0 0.5 1.0 1.5 2.0

Initial Velocity (rad/sec)

 

Radians/sec

 

 

 

 

Figure 6.3: The work and 0f — do plotted against initial velocity for a torso angle of
1.5 deg using the nominal trajectory.

6.3 Optimized Trajectories

We can choose better parameterized gait functions that will allow us to further un-
derstand the relationship between the uncontrolled and controllable states. First, we
will derive parameterized gait functions that allow zero total work over one step and
prove that it will always be zero work regardless of the initial velocity of the system
or the overall step size. We then can change velocity by changing the parameters
of the parameterized gait functions, allowing us to speed up or slow down the robot

while eliminating collision at the end of the step.

61

6.3.1 Constraints for Steps with Zero Total Work

For all of the previous sections we assumed the torso would lean forward, and therefore
the robot was constantly accelerating. This is shown by the increase in kinetic energy
at the end of all of the valid steps in the previous section. This system, when modeled
without friction as we have, is lossless, and therefore any positive work that enters
the system is manifested as an increase in kinetic energy at the end of the step.

We can also show that if we allow our controllable states to be constrained to
functions in 6 of a certain type, we can guarantee that uncontrollable states will do
what we would like them to, primarily progress from —6 to 6, and have 6(0) = 6(T),
where T is the time at the end of the step; and thereby ensure the same position and
velocity of the robot at the beginning and end of the step, but with legs exchanged.
This results in net zero total work over the step, because it has the same kinetic and
potential energy at the beginning and end of the step. This simulation requires the
ability to capture 100% of the negative work done by the system which, while not
really practical, shows that the system could ideally require no work to walk, but still
satisfy our desired boundary conditions of an arbitrary, but equal, step length, and

zero velocity in 16 at the end of the step.

6.3.2 Equations For Gait Requiring Zero Total Work

We would like to develop a constraint for the actuated degrees of freedom of our
system that will guarantee that the underactuated degree of freedom will be in the
correct place and velocity at the end of the step. Because any valid step will allow the
stance foot to rotate from —3 to )8, by definition the kinematics of the underactuated
state will satisfy the boundary condition of Equation 6.1. We need to ensure that
6(0) = 6(T). To prove this, it is sufficient to require 6 = .7: (6) where J'- (6) is an even

function in 6. Because the derivative of an even function is an odd function, it suffices

62

to say that 6(6) is an odd function in 6.

We begin with the first equation from the equations of motion, Equation 6.9. We

substitute into this equation our new gait functions, which have the following forms,

A

65(6) = (a6 + b) + o (6.16)

A

1M6) = (c6 + d) + 1/2 (6.17)

where (6 and 26 are any odd functions in 6. By solving the equation for 6, we find

6': 32% (6.18)
where
N(6) = — 65(10 + m0(d0 — R)(d0 — R + R cos [6 + (DD) (Term A)
— tllUsw + msw(dsw — R)(dsw — R + R cos [6 + 10]» (Term B)
+ m3t(d3t — R)(g + R 62) sin [6] (Term C)
+ m0(d0 — R) (g + R(6+<13)2) sin [6 + 6] (Term D)
+ m3w(dsw — R)(g + R(6 + 6)?) sin [0 + 6]) (Term E)
10(6) =10 'I' Ist + 13112 (Term F)
+ m3t((d3t — R)2 + R2 — 2R(—dst + R) cos [0]) (Term F)
+ m0((d0 — R)2 + R2 — 2R(—d0 + R) cos [0 + 6]) (Term G)
+ m3w((dsw — R)2 + R2 — 2R(—dsw + R) cos [9 + 6]) (Term H)

63

We will show, term by term, that N(6) is odd and 10(6) is an even function, and

therefore 6 is an odd function.

