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This is to certify that the
dissertation entitled

CONTROL OF MOBILE MANIPULATORS

presented by

JINDONG TAN

has been accepted towards fulfillment
of the requirements for

PH. D degree in W COMPUTER
ENGINEERING

 

 

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CONTROL OF MOBILE
MANIPULATORS

BY

J INDONG TAN

A Dissertation

Submitted to
Michigan State University
in partial fulfillment of the requirements

for the degree of
DOCTOR OF PHILOSOPHY

DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING

2002

ABSTRACT

Control of Mobile Manipulators
By

Jindong Tan

This dissertation presents a new class of control and coordination schemes for mo-
bile manipulators. A mobile manipulator, which consists of a mobile platform and
a robot arm, provides a new concept and direction in robot applications due to its
capability of infinite motion in a large workspace and dexterous manipulation. Mo-
bile manipulators have various applications in rescue and hazardous fields, battlefield,
homes, offices, hospitals, etc. However, motion planning and control of such a system
is challenging. This dissertation first presents the design of non-time based tracking
controller and its application to nonholonomic mobile platforms. The non-time based
tracking control is a key step to ensure the robustness of the mobile manipulator to
unexpected obstacles and coordination errors. Then, singularity analysis and motion
control of robot arms is discussed. Robot arm singularity avoidance is considered as
one of the criteria for coordinating the motion of the mobile platform and the robot
arm. Using a hybrid system approach, a singularity-free robot motion controller
has also been deve10ped for controlling the robot arm. Further, a unified dynamic
model for the integrated mobile platform and on—board manipulator arm is developed.
This provides an efficient and convenient framework to design a mobile manipulator
controller as well as plan its motions. Combining the non-time based planning and
control method with the nonlinear feedback technique, motion and force controllers
at task level are designed. An online non-time coordinating scheme is developed to

resolve the kinematic redundancy. Based on the decoupled and linearized model, this

dissertation discusses an online non—time based coordination scheme. Force/ position
control along the same task direction is originally proposed in this dissertation. The
proposed force/ position control scheme is illustrated using a cart pushing control ap-
plication, which requires both force control and motion control along the same task
direction. An integrated task planning and control scheme while the mobile manip-
ulator interacts with such moving objects is proposed. The proposed approaches are
implemented and tested on a mobile manipulator consisting of a Nomadic XR4000

and a Puma 560 robot arm.

Copyright by@

J INDONG TAN
2002

To my mother, my grad parents, my brother and for the
memory of my father

ACKNOWLEDGEMENTS

Foremost, I gratefully acknowledge all the invaluable help from my advisor, Dr.
N ing Xi, without whom this dissertation would not have been possible. This disserta—
tion comes from numerous discussions in his office, from his keen insight knowledge,
from his guidance to a fruitful research area, and his perseverance and support. Not
only the knowledge I learnt here, but also the research experience and friendship I
gained here will be beneficial to the rest of my life.

I would like to thank all my graduate committee members, Dr. Hassan Khalil,
Dr. Fathi Salam, Dr. Robert Schlueter and Dr. Zhengfang Zhou. In addition to their
invaluable time they spend on the committee meeting, their insightful comments and
suggestions will enhance the technical soundness of this dissertation. I also would like
to thank Dr. Wei Kang from Naval Postgraduate School for his valuable help.

I am grateful to my friends and colleagues from Robotics and Automation Lab.
The discussions with Weihua Sheng, Imad Elhaj j, Jizhong Xiao, Amit Goradia, Hen—
ning Hummert, Yu Sun and Heping Chen substantially contributed to my work and
broaden my knowledge.

Finally, I would like to reiterate my gratitude to my parents, my grandparents,
my brother and sister-in-law. Without their encouragement and continuous support

for years, I would not have achieved what I have done today.

vi

TABLE OF CONTENTS

1 INTRODUCTION 1
1.1 Motion Tracking Control of Mobile Robots ............... 2
1.2 Singularity Analysis of Robot Arms ................... 4
1.3 Motion Planning and Control of Mobile Manipulators ......... 7
1.4 Mobile Manipulation Using Force Control ............... 12
1.5 Contributions and Outline of this Dissertation ............. 14

2 MOTION REFERENCE PROJECTION 17
2.1 Introduction to Event-Based Planning and Control .......... 17
2.2 Reference Projection Method ...................... 20
2.3 Stability .................................. 23

2.3.1 Example .............................. 24
2.3.2 Error dynamics and stability ................... 26
2.3.3 Feedback design .......................... 31

3 TRACKING CONTROL OF THE MOBILE PLATFORM 35
3.1 An Introduction to the Control of Mobile Platforms .......... 35
3.2 Path Tracking Control for Mobile Robots ................ 36

3.2.1 Mathematical Model of the Robot ................ 36
3.2.2 Tracking Curves with no U-turn ................. 38
3.2.3 Tracking Circles and Ellipses ................... 42
3.3 Experiments and Results ......................... 45
3.3.1 Tracking Sinusoid Waves ..................... 45
3.3.2 Tracking Circles .......................... 48
3.3.3 Tracking Ellipses ......................... 48

vii

4 SINGULARITY ANALYSIS AND SINGULARITY-FREE CONTROLLER

OF THE ARM 53
4.1 Introduction ................................ 53
4.2 Robot Arm Singularities ......................... 56
4.2.1 Non-redundant Robot with Spherical Wrist ........... 57
4.2.2 Non-redundant Robot with Non-spherical Wrist ........ 61
4.2.3 Singularity Analysis for a Redundant Robot .......... 62
4.2.4 Workspace and Singular Conditions ............... 63
4.3 Hybrid Robot Motion Controller ..................... 64
4.3.1 Hybrid System Model ...................... 64
4.3.2 Design Procedure of a Hybrid Robot Motion Controller . . . . 65

4.3.3 Hybrid Motion Controller for a PUMA-like robot manipulator 68

4.4 Stability Analysis of the Hybrid Motion Controller ........... 78
4.5 Experimental Implementation and Testing ............... 83
5 CONTROL OF MOBILE MANIPULATORS 89
5.1 Introduction ................................ 89
5.2 Unified System Model .......................... 91
5.2.1 Kinematic model ......................... 91
5.2.2 Dynamics Model ......................... 93

5.3 Motion Control and Redundancy Resolution .............. 95
5.3.1 Tracking Controller ........................ 95
5.3.2 Singularity Analysis and Redundancy Resolution ....... 96

5.4 Experimental Implementation and Results ............... 100
5.4.1 Hardware and Software Implementation ............ 100
5.4.2 Path Tracking and Tele-operation ................ 101
5.4.3 Path Tracking with Unexpected Obstacles ........... 102

viii

6 INTEGRATED SENSING AND CONTROL OF MOBILE MANIPULATOR108

6.1 Introduction and Force Control Schemes Revisit ............ 108
6.2 Output Force/ Position Control of Redundant Robots ......... 110
6.3 Integrated Task Planning and Control ................. 113
6.3.1 Motion Planning of the Cart ................... 114
6.3.2 Force Planning of the Cart .................... 116
6.3.3 Integration of Sensing, Task Planning and Control ....... 119

6.4 Experimental Implementation and Results ............... 121
6.4.1 Pushing Along a Straight Line .................. 122
6.4.2 Turning the Cart at a Corner .................. 123
6.4.3 Pushing Along a Sine Wave ................... 124

7 CONCLUSIONS 126

ix

LIST OF TABLES

1.1
1.2
1.3

2.1

3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8

4.1
4.2
4.3
4.4
4.5
4.6

4.7

4.8

4.9

LIST OF FIGURES

A mobile manipulator ..........................
Path tracking of mobile manipulators ..................

The mobile manipulator and nonholonomic cart system ........
The comparison ..............................

The configuration of a mobile robot. ..................
Tracking a sinusoid curve with initial error ................
Time based control vs Non-time based control ..............
Tracking a circle without initial error ..................
Tracking a circle without initial error ..................
Tracking an ellipse without initial error .................
'Itacking an ellipse with initial error ...................

Tracking an ellipse with present of unexpected obstacle ........

Singular configurations of Puma 560 ..................
Illustrative plot of the robot workspace and subspaces ........
Hybrid system controller framework ...................
A typical switching sequence of the hybrid motion controller .....
Experimental Setup ............................
At boundary singular configuration, with DLS method only. Dash line:
Desired; Solid line: Actual ........................
At boundary singular configuration, with hybrid motion controller.
Dash line: Desired; Solid line: Actual ..................
At interior singular configuration, with DLS only. Dash line: Desired;
Solid line: Actual .............................
At interior singular configuration, with hybrid motion controller. Dash

line: Desired; Solid line: Actual .....................

xi

47
49
49
50
50
51

75

81

83

84

84

85

85

4.10 With both interior and wrist singular configurations. Dash line: De—

sired; Solid line: Actual .......................... 86
4.11 With wrist singularity. Dash line: Desired; Solid line: Actual ..... 86
5.1 Mobile manipulators ........................... 90
5.2 Event-based coordination between the arm and the mobile platform . 98
5.3 Hardware structure of the mobile manipulator ............. 101
5.4 Software structure of the mobile manipulator .............. 104
5.5 Multiple tasks path tracking ....................... 105
5.6 Multiple tasks path tracking ....................... 105
5.7 Tele—operation ............................... 106
5.8 Tele-operation ............................... 106
5.9 Teleoperation with unexpected obstacles ................ 107
6.1 Hybrid Force/ Position Control ...................... 109
6.2 Impedance Control ............................ 110
6.3 Hybrid Force/ Position Control for Redundant Robots ......... 111

6.4 Integrated Task Planning and Control for the Mobile Manipulation of

a Nonholonomic Cart ........................... 114
6.5 A mobile manipulator and a nonholonomic cart ............ 115
6.6 Configuration of the cart in the moving frame ............. 120
6.7 Laser range data ............................. 121
6.8 Pushing the cart along a straight line .................. 122
6.9 Pushing with unexpected obstacle .................... 123
6.10 Turning the cart at an corner ...................... 124
6.11 Pushing the task along a sine wave ................... 125

xii

CHAPTER 1
INTRODUCTION

A mobile manipulator, which generally consists of a robot arm and a mobile platform,
provides a new concept and direction in robot applications due to its capability of
large workspace and dexterous manipulation. The robot arm is mounted on a mobile
platform, as shown in Figure 1.1. A mobile manipulator combines the mobility of the
platform and the dexterous manipulation capability of the robot arm. The manipu-
lation capability on a mobile platform is called mobile manipulation[37]. The mobile
manipulation capability is important for some autonomous mobile robot applications,
which require both a large workspace and dexterous manipulation capability, such as
reconnaissance and surveillance in rescue and hazardous fields, battlefield, homes, of-
fices, hospitals, etc. The mobility of the (mobile) platform substantially increases the
performance capabilities of the system. For example, the platform increases the size of
the manipulator workspace and enables the manipulator to position the end—effector
for executing the task efficiently, while a fixed base manipulator arm is limited by
the size of its workspace. From a different point of view, the mobile manipulator has
high degree of kinematic redundancy. Because of the kinematic redundancy, the func-
tionality of the mobile manipulator is increased to a greater extent. The redundant
degrees of freedom can be used to accomplish secondary tasks. For example, joint
torques can be optimized, singular configurations of the manipulator can be avoided

and hybrid force/ position control can be achieved.

The objective of this dissertation is to investigate the modeling, redundancy res-
olution and control, integrated task planning and control of mobile manipulators.
The unique force control prOperties while the mobile manipulator interacting with
moving objects has also been probed. Among others, the following issues of mobile

manipulators will be discussed:

-x

r

 

Figure 1.1: A mobile manipulator
0 Motion tracking control of mobile robots
o Singularity analysis of robot arms
0 Singularity-free tracking control of robot arms.
0 Unified model of mobile manipulators and online redundancy resolution
0 Decoupled force/ position control along the same task direction
0 Integrated tasking planning, sensing, and control of mobile manipulators

A quick survey of the ongoing research on related topics is given bellow.

1.1 Motion Tracking Control of Mobile Robots

Most of the tasks of a mobile robot, which generally equipped with a computer and
sensors, are to move from one configuration to another configuration, or track certain
path in its workspace. The research on mobile robots is generally focused on motion
planning, localization, navigation, sensor fusion, tracking controller design. In this

dissertation, a non-time based approach for the path tracking controller design of

mobile robots will be discussed. The non-time based robot controller enables the
robot to handle unexpected events while tracking certain path. The non-time based
approach can also be used for the coordination of the mobile platform and the robot
arm.

In the control of autonomous mobile robots, unexpected events may change the
speed and path of a vehicle during a mission (such as traffic light, unexpected obsta-
cles, and system malfunction). When the unexpected event is over, the robot should
recover its normal performance smoothly. A time-dependent trajectory controller
drives a vehicle to its desired place at any given time. It speeds up the vehicle after
a temporary slow down to catch up with the schedule. However, in many applica-
tions the important task is to track the desired path with the desired speed, keeping
up with the schedule is not a first priority or not desired. A common approach for
this problem is to use a high-level rule-based controller to decide the speed and the
reset time. However, the non-time based controller proposed in this dissertation is a
different approach. The proposed controller is independent of time. For the purpose
of path tracking (with a desired speed depending on the location of the path), the
controller recovers the normal path and speed from unexpected event without reset-
ting time or speed. Furthermore, the algorithm introduces a reference projection, a
function of state variables that replaces the time variable.

When multiple vehicles are involved in the same mission (such as formation con-
trol of multiple agents), reference projection is a tool that can be used for cooperated
motion control of multiple vehicles. This idea of using reference projection for coop—
erated control was used in the formation control of multi—satellite systems ([71] and
[34]). For the mobile manipulator, instead of using time as the reference, the motion
of the mobile robot is referred to the trajectory of the end effector of the robot arm.
The motion reference will be developed to coordinate the motion of the platform

and the robot arm according to certain criteria such that the robot arm will always

working in a dexterous workspace, as shown in Figure 1.1.

Developing non-time based controller has attracted the attention of researchers
from different fields. In [81], a non-time based controller called event-based design was
introduced. It has successful applications in a variety of robot control problems ([81],
[80], [76], [83] and [78]). In addition to robot control, [32] studies the tracking of a
trimming trajectory for unmanned air vehicles using non—time based control. In [51],
non-time based control of autonomous underwater vehicles is discussed. In [34], a time
varying feedback is converted to a non-time based controller to achieve coordination
among multiple satellites flying in formation. Although different terminologies were
used in different applications, the author found that finding a path tracking controller
independent of time is a common interest among a variety of application areas (see
additional publications [22], [57], [59], [28], [25], [63], [33] and the references therein).
The contribution of our work is to introduce a reference projection method in the

design of non-time based controller.

1.2 Singularity Analysis of Robot Arms

A robot task is usually described in its task space. The direct implementation of a task
level controller can provide significant application efficiency and flexibility to a robotic
operation[68]. Implementation of task level controller becomes even more important
when robots are working in coordination with humans[79]. The human intuition on a
task is always represented in the task space. The existence of kinematic singularities
is a major problem in the application of task level controller. While approaching
a singular configuration, the task level controller may generate high joint torques,
which result in instability and large errors in the task space. The task level controller
is not only invalid at the singular configurations, but also in certain neighborhood of
the singular configurations. Therefore, the singularity of robot arms is analyzed and

a singularity-free path tracking controller is proposed. Different from a nonredundant

robot arm, a specific task for mobile manipulators can be performed by moving only
the mobile platform or the manipulator, or moving the platform and the manipulator
simultaneously, as depicted in Figure 1.2. It is ideal to coordinate the motion of
the platform and the arm in such a way that tasks can be performed by moving
them simultaneously. This involves the kinematic redundancy resolution of mobile
manipulators. The objective to resolve the redundancy is to position the mobile base
such that the robot arm will always be in a dexterous configuration and avoid singular
configuration. The singularity analysis of robot arms is then used in the coordination
of the mobile platform and the robot arm. A brief review of the research around this

topic is given below.

 

 

Va

A
7

Figure 1.2: Path tracking of mobile manipulators

The research in robot arm singularities has attracted researchers’ attention. The
Singularity-Robust Inverse(SRI) [15, 17, 41, 47] method was developed to provide an
approximate solution to the inverse kinematics problem around singular configura-
tions. Since the Jacobian matrix becomes ill-conditioned around the singular config-
urations, the inverse (pseudoinverse) of the Jacobian matrix results in unreasonable
torques in task level controller. Instead of using exact motion control resulting in large
torques, the SRI uses Damped Least-Squares(DLS) to provide approximate motion
control close to the desired Cartesian trajectory. Chiaverini, Siciliano and Egeland

[17] refined the basic DLS method to improve tracking errors from the desired path by

varying the damping factors. Caccavale, Chiaverini and Siciliano elaborated the DLS
method by considering the velocity and acceleration variables [11]. These methods
reduce the torque applied to individual joints and achieve approximate motion. By
allowing an error in the motion, the SRI method makes the robot pass close to the
singular point but not go through it. Actually, it can be shown [41, 50] that the sys-
tem is unstable at the singular points, which means either the singular points can not
be actually reached or the robots can not start from the singular points. Obviously,
this creates difficulties for many applications. Kiréanshi et al. [41] theoretically proved
that at the singular points, the resolved acceleration control of robots has unstable
saddles. The instability of this method for redundant mechanisms is discussed in [50].

The path tracking approach [38, 39, 48] augments the joint space by adding virtual
joints to the manipulator and allowing self motion. Based on the predictor-corrector
method of path following, this approach provides a satisfactory solution to the path
tracking problem at singular configurations. Timing is not considered at the time of
planning and it is reparameterized in solving the problem. However, when a timing
is imposed on the path, it forces the manipulator to slow down in the neighborhood
of singular configurations and to stop at the singular configuration.

The normal form approach in [46] provides a solution of inverse kinematics in
the entire joint space. With appropriately constructed local diffeomorphic coordi-
nate changes around the singularity, the solution of inverse kinematics can be found
and then transformed back to the original coordinates. The normal form method in-
volves heavy computation. Hence is extremely difficult to implement it in real-time.
Duleba[23] proposed the the channel algorithm of transversal passing through singu-
larities for non-redundant robot manipulators. The basic idea of channel algorithm
is to smoothly change a component of the joint space as the neighborhood of singu-
larity is entered. Thus a continuous joint coordinate can be obtained while tracking

a desired path in a task space.

Another method has also been developed to incorporate the dynamic poles of the
system by Sampei and Furuata [56]. They proposed a time re-scale transformation
method for designing a robot controller, which achieves slow poles in the vicinity
of singularity and fast poles in the regular area. It can be shown that this method
results in similar error dynamics to the SR1 method. For the application of switching
control to the robot manipulators, Bishop and Spong [7] proposed a switching control
method for redundancy resolution of redundant robots. The damped velocity method
was used to maximize the manipulability. Torques and joint velocities are limited

utilizing the redundancy of the robot.

1.3 Motion Planning and Control of Mobile Manipulators

Mobile manipulators provide numerous advantages compared with the fixed base
manipulator, path planning and control of such a system are challenging. First,
there is a redundancy resolution problem. Secondly, the dynamic model of a mobile
manipulator is complex, and there is a high degree of coupling between the platform
and the arm. The dynamics of the mobile platform and the manipulator arm are
generally different. The former is heavy and has slow dynamics. The later is relatively
light and has fast dynamics, 216., a wider bandwidth. Thirdly, many applications of
the mobile manipulator require the robot to interact with its environment dynamically
while providing motion, such as pushing, pulling, cutting, excavating. Implementation
of these tasks demands the mobile manipulator to provide both output force control
to overcome the resistance and friction of the objects, and motion control to track
certain trajectory. Force control schemes for the mobile manipulator are therefore to
be considered. The myriad use of the mobile manipulator and the difficulties in its
control and motion planning bring a lot of research issues to light. Motion planning,
modeling and control, coordination between the mobile platform and the manipulator

arm, force control are tOpics of this dissertation. Previous work related to these topics

in the literature is reviewed.

A: Motion Planning. A solution to kinematic redundancy is to coordi-
nate the mobile platform and robot manipulator to perform secondary tasks. Seraji
[62] proposed an on—line coordinated motion control method by treating the mobile
manipulator as a redundant mechanical systems. The mobility and manipulation
are treated in a common framework. Secondary tasks for redundancy resolution are
selected based on task specifications.

Egerstedt and Hu [24] proposed a coordinated virtual vehicle solution to the tra-
jectory following problem for mobile manipulators. Given a desired trajectory for the
end-effector, a trajectory for the base is planned in such a manner that the desired
end effector position is within the specified workspace of the arm. The mobile plat-
form and the manipulator then follow their respective reference trajectories. The two
subsystems are controlled separately and coordinated.

The kinematic redundancy can also be resolved using criteria which are used to
optimize application specific requirements, such as minimum torque output, minimum
distance, etc. Chen and Zalzala [13] discussed trajectory generation while taking
the dynamics of the system into consideration. The optimal trajectory generation
problem with nonholonomic constraints and obstacle avoidance were formulated as
a nonlinear multi-criteria Optimization problem. Given the trajectory of the end-
effector, near optimal trajectories for the mobile platform and manipulator joints are
obtained by an efficient genetic algorithm. The algorithm considers the optimization
of torque and obstacle avoidance.

