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dissertation entitled

ADAPTIVE CONTROL SUBJECT TO INPUT CONSTRAINTS:

MANEUVERING WHEELED PLATFORMS USING A
HOLONOMIC MOBILE ROBOT

presented by

Nandagopal Methil-Sudhakaran

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ADAPTIVE CONTROL SUBJECT TO INPUT CONSTRAINTS:
MANEUVERIN G WHEELED PLATFORMS USING A
HOLONOMIC MOBILE ROBOT

By

Nandagopal Methil-Sudhakaran

A DISSERTATION

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

DOCTOR OF PHILOSOPHY
Mechanical Engineering

2009

 

 

ABSTRACT

ADAPTIVE CONTROL SUBJECT TO INPUT CONSTRAINTS:
MANEUVERING WHEELED PLATFORMS USING A
HOLONOMIC MOBILE ROBOT

By
N andagopal Methil-Sudhakaran

The problem of adaptive control in the presence of input constraints is addressed. The
new control laws are presented in the context of a holonomic mobile robot pushing and
steering a wheelchair using a single manipulator. Adaptive control is employed to estimate,
in real time, the inertia characteristics of the wheelchair and its occupant, and friction be-
tween the wheelchair and the surface that it traverses. The input constraints considered are
actuator saturation and actuator dynamics. Saturation of control input limits the maximum
input that can be provided to the system and input actuator dynamics dictate the speed
with which the control input can be realized. If left unaddressed, these constraints can
result in estimation windup, poor tracking performance and/or instability. Additionally,
in this case the problem is further exacerbated due to the fact that only a single manip-
ulator is to be used for both pushing and steering due to weight and cost considerations.
The control designs presented here complement the mechanical design of the manipulator
such that a holonomic mobile robot employing the manipulator can maneuver a wheelchair
asymptotically along a reference trajectory. The reference trajectories can be generated au-
tonomously or provided by a human operator. Apart from the significant improvements in
the performance of adaptive control subject to input constraints, this work has the potential
to provide greater mobility to residents of Long Term Care facilities, which are currently
experiencing an acute shortage of nurses and an increasing number of inmates. In general it
can be used for maneuvering wheeled platforms. Experimental results employing an RP-6

mobile robot platform and a wheelchair are presented to validate the theoretical results.

To my parents and my wife.

iii

ACKNOWLEDGMENTS

I express my deepest gratitude to my major advisor and mentor, Dr. Ranjan Mukherjee,
whose invaluable guidance and support made this work possible. 1 thank the members of
my Ph.D. committee: Dr. Hassan Khalil, Dr. Alan Haddow and Dr. Alejandro Diaz, for
their suggestions and inputs.

I thank all my friends and family who have been supportive in every way.

Finally, I thank my father who has been a great source of inspiration, mother, for her

relentless hardwork and prayers, and my wife, for all her love and patience.

N andagopal Methil-Sudhakaran

iv

 

TABLE OF CONTENTS

LIST OF FIGURES .................................. vii
1 Introduction .................................... 1
1.1 Background .................................. l

1.2 Preliminary Challenge 1: Design of a Robotic Arm ............. 3

1.3 Preliminary Challenge 11: Adaptive Estimation ................ 4

1.4 Main Challenges ............................... 5

1.5 Using a Holonomic Platform ......................... 6

1.6 Progression of the Thesis ........................... 7

2 Preliminary Work ................................. 8
2.1 Conceptual Design of the Single Arm .................... 8
2.2 Dynamics of Wheelchair ........................... 10

2.3 Problem Statement .............................. 13
2.3.1 Main Result .............................. 14

2.3.2 Simulations .............................. 17

2.4 Forward and Backward Motion of Wheelchair ................ 31
2.4.1 Controller Modification ....................... 31

2.4.2 Simulations .............................. 34

3 Saturation of 'Ihrning Moment .......................... 39
3.1 Introduction .................................. 39

3.2 The Control Problem ............................. 41
3.2.1 Main Result .............................. 41

3.2.2 Simulations .............................. 44

3.3 Resetting of Estimates to Prevent Mndup .................. 49

3.4 Addressing Loss of Turning Moment ..................... 55
3.4.1 Change of Reference Trajectory ................... 55

3.4.2 Simulations .............................. 6O

4 Effect of Dynamics in the Prismatic Joint .................... 66
4.1 Background ................................. 66
4.2 Motivation ................................... 67
4.3 Preliminary Result .............................. 69
4.4 Addressing Sign Change of Control Inputs ................. 73
4.4.1 Change of Reference Trajectory ................... 73

4.4.2 Main Result ............................. 76

4.5 An Alternate Approach to Dealing with Sign Change in Control ...... 81

4.5.1 Closed-loop Control of the Prismatic Joint .............. 81

4.5.2 Main Result .............................. 83

4.6 Resetting of Estimates to Prevent Wmdup .................. 89
4.7 Dealing with Loss of Tuming Moment when F = 0 ............. 91

5 Generation of Additional Turning Moment by Varying the Angle of Push . . 101
5.1 Motivation ................................... 101
5.2 Preliminary result ............................... 102

5.3 Generation of additional turning moment ................... 106
5.4 Effect of Robot Dynamics on the Angle of Push ............... 108

5.5 Addressing Saturation in Angle of Push ................... 117
5.6 Resetting of Estimates to Prevent Wmdup .................. 126

5.7 Addressing Zero Force condition ....................... 133

6 Addressing the Limitations on the Applied Force ................ 138
6.1 Introduction .................................. 138
6.2 Saturation of Applied Force .......................... 138

6.3 Addressing dynamics of Applied Force ................... 148

6.4 Resetting of Estimates to Prevent Windup .................. 156

6.5 Addressing the situation when P} = O .................... 161

7 Experimental Verification ............................ 166
7.1 Focus of the Experimental Work ....................... 166
7.2 Arm Description ............................... 167

7.3 Experimental Results: Constraints of the Prismatic Joint .......... 168
7.4 Experimental Result: Prismatic Joint and Elbow Joint ............ 169

8 Conclusion and Future Work ........................... 173
BIBLIOGRAPHY ................................... 178

vi

2.1
2.2

2.3
2.4
2.5
2.6
2.7
2.8
2.9
2.10
2.11
2.12
2.13
2.14
2.15

2.16
2.17

LIST OF FIGURES

Conceptual design of single arm for pushing and steering wheelchairs . . . 8
Pushing and steering achieved through elbow positioning in forearm pris-
matic joint ................................... 9
Definition of coordinate system for the wheelchair and forces of interaction
between robot and the wheelchair during pushing and steering ....... 11
Simulation 2.1: Plot of error variables ex and co, and control inputs, F and
d, with controller parameters K1 = 15, K2 = 15, 21 = 50, 22 = 50 ..... 19
Simulation 2.1: Plot of estimates of unknown parameters with controller
parameters K1 =15, K2 =15,21 = 50, L2 = 50 ............... 20
Simulation 2.1: Plot of errors with controller parameters K] = 10, K2 =
10, 2.1 =10,2/2 =10 ............................. 21
Simulation 2.1: Plot of estimates with controller parameters K1 = 10, K2 =
10, 21 = 10, A; = 10 ............................. 22
Simulation 2.1: Plot of errors with controller parameters K] = 50, K2 =
50, 21 = 50, M = 50 ............................. 23
Simulation 2.1: Plot of estimates with controller parameters K1 = 50, K2 =
50, 2.1 = 50, L2 = 50 ............................. 24
Simulation 2.2: Plot of desired trajectory ................... 25

Simulation 2.2: Plot of error variables ex and eg, and control inputs F and d 26

Simulation 2.2: Plot of parameter estimates ................. 27
Simulation 2.2: Overlapping plot of desired and actual trajectories ..... 28‘
Simulation 2.3: Plot of parameter estimates ................. 29
Simulation 2.3: Plot of error variables ex and eg ............... 30
Simulation 2.4: Plot of x with time ...................... 34
Simulation 2.4: Plot of parameter estimates ................. 35

vii

2.18
2.19
2.20
2.21

3.1
3.2

3.3

3.4

3.5

3.6

3.7

3.8

3.9

3.10
3.11

3.12

4.1
4.2

Simulation 2.4: Plot of error variable ex and force F ............. 35
Simulation 2.5: Plot of x with time ...................... 37
Simulation 2.5: Plot of parameter estimates ................. 37
Simulation 2.5: Plot of error variable ex and force F ............. 38
Plot of (is with respect to d .......................... 40

Simulation 3.1: Plot of error variables, ex and eg and control variables F
and d as defined by Eqs.(2.12) and (2.13) in Theorem 2.1 .......... 46
Simulation 3.1: Plot of error variables ex and eg and control variables F
and d with saturation in d with dm = 3.0 m ................. 47

Simulation 3.1: Plot of parameter estimates with saturation in d with dm =

Simulation 3.2: Plot of error variables, ex and eg and control variables F
and d3 without modification in estimates ................... 52

Simulation 3.2: Plot of error variables, ex and eg and control variables F

and (IS with modification in estimates ..................... 53
Simulation 3.2: Plot of parameter estimates without modification ...... 54
Simulation 3.2: Plot of modification in parameter estimates ........ 54

Simulation 3.3: Plot of error variables ex and egi, and control inputs, F and
d with E = 0.1 ................................. 61
Simulation 3.3: Plot of estimates of unknown parameters with e = 0.1 . . . 62

Simulation 3.4: Plot of error variables ex and em, and control inputs, F and

d; with e = 1.0 ................................ 64
Simulation 3.4: Plot of estimates of unknown parameters with 8 = 1.0 . . . 65
Dynamics of d, with mechanical saturation ................. 66
Dynamics of d,- with software saturation ................... 67

viii

4.3

4.4
4.5
4.6
4.7
4.8
4.9

4.10
4.11

4.12

4.13
4.14
4.15

4.16

5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8

Simulation 4.1: Plot of error variables ex and eg, and control inputs, F and

ds, and d; ................................... 72
Simulation 4.1: Plot of parameter estimates ................. 73
Simulation 4.2: Plots of ex, egi,,F, d3, dg, Bx and Bo ............ 79
Simulation 4.2: Plot of parameter estimates ................. 8O
Dynamics of the Prismatic Joint ....................... 82
Standard Position Control ........................... 82

Simulation 4.3: Plot of error variables ex and eg, and control inputs, F, (13
and d; ..................................... 88
Simulation 4.3: Plot of parameter estimates ................. 89
Simulation 4.4: Plot of error variables, ex, e2 and eg and control variables
F and d; without modification in estimates .................. 92

Simulation 4.4: Plot of error variables, ex, e2 and eg and control variables

F and d; with modification in estimates ................... 93
Simulation 4.4: Plot of parameter estimates without modification ...... 94
Simulation 4.4: Plot of modification in parameter estimates ........ 94

Simulation 4.5: Plot of error variables ex and em, and control inputs, F, d;

and d3 ..................................... 98
Simulation 4.5: Plot of parameter estimates ................. 99
Varying the angle of push, B, to achieve greater turning moment ....... 102
Simulation 5.1: Plot of error variables ex and eg ............... 104
Simulation 5.1: Plot of parameter estimates ................. 105
Simulation 5.1: Plot of F, d3, d; and B ................... 105
Simulation 5.2: Plot of error variables ex and eg .............. 109
Simulation 5.2: Plot of control inputs, F, ds, d; and B ............ 110
Standard closed loop position control of angle B .............. 110
Simulation 5.3: Plot of error variables ex and e9 ............... 116

ix

5.9 Simulation 5.3:

5.10 Simulation 5.3:

5.11 Plot ofB vs B5 .

5.12 Simulation 5.4:
5.13 Simulation 5.4:

5.14 Simulation 5.5:

Plot of parameter estimates ................. 116
Plot of control inputs, F, d;, and B’ ............. 117
................................ l 18

Plot of error variables ex and eg and parameter estimates . 127
Plots of F, d5, d;, B and B’ ................. 128

Plot of error variables, ex and co and estimates without

modification in estimates ........................... 130

5.15 Simulation 5.5:

Plot of error variables, ex and eg and estimates with modi-

fication in estimates .............................. 131

5.16 Simulation 5.5: Plot of control variables F and d; and B; without modification 131

5.17 Simulation 5.5: Plot of control variables F and d; and B; with modification

5.18 Simulation 5.6:
5.19 Simulation 5.6:
5.20 Simulation 5.6:

6.1 Simulation 6.1:
6.2 Simulation 6.1:
6.3 Simulation 6.1:
6.4 Simulation 6.2:
6.5 Simulation 6.2:
6.6 Simulation 6.2:

6.7 Simulation 6.3:

in parameter estimates ............................ 132
Plot of error variables ex and eg ............... 136
Plot of parameter estimates ................. 136
Plot of control inputs, F, B’ and actuator dynamics (1’, . . . 137
Plot of error variables ex and eg ............... 146
Plot of parameter estimates ................. 147
Plot of control inputs, dg, and B; .............. 147
Plot of error variables ex, e9, F, and F3 ........... 156
Plot of parameter estimates ................. 157
Plot of d, d;, B and B; .................... 158
Plots of ex, ea, F, Fg’, d; and B; ................ 160
Plot of estimates ....................... 161

6.8 Simulation 6.3:
6.9 Simulation 6.4:

and B; .....

6.10 Simulation 6.4:

Plot of error variables ex and eg and control inputs Fs’, d;
................................ 164

Plot of parameter estimates ................. 165

7.1
7.2
7.3
7.4
7.5
7.6
7.7

8.1
8.2
8.3
8.4

Experimental setup .............................. 167

Trajectories .................................. 169
Estimates .......................... - ......... 170
Control Inputs ................................. 170
Trajectories .................................. 171
Estimates ................................... 171
Control Inputs ................................. 172
Extension to other platforms .......................... 174
Path taken in the absence of actuator constraints. .............. 176
Path taken in the presence of actuator constraints. .............. 176
Path taken when using the modifications presented in the thesis. ...... 176

CHAPTER 1

Introduction

1.1 Background

Modern medicine and technology have increased the average human life expectancy. Con-
sequently it is also the responsibility of technology, to aid aging humans lead a quality life
as they increase in number. Facts indicate that over the next 10 years, the number of elderly
aged 85 and over will grow by 38 percent [21] in the USA. The number of seniors requiring
healthcare support will also double. There are already 400,000 unfilled nursing positions
and this will more than double by the end of the decade. With few people to look after the
elderly, the quality of life of the elderly will deteriorate. Clearly technological solutions
are required to alleviate this problem.

The quality of life problems is most relevant in institutions like Long Term Care (LTC)
facilities. Almost 50 percent of the elderly population in LTC facilities are wheelchair
bound due to cognitive and/or physical impairments [19]. These residents require complete
assistance with their Activities for Daily Living (ADL’s), like eating bathing etc.. and LTC
facilities are finding it increasingly difficult to provide quality care. A significant portion
of nurse time is used to provide assistance in ADL’s [20], and consequently, residents have
to wait to be transported to the dining room, to the activities center, to the room of another
resident for a social interaction. For most of the residents, who are largely confined to the

perimeters of LTC facilities, a daily recreational wheelchair ride is highly desirable and also

very important for their emotional well-being. It allows them to see the world outside the
confines of their room and as such can avoid clinical depression and other serious mental
health problems. A wheelchair ride, however, requires a LTC nurse or staff member to
completely devote his/her time to the resident; this is not only an expensive proposition
but it is also deemed as a “non-value” added task. With shortage in nursing staff, it is
felt that “value-added” tasks, such as feeding or supervision during an activity, would be a
better usage of nurse time. Considering the situation, we feel that robots with the capability
of pushing and steering wheelchairs within the bounds of LTC facilities will be a greatly
helpful and responsible application of Robotics.

Several LTC facilities have adopted robotic systems. For example, the RP-6 [23], de-
veloped by the Intouch Health, Inc. provides telepresence capabilities to caregivers such
that they can visit their patients more frequently. The HelpMate [24], developed together
by Helpmate Robotics, Inc. and NASA can help patients remember daily chores. The
Nursebot [25], developed together by researchers at Carnegie Mellon and University of
Pittsburgh, assists the elderly suffering from chronic disorders in their every day life. Paro
[26] is a robotic seal that has been documented to improve the mental health of the elderly.
There are several other similar robots with the purpose of assisting the elderly and the dis-
abled [36] but none of these robots have the capability to push wheelchair-bound people.
It is envisioned that humanoid robots will provide assistance to residents in LTC facilities
and to this end there have been efforts to enable such robots with the capability of pushing
wheelchairs [31].

The main goal of this research work is to design a synergistic controller/ arm duo for a
holonomic mobile robot so that it can be used for applications such as pushing wheelchairs
and other wheeled platforms without human assistance. In other words, the robot should be
able to push the wheelchair, using an arm, such that the wheelchair tracks a given trajectory
asymptotically. The term “synergistic” irnplies that the controller and the arm will be de-

signed in such a way that they optimally complement and enhance the effectiveness of each

other. While mobile robots with manipulators have been studied extensively in the litera-
ture with the objective of motion control of both the base and the arm, most of these robots
are mainly composed of off the shelf industrial manipulators mounted on mobile bases.
There have been no attempts to optimize the manipulators, their degrees of freedom and
the control to suit applications such as pushing wheelchairs effectively. Some of the earlier
work has focussed on dual arm robots and the challenges associated with their cooperation
and control [1], [2]. While a dual arm may be anthropomorphic, it increases cost and adds
unnecessary weight. A number of researchers have considered single-arm robots but their
main focus has been utilization of redundancy for avoiding obstacles and joint limits [3],
[4]. In addition to maneuvering wheelchairs, it would also be useful to have the ability
to perform ancillary tasks such as picking up objects opening doors and pushing elevator
buttons. These functionalities can be found individually in several developmental systems
which can be incorporated into our design. We can additionally consider tele-operation of
such a robotic system using a Haptic device. A haptic device will help the operator, realize
the forces on the arm as it interacts with the environment such as pushing wheelchairs, or
opening a door. Tele-presence using the RP-6 [23] is already in commercial use is several
LTC facilities. This will be taken to the next level by allowing family members transport

wheelchair bound loved ones in LTC facilities from a remote location.

1.2 Preliminary Challenge 1: Design of a Robotic Arm

There are several challenges to developing a robotic system discussed in the previous sec-
tion. For a better understanding we classify the problems faced into preliminary challenges
and extended challenges. For the task of pushing and steering a wheelchair, one of the
preliminary challenges is the design of an arm itself. Although simple devices can be used
to push and steer wheelchairs, an arm is proposed since it can be used to perform ancillary
tasks like opening and closing doors, pressing elevator buttons, picking up objects etc., for

greater access and utility. From a human point of view, pushing and steering wheelchairs

with a single arm is a formidable task, let alone for a robot. Nonetheless, we propose to

use a single arm since

0 A dual arm system will make the robot top-heavy. This means that the center of

gravity of the robot will be raised, which is unacceptable since it will be prone to

toppling.

o The above issue will have to be addressed by suitable increase in the base dimensions
and weight, a bigger power supply to propel the increased weight, a more rugged
platform, etc. This will make the robot heavier than the weight of the additional arm.

0 A second arm will increase the cost of the robot substantially and will seriously

reduce commercial viability.

Using a single arm has its own complications with regards to the control problem but this
is a price that we will have to pay for a significant reduction in the hardware, cost and
overall system complexity. In the next chapter we will present one candidate design for
the purpose of illustrating the basic idea of pushing and steering a wheelchair with a single

arm.

1.3 Preliminary Challenge H: Adaptive Estimation

Another preliminary challenge is the lack of information in regards to the model of the
wheelchair and the occupant. Since we assume that we do not have prior information
on the mass of the person seated on the wheelchair, or the surface characteristics of the
floor, it is not possible to obtain a model of the wheelchair and its occupant. An obvious
work around this issue is to opt for an adaptive control wherein the parameters like the
mass, moment of inertia and the surface friction, as the wheelchair traverses the floor, are
estimated in real time. Some research has been done on pushing objects by robotic means

employing real time estimation of parameters. Yoshikawa [27] developed a method for

__ u -.-.__I

pushing an object along a specified trajectory using a manipulator. The effects of friction
were modeled but inertial effects were neglected. The inertial effects were considered by
Yu, et a1. [28], where they proposed a method to identify the mass and inertia of an object
being pushed with robotic fingers. Model identification while pushing a nonholonomic cart
has been proposed by Xi [29]. However, in their paper, they do not consider friction or any
wheel resistance. A brute force approach to pushing, without knowledge of the moment
created by the difference in reactions at the right and left wheels, could generate substantial
torque on the robot arm. Lynch [30] conducted elaborate experiments to identify friction
of a pushed object but this approach is more suitable for offline estimation. Since there

is no solution for direct application in this case, adaptive estimation can be considered a

preliminary challenge.

1.4 Main Challenges

A major part of this thesis will deal with what are called extended challenges. The ex-
tended challenges boil down to saturation and dynamics of robotic actuators which will
provide the required forces and moment to steer the wheelchair. When the actuators on
the robot reach their maximum limit, the control inputs for the wheelchair will saturate.
Similarly, the dynamics of the actuators will affect the motion of the wheelchair unless it
is include in the control system model. Thus we will have to deal with input saturation
and dynamics, while employing adaptive control. A condition wherein a complete loss of
control occurs will also be addressed. These issues are in part a consequence of the single
arm design choice. In the beginning, while addressing the preliminary challenges several
simplifying assumptions will be made. We will ignore the saturation and the dynamics of
the actuators to obtain a basic controller. To make the controller practically feasible, we
will eliminate the assumptions one after the other in the consecutive chapters and show
asymptotic tracking of a trajectory by the wheelchair even in the presence of these control

constraints provided the trajectory is reasonable and certain conditions are satisfied. The

requirement for a reasonable trajectory is justified due to the fact that even for known plants
with actuator saturation, asymptotic tracking is not possible [5]. Actuator saturation is not
a new phenomenon, however it is not a simple issue and numerous researchers have tried to
show some kind of well behaved nature of the system under such constraints [6—12]. Most
of these papers impose strict limits on the pemrissible trajectory at any given time. This is
not always possible. Further all of these papers ignore actuator dynamics which has been
separately considered by other researchers [13, 14]. Both actuator dynamics and saturation
have been addressed [15] but this has been accomplished by manipulating the reference
trajectory to prevent saturation altogether. The work here will be presented in the context
of a wheelchair-pushing holonomic robot. However it can be extended to other applications

with similar conditions where actuator constraints are an issue.

1.5 Using a Holonomic Platform

Although the mobile robot that will be used in this work uses a holonomic platform, the
entire system comprising of the wheelchair and the robot becomes a nonholonomic system.
A good review of the literature on nonholonomic control systems can be found in the arti-
cle by Kolmanovsky and McClamroch [16]. As indicated therein the problems of feedback
stabilization and tracking for car-like systems have been successfully resolved [17], [18].
These control algorithms apply to cars with n-trailers wherein each trailer is itself a non-
holonomic system. As mentioned earlier, the mobile robot that is to be used is a holonomic
robot. However the wheelchair is a nonholonomic system. Thus the combined system will
have a different kinematic structure. The highly efficient control algoritluns develOped for
cars with trailers [17], [18] also require frequent forward and backward motion, which will
be unsuitable while pushing residents on a wheelchair. Finally most of the results in the
literature are based on kinematic model of the wheeled devices. We would like to design
the controllers based on the dynamic model of the wheelchair and the robot since the forces

of interaction between the wheelchair and the robot will have to be sensed in order to steer

the wheelchair properly. Although control based on force sensing has been proposed ear-
lier [29], there have been no attempts to simplify the control problem through innovative

manipulator design.

1.6 Progression of the Thesis

This thesis is organized as follows. The preliminary challenges of designing the arm and
the basic adaptive controller will be addressed in the second chapter. First, the equations of
motion for the wheelchair will be derived based on the force interactions of the wheelchair
with the floor and the robot. The forces applied by the robot will be considered as the con-
trol inputs for the wheelchair in the design of the adaptive controller. The third chapter deals
with saturation of the moment applied based on the design of the arm. Saturation of the
moment applied by the robot implies that the control input for the wheelchair is saturated.
We will provide conditions for asymptotic tracking of a trajectory by the wheelchair even
in the presence of such a saturation in the input. The fourth chapter deals with bandwidth
limitations of the actuator. Even if the adaptive controller for the wheelchair prescribes a
moment to be applied for a turn, the robot may not be able to apply this moment instanta-
neously due to limited actuator bandwidth. We will investigate this problem and provide
conditions for asymptotic tracking by the wheelchair in the presence of both saturation and
dynamics of the control input.

The fifth chapter deals with hardware and controller modification that significantly im-
proves the performance of the Synergistic arm/controller duo. The sixth chapter includes
saturation and dynamics of the applied force which aids in the forward motion. It should
be noted that thus far only the saturation in the applied moment was considered. The sev-
enth chapter will present the experimental setup, hardware discussion and experimental
results. Since the thesis deals only with the pushing and steering capability of the arm, we
developed a simplified version of the arm which can be used to demonstrate the control

algorithm and the functioning of the arm and the synergism between them.

CHAPTER 2

Preliminary Work

2.1 Conceptual Design of the Single Am

Our arm design has a single degree-of-freedom at the shoulder and three joints at the elbow,
of which only two will be actuated at any given time. Unlike the human arm we will have
two graspers, one at the elbow and the other at the wrist. Each grasper will have two
degrees-of-freedom in addition to grasping and releasing capability. Our conceptual design

is shown in Fig.2.].

 

Figure 2.]. Conceptual design of single arm for pushing and steering wheelchairs

 

elbow joint to the left
of the forearm center

 

 

of the forearm

elbow joint at the center
of the forearm center

Elbow joint to the right

 

 

 

 

    

fieering to the right I) l pushing straight I steering to the left

Figure 2.2. Pushing and steering achieved through elbow positioning in forearm prismatic
joint

    

 

A novel and unique feature of the design in Fig.2.l is its under-actuated elbow joint,
which will help in reducing cost and weight. It will use two actuators to drive two rotary
joints and a prismatic joint. One actuator will drive one of the two rotary joints (jointl)
about an axis perpendicular to the rear arm. The other actuator will drive the other rotary
joint (joint2) and the prismatic joint (joint3) in two discrete modes of under-actuation. In
the first mode of underactuation, power from the actuator will be transferred entirely to
rotary joint (joint2), which will result in rotation of the forearm about the axis parallel to
the rear arm. This mode of operation will change when the forearm becomes perfectly
horizontal (as a prelude to pushing wheelchairs) and then power from the actuator will be
transferred entirely to the prismatic joint (joint3). The change in the mode of operation
will be facilitated by a special mechanism inside the elbow joint. In the second mode of
under-actuation, the elbow joint can essentially translate with respect to the forearm. “With
the forearm attached to the wheelchair by graspers, as shown in Fig.2.2, prismatic motion

of the elbow joint in the forearm enables pushing as well as steering the wheelchair. By

properly positioning the elbow in the prismatic joint of the forearm and simultaneously
moving itself sideways (possible because of its holonomic kinematics structure), the robot
will be able to push the wheelchair along a straight line or steer it. In particular, if the
elbow location moves from the center of the forearm to one side, it will result in steering
action; the angle of steer can be accurately controlled based on feedback of sensory signals.
By using the concept of under-actuation we will reduce the number of actuators but also
enable our robot to push and steer a wheelchair with a single arm.

