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Robotic Manipulator Control: Fundamentals of Task Space Design
Peilin Song
A thesis submitted in conformity with the requirernents for the Degree of Doctor of Philosophy Depart ment of Mechanical Engineering
University of Toronto
@ Copyright by Song. Peilin 1997
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Robotic Manipulator Control: Fundamenfais of Task Space Design Peilin Song (Ph-Dl, Dept. of Mechanical Engineering, University of Toronto
1997
ABSTRACT
The task space-based approach to the design of robot control as an alternative
to the joint space-based approach is an important area of robotics research and de-
velopment . However, its theoretical foundation is less well established. This t hesis
addresses the fundamental issues related to the task space-based approach and con-
tributes to the development of its theoretical foundation. The focus is on two im-
portant task space-based control problems: simultaneous position and force control
(PIF controi) of non-redundant robots redundant robots respectively. The common
feature of the corresponding task space controllers is that each consists of two subcon-
t rollers. The design of t hese controllers requires decomposing the t ask space and/or
joint space into two subspaces. The central issues in the design of the task space
controllers st udied here are: (i) the decomposition and (ii) the interaction between
the sub cont rollers.
Three principle are proposed as the guides for design of the task space controllers:
0 The Principle of Invariance for the decomposition of task space;
The Principle of Minimum Interaction for the design of P/F control;
0 The Principle of Non-Interaction for the design of task space control of
redundant robots.
Two design frameworks are developed based on these principles. The first uses
the principle of invariance and the principle of minimum interaction. It consists of
a generic design methodology for P/F control design and a platform for the analysis
and evaluation of existing and future PIF control schemes. The second frarnework
proposes a generic design methodology for the design of task space control of redun-
dant robots based on the principle of non-interaction. Together with the established
task space approaches to position control of non-redundant robots, the methodol-
ogy proposed in t his thesis yields a complete framework for task space-based control
design.
Robotic Manipulator ControI: Fundamentais of Task Space Design P d i n Song (Ph.D), Dept. of Mechanical Engineering, University of Toronto
1997
The conclusions of this thesis are not primarily the results of empirical studies;
they are theoretical in nature and supported by logical reasoning.
Acknowledgements
It is my pleasure to acknowledge al1 those who provided me with help and encour-
agement during my Ph.D studies.
1 would first iike to thank Professor Andrew A. Goldenberg for providing me
with the opportunity to undertake this rewarding project. He contributed both as a
professor of robotics and an editor.
Financial support for this research was provided by the University of Toronto.
'ISERC and Professor A. A. Goldenberg.
I also thank my colleagues in the Robotics and Automation Laboratory at the
University of Toronto for the friendship they extended to me during in the last 1
years .
Special thanks to my wife, Kathy Y. Huang, for her love and encouragement and
for carrying most of the financial burden during my studies.
Contents
1 Introduction
. . . . . . . . . . . . . . . . . . . . . 1.1 Robot Control: A Retrospective
. . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Joint Space Control
1.1.2 Task Space Control . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Design of Position/Force Control for Xon-Redundant Robots . . . . .
1.3 Design of Task Space Controller for Redundant Robots . . . . . . . .
1.4 Contributions of this Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Research Outline
2 Invariance Issues in the Decomposition of Task and Joint Spaces
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction
2.1.1 The Principle of Invariance . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Chapter Outline
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Task Spaces
2.3 Screw Representat ion of Kinematics and Statics . . . . . . . . . . . .
2.3.1 Euclidean Group Represent at ion of Rigid Body Dis placement
2.32 Duality between Motion and Force-Ray and Xxis Coordinates
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . of a Screw
2.3.3 Properties of a Screw under Transformation . . . . . . . . . .
2.4 Screw Spaces and Their Properties: Orthogonality or Reciprocity? . .
2.5 Description of Constrained Tasks . . . . . . . . . . . . . . . . . . . . 2.5. 1 A Brief Review of the ReIated Li terature . . . . . . . . . . . .