6.3.3 Term by Term Evaluation of Even/Oddness

cos (6 + qb)

cos (6 + qb) = cos (6) cos ((13) — sin (6) sin ((25)

= cos 6[cos (a6 + 5) COS (<15) — 5111019 'I' b) sin (I’ll

— sin 6[sin (a6 + b) cos (gb) — cos (a6 + b) sin (6)]

and

sin (a6 + b) = odd, cos (a6 + b) = even (6.19)

if b = nrr, where n is an integer. This means that cos (6 + (b) is an even function if

b = n7r. This is the same as for 21) and therefore cos (6 + 1,6) is also an even function.

sin (6 + (1’))

sin (6 + (15) = cos (6) sin ((1)) — sin (6) cos ((25)

= cos 6[sin (a6 + b) cos (a5) + cos (a0 + b) sin (46)]

+ sin 6[cos (a6 + b) cos (6)) — sin (a6 + b) sin (6)]

Once again we let b = nvr such that the relationships in Equation 6.19 hold. This
means that sin (6 + (15) is an odd function if b = mr, and once again this also applies
to sin (6 + 1/1), meaning it is also odd.

Term A

64

Term A can be rewritten as

—(/5(C1 + 02(03 + cos (6 + 45))

and is therefore the product of an odd function (b, and an even function composed of
constants, and the function cos (6 + 65), resulting in an odd term.
Term B

Term B can be rewritten as

—2/1(C4 + C5(Cg + cos (6 + 16))

is odd because it is the product of an odd function 1,6, times a number of constants,
times cos (6 + 1b). 0

Term C

Term C can be rewritten as

62 C7sin(6)

which is an even term 62, times an odd term sin(6), resulting in an odd term.
Term D

Term D can be rewritten as
08(09 + (9 + is?) sin(6) + <I>)

and is an even constant times an even term (6 + (b)2, times an odd term sin(6 + ¢),
resulting in an odd term.

Term E

Term E can be rewritten as

016(011 + (9 + 6)?) sin(0 + II»)

65

and is therefore an even constant times an even term (6+ 11.02, times an odd term
sin(6 + ’16), resulting in an odd term.

Because Terms A, B, C, D and E are all odd, the numerator, N (6), must be odd.
Term F
Term F is a constant and is therefore even.
Term G
Term G can be rewritten as

C12(013 + 005(9))

and is therefore an even function.
Term H

Term H can be rewritten as
014(C15 + cos(6 + phi))

and is therefore an even function.
Term I

Term I can be rewritten as
016(017 + cos(6 + 1932))

and is therefore an even function.

Because Terms F, C, H and I are all even, the denominator, 10(6), must be even.

6.3.4 Proof of Zero Total Work Gait

We have proven that 6 is odd in 6, meaning 6 is even. Therefore, for any given
symmetric step, the velocity of 6 will be the same at the beginning and the end of

the step. This is true for whatever initial velocity we choose because the function 6 is

66

even in 6, and not in time. This requires the step to be a zero total work step because
the other constraints put on the system require the robot have the same potential

and kinetic energies at the beginning and end of the step, i.e.

6(0) = — 6(7) 6(0) = 6(T)
0(0) = a(T) 61(0) = d(T)
6(0) = — 6m 6(0) = 6m

Also note that the restriction on b, namely b = n7r, in Equations 6.16 and 6.17
have physical limitations. The function 65 must be limited to n = 1,3,5, 7... if the
torso is to be standing upright, and for 1/J, n must be limited to 0, 2,4,6... if the
legs are going to have a small step size. It is easiest to simply set these to 1 and 0
respectively.

When using the original parameterized gait functions from Equations 6.5 and 6.6,
our zero work constraints would require the parameters k = 0 and f = 0. It is also
necessary to make ad = 0 to fit the correct form. Substituting in these parameters

creates the new constraint functions

(I) = (7r — 6) + dsin (1;) (6.20)
for the angle between the stance leg and the torso and

R6

6 = —26 sin (7) (6.21)

for the angle between the stance leg and the swing leg.

Now d is the only parameter that can be varied and will decide if a step is com-
pleted in a reasonable time and within our motor capabilities. Note here that even

though the total work is zero, the motors have to provide the correct amount of work

67

at the beginning half of the step and extract exactly the same amount of work dur-
ing the second half. However, there is no guarantee that we will remain within the
instantaneous torque constraints of our physical motors.