The above discussion obviates that the two actuators at the elbow would drive the two
rotary joints (jointl and joint2) in the first mode of under-actuation when the robot intends
to perform ancillary tasks such as opening and closing doors, pressing elevator buttons,
etc. When the robot intends to transport a resident on a wheelchair, the two actuators at the
elbow would drive the rotary joint and the prismatic joint (jointl and joint3) in the second
mode of under-actuation. In the first mode of under-actuation, the grasper located at the
wrist will be used for all ancillary tasks but the grasper at the elbow will not be used. We
do not discuss this scenario any further. The manipulator joints will be driven by servo
motors and will be equipped with encoders for sensing. The servo motors and sensors are
not shown in Figs.2.1 and 2.2. At the prismatic joint, the manipulator will house a force-
torque sensor for sensing the forces of interaction with the wheelchair. The force/torque

measurements will be used to generate feedback for wheelchair steering.

2.2 Dynamics of Wheelchair

In this section we present a mathematical model of the wheelchair and the assumptions
made to make the model tractable for control design. Consider the wheelchair in Fig.2.3
with the robot positioned behind the wheelchair for pushing it. The wheelchair is a two
degree of freedom system. These degrees of freedom are x and 9, where x and 6 denote
the instantaneous forward displacement and rotation of the wheelchair center of mass, re-

spectively. Before we present our mathematical model, we state our assumptions:

10

 

 

 

center of

 

 

Figure 2.3. Definition of coordinate system for the wheelchair and forces of interaction
between robot and the wheelchair during pushing and steering

Al.

A2.

The force applied by the robot along the direction of motion is denoted by F. This
force is applied perpendicular to the forearm that grasps the wheelchair. This as-
sumption can be justified by the fact that the robot platform is holonomic and it can
move sideways (along the y direction in Fig.2.3.) to maintain the arm perpendicular
to the line joining the wheels of the wheelchair at all times. For this preliminary
work, we assume F > 0 at all times and the robot has unlimited force producing

capability with infinite bandwidth.

The distance between the point of application of the force, F, and the center of the
forearm is denoted by d. This distance, shown in Fig.2.3, equals the distance that
the prismatic joint moves away from the center so that a moment can be created on
the wheel chair for the purpose of steering. At present, we assume that there can be
unlimited motion in this joint and as a result d can assume very large values. It is

also assumed that the actuator driving the prismatic joint has infinite bandwidth and

11

A4.

A5.

as a result, d can be commanded to its desired value instantaneously.

. The reaction forces acting on the left and right wheels of the wheelchair are denoted

by R1 and R2 respectively. It is assumed that the wheelchair is always moving for-
ward and the direction of R1 and R2 are positive along the direction shown in Fig.2.3.
The values of R1 and R2 will depend on the weight of the wheelchair and the per-
son seated on it as well as on the coefficient of friction between the wheels and the
floor. In reality, R1 and R2 will be time-varying and will depend on the velocity and
direction of motion but for the sake of simplicity we assume them to be constant.
Similarly, M and J, the total mass and mass moment of inertia of the wheelchair

about the center of mass, are assumed to be constant.

We assume that the robot and the wheelchair are moving on a flat horizontal surface
and the center of gravity of the wheelchair is always located symmetrically between

the wheels.

The desired trajectories are assumed to be obtained from a dynamic path plan-
ning algorithm that takes into account the position and orientation of the robot and

wheelchair. In the last chapter the path planning algorithm will be discussed at length.

Under the above assumptions, the dynamics of the wheelchair can be described by the

equations

Mi = F—(Rl-l-Rz) (2.1)

Jé = —Fd-(R2—R1)w/2 (2.2)

where w is the width of the rear wheelbase shown in Fig.2.3. Let us denote the total

resistance to forward motion as R. Thus

R = R1+R2 (2.3)

12

Since R1 and R2 may have different magnitudes, they may generate a turning moment on

the wheelchair. If we denote this turning moment as T,

T = (R2 -R1)w/2 (2.4)

Using Eqs.(2.3) and (2.4), Eqs.(2.1) and (2.2) can be rewritten as

Mir” = F—R (2.5)

16

—Fd — T (2.6)

In our control problem statement, discussed next, we treat d and F as our control variables.

2.3 Problem Statement

Our goal is to design a controller for the robot such that it can push and steer a wheelchair
along a desired trajectory in x and 9 coordinates, namely, xd (t) and 9d (t). For the purpose

of asymptotic tracking, we first define the error variables

ex = (Id-x) (2.7)

89 = (Gd—9) (2.8)

and the sliding surfaces (Khalil, 1996), [34],

i3}: + 11 ex (2.9)

e1

82 8'9 +2.2e9 (2.10)

where 21 and 12 are positive constants. The sliding surfaces have the property that all
trajectories on these surfaces asymptotically converge to the origin, i.e., (ex,éx) = (0,0)

3119(39aé9)= (0:0)

13

It was stated in the assumption A3 that M, J, R and T are constants. However, these
values are unknown and therefore we use estimated values of these parameters, namely,
M, f, R and T, in our control design. Our goal is to correctly estimate these parameters
concurrently with asymptotic tracking of the trajectories. The unknown parameters, their

estimated values, and error in the estimates are defined as follows

. ¢=(¢—«i) (2.11)

flax:
WW>SS>
I

We now present our control design with the help of the following theorem.

2.3.1 Main Result

Theorem 2.1: Consider the dynamical system described by Eqs.(2.5) and (2.6). For this
system, the error variables, el and e2, and the parameter estimation error, 6, asymptotically

converge to zero for the following choice of control inputs

F = an+11Méx+R+K1el (2.12)
1 .. -
d = —;;(f9d+3rzfe'g+T+Kzez) (2.13)
and the estimation law
(fd‘I'lléxkl
$=-P“Q, Qé (9"+:Qe9)e2 (2.14)
l
32

where K1 and K2 are positive constants and P is any positive definite matrix.

Proof: Consider the Lyapunov function candidate

1 1 1 - -
v = Em? + 51% + §¢TP¢ (2.15)

14

which is positive everywhere except at the origin (e1, e2, 43) = (0,0,0), where it is zero. By

differentiating V with respect to time, we get

v = Me] e'1 +Je2e'2 + 671%?) (2.16)

We obtain the derivatives (3'1 and e‘z from Eqs.(2.9) and (2.10) as follows
e'l = e} + A] e}, (2.17)
e'2 = e’b + AQe'g (2.18)

By multiplying Eqs.(3.l7) and (3.18) with M and J, respectively, we get

Mél = Mx"d — (F —R) +M21éx (2.19)

Jéz 19;, — (—Fd — T) +12% ea (2.20)

Substituting for F and d in Eqs.(2.19) and (2.20) we get

Me'l = Mx}, —(1l‘4x;1+111f4e'x+R+K1e1—R)+Milléx

(it'd-I’ll éx) (M-M) +(R—R)—K1 el

= Axci'x—Klel (2.21)
where
Axé [163+21e‘x 1 ] (2.22)
.. A?
¢x '3 [ R ] . (2-23)
and

15

Jéz = Jég— (few/bier, +T+K2e2-T)+J/lgé9
(éd+42ée) (J-j) + (T-T) -K262

= 49479 -K292

where

A9é[9';1+k2e'9 1]

-A j
¢9=[7-]

Substituting Eqs.(2.21) and (2.24) in Eq.(3.16), we get

V = 9743 —K1e% —K2e§ + 6"?4‘5

Substitution of Eq.(2.14) yields,

v = —K1e%—K2e§ g 0

(2.24)

(2.25)

(2.26)

(2.27)

(2.28)

Since V is negative semi-definite, the equilibrium point (e1,e2,4~>) = (0,0,0) is stable.

Moreover, since V is uniformly continuous we can use Barbalat’s lemma (Khalil, 1996),

[34], to deduce

e1,e2—+0 as t—roo

(2.29)

From Eqs.(3.l7) and (3.18) we know that 5?] and e'z are uniformly continuous. Therefore,

using Barbalat’s lemma (Khalil, 1996), [34], we can claim

é1,é2——>0 as t-—t°°

Substituting Eqs.(2.29) and (2.30) into Eqs.(2.21) and (2.24), we get

16

(2.30)

(55d + a'1‘5’x)

wTifi = o, w e (94+1’laé9) (2.31)
1

By ensuring that the desired trajectories xd and 9d are sufficiently rich, W can be made
Persistently Exciting (P.E.) (Astrom and \Vrttenmark, 1995), i.e.
t+8

(12 I 2 t W(r)WT(r) dr 2 all (2.32)
where a1, or; and 6 are positive constants. This results in 43 —> 0 as t —» 00, i.e., the param-
eters defined in Eq.(2.11) will be correctly estimated.
Corollary 2.1: For the dynamical system described by Eqs.(2.5) and (2.6), the wheelchair
trajectories, (rcpt, 6, 9), asymptotically track the desired trajectories (xd,xd, 9d, 9d) for the
choice of control inputs Eq.(2.12) and (2.13) and the estimation law Eq.(2.14).
Proof: From the proof of Theorem 2.1 we have e1,e2 ——> 0 as t —> 00. This implies
ex, éx, e9 , ée ——+ 0 as t —> 00, which in turn implies that the wheelchair will asymptotically

track the desired trajectories.

2.3.2 Simulations

We present simulation results in this section to demonstrate the efficacy of the control
system. Three sets of simulation results are presented. The first set demonstrates basic
trajectory tracking and accurate estimation of the parameters for sufficiently rich desired
trajectories. This simulation strictly adheres to the conditions and results of Theorem 2.1.
The second set of simulations illustrate a practical situation wherein the desired trajectories,
xd and 9d, have been defined to replicate a scenario where a wheelchair bound person is
transported along a hallway of a long term care facility. The third set of simulations show
the adaptability of the algorithm to changes in the parameters, although Theorem 2.1 does
not claim this ability.

l7

SIMULATION 2.1

For this simulation, the desired trajectories were chosen as

it'd l + sin(0.02t) (2.33)

0,, = sin(0.2t) (2.34)
along with initial conditions xd = rid = 9d = 9d = 0. These trajectories are sufficiently
rich such that W in Eq.(2.31) is persistently exciting. Also, the trajectories ensure that the

force F will always be positive. The mass and mass moment of inertia of the combined

wheelchair and occupant were chosen to be

M = 60 kg, J = 2.25 1(ng (2.35)
We additionally assumed
R = 30 N, T = 5 Nm (2.36)

The initial estimates of the parameters in Eq.(2.11) were assumed to be
43T(0) = [ 40.0 1.00 10.0 0.00 ] (2.37)

in SI units. The controller gains and the sliding surface constants were chosen as K1 = 15,
K2 = 15, 21 = 50 and M = 50. The simulation results are shown in Figs.2.4 and 2.5 for
wheelchair initial conditions x = O and 9 = 0. It is clear from the first two plots in Fig.2.4
that the errors ex and eg converge to zero. From Fig.2.5, the estimates are also seen to
converge to their original values. The actual values of the estimates are indicated with the
horizontal dashed lines.

The choice of the controller gains K1, K2, 2.1 and 2,2, determine the rate of convergence
of the parameters and the errors. Higher values of K1, K2, 2.1 and 22 result in faster con-
vergence but slower adaptation of the parameters and vice versa. This is observed from
Figs.2.6, 2.7, 2.8 and 2.9. The results in Figs.2.6 and 2.7 correspond to lower values of
K1, K2, it] and 22, whereas the results in Figs.2.8 and 2.9 correspond to higher values of

these parameters.

18

 

ex ODHU
OBS

 

 

 

 

 

 

60 «kg)
u.

 

 

 

120 -
80

-0.04 W

-0.08 1 1
0 10 20 30 40 50 60

time (sec)

F(N)

1 1 l 1

 

 

 

 

d(m)

 

 

 

Figure 2.4. Simulation 2.1: Plot of error variables ex and eg, and control inputs, F and d,
with controller parameters K1 = 15, K2 = 15, 21 = 50, 2.2 = 50

19

 

 

(,0 _____________ ___ ___._

 

 

 

 

 

N
r1111

I

l
1111

 

 

 

 

 

 

 

 

.—————_—————-—_ ___—

? (Nm)

 

 

6

4

2 - .
o 1 . 1 .

0 10 20 30 4O 50 60
time (sec)

 

Figure 2.5. Simulation 2.1: Plot of estimates of unknown parameters with controller pa-
rameters K1 =15, K2 =15, 21 = 50,11Q = 50

20

 

 

 

ex GURU

 

 

 

 

 

 

 

 

 

 

8 - l l I l I _‘

r 'I

A 6 . q

3 _.

e 4 -

s 2 j
0

0 10 20 30 40 50 60
thne(mx»

Figure 2.6. Simulation 2.1: Plot of errors with controller parameters K1 = 10, K2 =
10,21=10,2Q =10

21

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

70 w I 1 i .
£060 ————————————— __ ______ _
(250' ..
40 1 1 1 l 1
('2? ' r ' I I
E3: «
on ___ __________________________ 3
a2) -
(HIP 1 1 L L 1 _
230: """""""""""""""""""" .
V20r -
<11. . _
10 1 J l 1 1
A6____ r I , , ,
E4» __________ “‘— r
E
<1--2F -
0O 10 20 30 40 50 60
time(sec)

Figure 2.7. Simulation 2.1: Plot of estimates with controller parameters K1 = 10, K; =
10, 21 =10, )2 =10

22

 

15 l l I I I

10" ‘

 

ex (mm)

 

 

 

 

1.2 ~
0.8 -
0.4 *

e9 (deg)

l 1 l 1 J l l l

 

 

 

1 l l l

O 10 20 30 40 50
time (sec)

 

8

Figure 2.8. Simulation 2.1: Plot of errors with controller parameters K1 = 50, K2 =
50, A] = 50, 2,2 = 50

23

 

 

 

 

I I I I

 

 

 

(N)
8

 

 

 

 

 

 

 

0 1 1 r
0 10 20 3O 4O 50 60
time (see)

Figure 2.9. Simulation 2.1: Plot of estimates with controller parameters K1 = 50, K2 =
50, A] = 50, 2,2 = 50

SIMULATION 2.2

For this simulation, the initial conditions, parameter values and initial parameter estimates
were chosen to be the same as in simulation 2.1. The controller parameters were chosen to

be K1 = 10, K2 = 10, A] = 10 and A; = 10. The desired trajectories were chosen as

24

x. _ 1.0m/sec2 O$t_<_l secs
‘1 0.0 m/sec2 t>1 secs

r

(2.38)

0.0 rads/sec2 O S t < 5 secs

0.5 rads/sec2 5 S t < 6 secs

0.0 rads/se02 6 g t < 10 secs

—0.5 r'ads/sec2 10 S t < 11 secs

Q, = t 0.0 rads/sec2 11 _<_ t < 35 secs (2.39)
—0.5 rads/sec2 35 S t < 36 secs

0.0 rads/sec2 36 S t < 40 secs

0.5 reds/sec2 40 S t < 41 secs

0.0 rads/sec2 t2 41 secs

 

25 I I I I

 

 

20L _

15. -

10» 3

 

 

l l

-15 -10 -5 X 0 5 10

 

Figure 2.10. Simulation 2.2: Plot of desired trajectory

In global coordinates, the desired trajectory is defined by Eqs.(2.38) and (2.39) and is
shown in Fig.2.10. The wheelchair is required to accelerate at 1.0 m/sec2 along the path
for 0 S t < 1 sees and then move with constant velocity of 1.0 m / sec thereafter. Over the

interval 5 S t < 6 secs, the angular acceleration is set to 0.5 rad/sec2 and this results in a

25

change of heading to approx. 1430 as shown in Fig.2.10. Over the time interval 35 S t < 36
secs, the desired angular acceleration is set to -O.5 rad/sec2 and this resets the heading of
the wheelchair to 0". This is clear from the Fig.2.10. In agreement with the assumption
Al, the trajectory in Fig.2.10 requires F > O for the entire duration of time. This can be
explained as follows: The continual forward motion along the trajectory necessitates R1
and R2 to be always positive. With this resistance always present, F has to be positive for

a constant velocity motion.

A 10 1 I r I 1
3
-e 5 _
V
8 0 A VH1,
_5 1 I 1 1 l

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1m I I I I T
5? 60 + -
E: I. _

20 1 1 1 1 1

O I I I I I
"-01 - ~
5 . nv n -
U -0.2 r U r

0 10 20 30 4O 50 60
finn26wc)

Figure 2.11. Simulation 2.2: Plot of error variables ex and eg, and control inputs F and d

The simulation results in Fig.2.11 show that the errors converge asymptotically to zero.
For comparison, the desired and actual trajectories in global coordinates have been plotted
together in Fig.2.l3 using solid and dashed lines, respectively. The error during the initial
stages can be in part attributed to the fact that T is nonzero whereas T = 0. This implies
that an external moment is applied on the wheelchair and it is initially not compensated.

The prismatic joint location, d shown in Fig.2.1l, generates the appropriate moment on the

26

11.4..

 

4L5 r r i I .

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

@041- ..
<2 405
40 1 1 1 1 1
@1-4 1 1 r r r
E -
$1.2 ,____/
(H1 1 1 1 1 1 _
A30_.____L____T r T r
520: i
«z _ .
10 1 m 1 1 1
i? ‘1 » _
:5 — _
<t-a : :
O 1 1 1 1 1
0 10 20 30 40 50
thne(ux)

Figure 2.12. Simulation 2.2: Plot of parameter estimates

27

 

25 r I l j

 

20- _

15— -

>110» _

 

 

5 1

15 10 5 X 0 5 10

r—
v-
.—

 

Figure 2.13. Simulation 2.2: Overlapping plot of desired and actual trajectories

wheelchair. From Fig.2.ll it can be seen that force, F, becomes almost constant and is
approximately equal to the value of R during the constant velocity period as expected. It
is clear from Fig.2.12 that without a sufficiently rich signal in rd and ed, M and f do not
converge to the actual values of M and J, as given in Eq.(2.35). The M value rises towards
the actual value during the interval when there is some acceleration (0 5 t < 1 sees) and then
falls back. Similarly, it can be seen that in the presence of angular acceleration (5 S t < 6,
35 g t < 36 and 40 S t < 41 secs), f approaches the value of J. R and T converge to
their original values but this cannot be guaranteed to work in the absence of persistency of

excitation.

SIMULATION 2.3

In conformity with the assumptions made in Section 2.2 and the results in Theorem 2.1, we

assumed the system pararnters to be constant in simulations 2.1 and 2.2. In this simulation,

28

we investigate the performance of the closed-loop system when the system parameters
undergo step changes in their values. The control variables are chosen in accordance with
Eqs.(2.12) and (2.13). The desired trajectories and initial conditions were chosen to be
same as in Simulation 2.1. The controller parameters are chosen as K1 = 15, K2 = 15,

A] = 50 and 2.2 = 50. The system parameter values are chosen as follows.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

M = 60Kg (2.40)
2.251(ng 0<r<120 secs
= _ 2.41
J {31(ng £2120secs ( )
30N 0<t<605ecs
{40N t260secs ( )
ONm 0<t<60secs
T - 2.43
{5Nm r_>_60secs ( )
a60____'___:___[ _____ r r r 1 1
{4.50: I
<2 - -
4O 1 l l 1 l 1 l l 1
(<1; 3P_—L—_l_—:— - __I———I—_—l—-—l——_I_A-
g“ 25 ---------------------------- :
<H 1 1 1 1 1 1 1 1 1 1 _
so I I I I l I I I I
40 _____________ -:
230-- _— —————————————————————
<n: 20- ‘
10 m 1 r 1 1 1 1 1 1
2
0
-2 l l l 1 l l

 

 

 

0 20 40 60 80 100 120 140 160 180 200
time (see)

Figure 2.14. Simulation 2.3: Plot of parameter estimates

29

 

 

ex (min)

69 (deg)

CV-‘NUJ-hUIGQ

I
_s

 

 

 

 

 

 

 

M ‘

“I .

 

 

 

 

2O 40 60 80 100 120 140 160 180
time(sec)

C

Figure 2.15. Simulation 2.3: Plot of error variables ex and e9

30

200

[L

The choice of system parameters in Eqs.(2.40-2.43) depict two scenarios. In the first sce-
nario, one of the wheels of the wheelchair moves on to a different surface, from tile to
carpet, for example. In our simulation, at t = 60 sees, the value of R and the value of T
change from a zero value to a nonzero value. This is simulated to happen at t = 60 secs
when the value of R changes from 3ON to 4ON and the value of T changes from 0 Nm to
5 Nm. The other scenario depicts the situation where the person seated on the wheelchair
changes his/her posture, as a result of which the value of J changes from 2.25Kgm2 to
3.00Kgm2. This is simulated to happen at t = 120 secs. From the simulations shown in
Figs.2.14 and 2.15, it can be seen that the estimates converge to their original values irri-
tially and each time the parameter values change. The error plots in these figures indicate
that errors are generated after each of the changes in the parameter values, i.e, after t = 60
sees and t = 120 secs. The change in the value of R affects the error in x-coordinate, ex, and
estimation of the mass parameter, M. Similarly change in the value of T affects the error
in the 9 coordinate, eg, and estimation of the inertia parameter, f. The change in the value
of J does not seem to affect the estimation of T, T, although it does affect the tracking
error in 0, e9. Importantly, both trajectory tracking errors and parameter estimation errors

ultimately converge to zero.

2.4 Forward and Backward Motion of Wheelchair

2.4.1 Controller Modification

Thus far, we have assumed that the wheelchair is always moving forward (see assumption
A3 in section 2.2) and as a result the reaction forces R1 and R2 in Fig.2.3 are positive in
the direction shown. In this section we remove this assumption in order to consider both
forward and backward motion of the wheelchair. The dynamics of the wheelchair can be

written in the form:

31

M)? = F —ngn(x) (2.44)
.19 = -Fd—ngn(x) (2.45)

which are very similar to Eqs.(2.5) and (2.6) in Section 2.2 where the sgn function is defined

as follows

1.0 x > 0
sgn(x) = 0.0 x = 0 (2.46)
—l.O x < 0

We now present the main result which is similar to Theorem 2.1 in Section 2.3.
Theorem 2.2: Consider the dynamical system described by Eqs.(2.44) and (2.45). For this
system, the error variables, el and e2, and the parameter estimation error, if, asymptotically

converge to zero for the following choice of control inputs

F = ma + 21 Me}; +ngn(x) + K1 e1 (2.47)

d = "%~'(de + Mfeg -I- ngno't) +K2 62) (2.48)

and the estimation law

(xd + 1] éx)el
«ii = —P‘1 Q, Q 9- (9" Magi)” (2.49)
e1 sgn(x)
82 WW?)
where K1 and K2 are positive constants and P is any positive definite matrix.

Proof: We use the Lyapunov funtion candidate V used in the proof of Theorem 2.1, given
in Eq.(2.15). The derivative of V is equal to

V = Me] 6] +Je2e'2 + $TP$ (2.50)

Using Eqs.(2.9) and (2.10) and Eqs.(2.44) and (2.45), the first two terms on the right hand
side of Eq.(2.50) can be written as follows

32

Mél = ng—[F—ngn(x)]+Mlléx (2.51)

Jéz 10;, — [—Fd — ngn(x)] +JAQ ea (2.52)

Substituting for F and d from Eqs.(2.47) and (2.48) in Eqs.(2.51) and (2.52) we get

Me'l = Mx'g—(ng+/I.1Méx+ngn(x)+Kle1—ngn(x))+Mlléx

(564+Al éx) (M—M) + (R —R) sgn(x) - K1 e1

 

mi—MQ on)

where Ax is redefined as,
Ax é [ x}, +21e'x sgn(x) ] (2.54)

and,

19;, — (is, which +ngn(x) +K2e2 — T) +JAQé9

(éd+42ée)(J-j) + (T-T)Sgn(i) -K2¢’2

= 49459 — Kzez (2.55)

Jéz

where A9 is defined as,
A9 2 [ 9;, +51er sgn(x) ] (2.56)

Substituting Eqs.(2.53) and (2.55) in Eq.(2.50), we get

V = Q76 — K] e? — Kzeg + 43TH; (2.57)

Substitution of Eq.(2.49) yields,

v = —K1e%— Kzeg g 0 (2.58)

33

Following the footsteps of the proof of Theorem 2.1 in Section 2.3, we can show that
the trajectory tracking errors .9] and e2 and parameter tracking error, ()7, asymptotically

converge to zero as t '—> 00.

2.4.2 Simulations

In this section we present simulation results to compare performance of the closed—loop
system based on the control designs in Theorem 2.1 and 2.2. The first set of simulations
is based on the controller in Theorem 2.1 and the second set is based on the controller in
Theorem 2.2. Both simulations use the same desired trajectory, initial conditions, system
parameters, and controller gains. For simplicity, the desired trajectory involves forward
and backward motion along a straight line. This ensures that the required turning moment
is uniformly zero and therefore d is uniformly zero and there will be no problem if the
pushing force equals zero. If the required turning moment is nonzero but the pushing force
is zero, d will assume a very large value. We wish to avoid this situation here. It will be

addressed in the next chapter.