When a mobile manipulator performs multiple tasks in a sequence, the final config-
uration of each task becomes the initial configuration of the subsequent task. The con-
figuration between two tasks is called commutation configuration [54]. In the motion
planning for a sequence of tasks, the motion trajectories together with commutation

configuration need to be considered simultaneously. If the commutation configura-

tions are determined without considering the current task in a global perspective, a
subsequent task may be rendered difficult. Most of the redundancy resolution ap-
proaches for executing a sequence of tasks are criteria based. For instance, Pin and
Culioli [54] used multi-criteria Optimization schemes for motion planning. The com-
mutation configurations become boundary conditions for each task in the sequence. A
variety of optimization criteria such as obstacle avoidance, reach and maneuverability
are developed and the algorithms for optimization problems are discussed. However,
the algorithm is off-line and nonholonomic constraints were not considered. Zhao,
Ansari and Hon [85] formulated multiple task path planning as a nonlinear and non-
polynomial optimization problem. The optimal sequence of platform positions and
manipulator configurations are obtained by using genetic algorithms. Lee and Cho
[42] pr0posed a different motion planning method for mobile manipulators to execute
a sequence of tasks. They considered the motion trajectories and the commutation
configurations simultaneously in a global cost function instead of treating them differ-
ently. A global optimization problem using the necessary conditions, which minimize
the objective function, can be applied to both holonomic and nonholonomic systems.

From the above review, it can be seen that motion planning of the platform and
the manipulator arm is crucial for the coordination of the mobile manipulator. The
existing approaches to coordinate the platform and the robot arm, or in the other
world, to resolve the kinematic redundancy, are either based on additional tasks or
based on the optimization of criteria. For the additional tasks based approach, the
end effector tasks may require an extension of the end effector out of its workspace.
Since the motion of the platform and the manipulator is treated equally, mobility and
manipulation ability are not optimized. For the criteria optimization based approach,
a near optimal path for both the platform and the manipulator arm can be obtained.
However, it is generally computationally intensive and not eligible for realtime control.

Since most of the optimization algorithms for redundancy resolution is computation

intensive and have to be used off-line, this dissertation presents an online coordination
scheme based on the event-based approach. It optimizes the manipulation capability
of the robot arm and can avoid unexpected obstacles. The kinematic redundancy
resolution will be investigated based on the analysis of the robot arm singularities in
this dissertation.

B: Modeling and Control. To achieve effective and practical control, vari-
ous dynamic models and control schemes are proposed in the literature. The control
strategies are generally classified into two categories. One is decentralized control
of the mobile platform and the manipulator arm. The controller for each part is
constructed separately and then the interaction between the platform and the ma-
nipulator is considered. The other is a unified approach for the control of both the
mobile platform and the robot arm. In this case, the mobile manipulator is treated
as a redundant manipulator.

By considering the mobile robot and the manipulator as two subsystems, Liu and
Lewis [43] developed a decentralized robust controller for a mobile manipulator. The
reaction forces are considered in the dynamic models of the two subsystems. However,
only the translation of the mobile robot is considered in the paper. Chung and
Velinsky [18, 19] also derived the dynamic model of nonholonomic mobile robots and
manipulators separately. The kinematic redundancy is resolved by decomposing the
mobile manipulator into two subsystems, the mobile platform and the manipulator.
A robust interaction control algorithm, in which suitable controllers are designed for
the two subsystems, is developed. The algorithm is employed to minimize the adverse
effect of the wheel slip on the tracking performance.

Yamamoto and Yun [84] studied a two—linked planar mobile manipulator subject
to nonholonomic constraints. The dynamics model was derived and reformulated
according to the nonholonomic constraints. Output feedback linearization is used

to simplify the nonlinear model and derive the controller. The mobile platform and

10

the manipulator are coordinated based on the concept of a preferred operating re-
gion. However, if the manipulator was commanded to follow a spatial trajectory in
the task space, it is difiicult to keep the arm in a preferred region during a task
execution. Colbaugh, Trabatti and Glass [20] considered the dynamics model for
redundant nonholonomic mechanical systems and reformulated it into two parts, a
reduced dynamics model and a kinematic relationship which considers the nonholo-
nomic constraints. Adaptive control schemes were developed based on the two parts.
The mobile platform and the arm are coordinated such that the distance between the
end-effector and the mobile base does not exceed a pre-specified value.

Two problems are brought to light for the control structure of the mobile ma-
nipulator system, one is to ensure the system output regardless of the output of the
subsystems. It is therefore desirable to model the mobile manipulator such that mo-
tion planning and force/ position control can be considered in a unified task space
coordinate frame. The other is the system stability in the case of unexpected ob-
stacles or desynchronization of the two subsystems. The non-time based controller
design for the coordination is investigated for this purpose.

C: Force Control Schemes. Robot manipulator force control schemes such
as external force control, hybrid position/ force control, impedance control have been
applied to the force control of the mobile manipulators. Umeda et. al. [73] discussed
the hybrid position / force control of mobile manipulators. In order to obtain desired
hybrid control with respect to the end-effector’s tasks, the concept of performance
indices based on an equivalent mass matrix is introduced. A weight matrix is also
proposed to obtain adequate torque distribution. Antonelli, Sarkar and Chiaverini [3]
presented a force control scheme for underwater vehicle manipulator systems. The
environment is modeled as a frictionless and elastically complaint plane. External
control structure is used and kinematic redundancy is considered. Perrier, Cellier

and Dauchez [53] compared the robustness of hybrid postion/ force control structure

11

and external force control structure with respect to external disturbances for a mobile
manipulator. The effect of mobile platform velocities on the force control structures
were discussed.

Due to the kinematic redundancy, the force control and position control of mobile
manipulators can be decoupled along the same task direction. This dissertatoin will
discuss the decoupled force/position control based on a linearized and decoupled

model.

1.4 Mobile Manipulation Using Force Control

The integrated task planning and control of a mobile manipulator performing the
task of a moving object has relatively received less attention. Based on the decoupled
force/psoition control of mobile manipulator, this dissertation presents an integrated
task planning and control approach for the task of the mobile manipulator pushing

a nonholonomic cart. As shown in figure 1.3, the mobile manipulator and nonholo-

 

Figure 1.3: The mobile manipulator and nonholonomic cart system

nomic cart system is similar to a tracker-trailer system. In a tracker-trailer system,

there is generally a steering mobile robot and one or more passive trailer(s) connected

12

together by rigid joints. The tracking control and open loop motion planning of such
a nonholonomic system has been discussed in the literature. The trailer system was
controlled to track certain trajectory using a linear controller based on the linearized
model[22]. Exact linearization and nonlinear control of the track-trailer system has
been discussed in [58] and experiments on backward steering the trailer have been
reported in [40]. Instead of pulling the trailer, the tracker pushes the trailer to track
certain trajectories in the backward steering problem. Fire truck steering is another
example for pushing a nonholonomic system. A chained form has been used in the
open loop motion planning of such a system [10]. Based on the chained form, motion
planning for steering a nonholonomic system has been investigated in [45, 70]. Time
varying nonholonomic control strategies for the chained form can stabilize the tracker
and trailers system to certain configurations[60].

The planning and control of the mobile manipulator and nonholonomic cart system
is different from a tracker-trailer system. They are not linked by a rigid joint, but the
robot arm. Therefore, the mobile manipulator has more flexibility and control while
maneuvering the nonholonomic cart. In a tracker-trailer system, control and motion
planning were considered based on the kinematic model, and the trailer is steered by
the motion of the tracker. In a mobile manipulator and nonholonomic cart system,
the mobile manipulator can manipulate the cart by a regulated output force. The
nonholonomic cart can be controlled at a dynamic level by the output force of the
mobile manipulator. However, this requires full integration of the motion planning
and force planning of the cart, and the planning and control of mobile manipulators.

Though ample literature discusses force control in the context of manipulator
arms, little of it considers the force control of mobile manipulator and none of it
discusses the out force and position control along the same task direction. This
dissertation considers the output force control and position control simultaneously

making use of the kinematic redundancy.

13

1.5 Contributions and Outline of this Dissertation

This goal of this dissertation is to investigate new control and coordination algorithms
for a mobile manipulator. The contributions to the control and coordination strategy
of mobile manipulators can be grouped into four parts. The first part discusses a
motion reference projection method for non-time based tracking controller design
and its application to nonholonomic mobile platforms. The proposed design method
for non-time based tracking controller converts a controller designed using traditional
time-based approach to a new non-time based controller. This approach significantly
simplifies the design procedure to achieve a non-time based controller. This method
is applied to tracking controllers of mobile platforms. The experimental results have
clearly demonstrated the advantage of the non-time based tracking controller in the
presence of unexpected obstacles. Applying this method to the mobile manipulators,
a common motion reference based on the measurable system output and desired path
can coordinate the motion of the platform and the manipulator arm. The advantage
of the non-time based tracking controller while coordinating with robot is also shown.
More importantly, the results provides an efficient and systematic approach to design
non-time based tracking controller for general nonlinear dynamical systems. This
part is discussed in Chapter 2, Chapter 3 and part of Chapter 5

The second part presents singularity analysis of manipulator arms and a hybrid
system design of singularity-free motion controller for manipulator. Since the mobile
platform creates redundancy to mobile manipulator system, the redundancy enables
the manipulator arm to keep its optimal configurations and avoid singular config-
urations. Chapter 4 analyzes the singular configuration of robot arms, which will
provides a mechanisms for the coordination of the mobile platform and the robot
arm. A singularity-free motion controller is also designed for the manipulator using
a hybrid system approach.

The third part formulates the unified control of mobile manipulators and discusses

14

the redundancy resolution. A unified dynamic model for the mobile manipulator is
derived and nonlinear feedback is applied to linearize and decouple this model. The
advantage of this model lies in that the kinematic redundancy and dynamic properties
of the platform and the robot arm have been considered in the unified model. Based
on this model, mobile manipulator tasks such as path tracking control, force / position
control can be easily designed. Based on the analysis of the robot arm, an online
event-based coordination scheme is proposed for the cooperation of the platform and
the robot arm. This online approach is suitable for a point-to—point task, multiple
segment tasks and teleoperation. The mobile manipulator system is stable even in
the case of appearance of unexpected obstacles and the manipulation capability can
adapt to a best possible values during the operation. This part is discussed in Chapter
5.

The forth part discusses the force/ position control while the mobile manipula-
tion interacting with moving objects. Based on the decoupled and linearized model,
force/ position control along the same task direction is discussed utilizing the kine-
matic redundancy of mobile manipulators. For tasks that require both force and
position control along the same task direction, such as pushing a nonholonomic cart,
an integrated task planning and control for mobile manipulators is proposed. The
approach for manipulating the nonholonomic cart integrates the planning and control
of the mobile manipulator, the motion planning and force planning of the nonholo-
nomic cart, and the estimation of the cart configuration based on the sensors of the
mobile manipulators. The integrated task planning and control enables the robot
to interact with moving objects and complete complicated tasks by the output force.
The approaches discussed in this dissertation can be used for the motion planning and
control of mobile manipulators which can find applications in manufacturing, hospi-
tals, homes and offices. Experimental results have demonstrated the advantage of the

integrated task planning and control approach. This part of the work is presented in

15

Chapter 6

Chapter 7 concludes this dissertation.

16

CHAPTER 2
MOTION REFERENCE PROJECTION

2.1 Introduction to Event-Based Planning and Control

The basic idea of event-based planning and control [77] is to introduce the concept
of a motion reference, a parameter of the desired path, as shown in Figure 2.1(b).
This parameter is directly relevant to the sensing measurement and the task. Instead
of time, the control input is parameterized by the action reference parameter. Since
the action reference is a function of the real time measurement, the values of the
desired vehicle states are functions of the measured data. As a result, for any given
instant of time, the desired input is a function of system output. This creates a
mechanism to adjust or modify the plan based on the measurements. Thus, the
planning becomes a closed loop real-time process. The plan (desired path) can be
perceived as an “abstract” entity, like a function. It only has a “real” value when
measurements are made. The non-time based planning and control scheme and the
traditional time-based planning and control scheme are compared in Figure 2.1(b).
In this dissertation, event-based and non-time based are used alternatively.

In the non-time based planning and control scheme, the function of the Motion
Reference block in Figure 1(b) is to compute the action reference parameter s on-
line, based on sensory measurements. The planner generates the desired value to the
system, according to the on-line computed action reference parameter s. In addition,
the motion reference parameter is calculated near or at the same rate as the feedback
control. In other words, the action plan is adjusted at very high rate, which enables
the planner to handle unexpected or uncertain discrete and continuous events. For
example, the event-based planning and control scheme has been successfully applied
to deal with unexpected obstacles in a robot motion [81]. If the robot is blocked by

obstacles, the action reference also stOps evolving. As a result the desired input will

17

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

d(t) cm W)
Planner y + Controller Robot 4
(a) Traditional control
[dm
d(S) e(5) y(t)
Planner y + Controller Robot :
8 Motion Reference
(b) N ontime based control

Figure 2.1: The comparison

be constant and the error will remain unchanged. In this case, if the conventional
time-based motion plan were implemented, the system error would keep increasing
as the planner generates the desired input according to the original fixed plan. The
system would therefore become unstable. To solve this problem with the traditional
method, one has to incorporate a higher-level monitoring function such as “move
until touc ” or some kind of compliant motion strategy, both requiring extra sensing
(force/ torque or touch/contact sensing, see [44] and [16]). Furthermore, once the
obstacle is removed, the traditional methods require planning again to complete the
task. But our non-time based planning and control in this case does not require any
extra force-torque or contact sensing, or planning again. The experimental results in
[81] clearly demonstrate the feature.

For a path tracking task, the most natural thought is to pick the distance traveled

along the given path as the motion reference 3. After .9 is chosen, the motion along

18

the given path can be described as [77]

dif—

2.1
fl_ ( )
(It—a

where v and a are velocity and acceleration along the motion reference .9. In order
to obtain an non-time based trajectory plan, it is necessary to perform a transfor-
mation of variable by replacing the independent scalar variable t in eq. (2.1) by the
independent scalar variable 3, which is computed from system output. Basically, the
event-based trajectory planning is to find the motion profile as a function of the given
event-based motion reference 3, v 2 12(3) and a = a(s), subject to the kinematic and
dynamic constraints of the robot.

Different from [7 7], which focuses on applications of robotic arms, the framework
here is based on general dynamical systems of moving vehicles, including robotic
arms, ground vehicles, aircraft, spacecraft, and surface and underwater vehicles. Ex-
cept for some ideas used in [77], the viewpoint adopted in this dissertation is different
from some existing non-time based approach. Instead of developing design algorithms
based on the specific features of the desired path ([32]), the controllers are first de-
signed in the time domain using any existing tracking controller design method. Then,
the time variable in the feedback is substituted by a special transformation called a
motion reference projection. The final resulting feedback becomes non-time based
feedback. It drives the system asymptotically approaching the desired path.

This approach has the following four advantages. First, the model of the system
is a general nonlinear dynamical control system. The method is applicable to a wide
range of tracking control problems of autonomous vehicles or robotic arms. It has the
potential of performing cooperated control of multiple vehicles with different models

moving in formation ([71] and [34]). Second, a transformation is provided to transform

19

a time dependent controller into a non-time based controller. It bridges the traditional
state trajectory tracking with the non-time based design. The results enable us to
design non-time based feedback using the existing, well known methods of feedback
design in time domain such as LQR and H00 control, and feedback linearization. The
third advantage is the flexibility in choosing motion references. The arc length of the
desired path is often used as the motion reference. However, there are many curves for
which the formula of computing the arc length is complicated. Parameters other than
arc length can be used as motion references in the design to simplify the controller.
In the proposed method, the motion reference can be any parameter. The fourth
advantage of our method is that the transformation from the state space to the action
reference needs not to be the orthogonal projection. For many curves, orthogonal
projection requires on-line numerical solution of minimization. Our method allows
the designer to use any mapping satisfying a projection condition. A good choice of
the reference projection can dramatically reduce the on—line computational load.

In the following, the concept of reference projection method is first introduced.
The stability of the non-time based controller is studied and a general method of

non-time based tracking control feedback design is developed.

2.2 Reference Projection Method

Mechanical systems and electrical systems are modeled by differential equations in
which the free variable is time t. A desired trajectory is usually modeled as a function
of time, which is generally denoted by :rd(t). The variable :1: represents the state
of the system. Controllers are designed so that the trajectory of the system :z:(t)
asymptotically approaches the desired trajectory :rd(t). This is a typical tracking
control problem. There exist a lot of control design methods in the literature of
tracking control. Since the design is based on the model driven by t, the controller is

likely to be time dependent. However, in many applications of autonomous vehicles,

20

a feedback controller independent of time is preferred. Furthermore, the desired path
is not necessarily defined as a function of time. It is defined by parametric equations.
The parameter is called motion reference or action reference. It is denoted by 3. For
instance, a desired path can be defined by a parametric equation :cd(s), where s is the
arc length, or s is the projection of :54 to a coordinate axis. In this section, we prove a
result, which transforms a time dependent feedback controller into a time independent
controller to asymptotically track a desired path driven by action reference. A system

is defined by the equation

a: = f(:1:,u), :1: E 3f",u E 33’",

y = h(:r), y E iii"

(2.1)

where u is the control input, and :1: is the state of the system. The desired path usually
does not involve all the state variables explicitly. For example, to make a mobile robot
following a curve, the desired path is defined by the position of the system. The state
variables such as the orientation of the vehicle, the angle of the front wheels do not
appear explicitly in the desired path. Suppose that y = h(:r) represents the output

of the system. The desired path is given by a parametric equation

31 = MKS),

where s is the motion reference.

The first step in the controller design is to define a corresponding path in time
domain. If s is not time, then a strictly increasing function 3 = v(t) is assigned. How
to pick the function v(t) depends on the desired speed. For example, if the problem
requires that the system is operated so that s is increasing at a constant speed v0,
then

DUI) = ’Uot.

21

Notice that the desired trajectory is defined by two functions, yd(s) and s = v(t), not
by a single desired trajectory yd(t). We do this on purpose because moving a vehicle
along a curve, and how fast a vehicle moves are two relevant but different issues in the
planning. They should not be mixed together. In the implementation of the tracking
control, v() can be a proportional to time t, such as v(t) = vot. It can also be related
to the present of obstacles and the task specifications. Denote d as the distance form
the robot to the obstacle in the moving direction, v(-) can be designed as v(d, t) such
that obstacle avoidance can be considered.

The second step is to find a feedback, u(a:, t), to track the path yd(v(t)). There
exist many well-known design algorithms in the literature of control theory. The

feedback satisfies
1im<h<x<t>> — mm» = 0.

t—aoo

Furthermore, if the initial condition is on the desired path, then the trajectory of the

controlled system follows the path. More specifically, there exists an initial condition

of the system x0 such that h(;r(t)) = yd(t). Denote this path by 32,;(3) or zd(v(t)).
The third step is to find a suitable transformation, .9 = 7(3), from the state space

to the reference 3. The transformation satisfies

«was» = s (22)

For example, given any state .100, let :rd(so) be the orthogonal projection from $0 to
112.1(3). If 7(320) = so, then it satisfies (2.2). The transformation satisfying (2.2) is
called a reference projection. However, orthogonal projection is not the only way to
define 'y. In section 4.1, the projection to :r-axis is adopted as 7, which is a simpler
formula than orthogonal projection for the specific application studied in that section.

The last step is to construct the non-time based feedback. A virtual time is defined

22

T(:r) = v‘1(7(:r)). (2-3)

It is worthy noting that virtual time is given as a function of system output. The
system output is finally a function of time, the actually system operation is considered
in the virtual time T(;r). By substitute the virtual time T(:z:) for real time t, the non-

time based feedback is

u(:1:) = u(:r, T(:r)) (24)

where 21(23, t) is the feedback found in the second step, 7(a) is a reference projection.

The closed-loop system is

Ii“ = f(:1:,u(a:,T)). (2-5)

2.3 Stability

In the above design process, the feedback u(a:,t) drives the system asymptotically
approaching xd(t). The non-time based design is given by substituting the time
variable t in u(:r, t) by the virtual time v’1(7(:r)). Because of (2.2), the trajectory :r(t)
of (2.5) equals :rd(t) if .r(0) is on the desired trajectory rd. Based on our analysis and
simulations, the non-time based feedback is likely to drive the system asymptotically
approaching $d(s). However, the substitution t = T(:r) may change the stability of
the system in some cases as shown in the following example. In this section, sufficient
conditions are given to guarantee the asymptotic stability of the tracking error under

non-time based feedback.

23

2.3.1 Ezrample

Two feedbacks are given in this example for the same tracking problem. One feedback
result in a stable non-time based controller, and the other one result in an unstable

controller. The control system is linear,

271 —2 —2 $1 1
= + u (2.6)
(I12 2 1 $2 —I
It is a controllable system. The desired path is the line m2 = —1, i.e.
s
5,3,,(3) = , (2.7)
—1

Suppose that the velocity on the desired path is 1, so 3 = t. A time dependent
feedback is

u 2 2t — 1. (2-8)

It is easy to check that the error :r(t) — :rd(t) satisfies

e = —2 _2] x1 ] e. (2.9)

2 1.[_$2

It is asymptotically stable. The simplest reference projection for the desired path is

the orthogonal projection

7(r) = 171.

24

Therefore, the virtual time is T = 1:1. The control system under non-time based

control is
$1 —2 —2 $1 1
= + (2131 — 1)
$2 2 1 $2 —1
: (2.10)
—2$2 —- 1
$2 +1

 

The second equation in = $2 + 1 is unstable. So, the system does not approach
the desired path under the non-time based controller, although the non-time based
feedback is converted from an exponentially stable feedback (2.8).