Figure 6.4 shows the plots for 6, (,b — 7r, and 16 for a step starting with a slow initial
velocity of 0.1 rad/ sec and a d value of —0.06 rad. The traces look dramatically

different from the nominal trajectory (compare to Figure 6.1).

 

0.6

Radians

 

 

 

 

 

 

 

 

0.0 ' 0.5 1.0 1.5 2.0
Time (s)

Figure 6.4: Optimized trajectory simulation of 6, go — 7r, and 1,6 over one step starting
from 0.1 rad/ sec and torso amplitude of —0.06 rad.

The torque traces in Figure 6.5 are also odd over the course of the step. Because
of the odd nature of 6, d) and 1b the resulting work is zero. Note with this trajectory,
we continue to have a jump in the torques at the end of the step to meet the torque
requirements of the next step.

The velocity at the end of the step always matches the initial velocity independent
of our choice of initial velocity. To illustrate this, we have plotted 6 vs. 6 for different

initial velocities in Figure 6.6. The torso amplitude was set at —0.06 rad for all of

these calculations.

68

 

Torque (Nm)

 

 

 

 

 

 

 

0.0 0.5 1.0 1.5 2.0

Time (s)

Figure 6.5: Zero work trajectory torques 1'1 and r2 corresponding to general coordi-
nates 65 and 1,6 respectively. Calculations are over one step starting from 0.1 rad/ sec
and torso amplitude of -0.06 rad.

 

0.7

0.6 -

 

 

 

 

 

 

 

 

-0.2 -0.1 0.0 0.1 0.2

Figure 6.6: 6 is shown to be the same at the beginning and end of the step, regardless
of the initial velocity, 60. The torso amplitude is constant at —0.06 rad.

69

 

LB

 

 

 

 

 

 

 

 

 

 

 

 

1.6 " = -.01 /\
_ ........... d = -.015 l
1.4 d = -.02 _/ \\
1.2 - ———— = "03 -’ I
........... d = '.% I \
1.0 « ~— ----- d = -.1 ./ 1
<1: 0.8 4 I! \-
- \
0.6 - 'l '\
./ '\
Q46 ./ .u ........... - '\
‘/ ..0.. .0.... .\
0.2 1 /-.( ................... '..\.
(.332..-——" TTT“:"}3“
-o.2 T . ' r
-03 -o.2 -o.1 0.0 0.1 0.2
6

Figure 6.7: 6 is shown to be the same at the beginning and end of the step, regardless
of the amplitude of the torso angle as long as the robot is able to complete a step.
For small, or positive, torso amplitudes, the robot is not able to reach the length of
a full step and simply wobbles on the stance leg.

The amplitude of the torso has a large effect on the velocity of the system during
the step. This corresponds to injecting energy into the system during the first half
of the step and then taking it back out during the second half of the step. While
the initial velocity stays constant at 0.05 rad/ sec, the parameter d is varied. The
resulting phase portrait is shown in Figure 6.7. Note that for small values of d the
robot is not able to complete a full step and will wobble on the stance leg forever.
This is shown as a loop in the phase portrait that grows in size until a suitable d

allows 6 to reach the boundary of the step.

The conditions for the robot to wobble on its stance leg rather than taking a step
is a complicated mixture of inertial properties and initial velocity. The conditions can
be calculated, but we have chosen to simply exclude simulations that do not make a

step within 5 sec, with the rationale that if a step is to be useful, it must be completed

70

in a reasonable time. This is an arbitrary time limitation, but enables us to speed
the calculation of our simulation. We should also note that because our system is
periodic, if the velocity in 6 changes direction within a step before it reaches the end,

it should be considered a failed step.

6.3.5 Non-Zero Total Work Gait

All gaits that do not follow the form of the zero work gait discussed in Section 6.3.1
end up with either a higher or lower kinetic energy depending on the direction of lean
of the torso and the parameters used in the parameterized gait functions. We can
now map parameters from the parameterized gait functions Equations 6.5 and 6.6 to
best satisfy the new velocities we desire.