2 Previous Approaches to the Decomposition of the Task Space
2 - 5 3 Non-Invariance of the Orthogonality- Based Decomposit ion and
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Filters 29
5.4 Description of the Task: Decomposition of Task Spaces . . . . 3'2
2.6 Kinematic and Static Filters for Twists and LVrenches . . . . . . . . . :35
2.7 Invariance of the Reciprocity-Based Decompositions and Filters . . . 37
2.8 An Equivalent Approach: Indefinite Inner Product-Based Decomposition 39
2.8.1 OtherKinestaticFiltersandDecompositions . . . . . . . . . . 39
. . . . . . . . . "8.2 lndefinite Inner Product-Based Decomposition 40
?.Y -3 Equivalence of Indefinite Inner Product- Based and Reciprocity-
Based Decompositions . . . . . . . . . . . . . . . . . . . . . . 43
. . . . . . . . . . . . . . . . . . . 2.9 Decomposition in the Joint Domain 45
2.9.1 Decomposition in the Joint Domain: Non-Redundant Robots . 45
2.9.2 Invariance Issues in the Mappings Between Different Spaces . 47
2.9.3 Decompositioo in the Joint Domain: Redundant Robots . . . 51
. . . . . . . . . . . . . . . . . . . . . . . . . . . 2.10 Concluding Remarks 59
3 The Principle of Minimum Interaction and Its Application to P/F
Control Design 60
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.1 Introduction 60
. . . . . . . . . . . . . . . . . . . . 3.2 Principle of hIinimum Interaction 61
3.3 Dynarnics of Const rained Robotic Systems . . . . . . . . . . . . . . . 63
3.3.1 .A Brief Review of Dynamics of Constrained Robotic Systems . 63
. . . . . . . . . . . . 3.3.2 Mode1 of Robot Systems for PIF Control 65
. . . . . . . . . . . . . . . . . . . . . . . . 3.1 Design of P/F Controllers 69
. . . . . . . . . . . . . . . . . . . 3.5 Analysis of Iinown P/F Controllers 74 -c . . . . . . . . . . . . . . . . . . 3.5.1 Decomposed Dynamic Mode1 r û
. . . . . . . . . . . . . . . . . . . . . . . . . . 3 5 . 2 Hybrid Control 79
3.5.3 Resolved Accelerat ion Force Cont rol (RAFC) . . . . . . . . . Y O
. . . . . . . . . . . . . . . . . . . . . . . . . 3 - 5 4 Stiffness Control S4
3 . 5 . Analysis of the Proposed P/F Controller . . . . . . . . . . . . 86
. . . . . . . . . . . . 3.5.6 Cornparisons of Different P/F Controllers S i
.3.6 Indirect Interaction Between the Position and Force Controllers . . . 88
3.7 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
3.S Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
4 The Principle of Non-Interaction and Its Application in
Control Design of Redundant Robots
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . .
Task Space
100
. . . . . . 100
4.2 Task Space Controller Design For Redundant Robots: Position Controller 102
4.2.1 Previous Work . . . . . . . . . . . . . . . . . . . . . . . . . . 102
-1.2.3 Task Space Position Controller Design . . . . . . . . . . . . . 103
1.3 Design of P/F Control for Redundant Robots . . . . . . . . . . . . . 111
4.3.1 Design of P/F Controllers in the Task Space . . . . . . . . . . 112
4.3.2 Model-Based Hybrid Controllers for Redundant Robots . . . . 119
4.3.3 Interaction .A nalysis of P/F Controllers for Redundant Robots 1 19
4.4 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
5 Conclusions and Suggestions for F'uture Research 124
5.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
5.2 Suggested Future Research Topics . . . . . . . . . . . . . . . . . . . 126
A Position Cont ro t of Non-Redundant Robots 127
.A . 1 Position Controllers without Consideration of Dynamic Effects . . . . E S
A . 2 Illodel- Based Position Cont ro t lers . . . . . . . . . . . . . . . . . . . . 130
List of Figures
2.1 Line Representation of A Screw . . . . . . . . . . . . . . . . . . . . . 16
2.2 -4 Screw in Different Frames . . . . . . . . . . . . . . . . . . . . . . . 18
. . . . . . . . . . . . . . . 2.3 A Typical Constrained Task: Peg-in-Hole 26
. . . . . . . . . . . . . . . . . . . . . 2.4 Decomposition of Task Spaces 3.1
2.5 lndefini te Inner- Product Based Decomposition . . . . . . . . . . . . 42
'2.6 Transformation from Task Domain to Joint Dornain . . . . . . . . . 49
2.7 Decomposition of Joint and Torque Spaces for Redundant Robots . . 55
2.8 The Decomposed Joint and Torque Spaces for Redundant Robots Sub-
. . . . . . . . . . . . . . . . . . . . . . . . ject to Constrained Tasks 58
. . . . . . . . . . . . . . . . . . . . . 3.1 The Joint Robot with 2 DOFs 9'2
. . . . . . . . . . . . . 3.2 The Simulated Trajectory for the Joint Robot 94
. . . . . . . . . . . . . . . . . . . 3.3 The Cartesian Robot with 2 DOFs 9.5
. . . . . . . . . . 3.4 The Simulated Trajectory for the Cartesian Robot 97
vii
Terminology and Notation
position vector used in existing literat ure
joint vector
the Jacobian
inertia matrix of robotic manipulators
torque vector
torque vector for classicd joint control
position feedback matrix for classical joint control
velocity feedback matrix for classicai joint control
F, Fr task space controller designed for task space