On the other hand, if the time-variant feedback is

3 3
u=2t~1+§($1—t)+§($2+1).

It is easy to check that the time dependent feedback drives the system approaching

xd(t). Furthermore, the non-time based feedback is
3
u(:1:)= 2$1 — 1+ 5(1‘2 +1).

The system under the non-time based control is

[.1 [am

. = l 12 2 (2.11)
. __ , 1
[_ $2 [- 2(152 + l )
The second equation guarantees that lim 232(t) = —1. So, the non-time based control
drives the system asymptotically approaching :rd.
The example implies that not all the stable feedback u(.r, t) can be converted into

stable non-time based controllers. In the following, we derive conditions under which

25

the non-time based controller is guaranteed to be stable.

2.3.2 Error dynamics and stability

Two error dynamics are derived in this section for the stability analysis. The first

error dynamics is under the time-variant feedback u(a:, t). The error of the system

i: =2 f(as,u(a:,t))

is defined by

e = a:(t) — :cd(t). (2-12)

It satisfies

é : f(rc,u(:r,t)) — f(xdi “'(xdvtD
= f(e + 1rd, u(e + and, t)) — f(e + 33.1, u(e + and, t)).

aim

The desired trajectory :rd(t) can be considered as a function of time t. The error

dynamics is a time-variant system. Define

E(e, t) = f(e + and, u(e + rd, t)) - f(e + 27.), u(e + 13.1, t)). (2-14)

The error dynamics under u(:r:, t) is

é=E@¢) aim

Under the non-time based feedback u(.r, T(r)), the error is defined by

e = :r(t) — :rd(T). (2'16)

26

The system is not tracking the trajectory as a function of real time. The controller
suppose to drive the system toward the desired state at the virtual time. The error

dynamics is changed due to the derivative of T(:1:).

é = f(x,U($.T)) —fear).u<zd(T),T>)%§f<x.u(x.T))
= f($, U($,T)) _ f($d(T), “($d(T)1T)) _ f($d(T)7 U($d(T),T))(?I-f($, U($, T» — 1)

031;
(2.17)
From (2.14) and the relation
71(3): a—f($1u($vT))a
we have
é= Ear) — fear). .1..(T),T))(T(.) - 1). (218)
:i: = f(rr,u(:1:,T(:1:))) (2.19)

System (2.19) is treated as part of the error dynamics. System (2.18) is not a complete
system because T(:1:) and T depend on (2.19), which is the vehicle dynamics under
non-time based feedback.

The two errors e and 6 defined by (2.12) and (2.16) follows different dynamics.
The stability of e does not automatically imply the stability of e'. If we treat T as an
output of (2.19), them its performance is critical for the stability of (2.18). In this
dissertation, a 6 neighborhood of :rd(s) is the set of all :1: such that the distance from

the point :1: to the curve rd is less than 6. In this dissertation, we assume

0f 8f (911

Assumption A1: The vector fields and functions f(at,u(:r,t)), u(.r, t), 2);, a, a,

27

BuaT

79—, 5; are all bounded for t > 0 and a: in a 6 neighborhood of 3:0,.

The nonlinear function f (:17, t) represent the rate of change of the state variable.
It is reasonable to require in A1 that the speed, acceleration of the state variables of
a mechanical system is bounded near the desired path. A1 also assumes that u(a:, t)
and its derivative are bound. It implies that the controller can react to the change of

state and time variable at a limited rate.

Theorem 2.3.1 Suppose (2.15) is uniformly exponentially stable in He” < 6. Sup-

pose the system {2.19) has solution fort > 0 and the solution satisfies

lim T(:1:(t)) -— 1 = 0

t—roo

uniformly and exponentially if ||é(0)|[ < 6. Then, there is a 61 > 0 so that

lim E(t) = 0

t—+oo
provided ||é(0)|| < (51.

Proof. From the converse theorem of stability, there exists a Lyapunov function

l/(e,t)

aillellz S V(e,t) S azllellz, t2 0,

 

3V

EM] 3 asllell, t2 0. (2.20)
av av ,

_ _ = — > .
&+%flm)lw.u0

in a neighborhood |e| < 6. Under the non-time based controller, the error is replaced

by 6 in (2.16). The Lyapunov function for e is V(é',T(.r(t))). Its derivative in the

28

direction of (2.18)-(2.19) is

War) = gm 96—:(E(é.T)—f(xd,u(ard,T))(T-1))
3V 6V 6V 8V °
= EFL 8_E(e e,T)+ (ET—T93 (121. (12.13») ("F-1) (2.21)
= -||e ”2+ +(gTK—Z—‘ffa. M<x.,r)>)< -1>

From Al, we know that E (e, t) is bounded in Hell < 6. So, the third equation in

(2.20) implies that gift/T

V
equation of (2. 20), it is known that 33—2: (6, t) is also bounded. Therefore,

(6, t) is bounded in 0 S t _<_ co and Hell 3 6. From the second

2%?) — %—‘1<T>f<xd. new) (222)

is bounded in 0 g t S 00 and 'Iéll _<. 5. Since T(:c) — 1 is uniformly exponentially

convergent to zero, we have

8V 8V ))(\ _M
(8—7,- - T95 (1311,71 (”(11W) — 1)< Me (2'23)

for some M > 0 and A > 0, provided [lél' < 6. Equations (2.21) and (2.23)implies

V(e,T) < —[Iéll2+llfe"\t

1 (2.24)
< —a—V(é,T) + Me‘”, for He“ < 6.
2
At any fixed time to, if E(to) < 6, then
- 1
V(e, T) < ——V(é,T) + Me‘Moe‘Mt—m), for t> to (2.25)

012

29

This implies

d

_ (+2040) _ —).to (-X+;’-)(t—to)
(“(6 V(e,T))<Me e 2 . (2.26)

The integration of the inequality over the time interval [to, t] yields

; _ 1 _ _ _
e02“ “Wu-er) — vaccinate») s Mere—3:1: (e‘ “32’“ ‘°’ — 1)
— 1

(2.27)

It is equivalent to

— —i(t—to) — NIB—Mo -A(t—t ) ——‘—(t—to)
< .. —— 0 a2
V(e,T) _ e 2 V(e(to),T(:r(to))) + —). + 0;, (e + e ),
(2.28)
provided t 2 to and ||é(t)|| < 6. There exists to > 0 so that
Me‘M" ,\ _m _ a
— (t—to) 0.2“ t0) _1 2
——_)\ +01—1 (6 +6 ) < 2 6 , (2.29)
for t 2 to and ”(all < 6. If [|é(t0)|| < 20—162, (2.20) and (2.28) imply
0t2
al||é||2 S V(e,T()
s waitress» + $5?
a
s a2|lé(to)ll2 + 7‘62 (2.30)
(11 2 O1 2
< ——6 —6
_ Q2202 + 2
_<_ 0162.
Therefore, ]|é(t)|| S 6 for all t 2 to, provided
a
lléltolll < 2—162. (2.31)
0’2

30

Thus, (2.28) holds for t Z to if the initial error satisfies (2.31). From (2.28), it is
easy to derive that 3111.1) V(e,T) = 0, provided (2.31) holds. Therefore, 11120 E(t) = 0.
Because the right side of (2.18) is uniformly bounded in HE?” < 6, there exists 61 > 0
so that ||é(0)|[ < 61 implies (2.31). This proves that

lim E(t) = 0

t—roo

if new)“ < 5.. 9

2.3.3 Feedback design

Given a time-variant tracking feedback u(;1:, t) and a reference projection, one needs

to check the performance of the following system with dummy output y“ = T(:c) — 1,

:i: = f($,U($,T($)))

(2.32)
y* = gramme)» —1

If the dummy output y“ exponentially converges to zero, the non-time based control
drives the vehicle approaching the desired path. In this section, a design method is in-
troduced if T(:r) has relative degree. The feedback guarantees that y“ is exponentially

stable. In the following, we assume that the desired path satisfies

ydi = 8

= s
31112 yd2( ) (2.33)
ydk = ydk(3)-

31

Or equivalently, the first output variable ydl serves as the action reference. This is true

for many applications as shown in the next section. A natural reference projection is

V(l’) = 311(3) (234)

where yl is the first output of the original system (2.1). We also assume that in a
neighborhood of the desired path, reference projection 7(:1:) = y1 has relative degree,

i.e. 7(23) satisfies

a - .
aL}(i,u)(7($)) = 01 1 S ’l S 7',
a (2.35)

where L f(xm) is the Lie operator defined by

name» = gig-rm).

The following design method follows the idea of feedback linearization (see, for in-
stance, [30]).

If the desired speed on the desired path is defined by s = v(t), then the virtual
time is T = v‘1(7(:1:)). The feedback is defined based on the virtual time T(:r:). It
is easy to check that T(:1:) also has relative degree. Define 5,- = L}?1 (T(:1:)) for

1,11)

1 S i S r. The equation (2.35) implies

a. = a
.52 2 63 (2.36)

5} = b(a:) + alts->211 + (12(2)... + - - - + a...(a:)um, (11(1) a o.

32

The feedback for ul has the form

u1(:r,t, 11.2, - -~ ,um) = _a1(:c) (b(.1:) + a1(:l:)u1+ a2(:r)u2 + - - - + am(:z:)um)

+k1(€1—t)+ k2(§2 — 1) + (9353 + ' ° ' + krér.

 

(2.37)

The gain k1 k2 . .. kr satisfies that
3’ — krsr—l — 19,45"? — — k1 is Hurwitz, (238)
s"1 — krsr‘2 — k,_lsr‘3 — - - - — k2 is also Hurwitz if r > 1. (2-39)

In this system, the dummy output y“ = T — 1 is completely determined by (2.36)-
(2.37), which has the following form

£1=£2

(52 2 53 (2.40)

G = k1(€1-t)+ k2(€2 - 1) + 16363 + - - ' + krér

Because(2.38) is Hurwitz, the control guarantees that both {1 — t = T(z) — t and

{2 - 1 = T - 1 exponentially approach zero. If t is replaced by T, the variable 51 — t

is constant zero. The closed-loop system (2.40) becomes

£1=§2
52:53

{r = k2(€2 - 1) + 16363 + ° - - + krér

33

Because (2.39) is Hurwitz, {2 — 1 = T — 1 exponentially approach zero under u1 =

u1(:c, T, U2, - -- ,um). Furthermore, the tracking error yl — ydl(T) satisfies

yi — yd1(T) 2 7(17) — v—I(T)
= v‘1(T) — v‘1(T)
= 0

Therefore, the tracking of y1 is guaranteed. Applying (2.37) to the original system,

we have

i. : f(x,u1($,t,u2,--- yum))u27” ° yum) (241)

For this system, the control inputs are U2, u3, - . - , um and the desired path to track
is {yd2(8),yd3(8), - -- ,ydk(s). The feedback for u,, i Z 2, can be designed using any
tracking control design algorithms. Then, replace the time variable t in the feedback
to obtain a non-time based controller. The stability is guaranteed because (2.37)
drives T — 1 approaching zero.

To summarize, we assume that yd1(s) = s is the action reference. The first output
y1(a:) is defined to be the reference projection 7(23). Assume that 7(cc) has relative
degree, then feedback ul is defined by (2.37) satisfying (2.38)-(2.39). The other
feedbacks are designed to tracking y,, for z‘ 2 2. The final non-time based feedback
is to replace t by T(:z:) in all the feedbacks. It is guaranteed that the non-time based

feedback drives the tracking error to zero.

34

CHAPTER 3
TRACKING CONTROL OF THE MOBILE PLATFORM

3.1 An Introduction to the Control of Mobile Platforms

This chapter presents a novel method of non-time based controller design for path
tracking of moving platforms. The purpose of this proposed controller is to design
an event based controller which can coordinate the motion of mobile platform and
manipulator arm. In this chapter, the state variables of the mobile platform are
adOpted as the motion reference. In the control of mobile manipulators, a global
state variable is used as the motion reference of both the mobile platform and the
manipulator arm.

The key to the proposed control method is the introduction of a suitable motion
reference other than time, which is directly related to the desired path and the mea-
surable system output. It enables the construction of control systems with integrated
planning capability, in which planning becomes a real-time closed-loop process. As
a result, the control system is capable of coping with unexpected events and un-
certainties. This new design method converts a controller designed with traditional
time-based approaches to a non-time based controller using a reference projection.
It significantly simplifies the design procedure for a non-time based controller. The
method is exemplified by the design and experiment for the tracking control of car-like
mobile robots. Based on the properties of the desired path, two car-like mobile robot
controllers are proposed. According to the type of the desired trajectory, the time
based controller is mapped to a non-time based controller by choosing appropriate
motion reference. The controllers are implemented and tested for different trajecto-
ries such as sine waves, circles and ellipses. Lab experiment on mobile platform are
carried out in an environment with unexpected obstacles and unexpected stop for at

arbitrary time.

35

3.2 Path Tracking Control for Mobile Robots

3.2.1 Mathematical Model of the Robot

In this section, the general design algorithm is applied to the model of mobile robot
shown in Figure 3.1. The rear wheels are aligned with the vehicle while the front
wheels are allowed to spin about the vertical axes The robot is a rear wheel driven
robot. The system is constrained by allowing the wheels to roll and spin, but not slip.
Let (x, y,0, ¢) denote the configuration of the robot, parameterized by the location

of the rear wheels. The dynamical model of the mobile robot can be represented as

[45]:
i: = ulcosél
g = u sin6
. ”I (3.1)
6 = jitanqb
(IS: “’27

where ul corresponds to the forward velocity of the rear wheels of the robot and U2
corresponds to the angular velocity of the steering wheels, the angle of the robot body
with respect to the horizontal axes is 6, the steering angle with respect to the robot
body is ¢, (1:, y) is the location of the rear wheels, l is the length between the front
and the rear wheels.

Let Xd = [ctd(t), yd(t), 64(t), ¢d(t)] denote the desired states of the mobile robot at
time instance t, the purpose of the path tracking control is to design a controller ul é
u1(Xd,t),u2 = u2d(Xd,t) such that the robot will asymptotically track the desired
path yd = f (1.3). By using the reference projection method, the time based feedback
laws can be converted into the time invariant feedback 211(Xd, T(-1:)) and 112(Xd, T (13))
The virtual time T(a:) is designed by the non-time based motion reference 5.

The path control of a nonholonomic robot has been discussed in the literature.

36

 

 

 

 

>

X

Figure 3.1: The configuration of a mobile robot.

The system model can be linearized based on certain path and a feedback control was
designed using the LQR approach[22]. The disadvantage for this class of methods lie
in that the controller is path specified. The controller is dependent on the charac-
teristics of the path. Exact linearization approach, such as the chained form, has
been developed and used in the open loop path planning [45] and feedback control
design [21, 60]. However, singular points exist in the derivation of chained form, the
feedback controller based on the chained form therefore cannot track certain path.
In this section, two classes of paths are discussed. One is curve without U-turns, an
exact linearization and feedback control design similar to chained form is described.
For a curve with U turns, an exact linearization is presented and feedback controller is
designed. Both the time based controllers are converted to non-time based controllers

using the reference projection approach.

37

3.2.2 Tracking Curves with no U-tum

Suppose that the desired path is the following function of time

:6 = wt), 31 = yd(t)-

In this section, a desire path without U-turn is assumed, i.e. for any value and, there
is at most one value yd such that (:L'd,yd) is on the desired path. This implies that

id(s) > 0. Let’s first find a feedback for ul. Since

d
—(a: — $01) = ulcosfl — std,

dt

the control input ul can be defined by

u1 = —1—(:'cd - (a: —— 1:01)). (3.2)

Under (3.2), the first equation of (3.1) becomes

i(:1: — 23(1): —(:23 — 13d).

dt

Its solution asymptotically approaches to zero as t ——> 00.

The following change of coordinates is defined to achieve the linearization of the

38

last three equations in (3.1),

21 = y
22 = sin6
z3 = leosOtangb
1 m (3.3)
01 = _27 sin 0(tan W2
[3 __ c050
1 — lcos2¢

U: (11+B1U2

The last three equations in (3.1) are transformed into

21 = ”“122
22 = U123 (34)
33 = ’0

Define

€1=Zi—yd

82:22—%{ (3.5)
1
1d yd

6 =2 ————

3 3 Uldt 11.1

the error dynamics of the last three equations is represented by

B1 = U182

é2 = U163 (36)
é _ v d 1 d E

3 — dt uldt ul

The right side of the last equation in (3.6) satisfies

(1 1 d yd _ Jd _ 3.2/am + 3mm _ WI (3.7)

—__— — 3

_ 2 3 4 :
dt uldt ul “'1 a1 “-1 u1

 

 

39

Notice that ill contains the term (f), which depends on 112. Therefore, the expression

(3.7) has the form

a2($9y363¢9 t) + fi2($3yi61¢’7t)u2 (3'8)
To stabilize (3.6), v is defined by

’U = a2($iy36)¢at) + [82(xvy363¢at)u2

(3.9)
+U1(0181 + 0.282 + (1383)
Then, (3.6) is equivalent to
81 = 21.182
é2 : 11.183 (3.10)

83 = U1((1.181 + 0282 + (1383)

Select the values of a,,z’ = 1,2,3, so that all the eigenvalues of

l010
A=001

(11 a2 03

 

 

d

are on the left half plane. Since :isd(s) > 0, ul (t) > 0 in a neighborhood of (:rd(t), yd(t)),

fot u1('r)dT increases. The solution of (3.10) is

e(t) 2 e0 exp(A/0 u1(T)dT).

Since A is stable, tlime(t) = 0 as fot u1(T)dT increases. Therefore, y — yd(t) ——) 0 when

t ——> 00. By solving for 71.2 from (3.9) and the last equation in (3.3),u2 can be found.

40

The feedback is

_ (I2 — 01+ 01(0181‘l‘ 0282 + 0383)

 

 

 

u 3.11
2 a — a l )
In summary, suppose the desired path is
$d(v(t)) = t. yd(v(t)) = $th (3-12)
The non-time based control ul and 112 can be computed as following
id - (:17 - (L‘d)
a1 : cos6 ’
u—a2—a1+ ul (ae+ae+ae) (3.13)
2 a-a a—a “ 22 33

The coefficients a1, a2 and a3 are obtained from the linear control design.
Remark. A circle or an ellipse does not satisfy the condition id > 0. 6 = 3:32: are
the singular points of the system. The controller (3.11) does not apply. This case is

discussed in the next subsection. <1

The feedback (32,311) depends on time. However, it can be converted into a time-
invariant feedback using reference projection as proved in § 2.2. Using the method
developed in § 2.2, the feedback (32,311) is changed to time-invariant feedbacks for
the tracking of the curve y = sin :r. The key step is to find a motion reference for
the task. The'design method proposed in this dissertation is not based on a specific
motion reference. It works for arbitrary choice of the reference. Different from circles
and lines, arc length and orthogonal projection are not the best choice if the desired
path is y = sin 1‘, because the computation of arc length for this curve is complicated.
In this case, it will be shown in § 3.3 that s = a: is a much simpler choice of the action

reference. The lab experiment based on this controller is shown in § 3.3.

41

3.2.3 Tracking Circles and Ellipses

As remarked in the previous section, the controller (3.13) fails to track trajectories
having U-turns. For a class of trajectories such as circles and ellipses, another con-
troller is proposed in this dissertation for the mobile robot. Based on the same state
equation (3.1) and the desired trajectory ($d(t), yd(t)), the system is first transformed
into a polar form. A controller is then derived based on the polar axis.

By defining

mm

212=tang
:c

(3.14)

which is equivalent to

a: = rcoszl) (315)

y = r sin 11)
the system dynamic model in polar axis is
. 1 . .
r = ;(m‘ + 313/)
1
= ;(r cos will cos 0 + r sin 21ml sin 6) (3-16)

= ul cos(0 — 111)

1 gm — (by
1 + (31/102 $62

 

1
= T—2(u1 sin 0r COSI/J — ul cos 6r sin 21') (3-17)

U1

= sin(0 — 1(1)

7.

42

System model (3.1) is then transformed into

”(A = % 3111(6 — 1(1)
7': = :1 cos(9 - 10) (3.18)
6 = 71 tan (12

ct}: w
The desired trajectory in Cartesian space is also transformed into polar axis and

denoted by rd, 112d, 6d, 96,) respectively. Considering only the first equation of (3.18), ul

is designed to stabilize i/J. By defining

7‘ .

= mWa - kiw - 1%)), (3-19)

111

where k1 is greater than zero, the first equation (3.1) becomes

in - w.) = —k1(w — a), (320)

which is asymptotically stable.
Substituting (3.19) into the second equation of (3.18) and defining e, = r — rd, it

becomes

i~ = r ctan(6 — maid — k1(zp — 31.1)). (3-21)

Taking ctan(6 — #1) as the control input of this equation, a feedback controller which

stabilize (3.21) could be

1

ctan(9 _ 1(1) : f(d’d _ [910,0 — 7(3))

 

(Td — k28r). (3.22)

43

where k2 > 0. Therefore,

6 = 11) + arcctan( . 1 (7'31 - (926%))

red - (WW) — 1a)) (3.23)

 

= 910311515)

For the third equation of (3.18), defining e9 = 6 — 91, the error dynamics of 0 can be

derived,

9 = :ul—ltanqb

11.1

l

(3.24)
tan 03 — 91.