The parameter space is large and needs to be simplified for a global parameter
search. We selected boundaries for our parameters primarily based on physical limita-
tions of our robot, in addition to preliminary low resolution searches of the parameter
space. For example, by restricting ()6 to be bounded by i175, the parameter k is con-
strained to the region —55 < k < 55. Similarly, a bound on parameter d can be
chosen as it corresponds directly to the amplitude of motion around the torso set-
point. Here we chose to bound (1 by —0.2 < d < 0.2, corresponding to approximately
$11.5 deg of motion around the desired setpoint, ad.

We can also declare that any steps that require instantaneous torques above what
our motors can individually provide (150W) are not useful for this particular opti-
mization and can be disregarded. Once again we limit step time to 5 seconds for
reasonable steps.

For an illustration of the method, we also calculated solutions for a given torso
angle that was similar to the angles used for experiments with the coupled controller
(Section 3.3) and nominal gait (Section 2.4). The best angle was empirically deter-

mined to be around 1.5 deg for that setup. We begin the search for a better trajectory

71

by covering the parameter space determined in Section 6.3.5. Keeping the average
torso angle at ad = 1.5 deg and restricting k to the region —55 < k < 55, d to the
region —0.2 < d < 0.2 and f = 0 results in solutions in the shape shown in Figure

6.8.

0.2 —
0.15—
o.1 —
0.05—
0 _
0.05— g
0.1 — E
0.15—

0.2 I I
60 4O 20 0 20 40 60
Parameter k

 

 

 

 

 

Parameter d

 

 

Figure 6.8: Parameter space in k and d. A rough estimate of the valid space of the
parameters.

Figure 6.8 is a two-dimensional plot that gives a rough estimate of the valid region
of the parameter space. We can use this plot to determine what area of the parameter
space to focus our search. Looking at these values it is apparent that the final velocity
does not change much with the parameter k, and therefore we can choose a fixed value
for k , such as k = 10, that allows a large variety of possible values of d. Plots of the
change in velocity from an initial velocity of zero against different values for parameter
d are shown in Figure 6.9. As we change the torso angle setpoint, ad in degrees, we
can see that the robot accelerates faster over the course of one step. This allows us
to choose a final velocity for the robot after one step if the robot starts from rest.
Similar maps can allow different initial conditions, such as the map slowing down the

robot in Figure 6.10.

72

 

1.6
1.4 -
1.2 -
1.0 -
0.8 -
0.6 -
0.4-
0.2 «
0.0 -

Initial velocity = 0 rad/sec 06 = 5

 

9 final — 6initial

0rd = .1 degrees

 

 

 

 

I T l I I

-o.'04 ' 0.60 0.04 0.08
Parameter d

-o.12 ' -o.'08

Figure 6.9: Starting initially from rest, relationships between the parameter d and
the final velocity at different torso angle setpoints, ad.

 

 

 

 

 

 

0.2
Initial velocity = 1 rad/sec
0.0- =— \
do = -.1
E
E '0-2‘ ad =-.5
'Q
I
E -o.4l 0‘0“"
as
Old = -1.5
-0.6,
ad=-2
-O 8 ad = -5 degrees
-0.06 -0.04 -0.02 0.00 0.02 0.04 0.06 0.08 0.10

Parameter d

Figure 6.10: Starting initially from 1 rad/ sec, relationships between the parameter d
and the velocity difference at different torso angle setpoints, ad.

6.3.6 Discussion of Optimization

Our optimization method attempts to address the issue of velocity mismatch at the
end of each step, but it has not been implemented. We present a method to constrain
the controllable degrees of freedom to functions of the uncontrolled degree of freedom
and also determine a gait for which the total work is zero over a step and is collision-
less at the switching time. We also describe a method of controlling the torso to

change the velocity of the robot over one step.

The optimization method has a few issues that we choose not to address at this
time. This is a simple simulation, and so does not include the controller dynamics. We
assume that for each step the controller is able to track perfectly, which is impossible
in practice, but especially hard for our robot to approximate because of inability to
use large control gains due to nonlinearities, such as backlash, in the physical system.