60 . r l l I
50- -
E40- —
N3o. _
20- -
10- .

o 1 1 1 1
0 20 40 60 80 100 120 140

time(secs)

 

 

 

 

Figure 2.16. Simulation 2.4: Plot of x with time

SIMULATION 2.4

For this simulation, the mass and mass moment of inertia of the combined wheelchair and

occupant were assumed to be

34

<

<

M (Kg)

R.(bD

 

 

 

 

 

 

 

 

 

ex,(nun)

F(N)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

48 . -
46 - -
44.. _
42 - -
4
4O 1 1 1 1 1 g
100 120 140
tuneemcs)
Figure 2.17. Simulation 2.4: Plot of parameter estimates
2m I I T I I I
0 K ‘
i- -<
-200 _ _
‘4“) 1 1 1 1 1 1
1% L I I I I I I
0 ~ _
_ [T _
-100— a
_2 1 l 1 l l 1
0 20 40 60 80 100 120 140
time (sec)

Figure 2.18. Simulation 2.4: Plot of error variable ex and force F

35

M = 60 kg, J = 2.25 1(ng (2.59)

The reaction force and reaction moment were assumed to be
R=30 N, T=5Nm (2.60)
The initial estimates of the parameters in Eq.(2.11) were assumed to be
437(0) = [ 40.0 1.00 10.0 0.00 ] (2.61)

in SI units. The controller gains and the sliding surface constants were chosen as K1 = 15,

K2 = 15, 21 = 50 and 2,2 = 50. The desired trajectories were chosen as

1.0m/sec2 OSt<1 secs
x, _ 0.0 m/sec2 1$t< 60 secs (262)
d — —2.0 m/sec2 60£t<61 secs '

0.0 m/scc2 61 St 3 120 secs

6d 0.0 rads/sec2 t Z 0 secs (2-63)

with initial conditions xd = rd = 9d = 9d = 0. The desired trajectory is along a straight
line, moving forward first and then moving backwards. Figure 2.16 shows the x position
of the wheelchair with respect to time. Figure 2.17 shows the estimates of M and R. It is
seen that R is being estimated although M is not. This is similar to the case in Simulation
2.2 where there was not enough persistency of excitation. It can be seen that the value
of R changes from +30.0 N to -30.0 N as the wheelchair starts moving in the backward
direction. This is because the direction of R is reversed when the wheelchair moves in the
backward direction. Since the wheelchair moves with constant velocity for a majority of
the time interval, the force applied on the wheelchair, F, converges to the value of R. This

can be verified from Fig.2.21.

36

 

30’
20“
10'

x (m)

 

l I 1 1 1

 

 

41.8

41.4

<

M (Kg)

40.6

40.2-

35
30
25
20
15
10

R (N)

20 40 60 80 100 120
time(secs)

Figure 2.19. Simulation 2.5: Plot of x with time

140

 

 

 

 

 

 

 

 

l L 1

 

 

0 20 40 60 80 100 120

time (secs)

Figure 2.20. Simulation 2.5: Plot of parameter estimates

37

140

 

 

ex (mm)
‘6

 

 

 

1
M
o o
r
1 1

 

 

 

 

 

 

 

_1m 1 1 1 1 1 1
100 r r 1 r 1 r
50 {I J
g 0- J
a, -50 - J
-100. -
-150- -
_2m 1 L 1 1 1 1
0 20 40 60 80 100 120 140

time (see)

Figure 2.21. Simulation 2.5: Plot of error variable ex and force F

SIMULATION 2.5

The conditions and trajectories for this simulation have been chosen to be identical to Sim-
ulation 2.4. The control and estimation laws from Theorem 2.2, described by Eqs.(2.47)
and (2.48), were used for this simulation. Since the direction of R was explicitly taken into
account in the dynamic model of the wheelchair in Eqs.(2.44) and (2.45) and the control
and estimation laws in Eqs.(2.47), (2.48) and (2.49), R does not require re-estimation upon
change in direction of the wheelchair. Consequently, the trajectory tracking errors are con-
siderably smaller for Simulation 2.5 in comparison to Simulation 2.4. This can be observed
from Figs.2.20 and 2.21. It is reasonable to claim that the controller in Theorem 2.2 results

in improved performance and less errors in comparison with the controller in Theorem 2.1.

38

CHAPTER 3

Saturation of Turning Moment

3.1 Introduction

Several assumptions were made in Chapter 2 and we begin this chapter by revisiting one of

them, namely, assumption A2.

A2 The distance between the point of application of the force, F, and the center of the
forearm is denoted by d. This distance, shown in Fig.2.3, equals the distance that
the prismatic joint moves away from the center so that a moment can be created on
the wheel chair for the purpose of steering. At present, we assume that there can
be unlimited motion in this joint and as a result d can assume any value. It is also
assumed that the actuator driving the prismatic joint has infinite bandwidth and d can

be commanded to its desired value instantaneously.

In A2 it is assumed that the distance (I moved by the prismatic joint can be unlimited.
While this can be used to handle situations where the required turning moment is large but
the required pushing force is small, it is not practical. In reality, the maximum value of d
will be limited by half of the length of the forearm, shown in Fig.2.]. We choose to denote
the physical limit of d as

ds = sat(d, dm) (3.1)

where dm denotes the maximum absolute value of the moment arm. Figure 3.1 shows a

plot of d,- with respect to d. A saturation in the value of d implies that the moment applied

39

by the robot on to the wheelchair will be saturated.

11 (13

am.--—

-dm

o.
B
Q]?

:____ -dm

 

Figure 3.1. Plot of d,- with respect to d

We begin our investigation by considering the force and moment required by the adap-
tive algorithm for asymptotic trajectory tracking, discussed in the previous chapter, and the
actual force and moment that is applied by the robot. Let us denote the actual force applied
by the robot as F. Then the actual moment applied by the robot, measured in the positive
2 direction, is equal to —F ds, where d, is measured positive in the y direction. From our
results in Chapter 2 we denote the force and moment required for asymptotic stability, in
the absence of saturation in d, as F, and M,, respectively. Then, from Eqs.(2.12) and (2.13)

we have

F, = ng+tlrirey +R+K1 e1 (3.2)

Mr=-F,-d = féd+aqfeg+T+K2e2 (3.3)

The dynamics of the wheelchair, when d can saturate, can be described as follows

M)? = F—R (3.4)

10' = —Fds—T (3.5)

40

511m:
___;

 

Note that the above equations are very similar to Eqs.(2.5) and (2.6) in Chapter 2. Now,
subject to saturation in d, the control problem refers to redesign of the inputs F and d for

asymptotic tracking of the desired trajectories in the x and 9 coordinates.

3.2 The Control Problem

3.2.1 Main Result

Due to saturation of d, the moment applied on the wheelchair will be equal to —F d3, where
F is the force applied on the wheelchair. This may not equal the desired moment, Mr, and
therefore we introduce Be, the ratio of the saturated moment to the required moment

—-F d3

30 = Mr (36)

The negative sign in the expression for Ba accounts for the fact that a positive force F

 

(along the x direction in Fig.2.3) and a positive displacement ds (along the y direction in
the Fig.2.3) results in a moment in the negative z direction whereas M, is positive in the
positive 2 direction.

From Eq.(3.6), we can express the applied moment as

—Fds = 39M, = 89 f0}, +13,9 Mfeb +39 T +39 Kzez (3.7)

Since only a ratio of the estimated values are used in the applied moment, we redefine the

estimation errors in Eqs.2.23 and 2.26 as follows:

.. - M . M
¢x=(¢x_¢X)r ¢x:[R:lr ¢X=I:R] (38)
Ve=(¢e-Beli39), ¢e=[;], $e=[;] (3.9)

The definition of the errors el and e2 remain unchanged from that given by Eqs.(2.9) and

(2.10). We are now ready to present the main result of this chapter.

41

 

Theorem 3.1: Consider the dynamical system described by Eqs.(3.4) and (3.5). For this
system, the equilibrium point (e1, e2, 6;, $9) = (0,0, 0,0) is asymptotically stable when 39

is greater than zero for the following choice of control laws

F = F, (3.10)
d3 = sat(d, dm) (3.11)
d = —i(Mr) ' (3.12)
r
and estimation laws
«5x: —P“Afe1, (3.13)
(is = -P"A€e2 (3.14)

where F, and M, are defined by Eqs.(3.2) and (3.3), respectively and dm 6 (0, d].
Proof: Consider the Lyapunov function candidate

1 1 1.. .. 1~ -
V1 = 5m? + 51.2% + 503%,, + 503134)], (3.15)

which is positive everywhere except at the origin (e1,e2, 6x359) = (0,0, 0, 0), where it is
zero. Although the expression of V1 is similar to that of V in Eq.(2.15), it should be noted
that 439 in the expression of V1 is different from that in V. By differentiating V1 with respect

to time, we get

V1 = Mele'1+Je2e'2 +6IP$X+63P$9 (3.16)

We obtain the derivatives e'l and e'2 from Eqs.(2.9) and (2.10) as follows

e] = é}; + 2.1 e'x (3.17)

e'2 = e'b + Aqeb (3.18)

42

By multiplying Eqs.(3. l7) and (3.18) with M and J, respectively, and substituting Eqs.(3.4)

and (3.5), respectively, we get

Mél = Mia-Mf-I-MAléx
= Mia-(F-R)+Mlléx

(It'd-I'll éx) (M—M) +(R—R) -K] e1

Jéz = Jag—mum,
= Jé,-(—Fd,—T)+Jaqé9
= Jéd—(BgM,—T)+Jfiqe9

= (9.1+32é9) (J-Boj) + (T-Be T) -BeK282

(3.19)

(3.20)

We obtain Eq.(3.19) by substituting Eq.(3.10) and the expression for F, from Eq.(3.2). We

obtain Eq.(3.20) by substituting the expression of 89 from Eq.(3.6) and the expression for

M, from Eq.(3.3). Equations (3.19) and (3.20) can be written as

Me] =14xe —K1e1
162 =Ae¢6 4% K282

Substitution of the above equations in Eq.(3.16), yields

. ~ ~ .. T .:. ... T .2
V1 = erAx¢x + 8249459 - K1 8% -Be K283 + ¢x 1%: + ¢e P¢0

Using the estimation law in Eq.(3.13) and (3.14), we get

V} = —K1e%— 39 [(28%

43

(3.21)

(3.22)

(3.23)

(3.24)

from Eqs.(3.3) and (3.6) we can show thath is a positive number and claim V1 is negative
semi-definite. Following the footsteps of Theorem 2.1, we can show that the error variables,
el and e2, and the parameter estimation errors, 43,, and 439, asymptotically converge to zero. —

Remark 3.1: Theorem 3.1 claims asymptotic stability of the equilibrium point
(e1, e2, Vac, $9) = (0, 0,0, 0). This implies that 69 —> 0 as t —-> co and estimated values of
the parameters J and T, denoted by $9, converge to (419/89) > 1119. This can be explained
as follows: The algorithm is developed on the presumption that the required moment M,
can be always applied. When the applied moment is saturated and has a lesser value than
the required moment due to saturation, the convergence of tracking error, e2, to zero be-
comes slower. The algorithm interprets this as increased resistance to turning and increases
the parameter values thus leading to higher parameter values.

Remark 3.2: Theorem 3.1 is applicable when Be > 0. To investigate when Theorem
3.1 is not applicable, we analyze the expression of 39. From its definition in Eq.(3.6), we

can show

 

 

= _ (3.25)

where the expression M, = —F,d was substituted from Eq.(3.12). As seen from Eq.(3.25)
the value of Be can never be less than or equal to zero, as long as F = F, 75 0. Therefore

Theorem 3.1 is applicable for situations where |F,| > 0.

3.2.2 Simulations

In this section we compare the results of Theorem 3.1 with the results of Theorem 2.1 in
Chapter 2, using simulations. The simulation results are presented in Figs.3.2, 3.3 and 3.4.

The maximum value for d, at saturation has been chosen to be dm = 3.0m. This choice

44

 

of a rather large value, although not practical, has been made due to the rich nature of
the trajectory with high rates of acceleration and turning. In Chapter 5, we will discuss
another method for generating larger turning moment even when d gets saturated to a more
practical value.

SIMULATION 3.1

For this simulation, the mass and mass moment of inertia of the combined wheelchair and

occupant were assumed to be

M = 60 kg, J = 2.25 1(ng (3.26)

The reaction force and reaction moment were assumed to be

R=30 N, T=5 Nm (3.27)
The initial estimates of the parameters in Eqs.(3.8) and (3.9) were assumed to be

«if (0) = [ 40.0 10.0 ] (3.28)

(a; (0) = [ 1.0 0.0 ] (3.29)

in SI units. The controller gains and sliding surface constants were chosen as K1 = 10,

K2 = 10, 2.1 = 10 and 2x) = 10. The desired trajectories were chosen as

red sin(0.lt) (3.30)

0,, = sin(0.2t) (3.31)
along with initial conditions xd = xd = 9d = 9d = 0. These trajectories are sufficiently
rich such that W in Eq.(2.31) is persistently exciting. For the trajectories in Eqs.(3.30) and

(3.31), the value of F, is computed using Eq.(3.2) and plotted in Fig.3.2. \Vrth F, changing

sign, we assume that F, assumes values arbitrarily close to zero but is never exactly equal

45

 

 

 

ex ODHD

 

 

 

 

«new

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

E
d:
ll
LL.

4m- -
gzm— -
'U 0 i

-2“) 1 1 1 1 1 1 1 n 1

0 10 20 30 40 50 60 70 80 90 100
thne(mx9

Figure 3.2. Simulation 3.1: Plot of error variables, ex and eg and control variables F and
d as defined by Eqs.(2.12) and (2.13) in Theorem 2.1

46

 

ex (mm

 

 

 

 

 

 

 

60 (deg)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

”2.:
IL.
4 I I r r r r r r I
A 2 - _
5 0 —
e -2 — r
_4 1 1 1 1 1 1 4 1 1
0 10 20 30 40 50 60 70 80 90
time (see)

Figure 3.3. Simulation 3.1: Plot of error variables ex and e9 and control variables F and d

with saturation in d with dm = 3.0 m

47

 

 

M (Kg)
U1
0

 

 

3.3..

 

 

1 (Kg m2)

 

 

 

 

MN)

 

 

 

 

’1“ (Nm)

 

 

BOBSSSBBSO
4

 

 

Figure 3.4. Simulation 3.1: Plot of parameter estimates with saturation in d with dm = 3.0
m

to zero. The problem where F, = 0 will be addressed in the next section. Figure 3.2 shows
the simulation results in the absence of saturation in the value of d, i.e., using the control
variables defined in Theorem 2.1. It can be seen that as the force becomes very small, the
value of d assumes very high values. This occurs twice, at approximately I = 37 sees and
t = 58 secs. Figure 3.3 shows the simulation results with the saturation in the value of d.
For this simulation, F and d, are chosen according to Eqs.(3.lO) and (3.11) in Theorem
3.1. It can be seen from Fig.3.3 that d, has a maximum value of 3.0 m whereas d in Fig.3.2
assumes very large values. When ds is at its maximum value, the error variable e9 jumps
because the robot is unable to produce the required turning moment. The behavior of the
estimates have been presented in the Fig.3.4. It can be seen that the values of the estimated
parameters associated with the 6 coordinates increase as (1 gets saturated but converge to

their actual values when d is no longer saturated.

48

 

3.3 Resetting of Estimates to Prevent Windup

It was noted in Remark 3.1 that the estimates increase during the period that the actuator
is saturated. This is known as estimation Windup in the literature (Karson and Annaswamy
1994). This nature of the estimates could lead to poor or unnecessary transient behavior in
the system. It also necessitates that we impose the condition that the trajectory is such that
saturation will occur only for brief periods at a time. In this section we intend to address
this excessive increase in the estimates and curtail their negative influence on the system.
The idea is to bring back the estimates to normal values as e2 = 0 and thereby reduce
overshoot due to estimate windup, if any, in e2. Such an action can ultimately result in
better behavior of the physical system, such as reduction in any oscillatory nature.

Consider the parameter error

‘50 = (4’0 -Be430) (3.32)
During saturation, Be decreases and 43 takes large values. In order to set 43 to a smaller
value we intend to multiply (I; with 39 when ever the condition e2 = 0, 89 < 1 is satisfied.
In physical terms the condition, 89 < 1, implies that there is saturation in the actuator and
e2 = 0 implies that the estimation law is zero and $9 is momentarily constant. Overshoot
in e2 may occur depending on the difference between the required moment and the applied
moment. Let us analyze the system when we perform this resetting of R- We will show that
the value of (59 does not change during this process. 3
Let tj, j = 1,2,3... be the instants in time when the condition e2 = 0, 89 < l is satisfied.
We assume that this condition occurs only a finite number of times. When e2 = 0 it can be

seen that the required moment in Eq.(3.3) reduces to

A

= 4,0 (3.33)

A reset in the values of the estimates at t = t j, given by the expression

49

430;) = 39 (17(0) (334)

results in

MAI?) = BO Mr(’j) (335)

We know from Eq.(3.6) that

—Fds :39 M, (3.36)

The resetting of estimates in Eq.(3.34), in addition to. reducing 439, results in

M,=—Fds => 39:1 (3.37)

This follows from the definition of Be, namely, 39 = 216,41. From the above analysis it is

clear that the product

(30 439W?) = (39 439W)? (338).
although the values of Be and 1139 changed discontinuously. Thus we achieved in resetting

43 to a smaller value without discontinuity in the parameter error

59 = (4’9 —Be439)
The following observations can be made from the above analysis.

1. It should be noted that the condition e2 = 0, 39 < 1 is satisfied only for an instant at

a time, since the switching immediately renders 39 = 1.

2. Since switching happens at e2 = 0 and when 69 is constant, there is no change in

the estimation law, namely (59 = —P‘1Age2.

50

3. V1 and V1 are continuous since e2, 69 and 69 are continuous and thus the proof

of Theorem 3.1 is still valid for the theorem we present next, which includes the

resetting of estimates.

Theorem 3.2: For the dynamical system described by Eqs.(3.4) and (3.5), the equilib-

rium point (e1, e2, 6,, 69) = (0,0,0,0) is asymptotically stable for the following choice of

control laws

F = Fr
4 — 1 (M)
_ Fr ,
d3 = sat(d, dm)
and estimation laws
(15', = —P—1A,{e1,

$9 = 4044582

$0=30¢39(tj), e2=0mldl319<1

provided F, 51$ 0.
Proof: The proof of Theorem 3.1 is still valid.

SIMULATION 3.2

(3.39)
(3.40)

(3.41)

(3.42)

(3.43)

(3.44)

The conditions for this simulation are chosen to the same as in Simulation 3.1. The only

change is in the choice of trajectory and the value of dm. The trajectory is chosen as follows

x _ 1.0 m/sec2 03131 secs
d — 0.0 m/sec2 t > 1 secs
0.0 rads/sec2 0 S t < 20 secs
0,; = 0.5 rads/sec2 20 g t < 21 secs

0.0 rads/sec2 t 2 21 secs

51

(3.45)

(3.46)

1??"

This trajectory is similar to the one chosen in Simulation 2.2 (which also has same condi-
tions). However we have limited the 0 trajectory. Therefore such a choice implies that the
wheelchair moves forward in the x coordinates at a constant velocity. After the initial 20
sees, it goes through an angular acceleration of 0.5 rn/sec2 when 20 < t g 21 sec and then
starts turning with a constant angular velocity until end of simulation. The maximum value
for d5, dm is chosen as 0.172 m. The reason for such a choice is as follows: It can be seen
in Simulation 2.2 that the value of d required during steady state was 0.167 m. This is the
minimum value of d required to counter the effect of T (5 Nm), when F is almost constant
around 30.0 N. Thus 0.172 m is a tight saturation limit with 5.0 mm to spare. Further it is
also seen that during the period of angular acceleration in Simulation 2.2, d reaches a max-
imum of approx. 0.22 m. Since the conditions and the angular acceleration rate are similar,

it can be assumed that this is the same for this simulation too if there were no saturation.

 

cum!!!)
MI—ii—l

CDC=88

111

 

 

1'11
0

 

 

 

30 (deg)

 

 

 

F(N)
o seeeg asses

 

 

 

 

 

-02 1 1 1 1 1

0 20 4O _ 60 80 100 120
trme (sec)

 

Figure 3.5. Simulation 3.2: Plot of error variables, ex and eg and control variables F and
ds without modification in estimates

52

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

l l l 1

0 20 40 60 80 100 120
time (sec)

 

Figure 3.6. Simulation 3.2: Plot of error variables, ex and eg and control variables F and
d, with modification in estimates

53

 

 

 

 

 

 

 

 

 

 

 

 

09

N

M
1111

l l 1 l

 

 

 

 

:5 100- .

.4

A

T‘
M
C
r

 

 

0 1 . - 1
0 20 40 . 60 80 100 120
trme (sec)

 

Figure 3.7. Simulation 3.2: Plot of parameter estimates without modification

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

,N415 , . . . .
§§41 _ -
6,5405 4
40 1 1 1 1 1
"150 , , , , ,
(E 100
go _ _
(H 50 - -
0 l 1 l l
35 I 1 I I I
30 ~ 4
33‘ ‘
«a 15 : :
10 1 1 1 1 1
E150 , . 1 1 1
25100- .
<1— 50 — .
0 __L 1 1 J
o 20 40 _60 80 100 120
t1me (see)

Figure 3.8. Simulation 3.2: Plot of modification in parameter estimates

54

The simulation results are presented in Figs.3.5 through 3.8. Figs.3.5 and 3.7 are the results
of the simulation without the new modification and Figs.3.6 and 3.8 show the results of
simulation with the new modification in place. In both cases it can be seen that the errors
converge to zero. However in the second case the convergence is faster around 30 < t < 40
see as against 50 < t < 60. It can also be seen that the estimates have been brought back to

lower values during the resetting.

3.4 Addressing Loss of Turning Moment

3.4.1 Change of Reference Tiajectory

For the preliminary work in Chapter 2, it was assumed (see assumption A.1) that F > 0. For
this reason, we used trajectories that would keep F = F, > 0. This is not practically possible
since the force required, F,, can be negative or even zero, depending on the trajectory.
Theorem 3.1 can deal with positive and negative values of F, provided F, does not change
sign. In cases when F, becomes zero, the moment applied by the robot becomes zero
despite the fact that the required turning moment may be nonzero. This can be verified from
Eq.(3.5) where the turning moment applied by the robot is given by the expression —F d,
which assumes a zero value when F = F, = 0, irrespective of the value of (1,. This results
in 89 = 0, a situation where we do not have control over the turning of the wheelchair. As
a result, the errors in the 9 coordinate can increase significantly depending on the desired
trajectory. This is explained next. Consider the derivative of the sliding surface for the 9
coordinate in Eq.(3.18)

6'2 = e’b+42€'0
= (9d-é)+42e'e
= (éd—Jé/J)+2Qe'9 (3.47)

Substituting for J 6 from Eq.(3.5) we get

55

e: = (éd—(Fds—T)/J)+Meb (3.48)

Since F = 0, this becomes

e'2 = éd+T/J+2Qe'9 (3.49)
The desired trajectory, 0d, will be provided by a path planning algorithm (see assumption
A5 in Chapter 2) that does not take into account the loss in turning moment and therefore e2
can increase or decrease. It is prudent to suspend trajectory tracking in the 0 coordinate un-
til the control effort is regained. To this end, we propose tracking an intermediate reference
trajectory 0d,, which follows the original desired trajectory when |F,| > s, where E is some
small number, and maintains a constant error in the 9 coordinates, when |F,| g 8. Taking
the liberty of slight misuse of notation (mixing time and Laplace domains), we define 9di

as follows

.. __ F
9.1: = gs+1’ I 'l > 8 (3.50)
e-A’Zebia IFrI _<. 3
The errors and sliding surfaces are redefined based on the intermediate reference trajecto-

ries as follows

89" = 9d,: — 9 (3.51)
6’21 = 5’91 +42891 (352)

Indeed, for |F,| S 8,
(921°: (édi_ 6) +42%: = 0 (3-53)

This implies that e2,- is constant during this period and consequently égi = 0 and e9,- is
constant. Based on the definition of the errors and sliding surfaces in Eqs.(3.51) and (3.52),

we redefine the required turning moment, given by Eq.(3.3) as follows

56

M, = f ed,- +Mfebi +T +K2 e2, (3.54)

In addition to the trajectory issue, we stated in Remark 3.1 that the estimated values of the
parameters J and T, denoted by 69, converge to (69 / Be) > 69. This would mean that as
89 becomes very small, the values of the estimates will become very large and will become
undefined when 39 = 0. This is another issue that needs to be addressed to deal with loss
in tuming moment. The next theorem presents a method to deal with this problem. We
have however chosen not to include the resetting presented in Theorem 3.2 for clarity in
the results.

Theorem 3.3: Consider the dynamical system described by Eqs.(3.4) and (3.5). For
this system, with estimation errors defined by Eqs.(3.8) and (3.9), the equilibrium point
(e1,e2,-,6x,69) = (0,0,0,0) is asymptotically stable when |F,| > e and stable otherwise,

for the following choice of control inputs:

F = F, (3.55)

Fr
(13 = sar(d,dm) (3.57)
(3.58)

and estimation laws:

$1: = -P“AIe1 (3.59)

"' _ _ —1 T ,

¢01 ¢0:01 IFrISE

where 8, 8 > 0 is some small number.

Proof: Consider the Lyapunov function candidate

1 1 1.. .. 1.. ..
v2 = -2-Me% + 51.3%,. + 593%: + 56311419 (3.61)

57

which is similar to the expression of V1 in Eq.(2.15). The only difference is that the ex-
pression of V2 replaces e2 in the expression of V1 with e23. As a result we can show that V2
satisfies

V2 = —K1e§ — Be Kzeg, (3.62)

when |F,| > 8. Since |F,| > 8 ensures Be > 0, we have V2 3 0. Following the same logic
as in Theorem 2.1 in Chapter 2, we can show that the equilibrium point (e1 $2,361, 69) =
(0, 0,0, 0) is asymptotically stable.

When |F,| S 8, the derivative of the Lyapunov function, through direct differentiation is

given by the expression

V, = Me, é, +Je2é2, + 617106, + «5'9 Trio (3.63)

which then takes the form

V2 = -—K1e% S 0 (364)

Equation (3.64) is obtained from Eq.(3.63) by substituting 69 = 0, which follows from
Eq.(3.60), and é2, = 0, which follows from Eq.(3.50) and (3.52). From Eq.(3.64) we claim
that the equilibrium point (e1,e2,-, 6.69) = (0,0,0, 0) is stable.

Remark 3.3: We need to stress here that we do not expect the condition |F,| g 8 to
come into play often. Then the result may be useful because it shows that the system will
remain well behaved in some sense. However if the condition |F,| S 8 is satisfied frequently
the switched system can become unstable. The above theorem does not comment on the
properties of the equilibrium for the switched system but only the sub systems subject to
the conditions. It should also be noted that the switching in the estimates in this section is
not the same as that presented in Theorem 3.2
From Theorem 3.2, an useful observation is made with the help of the following corollary.

Corollary 3.2: Consider the dynamical subsystem described by Eq.(3.4). For this sys-

tem, the equilibrium point (e1,6x) = (0, 0) is asymptotically stable for the following choice

58

of control input and estimation laws

F = Fr (3.65)
6', = —P-‘Afe1 (3.66)
Proof: Consider the Lyapunov function
1 2 1 T ..
V, = -2-Me1 + 56, P6, (3.67)

This Lyapunov function contains the terms of V2 in Eq.(3.61) that correspond to the x

coordinates. The derivative is given by

v, = Melél + 6&6, (3.68)

Proceeding in the same manner as in Theorem 2.1 we can show that

Vx S _Kle% _<_ 0

Following the footsteps of Theorem 2.1 we can show that e1 -» 0 as t —» co. By ensuring
that the desired trajectory xd is sufficiently rich, Ax defined in Eq.(2.22), can be made
persistently exciting. (Astrom and \Vlttenmark, 1995). This will ensure 6x -+ O as t -—> oo,
which implies that the parameters defined in Eq.(3.8) will be correctly estimated. Thus, the
equilibrium point (e1, 6,) = (0,0) will be asymptotically stable.