Taking ()5 as the control input, the feedback controller for (3.24) is

(b = arctan(ifih — k389))
lsin(0 - 11))
f(lbd — (91(1)!) - 1%))

:2 92(T, 0, 1,0, t).

 

= arctan( (g. — k3e9)) (3.25)

For the fourth equation, let 64, = (b - 92,

62¢ 2 112-95 (3.26)

the controller of the system is finally designed by

112 : 92—k4e¢ (3.27)

44

To summarize, the time depended feedbacks of system (3.18) are

 

r .
u. = ma. — W — a»
U2 = 92 — (“46¢
sin(6 — w) . (328)
= . — k
92 arctan(rwd _ 191(1/1 _ 1%)) (91 389))
91 = ’1/1 + arcctan( . 1 (7'11 — Merl),

 

TWM — k1“? “ 11101))

where khi = 1, 2, 3, 4 are greater than zero.

The feedbacks in (3.28) are also time dependent controllers. However, it can be
converted into a time—invariant feedback using reference projection as proved in § 2.2.
Since the design method proposed in this dissertation is not based on a specific motion
reference, different references according to different curves are used. As it is shown

in § 3.3, a combination of the state variables can be used for convenience.

3.3 Experiments and Results

The non-time based controllers have been implemented on a Nomadic XR4000 mobile
robot. The mobile robot has four wheels. Each wheel can be individually steered.
However, in order to implement the above control scheme, only two front wheels are
steered in the experiments. The other two wheels maintain Straightforward config-
uration. The distance between front and rear wheels is 0.3053 m. In the following
subsections, three experimental results are given. In § 3.3.1, the results of controller

(3.13) are shown. In § 3.3.2 and § 3.3.3, the results of controller (3.28) are reported.

3.3.1 Tracking Sinusoid Waves

The desired trajectory is given by:

yd = sin(s) 1rd = s,

45

Figure 8 (Y Trajectory) Home O (X Trajectory)

 

 

 

 

 

 

 

    

 

 

 

 

 

 

0.8' 7.

A . get-

i . 25
E . 151.».

§

§ 23*

E - 32*

2 B, ,

-o.e 4 4 4 . o - - ‘

0 20 40 60 80 O 20 40 60 80
((Soc) f(Sec)
FigurecWEwor) “04 medMEmr)
005 f ' ' 7 ' ‘.' .'
o». 6»

-o.05» 5»~ - 1
E —o.1) EA» .
r r
é-OJS) 331

-0.2* 2 -

—0.25*~ , 1’ ‘1

-o.3 - A . o 4 - -

0 20 40 60 00 O 20 40 60 80
t(Sec) f(Sec)

Figure 3.2: Tracking a sinusoid curve with initial error.

where s is the motion reference. The velocity along the desired path is defined by
s 2 vi, where v is a constant. Notice that the desired velocity is a constant in the
direction x. But the desired velocity on the sinusoid wave is not a constant. This
change of desired velocity can be tracked by a non-time based controller using the
reference projection method from § 2.2.

In the controller, the closed loop poles are assigned to be —3, —5 + i, —5 — i. The
reference projection is s = x, and the synthetic time is t = $23. It is easy to check
that the relative degree of the reference projection is 1. Therefore, the feedback in
§ 3.2.2 satisfies the design requirements of § 2.3.3. It is converted to a non-time based
controller and then applied the N ormadic XR4000 mobile robot. If the initial position
of the robot is not on the desired path as shown in Figure 3.2, the controller is able
to command the robot to approach the desired path. There exists an initial error
of 0.3m in the experiment. But the tracking error converges to zero after the robot
starts moving. This experiment demonstrates the ability of the controller to rapidly
overcome initial tracking error. It is worthy noting that the XR4000 is a holonomic

robot, and it is simulated as nonholonomic robot in the experiment.

46

Figure a (Y Trajectory) Figure b (Trajectory of X)

 

 

 

 

 

 

 

 

 

 

3..
E “2.5
3. i ,
go's» E 2}. ./_/....
‘O
X15 .fl ,.
S if I
g o. g 1,. l/
E E _ _____ ~
31 :05 .-’_.,
—o.5 o i
20 0 1O 20 30
N300) ((Sec)
Home“ Traiectory) Figuredot Trajectory)
750.4 E 3»
)5: 0.2’ >525»
K 0“ Y 2
54,2... 5,5
3 ' i~ 3'
E : 5'
-°_8b . ”0.5’
-O.8 ‘ 0
O 10 20 30 40 50 0 1O 20 30 40 50
”$06) t(Sac)

Figure 3.3: Time based control vs Non-time based control.

The interesting result of the experiment is presented in Figure 3.3. While tracking
a pre-planned desired path, the robot was stopped by an unexpected obstacle at the
10th sec. Figure 3.3a and 3.3b show the results of a time-based controller. Its
desired input was a function of the time. Therefore the desired position would keep
changing even the robot was blocked. As a result, the tracking error kept increasing
until the system became unstable. However, for the non-time based controller, the
results are completely different. As shown in Figure 3.30 and 3.3d, when the robot
is blocked, the action reference remains constant because the reference projection
depends on the position of the robot. Therefore, the tracking error remains constant.
Furthermore, after the obstacle is removed at the 20th sec, the robot automatically
resumes the original motion without re—planning. This experiment demonstrates the
capability of the non-time based controller for coping with unexpected events. It
should be noticed that the non-time based controller is not simply replacing the time
by a non-time based motion reference. The new motion reference is determined by

sensory measurements. It reflects the key information of the state of the system. The

47

reference projection acting as an on-line real—time planner.

3.3.2 Tracking Circles

This experiment is done under the control of (3.28) with feedback gains k1 = 1, k2 =

0.4, kg = 8, k4 = 3. The desired path is

2

2 2
(rd + yd=a

yd = asin(s),:1:d=acos(.s)

where s is the non-time based reference. It represents the angle corresponding to the
tracked arc of the robot. The reference is computed directly by the location of the
robot, (2:, y). In the polar coordinates model of the vehicle, the reference projection
has relative degree 1. Given a constant relation between the reference 3 and the
time t, s = at, the synthetic time is t = iatan2(y,:c), where atan2(-, ) maps (y,:r)
to (0,27r). The radius of the circle in the experiment is 0.5m. Figure 3.4 and 3.5
recorded the experimental results of tracking a circle by the controller ( 3.28 ). From
Figure 3.4a-3.4d and Figure 3.5a—3.5d, it can be seen that the static tracking error is

less than 1mm.

3. 3. 3 Tracking Ellipses

This experiment is also done under (3.28). To track an ellipse, the desired path is

given by
E: + y_3 1
a2 b2

48

Figure a‘ Actual and Desired Trajectory Figure b: X and Y Trajectory

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0.6 A . j 06 ' 3’ ."
A E
E 0.4 -
x w 1%
0.2 -
0
—0,2
-0.4 - '
-06 1 ~ ,
-0.5 0 0.5
X0") «3)
Figure c: X Error Figure G: Y Error
0.01 . - - 7 001 - fi r
E E . ; .
o W
_0005 . _0005 ................................. ..
-°°1 * ~o.01 A -
10 20 30 40 10 20 30 40
t(s) t(s)

Figure 3.4: Tracking a circle without initial error

FrmreaActualandDesiredTraiectory Figure b: XandYTraiectory

 

 

 

 

 

 

 

 

 

 

 

0.6 , 0-6 T
A \
’5‘ 0.4 E 0-4 ‘
3' I?
0.2 0.2’ - r
0 o ......
—0.2 -O.2 ‘
.04 '0-4’ ..... , ..-. . .... . .4
-0.5 ' ’ ' _o 6 . . . . A _ . . ‘ .
-0 5 0 0.5 0 10 20 30 40
x(m) «3)
Figure G: Y Error
004 r i T

   

 

 

 

 

 

 

 

A I I A ‘ A > - L i _o M A A A
O 10 20 30 40 0 10 20 30 40
K3) 1(3)

 

Figure 3.5: Tracking a circle without initial error

where s is the non-time based reference. Instead of the original state variables in the
system, 3 = w is chosen as the motion reference in this experiment. It is computed by
the measured output :5 and y by d) = atan2(y, at). For simplicity in the lab experiment,

a constant relation between the reference and the time is chosen, though a variable

49

Figure a; Actual and Desired Trajectory

 

04>-

Y0“)

0.2 ‘
0
-0.2

 

—O.4 ~~

-O.6

 

 

 

-0.5 0 0 5
x(m)

Figure c: X Error

 

0.01 r v , v

(m)

 

 

 

 

t(s)

(m)

 

 

Figure b: X and Y Trajectory

 

 

 

 

10 20 30 40
t(s)
Figure d: Y Error

 

 

f

 

 

«3)

Figure 3.6: Tracking an ellipse without initial error

Home a- Actual and Deeimd Trajectory

 

0.3
E My, _
0.2 ~

0
-0.2 .
.04
-0.6

 

 

 

 

-o.5 o 0.5
X(m)

Figure c: X Error

 

(m)

 

 

 

 

0 10 2O 30 40
t(s)

IWU")

(m)

0.6
0.4

0.2 >

—0.2

-0.4 .

—06

Figure b: x and Y Trajectory

 

 

 

 

 

«3)
Figure (1: Y Error

 

 

 

 

A A A

 

10 20 30 40
t(s)

Figure 3.7: Tracking an ellipse with initial error

relation depending on the curvature of the path would be better. The relation is

defined by 112 = vt. The synthetic time is t = £21). It should be noticed that the

motion reference here is not a existed path parameter of the ellipse,i.e., the relation,

:1: = acosr/2,y = ()8in are not true.

50

Figure a: trajectory with respect to s Figure b: trajectory with respect to time
. . . 1 .

a

 

 

1

0.5- .

0.1V,

x(solid).y(dotted)
x(sotid).y(dotted)

 

_0_5...... . . ‘

 

 

 

 

 

 

 

_1 .9 i i _1 ; ;
0 2 4 6 8 0 20 40 60
motion refemeoe t(8)

Figurerxcontrolutwithreepecttotime Figuredzcontrolu2withreepecttotime
f 0.6 r. .

  
  

 

 

v

‘3 0,4

0.2

 

 

 

 

 

0 2‘0 ab so
t(s) t(s)

Figure 3.8: Tracking an ellipse with present of unexpected obstacle

To test the control law on a trajectory with large curvature, a small ellipse with
a = 0.6m, b = 0.5m is used. The feedback gains are selected as k1 = 1, k2 = 0.4, kg =
8, k4 = 3 in the experiments. From the experiment result (Figure 3.6 and 3.7), it
can be seen that the robot approaches the desired trajectory asymptotically and the
static tracking error is bounded by 5mm.

An interesting result is shown in Figure 3.8. A unexpected obstacle occurred at
time t = 16sec. When an obstacle is detected by the laser sensor equipped on the
mobile robot, it is seen that the robot is slowed down. This is done by relating
the motion reference to the distance with respect to the obstacle. Instead of using a
constant 220 to relate the motion reference and time t, the distance from the obstacle to
the robot, (1, is also considered into the relation. For simplicity of the implementation,
three situations are considered. When d is large enough, the distance to the obstacle
is not considered in the design of the motion reference. When d is small enough, 8 will
keep constant to suspend the system. It is scaled down gradually when the obstacle is

approaching. Of course it can also be designed as continuous function of the distance

51

to the obstacle. From figure 3.8(a) and (b), it is seen that the mobile robot almost
stopped when an obstacle is close. When the obstacle is removed gradually, the robot
resumes its action and the originally path is tracked. Figures 38(0) and (d) show the
control commands. This experiment has demonstrated that the relation 3 = e(t) can
be designed to accommodate the sensor information without any effect on the design

of the time based controller.

52

CHAPTER 4
SINGULARITY ANALYSIS AND SINGULARITY-FREE
CONTROLLER OF THE ARM

4. 1 Introduction

One advantage of the mobile manipulator is that the kinematic redundancy can be
used to avoid singular configuration of the manipulator arm. The purpose of inves-
tigating robot arm singularity is to ensure the robot arm dexterity by positioning
the mobile platform and/or ensure the stability of the robot arm while approach-
ing singular configurations. This chapter first analyzes singular configurations of
the manipulator arm, and then presents a hybrid system design of robot arm con-
troller for singularityless robot motion control. To ensure feasible robot motion in
the neighborhood of kinematic singularities as well as at the singularities, a hybrid
robot motion controller is designed to integrate joint level control and task level con-
trol. The hybrid control system has a two-layered hierarchical structure, a discrete
layer and a continuous layer. The workspace is partitioned into subspaces based on
the singular configurations of the robot. Switching between continuous controllers is
involved when the robot travels across subspaces. The discrete layer is comprised of
the switching logic and the continuous layer includes the controllers used in various
subspaces. With the hybrid controller, the robot not only can pass by the singular
configurations, but also can go through and stay at the singular configurations. The
stability of the hybrid system is analyzed using Multiple Lyapunov Function theory.
The singularity-free motion control for mobile manipulators will be presented in the
Chapter 5.

The existence of robot arm singularities can be seen and analyzed from its task

level controller. The dynamic model for a robot arm with n joints in the joint space

53

can be written as

D(q)i1' + C(q, (i) + 9(a) = u, (41)

where q = {q1, q2, - - - ,q,,}T is the n x 1 vector of joint displacements, u is the n x 1
vector of applied torques, D(q) is the n X n positive definite manipulator inertia
matrix, C(q, q') is the n x 1 centripetal and coriolis terms, and g(q) is the n x 1 vector
of gravity term. A robot task can be defined in either the joint space or the task
space. Robot controllers can be designed accordingly.

For a robot task specified in the n dimensional joint space, denoted by qd, 4", if,

a computed torque controller at joint level can be formulated as:

“'1 : D(qqu + KquqZ + KPqeql) + C(Qa‘l) + 9(‘1)a (42)

where Kv and K p are feedback gain matrices, e 1 = qd — q, and e 2 = rjd — (j. The
‘1 q q ‘1

error dynamics in joint space can be described by

éql ___ W (4 3)

éqg = —qu8q1 — ququ.

It is straightforward to prove that system (4.3) is asymptotically stable for appropriate
gain matrices 1(1),),qu [6]. Irrespective of the current robot configuration, the system
(4.3) can track any reasonable trajectories given as qd, 4", if. In other words, controller
(4.2) is valid in the entire workspace.

However, a robot task is generally represented by the desired end-effector position
and orientation, a robot controller in task space is therefore desirable. To develop a
task level controller, the robot model (4.1) needs to be represented in the form of task

level variables. Let me = {2:1, 2:2, - -- ,xm}T represent the position and orientation of

54

the end effector in the task space, the relation between joint space variable q and task
space variable me can be represented by me = he(q), where h denotes a vector consisting
of m scalar functions. In general case, m = 6 and me E W" under certain assumption.
The n dimensional joint velocity and m dimensional end effector task velocity can be
related by the Jacobian equation: is = e(j, where Je is the Jacobian matrix. For
a nonredundant robot, n = m, Je is a square matrix and the relation between joint
level and task level variables is quite straightforward. For a kinematically redundant
manipulator, n > m, which means the manipulator has more degrees of freedom than
necessary to execute a task. The concept of extended task space for kinematically
redundant manipulators was introduced in [12]. Based on the kinematic redundancy
resolution algorithms, an independent constraint task vector 1:6 = hc(q) E Rn‘m
can be chosen. Therefore, for a redundant robot, the task involves not only the
end effector motion, but also the achievement of additional performances by utilizing
the kinematic redundancy. In the mapping 236 = Jc (j, the constraint task Jacobian
matrix Jc can be chosen as a matrix consisting of the orthonormal vectors that span
the null space of JC. The Jacobian equation in the extended task space can then be

TxTT

83C

represented by i‘ = Jq', where a: = {x is task level variables in the extended

task space and

is the Jacobian matrix in the extended task space. It is worth noting that .1le =
0[12]. In the extended task space, the redundant manipulators can be treated as
nonredundant manipulators. Therefore, the acceleration in task and joint space can
be related by :‘c' = id + Jr'j. Considering (4.1), an robot dynamic model in the term

of extended task space variables can then be described as follows:

DJ‘ICE - 1(1) + C(q, (i) + g(q) = u (44)

55

where the Jacobian matrix is n x n. Given the desired path in task space, and, the

robot controller in task space can be described as:

“*2 : DJ‘lci‘d — Jq + KVxer2 + KP1811)+ C(qr q) + g(CI), (45)

where

8:1 =$d—ar,e$2 zid—i.

The error dynamics in task space can be described as:

é:rtl = 6:2
(4.6)
écr2 : —KP:re;rl — erel‘2)

It is also easy to show that system (4.6) is locally asymptotically stable for appropriate

gain matrices K p1, K van [6].

4.2 Robot Arm Singularities

By comparing the controllers in (4.2) and in (4.5), it can be seen that the existence of
controller (4.5) depends on the existence of J ‘1, while (4.2) does not. At the singular
configuration or in its vicinity, the Jacobian J is singular or ill—conditioned. The
determinant of J ‘1 could be very large, which will result in unreasonable large joint
torques. The robot configurations such that |J| = 0 are called robot singular con-
figurations. Theoretically, the robot singular configurations and their corresponding
analytical singularity conditions can be obtained based on the analysis of the Jaco—
bian J. To better understand the kinematic redundancy and singularity, consider the

Singular Value Decomposition(SVD) of the matrix J

J = UZVT = Zaiuwf, (4.1)

1'21

56

where U is the (n x n) orthonormal matrix of the output singular vectors ui, V is
the (n x n) orthonormal matrix of the input singular vectors 2),, and E is a diagonal
matrix whose elements are the singular values of J, 01,02,--- ,0”. At a singular

configuration, one of the singular value a,- is 0. Thus,

u,TJ(q) = uTUsz = 0.

This means that motion in the joint space will result in zero motion in the output
direction u, of the task space. This direction in the task space is called degenerate
direction.

Each singular configuration of the robot can be represented by a singular condi-
tion in terms of the robot parameters and joint variables. The singular conditions can
be used to check the proximity of robot configuration to the singular configuration.
Since singular configurations may varies for mechanical structure, the singular con-
ditions of manipulators are analyzed based on three classes of mechanical structures:
(a) nonredundant manipulator with spherical wrist; (b) nonredundant manipulator
with non-spherical wrist; (c) redundant manipulators. Corresponding to the singular

configurations, a vector of singular conditions, denoted by I‘, can be derived.

4.2.1 Non-redundant Robot with Spherical Wrist

A general task for an end effector motion requires six degrees of freedom, therefore,
m = 6 can be assumed. For a 6-DOF robot manipulator, which consists of a 3—DOF
forearm and a 3-DOF spherical wrist, J 2 J6 is a 6 x 6 matrix. Due to the mechanical
structure of the spherical wrist, the motion of the position and orientation can be
decoupled. As a result, the Jacobian can be decoupled in such that the singular

configurations resulting from the arm and the wrist are separated. According to

57

[15],[67], the Jacobian J for a PUMA-like robot can be decoupled into:

 

I W
Je =
0 I 0 \I’
. - .. .J
I W
0 I 0 \Il
_ - L -
where
Jw = J11 0 )
J21 J22
' _SOSA 608A
CA CA
‘1’ = —c0 —30 0
30 Co
. CA CA
0 d5 ‘ 8};
W 2 —d6 ° 6],; 0
L d6 ' 6, —d6 ' 61'
—d481823 — 018102 — 0331023 - d261
J11 — d4c1323 + a1C1C2 + a3c1c23 — r1231

 

 

 

 

 

 

 

0

58

0
J22

J11
J21

Jw

 

_d6'ej
de-e, 7
0

 

d401023 — 020132 — (1301823
d481023 — 028182 — 0351823
—d4823 — 02 — a3€23
61(d4623 — 03823) A
81(d4623 - 03323) ,

—d4 323 — £13623

 

0 —81 —81

J21: 0 —c1 —c1 ,

 

01823 —01€2384 " 8104 016402385 — 818485 + 6132305

J22 = 81823 —8162334 — €104 3104C2385 + C18485 + 81323C5 a

 

 

C23 32334 C2305 — 823C485

er = 01(0236435 + 82365) - 318455,
6]- : 31(C23C485 + 82365) + €134.95,

8k = C2365 - 8230485-

Here s,-, Q, 5,,- and a,- stand for sin(q,-), cos(q,), sin(q,- + q,) and cos(q.- + q,-) re-
spectively. a2,a3,d2,d4 are robot joint parameters, (0,14,T)T denotes an orienta-
tion representation (Orientation, Attitude, Tool angles) [67]. Since the singularity
of \II caused by CA = 0 can be overcome by choosing other definition of the angle
representation([76],[26]) and W is not singular, the singularity analysis can be ob-
tained by checking Jw. Observing J11 and J22 of Jw, it can be seen that J11 involves
only q1,q2,q3, which correspond to the arm joints, and J22 involves only q4,q5,q6,
which correspond to the wrist joints. Singular configurations caused by J11 = 0 are
called arm singularities and singular configurations caused by J22 = 0 are called wrist

singularities.