Including the controller dynamics would be an obvious extension of this analysis.

Another thing we do not account for is the inability of our system to recapture
100% of the negative work done by the system. Using a weighting method for neg—
ative work would be a good way to attempt to optimize the trajectories further.
Adding friction and motor dynamics to our model would also be a way of getting the

simulation to closer match real life performance of our system.

Another method for doing this would be to simply relax our boundary conditions
and let the simulation run until the swing leg naturally matches velocity with the
stance leg. If there is a zero torque solution that results in an end state that matches
the initial state, as in the method of McGeer [26], we could track the resulting trajec—
tory, knowing that it is a minimum work trajectory for the passive system. Another
good aspect of this type of controller would be that the desired torque would be
necessarily zero at the beginning and end of the step, meaning there would be no

jump in the required torque of the motor. One major problem with this is that some

74

initial conditions will not have suitable zero torque trajectories, and does not provide
a method to rectify this in an efficient way. For example, for a small foot size, at
higher velocities the required step length might be beyond the physical foot size, re-
quiring an undesirable gap between the feet during transfer. In cases like these, our

constrained method would be superior.

75

Chapter 7

Conclusion and Future Research

We have presented a new simple paired leg, three-link biped which we have called
the MSU Active Synthetic Wheel Biped. This biped has legs that have solid arc feet
that have the same radius as the leg length, allowing it to place one foot in front of
the other at the right times with the correct velocity to simulate the rolling motion
of a wheel. We have developed a general dynamic model and imposed constraints to
create gaits. Using these gaits, we designed and implemented two different controllers
successfully. The first was a decoupled controller, where the swing leg and stance leg
torques were determined independently of each other without considering the coupled
dynamics of the system, and the second was a full dynamics controller utilizing feed-
back linearization. The experimental results of both of the controllers were presented
and finally, we described some ways of improving upon the techniques, primarily in

the methods to find better reference trajectories for the machine.

The robot takes advantage of its synthetic wheel configuration through the use of
its simple stability criterion, namely, if there is a foot below the hip at all times it will
remain stable. This guarantees that as long as we have motor bandwidth sufficient to
satisfy this condition, we can remain walking. We have not yet taken advantage of the

fact that a robot of this type could roll without impact during the switching between

76

legs by matching the swing leg velocity with the stance leg velocity in experiments.
Implementing different reference trajectories should allow for a decrease in mechanical
work per step through the reduction of impact at the end of each stride and through
better control of velocity of the robot. We plan on implementing new reference
trajectories in an effort to show this reduction experimentally, as well as attempt to
produce a more pleasing, smooth gait.

Our method works for our synthetic wheel design, and we feel confident that we
can expand the method to robots with more degrees of freedom and different body
shapes. Expanding the method to include robots that have feet with a small radius
and more degrees of freedom will also require the inclusion of collision dynamics, or at
least a different method to avoid collision at the foot fall in at least two dimensions.
Inclusion of all of these topics should provide a much better understanding of how to

produce stable, controllable, and efficient bipeds for general use.

77

Appendix A

Impulse Constant 16

While calculating the change in the velocity of the biped system due to impulse at
foot switching (Section 2.3), we encounter a large constant which we defined as It.
This constant is comprised of the kinematic and dynamic parameters of the biped

and is defined as follows.

n =[sz + m,w(d§w — 26sz + 2R2) + msw(dsw - R)((—dst + R) cos (6)
+ 2Rcos (6 + 10+ (6.. — R) cos (26 + 16) - Rsin (9 + 16)]
/[.1,, + sz + dgwmsw — 263.061sz + (mst + 26131161?2 + mst(dst — 10260809)
+ Rcos (6)(2m3t(dst — R) + msw(dsw - RX? COS (w) - sin (16»)
+ sin (6)(m5tR(—dst + R) + msw(dsw - Ell-5160806) + 2 sin (16))

+ 2(—d3t + R) sin (6 + 16)))]

78

[ll

[2]

[3]

I4]

[5]

[6]

l7]

[9]

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