Remark 3.4: Corollary 3.2 establishes that trajectory tracking in the x coordinate is
achieved at all times including times when there is a loss of turning moment that cre-
ates problems in tracking in the 0 coordinate. This is expected because the applied turning
moment is dependent on the pushing force but the reverse is not true. The problem of track—
ing of tracking in the 9 coordinate occurs when |F,| 5 8, which is warranted by trajectory

tracking in the x coordinate.

59

3.4.2 Simulations

We present simulation results to verify the stability property of the equilibrium, claimed
in Theorem 3.3. TWO sets of simulations are presented. The desired trajectories, initial
conditions, parameter values, initial parameter estimates, controller gains, sliding surface
constants, and the value of dm for the two simulations are the same and equal to those used

in Simulation 2.1. The difference is in the choice for the value of 8.

SIMULATION 3.3

This simulation was performed with an 8 value of 0.1. The results are shown in Figs.3.9
and 3.10 where the vertical lines show the times when the condition |F,| S 8 is satisfied.
The period that |F,| remains less than 8 is very small due to the choice of our trajectory and
the small value of 8 (see next simulation for a more defined period of |F,| S 8). Thus the
time periods have been reduced to instants of time, which are denoted by the vertical lines
at approximately I = 37 secs and t = 57 secs. Corollary 3.2 can be immediately verified
since the error in the x coordinates, ex, and estimation error, 6, = [M ,R]T, converge to zero
asymptotically irrespective of tracking and estimation errors in the 9 coordinate. The same
is true for the next simulation where we used a higher value of 8.

The error in tracking the 9 coordinate, egi, eventually converges to zero but undergoes
a significant deviation from zero when F, becomes very small as shown in Fig.3.9. As
expected, the value of d is seen to saturate to d, = dm = 3.0m when F, becomes very small
and changes sign. This can be verified from Fig.3.9. Figure 3.10 shows the plot of the
estimated values. It can be seen that both f and T are set to zero for the short durations
of time when |F,| g 8. After the brief period at zero, the values increase significantly
but eventually converge to the actual parameter values. The significant increase can be
attributed to the fact that when |F,| is a small number approaches epsilon while satisfying
|F,| < 8, the value of 39 becomes small initially and consequently the estimated values

approach (169/B9), assume large values consequently.

60

 

ex ODHD

l l l l

 

 

 

 

 

 

 

 

 

601 (deg)

 

 

 

 

F(DU
c> g; g;
)1

 

 

 

 

 

 

 

 

 

_4 1 L L 1 1
0 10 20 30 40 50

 

70 80 90 100

 

ds (m)
|b<DtQ-h
7‘53"] -

Figure 3.9. Simulation 3.3: Plot of error variables ex and em, and control inputs, F and d
with 8 = 0.1

61

 

 

 

— ‘7 q
,0. _
9

XV.

 

 

X: 89.36
Y: 4.785.

 

100

80

70

 

 

 

50 (60

time (sec

40

 

 

P p p

 

 

 

505
554

8% d

b b n

1.

 

 

 

 

_
_
_
_

 

 

 

 

 

w wwwmom. 104
@836“

O
3
AZ

 

10

30

20

10

Figure 3.10. Simulation 3.3: Plot of estimates of unknown parameters with 8 = 0.1
62

SIMULATION 3.4

This simulation was performed with an 8 value of 1.0. The results are shown in Figs.3.ll
and 3.12. As in the previous case, the vertical lines at approximately t = 37 secs and t = 57
secs, show the duration of time when force |F,| g 8. Clearly, the duration of time when
the condition |F,| < 8 is satisfied is longer in this simulation. An overall first impression
obtained from Figs.3.ll and 3.12 make on comparison with Figs.3.ll and 3.9 is that a
higher value of 8 results in lower tracking and estimation errors. A comparison of Figs.3.11
and 3.9 indicates that the error in 9 coordinate, egg, is less and converges to zero faster for
larger value of 8. The brief period where e9,- remains constant is more discernible between
the vertical lines in Fig.3.11 than in Fig.3.9. The estimated values of J and T also remain
constant at zero for a longer duration of time. This can be verified by comparing the plots
in Fig.3. l2 and 3.10. The error and estimates corresponding to the x coordinate converge
unaffected, as was the case with the previous simulation. The plot of d, changes sign
exactly between the marked lines which is expected. Overall, the system is better behaved

when the value of 8 is larger.

63

 

F

ds (m)

 

10 20 30 40 50 60 70 80 90 100
time (sec

Figure 3.11. Simulation 3.4: Plot of error variables ex and em, and control inputs, F and
ds With 8 = 1.0

 

0 10 20 30 40 50 60 70 80 90 100
time (

Figure 3.12. Simulation 3.4: Plot of estimates of unknown parameters with 8 = 1.0

65

CHAPTER 4

Effect of Dynamics in the Prismatic Joint

4.1 Background

It was assumed in Chapter 3 that the prismatic joint variable denoted by (1, shown in Fig.2.3,
saturates to an absolute maximum value of d,. The bandwidth of the actuator driving this
joint was assumed to be infinite. This is clear from the assumption A2 (see Chapter 2)
which states that the commanded position of d is achieved instantaneously. In reality, the
position of the prismatic joint is affected by the dynamics of the actuator. If we denote the
actual position of the prismatic joint as d;, then dz. can be considered to be the output of the

block diagram shown below:

 

 

d ——>ds
rs+l

 

 

 

 

 

 

 

 

 

Figure 4.1. Dynamics of d, with mechanical saturation

According to the above block diagram, d is a commanded signal obtained from the adaptive
algorithm. The first block is the transfer function of the actuator. Thus a" is the actual
location when the actuator dynamics is considered. Once (1’ reaches a physical limit, it gets
saturated to d; and cannot move further. This is denoted by the second block. If the above
block diagram is implemented, the saturation block has to be implemented by mechanical

stops or the like. This will result in additional complexity and create impact on the ends

66

of the forearm which is not desirable. To avoid this scenario, we propose to implement the
saturation block in software and control the actuator with the saturated signal d, as shown

in the block diagram below:

 

 

 

 

 

 

rs+1

 

 

 

 

 

 

 

 

Figure 4.2. Dynamics of d, with software saturation

We discussed saturation of d in the previous chapter and in this chapter we address the
effect of actuator dynamics. The above block diagram models the actuator dynamics as a

first order system.

4.2 Motivation

If we model saturation and actuator dynamics of the prismatic joint, the equations of motion

of the wheelchair take the form

M)? = F—R (4.1)

10' = —Fdf,—T (4.2)

It can be seen that the actual applied moment is now given by —F d;. In Chapter 3, where
saturation of the prismatic joint was only taken into account, the applied moment was given
by —F d,, and 89 was defined as the ratio of the saturated moment to the required moment,
M,. Since the applied moment will now include the effects of actuator dynamics, we define
B’O as the new ratio of applied moment to the required moment as

_ng

B’:M,

 

(4.3)

Thus far we have always chosen F = F,. Therefore,

67

_ _Fdi _ ‘Frdi _ dis
9 “ M, — —F,d _ 17 (4'4)

Since d; follows the commanded signal d, with a phase lag 1, the applied moment, —F d;,
may have opposite sign of the required moment, M, = —F d. Typically, this will happen
after d passes through zero and changes sign, or when F, passes through zero and changes
sign. Due to actuator dynamics, d; will not change sign instantaneously when d (and hence
(13) changes sign, and as a result B’e will be negative.

In Chapter 3 it was noted thath is positive or equal to zero. In fact, a nonnegative Be
is also a requirement for Theorem 3.1. With the inclusion of actuator dynamics, Bo will
be replaced by 3’9 but since a non negative 3’9 cannot be guaranteed, the wheelchair may
turn in the opposite direction to what is required to reduce the errors if the control laws in
Theorem 3.1 are used. To address this problem we first introduce the variable B, as the

ratio of the applied force to the required force i.e.

F

B:—
x Fr

(4.5)

Thus far, we have used F = F, or B, = 1. After the preliminary result, which we present
next with the help of Theorem 4.1, we will not constrain F = F, at all times. Manipulating
F to be different from F, will be required to avoid a negative 8"). Consequently B, will
not be restricted to unity. The applied force F can then be thought of to be a fraction of
the required force F,, denoted by B, F,. In the previous chapter we modified the estimation
errors in the 0 coordinates to take into consideration the fraction of the required moment
applied - see Eq.(3.9). Here we update it to 3'9 and further redefine the estimation errors in
the x coordinates in Eq.(2.11) to reflect the applied force as F = B, F,. Thus the estimation

errors are redefined as follows.

Vx=(¢x-Bx¢;x)t ¢x= [it], Rx: [1;] (4-6)

 

1There is a phase lag between the input and output of a proper transfer function

68

60 = (¢e - 9‘59), (Pa = (4-7)

 

J . f
T], ¢0=[T

The definition of the errors, el and e2, remain unchanged from that given by Eqs.(2.9) and

 

(2.10) respectively. We are now ready to present the first result of this chapter.

4.3 Preliminary Result

Theorem 4.1: Consider the dynamical system described by Eqs.(4.l) and (4.2). For this
system, the equilibrium point (e1,ez,6x,69) = (0,0,0,0) is asymptotically stable for the

following choice of control inputs:

F = F,- (4'8)
1

d = —— M 4.
Fr( r) ( 9)

ds = sat(d,dm) (4.10)

d} = ',l;(ds‘d.’t) (4.11)

and estimation laws:
6x = —P—1A£el, (4.12)
«is = -P“A€e2 (4.13)

provided B’e > 0.

Proof: Consider the Lyapunov function candidate
12121~T~1~T~

which is positive everywhere except at the origin (81,82, 6x,69) = (0,0,0,0), where it is

zero. By differentiating V3 with respect to time, we get

V3 = Mele’1+Jeze'2 +433P¢Tx+6gp$9 (4.15)

69

We obtain the derivatives e'l and e‘z from Eqs.(2.9) and (2.10) as follows

8'1 = 8‘}, + 51.1 e', (4.16)
e, = e}, + Me's (4.17)

By multiplying Eqs.(4. l6) and (4.17) with M and J, respectively, and substituting Eqs.(4. l)

and (4.2), respectively, we get

55.
II

= Mx'a—(F—R)+Mlléx

Mxy — (13,r F, — R) + M21 8,, (4.18)

= (std +2.1éx)(M—BxM) +(R—BxR) —B,,K1e1 (4.19)

Jéz = 19d —Jé +JAQég
= 10;, — (—F d; — T) +1328,
= Jéd - (BIOMr - T) +JAI289
= (éd+}IQé9)(J—B’GJ) +(T—B’9 T)—B'9K2e2 (4.20)
We obtain Eq.(4.l9) by substituting for F = B, F, from Eq.(4.5) and the expression of F,

from Eq.(3.2). Similarly we obtain Eq.(4.20) by substituting the expression for B’e from
Bq.(4.3) and the expression for M, from Eq.(3.3). Thus we get,

Me'1=Ax6x —BxK1e1 (4.21)
Je‘2 = A9 (Ii-9 — BIO K262 (4.22)

Substitution of the above equations in Eq.(4.15), yields

70

. ~ ~ 1... T 1:. ~ T 1:.
V3 = ele¢x + €2Ae¢e - Bx K1 9% - 3’9 K283 + 9x P¢x + lie “is (423)

Using the estimation law in Eq.(4.12) and (4.13), we get

V3 = -B, K112? — 3’9 Kzeg g 0 (4.24)

since B’e > 0 and B, = 1. Following the footsteps of Theorem 2.1, we can show that the
error variables, el and e2, and the parameter estimation errors, 6, and 69, asymptotically
converge to zero.

Remark 4.1: Theorem 4.1 is similar to Theorem 3.1 except that it includes the dynam-
ics of the prismatic joint. However, it only considers the case where (I does not change
sign, and as a result of which 32, remains positive at all times.

SIMULATION 4.1

We present simulation results to validate the claims of Theorem 4.1. We assume the

mass and mass moment of inertia of the combined wheelchair and occupant to be

M = 60 kg, J = 2.25 kgm2 (4.25)

The reaction force and reaction moment are assumed to be

R = 30 N, T = 5 Nm (4.26)
We assume the initial values of the parameter estimates in Eqs.(4.6) and (4.7) to be

(133(0) = [ 40.0 10.0 ] (4.27)

433(0) = [ 1.0 0.0 I (4.28)

in SI units. The controller gains and sliding surface constants were chosen as K1 = 10,

K2 = 10, 2.1 = 10 and A; = 10. The desired trajectories were chosen as

71

 

 

 

 

 

 

691 (deg) ex (mm
o '5’
‘1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

-20 1 1 l 1 1
150 , t t I
2 100 i -
LL 50 1 1 1 1 1
:5:
:3
E
'U
.0.2 l 1 1 1 1
0 10 20 30 40 50 60

time (sec)

Figure 4.3. Simulation 4.1: Plot of error variables ex and 89, and control inputs, F and ds,
and d;

id = sin(0.02t) (4.29)

04 = sin(0.2t) (4.30)

along with initial conditions xd = 2,, = 9d = 0d = 0. These trajectories are sufficiently rich
such that W in Eq.(2.31) is persistently exciting. This will ensure convergence of parameter
estimates to their true values.

Figures 4.3 and 4.4 show the simulation results for T = 0.25 secs. In this particular
simulation, the commanded value of the prismatic joint did not exceed the saturation limit
and therefore we had d3 = d. The variable d; is however different from d, = d. In the pres-
ence of actuator dynamics, it can be seen that the errors converge to zero and the estimates
converge to their actual values given by Eqs.(4.25) and (4.26). In this particular example,
3’9 was always positive, a requirement for Theorem 4.1. This was achieved through proper

72

 

M (K8)

 

 

 

 

A

 

 

’1‘ (Kg m2)

 

 

 

1 l l 1

 

 

 

R(N)
8

 

 

 

 

__————_——-——

 

 

A
T flqnn
curses

 

0 10 20 30 40 50 60

choice of the

4.4 Add

It is obvious

intermediate

was equal to

time (sec)

Figure 4.4. Simulation 4.1: Plot of parameter estimates

desired trajectories and a non zero value of the reaction torque T.

ressing Sign Change of Control Inputs

4.4.1 Change of Reference Trajectory

that changes in the sign of either F, or d will make B’o < 0 for some period
of time and will render the result of Theorem 4.1 inapplicable. To avoid this situation, we
propose to enforce F = 0 irrespective of the value of F, for the duration of time when B’s is
negative. This will keep both 3, and 3’s at zero value since F = 0 implies that the applied
force and the applied moment are zero. In the previous chapter we proposed tracking an
trajectory, 0,1,, which followed the original desired trajectory 6,, when 89 > 0
and maintained the errors at their current values ((12, = ég; = 0 - see Eq.(3.53)) when 3,;

zero. Here we modify this to include 3’6 and further introduce an intermediate

73

trajectory for the x coordinates. This intermediate trajectory xdi will track the original

desired trajectory xd when 8, > 0 2 and will maintain a constant error when Bx = 0. The

intermediate trajectories are defined as follows

rd: *d-fdt, Bx>0
l f-Ale'xi, Bx=0

-- {Gd-9.11, |F|>8
9dr:

9-346301, IFI .<_8

The error variables and sliding surfaces are redefined accordingly

exi = xdi _x

311' = éxi+ll exi

and

em = 9dr- 9

621 = éei+32€91

The required force and moment expressions in Eqs.(3.2) and (3.3) are modified to

F, = Midi+llMeii+R+Kle1i

M, = fédi+42febi+T+K2e2i

(4.31)

(4.32)

(4.33)

(4.34)

(4.35)

(4.36)

(4.37)

(4.38)

The following corollary uses Theorem 4.1 to show asymptotic stability while using the

redefined trajectories.

Corollary 4.1: Consider the dynamical system described by Eqs.(4.l) and (4.2). For this

system, the equilibrium point (e1,,e2,-, 6x, 69) = (0,0,0,0) is asymptotically stable for the

 

2B,.,>0-—»B,,=1fornow

74

following choice of control inputs:

F = F, (4.39)

1
d = -;(M) (4.40)

,
ds = sat(d, dm) (4.41)
at: = ital. — dé) (4.42)

where F, and M, are given by,
F, = Mica-+711 Me},,+R+K1 e1,- (4.43)
M, = fédi+lquebi+T+K2e2i (4.44)
and estimation laws:

(it = —P“A§ert. (4.45)
$9 = 44115122,- (4.46)

if the condition B’e > 0 is satisfied.
Proof: Consider the Lyapunov function candidate
1

l 1 .. - 1 - ..
V4 = 5Mefi. + 51.2%,. + 5411104», + 541510419 (4.47)

The Lyapunov function candidate V4 is similar to V3 in the proof of Theorem 4.1 except
that e] is replaced by 81,-. Since F, and M, are now functions of e1,- and egg, respectively,
the proof is very similar to that of Theorem 4.1.

We are now ready to state the final result of this chapter. Since the error in parameter
estimates corresponding to the x coordinates have been redefined to include 8,, we will
also be addressing the issue of the estimates becoming undefined when 8, becomes zero.
This is similar to the case with the parameters estimates in the 6 coordinates in Chapter 3.

We present the result next.

75

4.4.2 Main Result

Theorem 4.2: Consider the dynamical system described by Eqs.(4.1) and (4.2). For this

system, the following choice of F and d;

Fr, sgn(d§) = sgn(ds)

F = 4.48
{ 0. sgn(d§) 9* 58"(ds) ( )

1
d = ——(M,) (4.49)

Fr
d, = sat(d,d,,,) (4.50)

. 1
d; = ;(d. — dé) (4.51)
where F, and M, are given by,

F, = M8,,- +1.1 Meg,- +R+K1 e1,- (4.52)
M,- = fédi+AQfebi+T+K2 821' (4.53)

renders the equilibrium point (81 113213620 69) = (0,0,0, 0)- asymptotically stable when

|F | > 8 and stable otherwise for the following choice of estimation laws

‘" _ _ -1 T , I =
4;. — P 4, .,,, sgnrdf) sgnrd.) (4.54)
¢x = 0, Sgn(ds) 75 Sgn(ds)

($9 = —P_1AgeZir IFI > 8

69 = 0, IF I S 8, sgn(d(.) 75 sgn(ds) (4.55)

439, $9 = 0. IFI S 8, 38"(di~) = sgn(ds)
Proof:

There are 3 cases to be studied in this theorem.
1. |F| > 8
2- IF I S 8 and S8n(d.’t) 79 sgn(ds)
3. [F] g 8 and sgn(d’.) = sgn(ds)

76

Consider the first case where IF I > 8. We know

|F| > 8 —> sgn(d_(.) = sgn(ds) (4.56)

since from Eq.(4.48) it can be seen that

sgn(d;) ,4 sgn(d,) .—+ |F| = o < s (4.57)

In Theorem 3.2 we have shown

|F| > 8 —> B9 > 0 (4.58)

From Eq.(4.3), B’9 can be written as

 

 

3’9 = 72"; (4.59)
_ I _

= 13:71:? (4.60)

= gig? (4.61)

= $3139 (4.62)

> 0 (4.63)

since d; and d, are of the same sign and B9 > 0. Further B, = l > 0 during this period
since F = F,. Thus since 8,, and B’e are both greater than zero, using'corollary 4.1 we can
say that the equilibrium points are asymptotically stable.

For the case |F | g 8 there are two scenarios. The first one being sgn(d;) = sgn(ds) and the
other sgn(d§) 7£ sgn(ds). Let us first consider the case when sgn(d§) 75 sgn(ds). Consider

the derivative of the Lyapunov function candidate in corollary 4.1
- . ~ T =- . ~ T .
V4 =Melieli+¢x P¢x+Jeztezi+¢9 P¢0 (4-64)

77

From Eq.(4.32) and (4.36), we know e2,- is constant. Similarly from Eq.(4.31) and (4.34)
we know 911’ is also constant. From Eq.(4.55), we have 69 is constant. 3’0 and B, are
zero during this period, since F = 0. So we can see that 69 = (69 -B"9 69) = 69 and
6x = (6,, — B, 6,) = 6,, are also constant. thus

- . ~ T :- . ~ T =-
V4 =Melieu+6x P¢x “Flea-8214109 P439 = O (4.65)

Therefore the equilibrium points are stable for this case.

Now let us consider the next case wherein sgn(d_’,) = sgn(ds). For this case it can be shown

that

v, = —B,K1e%,. g 0 (4.66)

since 3, > 0 when F = F, and moreover e2,- is constant and 69 is also constant since
69, 69 = 0. Therefore the equilibrium point is stable.
It can be noted that if sgn(d§) = sgn(d,) for a sufficiently long time and sgn(d_(.) 75 sgn(ds)
only a finite number of times, then the errors in the x coordinates converge to zero asymp-
totically. In addition to the above condition if |F | g 8 for a finite number of times and
remains greater than 8 for a sufficiently long period then the errors in 6 coordinates also
converge.

SIMULATION 4.2

Consider the following scenario similar to Simulation 4.1. The mass and mass moment

of inertia of the combined wheelchair and occupant is chosen to be

M = 60 kg, J = 2.25 kgm2 (4.67)

The reaction force R and reaction moment T are assumed to be

R = 30 N, T = 5 Nm (4.68)

78

As in Simulation 4.1, we assume the initial values of the parameter estimates in Eqs.(4.6)

and (4.7) to be
43,? (0) = [ 40.0 10.0 ] (4.69)

a; (0) = [ 1.0 0.0 ] (4.70)

in SI units. The controller gains and sliding surface constants were chosen as K1 = 10,
K2 = 10, A] = 10 and 2,2 = 10. The desired trajectories were chosen so as to allow for

changes in sign in ds = sat (d ,dm) and consequently d; as

200 -

100 /\
0

-100
50 I

0 FA.
-50

lm I I I I I
50 . .
0 ~ 4

-50 4 .
10 . .

-10 b . . . 4:

0 [I I 1 m

5 t 1 t t

0 P‘s J“ M

L L 1 l l

0 10 20 30 40 50 60 70
time (sec)

 

 

 

6x (mm)

 

 

 

 

 

 

 

 

eei (deg)

 

 

F (N)

 

 

 

ds (m)
o

 

 

 

 

d's (m)
o

 

 

 

 

Bx
9
M

I

1

 

 

I
P 1 l
l

 

 

BO

 

Figure 4.5. Simulation 4.2: Plots of ex, e9i,,F, d5, (1;, B, and B9

79

S

M (Kg)
8

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0 10 20 30 40 50 60 70
time (see)
Figure 4.6. Simulation 4.2: Plot of parameter estimates
it'd = sin(0. l t) (4.71)
("id = sin(0.2t) (4.72)

along with initial conditions xd = id = 99 = 9d = 0. These trajectories are sufficiently rich

such that W in Eq.(2.31) is persistently exciting.

The simulation results are presented in Figs.4.5 and 4.6. The value of r: was kept same at

0.25 secs and the epsilon was chosen to be 0.1. Figure 4.5 shows the error in x coordinates

8,9, the error in 0 coordinates 89,-, the control input F, the saturated signal (1,, control input

11;, B, and B’9 in that order. It can be seen as in previous simulations that the errors go

to zero. The vertical lines show some of the instances when they remain constant. These

correspond to situations when sgn(d_(.) 74 sgn(ds) for ex,- and F S 8 for e9 i- It should be

noted that sgn(d§) yé sgn(ds) will also affect e9,- indirectly, since F is affected by sgn(d_(.) 79

sgn(ds). Going on to F, which is the applied force, we can see that F is set to zero at certain

80

points. These points correspond to the period of time when d; and d, are not of the same
sign. This is observable from the next two plots. The value of d, has been saturated to a
maximum of 10.0 m. This is obviously an impractical value, however it has been chosen so
as to make the graph more readable and observe the convergence faster. Further the fact that
the trajectory is chosen to be very rich with very high rates of turning and acceleration also
necessitate such a large maximum value. In the succeeding chapters we will be discussing
methods to provide more torque when d gets saturated at a more practical value and in the
final chapter, we will be providing more practical scenarios with experimental results. The
next plot shows 8,, which can be seen to become zero each time d; and d, are not of the
same sign ( when F = 0). Otherwise Bx remains at 1.0. The plot of 3:9 is more varied.
However it has been prevented from going negative when d; and d, are of different sign by
enforcing the F = 0 condition.

Figure 4.6 shows the plots of the estimates. It can be seen, similar to the previous sim-
ulation, that the always tend to go toward the original parameter values when the condition
F > 8 is satisfied. It can also be seen that they are set to zero at certain points. Special
note should be made however that these values are set to zero only when the conditions
IF I g 8 and sgn(d§) = sgn(ds) are satisfied. Otherwise, when the conditions are IF I S 8

and sgn(d§) 96 sgn(ds), they are maintained constant. This is also seen in the graph.

4.5 An Alternate Approach to Dealing with Sign Change in Control

4.5.1 Closed-loop Control of the Prismatic Joint

In the previous sections, we tried to address the problems related to the dynamics asso-
ciated with the actuator of the prismatic joint, by manipulating the control inputs of the
wheelchair. For example setting F = 0 when ti; and d, have opposite signs. This was done
from the wheelchairs perspective, using an arbitrary first order system to represent the dy-

namics of the prismatic joint. From the perspective of the robot pushing the wheelchair, the

81

control inputs for the wheelchair, namely F and d3, can be viewed as the reference inputs.
The robot has a controller that works to meet the reference that is provided to it in terms of
F and ds. In this chapter we are concerning ourselves with only the controller of the pris-
matic joint. We intend to control the robot such that the equilibrium point of the closed loop
system consisting of the wheelchair and the prismatic joint of the robot is asymptotically
stable. The block diagram in Fig.4.7 is the dynamics part of the Fig.4.l that was presented
earlier in the chapter. Since this is the arbitrary controller/actuator dynamics used in Theo-
rem 4.1 and 4.2, the block diagram can be expanded out to look like that in Fig.4.8, where
rm is the time constant of the motor’s approximated first order transfer function and u is the
control input to the motor.We intend to define it at the end of the theorem presented next
to achieve our objective. This theorem can be considered to be the counterpart of Theorem
4.1, which was using a standard position controller. Before we present the theorem we

would like to layout some basics in the next paragraph and introduce a new variable, 89.