Since the determinant of J11 is (d4c3 — a353)(d4323 + 0,202 + a3cQ3), two singular

configurations can be obtained. One is called boundary singular configuration, as

59

 

(b) Interior singular configuration (c) Wrist singular configuration

Figure 4.1: Singular configurations of Puma 560

shown in Figure 1, (a), which corresponds to:

7b = d403 — a333 = 0. (4.3)

This situation occurs when the elbow is fully extended or retracted. When fully
extended, there is no way to move the end effector forward, this is the degenerate
direction. The other situation is called interior singular configuration, as shown in

Figure 1, (b), the corresponding singular condition is

75 = £14523 + 0202 + (13023 = 0- (4.4)

In addition, the wrist singularity, as shown in Figure 1, (c), can be identified by

checking the determinant of the matrix J22. The wrist singularity happens when two

60

wrist joint axes are collinear and the corresponding singular condition is

7112 = _35 = 0- (4.5)

Therefore, a vector of singular conditions for a PUMA-like non-redundant arm with

spherical wrist can be given as:

 

 

d463 - 0383 d
F = d4823 + a2c2 + a3c23

By checking the value of singular conditions ’7 E F, one can decide the proximity to
the robot singular configurations. Denoting the manipulatability as v J JT , it can be

seen that \/ J JT is proportional to I71; - 7,- - 7w].

4.2.2 Non-redundant Robot with Non-spherical Wrist

A robot manipulator with non-spherical wrist generally does not have a decoupled
mechanical structure, though constructing a non-spherical wrist is easier. The inverse
kinematics of a manipulator with non-spherical wrist generally cannot be computed
in closed form and numerical computation needs to be applied. However, the singular
configurations of a non-redundant robot with non-spherical wrist can also be obtained
by considering the Jacobian [72]. For instance, the determinant of the Jacobian of a

manipulator with non-spherical wrist can be derived as given in [72]:

IJI = ka2a35385(a1 + (1262 + 0.3623 _ (158234),

where a1, a2, (13, d5 are robot parameters. The singular configurations can be decou-

pled into three kinds of singular configurations: (i) shoulder singularity, (11 + (12C2 +

61

a3c23 — (153234; (ii) elbow singularity, 33 = 0; and (iii) wrist singularity, s5 = 0. The

singular condition vector is represented by

r- -j

01 + (1202 + 03023 — d53234

F: 83 7

 

 

85

which is used to check the proximity to the singular configurations.

4.2.3 Singularity Analysis for a Redundant Robot

Generally, a redundant manipulator which has more than necessary degree of free-
dom is more flexible and dexterous than a nonredundant manipulator, though the
singularity analysis of a redundant manipulator is more complex. A singularity could
be caused by the singular value of .18, which is called a kinematic singularity, or the
singular value of JC, which is called an algorithm singularity.

For a redundant manipulator with spherical wrist, the singular configuration of
robot arm can be obtained by checking the Jacobian matrix J8 [14]. The position
singularity and orientation singularity can also be decoupled due to the mechanical
structure. Some of the configurations are boundary configurations, which are non-
escapable, whereas some are interior singularities, which are escapable [5]. However,
a singular condition vector P can still be obtained by analyzing the analytical repre-
sentation of the Jacobian matrix [14].

For a given constraint task, the singular configurations of Jc can also be determined
in advance. However, with a full rank Je and JC, there are robot configurations for
which J = [JZ, JZ]T loses rank. All these singular configurations caused by the choice
of JC are called algorithmic singular configurations. For a given Jc, the algorithm
singular conditions can be determined. Therefore, given a redundancy resolution

algorithm and an extended task space, the singular condition of matrix J can be

62

determined.

4.2.4 Workspace and Singular Conditions

The workspace of a robot, which is a complex volume calculated from the limit val-
ues of the joint variables, can be described by two concepts: reachable workspace
and dexterous workspace. A reachable workspace is the volume in which every point
can be reached by a reference point on the manipulator end effector. The dexterous
workspace is a subset of the reachable workspace, and it does not include the singular
configurations and their vicinities. The dexterous workspace is therefore a volume
within which every point can be reached by a reference point on the manipulator’s
end effector in any desired orientation. Unfortunately the dexterous workspace is not
a connected space. The singular configurations partition the dexterous space into dis-
connected subspaces. For example, a wrist singular condition (4.5) of a nonredundant
robot can be satisfied at almost any end effector position in the workspace. Thus, for
almost any position in the workspace, there exists an orientation of the end effector
which satisfies 7,” = 0. A robot path may therefore is composed of several segments
of paths in the dexterous subspaces and several segments of paths in the vicinity of
singular configurations. In the dexterous subspaces, there is no problem to control
the robot at task level. When the task requires the robot to go from one dexterous
subspace to another dexterous subspace, the robot end effector needs to go through
the vicinity of singular configurations. At these configurations, the manipulator loses
one or more degrees of freedom and the determinant of J approaches zero. In order
to generate a reasonable motion in task space, the joint motion constraint may be
violated.

Based on the above analysis, the robot workspace, denoted by Q, can be divided
into two kinds of subspaces, the dexterous subspace denoted by $20 and the vicinity

of singular configurations denoted by {21 and (22. ()2 specifies a smaller vicinity of the

63

singular configuration. The definition of S21 and {22 are given as follows:

92={q€§R",7€I“I |i|<a}
01={q€§R",7€I‘| aSlvlSfl.7<i/Q2},

where 6 > a > 0. Figure 4.2 shows an illustrative plot of the subspaces.

 

Figure 4.2: Illustrative plot of the robot workspace and subspaces

It is worth noting that 90 U 01 U 02 = Q and S20 f] {21 fl 92 = O. A A region
called dwell region is defined in 91 to avoid the chattering when switching occurs.

The definition of a dwell region is given as:

A={q€§ft",7€f‘| a<|7|<a+6,76’92}, (47)

where 6 is a constant greater than zero.

4.3 Hybrid Robot Motion Controller

4.3.1 Hybrid System Model

Hybrid control systems involve both continuous and discrete dynamic systems. The
evolution of such systems is given by equations of motion that generally depend on
both continuous and discrete variables. The continuous dynamics of such systems

is generally modeled by several sets of differential or difference equations while the

64

discrete dynamics describes switching logic of the continuous dynamics. For example,
robot controllers (4.2) and (4.5) can switch in the workspace according to certain
switching logic.

A general form of the hybrid robot motion controller is defined as follows. The
continuous state variable is joint angle q, or the end-effector position and orientation
:r. The discrete state variables, denoted by m E 32’ or m 6 {m1, mg, ...m;}, represent

the switching conditions. The hybrid system model of the robot can be described by

D(q)c'1' + C(q, (i) + 9(a) = u
(4.1)

1105) = u(113(t),q(1t),m(t))- (4-2)

The dynamics, f, is governed by the singular conditions. Depending on the cur-
rent discrete state of the robot and continuous states, f generates the next discrete
states. t' denotes that m(t) is piecewise continuous from the right. f discretizes the
continuous states and switches the local controllers. Max-Plus algebra [4] is used to
describe the discrete event evolution. It can provide an analytical representation of

a discrete event system. The detail design of (4.2) is given in the next subsections.

4.3.2 Design Procedure of a Hybrid Robot Motion Controller

The first step in the hybrid motion controller design is the singularity analysis of
the manipulator and definition of the singular and nonsingular regions. For me-
chanical structures, the kinematic and algorithmic singular configurations and the
corresponding singular condition vector P can be obtained. The analysis of the sin-

gular conditions F and the singular values is an important step in the design of the

65

switching conditions.

The second step is the design of the switching functions based on the definition
of the subspaces. Two switching function vectors are generally necessary. One is the
switching conditions between the regular region 90 and region 91, the other is the
switching function between region 91 and region {22. The switching functions can be
formulated as:

"131: sgn(fi - lrl) EB 0, V7 6 F
(4.3)

m32 = ms2(t‘)sg“(°+6""'@°) EB sgn(a — lvl) 69 0, V7 6 P

where sgn(-) is a signum function. The switching function is a function of the singular
conditions represented by Max—plus Algebra. The Max-Plus algebra, widely used in

the Discrete Event Systems, is defined as m(t) 6 ER" where

max

Rm”: : m U {—00}
EB : Max operation

(8) : Plus operation.

Here the operations, 69 and <8) are defined as: a6} b = ma${a, b}, and a®b = a + b. It
can be proved that {Sign : 69, (8)} is an idempotent and commutative semi-field with
zero element a 2 -—oo and identity element 6 = 0 [4]. It can be seen that mu = 0
in 90, mal = l in 91 and msg = l in 522. It is worth noting that 777.32 is not only
a function of singular condition vector I‘, but also a function of the current status
of m32(t‘). It means that the switching condition for switching into a subspace and
switching out a subspace may have different values. This is designed for eliminating
the chattering characteristic of switching controller.

The third step is the design of the continuous controllers. Three continuous con-

trollers are to be designed. In subspace $20, the inverse of J always exists and the

task level control (4.5) is effective in this subspace. In subspace {22, the joint level

66

controller is used.

In region (21, though it is close to a singular configuration, a feasible solution of
inverse Jacobian can be obtained by pseudo-inverse or singular robust inverse(SRl).
Controller (4.5) can still be used after substituting J # for J ‘1, where J # is a kind of

pseudo-inverse represented by:
J# = (JT - J + Amsl)‘1JT, (4.4)

where A is the damping factor. Instead of using a constant damping factor, A is

generally defined a function of the singular value 0[17]:

A771(ng
A = A(o) = — o + Amm-

0min

 

It is generally hard to compute the singular value online, the singular condition 7 E I‘

is used in this design:

Amal-
A 2 A07) = _ 7 + Amaxi

7min

 

where the parameters Am” and 7",," are chosen experimentally. The controller in

region (21 can then be represented by

“3 = 1.9de — 14+ Klee... + Km...) + c + g. (45)

The stability analysis of controller (4.5) is discussed in [11].

However, it can be proved that the controller based on pseudo-inverse method will
cause instability in subspace (22 [41, 50]. Though subspace {22 could be very small, it
cuts the dexterous workspace into pieces. If a controller can not make the robot travel
through subspace 92, the singular configurations will greatly restrict the dexterous
workspace. Thus the joint level control is enabled in subspace {22.

With the continuous controllers in different subspaces and the switch functions,

67

the next step is to formulate the hybrid robot motion controller in the entire workspace

utilizing the switching conditions msl and 771322

u = (I — m32)u3 + msgul, (4.6)

where the switch function msl is within u3. Based on the stability analysis of the
continuous controllers in their respective region, the stability of the hybrid motion
controller in the entire workspace should also be investigated. Section 4.4 discusses
the stability of such a system.

The last step of the hybrid robot motion controller design is the path planning of

d, controllers (4.5)

the continuous controllers. Given desired path in the task space, a:
and (4.5) are complete. To complete controller (4.2), rd should be converted into qd
in region {22. The mapping should be considered in the context of the mechanical

structure.

4.3.3 Hybrid Motion Controller for a PUMA-like robot manipulator

For a Puma-like robot manipulator, the continuous controllers (4.2) and (4.5) can be
used in region 522 and (20 respectively. The controller design in region (21 and the
switching functions are considered based on the characteristics of a PUMA-like robot

manipulator.

Continuous controller design in $21

In this subspace, the robot is in the neighborhood of singular configurations. Using
direct J '1 in task level will result in large torques. The modified damped least squares

method is used in the hybrid robot motion controller in this subspace. According to

68

(4.2), the inverse Jacobian J“1 in (4.5) can be represented by

—1 —1

I 0 I W
J-1 = 111-)1. . , (4.7)
0 \II 0 I
where
J 0
Jw = u . (4.8)
J21 J22

In the neighborhood of singular configurations, Jw is ill-conditioned and the inverse of
Jw can be computed by the damped least squares technique [47]. Since a uniformed
damping factor A in the entire workspace may cause unnecessary position and ori-
entation errors in the regular region, a damping vector based on the analysis of the

inverse of Jw[15] [47] can be defined as:

 

 

F A1(t) S P k.‘ 69 0 q
l A2(t) [€169 kb 69 0
A3“) . kb 69 0 ,
A(t) = H = (4.9)
A4(t) [ . kw GB 0
W] 0
[ A6(t) J l k... a 0 _w

 

 

where

kb = kw“ — heal/5b)
k1: k,0(1— [Til/l3?)
kw = kw0(l — hurl/113w)-

69

A is a vector of the positive damping factors depending on the proximity to the

singular configurations. Denoting the singular value decomposition of Jw as
6
J... = Era-um? = U - 2, - VT (4.10)
i=1

where 2),, u,-,i = 1, ..., 6 orthonormal bases of R6. 0,,i = 1, ..., 6 are the singular values
of Jw. U and V are orthonormal matrices and 21 is diagonal. The pseudoinverse of
Jw is:

Jr = (1.:- J... + vaiagrA)m.1(t)VT>-l - fl (4.11)

w

where m,1(t) is the switching condition. The inverse of J based on Jf is called
J#. Here diag(v) denotes a matrix whose diagonal elements are vector v, and the
remaining elements are zeroes. It is worth noting that (4.11) is different from (4.4)
to reflect the characteristic of Puma-like robot.

The controller of the robot in subspace $20 and 521 can be synthesized by:

U3=DJ#(id—jq+KVIeIQ+KPIeII)+C+g' (412)

Here a: = {191, pg, Pz,0,A,T}, where (p,,,,py,pz)T denotes the position of the end-
effector in the task space.

Substitute (4.12) into (4.4), the error dynamics of the system can be obtained as:

6.1-1 : 61'2

_ (4.13)
612 = (I - JJ#)($d — JQ) — JJ#(KPI€J;1 + erexg)

To analyze the stability of the system, the term J J # in the error dynamics (4.13) is

a key component. Considering (4.2), (4.7) and (4.10), .]J# can then be simplified as

70

follows:

 

 

k = J J#
' T
d, 0312 a; 03 a: cg 03,
= lag
2 2 2 2 2 2
01 + ml 02 + mg 03 + mg 04 + m4 05 + ms 06 + m5
Defining

 

. 0,2
km,” = min 2 ,
(1,1 0": + mi

it can be seen that km," becomes smaller and smaller when the robot configuration is
approaching singular configurations. The stability of the controller (4.12) in subspace
90 and 01 depends on k. In subspace {20, msl = 0, k = I , the error dynamics (4.13)

in region (20 becomes

érl = 612
(4.14)
éz2 = -KP:re.rl — KV1'612

It is obvious that system (4.14) is asymptotically stable in subspace $20 for apprOpri-
ate gain matrices. While in subspace {21, the error dynamics of the system can be

described by

éxl = 63:2
(4.15)
(2,2 = (I — wad — Jq) — k(prex1 + erexg)

According to the discussion in [11],[4l], a candidate Lyapunov function of (4.15) cab
be defined as

l
V = ~2-[eflKerl + (,uexl + e12)T(,uerl + eI2)] (4.16)

where K is is positive definite matrix and ,u is positive constant. The derivative of

71

the candidate Lyapunov Function can be derived by

V = efiKes + (#611 + 8x2)T(#€I2 + (I — W — Jr) — when + Khan»
= -—p6:1kKPreIl — 8:2(kKVI — ””812

+(uex1 + e12)T(I -— k)(:’td —— Jq) + 9:10 — k)(pr + ,qux)ex2

Here K = K P15 + qul. — #21. a can be chosen such that K v: — ,uI and K are positive
definite. Denoting vector v = Jq, an upper bound of the vector term can be given
by[11]:

7'
llvll S -2—(vfu + 2vmllex2l| + llex2ll2),

min
where ’UM denotes the maximum norm of the desired end effector velocity, om," denotes

the minimum of the singular value of matrix J, and

6
7" = max ZHNJP,
q i=1

1'

8q and J,- is the ith row of matrix J. Thus

 

where N, =

V < _kmin)‘minllean“2 +(1_ kmin) ' ||€x||(k1 + k2||exll + k3llexll2)
= —(1 — 6)k,,,,-,,A,,,,An|leiII2 + (1 - kmin)' ||€x||(k1+(k2 — k4lllexll + k2l|3xll2l

= -(1-0)kmm/\mmllezll2. v (k1+(k2-k4)llezll+k2|lexll2)$0-

where

 

 

r 2r
k1: aM + \/1+j1.2 22,2”, [:2 = (/1+11202 UM + “(I —- k)(KpI +#sz)||,

0.2

min min

72

 

T akrnin Amin
k3 = WUQ 1 k4 :- 1

min 1 — kmm

and Am," is the minimum singular value of uK p1 and KVJ, — pI/km,,,. It can be
shown that (k1+(k2 — k4)||e1[| + kglleIIP) g 0 can be satisfied by regulating k4. From
the above derivation, it can be seen that the stability depends on the value of km".
In subspace 520, km," 2 1, V is negative definite and the system is asymptotically
stable. In subspace S21, kminAmm — (1 — lam-"MW S 0, the system is uniformly
asymptotically ultimate bounded. The bound is (1 where (1 < (2 are the root of
km,nAm,-,, — (1 — mam/17? = 0.

However, in subspace (22, the value of km," is close to zero, and —km,nAm,A,.| leg,“2 +
(1 — lam-n) - ||ex||(k1 + k2||e1|| + k3||ez||2) may be greater than zero. The system sta-
bility can not be ensured. When the robot approaches singular configurations, i.e.,
in region (22, some of the elements of k become very small. Due to this, the corre-
sponding elements of the gain matrices become small. Furthermore, at the singular
configurations, lam-n = 0. Assuming 0,- is zero, the ith element of (4.13) can be written

as:

6:11 = 8:21

651-21“: 513:1— («1(1):

It can be seen that the system is an open loop system. Though it can be proved the
system is ultimately bounded[41], large errors in the task space are expected in the

vicinity of singular configurations.

Controller design in subspace 92

To achieve stable control in region 92, a switching control is needed so that joint level

controller (4.2) can be enabled in {22. Two discrete variables, m7 and ma, are defined

73

to represent arm and wrist singularities respectively,

m7(t) = m7(t—)sgn(ai+6—l7'b|)®sgn(0b+6—l7bl)®0 EB sgn(ai _ Ira/1]) ® sgn(ab _ Irybl) 69 0
(4.17)

m(t) = m(t—)“’g“("“'+“"l“’D630 EB sgn(aw - In!) 69 0-

Here m7(t‘) and m3(t‘) represent the values of my and ma before a time instance t
respectively. a,, a), and aw are used to specify subspace $22. In this design, a and 6
take different values for different singular configurations. Depending on the values of
7b, 7, and 7w, m7 and ms will determine what kind of continuous controllers should
be used. It is worth noting that in region A, the values of m7 and ms not only depend
on '75, 7,- and 7,0, but also depend on the previous value of m7(t‘) and m3(t‘). The
A region, which is defined in (4.7), is called a dwell region. It is introduced to avoid
chattering along the switching surface. In other words, the switching surfaces for
switching into joint level control and switching out of joint level control are different.
After the controller switches into joint space control, it can not switch back to task
space control unless the robot reaches a larger region 02 U A. This strategy can
effectively avoid chattering and ensure the system stability.

A switching matrix, 77132, is defined based on m7 and mg as:

 

m t I 0
ms2(t = 7( ) 3‘3 (4.18)
0 (m(t) <9 ms(t))13x3
Hybrid robot motion controller
The singularityless robot motion controller can be represented by
’U. = (I — m32)1t3 + msgul (4.19)

The above controller involves both continuous and switching controls, as shown

74

in Figure 4.3. It forms a two-layered hierarchical hybrid controller, a continuous
layer and a discrete layer. The switching conditions are dependent on the proximity
to the singular configurations and the previous controller status. When the robot
approaches the singular configurations, the hybrid controller first uses damped least
squares to achieve an approximate motion of the end effector. It will then switch into

joint level control if the robot reaches the singular configuration.

 

 

Robot ]——._+

Gravity. Coriolis and
Centrifugal terms

Task Level Council——

Task Level Control
- with DLS

Joint bevel Control [—

+

 

 

—

 

 

 

 

‘—

 

 

 

l

 

Singular Conditions ]¢—-

 

 

Figure 4.3: Hybrid system controller framework

For a robot with a spherical wrist, it is worth noting that if the position and orien-
tation control are decoupled, joint level control and task level control can be enabled
simultaneously. This happens when m7 = 0 and m3 = 1. It means that the position
is controlled at task level and the orientation is controlled at joint level. However, the
position and orientation are generally coupled if the tool center of the end effector is
used to indicate the desired position. For a given task and = {p3, pg, pg, 0", Ad, Td},
the desired position {pi pg, pf} is coupled with the desired orientation {01, Ad, Td}.
However, it can be easily seen that after proper transformation of a given task, the
desired position at the wrist center can be taken as the desired position at task level.
It is only related to the joint angles q1,q2,q3 and can be obtained from rd [67]. The
orientation control will only depend on qff, qg, qg. The end effector position and ori-
entation can therefore be decoupled and can be controlled separately. Since the wrist
singular configuration may happen at almost any end effector position, the separated

position control and orientation control can ensure a relatively larger continuous dex-

75

terous workspace for position control.