position feedback matrix for task space control
velocity feedback matrix for t ask space control
torque vector for task space control
eigenvalue matrix of the inertia matrix
eigenvector matrix of the inertia matrix
torque vector for model-based joint control
inertia matrix in the task space formulation
non-linear and gravitational force in the task space
torque vector for model-based task space control
desired position and orientation in the existing literature
translation between two coordinate fiames
3 x 3 rotation matrix
the skew-symmetric matrix associated with p
translation vector
energy
the twist of motion
the wrench of force and moment
the angular velocity of a rigid body
the velocity of a rigid body
the moment applied to a rigid body
the force applied to a rigid body
the screw in ray coordinates
the screw in axis coordinates
the unit matrix of ray coordinates
the unit matrix of axis coordinates
the 6 x 6 matrix for a ray coordinate screw
the 6 x 6 rnatrix for a axis coordinate screw
6- dimension space
the subspace (of R6) of dimension 6 - rn the subspace (of Rb) of dimension rn
the twist subspace of dimension 6 - rn the twist subspace of dimension m
the wrench subspace of dimension 6 - m
the wrench subspace of dimension rn
the 6 x 6 identity matrix
the subspaces of task space resulting from orthogonal decomposition
the kinematic and stat ic filters result ing from ort hogooal decomposi tion
the space of twist
the subspace of twist of freedom
the subspace of twist of constraint
the space of wrench
the base of the subspace of wrench of freedom
the base of the subspace of wrench of constraint
the component of twist t in the subspace of twist of freedom
the component of twist i in the subspace of twist of constraint
the coordinates of tI in the subspace of twist of treedorn
the coordinates of ij in the subspace of twist of constraint
kinematic filter for the twist of freedom
kinematic filter for the twist of constraint
the component of wrench w in the subspace of wrench of freedom
the cornponent of wrench w in the subspace of wrench of constraint
the coordinates of wF in the subspace of ivrench of freedom
the coordinates of in the subspace of wrench of constraint
static filter for wrench of freedom
static filter for wrench of constraint
the base of the joint subspace of freedom
the base of t h e joint subspace of constraint
the base of the torque subspace of freedom
the base of the torque subspace of constraint
the base of joint space
the base of torque space
the image of the Jacobian in abstract space
the base of the external joint subspace
the base of the internal joint subspace
the base of the external torque subspace
the base of the internal torque subspace
the base of the joint subspace of freedom for redundant robots
the base of the joint subspace of constraint for redundant robots
the base of the torque subspace of freedom for redundant robots
the base of the torque subspace of constraint for redundant robots
the actual contacting wrench
the torque induced by t h e contacting wrench
the torque in the modified dynamics of robots
the orientation and position error
the component of ê in the joint subspace of freedom
the component of ë in the joint subspace of constraint
the coordinates of ê1 in the joint subspace of freedom
the coordinates of ë, in the joint subspace of constraint
the proposed PIF controller for modified dynamic mode1
the proposed position su bcont roller
the proposed force subcont roller
the position subcontroller in exist ing P/F controllers
the force subcont roller in exist ing P/ F controllers
position. velocity and force feedback matrices in Hybrid Control
position, velocity feedback matrices in resolved acceleration force control
position, velocity and force feedback mat rices in St iffness Control
the controller for the modified dynarnic mode1 of redundant robots
the external controlier in r,
the interna1 controller in TV
the external controller designed in task space
the position. velocity feedback matrices in Ft
the position and orientation error used in Chapter 4
Chapter 1
Introduction
1.1 Robot Control: A Retrospective
The control of robotic manipulators is a challenging and productive area of robotics
research and development. In recent decades. much effort has been devoted to this
area. and fruitful and diverse results have been obtained. A variety of robot con-
trollers has been proposed for different types of tasks and robotic systems. Position
controllers have been developed for free motion control and position/force (PIF) con-
trollers for constrained tasks. Classical compensators are used to improve the dynamic
performance of a robotic system, and modern controllers such as adaptive controllers
are used to d e d with uncertainties of the system. The theoretical domain in which
controllers are designed can be divided into joint space-based and task space-based
controllers. This thesis deals with fundamentals related to task space approaches.