 

ds 1

———> ds'
ts+l

 

 

 

 

Figure 4.7. Dynamics of the Prismatic Joint

 

 

 

+ u .
ds—-—>O—-I Controller TmS + 1 - ds

 

 

 

 

 

 

 

 

 

Figure 4.8. Standard Position Control

The model of the wheelchair is unchanged from Eqs.(4.1) and (4.2). The idea behind
this method is to build upon the result obtained in Theorem 3.1. Hence we will be using

the B9 defined as the ratio of the saturated moment to the required moment, in Chapter

82

3, Eq.(3.6) instead of B’9 from Eq.(4.3) which was introduced earlier this chapter. Corre-
spondingly the estimation errors are also defined the same way as in Chapter 3, Eq.(3.8)
and (3.9). Further the definition of the errors, 81 and 82, also remain unchanged from that
given by Eqs.(2.9) and (2.10) respectively. We introduce the error between the commanded

location of the prismatic joint, (1,, and the actual location of the prismatic joint d; given by

ed = d, — d; (4.73)

We also replace the saturation function with a hyperbolic tan function so that the derivative

of d, is continuous and obtainable. We are now ready to present Theorem 4.3.

4.5.2 Main Result

Theorem 4.3: Consider the dynamical system described by Eqs.(4.1) and (4.2). For this
system, the equilibrium point (e1,ez,6x,69,ed) = (0,0,0, 0,0) is asymptotically stable for

the following control law:

F = F, (4.74)

d = _ir(Mr) (4.75)

d, = dmtanh(d) (4.76)

d", = { ds+Kid[—F]ez+ed, [—F]ezed >0 (4.77)
Js+ed, [—F]82ed 30

where F, and M, are given by,

F, = Mid-FAIMe'x-I-R-I-Klel (4.78)
M, = f0d+lgfe9+T+Kze2 (4.79)
and estimation laws:
«ix = -P“Afet, (4.80)
6”,, = —P-‘Age2 (4.81)

83

provided F 75 0.
Proof: Consider the Lyapunov function candidate

1 1 1 - - 1 .. .. 1
V5 = 5M8)? + 518% + 5(1ng + 5930119 + 5K9 e}, (4.82)

which is positive everywhere except at the origin (81 ,ez, 6,, 69,801) = (0,0,0,0,0), where

it is zero and K9 is a positive constant. By differentiating V5 with respect to time, we get

V5 = Mele'l + J82e°2 + 61P$x + égptlig + Kd ed éd (4.33)

We obtain the derivatives e‘l and 8'2 from Eqs.(2.9) and (2.10) as follows

8'] = 83, + 2.1 e, (4.84)
8'2 = 8'9 + 2989 (4.85)

By multiplying Eqs.(4.84) and (4.85) with M and J, respectively, and substituting Eqs.(4.1)

and (4.2), respectively, we get

Mél = Mia—Mf-I-MAIéx
= Mx'g—(F—RHMAIé,
= Mtg—(F,—R)+Mlléx (4.86)

= (id-FA] ex) (M—M) + (R—R) —K1 1.», (4.87)

84

J82 = 10;, — 18 +ng e,

= 109— (—F d; — T) wage,

= 10;,— (—F d,+Fd,-Fd§—T)+JAQég

= 19;,— (—F d, — T) +JAQe'9 + [—F](ds — (1;)

= 1911 — (—F ds — T) +Jfiaéo + I-F] (ed)

= 16,, — (Be M, — T) +171an + [-F I (ed)

= (0,, +1Qe9) (J—B9J) + (T -39 T) -39 Km + l-Fl (ed) (438)
We obtain Eq.(4.87) by substituting for F = F, from the expression of F, from Eq.(4.78).

Similarly we obtain Eq.(4.88) by substituting the expression for B9 from Eq.(3.6) and the
expression for M, from Eq.(4.79). Thus we get,

Me'1=Ax6x—Klel (4.89)
1‘32 = Ae¢~o - 39 K282 + I-F I (88) (4-90)

Substitution of the above equations in Eq.(4.83), yields

. ~ 1.. ~ T .5 .. T 1:. .
V5 = ele¢x +8240¢9 - Klei -Be K283 + ¢x 1%: + ¢9 P¢o + [4’] (ed)82 +Kdeded
(4.91)
Using the estimation law in Eq.(4.80) and (4.81), we get

V5 = --K1 8% — B9 Kzeg + [--F] (ed) 82 + Kd 8989 (4.92)

If we substitute the control law of the prismatic joint in Eq.(4.77) depending on whether

[—F] e2 ed > 0 or not, the derivative will take the form

85

V __ —Klei—B9K2e2—Kde§, [—F]ezed >0
5 —Klel —Bg K2e2 - Kded + [—F] e2 89, {—F1e2ed _<_ 0

g —K1e§ —Bg K2123 —Kde§ (4.93)

Since F 79 0, B9 > 0 and V5 is negative semi definite. Following the footsteps of Theorem
2.1, we can show that the error variables, 81 , e2, 89 and the parameter estimation errors, 6,,
and 69, asymptotically converge to zero.

Remark 4.2: The reason for using a switched controller is to contain d; within the
maximum bounds. Other methods to address the issue are to provide a sufficiently high
gain [(9 depending on the application. Higher values of [(9 will reduce the dependency of
the control law on the value of 82 and F and add weight to the error ed and vice versa.
Reducing the maximum allowed for d, is also a possible method. For situations where

these are not possible we can use the hybrid controller in the above theorem.

Remark 4.3: The control law for d} is obtained from the following definition of u:

Tm(ds+R1;[—F]ez+ed)+d;, [—-F]ezed > 0 (4.94)
Tm(cis+ed)+d§, [—F]e28d SO

A quick comparison between Theorems 4.1 and 4.3, show several benefits with the
new controller. It renders the equilibrium point of the closed loop system, comprising of
the robots’ prismatic joint and wheelchair, asymptotically stable. The necessary condition
for asymptotic stability has been relaxed by the new method. Earlier, it was required that
B’e = (—F d§)/Mr has to be positive. This implies that neither F nor ds can cross over
zero risking in triggering a momentarily negative B’e. This condition has been relaxed
to only F 51$ 0, implying only F cannot cross zero. For several practical purposes, like
for the wheelchair for example, which moves with small, near constant speeds, in one

direction (either forward or backward), this controller provides an adequate solution. Thus

Theorem 4.3 can be applied by itself for many situations, whereas Theorem 4.1 applied

86

by itself imposes the addition restraint of turning in only one direction which is not very
practical. Simulation 4.3 shows the benefits of the new controller. Although these are
significant improvements, the question of what happens when F = 0 still remains. This
will be addressed in Section 4.7. In the next section we will address the winding up nature
of estimates in presence of saturation and dynamics of the actuator.

SIMULATION 4.3

Simulation was performed under the following scenario. The mass and mass moment

of inertia of the combined wheelchair and occupant is chosen to be

M = 60 kg, J = 2.25 kgm2 (4.95)
The reaction force R and reaction moment T are assumed to be
R = 30 N, T = 0 Nm (4.96)
We assume the initial values of the parameter estimates in Eqs.(3.8) and (3.9) to be
if (0) = [ 40.0 10.0 ] (4.97)

433(0) = [ 1.0 0.0 ] ' (4.98)

in SI units. The controller gains and sliding surface constants were chosen as K1 = 10,
K2 = 10, [(4 = 1, 21 = 10 and 22 = 10. The desired trajectories were chosen so as to allow

for changes in sign in ds = dm tanh(d) and consequently d; as

it'd = 1 + sin(0.02t) (4.99)

('9', = sin(0.2t) (4.100)
along with initial conditions xd = xd = 99 = 09 = 0. These trajectories are sufficiently rich
such that W in Eq.(2.31) is persistently exciting.

The simulation criterion are very similar to that of Simulation 4.1 except that a value

of zero has been chosen for T. The value of T was chosen to be zero so that d, may go

87

3m 1 r 1 I I
200
100

 

ex (mm

 

 

 

-100

06
0.4
0.2

 

 

69 (deg)

 

 

-02 .
150

 

 

 

F (N)
8'

 

 

Ur
O

 

8

 

 

d’s (m)
e§§°§9

 

 

 

I
.9

 

 

ds (m)
ebb co
882°28

 

 

 

30 40 50 60
time (sec)

O
n
O
N
0

Figure 4.9. Simulation 4.3: Plot of error variables ex and 89, and control inputs, F, d, and
4

88

 

 

 

 

 

 

 

 

 

 

 

U)
C
111r

1 1 1 1

 

 

 

 

 

 

 

    

. 30
t1me (sec)

Figure 4.10. Simulation 4.3: Plot of parameter estimates

both positive and negative depending on the trajectory. Otherwise with a non zero T, d,
could more or less be restricted to one side to counter its effect. The other difference is
the introduction of the gain Kd = 1. The simulation results are presented in Figs.(4.9) and
(4.10). Fig.(4.9) shows the familiar plots of ex, e9, F, d; and (is, in that order. As always the
errors are seen to converge to zero. The force does not cross over zero. It can be seen that
d, and d; traverse on either side of zero without problems. It is also seen that d; achieves

asymptotic tracking of ds. Fig.(4. 10) shows the estimates going to their actual values.

4.6 Resetting of Estimates to Prevent Windup

In this section we will present the results with the resetting of estimates when the condition
82 = 0, B9 < 1 is satisfied. The analysis in Section 3.3 still holds true since none of the

variables have changed. The additional term in theLyapunov equation, 89, can also be

89

shown to be continuous since there is no change in d, and dg. The next theorem provides

the technique for resetting of estimates in presence of actuator saturation and dynamics.

Theorem 4.4: For the dynamical system described by Eqs.(4.l) and (4.2), the equi-

librium point (81,82,6,,69,ed) = (0,0,0,0,0) is asymptotically stable for the following

control law:

F = F,

d = ‘i(Mr)
Fr

ds = dm tanh(d)

- 1

(15+ F[—F]ez+ed, [—-F]e2ed > 0
, d

ds+ed, [—F]ezed S 0

where F, and M, are given by,

F, = Mfd+llfiéx+R+Kl e1

M,- = féd+AQfe9+T+K2e2

and estimation laws:

(I, = -P—1.4§e1,
(59 = -P_lAg€2

430=Be<50(tj), €2=Oand39<1

provided F 75 0.
Proof: The proof of Theorem 4.3 is still valid.
SIMULATION 4.4

(4.101)
(4.102)

(4.103)

(4.104)

(4.105)

(4.106)

(4. 107)

(4.108)

(4.109)

The conditions for this simulation were chosen as follows. The mass and mass moment

of inertia of the combined wheelchair and occupant is chosen to be

90

M = 60 kg, J = 2.25 ltgm2 (4.110)

The reaction force R and reaction moment T are assumed to be

R=30N, T=5Nm (4.111)
We assume the initial values of the parameter estimates in Eqs.(3.8) and (3.9) to be

63‘ (0) = [ 40.0 10.0 I (4.112)

1133(0) = [ 1.0 0.0] (4.113)

in SI units. The controller gains and sliding surface constants were chosen as K1 = 10,

K2 = 3, [(9 = l, 111 = 10 and Ag = 10. The trajectory is chosen as follows

(4.114)

,, { 1.0 m/sec2 ogtSl secs
xd =

0.0 m/sec2 t > 1 secs

0.0 rads/sec2 0 S t < 20 secs
0,, = 0.5 rads/sec2 20 s t < 21 secs (4.115)
0.0 rads/sec2 t Z 21 sees

and the maximum value for d3, (1,, is chosen as 0.175 m.

The simulation results are presented in Figs.4.11 through 4.14. Figs.4.ll and 4.13 are
the results of the simulation without the new modification and Figs.4.12 and 4.14 show the
results of simulation with the new modification in place. In both cases it can be seen that
the errors converge to zero. However in the second case the convergence is faster around
30 < t < 40 sec as against 60 < t < 70. It can also be seen that the estimates have been

brought back to lower values during the resetting.

4.7 Dealing with Loss of Turning Moment when F = 0

In this section we address complete loss of turning moment due to F = F, = 0. The treat-

ment of this situation is different from that in Theorem 4.2, since we will not forcibly set

91

U!
0

ex (nun)
8§T

 

 

 

 

I
M O
O

)-

 

p—r
O

 

32
CU!

 

 

I
M
1

 

 

 

.L.
c:

 

 

88

 

 

69 (deg)

I
N
O O

 

 

 

F(N)

'8 )6: ..
age-go 88888 8

 

 

 

 

 

 

L ’\/\ ' i
I
SI

 

 

l L 1 I L

20 30 40 50 60 70
time (sec)

Figure 4.11. Simulation 4.4: Plot of error variables, ex, 82 and 89 and control variables F
and d; without modification in estimates

92

 

Hfl
M
O

—(
-1

ex OHHD
c>gggg

 

 

 

1

I~JI

C
P

 

 

 

 

 

 

 

 

 

 

 

 

 

 

I T I I I I

 

 

 

 

No.05 - 4

37,—0.1 —
.11... .1 g _ [1 -
-02 1 1 1 1 1 1
0 10 20 30 40 50 60 70
thne(mu»

 

 

Figure 4.12. Simulation 4.4: Plot of error variables, ex, e2 and e9 and control variables F
and d’, with modification in estimates

93

 

 

 

 

 

 

 

 

 

 

 

-20

I I I I

 

 

 

 

 

 

 

 

 

 

10 20 30 40 50 60 70
time (sec)

Figure 4.13. Simulation 4.4: Plot of parameter estimates without modification

41.5
1

A

M (Kg)
g

I (Kg 1112)

MN)
8.

T (Nm)
8

 

 

 

 

 

 

 

 

 

I I I I

 

 

 

 

 

 

 

 

 

L I 1 l I T

 

10 20 30 40 50 60 70
time (sec)

Figure 4.14. Simulation 4.4: Plot of modification in parameter estimates

94

F = 0 at any time. Hence the only situation F = 0 will be when F, is also zero. Thus
this problem is more closely related to the loss of turning moment without actuator dy-
namics that was discussed in Section 3.4. Please refer to Section 3.4 for more details. All
information in Section 3.4 until start of Theorem 3.3 is pertinent for this case also. In this
section however we will be presenting a new theorem that addresses this case along with
the dynamics of the prismatic joint included. This theorem can be considered to be the
counterpart of Theorem 4.2, which was presented without the closed loop Controller. As
was the case in the previous chapter, we will not include the resetting presented in Theorem
4.4.

Theorem 4.5: Consider the dynamical system described by Eqs.(4.l) and (4.2). For
this system, with estimation errors defined by Eqs.(3.8) and (3.9), the equilibrium point
(121,122,, 6,, 69,190,) = (0,0,0,0,0) is asymptotically stable when |F,| > s and stable other-

wise, for the following choice of control inputs:

F = F, (4.116)
1
d = ---(Mr) (4.117)
F,
d, = sat(d,dm) (4.118)
. l
. d — —F - , —F - >0
ds+ed, [—-F] e21-ed S 0
where F, and M, are given by,
F, = Mid+llMe°x+R+K181 (4.120)
M, = fed,+42fe9,-+T+K2e2, (4.121)
and estimation laws:
('5, = —F‘1A,{el (4.122)
‘ ____ -—1 T ,
‘39 . P A982" W" >8 (4.123)
¢91¢9=0r IFI'ISE

where 8, 8 > 0, is some small number.

95

Proof: Consider the Lyapunov function candidate

V6 = gm? + $1.3, + £61156: + £63 P69 + $423, (4.124)
which is similar to the expression of V5 in Eq.(4.82). The only difference is that the ex-
pression of V6 replaces e2 in the expression of V5 with 321'. As a result we can show thath
satisfies

V6 = —K1e% — B9 kzeg, — a}, (4.125)

when |F,| > 8. Since |F,| > 8 ensures B9 > 0, we have V9 5 0. Following the same logic as
in Theorem 2.1 in Chapter 2, we can show that the equilibrium point (e1 ,ez,-, 6,, 69, ed) 2
(0, 0, 0, 0, 0) is asymptotically stable.

When |F,| S 8, the derivative of the Lyapunov function, through direct differentiation is

given by the expression

V6 = M8181 +Jezé2i + 6xTP$x + Jgrptfg +éd ed (4.126)

which then takes the form

v6 = —K1e3f - e}, g 0 (4.127)

Equation (4.127) is obtained from Eq.(4.126) by substituting 69 = 0, which follows from
Eq.(4.123), and 321' = 0, which follows from Eq.(3.50) and (3.52). From Eq.(4.127) we
claim that the equilibrium point (el , 82,-, 6,, 69 , ed) = (0,0,0,0, 0) is stable.

Remark 4.3: We need to stress here that we do not expect the condition |F,| S 8 to come
into play often. Then the result may be useful because it shows that the system will remain
well behaved in some sense. If the condition |F,| g 8 occurs a finite number of times,
then the equilibrium points will asymptotically converge to zero. However if the condition
|F,| g 8 is satisfied very frequently the switched system can become unstable. The above

theorem does not comment on the properties of the equilibrium for the switched system

96

but only the sub systems subject to the conditions. It should also be noted that the resetting
presented in Theorem 4.4 was not included for clarity and to avoid confusion with the
current resetting to zero.
Remark 4.4: The controller in Theorem 4.5 has some benefits over the controller presented
in Theorem 4.2. Here, there is no redefinition of trajectory in the x coordinates. Thus
Corollary 3.2 is still valid, which would imply that the trajectory tracking in x coordinates
is still accomplished without problems. The condition for asymptotic stability has also
been relaxed to only being |F,| > 8.

SIMULATTON 4.5

The mass and mass moment of inertia of the combined wheelchair and occupant is

chosen to be

M = 60 kg, J = 2.25 kgm2 (4.128)
The reaction force R and reaction moment T are assumed to be
R = 30 N, T = 5 Nm (4.129)
We assume the initial values of the parameter estimates in Eqs.(4.6) and (4.7) to be
(133(0) = [ 40.0 10.0 ] (4.130)

133(0): [ 1.0 0.0] (4.131)

in SI units. The controller gains and sliding surface constants were chosen as K1 = 10,
K2 = 10, K9 = 1, 21 = 10 and 51/2 = 10. The desired trajectories were chosen so as to allow

for changes in sign in ds = (1,, tanh(d) and consequently d; as

it, = sin(0.1t) (4.132)

0,, = sin(0.2t) (4.133)

97

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

gm: :
6‘ o _ q
-50 1 l 1 1
40 1 1 1 1 1 1
a -4
0
3 0 i” 4th—
§ " -
_40 1 1 1 1 1 1
1m 1 1 1 1 1 .' 1
g 0 1— HVI —i
-50 1 1 1 1 1 1
10 1 1 1 1 1 1
A 5 ” "
E 0 M e1,...11......_.__
as -s— —
-10 1 1 1 1 l 1
20 r 1 t t t 1
€- 10 — -
v o ,- 22‘
3 -10 - i 4
-20 m 1 _L 1 _l 1
0 2 60
time (see)

Figure 4.15. Simulation 4.5: Plot of error variables 8, and 89,-, and control inputs, F, d;
and ds

98

i‘ (Nm)

 

Figure 4.16. Simulation 4.5: Plot of parameter estimates

99

 

along with initial conditions xd = id = 9d = 9d = 0. These trajectories are sufficiently rich
such that W in Eq.(2.31) is persistently exciting.

The simulation results for 8 = l are presented in Figs.4.15 and 4.16. Figure 4.15 shows
the error in x coordinates ex, the error in 9 coordinates egi, the control input F, control
input d; and the saturated signal ds in that order. It can be seen as in previous simulations
that the errors go to zero. The region between the vertical lines show the instances when
egi remains constant. These correspond to situations when F g 8 at approx. t = 37 secs
and t = 57 secs. These periods are slightly larger than the ones in simulation 4.2, where
we had used 8 = 0.1. It can also be seen that ds changes sign during these instances. The
value of (is has been saturated to a maximum of 10.0 m. This is obviously an impractical
value, however as it has been already said, it has been chosen so as to make the graph more
readable and observe the convergence faster. Further the fact that the trajectory is chosen
to be very rich with very high rates of turning and acceleration also necessitate such a large
maximum value. In the succeeding chapters we will be discussing methods to provide more
torque when d gets saturated at a more practical value.

Figure 4.16 shows the plots of the estimates. It can be seen, similar to the previous sim-
ulation, that the always tend to go toward the original parameter values when the condition

F > 8 is satisfied. It can also be seen that they are set to zero when IF I g 8.

100

 

CHAPTER 5

Generation of Additional 'lhrning Moment by
Varying the Angle of Push

5.1 Motivation

The results obtained in chapters 3 and 4 are based on a dynamic model of the wheelchair
where the force applied by the robot is perpendicular to the forearm that grasps the
wheelchair - see assumption Al in Chapter 2. This assumption was justified by the fact
that the robot platform is holonomic and it can move sideways to maintain the arm perpen-
dicular to the line joining the wheels of the wheelchair at all times. As a direct consequence
of this assumption, the turning moment applied to the wheelchair is completely dependent
on the product of the control variable d and the pushing force F. If the trajectory requires
a small magnitude of pushing force, the turning moment capability is significantly reduced
since the control variable d has a physical limit. This will cause errors in trajectory tracking
when d reaches its physical limit.

In this chapter we propose to vary the angle of push to provide greater turning mo-
ment when required, for the same level of pushing force. The angle of push, defined by
{3 E {-72:40 g] in Fig.5.], was maintained at zero value for the results presented in chap-
ters 3 and 4 to comply with assumption A1. By removing this assumption and actively

controlling fl at a value other than zero, the additional moment obtained is given by the

101

 

relation —than ,6, where h is the horizontal component of the distance between the point
of application of the force and the point of contact of the wheels with the floor, as shown
in Fig.5.]. The negative sign in the term —than[3 is due to the fact that a negative angle
in B results in a positive turning moment. This is similar to the notation used in Chapter
3, where a positive displacement in d was defined to generate a negative turning moment.
Mth the additional moment obtained by varying the angle of push, the dynamics of the

wheelchair is described by the equation

M5: = F—R (5.1)
16' = —Fd§—than(B)—T (5.2)
1——‘”——-1

    

robot platform

 

 

(a)

Figure 5.1. Varying the angle of push, B, to achieve greater turning moment.

5 .2 Preliminary result

Consider the following choice of [3 '

102

 

d—d;

 

B = arctan (5.3)

Substitution of the above expression for B in Eq.(5.2), results in the wheelchair dynamics

M56 = F—R (5.4)

16 = -—Fd—T (5.5)

which are identical to Eqs.(2.5) and (2.6) in Chapter 2. If we assume IF I aé O, the results
of Theorem 2.1 become applicable. The choice of B in Eq.(5.3) indicates that the angle of
push is only varied when there is discrepancy between the commanded value of d and the
actual value, d;. If we ignore the dynamics of the prismatic joint, for example, the angle
of push will be maintained at zero value till it reaches its physical limit. In the sequel we
present simulations with the additional control provided by the B.

SIMULATION 5.1

The objective of this simulation is to show the effect of actively controlling the angle
of push, B, to counter the effect of saturation and dynamics of the prismatic joint actuator.
The results of the simulations are presented in Figs.5.2 through 5.4. The conditions for the
simulations are as follows: The mass and mass moment of inertia of the wheelchair and

occupant combined are assumed to be

M = 60 kg, J = 2.25 11ng (5.6)

The reaction force and reaction moment are assumed to be
R=30 N, T=5Nm (5.7)
We assume the initial values of the parameter estimates to be

43,3”(0) = [ 40.0 10.0 ] (5.3)

103

«135(0) = [ 1.0 0.0 ] (5.9)

in SI units. The controller gains and sliding surface constants were chosen as K1 = 10,

K2 = 10, A] = 10 and L2 = 10. The desired trajectories were chosen as

 

 

 

 

 

 

 

 

 

 

 

2m I I I I I I I I
A 150L a
Em- .
K
0 50 ..
O .
-50 4 1 1 1 1 1 1 1
8 F I T T I I I I I .1
’3? 6 ‘ ‘
1: L -
v 4 1- _
3 . _
2 . -
o L! l
O 10 20 3O 40 50 60 70 80 9O
time(secs)
Figure 5.2. Simulation 5.l: Plot of error variables ex and eg
it‘d = sin(0. l t) (5.10)
6,, = sin(0.2t) (5.11)

along with initial conditions xd = id = 9d = Gd = 0. These trajectories are sufficiently rich
such that W in Eq.(2.31) is persistently exciting. This will ensure convergence of parameter
estimates to their true values.

Figure 5.2 shows that errors in the x and 9 coordinates converge to zero. Similarly,

Fig.5.3 shows the convergence of the parameter estimates to their original values given in

104

 

 

M (Kg)

 

 

 

 

 

 

1(Kgm2)
N
I ill I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
1
I
I
I

 

 

 

 

<

 

R(N)
U)
o
r
1
1
1
1
1
1
1
1
1
1
1
1
1
1

 

 

 

 

T (Nm)

<

l l l l l I l l

 

 

 

0 10 20 30 4O 50 60 7O 80
time (sec)

Figure 5.3. Simulation 5.1: Plot of parameter estimates

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

time (sec)

Figure 5.4. Simulation 5.]: Plot of F, d5, d; and B

105

 

 

Eqs.(5.6) and (5 .7). The smooth convergence of the error variables and parameter estimates
in Simulation 5.1 is similar to that of Simulation 2.1 in Chapter 2 (see Figs.2.6 and 2.7).
This is not surprising since the choice of B in Eq.(5.3) cancels the effects of saturation and
dynamics in the prismatic joint actuator and reverts the dynamic equations of motion to
those given in Eqs.(5.4) and (5.5) which are identical to Eq.(2.5) and (2.6) in Chapter 2.

Figure 5.4 shows the plots of the variables associated with pushing and turning of the
wheelchair, namely, F, ds, d; and B. It can be seen that the angle of push, B, orients itself
to cancel the effects of saturation and dynamics of the prismatic joint. As F becomes small
and changes sign d and B reach their maximum values and change sign. It can be observed
that the saturation value of d is set at a practical value of d, = dm = 0.15 m. This is different
from simulations in Chapters 3 and 4, see Fig.3.2, for example, where large values were
chosen to allow for the high rates of turning in the trajectory.

Although there are small movements in B throughout the simulation, large changes
occur only after saturation of the prismatic joint. This can be seen at approx. t = 36 sec
and t = 56 sec. This implies that the additional turning moment is not generated unless the
prismatic joint reaches its physical limit. In the next section we redefine B to exploit the

complete potential of this degree of freedom.