Path planning for switching

In the regular region 90 and region 521, the task is represented in Cartesian task
space. The task, however, in region {22 needs to be transformed into joint space since
a joint level controller is used. Given an" in task space, joint level command qd needs
to be calculated in subspace S22. Substantial research has been done on this topic

[15, 47, 75, 56, 49]. By the relation between task space and joint space variables,

x = (1(9),

a first order approximation of the kinematic function at a configuration q can be

written as:

div = J (q)dq,

where do: and dq represent motion increment in task space and joint space respec-
tively. At robot singular configurations, rank( J (q)) < n. When a manipulator passes
through a singular configuration, a certain degrees of freedom in the singular direction
or the degenerate direction is lost. Therefore, there would be no feasible solution for
the manipulator to move into the singular direction. For a path close to a singular
configuration, an exact solution is possible but very large joint motions are required
if the assigned dx has components along directions which become degenerate at the
singular configuration. To balance the accuracy and feasibility of the joint space

motion, the following criteria is used to find the inverse kinematics:

mqinfwllldrr - J(q)alq)||2 + wzllqd - q||2}-

76

where w, and 7112 are positive weight factors. This criteria minimizes the deviation
after the task is mapped into joint space, and tries to find qd such that the least joint

movement is needed. An optimal solution to this can be written as [47]:
dq = w1(w1JT(Q)J(€1)+ wzl—1J(Q)Td33

In the computation of dq, it can be found that the joint space variable can be divided
into two subspaces: joint variable that can be computed by normal inverse kinematics
and joint variable that cannot[15, 75]. For example, at the wrist singular configura-
tion, Figure 4.1(c), only q}; and qg cannot be obtained by the inverse kinematics; they
are gal and q‘3’ at the boundary singularity, Figure 4.1 (a); and qf and q; at the interior
singularity, Figure 4.1 (b). Thus, the desired joint level command can be grouped
into two subsets: q“) denotes the joints that can be obtained by inverse kinematics,
(1(2) denotes the joints that cannot be obtained by inverse kinematics. By checking
the values of singular conditions 7,,, 74,711,, the elements of q(2) can be determined. To

compute these joint level commands (1(2), the following criteria are considered:
ngpiwludz — J(q)dq)||2 + w2|lqd — 4112}.
‘1

Motion planning should also be considered in the joint space. In the vicinity of a
singular configuration, a pure geometrical path can always be tracked, but a trajectory
which includes both geometrical and temporal specifications cannot be tracked. In the
other word, the geometrical parameter and the temporal parameter can be considered
separately, as discussed in[56, 49]. It is easy to understand that an appropriate do:
would result in a large dq in the neighborhood of singular configurations. A new path
can be planned to satisfy the velocity and acceleration constraints. The continuity
of joint velocities and task level velocities are also considered in the planning. After

switching from task level control to joint level, the initial velocity for every joint is

77

the joint velocity prior to switching. If the control switches from joint level control
to task level control, the velocity continuity can be ensured by taking :i: = Jr} as the

initial velocity in task space.

4.4 Stability Analysis of the Hybrid Motion Controller

In Section 4.3, it has been shown that task level controller (4.12) is uniformly ultimate
bounded in the workspace except subspace $22. And joint level controller (4.2) is
asymptotically stable in the entire workSpace. This section discusses the stability of
controller (4.19) when switching between (4.12) and (4.2) is involved.

Theorem Suppose controller (4.2) is asymptotically stable in workspace and
(4.12) is uniformly ultimate bounded in region 01, there exists a constant 6 > 0, as
shown in Figure 4.2 and (4.7), such that the hybrid robot motion controller (4.19) is
ultimate bounded in the entire workspace of the robot. <1

Proof The state variables in the error dynamics in joint space and task space
are different, due to which the stability analysis of the hybrid motion controller is

difficult. The errors in joint space and task space are defined as:

The error dynamics of the manipulator in joint space can be written by

éql = 8"2 (4 20)

eq2 = —KPqeq1 _ KVq€q21

78

or éq = fq(eq). The error dynamics in the task space are

611 = 61:2

(4.21)
(312 = (I — k)(;L‘d — Jq) — [C(Kpl-erl + Kvxerg),

or 6'2; = Mex)-

It is easy to prove that system (4.20) is stable. (4.21) has been proved to be
ultimately bounded in 00 U 01. However, if the controller switches between 111 (4.2)
and an (4.12), the system will switch between (4.20) and (4.21). Since the state
variables in dynamics model (4.20) and (4.21) are different, the relation between the
state variables in model (4.20) and (4.21) is derived first.

It is easy to prove that :1: = h(q) and i: = Jq are globally Lipschitz continuous in

their defined domain, i.e.,

llexlll = Hill“l - xll = “h(qd) - h((1)|| S Lillq" — all = Lilleqlll

”622“ = Hid - dill = ||J(qd)qd - J(q)éll S fella“ - (1|! = L2||€q2||

where L1 and L2 are constants. In summary, the following relation between joint

space error and task space error can be found:

llexll S Lolleqll

where Lb is a constant. Similar to the above inequality, the following relation can also
be obtained based on the Lipschitz continuity of the function q = h‘1(a:),q = J ’14:
in 00 U 91.

lleqll S Lallexll
Therefore, the errors in joint space and task space satisfy the following inequality in

79

DOUQI.
lleqll S Lallerll, llerll S Lblleqll (422)

Defining the Lyapunov function of system (4.20) and system (4.21)

1
V1 = 5(elepqeql + eggqu)

1
V2 = §[€:1K81-1 + ([1811 + €12)T(/1€1-1 + 612)].

Since the system (4.20) has been proved asymptotically stable in the entire workspace,
and (4.21) has been proved to be uniformly ultimate bounded, the Lyapunov functions

satisfy the inequalities in region {20 U (21 [35]

allleqll S “(6,1) S bllleqll

(9V e .
gigglme.) s -61lleql.

 

a2||€4|| S V2091) S b2||84~||

8%(81)
Be;

 

fx(er) S —02||€z||

where a1,b1,c1,a2,b2,c2 are positive scalars. For an initial time to, the following

inequalities can be obtained.

V1(eq(to + 7)) S e‘*"‘%(€q(to))
(4.23)

V2(er(to + r)) g e‘A2721/2(e1(t0))

where A, = c, / b1 and A2 = c2/b2 are positive scalars. In order to prove the stability
of the hybrid controller (4.19), it is essential to show that V, is monotonely decreasing

at all odd number of switch instances, t1, t3, t5,- - -, and V2 is monotonely decreasing

80

at all even switching instances, t0, t2, t4, - - -.

VA

 

 

Figure 4.4: A typical switching sequence of the hybrid motion controller

Based on the switching sequence in Figure 4.4,

V1(€q(t1)) S b1||64(t1)||SbiLallt‘ix(tl)ll

 

_<_ b:"%(ex(t1)) g %e“*2(tl—‘0)V2(ez(to))

V2(er(t2)) S b2||e,(t2)||sbnglleq(t2)|l (4.24)
g fiffmeqam g E’jTII’Be.>-*1“2“”i/1(c.1050)
S bi?11%,?““2‘“)e_A2(t‘_t°)V2(€x(to))

Form the above inequalities, it can be seen that

b L b La
V2(er(t2)) S :5 ; e-MUz-ti)e—A2ft1-to)v2(ex(t0))
1 2

_ C en(—A,(t2—t1)-A2(tr-to)) e(l—u)(-»\1(t2—tr)—A2(t1—to))V2(ex(to)) (4-25)

|/\

e(l—p)(—A1(tg—tj)—A2(t1—to))V2(eI(t0)), for c ejr(-A1(t2—11)-/\2(tl—t0)) S 1

Here
_ b2 Lb b, L,

(11 (12

 

 

C

81

is a constant and 0 < ,u < 1. Based on Multiple Lyapunov Function method [8],
[52],[69], the system can be proved to be stable as long as (4.25) holds. It means
that c e"("\1(‘2“1)"‘2(‘1“°)) S 1 is a sufficient condition for the stability of the sys-
tem during switching. This can be achieved by regulating two sets of parameters,
A1 and A2; t2 — t1 and t1 — to. The values of A1 and A2 are related to the gain
matrices Km, Kw, Km and qu. In the implementation of the controller, high gains
and high sampling frequency are chosen to ensure proper A1 and A2. The values of
t2 — t1 and t1 — to are also related to 6. The boundary for switching into and out
of a continuous controller is not the same. This forms the dwell region A, and can
effectively avoid chattering at switching. From the definition of (4.7), the value of
mag cannot be changed immediately. Since t2 — t1 and t1 — to are time intervals when
a continuous controller is enabled, they can be regulated by the value of 6. There-
fore, the gain matrices and the size of the dwell region A are used to ensure that
c e”(‘61(‘2—t1)-62(tl-to)) S 1. The above analysis is valid for He, > C1”, the switching
system is uniformly ultimate bounded.

In the design of the controllers, the size of the dwell region can be chosen based on
two situations. The first one is the ultimate bounded. Considering controller (4.12)
is ultimate bounded by (1, therefore “ex” 2 ”6:1” 2 C1 should be ensured by 6. The
other one is the the gain matrix and motion planning. It can be proved that the
transition of ex and eq are bounded by their initial values during switching. And the
bound is related to the gain matrices and the desired trajectory. It can be assumed
that the initial value of errors are also bounded. Therefore, e,t and eq are bounded.
Thus, 6 can be chosen that eq(t) E (22 for t > T and T is a time instance after
finite switches between joint level and task level control. By choosing an appropriate
region, the switching sequence can also be completed in one cycle.

The same procedure can be applied to time instances t4, t6, for V2. Therefore, V2

is monotonely decreasing at all even switching instances. Similarly, V1 is monotonely

82

 

decreasing at all odd switching instances t1, t3, t5,- - ~. Thus, the switching system is

stable. <1

4.5 Experimental Implementation and Testing

The proposed hybrid robot motion controller was implemented on a PUMA 560 robot.
Figure 4.5 shows the experimental setup. The controller ran under the real-time op-
erating system QNX [1]. The parameters in the experiments are chosen as 6, = 6,, =
015,61, = 025,6 = 004,01, = 008,01), = 0.08, am = 0.03,k,0 = kwo = 02,117“) =
1, w, = 1112 = 1, pr = diag(4800.0,6000.0, 4800.0, 200.0, 200.0, 280.0), Ky, =
diag(80.0, 88.1, 88.2, 22.0, 24.0, 25.6), qu = diag(2400.0, 2400.0, 2400.0,11000, 1100.0,
1100.0) and qu = diag(50.2,50.5, 50.1, 360,360,360). All the parameters are de-
termined experimentally. The sampling period is set as 1ms. Given a path, the
trajectory was computed based on the path planning approach in [26]. A trajectory
is them formed by integrating temporal parameters with the given geometrical path
considering the velocity and acceleration constraints. In the figures, the unit for posi-
tion is meters. For angles, it is radians. The dash line denotes the desired trajectory

and the solid line denotes the actual trajectory.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

QNX PC ServoToGo A ND
coder .
Controller (1:) Interface <E Amplifier <:[> Puma 560
User Interface card five“
‘ Power Supply

 

 

 

Figure 4.5: Experimental Setup

Figures 4.6 and 4.7 compares the DLS method with the hybrid motion controller

when the robot approached its arm singular configurations at a: = 0.9, y = 0.15, z =

83

Figure 4.6: At boundary singular configuration, with DLS method only. Dash line:

(a) x position

 

 

 

 

 

 

_o_31.. ..

 

-0.32
0

 

 

 

Desired; Solid line: Actual

A

E

v

Figure 4.7: At boundary singular configuration, with hybrid motion controller. Dash

-0.27

-0.28

—0.31

-0.32

(a) at position

 

 

V

 

 

 

 

 

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

 

 

 

0 2 4 6

t(sec)

line: Desired; Solid line: Actual

84

(b) y position

 

 

 

    

 

 

 

 

 

 

 

0.16 4
£0.14»
>
012. ....... ,,
0.1' ~(
0.08 * ‘
0 2 4 6
t(sec)
(d)rneasurementotrb
0 : -
~(
..................... l
—0.5 .1
0 2 4 6

t(sec)

 

 

 

 

 

(b)ypos‘rtion
0.16 : :
£0.14»
>~
0.12’
01 t. .. ................. ',. ........... 4
0.08 *
0 2 4 6

t(sec)

 

 

 

(d) measurement ot rb

........

 

 

 

 

(exposition 4x103 - (0)yposjtion - -
02 . .......
:5; 0, .......................... E 2 ------- .
0». ....... . 0 .......... ‘
-04) ..... .
_02 _2, ..............................
_4 - - -

 

 

 

 

 

 

 

0.4

 

 

 

 

(0) measurement at ri

 

 

 

 

 

g 05
77 .
0.49, «
0.48» 1
o. ....................
0.47A ---l _01: .............. .
0.48 -o.2 A A A
5 10 15 20 25
time)

Figure 4.8: At interior singular configuration, with DLS only.

Solid line: Actual

 

 

 

 

 

 

 

 

 

Dash line: Desired;

 

 

 

 

 

 

 

 

 

 

(a) x position x 104 (b) y position
' ' Y ' a 4 ' . T
g g 2 .. .............
x 0.1 ..................... , ., >
0 »- 0 ’
-0.1 ................. .. .
-2 i ......... ............... . . l
_02 ........... .
A A A ‘ _4 A A A A
0 5 10 15 20 25 0 5 10 15 20 25
t(sec) t(sec)
(c) 2 position
0.51 . - - -
5 0.505 »» .
N
0.5: g
0.495 f , 3
0.49 A A -0.2 A A A A ‘A
0 5 10 15 20 25 S 10 15 20 25

1(566)

Dash

 

Figure 4.9: At interior singular configuration, with hybrid motion controller.
line: Desired; Solid line: Actual

85

 

0.2
0.1

—0. 1
-0.2
-0.3
-0.4

 

-0.5

 

 

0.535

 

0.53 .
0.525
0.52

0.515'

 

 

 

 

0.51
0

x 10" (b) y position

 

v f

 

 

 

 

 

 

 

-5 .
0 10 20 30
t
singular conditions. dotedzrl; solidzrw
0.4 A .
0.2 r AAAAA
o > . \ x ...... ‘
_0'2 . ........... . \ \. . _ .
. \ \
-0.4 f """"" ' ' ' ' 1‘ \ .,
—0.6 A ;
10 20 30

Figure 4.10: With both interior and wrist singular configurations. Dash line: Desired;

Solid line: Actual

(a) x position

 

-0.505

 

 

 

 

 

 

 

 

-0.51 .
0 10 20 30
t
(c) 2 position
0.3 f .
0.2 .
o1 , . ..............
o . ........
-0.1 . -----
—0.2>
-0.3 .
-0.4 .
0 10 20 30
t

0.286

0.284 '
0.282 r ‘ ‘
0.28 >

0.278
0

0.8
0.6

0.4 r
0.2 r

—0.2
-0.4
-0.6

(b) 1! position

 

 

 

 

 

10 20 30

singular condition. rw

v

 

 

 

 

 

Figure 4.11: With wrist singularity. Dash line: Desired; Solid line: Actual

86

 

—0.297. The task of the robot was to move along x direction from 0.7m to 0.9m. In
Figure 4.6, it can be seen that the error in y and z direction became bigger and bigger.
In Figure 4.7, the hybrid robot motion controller was used. It can be seen that the
tracking errors in y and 2 directions were much smaller comparing with Figure 4.6
and the task was completed. It is worth noting that the path was replanned in joint
space after the controller switch into joint level control. The trajectory was planned
in joint space such that the joint motion constraints were satisfied. The end effector
is slowed down to avoid unreasonable joint velocities.

The desired trajectory for the next two experiments was —0.3m to 0.25m in 2:
direction so that interior singular configuration would be reached. Figure 4.8 shows
the trajectory of the robot under task level controller u3. When the end-effector
entered the interior singular configuration at a: = —0.06, y = 0, z = 0.45, a very large
deviation from the desired path was observed. Figure 4.9 shows the experimental
results under the hybrid robot motion controller. Although the deviation from the
desired path is still observable, it was smaller as compared to the results in Figure
4.8. The hybrid controller switches into joint level at t = 143, where 7, = 0.08; and
it switches back to task level at t = 193, where 7, = —0.08.

Figure 4.10 shows the position tracking result when both interior and wrist singu-
larities were reached in the motion. At time t = 888C, both interior and wrist singular
configurations were reached, the controller switches into joint level. Errors caused by
switching at t = 123cc can be seen from Figure 4.10 (a) (b) and (c). This also shows
that the hybrid system was stable when double singular configurations were reached.

Figure 4.11 shows that the hybrid motion controller can stably pass the singular
configurations with certain speed. In this experiment, :c,y, O,A,T were fixed. The
end effector position 2 changed from —0.3m to 0.3m. The average velocity is 0.2m / s.
The end-effector passed the wrist singular configuration at about t = 15sec and in the

middle of the path at certain speed. Small deviations from the desired path can be

87

 

is.

seen from the results, but the system was stable. This experiment has demonstrated
that the end-effector can pass the singular configurations using the hybrid motion

controller.

88

CHAPTER 5
CONTROL OF MOBILE MANIPULATORS

5.1 Introduction

In Chapter 3 and Chapter 4, control of mobile platforms and singularities of robot
arm are discussed. This chapter discusses the control of mobile manipulators, which
consists of a mobile platform and a manipulator arm. To take the advantage of large
space mobility of the mobile platform and dexterous manipulation of the manipulator,
path planning and control of mobile manipulators are discussed. The kinematic and
dynamic model of mobile manipulators are derived first. Tracking control and force
control are then discussed for holonomic mobile manipulators.

A mobile manipulator usually consists of a mobile platform and a robot arm.
Figure 5.1 shows the associated coordinate frames of both the platform and the ma-

nipulator, which is the top view and side view of figure 1.1.

a World frame 2: Inertial frame where robot tasks are generally defined,
0 Moving frame Eb: Frame attached on the mobile platform,

0 Virtual moving frame 2;: Frame attached on the mobile platform, parallel

to 2.

Since the tasks of a mobile manipulator are generally given in task space, i.e.,
in the world frame, a unified model of the mobile manipulator in the world frame
is preferable for control and path planning. For easy reference, all the definitions of
the state variables for mobile manipulators, mobile platform, and the robot arm are

listed as follows:

89

 

 

 

 

 

o
I
U Ll ; 4
r
I

 

 

 

 

(a) side View

Figure 5.1: Mobile manipulators
q = {q1, Q2, q3, q4, q5, q6}T is the joint angles of the robot arm,
Y, = {2:5, yb, 6b}T is the configuration of the mobile platform in the world frame,

q’ = {q1 + 01,,qo,q3,q4,q5,q6}T considers the orientation of mobile platform as

an addition of the first joint angle of the robot arm,
8,, = {6b,0, 0,0,0, 0}T is the orientation vector of the mobile platform,

P = {‘11, (12, (13, q4, q5, q5, 2:1,, yb, 6),}T is defined as the joint variables of the mobile

manipulator,

yf, = {235,315, 115,0, 0,0}T is the origin of 2,, and )3; where hb is the height of the

mobile robot,

ym = {19,,,,.,py,,,,pm,0,,,,A,,,,Tm}T defines the end effector position and orien-

tation in frame 2),,

yin = {p;,,,, me, p’zm, 0;", Afin, T,’,.,}T defines the end effector position and orien-

tation in the virtual moving frame 2;,

y = {193, pg, pz, 0, A, T}T defines the end-effector position and orientation in the

world frame, which is also the output of the mobile manipulator,

f = { f1, fy, f2}T defines force applied on the end effector of the mobile manip-

ulator.

90

5.2 Unified System Model

5.2.1 Kinematic model

From Figure 5.1, it can be seen that the translation that relates the virtual moving

frame 2;, and the moving frame 2,, can be defined by T3

Cb —Sb 0 0
85 Cr, 0 0

0010

 

 

0001]

where cb and Sb stand for cos 9b and sin 0,, respectively. Since only translation is
involved between moving from 2 and Z33, it make it easier to derive the kinematics of
the arm in frame 2;, and then transform it to frame 2. The kinematics of robot arms
in frame 2,, and the kinematics of mobile platforms in the world frame 2 are used in
building the kinematics of mobile manipulators in frame 2. In moving frame 2),, the

relation between the end effector and the joint angles of the arm can be defined as

9711 = h(Q) (5'1)

where h is a nonlinear transformation. The velocity and acceleration relationship

between the task space and joint space can then be derived.

gm : Jq
17m = J4+Jq

where Jacobian J denotes the relation between joint space and task space velocities.
The kinematics of the manipulator in frame 2,, can be transformed into frame 2;,

using the transformation T l" The end-effector in the moving frame 2;, can be proved

91

to be

yin = h((1’)

where h is the same as the transformation in (5.1). In 2’ , the velocity and acceleration

relationship between the task space and joint space can then be derived as:

01.. = Jci’
17:. = J4'+iq'

Here J denotes the velocity relation between task space and joint space in frame 2],.
Since only a translation is involved between the world frame 2 and frame 2;, the end

effector position and orientation in the world frame can be expressed as

y = yi + 311..-
The velocity and acceleration relation in the world frame 2 can then be derived as:
4'1 = 4:. + f4’
0 = 173+ Jci’ + ici’ (5-2)
= yg+Jrj+Jéb+j¢

Actually, by relating the task space velocity and the joint space velocity of the

92

mobile manipulator in matrix form, the first equation in (5.2) can also be written as

1 0 J11
1 j21
jar
‘7‘“ (5.3)
j61
jar

OOOOO
COCO

 

 

 

= Jhp

where J), is defined as the Jacobian of the mobile manipulator. It can be seen from
J), that the manipulator Jacobian is augmented by a submatrix whose rank is 3(J61
equals 1). The Jacobian can be full rank even when J is singular. Thus the mobile
platform increases the capability of manipulation. It is worth noting that the virtual
frame 23;, will ease the modelling of the mobile manipulator. The existing models of

the robot arm and the mobile platform can be used.