in this chapter, a systematic overview is given of the joint and the task space
approaches in the context of robot control design. The research topics addressed in
t his thesis are also identified.
1.1.1 Joint Space Control
Robots are programmed and controlled to execute specific tasks defined in the task
space. In the joint space-based control. tasks are first mapped into joint space through
inverse kinemat ic techniques. and t hen the controller is designed in torque space
based on the information about the joint space. The robot is controlled to track
the trajectory in joint space. and the trajectory in task space can then be followed
because the mapping from joint space to task space is injective.
Joint space controllers can be further divided into two groups:
a classic joint control
a model- based control
Generic classic joint control is discussed in [XI. This approach does not consider
the effect of robot dynamics. During the early days of robotics. even the interaction
between the joint variables was ignored. This approach can be easily designed, ana-
lyzed and implemented using the single-inputfsingle-output classical PID technique
because no computation is needed for complicated kinematic and dynamic effects.
Classical joint control is not only simple to implement, but also stable and robust
[ 3 ] . To date. this controiler is the most popular because i t is the one best suited to
practical applications. and modern nonlinear control schemes are too complicated for
industrial robotic applications.
Howerer. the robotic system is a highly nonlinear dynamical system with multi-
ple coupling variables. In applications where either high speed or high precision is
required. nonlinear, dynamic and coupling effects play such a significant role that
ignoring them in controller design will cause robot performance to deteriorate. In
order to design high performance controllers. dynamic effects must be taken into
consideration. There are two steps in the design approach: the inner feedforward
loop design for eliminating dynamic effects; and PD(or PID) outer loop designs for
feed back control.
The inner feedforward loop is designed to eliminate dynamic effects, nonlinearity
and couplings based on the availability of a perfect dynamic model. It is realized by a
computed torque controller which linearizes robotic systems. In the computed torque
approach. knowledge of the dynamic information of robotic systems is a prerequisite.
This method is known as the model-based approach. Typical works in this area include
[30] and [16].
1.1.2 Task Space Control
The task space controller is designed different ly from the joint space controller;
It is first designed in task space and then transformed into torque space. Task space
control has some advantages over joint space control:
Because cont rollers are designed based direct ly on information in t ask space.
they are more sensitive to environments and can be made more intelligent.
Because no inverse kinematics is oeeded, robots can be properly controlled even
in the case of kinematic singularity as long as no predefined motion exists in
the singular directions.
Robot programming can be faster.
The geornetrical data from CAD
robots. greatly reducing the compll
packages can be directly used to program
exity of robot programming.
However, task space control is not as well understood as joint space control and
its design methodology is less established. The feedback gains in task space are not
intuitive and joint space behavior is difficult to predict.
The focus of this thesis is on the design of task space controllers. We will begin
with a review of known research in this area.