5.3 Generation of additional turning moment

We modify the definition of B in Eq.(5.3) to

I
l3 = arctan [Ed—hi] (5.12)

where K4 is some positive constant. Substitution of the above equation in Eq.(S .2), results

in

106

 

Mic'

F—R (5.13)

19' = —K4Fd- T (5.14)

From the above equations, that describe the dynamics of the wheelchair, it can be seen that
the applied moment is —K4 F d. For values of K4 greater than unity, it can be seen that the
applied moment is greater than —F d, which is the moment applied when the angle of push
is maintained at zero value. To investigate the effect of variation in the angel of push, and
in the new choice of beta in particular, we compute Be, which is the ratio of the saturated

moment to the required moment, as follows

 

_—K4Fd _ K4, F750
39— Mr _ {0, F=O (5.15)

If F 75 0, 89 = K4 > O and the result of Theorem 3.1 becomes applicable. This implies that
the additional degree of freedom associated with B can compensate for the negative effects
of saturation and dynamics associated with the prismatic joint. Furthermore, for 89 > 1, a
significant portion of the moment required is provided by varying the angle of push without
relying solely on the prismatic joint displacement d. The case of F = 0 will be addressed
later in the chapter.

Before we present the simulation results, we list the advantages of the choice of B in

Eq.(5.l2).

1. Since 39 = K4, the speed of convergence of the error in the 9 coordinate can be
increased by choosing K4 > 1. This follows from the expression of the derivative of

the Lyapunov function in Eq.(4.93).

2. As already stated above, the turning moment is produced by both the control variables
B and d, even when d is not saturated. This reduces reliance on the prismatic joint

alone and therefore saturation time of d can be reduced or even eliminated.

107

The simulations presented next show the effect of using a gain K4 = 10.

SIMULATION 5 .2

The initial conditions and parameter values for this simulation are identical to those
used in Simulation 5.1. The results of Simulation 5.2 are presented in Figs.5.5 and 5.6 and
compared with the results of Simulation 5.1 in Figs.5.2 and 5.4. A comparison of Figs.5.5
and 5.2 indicates that the speed of convergence of errors in the x coordinate are similar.
However the error in the 9 coordinate converges faster for Simulation 5.2 in comparison to
Simulation 5.1. This can be attributed to the choice of K4 = 10 which results in a tenfold
increase in the effective turning moment. Another important observation can be made by
comparing Figs.5.6 and 5.4. In particular, d remains saturated for a shorter period of time
for Simulation 5.2 in comparison to Simulation 5.1 (at around I = 36 and 56 sec). This is
due to the fact that the turning moment produced by the control variable B augments that
produced by the control variable d. Furthermore, the control variable B is active all the
time, unlike in the previous simulation where it is active only when d is saturated.

Thus far we have assumed that the commanded angle B can be realized instantaneously.
This is clear from Simulations 5.1 and 4.2 (Figs.(5.4) and (5.6)) where B is seen to jump
between +90° and —90°. This will not be feasible in hardware implementation due to

finite bandwidth of actuators. In the next section we address this issue.

5.4 Effect of Robot Dynamics on the Angle of Push

In the previous section it was assumed that the commanded value of B can be realized
instantaneously. In reality, the actual value of B, B’, will be different from the commanded
value of B due to the dynamics of the robot in the lateral direction]. The commanded value

of B in Eq.(5.12) will serve as a reference signal and the controller in the robot will strive

 

'The angle of push, B, is measured by an encoder. In the absence of an actuator, it is a
free joint but it is indirectly controlled by the lateral motion of the robot (y direction) which
is possible because of the holonomic structure of the robot.

108

 

200 I T 1 1 r I T I
150_ .

ex Onnfi
§

 

 

 

 

1'4 I I I I r I T T
L2- -

03_- -
015- 4
04-. a
02 - -

69 (deg)

 

 

 

.O_2 1 1 L 1 r 1 1 1

0 10 20 30 40 50 60 70 80 90
finuaamxs)

 

Figure 5.5. Simulation 5.2: Plot of error variables ex and eg

to reduce the error e I3 , defined as

313 =fi_fi’ (5.16)
to zero. A schematic of the controller is shown in Fig.5.7. A first-order transfer function
is used to model the lateral dynamics of the robot and the time constant of the dynamics is
assumed to be r3. The goal of this section is to design the controller in Fig.5.7 that will
guarantee asymptotic tracking of the reference trajectories in the presence of saturation and
dynamics of the prismatic joint actuator and lateral dynamics of the robot.

If we take the dynamics of the robot in the lateral direction into account, the equations

of motion of the wheelchair can be expressed in the form

109

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

    

 

 

 

A 0.2 l 1 1 1 1 I I I
g 0.1 -
=1: 0

or -

-02 1 1 L r L 1 1 1
a 1m 1 f 1 1 1 l I I
g 50
n 0

-50

.1w 1 1 1 1 1 1 1 1

O 10 20 30 40 50 60 7O 80
time (see)

Figure 5.6. Simulation 5.2: Plot of control inputs, F, d3, d; and B

Robot dynamics
us 1 3'

133+]

 

 

 

Controller

 

 

 

 

 

 

 

 

Figure 5.7. Standard closed loop position control of angle B

110

M56 2 F-R (5.17)

16' = _ng —than(B’) — T (5.18)

The above equations are very similar to Eqs.(5.l) and (5.2) and the difference is that B is
replaced by B’. Using the next theorem we will show that the following choice of nonlinear

controller

I-Fhl(1+tan(l3)tan(fi’))ez
KB (1 +tan2(e,3))

in Fig(S .7) will result in asymptotic tracking of the reference trajectories, provided F 99 0.

 

uB = 15 5+ +tan(e5) +B’ (5.19)

Theorem 5.1: For the system given by qu.(5.17) and (5.18), the equilibrium point

(e1,e2,$x,69,ed,ep) = (0,0,0, 0,0,0) is asymptotically stable for the reference control

inputs for F and d :
F = F, (5.20)
d = —-£,;(Mr), (5.21)
the control law for d:
d; = d“, + ed (5.22)
the reference control input for B:
B = arctan 55517131; (5.23)

the control law for B:

{-Fh](1+tan(fi)tan(l3'))e2

‘3 = [3+ Kfi(l+tan2(ep))

 

+tan(eB) (5.24)

and estimation laws:

«1. = —P"A?£ e1, (5251

111

$9 = —P"A€e2 (5.26)

where F,, M, and (Is are given by,
F, = de+21Mex+R+K1e1 (5.27)
M, = féd+AQfeb+T+K2e2 (5.28)
provided F 7e 0.

Proof: Consider the Lyapunov function candidate

1 1 1- - 1- .. 1 1
v7 = Em? + 512% + 54131041, + 5413mm + 5K, e3, + 5K3 m2 (ep) (5.29)

which is positive everywhere except at the origin (e1,e2,<§x, $9,ed,efl) = (0,0,0, 0,0,0),
where it is zero and K4, K5, are positive constants. By differentiating V7 with respect to

time, we get

V7 = Mele'1+Je2e'2 + 631%; + égpé'g + ed éd + KB tan(ep) secz (e5) éB (5.30)

We obtain the derivatives e‘l and e'2 from Eqs.(2.9) and (2.10) as follows

e'l = éjt+lléx (5.31)
8'2 = éb + 1289 (5.32)

By multiplying Eqs.(5.3l) and (5.32) with M and J, respectively, and substituting
Eqs.(5.l7) and (5.18), respectively, we get

Mél = Mx'g —Mjc'+MAI e,
= my — (F, —R) +1421 2, (5.33)

= (xd+21é,)(M—M) + (R—R) -K1e1 (5.34)

112

Jé2 = 19;, —Jé +JAQé9

= 10;, — (—F d; —Fh tan(B') — T) whee

= 19;, — (—F d; —than(B) +Fh tan(B) —Fh tan(B’) - T) whee

= 19L: - (-K4Fd- T) +132ée + I-Fh](tan(fi) -tan(13'))

= 19L: - (-K4F d - T) +1124) + {-Fh](tan(13) -tan(l3'))

= 19d - (39 Mr - T) +132 89 + H” h] (tan(13) - t2111(13’»

= (éd+3/zé9) (J-Boj) + (T-Be 7) —Be K282

+I—F h] (mum) —tan(B’)) (5.35)

We obtain Eq.(5.34) by substituting for F = F, from the expression of F, from Eq.(5.27).

Similarly we obtain Eq.(5.35) by substituting the expression for 89 from Eq.(5.15) and the
expression for M, from Eq.(5.28). Thus we get,

Me'1=Ax¢7x —K1e1 (5.36)
162 = A9159 — Be K282 + {-Fh] (WW) - ta1103'» (5-37)

Substitution of the above equations in Eq.(5.30), yields

. ~ ~ n. T .1. r. T :. .
V7 = €1Ax¢x+82Ao¢e-K1ei—30K28%+¢x P¢x+¢9 P¢e+€ded

+[—Fh] (tan(B) — tan(B'))e2 +Kfl tan (e3) see2 (8B)éfl (5.33)

Using the estimation law in Eq.(5.25) and (5.26), we get

V7 = —K1e%—Bg K263+Kd ed éd-I- [—Fh] (tan(B) —tan(B’))e2+KB tan(eB) SIBC2 (efi)éfi
(5.39)

Substituting for the control law for the prismatic joint ((11) we get,

113

177 = 414—39512; — 19,123+ [—Fh] (tan(B) —tan(B’))e2 +K,3 tan(e,,) sec2(e5)éfi

 

 

(5.40)
Using the trigonometric identity,
I t2111(13)-tan(l3’)
- = 5.41
““3 B) 1+tan<13>tan<180 ‘ ’
in above equation, we get,
V7 = —K,e% — Be Kzeg — K, e}, + [—Fh] tan(B — 13’) (1 + tan(B) tan(B'))e15.42)
+KB tan (e3) 8602 (eB) 8B (5.43)
Using the trigonometric identity,
sec2(efi) = 1+tan2(efi) (5.44)
we get,
V7 = —K1e% —Bo K24 — Kue§+1—Fh1tan<13 —18') <1+tan<13> tan<13’)>e2
+K5 tan(eB) (1 +tan2(el3))é5 (5.45)
which can be written as
V7 = -K1 e? — Be [(28% — Kde5+tan<es)(1—Fh1(1+tanw) tan<16’>)e2 (546)
+193 (1 +tan2(e,3)) (B — B’) (5.47)
Substituting the control law for 13',
'1 _ - I-Fhl(1+tan(l3)tan(fl’))ez 5 48
fl _fi+ K3(l+tan2(e’3)) +tan(e )7 ( - )

114

in Eq.(5.47), we get

v7 = —K1e%—39K2e§—Kde§- K5 tan2(efl) (5.49)

Since F 79 0, 89 = K4. So V7 is negative semi definite. Following the footsteps of Theorem
2.1, we can show that the error variables, e1, e2, ed, e 5 and the parameter estimation errors,
5x and 69, asymptotically converge to zero.

Remark 5.1: The controller for the prismatic joint in the above theorem was chosen as
4=4+Q am)

This is different from the switched controller used in Theorem 4.4 in Chapter 4 in that
the term containing —F e2 has been excluded. With B compensating for saturation and
dynamics of d, the term containing -F e2 has been included in the controller for the angle
B. The reason for using a switched controller for the prismatic joint in Theorem 4.4 is to
avoid d; exceeding the physical joint limit. In the present theorem, this problem has been
addressed by generating additional turning moment using B and avoiding the tendency of
the prismatic joint to overshoot the maximum limit. Later, we will use a switched controller
for B to impose a limit on the maximum value that it can assume.

SIMULATION 5.3

The initial conditions, parameter values and the controller gains were chosen to be the
same as in Simulation 5.2 with the exception of the trajectory and the value of T. The

trajectory is given by

it'd = 1+sin(0.02t) (5.51)

6,, = sin(0.2t) (5.52)

and the value of T is assumed to be zero. These changes are made to ensure that F does

not cross zero but (I and B do cross zero. Figures 5.8, 5.9 and 5.10 show the results of the

115

 

250 T I I I I
200
150

ex (mm)

 

 

 

 

 

 

69 (deg)

 

 

.O_] 1 1 1 1 1
0 10 20 30 40 50 60
time (sec)

 

Figure 5.8. Simulation 5 .3: Plot of error variables ex and eg

 

 

___-_— __————_— ___ ——Ia—r—

M(Kg)
$383

 

 

 

 

 

 

 

 

(H

 

 

(Kg m2)
-- N

 

 

 

 

 

 

 

 

 

 

. 3O
t1me (sec)

Figure 5.9. Simulation 5.3: Plot of parameter estimates

116

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0.02 . , . . .
go? 0 _ V
B -002: 3

1 I I I I I
<33 __ -1
3 0 _
Era. 1 -
o 10 20 30 40 50 60

time (sec)

Figure 5.10. Simulation 5 .3: Plot of control inputs, F, d_(., and B’

simulation. The errors in Fig.5.8 and the parameter estimates in Fig.5.9 converge to zero
and the original parameter values, respectively. The plots of F, d; and B’ are shown in
Fig.5.]0. The values of d and B are quite small - this can be attributed to the large positive
value of F. In the succeeding sections we will deal with situations like saturation in B and
loss of turning moment due to F becoming zero wherein the plots become complex than

that presented in this simulation.

5.5 Addressing Saturation in Angle of Push

In the previous section, the range of B was assumed to be [— 32‘ , 329] for the sake of simplicity.
In this section we impose a joint limit on B such that —Bm _<_ B S B)... B". < g. The
motivation for imposing the joint limit on B is that when B is close to 325, the robot may
be very close to the wheelchair and may even come in contact with it, depending on the
length of the rear arm. Also, in such a configuration, the forward pushing force (given by
the expression F / cos B) that the robot will have to generate for producing a desired force

on the wheelchair (F = F,) will be very large. Although the limit is externally induced

117

 

through software, it can still be modeled as saturation. Thus we introduce

13s = saw, Bm) (5.53)
11 Bs

Bm»---.

 

-Bm

B
to v

:____ ‘Bm

 

Figure 5.11. Plot of B vs BS

Figure 5.1] shows a plot of B, with respect to B. From this point onwards, B,- will be the
reference signal for the robot and B; will be the actual angle of push. The equations of

motion can therefore be modified to

M56 = F—R (5.54)
10' = —F(d;+htan,6,’)—r (5.55)

To address the control problem in the presence of B saturation we redefine some of the

variables as described below.

1. Equation (5.12) provides our earlier choice of B. It is now chosen as

M (5.56)

=tan
Barc h

In Eq.(5.12), B was a function of d and dg. In the above choice, it is a function of d
and d; which implies that the angle of push only compensates for the saturation but

not for the dynamics of the prismatic joint.

118

 

2. In Eq.(3.6) in Chapter 3, 39 was defined as the ratio of the saturated moment to the
required moment. When the angle of push is not identically zero, the definition of

Be is modified to the form

_ —Fds — Fh tan(B_,-)
_ Mr

 

39 (5.57)

In the sequel we will establish thath > 0 provided F 75 0. Recall, a positive value
of Be is required for asymptotic stability.

3. The error between the commanded and the actual angle of push is modified to the

form

es = 18.. — 13; (5581

Assuming F 75 O, Eq.5.57 can be written as

—Fds-thanBs
—Fd
d, htanBs

= 7 + d (5.59)

 

The first term on the right hand side of the above equation is positive since d and d; are the
same sign. Using the fact that B and B, have the same sign and Eq.(5.56), we investigate

the sign of the second term as follows

 

thanBs _ 'thanB
sgnI— ] — sea F. I
= sg11-————K4dd_ds]
r d,-
= sgn K4—g] (5-60)
1

 

Clearly, the second term on the right hand side of Eq.(5.59) is positive or zero provided

K4 2 1. This implies thath > 0 if K4 2 ]. We now present a control design to provide

119

asymptotic tracking of wheelchair reference trajectories in the presence of saturation and
dynamics of the prismatic joint and angle of push.

Theorem 5.2: Consider the dynamical system described by Eqs.(5.54) and (5.55). For
this system, the equilibrium point (e1 ,e2, (15x, 69,ed,el3) = (0,0,0, 0,0,0) is asymptotically

stable for the reference control inputs, F and d :

F = F, (5.61)
l
d = ——(M,), (5.62)
Fr
the control law for d:
. 1
. d — —F , —F 0
d; = s+Kdl 162+ed [ 1828.: > (5.63)
ds+ed, [—F]e2ed 50
the reference control input for B:
[3 = arctan 5:425:45 (5.64)
the control law for B:
B; =

 

B. + ”"1“ “memWez +tan(ep). 1411104108.) -tan<13.:»e2 > 0

KB (1 +tan2(efl))
3s +tan(ep), [‘F 17] (tan(13s) —tan(B§)) 92 S 0
(5.65)
and estimation laws:
615'. = —P“Ai e1, (5.66)
$9 = -P“A£ez (5.67)

120

where F,, M,, d,- and B5 are given by,

F, = Mid+llMéx+R+Kle1 (5.68)
M, = f9d+hfeb+T+K2e2 (5.69)
d, = dmtanh(d) (5.70)
B, = Bmtanh(B) (5.71)

provided F aé 0.
Proof: Consider the Lyapunov function candidate

1 1 1 - .. r .. .. 1 1
V3 _—_ 5M6} + 51.3 + 5111me + 51113101119 + EK" e}, + 5K5 tan2 (eg) (5.72)

which is positive everywhere except at the origin (e1,e2,$x,$9,ed,efi) = (0,0,0,0,0,0),
where it is zero and Kd, K5, are positive constants. By differentiating V8 with respect to

time, we get

V8 = Mele'l +Je2e'2 + 4595, + 1531069 +Kd ed ed +Kp tan (eg) see2 (e5) .2, (5.73)

We obtain the derivatives e'l and e'2 from Eqs.(2.9) and (2.10) as follows

e'l =ég+lléx (5.74)

e‘z =éb+lgeb (5.75)
By multiplying Eqs.(5.74) and (5.75) with M and J, respectively, and substituting
Eqs.(5.54) and (5.55), respectively, we get

Mél = Mx}, —Mx'+Mll e,
= Mr}, — (F -—R) +M/‘Ll e,

= (xd+21é,)(M-M) + (R—R) —K1 e1 (5.77)

121

 

Jéz = Jag—Jéwageg
= Jé,-(-Fd;—than(13,’)—T)HAM,
= 10;,—(—Fd,—than(13,)+Fd,+than(13,)—Fd;—than(13,’)—T)+J)12é9
= 198-(-K4Fd-T)+chée+I-Fhl(tan(l3s)-tan(fis’))+[-F1(ds—d§)
= Jéa-(-K4Fd-T)+J/‘2éo+[—Fh] (tan(18.)—tan(13;))+1—F1 (e1) ‘
= 194 - (30 M, - T) +142 80 + l—Fh] (“HUM - tan(f5§))+[-1"‘](6’d)
= (éd+42é0) (J-Bej) +(T-BOT)-BeK282+[-Fh1(tan(l3s) -tan(l3§))

+[—-F] (ed) (5.78) 1 '

 

We obtain Eq.(5.77) by substituting for F = F, from the expression of F, from Eq.(5.]16).
Similarly we obtain Eq.(5.78) by substituting the expression for 89 from Eq.(5.57) and the
expression for M, from Eq.(5.117). Thus we get,

Me°1=Axox -K1e1 (5.79)
J 6'2 = 40156 '- 30 K282 + [‘1’ h] (tan(Bs) - ta1103.9) + [*F] (ed) (5-30)

Substitution of the above equations in Eq.(5.73), yields

. .. . .... .:. .. T .1. '
V8 = 81 Ax¢x + €240¢0 - Krei -Be K283 + ¢xTP¢x + 4’0 P¢0 +Kd ed ed
+I-Fh] (tan(l3s) - tan(B§))e2

+KB tan (8p) 8602 (efl)éfi + [—F] (64) 82 (5.81)

Using the estimation law in Eq.(5.113) and (5.114), we get

V8 = -K1€i-30K2€%+Kdedéd+l—Fh](tan(fis)-tan(l3s'))82

+193 tan(e,,) see2 (eye, + {—F] (ed)e2 (5.82)

122

The rest of the proof will proceed according to the conditions whether [—F] (ed) e2 > 0 or
not and whether [—F It] (tan(Bs) — tan(B§)) e2 > O or not. In order to simplify things we

propose breaking down of the Lyapunov derivative, V3 into three parts, namely

V3 = V30 + V31, + V3,: (5.83)
V80 = —K1€%-Bg [(28% S 0 (5.84)
V86 = Kd ed éd + {-17} (ed) 82 (5-85)

V8c = {—Fh] (mums) - t3111031))6’2 +Kp tanks) 8602_(e)3)ép (5.86)

Since F 9’: 0, 39 > O and V3,, 3 0. We next analyze 17%. Substituting the control law for

the prismatic joint in ng we can get,

V86 = Kd ed éd + {—F] (ed)ez (5-87)
_ 2 _
= Kde , [ F]e2ed >0 (5.88)
"Kd 8,; +1‘F1(ed)321 {—Flezed S 0

3 —K4 e5 (5.89)

Next lets consider the case when [—F h] (tan(Bs) — tan(B§)) e2 > O in the definition of Vgc,

Using the trigonometric identity,

 

tanws — [3,!) = 3:513: 13: ; :15? 3;) (5.90)
in V3,, we get,
V8c = [‘F h] tan(l3s " fig) (1 +tan(Bs) tan(13s')) 32
+KB tan(e,3) 8602 (efi)éfl (5.91)

Using the trigonometric identity,

sec2(efl) = 1 +tan2(efi) (5.92)

123

 

we get,

V8c = {-Fh]tan(l3s-I3§)(1+tan(13s)tan(fis'))62

“‘13 tan(e,3) (1 +tan2(ep))éB (5.93)

which can be written as

V... = tan(ep)(I-Fh](1+tan(fis) wanes

+193 (1+ran2(es))(Bs—B;) (5.94)

Substituting the control law for B for this case given by,

-r_ - [—Fh](1+tan(Bs)tan(B§))e2
fis fis‘l' Kfl(1+tan2(el3) +

 

tanep,) (5.95)

in Eq.(6.93), we get

v3, = —K5 tan2(e,3) (5.96)

Next lets consider the case when [—F h] (tan(Bs) — tan(B§)) e2 5 O in the definition of V3,.

Substituting for the corresponding control law of B we can obtain

Vsc = {-Fh](tan(Bs)-tan(fi§))e2—Kp tanzte31sec21eg) (5.97)

Thus Vgc can be written as

V80 = Kd ed éd + {—F] (ed) 82 (5.98)
{ 4% ta112(88), {—Fh1(tan(l3s) -tan(l3s'))ez > 0

= -Fh (mums) - t2W 1))82 (5-99)
4% tanz(€13) 8602071): {-Fh1(tan(fis) — t3111031))9'2 S 0

g 41,, tan2(eB) (5.100)

124

 

 

Thus assembling all the parts we get

V3 3 —K1e%—39K2e§— 1rd,; -Kfl tan2(efl) g o (5.101)

Following the footsteps of Theorem 2.1, we can show that the error variables, e1, e2, ed,
e B and the parameter estimation errors, 5, and (139, asymptotically converge to zero.
SIMULATION 5.4
Simulation 5.4 is performed to verify the results of Theorem 5.2. The conditions for
the simulation are as follows: The mass and mass moment of inertia of the wheelchair and

occupant combined are assumed to be

M = 60 kg, J = 2.25 kgrn2 (5.102)
The reaction force and reaction moment are assumed to be
R=30 N, T=5 Nm (5.103)
We assume the initial values of the parameter estimates to be
“1(0) = [ 40.0 10.0 ] (5.104)

(33" (0) = [ 1.0 0.0 ] (5.105)

in SI units. The controller gains and sliding surface constants were chosen as K1 = 10,

K2 = 10, 3.1 = 10 and IL; = 10. The desired trajectories were chosen as

2
. < <
55d = lOm/sec2 O_t_1 secs (5.106)
0.0 m/sec t > 1 secs
6,, = sin(0.2t) (5.107)

along with initial conditions xd = 1,, = 9d = 9d = 0. It was noted in Simulation 5.3 that the
values of d and B were very small. This was attributed to the large values of F. Therefore,

in order to be able to observe the effect of saturation, the trajectory for x has been toned

125

 

down to a constant velocity motion. This is observable from Eq.(5.106). According to this
equation the wheelchair accelerates at the rate of 1.0 m/s2 for 1.0 sec and then follows a
constant velocity forward motion. The saturation values of d and B have been set at 0.15
m and 25°, respectively, which is observable in Fig.5 .13. Further the value of K4 has been
chosen to be unity. This is done so that the effect of saturation is more observable in the
plots. As a result of this, in Fig.5.l3, it can be seen that B, is activated only when d is
saturated and not otherwise. This follows from the discussion in Section 5.2, where it is
noted that, K4 = 1, implies that there is no simultaneous application of moment by use of
the additional degree of freedom, B. The angle of push is used to produce moment only
when d is saturated. In this simulation, B does not directly depend on d;, which is the
actual position of the prismatic joint, but rather depends on ds. This is unlike Simulation
5.] where B was used to cancel out the effects of the saturation as well as the dynamics
of the prismatic joint. The saturation of B will be more pronounced when we consider the
case when F = 0. Figure 5.12 shows the errors. ex and eg, converging to zero and the
estimates, f, R, and T, converging to the original values. 117! does not converge to the actual

value of M due to unavailability of PE. in the x trajectory.

5.6 Resetting of Estimates to Prevent Windup

In this section we will present the results with the resetting of estimates when the condition
e2 = 0, 39 < l is satisfied. The analysis in Sections 3.3 and 4.6 still hold true since none
of the relevant variables have changed. The additional term in the Lyapunov equation, e3,
can also be shown to be continuous since there is no change in B5 and 13;. The next theorem
provides the technique for resetting of estimates in presence of B saturation and dynamics.