5.2.2 Dynamics Model

The dynamics of the robot arm in frame 2,, can generally be described by

Mr(¢1)r'1' + 0101.0) + 91(0) = 7'1 (5-4)

where r, is the 6 x 1 vector of applied torques, M((q) is the 6 x 6 positive definite
manipulator inertia matrix, c1(q, q) is the 6 x l centripetal and coriolis torques, and
g,(q) is the 6 X 1 vector of gravity term. Substituting (5.2) into the dynamics (5.4),

the robot dynamics in the form of world frame variables can be obtained:

CJ'I
C}!
v

Mlj—lai—ii—Jéb—J‘q')+c,+9, =71. (

93

Defining the output of the mobile manipulator as :1: = {2:1, .132}, where 2:, = y, 2:2 = Y,,,

this equation can be rewritten in the form of

Mlj‘lii, + N143, — MIJ-lliq' + 01+ 91 = r1 , (56)

where N122 combines terms —M1J‘1JO,, and —M1J‘1y'b. Following the work of Liu
and Lewis [43],Yamamoto and Yun [84], Morel and Dubowsky [27], the dynamics of

a holonomic mobile platform can be represented by

N2(P)i1+ M20042 + 02(1),?) = T2

where N2(p) stands for the interaction between the mobile platform and the robot
arm, M2(p) is the inertia matrix of the mobile platform, and 72 = {71,7'yfl’o} is
the input torques applied on the mobile platform. The dynamic model of mobile

manipulators with holonomic mobile platform can then be described by

l f

Mlj—l N1 ] [ (I31 ]—A[1j-1jQI+C1 +91] 7'1
l- .. + ,= (5.7)
N2 M2] (132 C2 7.2
Let
011.j—1 N1]
M =
N2 Mg]
F
7’1
T :
T2
—M,J-liq’ + c1 + 91
C ':
1 C?

 

94

(5.7) can be written in a concise form as

Mat + c = r. (5.8)

It is worth noting that both the kinematic model and the dynamic model is described
in a unified coordinate frame. Since the end effector and mobile platform variables
are described in unified frame, it will ease the resolution of kinematic redundancy

and the force/ position control.

5.3 Motion Control and Redundancy Resolution

5. 3.1 Tracking Controller

Applying the following feedback control to (5.7)

MJ-1 N1 ' -—1mJ-1J'q'+c1 +91
7' = -u + (5.1)
N2 A12 L C2

the dynamics model of the mobile manipulator can be simplified and linearized as
if = u, (5.2)

where 7' = {71, 72}T, the dynamic model of the mobile manipulator can be linearized

and decoupled as
515 = u, (5.3)

where the vector x 2 {p1, pg, 19;, O, A, T, 161,, yb, 61,}T is the mobile manipulator output
and u is a linear control vector. Eq.(5.2) forms a unified model for the mobile manip-

ulator. It is worth noting that vector :1: not only contains task level variables y, but

95

also part of the joint level variable Yb. Since y is defined in task space, direct control
of the end effector position and orientation has been implemented. As a result, the
mobile manipulator controller has the advantage of the task level controller. At the
same time, the joint level redundancy can also be resolved easily.

Given 27" = {193,193,131 0", Ad, Td, :55, yg, 031V as the desired position and orienta-

tion of the mobile manipulator, the linear system feedback for model (5.2) can be

designed as:

u = id + and — sic) + and — 2:). (5-4)

It is easy to prove that the above control algorithm (5.4) is asymptotically sta-
ble. However, the task of a mobile manipulator is generally given in the form of
yd = {pipipi Od,Ad,Td}T. The desired mobile robot position and orientation
de = {223,313, 0;}? should be obtained by considering the mobile manipulator config-
uration and bandwidth of the mobile platform and the arm. By properly positioning
the mobile platform, singular arm configurations can be avoided, i. e., the coordinated
motion of the mobile platform and the arm can maximize the capability of manip-
ulation. In the following subsection, we give a coordination scheme considering the

robot arm manipulation capability.

5. 3.2 Singularity Analysis and Redundancy Resolution

Form (5.1), it can be seen that no matter how the linear controller u is designed, the
feasibility of the mobile robot controller (5.1) depends on the existence of matrix j‘l.

From the singularity analysis of the manipulator, the following singular conditions can

96

be obtained [65]:

7b = ld4C3 — a333|
72‘ = |d4823 + (1262 + a3023| (5.5)
7w = l — 35l

Here 7;, = 0,7,- = 0,7,” = 0 stand for certain arm singular configurations. Here
d4,a2, a3, a4 are robot arm parameters and 3,,0, stands for sin q.- and cos q,- respec-
tively. When the robot arm approaching a singular configuration, the value of at
least one of the three expressions will become smaller. If the manipulator is put on a
fixed platform, singular configurations can not be avoided for some tasks. However,
singular configurations of the arm can be avoided by appropriately positioning the
mobile platform for a mobile manipulator. Given a path for the end effector of the
mobile manipulator, a path for the mobile platform should be calculated online in
such a way that manipulator is always in a dexterous configuration, i.e., not near a
singularity. The desired mobile platform path is therefore dependent on the system
output. An event based planning approach [79], as shown in Figure 5.2, is used for
online coordination control of the mobile platform and the manipulator.

The basic idea of event based planning is to introduce a new motion reference vari-
able, 3, which is different from time and directly related to the sensory measurement
of system output. The motion information of both the arm and the manipulator is
passed to the planner by the motion reference 3. For coordinated control of multiple
robots, the planner of each robot should take the common motion reference 3 as an
input. Thus the robots are coordinated by a common motion reference.

The objective of coordinated control can be expressed by maximizing the ability
of manipulation, which also means maximize the values of the expressions given by
(5.5) and taking advantage of the high bandwidth of the manipulator. However, it

is not necessary to keep the capability of manipulation at its maximum value. If the

97

conditions in (5.5) are always maximized, the manipulator will be fixed at a certain
configuration and lose its dexterous manipulation capability. As long as the values
of yum-,7“, are greater than certain prespecified minimum values, the manipulator
will certainly have the ability to move. The desired position and orientation of the
platform should be determined by the trajectory of the end effector and the conditions

given by (5.5), which are also the system output. Define the motion of the end

    
    

   

Controller

 

 

 

 

: ‘ Mobile ‘
Controller Platform

 

 

 

 

Logic
Command
i .
Motion l
Reference

 

 

 

 

 

 

 

 

 

 

 

 

Figure 5.2: Event-based coordination between the arm and the mobile platform

effector as the motion reference, the desired trajectory of the mobile platform is a

smooth function of s and 7b, 7,, 7,“.

(Kb = 9109,71): 7i77w)
yb = gy(s,%,%,7w) (5-6)

6:, = go(s,n,vm’w).

Functions gz, g,,, 99 are continuous and piecewise differentiable. Here the motion ref-
erence s can represent the motion of both the platform and the manipulator,i.e., a

function of the augmented output :10. Let 9 represent 91,, gy,or go and 7 represent

98

 

7b, 7,- or 7,0, the requirement of the definition of coordination can be represented by

39 . ('37
5—0 lf|7|>7m1m0rg>0
39 . 87

_ < . __

as > 0 1f |7| _ 7mm and ('33 < 0

where 7mm stands for the allowable minimum value of 7. If the value of 7 is greater
than 7min or it is increasing with respect to s, the platform can choose to keep the
current position. The mobile platform should be commanded to a position such that
the value of 7 will increase. An example implementation of g considering only the

arm singular configuration 71, is shown here:

323 = an, + (xfls) —- x1)r
y? = 316 + (1173(3) - 1‘2)?
0;: = 91, + (51",

where (5 = atan2(:rg — 315,931 — mb) and

0, if (|7| > 7mm or g:- > O) and r' = 0

r = a1, if [7| > 7m,-n and r" > O
67

as <0.

(12, if Ivl S 7mm and

Here r is designed as a hybrid system [65] and it is piecewise continuous from the
right, r' and r+ stand for the values of r before and after a time instant, 012 > (11 > 0
are constants. The values of a2 and an > O are obtained adaptively in the experiment.
r can take more discrete values in implementation. By this design, the end-effector
will go into certain range of 7 no matter what the initial value of 7 is. Using this
strategy, coordination can be achieved even when obstacles block one of them. When
the end effector is blocked by obstacles but the mobile platform is free to move, 3

stops evolving and maintains a constant value. The desired trajectory of the mobile

99

 

platform will therefore not change and the whole system is stable. If the mobile
platform is blocked but the arm is not, the arm can still continue performing its task
without the help of the platform. If now a singular configuration is reached, the
desired trajectory of both the arm and the platform will remain constant. Thus the

system stability is not affected by obstacles or singularities.

5.4 Experimental Implementation and Results

5.4 .1 Hardware and Software Implementation

The motion planning and control approaches for a mobile manipulator have been
implemented on a mobile manipulator consisting of a Nomadic XR4000 mobile robot
and a Puma560 robot arm, as shown in Figure 1.3. For the mobile manipulator, there
are two PCs in the mobile platform, one uses Linux as the operating system and runs
the mobile robot control software and the other uses a real time operating system
QNX and runs the Puma 560 control software. The two computers are connected via
an Ethernet connection and communicate at a frequency of 300—500Hz. The sampling
period for the Puma 560 control software is 1ms.

The hardware structure is shown in Figure 5.3. The control softwares was devel-
oped for the experiments in this chapter and chapter 6. The software components for

the experiments are listed as follows and a software structure is shown in Figure 5.4:

o Non-time based mobile robot motion control software
0 Singularity-free robot arm motion control software

0 Mobile manipulator motion control and force control software

100

 

 

Puma 560:

 

 

 

Pentium PC force/torque, acceleration, fl
RTOS QNX encoder, actuators, .» ' ~
joystick. ' . ”k
Mobile Robot:
Laser, Sonar,

Infrared, bumper,
encoder. actuators

 

 

 

   

Figure 5.3: Hardware structure of the mobile manipulator
5.4.2 Path Tracking and Tele-operation

In Figure 5.5 and 5.6, the results of path tracking are presented. A sequence of tasks
are given in the form of desired end effector position. In Figure 5.7 and 5.8, the
results of tele—operation are recorded. The desired end—effector position is generated
by Microsoft Joystick connected to a remote PC and sent to robot controller via
wireless Internet.

Plots (a) and (b) of figures 5.5-5.8 show the end-effector position in the world
frame. The solid curve is the end-effector position in world frame and the dash line
is the mobile platform position in the world frame )3. The end—effector position in
the moving frame Eb is shown in plots (c) of figures 5.5-5.8. The solid curve is mm
and the dash line is ym. The measurement of the capability of manipulation is shown
in plots ((1) of figures 5.5-5.8. Its maximum value is 4.4. For a sequence of tasks,
as shown in Figure 5.5 and 5.6, the capability of manipulation is kept at certain
value, which is larger than 0.4. The trajectory of the mobile platform is given on line

according to the measurement of the capability of manipulation and the end-effector

output. For tasks not known a priori, as shown in Figure 5.7 and 5.8 (a)(b)(c), the
capability of manipulation can adapt to a higher value, as shown in Figure 5.8(d).
This phenomena demonstrates that the robot arm is kept in a dexterous workspace
during the operation and this kinematic redundancy resolution scheme is suitable for

a point-to—point task, a sequence of multiple tasks, and task unknown a priori.

5.4.3 Path Tracking with Unexpected Obstacles

To test the path tracking controller and the event based coordination scheme when
tasks are not known a priori and unpredictable obstacle is present, the results of
teleoperation are presented in Figure 5.9. The desired position of the end effector is
generated by a joystick. Figures 5.9 (a) (b) depict the end effector position in :r and
y direction of the world frame respectively. Figure 5.9 (c) shows the position of the
platform 23;, and yb. The Operation can be divided into four parts by t1 = 15sec, t2 =
44.4sec and t3 = 54.93ec, as shown in the figure. Before t1, the mobile manipulator
operates normally. At time t1, the platform meets a unexpected obstacle and can
not move any more, as shown in Figure 5.9 (c). The tasks assigned to the mobile
manipulator can still be performed by the robot arm without the help of platform.
It can be seen that the value of manipulation capability measurement becomes very
small around time instance 383cc. At t2, the obstacle is removed, the platform begins
to cooperate and the manipulation capability adapts to a higher value. At time
instance t3, the platform is blocked again, the robot arm perform the task alone
and the mobile manipulator stops evolving after the manipulation capability drops
to a very low value. By the event-based coordination scheme and path tracking
controller based on the unified model, the mobile manipulator system is stable even
when unexpected obstacles occured. capability is especially useful when the mobile
manipulator works in a highly unstructured environment. This experiment also shows

that the kinematic redundancy is resolved online and the manipulation capability is

102

Optimized during the operation.

103

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

‘ I 19%

 

 

mo m feted
] [Actuate oiled Pot 'm s a I

it ‘
COD
Cam-n Comm “More

2

 

Wheel

 

.l

 

(“—‘fi
Allow

Prowl

“MOM”

 

Figure 5.4: Software structure of the mobile manipulator

104

(a) ”b (b) 3WD

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0.1 - 1.5 -
A 005» 1 A 4
.5. E:
o 4
_O_05r 4 4
-0.1 * -O.5 ‘
20 4O 60 O 20 40 60
f(aec) t(sec)
(C) MM" 0 (d) ManipulationAblflty
' 7 5 f V.
_ __ W
I \ .4’ .. .. .,
AOBl—r- \_ ___ -,-__ , 0 . :
g _-‘
0.8» t 0.3»
0.4" ' ~ ‘ 0'2
0 a E :
- o A ;
0 20 40 60 0 20 4O 60
New) t(sec)
F igure 5.5: Multiple tasks path tracking
(a) mm (b) m0
015 - : - . e
g 00.5; . . .. ., . 4
0 I V 4
—o.1’ - » ‘ ,
—0.15> +
412 . - - ., ‘- . - .
0 1O 20 30 ‘0 0 10 20 30 40
f(sec) t(sec)
(c) xm,ym (d) Manipulation Ability
1.2 , v . 05 ._
lg ‘ _ r . ._'—_—\.... .: / '. . .
V 08. I . J . , . . N ‘ '
0,6» ' ' . 03 - - . .
0.2, . _ .: . , _.. .. ,
, 0.1»
-0.2 ‘ ‘ ‘ 0 ‘ ‘ *
0 1o 20 30 40 0 10 20 30 40
((sec) t(sec)

Figure 5.6: Multiple tasks path tracking

105

 

(m)

0.2

 

 

 

 

60 80

40
1 (sec)
(C) NIH-YT"

 

 

 

a

 

 

l (860)

”390)
(d) MmipulationAbility

0.5 v v
0.4,, ..... .......
0.3.
02. ....
0.1.“.

0 i

0 10 20

 

 

 

 

 

 

 

 

t (SOC)

Figure 5.7: Tele-operation

 

(m)

-0.1

_03.. . .

-O.4 ’
—O.5
O

(m)

0.6’

0.4'

02*

 

 

 

 

 

08*

 

 

 

20 30

t (sec)

 

 

 

 

 

 

 

 

0.5 ' '
I I \\ l \
/ \ / \
0 h "‘ I ‘ ‘\ I" .
—O.5 ‘ ‘
10 20
t (sec)
((1) Manipulation Ability
0.5 f f
0.3 *
0.2 ’
0.1 >
o a k
0 10 20
1 (sec)

Figure 5.8: Tele-operation

106

 

 

 

 

 

 

 

(a) 0X60“) Blamed)

 

 

 

 

 

 

 

 

 

   

 

 

 

 

 

 

0.5
o...
-o.5 - A
80 80
0.45
0.4
0.35.. , ..
-0.5 t L i . i 4 0.25 1 ; l ' ‘ i
0 2o 40 60 80 o 20 40 60 so
Nose) t(aec)

Figure 5.9: Teleoperation with unexpected obstacles

107

CHAPTER 6
INTEGRATED SENSING AND CONTROL OF MOBILE
MANIPULATOR

6.1 Introduction and Force Control Schemes Revisit

Besides trajectory tracking, object manipulation is another kind of task for a mobile
manipulator. The mobile manipulator needs to interact with the environment, and
output force control is necessary in this case. There are generally two ways for
the end effector to interact with the environment. One is interacting with static
environments or objects, such as force tracking along hard surface. Force control
schemes such as hybrid force/ position control, impedance force control, explicit force
control and many others have been proposed in the context of fixed base manipulator.
The force control schemes are highly dependent on the environments of the robot.
In many cases, the environment is assumed as static and the directions for force
control and position control are separated. The end effector can also interact with
dynamic environments, such as interacting with moving objects. For some objects
that are too heavy or large for the mobile manipulator to carry, pushing or pulling is
an alternate choice to manipulate objects. Since the mobile manipulator can provide
both motion in large workspace and dexterous manipulation capability, we discuss
the force control of the mobile manipulator while pushing an object. This requires
the mobile manipulator to provide both output force and motion along the same task
direction. In this dissertation, the kinematic redundancy of the mobile manipulator
is utilized to decouple the force control loop and motion control loop in the same
task direction. To compare with, the force control schemes based on a decoupled
nonredundant robot model is revisited first.

Hybrid force/ position control was first proposed by Raibert and Craig [55]. The

workspace is divided into two orthogonal subspaces as shown in Figure 6.1. A selection

108

 

matrix S determines the subspaces for which force or position are to be controlled.
The control laws for position and force control can be designed independently to
satisfy the different control requirements of force and position. Generally, the force
control law is designed to interact with static environment. However, the motion of
the environment should also be considered when the robot interacts with a moving
object. To improve the performance of the force control law, Schutter [61] proposed
an approach to feed forward the object motion parameters such as object velocity (1'20
and acceleration Eta, as shown by the dotted lines in Figure 6.1. The desired output
force fd along the motion can be tacked. However, it can been seen that the motion

control along the same task direction is open loop.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

4 Position
x + Control Law
Linearized r j 5"
. 0 ’
Nonredundant Object
Robot

f d FOIOC f

+ I - 5 Control Law
f. t.

lxa I x a

 

 

Figure 6.1: Hybrid Force/ Position Control

For an object or environment, it is assumed that the end effector position and the
contact force with the environment along one DOF can not be controlled indepen-
dently. The force can then be regulated by controlling the impedance, or compliance
of the robot, as shown in Figure 6.2 [29]. The basic idea of this approach is to design
a control law which will function in accordance with f = M513 + Bi: + K ac, where the
constant matrices M, B, K represent inertia, damping and stiffness matrices of the

interactive system respectively.

Combining the hybrid control and impedance control, hybrid impedance control
was proposed [2]. The environment dynamics was also represented in an impedance

form and the interaction between the environment and the manipulator was studied.

109

 

 

 

 

d Position Linearized _—x' Object

Nonredundant
Control Law Robot '

 

 

 

 

 

 

 

 

 

 

 

 

 

1
Ms2+Ds+K f

 

 

 

 

 

 

Figure 6.2: Impedance Control

Since the robot may encounter different environments for various applications, control
gains of the robot should be tuned in accordance with the environmental characteris-
tics. This scheme also has a slow response to force perturbations and the performance
of the implicit force control is restricted by the bandwidth of the position controller

[74].

6.2 Output Force / Position Control of Redundant Robots

For a nonredundant robot arm, the directions for force control and position control
have to be orthogonal[36], as shown in Figures 6.1. Therefore the force and the
position can not be controlled along the same task direction. It is caused by the equal
number of control input and desired system outputs. For the hybrid force/ position
control scheme, the force and position control directions are generally separated by
a selection matrix S. For many cases of the impedance control scheme, the control
law is essentially a modified position controller. And the performance of the implicit
force control is restricted by the bandwidth of the position controller. However, the
force control loop and position control loop of a redundant robot can be decoupled
along the same task direction. Since the force control is immensely environmental
related, the hybrid force/ position control of a redundant robot is discussed from two
aspects, the capability of the robot and the constraints the environment.

Due to the kinematic redundancy, there are more control inputs than the desired

110

outputs. For the decoupled unified system model (5.3) of the mobile manipulator,
the linear control input is a 9 x 1 vector, while the desired end effector position and
orientation, {p35, pg, pg, 0“, Ad, Td}T, is only a 6 x 1 vector. The redundant degrees of
freedom, which correspond the extra linear control inputs, can be utilized to accom-
plish secondary tasks. For a path tracking task, the redundant degree of freedom can
be used to position the mobile platform such that singular configurations of the arm
are avoided. For a task to interact with the environment, output force of the end ef-
fector can be chosen as secondary tasks. For instance, the desired output position and
force of the mobile manipulator can be chosen as {ff, 5, f, 0", Ad, Td, pg, pg, 6:}T,
where ff, f: and ff are the desired output forces. As shown in Figure 6.3, a selection
matrix S is not necessary for the redundant robot. It is worth noting that along a:
direction of the world frame 2, both desired position and force, pi and ff, are chosen.

p3 and f: are chosen simultaneously in y direction.

 

 

 

 

 

 

 

 

 

x4 Position
Control Law x
Linearized L x x' x-
mum can C ’ C ’ C d'
Force R°b°‘ f f c Dynamics f

 

 

 

 

 

 

 

 

 

 

 

+
fd . Control Law '
+ T- fpf.

 

Figure 6.3: Hybrid Force/ Position Control for Redundant Robots

Since system (5.3) is decoupled, it can be divided into two subsystems, position
control subsystem and force control subsystem. The state variable space of the po-
sition control subsystem, mp, is a subspace of the state space m of system (5.3). Let
mp = {p;,,.,p,,,0,A,T,0b}T and denote its corresponding linear control input by up.
The force control subspace :cf is chosen as :1: f = {2:5, yb, pz}T and the corresponding

linear control input is denoted by Uf. System (5.3) can therefore be rewritten into

111

 

two subsystems:

a: = u
p ” (6.1)
5L"; = “f
The linear feedbacks for the two subsystems can be designed as
_ d . . ..
up —kpp($p - mp) 'l' knit”: — mp) ‘l’ 5’3: (6.2)

to =5; + kfp(fd — fHkrlfJ(fd(U)-f(0))d0

In the controller (6.2), the force control loop and position control loop along the
same task direction are decoupled due to the redundancy of the control inputs. And
explicit force/ position control of the end effector can be designed.

From equation (6.2), it seems that the desired position and force along the same
task direction can be tracked simultaneously. This is only mathematically true. Actu-
ally, the environment plays an important role in the force control schemes. The desired
output forces f: and f: in (6.2) are directly related to the environment dynamics. For
instance, zero force output should be commanded in the free space. For an object, the
force and position is related by object dynamics and the friction. The dynamics of
the object and friction determine the desired output force and the output position of
the end effector. However, force and position do not have to be related to the mass of
the object only. For some tasks, the force and position can be planned separately in
the same task directions[66]. For instance, while a robot cooperates with humans, the
output force of a human, fm, is independent of the robot. This kind of environment
can be considered as non-passive environments. In multi—robot coordination tasks,
more than one robot can share a task. The robots can be considered working in a
non-passive environment. The output force for individual robots can be considered
independently from its motion, as long as the composed force satisfies the constraints

of the environment dynamics. In brief, the environments will determine the desired

112

W!)

output force and position, and force and position can be planned independently for
certain applications, specially if the environment is non-passive or dynamic[66]. The
cart pushing task is a typical example of the non-passive or dynamic environments
since the payload and friction of the cart are unknown. In this case, the advantage
of decoupled force and position control lies in that the interacting force with the dy-
namic environment can then be regulated explicitly by considering the environmental
dynamics, as shown in Figure 6.3. The force does not have to be regulated implicitly
as it is done in the implicit force control scheme. This ensures the bandwidth of the
force control loop, and ensures that the force planning for manipulating the cart can

be tracked.

6.3 Integrated Task Planning and Control

Due to the large workspace and dexterous manipulation capability, mobile manip-
ulators can be used in a variety of applications. For some tasks like painting and
soldering, only the motion planning and control of the mobile manipulator are in-
volved. For some complicated tasks, such as pushing a nonholonomic cart, the mobile
manipulator maneuvers the object by the output forces. Therefore, the task planning
for manipulating a cart involves three issues: the motion and force planning of the
cart, the trajectory planning and control of the mobile manipulator, synchronization
of the motion planners for the cart and the mobile manipulator.

Figure 6.4 shows a flow chart for the integrated task planning and control for
pushing a nonholonomic cart. First, for a given task, the geometrical path needs
to be determined considering the nonholonomic constraint of the cart. Secondly, a
trajectory should be planned based on the geometrical path, and the input force for
the cart to track the trajectory should be planned. Thirdly, the trajectory planning
and coordination of the mobile manipulator [66] should be considered. Finally, the

output force control of the mobile manipulator and task planning should be integrated

113

based on the decoupled system model (5.3).

In the cart pushing task, the mobile manipulator and the cart compose a system.
Therefore, the trajectory planning of the mobile manipulator and the cart should
be synchronized. The event-based approach for the path planning and control of
multirobot coordination [82] can be used in integrating the planning and control of
the pushing task. A common motion reference 3 is chosen for the trajectory planning
of both the mobile manipulator and the cart. In other words, the evolution of the
cart trajectory and the mobile manipulator trajectory is based on the same motion
reference. It is worth noting that the common motion reference 3 is computed based

on sensory information of the mobile manipulator.

Tasks

 

Path Planning of the l
nonholonomic can ;

l

Trajectory/Force Planning of
the nonholonomic can ‘

l

Online trajectory planning of
the mobile manipulator

. l

Decoupled force/position
control of mobile manipulator

l

I Nonholonomic cart J

l

s Sensory information of the
can and mobile manipulator

 

 

 

 

 

 

 

 

 

 

 

 

 

 

aaualajal uonow

 

Can Configuration

 

 

 

 

 

 

 

 

 

Figure 6.4: Integrated Task Planning and Control for the Mobile Manipulation of a
Nonholonomic Cart

6.3.1 Motion Planning of the Cart

The cart shown in Figure 6.5(b) is a nonholonomic system. The path planning for
a nonholonomic system is to find a path connecting specified configurations that
also satisfy nonholonomic constraints. The cart model is similar to a nonholonomic

mobile robot except that the cart is a passive object. Therefore, many path planning

114

algorithms for a nonholonomic system such as steering using sinusoids [45] and goursat
normal form approach [70] can be used for the motion planning of the cart. In this
dissertation, the kinematic model is transformed into the chained form and the motion
planning is considered as the solution of an optimal control problem. Considering the

kinematic model of the cart as

:i:C = 121 cos 6,: (6.3)

g. = 111 sin Go (6.4)

a. = 112 (6.5)

:icC sin 6c — g. cos QC = 0, (6.6)

where 121 and 112 are the forward velocity and the angular velocity of the cart respec-
tively. An optimal trajectory connecting an initial configuration and a final configu-
ration can be obtained by minimizing the control input energy J: ft: l||v(t) szt.
Given a geometric path, an event-based approach can be used to determine the tra-

jectory [82].

 

 

 

 

 

 

 

 

(a) Mobile Manipulator (b) The nonholonomic cart

Figure 6.5: A mobile manipulator and a nonholonomic cart

115

6.3.2 Force Planning of the Cart

The nonholonomic cart shown in figure 6.5(b) is a passive object. Assuming that the
force applied on the cart can be decomposed into f1 and f2, the dynamic model of

the nonholonomic cart in world frame 23 can be represented by

 

Etc = —sin6C + £00366
me me

ya = —— cos 9(,. + fl sin BC (6.7)
f mc mC

9°C _ —2
Ic

where mayc and 6c are the configuration of the cart, me and Ic are the mass and
inertia of the cart. A is a Lagrange multiplier and 06 is the cart orientation.

As shown in figure 6.5, The input force applied onto the cart, f1 and f2, corre-
sponds the desired output force of the end effector, f:, f:. In other words, the mobile
manipulator controls the cart to achieve certain tasks by its output force. Therefore,
manipulating the cart requires the motion planning and control of the mobile manip-
ulator based on the decoupled model (5.3), and the motion and force planning of the
cart based on its dynamic model (6.7).

To track a given trajectory, the input forces applied onto the cart, f1 and f2, need
also be planned. The input force planning of the cart is equivalent to the output force
planning of the mobile manipulator. The control input of the nonholonomic cart is
determined by its dynamic model (6.7), the nonholonomic constraint (6.6) and its
desired trajectory {3:2, yg, 03}.

Due to the lack of a continuous time-invariant stabilizing feedback[9], output
stabilization is considered in this dissertation. Choosing a manifold rather that a
particular configuration as the desired system output, the system can be input-output
linearized. By choosing mayC as the system output, the system can be linearized

with respect to the control input f1 and v2. This can be explained by the following

116

 

derivations. From equation (6.3) and (6.4), it is easy to see that 121 = 236 cos 0C +
ye sin 06. Here v1 is actually the forward velocity along x’ direction. Considering the
velocity along y’ direction is inc sin 0c — 3). cos 0C 2 0, the following relation can be

obtained:

a, = ire cos BC + ye sin QC

Cg = éc
Suppose the desired output of interest is {:rc, ya}, the following input-output relation
can be obtained by the derivative of the equations (6.3,6.4):

.. 1 .
me = —cosBC-f1—vlsm6lc-v2
C

1
y, = ——sinHC-f1+v1cosfic-v2
C

Considering f1 and 122 as the control inputs of the system, the input output can be

formulated in a matrix form:

where

 

{ 00866 -v1 sin9c
G— l

211 cos 6C

 

where {101,w2}T = G{f1, u2}T. Given a desired path of the cart, 33g, yg, which satisfies

117

the nonholonomic constraint, the controller can be designed as

wl = 5155+ kp1(:rg —- (EC) + kd$(;i:f — irc)

2122 = 332' + lady? - ye) + my: - lie)

The angular velocity oz can then be obtained by {f1,v2}T = G‘1{w1,w2}T. The
physical meaning of this approach is that the control input f1 generates the forward
motion and 122 controls the cart orientation such that 1:3 and y3 are tracked. However,
control input '02 is an internal state controlled by fry, the tangent force applied onto
the cart. The control input f2 can then be computed by the backstepping approach
[35] based on the design of 1);. Defining v2 = q3(:rc, ye, 0C) and z = tic—d), then equation

(6.5) can be transformed into

. - L

2 = —¢ + Tf2 (6.8)
The control input f2 can then simply be designed as

f2 = [fob — kale. — o» (6.9)

It is worth noting that the desired cart configuration {wiyg} is the result of

trajectory planning. As shown in figures 6.5 (a) and (b), the output force of the
mobile manipulator corresponds to the control input force of the cart. Given the
force planning f1 and f2, the desired output force of the mobile manipulator would
be: ff = fl * sin 9,: — f2 * cos 60, f: = fl * cos 0c + f2 * sin QC. To integrate the task
planning and control, the decoupled force / position controller 6.2 can be used to track

certain trajectories.

118

 

6.3.3 Integration of Sensing, Task Planning and Control

To implement the force control of the cart, the actual configuration {336, yc,l9c} and
corresponding velocities need to be estimated. Since the cart has no sensors equipped
on it, sensors on the mobile manipulators, such as laser range finder, encoder and
force/ torque sensor are used to estimate the configuration of the cart. To estimate
the orientation of the cart in the moving frame attached on the mobile platform, laser
range finder is used. Figure 6.6 shows a configuration of the cart in the moving frame
of the mobile platform. The laser sensor provides a polar range map of its environment
in the form of (a, r), where a is a angle from [—7r/ 2, 7r/ 2] which is discretized into
360 units. The range sensor provides a ranging resolution of 50mm. Figure 6.7 (a)
shows the raw data from the laser sensor. Obviously the measurement of the cart is
mixed with the environment surround it. Only a chunk of the data is useful and the
rest should be filtered out. A sliding window, w = {011, 02} is defined. The data for
01 ¢ w are discarded. To obtain the size of w, the encoder information of the mobile
manipulator should be fused with the laser data. Since the cart is hold by the end
effector, the approximate position of l'r, y,., as shown in Figure 6.6, can be obtained.
Based on 23,, y, and the covariance of the measurement r, which is known, the sliding
window size can be estimated. The filtered data by the sliding window is shown in
Figure 6.7 (b).

In this applications, the cart position and orientation (me, ya, 66) are the parameters
to be estimated. Assuming the relative motion between the cart the robot is slow,
the cart position and orientation can be estimated by the standard Extended Kalman

Filter(EKF) technique[31]. The cart dynamics are described by vector function:

X. = F(Xc, f) +£ (6-10)

where f is the control input of the cart, which can be measured by the force torque sen-

119

sor equipped on the end effector of the mobile manipulator. X c = {mm .736, yo, ya, 06, 90}.
g is random vector with zero mean and covariance matrix 0:. Considering the output

vector as Z (t) = {a, r}T, as shown in Figure 6.6, the output function is described as:

Z(t) = out.) + c (6.11)

where C is a random vector with zero mean and covariance matrix 0,. It is worth
noting that nonholonomic constraint should be considered in (6.10) and the measure-
ment ((1, r) should be transformed into the moving frame. By applying the standard
EKF technique to (6.10) and (6.11), an estimate of the cart state variables Xe for XC
can be obtained. The dotted line in Figure 6.7 (b) shows an estimation of the cart
configuration in the mobile frame Eb. It is worthy noting that both the force/ torque
sensor information and the laser range finder information are used in the estimation

of the cart configuration.

 

 

 

Figure 6.6: Configuration of the cart in the moving frame

To interact with a cart with different weight and length, the cart parameters,
me, IC and cart length L can also be estimated based on the force/ torque sensor and

accelerometer. Details please see [64].

120

x 10 (8) Raw data

 

 

 

 

 

 

2 (rad)

(mm)
640

 

620

600

580

560

540

 

520

 

 

 

 

500 i L L 1 L 1 ;
-o.5 o4 -o.3 -o.2 -o.1 o 0.1 0.2 0.3(rad)

Figure 6.7: Laser range data

6.4 Experimental Implementation and Results

The path planning and control approaches have been implemented on a mobile ma-
nipulator consisting of a Nomadic XR4000 mobile robot and a Puma560 robot arm,
as shown in Figure 1.3. For the nonholonomic cart, the mass is 45kg and the cart
length is 0.89m. The gripper of the mobile manipulator held the handle of the cart

during the manipulation.

121

6.4.1 Pushing Along a Straight Line

(a) desired and actual forces (b) tome error in y direction
Y .' 7 6 ' ' Y

 

 

   

 

 

 

 

 

 

0 5 1 0 1 5 20 0 5 1O 15 20
{(806) ttaec)

(c) cart trajectory

(d) cartdtrectlon

 

g 0, .. . .. ,L.

    

 

 

 

 

 

’ o 5 10 15 20 o 5 10 15 20
((866) «800)

Figure 6.8: Pushing the cart along a straight line

Figure 6.8 shows the results of pushing a cart along a straight line using the mobile
manipulator. In this experiment, the task was to push the cart along y direction in
frame 2 from 1.675m to 2.675m. Figure 6.8 (a) shows the desired output force and
actual output force in y direction. Figure 6.8 (b) recorded the force tracking error. It
is worth noting that the static friction force was also overcome in the first two seconds
of the task. Figures 6.8 (c)(d) show the trajectory of the cart and the cart orientation
respectively. The position and the orientation of the cart were estimated by fusing the
sensor information on the mobile manipulator. The laser range sensor, encoders of the
mobile platform and the robot arm are used in the estimation. In this experiments,
the motion in y direction was chosen as the motion reference to synchronize the motion
planning of the cart and the mobile manipulator. This experiments demonstrated that
the desired force, the desired trajectory of the cart has been tracked with satisfactory
precision.

For the same task, figure 6.9 shows the experimental results when an unexpected

122

 

 

 

(a) desired and actual tomes (b) loroe error in y direction
j, * T '. 6 . f

actual

    

 

 

 

 

 

 

 

 

o 5 10 15 20 25 o 5 10 ’15 20
“006) K800)

(o) cantralecwrv . .

(meandreetlon

 

 

 

 

 

 

 

0 5 10 15 20 25
ttsec)

 

Figure 6.9: Pushing with unexpected obstacle

obstacle blocked the mobile manipulator along the straight line. At about 7.53, the
system is blocked by an obstacle, the motion reference stops evolving. As a result, the
desired output force was set to zero and the cart stopped moving. When the obstacle
was removed at about 143, the motion reference started evolving again, the desired
output force was computed based on the desired motion of the cart and the pushing
task is resumed. It can be seen that the motion of the cart and mobile manipulation

system was synchronized by a common motion reference.

6.4.2 Turning the Cart at a Corner

Pushing the cart along a straight line is relatively easy since the desired end effec-
tor position and output force is easy to plan. The second experiment considered a
complex task. The mobile manipulator first pushed the cart forward 0.4m along y
in about 20 seconds, made a turn in about 35 seconds, and then pushed the cart
forward again for 0.4m along a: direction. Figure 6.10(a) shows the trajectories of the

cart, :80 and ye, and the end effector trajectory, p1. and py. Figure 6.10(b) is the cart

123

 

 

 

 

 

 

 

 

 

 

 

(a) trlectories (b) cart direction
35 . . 2 . .
A 3 A
E,
>
2.5
2 .
1 ’ A A L _05 L L L l
—1 -o.5 0 0.5 1 1.5 O 20 40 50 80
X0") «800)
(c)xdirectlootorce (d)ydirectlonloree
8 Y ' r r 5 ' '. Y
6 .,
4 ...

 

 

 

 

 

 

_8 L - g L
O 20 40 80 80
“590)

 

 

Figure 6.10: Turning the cart at an corner

orientation 0,; with respect to time. The cart started from a configuration parallel to
y direction, and turned to a configuration parallel to a: direction. The output force is
planned based on the desired trajectory of the cart. Figure 6.10 (c)(d) are the force
applied onto the cart, f1 and fy. It is worth noting that the force are recorded in
world frame 2. It is seen that the force fr pushed the cart along a: direction in the
last 20 seconds, and fy pushed the cart along y direction in the first 20 seconds. This
experiment has demonstrated that a complex task can be completed by the integrated

task planning and control approach.

6.4.3 Pushing Along a Sine Wave

The task in this experiment was to push the cart forward along a sine wave denoted
by are = 0.2 sin(1.8yc)m. In this experiment, the force f1 generates the motion along
the since wave, the force f2 regulates the cart orientation. Figure 6.11 (a) shows
the desired and actual trajectory of the cart, and the desired and actual of the end

effector. The noise of the actual trajectory was caused by the noisy estimation of cart

124

 

 

 

   

 

 

 

 

 

 

(a) desired and actual trajectory (b) cart direction
‘ . . f 0.4 +4
V 3 l ;
. .J .
o .. . . . .
_0_2 . . . .
. J g
-o 2 ~ 4.4 4
1 2 3 4 5 1 2 3 4 5
vii") vim)

 

 

(c)loroeinxdrerfion (dlloroehydrecuon
.' Y 4 f r

    

 

 

 

 

o 50 160 150
um)

 

Figure 6.11: Pushing the task along a sine wave

orientation, which is shown in Figures 6.11 (b). Figure 6.11 (c)(d) show the desired
and actual force signal in :r and y direction respectively. f1 and f2 were transformed
to frame 2 and compared with the actual force signal in the same frame. It is seen
the desired force in both directions were tracked. At t = 258 or y6 = 1.9m, the cart
slipped on the floor and the cart orientation was changed, the mobile manipulator
still can manipulate the cart such that the desired trajectory was tracked. In this
experiments, the motion of the cart in y direction, ye, was chosen as the motion
reference. The experiments results has shown that the integrated task planning and
control approach enables the mobile manipulator to perform complicated tasks with
satisfactory performance. In all the experiments, both the desired cart trajectory and

desired output force were tracked.

125

 

CHAPTER 7
CONCLUSIONS

The major results of this work can be concluded in four parts, the motion reference
project method and its application in mobile robots, robot arm singularity analysis
and its application to singularityless tracking control and mobile manipulator coordi-
nation, the unified model of mobile manipulators and an online redundancy resolution
scheme, force / positon control along the same task direction and its application to ma-
nipulation of moving objects.

First, a new design method for non-time based tracking controller has been de-
veloped. It converts a controller designed using traditional time-based approach to
a new non-time based controller. This approach significantly simplifies the design
procedure to achieve a non-time based tracking controller. The results provides an
efficient and systematic approach to design non-time based tracking controller for
general nonlinear dynamical systems. It is an important step in the development of
a theoretical foundation for sensor-referenced intelligent control. The non-time based
approach is also a key step in design the coordination of the mobile platform and
the manipulator arm. In Chapter 3, this method is applied to tracking controllers of
nonholonomic mobile platforms. The experimental results have clearly demonstrated
the advantage of the non-time based tracking controller in the presence of unexpected
obstacles.

The singularity of manipulators and mobile manipulators are studied. For the ma-
nipulator arm, a singularity—free robot motion controller which can work at or near
singular configurations is presented. Furthermore, the stability of the hybrid con-
troller has been theoretically proven. The primary contribution of the hybrid system
approach is that the motion controller can satisfy different constraints in different
workspace of the robot. The continuous dynamics have different state variable and

the supervisory control is represented in Max-Plus algebra. The stability of such a

126

 

 

switching control system is analyzed by Multiple Lyapunov functions. For the mobile
manipulator, the kinematic redundancy is used to avoid singular arm configurations,
the redundancy resolution is therefore solved by the coordination scheme of the mobile
manipulator.

A unified dynamic model for the mobile manipulator is derived and nonlinear
feedback is applied to linearize and decouple this model. The kinematic redundancy
and dynamic properties of the platform and the robot arm have been considered
in the unified model. Based on this model, mobile manipulator tasks such as path
tracking control, force and position control can be easily designed and kinematic
redundancy can be used to accomplished secondary tasks, such as controlling force and
position along the same direction. An event-based coordination scheme is proposed
for the cooperation of the platform and the robot arm. This online approach is
suitable for a point-to-point task, tale-operation and a sequence of tasks. The mobile
manipulator system is stable even in the case of appearance of unexpected obstacles
and the manipulation capability can adapt to a best possible configuration during the
operation.

Based on the decoupled and linearized model, this dissertation further discusses
the force / position control along the same task direction utilizing the kinematic redun-
dancy of mobile manipulators. For tasks that require both force and position control
along the same task direction, such as pushing a nonholonomic cart, an integrated
task planning and control for mobile manipulators is proposed. The approach for ma-
nipulating the nonholonomic cart integrates the planning and control of the mobile
manipulator, the motion planning and force planning of the nonholonomic cart, and
the estimation of the cart configuration based on the sensors of the mobile manipu-
lators. The integrated task planning and control enables the robot to interact with
moving objects and complete complicated tasks by the output force. The approaches

discussed in this dissertation can be used for the motion planning and control of mo—

127

 

bile manipulators which can find applications in manufacturing, hospitals, homes and
offices. Experimental results have demonstrated the advantage of the integrated task

planning and control approach.

128

 

l1]
[2]

[3]

[4]

[5]

[6]

[7]

l8]

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