Theorem 5.3: Consider the dynamical system described by Eqs.(5.54) and (5.55). For
this system, the equilibrium point (e; , e2, (15,, 69,ed,ep) = (0, 0,0,0,0,0) is asymptotically

stable for the reference control inputs, F and d :

126

 

15 1 1 I I 7

 

 

ex (mm)
o ‘e’;
2
l I

 

 

-50 1 1 1 1 41

 

 

20 _

 

 

69 (deg)

 

 

411- -

 

 

M (K8)

 

 

 

 

(Kg m2)
U)

 

 

 

.I

<

 

 

I I I I
l

R (N)
8

<

 

 

 

 

 

 

 

 

T (Nm)

<

 

0 10 20 30 40 50 60
time (sec)

Figure 5.12. Simulation 5 .4: Plot of error variables ex and eg and parameter estimates

127

 

F (N)
8

ds(m)
. g o

 

 

 

 

 

 

 

d's (m)
E

 

 

 

I3 (deg)
888

 

 

 

 

 

 

fl'(deg)
888

 

 

O 10 20 30 4O 50 60
time (see)

Figure 5.13. Simulation 5.4: Plots of F, (is, dg, B and B’

F = F, (5.108)
1
d = 7,014,), (5.109)

the control law for d:

d; : 4s+fig[-F]ez+edi l-F1826d>0
ds+edi [—F]e2ed $0
the reference control input for B:

(5.110)

(5.111)

the control law for B:

B + I-Fhl(1+tan(l3s)tan(fié))e

f' K, (1 +tan2(efl))

1’3 Hanks), I—Fh](tan(18s) - tan(l3§))e2 S 0
(5.112)

2 +un(e,,), {—1721 (mass) - tan(fls’))e2 > o

128

 

and estimation laws:

47x = —P“Afe1. (5.113)
$9 = 4445122 (5.114)
$9=Be¢ie(tj), e2=0andB9 <1 (5.115)

where F,, M,. d, and B3 are given by,

F, = de+théx+R+Kle1 (5.116)
M, = féd+hgfeb+T+K2e2 (5.117)
d, = dmtanh(d) (5.118)
13, = pmtanhui) (5.119)

provided F 76 0.
Proof: The proof of Theorem 5.2 is still valid.
SIMULATION 5.5
The conditions for this simulation were chosen as follows. The mass and mass moment

of inertia of the combined wheelchair and occupant is chosen to be

M = 60 kg, J = 2.25 kgnu2 (5.120)

The reaction force R and reaction moment T are assumed to be
R=30 N, T=5 Nm (5.121)

We assume the initial values of the parameter estimates in Eqs.(3.8) and (3.9) to be

(13,7 (0)=[40.0 10.0] (5.122)
«173(0): [ 1.0 0.0] (5.123)

129

 

in SI units. The controller gains and sliding surface constants were chosen as K1 = 10,

K2 = 3, K4 = 1, K4 =1, Kfl = 111 =10 and M = 10. The trajectory is chosen as follows

1.0 m/sec2 O < 1 <1 secs
" = - - 5.124
xd { 0.0 m/sec2 t > 1 secs ( )
0.0 rads/sec2 0 s t < 20 secs
0', = 0.5 runs/sec2 20 g t < 21 secs (5.125)

0.0 rads/sec2 t Z 21 secs

 

 

100» _
0 r\ -

.1m 1 1 1 1 1 1 1 1

 

 

 

 

 

 

 

J (Kg 11121101 (Kg) 96 (deg) Cx (mm)
.5

 

 

A

 

 

 

 

 

 

 

 

A 30 r- .1
E 20 , 4
(m 10 1 1 1 1 1 1 g 1
’g 100. .
\Z/ 0 /\/-\ d
<[—0 .1m 1 1 1 1 1 1 r 1
0 10 20 30 40 50 60 70 80 90
time (see)

Figure 5.14. Simulation 5.5: Plot of error variables, ex and eg and estimates without
modification in estimates

and the maximum value for ds, dm is chosen as 0.10 m.

The simulation results are presented in Figs.5.14 through 5.17. Figs.5.l4 and 5.16 are
the results of the simulation without the new modification and Figs.5.15 and 5.17 show the
results of simulation with the new modification in place. In both cases it can be seen that

the errors converge to zero. However in the second case the convergence is faster around

130

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

E 1m N -
0 -
3.1“) 1 1 1 1 1 1 1 4

GB 50 1 1 1 1 1 1 1 1

Q

-o 0 |~ e _
‘a’: [\l

0 -50 1 1 1 1 1 1 1 1

E0 41 _ 1 1 If 1 1 1 1 1 _
<2 40 1 1 1 1 1 1 1 1 "
NE 1m _ 1 t/\1 r 1 1 1 1 —

OD .. ..
a O 1 1 1 1 1 L
(H

2 30 __ 1 1 1 1 1 1 1 1 _
(m 10 1 4 1 1 1 1 1 1 -
E 1m __ 1 1 1 1 1 1 1 1 _
E ' /1 ‘
(I— 0 . . 1 1 1 1 .
0 10 20 30 60 70 80

, 50
tune(8ec)

90

Figure 5.15. Simulation 5.5: Plot of error variables, ex and eg and estimates with modifi-

cation in estimates

100

80
goo
u.

40

20

(102
0
4102
4106
-OJ

d%(nfl

-5
-10
-15
-20
-25

3'8 (deg)

 

 

 

 

 

 

 

 

 

 

 

 

 

J
I I I I I I l I
10 20 30 40. 50 60 70 80 90
tune(mx9

Figure 5.16. Simulation 5.5: Plot of control variables F and d; and B; without modification

13]

 

 

F<N)
Beseé‘

 

 

 

 

4102

d's (m)

l l l I l

 

 

 

B's(deg)
'8

 

 

 

-25 1 1 1 1 1 1 1 1
O 10 20 30 40 50 60 7O 80 90

time (sec)

 

Figure 5.17. Simulation 5.5: Plot of control variables F and d; and B; with modification
in parameter estimates

132

 

30 < t < 40 sec as against 60 < t < 70. It can also be seen that the estimates have been

brought back to lower values during the resetting.

5.7 Addressing Zero Force condition

Until this point in this chapter, it has been assumed that F 91$ 0 and all the above theorems
are valid only when F aé 0. Since F = 0 implies that there is a complete loss in control
effort, we only expect to show some kind of a well behaved nature of the system during
the period that this situation occurs. This issue was also faced and addressed in Chapters 3
and 4. We intend to address this issue here in a similar fashion as in the previous chapters.
Please refer to Section 3.3 and Section 4.7 for further information on setting up of the basics
for addressing this issue. The theorem presented next can be considered the counterpart of
Theorem 4.4 with the included B saturation and dynamics.

Theorem 5.4: Consider the dynamical system described by Eqs.(5.54) and (5.55). For
this system, the equilibrium point (el 18211511, 69 , ed, e5) = (0, 0,0,0,0, 0) is asymptotically

stable when |F,| > 8 and stable otherwise for the reference control inputs, F and d:

F = F, (5.126)

1
d = —E(M,), (5.127)

the control law for d:

- 1
ds + K—[—F]82i +ed, [—F]e2,-ed > 0
d

d'; = . (5.128)
ds+ed, [—F]e2,-ed $0
the reference control input for B:
5 = mun—[(41:11, (5.129)
the control law for B:
13.! =

133

 

 

 

[3‘s + [—Fh](l +tan(Bs)tan(B£))ezi

 

+tan<ep), I-Fh](tan(18s) — masons.- > 0

KB (1 +tan2(efi))
9. wanes), I—Fh] (mums) — tan< mes.- s o
(5.130)
and estimation laws:
(5,, = 47143,}, (5.131)
" __ _ —l T ,
¢0, 4’0 = 0, lFrl S 8
where 8, 8 > 0 is some small number, F,, M,, d; and B5 are given by,
F, = de+21Mex+R+K1 c, (5.133)
M, = 104-+18%,” +K2e2, (5.134)
d3 = dm tanh(d) (5.135)
B. = fimtanhw) (5136)
Proof: Consider the Lyapunov function candidate
V —1Me2 +-1-Jez + 167F115, + 16TH); + 1e2 + 1K tan2 (8 ) (5 137)
9—212212x 2092d213 )3 '

which is similar to the expression of V3 in Eq.(5.72). The only difference is that the ex-
pression of V9 replaces e2 in the expression of V3 with egg. As a result we can show that V9
satisfies

V9 = -K1e% 439 K2133, — Kde¢21_ K5 tan2(e,,) (5.138)
when |F,| > 8. Since |F,| > 8 ensures 39 > 0, we have V9 3 0. Following the
same logic as in Theorem 2.1 in Chapter 2, we can show that the equilibrium point
(e1,e2,,o,,69,ed,efi) = (0,0,0,0, 0, 0) is asymptotically stable.

When |F,| g 8, the derivative of the Lyapunov function, through direct differentiation is

given by the expression

134

 

 

 

V9 = Melé1+Jezé2i+ Iii-xTPISx + 69TP$9 +Kd éd ed +KB tan (8B) StiC2 (efi)ép (5.139)

which then takes the form

V9 = —K1e§-efi—Kg tan2(eB) g 0 (5.140)

Equation (5.140) is obtained from Eq.(5.139) by substituting $9 = 0, which follows from
Eq.(6.139), and 852; = O, which follows from Eq.(3.50) and (3.52). From Eq.(5.140) we
claim that the equilibrium point (121,122,, (5,, 661311163) = (0,0,0,0, 0,0) is stable.
SIMULATION 5 .6
The initial conditions, parameter values, controller gains for this simulation are chosen
similar to the previous simulation except for the following: a trajectory which requires F,

to go through zero is used, namely

1', = sin(0.lt) (5.141)

0,, = sin(0.2t) (5.142)

The value of T has been changed back to 5.0 Nm. The saturation values for d and B were
chosen to be 0.15 m and 75° respectively. These changes are made to make the simulation
more practical. The results of Simulation 5.5 are presented in Figs.5.l8, 5 .19 and 5.20. The
switching between the two control laws is observable as F < 8 = 5, approx around 1‘ = 36
sees and t = 56 secs. This region is marked by the vertical lines in all the three figures.
The error in 9 coordinate, 361', reacts to the condition F < 8 by staying constant, which can
be seen in Fig.5.l8. The estimates, f and T, react to the condition F < 8 by going to zero
and staying constant during that period, which can be seen in Fig.5.l9. Finally, the control
inputs, d; and 13;, can be seen changing sign as F changes sign at around these times in

Fig.5.20.

135

 

 

 

200 . , , . , , , ,
150- _
100- _
50 _

 

ex (mm)

 

 

-50 l l 1 l 1 J I I

 

 

eei (deg)
8

 

 

 

 

 

_30 I l 1 I

0 10 20 30 4O 50 60 70 80 90
time (secs)

Figure 5.18. Simulation 5.6: Plot of error variables 8, and eg

 

 

 

 

 

 

 

 

 

 

 

-20 l l l I L 1 l l

 

 

 

 

 

 

 

 

 

T (Nm)
5

 

<

 

 

time (sec)

Figure 5.19. Simulation 5.6: Plot of parameter estimates

136

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0 10 20 30 4O 50 60 70 80 90
time (sec)

Figure 5.20. Simulation 5.6: Plot of control inputs, F, B’ and actuator dynamics d;

137

 

 

41'

CHAPTER 6

Addressing the Limitations on the Applied Force

6.] Introduction

In previous chapters, it has assumed that the required force prescribed by the adaptive
algorithm, F,, is always available for the wheelchair. This is reflective of Assumption A1, in
Chapter 2. However in reality, this cannot always be true. There is a limit on the maximum
force that can be conveniently exerted by the robot on the wheelchair. Furthermore, F,,
which is continuously varying, is a reference command for the robot. The robot by means
of its controller tries to achieve this force. Thus there could be an error between the desired
force and the actual force applied. This effect of this error on the entire system also needs
to be addressed.

In the next section we address the issue of saturation of the applied force. We introduce
a new variable, Fs, which denotes the saturation of the applied force and show asymptotic
stability of the equilibrium point of the system with F} as the force input. In later sections

we address the dynamics of the controller and the robot in producing this force.

6.2 Saturation of Applied Force

Since we are imposing a maximum limit on the applied force, we have to bring this into
account in the dynamic equations of motion of the wheelchair. Thus the equations of

motion of the wheelchair are given by

138

 

J6

—Isd;—Fihtan(fis’) —T (6.2)

Although the above equations are similar to the equations of motion defined in Eq.(5.]7)
and (5.17) of Chapter 5, it can be seen from the above equations that the applied moment
and the applied force are different here and use F} instead of F as the force. The represen-
tation of 39, which is the ratio of saturated moment to the required moment, as given in

Eq.(5.57) is no longer valid since F needs to be now replaced with F}. Thus

: _‘Fs d3 - Frgh tan(fis)

36 Mr

 

(6.3)

Further since the applied force is also not equal to the required force, in order to employ
a similar strategy as d and B saturation, we introduce variable Bx which is defined as the

ratio of the saturated force to the required force. Thus

_Fs

B—
xF,

(6.4)

and correspondingly, similar to the definition of the 6 coordinate estimation errors, f and
T, in Eq.(3.9), we redefine the estimation errors in the x coordinates, M and R, given in

Eq.(3.8), as follows

6.41.4.1». ¢x=l:1:] 13,432] (6.5)

The definition of the errors 81 and 82 remain unchanged from that given by Eqs.(2.9) and
(2.10). We are now ready to present the first result of this section.

Theorem 6.1: Consider the dynamical system described by Eqs.(6.1) and (6.2). For

this system, the equilibrium point (e1, 82, (5,, (13981, 813) = (0,0,0,0,0,0) is asymptotically

stable for the following reference control inputs Fs and d:

139

 

 

 

F, = F,,,tanh(F,) (6.6)

 

1
the control law for d:
. 1
d: = d's + E;[—Fs] 82 +ed, [—&1 828,1 > 0 (6.8)
(1; + ed, {-Rg] e2 ed S 0
(6.9)
the reference control input for B:
B = arctan K44139- (6.10)
the control law for B:
3.! =
' [—th](1+tan(Bs)tan(B;))e2 I
+ + tan , -F h tan — tan > O
18. K, “WW (4,) 1 .1< (13.) (mm
13s +tan(ep), {-Fsh] (tanws) - titm(l3s{))ez S 0
(6.11)
and estimation laws:
J5, = —P—‘AIe1, (6.12)
('59 = —F—‘Age2 (6.13)
provided F} 71 0, where F,, M,, d; and B,- are given by,
F, = de+21Mex+R+K1e1 (6.14)
M, = féd+hqfeg+T+K2e2 (6.15)
d3 = dm tanh(d) (6.16)
[is = fimtanh(fi) (6.17)

140

 

Proof: Consider the Lyapunov function candidate

1 1 1.. .. 1~ - 1 1
vlo = Emi- + 563 + 5413106,, + 5(1); P¢9 + 519,521+ 5K1, tan2 (e5) (6.18)

which is positive everywhere except at the origin (e1,e2,43x, $9,ed,efi) = (0,0,0,0,0,0),
where it is zero and K4, K5’ are positive constants. By differentiating V10 with respect to

time, we get

V10 = Mele'1+Je2e'2 + $3P$x+ digpé'g +Kd edéd +Kp tan(e,,) secz (efi)éfl (6.19)

We obtain the derivatives e'1 and e‘2 from Eqs.(2.9) and (2.10) as follows

e’1 = é}, + Ale, (6.20)
e'2 = e"9 + Age'g (6.21)

By multiplying Eqs.(6.20) and (6.21) with M and J, respectively, and substituting Eqs.(6.1)
and (6.2), respectively, we get

Mél = an —M)'c'+Mll ex
= Mx}, — (F, —R)+M/'L1(éJr

= (id + 21 ex) (M — 3x11?) + (R —BxR) — BxK1 e1 (6.23)

141

 

 

 

Jéz = 10'}, —Jé +122 ég
= 19;: — H} d; ‘Fsh ammo — T) +112éa
= 16;: — (—Fs ds —Fsh tanws) +6 «I.
+Fsh tanws) — F3 d; - m tan()3§) — T) +112 ée
= 19;, — (-—K4F} d - T) +112 ée + H} h](tan()6s) - tan(fis’)) + {-Fs] (ds — dé)
= J 9L1 - (-K4 Fs d - T) +132 ée + {-Fsh] (tan(l3s) - tan(B§)) + {-Fs] (ed)
= 15d - (39 M, - T) +132 £9 + {-Fs h] (mums) - ta“(13.0) + {—17,} (64)
= (éd +12%) (JrBej) + (T -Be T) -39 K282+ l-Fsh] (mums) 4311030)

+[—Fs] (ea) (6.24)

 

We obtain Eq.(6.23) by substituting for Bx and from the expression of F, from Eq.(6.14).
Similarly we obtain Eq.(6.24) by substituting the expression for 39 from Eq.(6.3) and the
expression for M, from Eq.(6.15). Thus we get,

Me‘l =Ax¢3x —BxK1e1 (6.25)
162 = A959 — Bo K262 + {-Fs ’4 0311033) - tan(13s')) + {—Fs] (ed) (626)

Substitution of the above equations in Eq.(6.19), yields

V10 = 81 Axéx + 8214953 —BxKle% - Be [(26% + 6xTP$x + ¢~9TP$9 + Kd ed éd
+[-Fs h] (tan(l3s) - ta110355)) (32 + Kp tan (313) 8602 (8(3) ép + {-Ec] (ed) e2

(6.27)

Using the estimation law in Eq.(6.12) and (6.13), we get

142

V10 = -BxK1 6% -Be K263 +Kd ed éd + [—F,h] (tan(l3s) - 1311039) 62
+16,3 tan (e5) .66:2 (efi)éfl + {—12,} (ed)e2 (6.28)
The rest of the proof will proceed according to the conditions whether {-17,} (ed) e2 > O or

not and whether [-Fs h] (tan(Bs) — tan(fi§)) e2 > O or not. In order to simplify things we

propose breaking down of the Lyapunov derivative, V10 into three parts, namely

V10 = V100 + VlOb + VlOc (6.29)
V100 = —Bx K16? - Be [(28% S 0 (6.30)
VlOb = Kd ed éd + {—Rg] (ed) e2 (6.31)

V100 = {—Fsh] (tan(fis) —tan(Bs’))e2 +Kfl tan(e’3) $662 (ep)é5 (6.32)

Since E, 76 0, 39 > 0. We also know thath > 0, thus V10“ 3 0. (We next analyze V10),

Substituting the control law for the prismatic joint in V106 we can get,

VlOb = Kd ed éd + {—175} (ed) 62 (6.33)

: _Kde21 {—EY] 82 ed > O (634)
-Kded+1—61<ed)e2. {—61% s o

s “K6163 (6.35)

Next lets consider the case when {—15} h] (tan([33) — tan(B§)) e2 > 0 in the definition of V100,

Using the trigonometric identity,

I _ tan(l3s)-tan(l3§)
mums-135)- 1 “311083) tan(l3s’) (6.36)

in V100 we get,

V10c = l-Fsh] tan([3,-B§)(l+tan(fis) t2111(l3s'))<?2

+Kfi tan (e5) 8602 (efi)éfl (6.37)

143

 

 

Using the trigonometric identity,
sec2(ep) = 1+ tan2(efl)

. we get,

V10c = {-Fshltan(l3s-fi§)(1+tan(fis) t2111014))62

+Kfi tan(ep) (1 +tan2(e]3)) éfl

which can be written as

Vloc = t2111(1313)([-Fsh](1'l't~'1m()3s) t2111(l3§))€2

+KB (1+ tan2(ep)) (BS _ 3;)

Substituting the control law for B for this case given by,

~1_ - [-Fsh](1+tan(]33)tan([3§))e2
3345+ Kfi(l+tan2(ep))

in Eq.(6.93), we get

 

+taneB,

V10c = 493 ta“203(3)

(6.38)

(6.39)

(6.40)

(6.41)

(6.42)

Next lets consider the case when [—Fs h] (tan(]33) — tan([3§)) e2 3 O in the definition of V100

Substituting for the corresponding control law of [3 we can obtain

V16. = {—6121 (mums) —tan<13.’))e2 — K6 tan2 (e6) .662 (e6)

Thus Vloc can be written as

144

(6.43)

V10c = Kdedéd+[-F3](ed)ez

—K5 ta“12(88): [-Fs h] (MCBs) -tan(l3§))82 > 0
= -Fsh 0311031) - tan(13§))¢32
-Kp “1112038) 8602 (3)3), l-Fs h] (tan(l3s) -tan(l3s')) £2 S 0
3 —K6 maze,» (6.44)

Thus assembling all the parts we get

V10 S -BxK1e% —Bg Kzeg ‘Kd e3 ‘Kfl tan2(ep) S 0 (6.45)

Following the footsteps of Theorem 2.1, we can show that the error variables, e1, e2, ed,
423 and the parameter estimation errors, 6,, and $9, asymptotically converge to zero.

Remark 6.1: It can be noted from the theorem statement that the law for obtaining d
has been changed from d = —7,1r-(M,) to d = —;1-s-(M,). This is based on the fact that the
location of pushing should be based on how much force is actually being applied.

SIMULATION 6.1

Simulation 6.1 is performed to show the effect of saturation in F. The results of the
simulations are observable from Figs.6.1 through 6.3. The conditions for the simulations
are as follows: The mass and mass moment of inertia of the wheelchair and occupant

combined are assumed to be

M = 60 kg, J = 2.25 1:ng (6.46)

The reaction force and reaction moment are assumed to be
R = 30 N, T = 5 Nm (6.47)
We assume the initial values of the parameter estimates to be

(3,7 (0) = [ 40.0 10.0 ] (6.48)

145

-—

 

(33‘ (0) = [ 1.0 0.0 ] (6.49)

in SI units. The controller gains and sliding surface constants were chosen as K1 = 10,

K2 = 10, I'll = 10 and IQ = 10. The desired trajectories were chosen as

0.15

0.05 - -
0
-005 1 1 1 1 r L

20 I r
10 r
0 .-. r _n L r -
-10 .4
-20 1 l l l l l

50 , .

 

ex (mm

 

 

 

69 (deg)

 

 

 

Fr (N)
8 ‘5’ 8

 

 

 

Fs (N)
338

 

time (see)

Figure 6.1. Simulation 6.1: Plot of error variables ex and eg

and reference control input F, and control input F}.

x", 0.1 ”61(1) (6.50)

(9}, = sin(0.2t) (6.51)

along with initial conditions xd = id = 9d = 9d = 0. These trajectories have been chosen
so that F remains positive always. It can be seen from Fig.6.l that the errors ex and eg tend
to converge to zero. This figure also shows a comparison between the required force, F,,
and the actual force applied F,. It can be observed that the applied force saturates at about

35 N. An important observation in this simulation is the repetitive saturation of the force.

146

 

41.5 , I 1 1

M (Kg)
.2

 

 

 

 

 

 

 

 

J (Kg m2)

1 l l l

 

 

 

 

fa (N)
a”).

l l l l

 

 

 

 

 

T (Nm)

 

 

I l I 1 l

 

time (sec)

Figure 6.2. Simulation 6.1: Plot of parameter estimates

 

-0.05 -

m - /V\__‘

d's (m)

 

 

 

 

 

B We)
'8

 

.40 1 1 1 1 g 1

O 10 20 30 4O 50 6O 70
time (sec)

 

Figure 6.3. Simulation 6.1: Plot of control inputs, dg, and B;

147

.‘n “‘PL‘.'

 

In accordance to our requirement of a reasonable trajectory, this repetition of the saturation
cannot continue infinitely, thus eventually sending the errors to zero. Figure 6.2 show
the parameter estimates. it can be seen that all the estimates except 1V! converge to their
respective original values given in Eq.(6.46) and (6.47). The reason for M not converging
to the actual value can be attributed to the non availability of RE. in the trajectory of 1:.
Figure 6.3 shows the actual location of d; and the actual joint angle 6;.

6.3 Addressing dynamics of Applied Force

The next issue that we will be addressing in this chapter is the effect of robot dynamics
on the adaptive controller. Until this point we had assumed that the force F,- could be
instantaneously realized by the robot. However like it was discussed in Chapters 4 and 5,
where the robot dynamics at the prismatic joint and at the angle of push were addressed,
this is not possible. The robot and its controllers have their own dynamics that affect the
robots ability to realize the force F. In this section we propose a controller for the actuators
responsible for the pushing force F and show asymptotic stability of the equilibrium point.
Before we propose the next result we present some modifications that need to be made to

address this issue.

1. The equations of motion of the wheelchair have to be modified to incorporate the
effect of the force dynamics. Thus the equations of motion of the wheelchair take the

form

M5: = Fs’ -R (6.52)

19'

—Fs’d§—E.htan(fis’) —T (6.53)

2. It can be seen from the above equations that the applied moment and the applied
force are different from Eqs.(6.1) and (6.2). This implies that 39, which is the ratio

of the saturated moment to the required moment, will be modified to

148

 

 

 

_ "Fs’ds “Erik tan(fls)

Be Mr

(6.54)

3. We change the law for obtaining the value for d from d = —}!;(Mr) to d = —;.1;,(Mr).
This will be seen in the forthcoming theorem. This is done based on the fact that the
actual location of force application should be based on the actual force being applied.
With this modification it can be seen thath > O as long as F; ¢ 0 (see Section 5.4

for more details).

4. We redefine the error e f as follows

ef = F,,—F; (6.55)

The definition of Bx does not change since the ratio of the saturated force to the required
force is still the same. We now are ready to propose the result for this section.

Theorem 6.2: Consider the dynamical system described by Eqs.(6.52) and (6.53). For
this system, the equilibrium point (e1,e2,6x,$9,ed,e,3,ef) = (0, 0,0,0, 0, 0, 0) is asymp-

totically stable for the following control laws:

F+-—1—e +e e e >0

F,’ = { ‘ Kfl f’ ‘f (6.56)
P}+ef, elefSO

(6.57)

O 1 I I
_ _ _. -F
d; : (1.3+Kd[ E9]82+ed, ] s]e2ed>0 (6.58)
ds‘l‘ed) [“queZed £0
(6.59)

149

 

 

— l-Fs’ h1(1 +tan(l3s)tan(l3s'))e

 

2 _ I _ I
fis‘l' Kfl (1+[an2(efl)) +tan(e[3)) [ Fs h](tan(fl3) tan(fis))92 >0
Bs+ran<epx 1—F.'h1 (tan(fis)—tan(B§))e2 so
' (6.60)
the reference control inputs given by:
F,- = Mfd+llfiéx+R+Klel (6.61)
P} = F,,, tanh(F,) (6.62)
Mr = féd-l-Aqfe'g'l-T-l-Kzez (6.63)
1
d = ‘E (Mr) (6-64)
(is = dm tanh(d) (6.65)
)3 = arctan ___—"4",: d‘ (6.66)
1% = Butanhw) (6.67)
and estimation laws:
6: = —P“A§ e1, (6.68)
479 = 44245632 (6.69)

provided Fs’ ¢ 0.
Proof: Consider the Lyapunov function candidate

1 l 1.. ~ 1.. ~ 1 l l
V11: §Me%+ Eleg+ §¢xTP¢x+§¢gP¢9 +§Kd€§+ 5K5 113112 (85) + EKfefi- (6.70)

which is positive everywhere except at the origin (e1,e2,6x,$9,ed,efi,ef) =
(0,0,0,0,0,0,0), where it is zero and K4, K5, K f, are positive constants. By differenti-

ating V11 with respect to time, we get

V11=Me1e'1+Je2e'2+6{P$x+$gP$9 +Kdedéd+Kl3 tan(e5) sec2(e]3)é5 +Kfeféf
(6.71)

150

We obtain the derivatives e'l and e'z from Eqs.(2.9) and (2.10) as follows

8'1 = e} + 11 ex (6.72)

e'z = e}; + Age'g (6.73)

By multiplying Eqs.(6.72) and (6.73) with M and J, respectively, and substituting

Eqs.(6.52) and (6.53), respectively, we get

Jéz

Mél = Mia-Mf+Mlléx
= an—(Fj—RHMhléx
= an—(Fs-FS+FS’—R)+Mlle'x
= Mrg-(R—R)+Mlléx+(Fs—F;)
= ng—(BxF,-R)+Mhléx+ef (6.74)

10;, —Jé +1126),

196- (-Fs' d; -Fs'h t211103.15) - T) +112é0

16;, - (—F;.’ d, -Fs’h mums) +12; ds

+Fs'h tan(13s) - Fs' di- - 1'1”! “10030 - T) +132 ée

196 - (39 Mr - T) +112 ée + l-Fs' h103111036) - tan(fi§))+1-Fs'1(ds - d6)
196 - (30 M, - T) +142é9 + l-Fs’ ’11 (“11103.0 - “111039) + l-Fs'l (8d)

16d — (30 M, - T) +132 89 +1-Fs'h] (mums) - t311413;)” l—F;](ed)
(éd+12ée)(J-Boj) + (T -39 T)-1319K2€’2+l-I’s'h](tam()5s)-t&m(l3§))

+l-Fs'l (ed) (6.76)

151

We obtain Eq.(6.75) by substituting the expression for Bx and F, from the expression of F,
from Eq.(6.13l). Similarly we obtain Eq.(6.76) by substituting the expression for Be from
Eq.(6.54) and the expression for M, from Eq.(6.133). Thus we get,

Me'l =Ax47x—BxK1e1+ef (6.77)
182 = Ae¢~0 - 30 K282 + {‘1'}, h] (Mums) - t8111(13.s'))+ {-Fs'] (8d) (678)

Substitution of the above equations in Eq.(6.71), yields

 

V11 = eletfx +82A9¢lvg —BxKle% —Bg [(28% + 6xTP$x+ (ferpfg +Kdedéd+Kfeféf
+81 8 f +1-Fk'hl(tan(fis) - “111030) 82 + Kp tan (85) 8882 (8)088 + l-Fs’] (8d) 82

(6.79)

Using the estimation law in Eq.(6.120) and (6.121), we get

Vn = —BxK1ei—BoK2e%+Kdeded +1—F.’h1(tan03.) - 6603;» 6
+Kfeféf + e] 8f + Kp tan (e5) 86¢2 (65) ép + [-F:,’] (ed) 82 (6.80)
The rest of the proof will proceed according to the conditions whether [—FS’] (ed) e2 > 0,

[—F,’h] (tan(Bs) — tan(]B§))e2 > 0 and elef > O or not. In order to simplify things we

propose breaking down of the Lyapunov derivative, V11 into four parts, namely

V11 = V116+V116+V116 (6-81)
V, ,a = —19,r K113? — 39 [(263 g 0 (6.82)
V1 lb = Kd 8d éd + l-Fs] (8d) 82 (6-83)

V11c = l-Fs h] (tan()3s) - t8111(13s'))82 + K8 t8l11(8]3) 8882 (88) 83 (6-84)

Vlld = Kfeféf+elef (6.85)

152

Since Fs’ 75 0, Bo > 0. We also know that B, > 0, thus V110 S 0. We next analyze V11 6-

Substituting the control law for the prismatic joint in V1 1b we can get,

V116 = Kd 8d 84 + PM (8.1) 82 (6-86)
_ 2 _
= Kde ) [ F31e2ed > O (6.87)
‘Kd 8,; +["E$1(ed)321 [—F,] 82 8d S 0
s —Kd e3 (6.88)

Next lets consider the case when [—Fs h] (tan(fis) — tan(16§)) e2 > 0 in the definition of 17116,

Using the trigonometric identity,

tan(5.5") - “I"(BD

 

”“03"” = 1+tan(13s)tan(fis’) “'89)
in V110 we get,
V1 IC = [“Fs ’1] WW: - 5;) (1 + tam(13s) ta“(13.0) 82
+Kfi tan(e]3) SCC2 (efl)éB (6.90)
Using the trigonometric identity,
secz(e,3) = l+tan2(efl) (6.91)
we get,
V11c = FR: ’1] “films - fii) (1 +tan(l3s) @1030) 82
+19, tan (e13) (1 +tan2(e]3)) a], (6.92)
which can be written as
V11c = t8111(813)([-1""sh] (1 +tan(l3s) t8110315))82
+186 (1 +tan2(ep))(Bs—B.’) (6.93)

153

 

Substituting the control law for B for this case given by,

l-Fsh1(1+tan()3s)tan(l3§))82
Kp (1+tan2(ep))

 

B;=Bs+

in Eq.(6.93), we get

+ tanefi, (6.94)

V1 IC = _KB tan2(efi) (6.95)

Next lets consider the case when [—F} h] (tan(Bs) - tan(B§)) e2 S O in the definition of V116.

Substituting for the corresponding control law of B we can obtain

V“. = {—8121 (tan()3s)-tan(5s'))82—Kp m2(8fi)secz(8p) (6.96)

Thus V1 1C can be written as

Vllc = Kd8d8d+[—Fs1(8d)82

_Kfi 16620,), H} h] (tan(l3s) —tan(fl§))e2 > 0
z ‘Fsh(tan(l3s) -tan(B§))e2
_Kfi tan2 (e5) Sec2 (85), {-Fs h] (mums) __ tan(fi;))ez S O
3 —K13 tan2(efi) (6.97)
Finally we analyze V] 1d
Vlld = Kfeféf+(ef)e1 (6,98)
_ “Kfez, e1 8 f > O
_ { —Kfe§+(ef)el, elef_<_0 (6.99)

_<_ —Kfe} (6.100)
Thus assembling all the parts we get
V <-B Kez-B Kez-K ez—K tan2(e )—K e2<0 (6.101)
11_x11922 dd B 13 ff_

154

 

Following the footsteps of Theorem 2.1, we can show that the error variables, e1, e2, ed,
e13 , e f and the parameter estimation errors, 6, and $9, asymptotically converge to zero.
SIMULATION 6.2
Simulation 6.2 is performed to show the effect of saturation and dynamics of F. The
results of the simulations are observable from Figs.6.4 through 6.6. The conditions for the
simulations are as follows: The mass and mass moment of inertia of the wheelchair and

occupant combined are assumed to be

M = 60 kg, J = 2.25 1(ng (6.102)
The reaction force and reaction moment are assumed to be
R = 30 N, T = 5 Nm (6.103)
We assume the initial values of the parameter estimates to be
«if (0) = [ 40.0 10.0 ] (6.104)

(35 (0) = [ 1.0 0.0 ] (6.105)
in SI units. The controller gains and sliding surface constants were chosen as K1 = 10,
K2 = 10, 11 = 10 and IQ = 10. Also, Kd = KB = Kf = K4 = l. The desired trajectories

were chosen as

it'd = 0.5+0.1*sin(t) (6.106)

("id = sin(0.2t) (6.107)

along with initial conditions xd = id = 9d = Q, = 0. These trajectories have been chosen
so that F remains positive always. It can be seen from Fig.6.4 that the errors ex and eg
tend to converge to zero. This figure also shows a comparison between the required force,

F,, and the actual force applied F,’ . It can be observed that the applied force saturates at

155

 

 

 

6x (mm)

 

 

 

 

 

 

e0 (deg)
C

 

 

 

 

 

Fr (N)

 

 

Fs (N)

 

 

 

time (see)

Figure 6.4. Simulation 6.2: Plot of error variables ex, 429, F, and F}

about 65 N. Figure 6.2 show the parameter estimates. it can be seen that all the estimates
converge to their respective original values given in Eq.(6.102) and (6.103). Figure 6.6
shows the actual location of d, d; and the joint angle commanded B and the actual B}. It is

seen that d is saturated for most of the time and B is saturated intermittently.

6.4 Resetting of Estimates to Prevent Windup

In this section we will present the results with the resetting of estimates when not only the
condition e2 = 0, 39 < l is satisfied but also the condition el = 0, B, < 1 is satisfied. The
analysis in Sections 3.3 and 4.6 still hold true and can be extrapolated to the case of force
saturation. The additional term in the Lyapunov equation, e f, can also be shown to be
continuous since there is no change in F5 and Fg’. The next theorem provides the technique

for resetting of estimates in presence of F, d and B saturation and dynamics.

156

 

 

120 , , . 1

 

f4 (Kg)
§

 

 

 

 

 

N
I
l
I
I
I
I
l
|
l
l
I
I
|
|
I
l
I
I
l
|
I
I
I
I
I
I
I
I

l I I l

J (Kg m2)
{336.

 

 

 

 

KW)
‘58
[I]
I

 

 

 

 

 

 

 

 

 

 

< 20 - -
O 1 1 1 g 1
A 6 l l l I I
g 4 I’ ___________ _
<1— 2 — -
0 1 1 1 1 1
0 10 20 30 40 50 60

time (sec)

Figure 6.5. Simulation 6.2: Plot of parameter estimates

Theorem 6.3: Consider the dynamical system described by Eqs.(6.52) and (6.53). For
this system, the equilibrium point (e1,e2,6x,69,ed,efi,ef) = (0, 0,0, 0,0,0, 0) is asymp-

totically stable for the following control laws:

F +ie +e e e >0
F; = S K, 1 f’ 1 f (6.108)
Eg+€f, e1 efSO
(6.109)
., { Js+-1—[—Fs']e2+ed, [—Fs’]e2ed>0
d, = Kd (6.110)
ds+8d, I—Fs’l828d S 0
(6.111)

157

I3 (deg) d's (m) d (m)

I3 's(deg)

 

 

 

 

 

 

 

 

 

 

 

 

Figure 6.6. Simulation 6.2: Plot of d, d;, p and 6'

158

time (sec)

 

S

 

 

 

 

 

 

B.+ [‘FS' ’1“ “”03”“(39VZ +tan<e,). 1—F;h1(tan<13.) -tan08.'))e2 > o

 

K13 (1 +tan2(e13))
Bs+mn(ep): l-F§h1(tan(fis)-tan(fi§))82 so

(6.112)

the reference control inputs given by:
F, = de+llMéx+R+K1e1 (6.113)
F, = F,,. tanh(F,) (6.114)
M, = f6}, +Agfe‘9 + T+K2 e2 (6.115)
d = -l(M,) (6.116)

F,
d, = dm tanh(d) (6.117)
B = arctan ——K4 d; d‘ (6.118)
B, = Bmtanh(B) (6.119)
and estimation laws:

6', = —P"‘AIe1, (6.120)
(39 = -P"A5e2 (6.121)
($9 :39 439(0), e2=0and B9 <1 (6.122)
(3,:B,43,(:j), e1=0ande <1 (6.123)

provided F} 76 0.
Proof: The proof of Theorem 6.2 is still valid.
SIMULATION 6.3
The conditions for this simulation were chosen to be the same as in Simulation 6.2. The

trajectory is chosen as follows

159

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

LO 2 <t<l
55d _ m/sec2 0 _ _ secs (6.124)
0.0 m/sec t > 1 secs
0.0 rads/sec2 0 s t < 20 secs
0d = 0.5 rads/sec2 20 S t < 21 secs (6.125)
0.0 rads/sec2 t 2 21 secs
0.1 fi 1 I I 7 I
A 0.05 b -
5:, 0- .
"U -0.05 .
_0. 1 1 1 L 1 1 1
4O 17 I I I I I
”6‘6 20 - _
1‘3, 0 I _
65”. -20 1 1
.40 l 1 I I l I .
0 20 40 60 80 100 120 140

time (sec)

Figure 6.7. Simulation 6.3: Plots of ex, ea, F, F}, d; and B55.

The simulation results are presented in Figs.6.7 and 6.8. The errors in x and 9 coordi-

nates are seen to converge to zero. It can also be seen that the estimates have been brought

160

 

 

 

 

M (Kg)
.§ 13

M

8
r

_1

 

 

 

 

A
J
§§§

 

 

 

 

(N
é

 

a
8

 

 

 

 

 

'1‘(Nm
egg;

1 l

0 2O 40 60 80 100 . 120 140
time (sec)

 

Figure 6.8. Simulation 6.3: Plot of estimates

back to lower values during the resetting. The results are in concurrence with the similar

situations in the previous chapters.

6.5 Addressing the situation when F,’ = 0

In this section we will deal with the situation of F,’ = 0 as was done in the previous chapters.
Please refer to Sections 3.3 and 4.7 for more information. The following theorem presents
the results inclusive of the situation when F,, = 0.

Theorem 6.4: Consider the dynamical system described by Eqs.(6.52) and (6.53). For
this system, the equilibrium point (e1,e2,-,6x,69,ed,efi,ef) = (0,0,0,0,0,0,0) is asymp-

totically stable when IFS’ | > 8 and stable otherwise for the control laws:

161

. F +ie +e e e >0
F; = { ‘ Kf' f’ ‘f (6.126)
Fs+ef, elefSO
(6.127)
.1; ___ { 465—81666. l—Fs’lezzepo (6.128)
J, +ed, {—Fs']e2,-ed S 0
(6.129)

- I-Fs' h1(1 +tan(13s)tan(f3§)) 82:

13::

 

‘8” K13 (1 +6620,»

BS +tan(e13),

+ tan(8p), l—F§h1(tan(l3s) - t81103.0) 821 > 0

{—671 (mass) —tan()3..’))e2. s o

the reference control inputs given by:

d5:

93 =

and estimation laws:

de+théx+R+K1 e1
Fmtanh(Fr)
féd+h2feg +T+K2e2,-
l
—— M
Fs( r)

dm tanh (d)

K4d—d,
h

BmtanhLB)

arctan

 

‘I’x = —-P“A£ e1

(if, = 44.45.22,, |F,| > a
$0, 4’0 = 0, IN S 8

162

(6.130)

(6.131)
(6.132)
(6.133)
(6.134)
(6.135)
(6.136)

(6.137)

(6.138)

(6.139)

 

$9 = 89 (139(9), e2 = 0 and B9 < 1 (6.140)

$,=B,43,(1j), e1=0ande <1 (6.141)

where 8, 8 > 0 is some small number.
Proof: Proof can be arrived following the footsteps of the proof of Theorem 5.3.
SIMULATION 6.4
Simulation 6.4 is performed to show the effect of the switched controller employed due
to the condition F,’ = 0. The results of the simulations are observable from Figs.6.9 and
6.10. The conditions for the simulations are the same as that of Simulation 6.3. The desired

trajectories were chosen as

id = sin(0.1t) (6.142)

6,, = sin(0.2t) (6.143)

along with initial conditions xd = id = 9d = Gd = 0. It can be seen from Fig.6.9 that the
errors ex and eg tend to converge to zero. This figure also shows a comparison between
the required force, F,, and the actual force applied FS’. It can be observed that the applied
force saturates at about 65 N. Figure 6.10 show the parameter estimates. Although the
parameter estimates of M and R have not yet converged, they are well behaved. Based
on previous simulations they will converge given unsaturated conditions or enough time.
The switching between the two control laws is observable as F,’ < e = 5, approx around
I = 36 sees and t = 56 secs. The error in 0 coordinate, e95, reacts to the condition Fs’ < e
by staying constant, which can be seen in Fig.6.9. The estimates, f and T, react to the
condition F < 8 by going to zero and staying constant during that period, which can be
seen in Fig.6.10. Finally, the control inputs, d; and B}, can be seen changing sign as FS’

changes sign at around these times in Fig.6.9.

163

 

 

6x (mm)

 

 

 

 

 

 

 

601 (deg)

 

 

 

F's (N)

 

 

 

 

 

 

d's (m)

 

 

 

 

 

 

 

’63
D
'3
66”.
.1“) 1 1 1 1 1 1 L 1 1
O 10 20 30 40 50 60 70 80 90 100
time (sec)

Figure 6.9. Simulation 6.4: Plot of error variables ex and eg and control inputs 1'}, d; and

fis’

164

 

M (K8)

(K8 m2)

(H

R(N)

c: to 4x as cagggggg

<

T Gflnfl

<

 

I I 1' I

 

I 1 l 1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1 4L I l l I l l

 

 

 

O 10 20 30 40 50 60 70 80

thne(&u9

Figure 6.10. Simulation 6.4: Plot of parameter estimates

165

 

CHAPTER 7

Experimental Verification

7.1 Focus of the Experimental Work

In this Chapter we provide a discussion on the experimental setup we used, to verify the
viability of our theoretical results. Before we begin let us go through a brief review of the
developments made in the previous chapters. After the introduction in Chapter 1, in Chapter
2, we developed a basic controller based on certain assumptions for pushing and steering
wheelchairs along a trajectory. We had also proposed an arm design which can be used for
pushing wheelchairs and also various other purposes like, opening doors, picking up objects
etc. In Chapter 3 and Chapter 4, while dedicating ourselves to the problem of pushing and
steering wheelchairs, we have removed some of the impractical assumptions by addressing
the saturation in the moment arm, the finite nature of the bandwidth of the actuator driving
the prismatic joint and the possibility of the force F being positive, negative and zero. The
first part of this chapter is dedicated to observing the efficacy of the results presented until
chapter 4. Later we show the experimental results obtained using the complete results in
the thesis. In order to conduct the experiments, we built a simple arm that can be attached
to a robot and provide the basic capabilities to maneuver a wheelchair, discussed in the
next section. The robot we used for this purpose was the CompanionTM robot developed

by Intouch Health Inc. The arm that was used in these experiments is discussed next.

166

 

7 .2 Arm Description

For the purpose of verifying the results we developed a simplified version of the arm pro-
posed in Chapter 2. This arm is shown in the inset of Fig.7.l. The arm has 2 DOF, the
first DOF being the prismatic joint and the second DOF, the elbow joint, corresponds to the
joint 1 of the Arm described in Chapter 2. The arm is positioned on the robot such that it
is at wheelchair height. See Fig.7.1. This is done so as to reduce the number of joints and
hence, complexity. The prismatic joint which is driven by the motor, shown in Fig.7.], lo-
cates itself on the forearm as commanded by d, and provides the moment arm. The second
DOF shown as Elbow Joint in the Fig.7.] is controlled by the robot itself and is used for
maintaining the angle that the rear arm makes with the elbow. This is possible due to the
holonomic nature of the robot. For the first part of the experiments this was maintained at

90° to the forearm. There are a total of four sensors used on the arm

detector (4) /ermtter (4

 

Figure 7.1. Experimental setup

167

1. A force sensor which measures the force applied by the robot on the wheelchair.

2. An encoder to measure the angle at the 2nd DOF at the junction of the rear arm and

the elbow (Elbow Joint).
3. An encoder to obtain the actual location of the prismatic joint on the elbow (dg).

4. An emitter/detector triangulation sensor to track the trajectory of the arm and hence

the wheelchair.

7 .3 Experimental Results: Constraints of the Prismatic Joint

Figure 7.2 shows the desired reference trajectory provided by the joystick (in solid lines)
and the actual trajectory taken by the wheelchair (in dotted lines) in x and 9 coordinates.
It can be seen that both x and 9 track xd and 9,], respectively, although slowly. The reason
for the slow convergence is discussed later.

Figure 7.3 shows the estimates of M, R, J and T, namely M, R, f and T. The value of M
was approximate 25 Kg. The first plot shows that the estimate M increases when the error in
x coordinate is large but it finally converges towards its true value as the error goes to zero.
Although it was assumed that F = F,, this was not the case during experiments and the
robot fell short of applying the required force. This is one reason for the slow convergence
of the trajectories and therefore rise in the estimates. The plot of R shows convergence to
approx. 45 N. However, it cannot be ascertained if this is the true value of R. The estimate
of f starts at 1.0 Kg.m2 and settles to approx. 1.5 Kg.m2 which seems to be a reasonable
number for the true value of J. Beyond 1 = 17 sees, the value of f starts rising. This can
be attributed to the transients in error in the 9 coordinate. The transients correspond to the
saturation and dynamics of the actuator of the prismatic joint. This can be independently
verified from Fig.7.4 from the plots of d, (in dotted lines) and d; (in solid lines). The plot of
I", shows signs of settling to the value of 8.0 Nm. It exhibits a transient for a brief interval

of time during 17 < t < 20 sec and this can also be attributed to saturation and dynamics of

168

 

the prismatic joint actuator. The positive value for the T could be due to the difference in

performance between the wheels.

 

I I I I

9

6

X.xd (m)

IIIITTIII

l 1 1 1 l l I 1 1 I

 

 

 

 

 

 

......

 

 

9,0,1 (rad)

0.4
0.8

1 1 1 I 1 l l l I

I I I I

 

 

 

O
U!
_-
O
31—.)-
M
N
O
N
M

Figure 7.2. Trajectories

From the above results an important observation to be noted is that, trying to turn using
only the location d; on the prismatic joint is not enough. Only small rates of changes in
orientation are possible with the prismatic joint. Therefore tracking in 0 coordinates show
room for improvement. It was noted during these experiments that the wheelchair seemed
to have an added advantage for faster turning the angle in the 2nd DOF was still catching
up to 90". In the succeeding sections this additional advantage is further exploited as the B

angle to produce extra moment.

7 .4 Experimental Result: Prismatic Joint and Elbow Joint

For this part of the experiments the elbow joint was no longer commanded at 90”. Rather

it was commanded by B,. The results are presented in figures 7.5, 7.6, 7.7.

169

 

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25

 

 

 

 

 

 

 

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170

 

 

 

 

 

 

time(secs)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

__ddq_——_ dud—qqqqdb 5
____ «q: _4q dj
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171

Figure 7.6. Estimates

 

P
a e
I
l

 

 

 

 

13' (deg)

 

I 1 l 1 l 1

 

 

 

 

 

 

 

time(secs)

Figure 7.7. Control Inputs

From the Fig. 7.5, it can be seen that a significant improvement has been made in the
tracking of 0 coordinates. This is due to the additional moment provided by the pushing
angle B, which has been plotted in Fig.7.7. The robot is capable of making faster and larger
turns when compared to the previous set of experiments with only the prismatic joint. The
prismatic joint movement has also been plotted in Fig.7.7. It can be seen that although B
assists d, the full potential is realized only when d saturates. A small improvement is also
seen in the tracking of the x trajectory which can be partly attributed to improved pushing
force in the forward direction due to the holonomic robots effort being efficiently used in
the sidewise motion. It has to be mentioned that the resetting has not been implemented in
these algorithms due to shortage of expensive hardware (sensor 4) and therefore space to

run the robot.

172

CHAPTER 8

Conclusion and Future Work

Robots have been used to replace or augment the performance of human work force in the
past. Increasingly complex control systems had to be developed to keep up with increasing
requirements. In this dissertation we built upon an adaptive control strategy to address
constraints in the control input while maneuvering wheeled platforms using a holonomic
mobile robot. The constraints considered were actuator saturation and actuator dynamics.
These constraints, when ignored, degraded the performance of the system due to excessive
adaptation which leads to estimation Windup. In the chapters so far these adverse effects
were demonstrated and consequent modifications in the algorithm were made and also
shown to be beneficial. The results obtained were verified with simulations and later with
experiments in the context of a robot pushing a wheelchair.

These results can also be easily extended to other wheeled platforms with non holo-

nomic constraints whose equations of motion can be simplified as follows:

(8.2)

where u is the input and r is the reaction force. For example, in the case of a stretcher, the
equations of motion are quite similar to the wheelchair. The major difference being that the

non-holonomic constraint is in the front wheels and the casters are the rear wheels. This is

173

shown in Fig.8.1. If we look at the corresponding equations of motion,

 

   

Stretcher

 

 

 

 

Figure 8.1. Extension to other platforms.

M)? F — R (8.3)

10 = —than(B)—T (8.4)

we can deduce that the movement of the prismatic joint does not produce any moment,
however, the results in Chapter 6 are still valid albeit this fact. Further the prismatic joint
can also be commanded to zero and the turning can be entirely done by the pushing angle
B.

One of the important assumptions in the beginning while developing the thesis was that
the trajectory was to be provided by a path planning algorithm. However it can also be
provided with a human in the loop using Haptic devices. A Haptic device is a joystick like
device that will enable the operator to send commands to a remote robot or similar device
and also receive force feedback. Thus it will enable him or her to experience the forces

that the remote device experiences. A typical application in our case would be doctors

174

doing their rounds, nurses moving wheelchairs from remote locations, relatives taking an
inmate in an LTC facility for a wheelchair ride, personnel moving hazardous materials in
hazardous environments using teleoperated robots, etc.

Finally we would like to conclude by reemphasizing the performance improvement ob—
tained with our algorithm, which we demonstrate in this section by looking at the overall
path/trajectory taken by a wheelchair. This provides a fresh perspective, different from the
state variable plots presented throughout the chapters. The figures presented below repre-
sent the top view in global (X,Y) coordinates, of a wheelchair being transported from point
A (2,2) to point B (14,14) and back to A. These are performed under different conditions
using a simple path planning algorithm that generates the trajectory based on the current
position of the wheelchair. The first condition shown in Fig.8.2 corresponds to the basic
adaptive algorithm we started off with. This assumes that the saturation and dynamic con-
straints are not present. This provides a comparison bench mark for the other plots. An
important observation to be made here are the tight turns taken by the trajectory curve in
both the directions. Next, Fig.8.3 shows the situation when saturation and dynamics in the
prismatic joint are included in the simulation. It is very obvious that the wheelchair takes
wide and excessive turns in both the directions. This is attributed to the actuator constraints
and estimation Windup. The final scenario shown in Fig.8.4 shows the same simulation us-
ing estimation reset and other modifications presented in this thesis. Although the turns are
not as tight as in the first case, the excessive turns have been eliminated. Thus the degrading
effect of the estimation windup, due to excessive adaptation, resulting in bad performance
or even instability, is avoided. We contend that the work presents a significant contribution

obtained in a novel and practical manner.

175

 

 

 

 

14 . __3
I-
/
//
r /
//

8. //
g /
>-- /

2, ('f/

2 ‘ ' X(m)‘ L 14

Figure 8.2. Path taken in the absence of actuator constraints.

 

Y(m)

 

 

 

 

Figure 8.3. Path taken in the presence of actuator constraints.

 

 

 

 

T fi
14 .
/
// '1
,1
)- ,/ ’1’
I/ It
1' 1’
.I
..§ ’ ,4?“
E . /’
v )- t
>‘ /’:I I
f (I
b “/11,"
4’
Fix"
2 ta" 4
l L l 18 4 i4
2
X(m)

Figure 8.4. Path taken when using the modifications presented in the thesis.

176

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177

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