Task space controllers can be further categorized into position controllers and PIF
controllers. The former refer to a robot whose end-effector moves freely in 3-D space,
and the latter refer to a robot whose end-effector is in contact with constraining en-
vironments so that some degrees of freedoms are eliminated. In the following section.
we will give an overview of developments of ta& space controllers research topics of
t his t hesis,
1.2 Design of Position/Force Control for Non-
Redundant Robots
Whether it applies to task space or joint space control. the task is always defined in
task space whereas control output is always realized in torque space. The fundamental
difference between task space and joint space controllers is in their mapping relations
from task to torque spaces. In joint space control. the robot's motion in task space
is first transformed into joint space and then the motion in joint space is mapped
into torque space to define the input torque. In task space control, the controller is
first designed in the space of wrench by defining a mapping relation inside the task
space. and then the controller in task space is transformed into the torque space.
The difference in design philosophy between joint space and task space controllers
is determined by the differences in mapping. The theoretical framework for error-
based and model-based position control of non-redundant robots is well established.
.A comprehensive review is given in Appendix A.
Position control is applicable to tasks in which the end-effectors can move freely
in 3 D space. However. there are tasks which require the end-effectors to interact with
the environment. In such cases. the motion of the end-effectors must be in cornpliance
wit h the const raining environment and its contacting states must be controlled. These
t asks include:
high precision assembly requiring information about the state of the contact
between mating parts;
O manufacturing tasks, such as grinding and deburring which require that contact
force and position be cont rolled simultaneously;
0 profile t racking in which the end-effector is required to move along certain trajec-
tories across some surfaces, in contact with the surface. but without application
of force: - Coordination of multiple robots arms ([46] and [lY])
For these tasks. position control is insufficient to realize control goals because
t here is no force feedback loop to control the contact states and the contacting forces
between the end-effectors and the environments. It is necessary to combine both
position and force control for robot manipulators executing the constrained tasks.
This field has attracted the attention of many researchers. and a considerable body
of literature exists on the subject. The fundamental works in this area are:
0 Hybrid Control [39]
Resolved Accelerat ion Force Control
1. Joint space formulation [U]
2. Operational space formulation ['II
Stiffness Control [40]
0 Impedance Control [20].
Referenced P/F controllers depend on the decomposition of task space which
defines the position and force control variables. and allows effective use of the exist ing
pure position and force control methods. When the end-effectors are free in 3D space.
motion space is intact and there is no external force applied to the end-effectors: only
position control is in effect. When the end-effectors are fully constrained, there is no
motion at all, and the space representing the contact force is six dimensional. The
design of force controllers is based on static relationships between forces in task space
and generalized forces in torque space. If the end-effectors are partially constrained,
bot h position and force controls should be employed simult aneously, so combining
t hem is a problem specific to P/F control design.
Because there is no theory guiding the design of PIF control, known PIF con-
trollers are exclusively combinations of pure force and position controllers based on a
variety of intuitive ideas. Thus, due to interactions between them. they often exhibit
unexpected behavior when implemented. For example, in [l] kinematic instability
in hybrid control is reported. The variation in performance and stability among the
published P/F controllers is perplexing. Guidelines are required for the design of
P/F control which allow position and force subcontrollers to cooperate to achieve the
desired performance. Also. wi t hout a consistent t heoret ical framework. i t is oftea
difficult to assess new developments and determine their impact on the field.
Overall. two problems are fundamentals to the theory of P/F control:
a The decornposition of task space t o represent constrained tasks and free motion
simultaneously:
O Design of the the two subcontrollers based on the task space decomposition.
In Chapter 2. representation of constrained tasks is presented based on the funda-
mental properties of task space. The Principle of Invariance is proposed as a guideline
for the description of constrained tasks. Based on this principle, a generic methodol-
ogy is developed for the decomposition of task space. In Chapter 3, the eEect of the
interaction between the two subcontrollers is investigated and the Principle of Mini-
mum Interaction is proposed as guideline for the design of P/F controllers. Overall.
the investigation of the two fundamental issues leads to the formulation of a design
framework for P/F cont rollen.
1.3 Design of Task Space Controller for Redun-
dant Robots
The design of 3 controller for redundant robots has traditionally b e n based on the
joint space approach. The Eoundation is the pseudo-inverse of the Jacobian. Numer-
ous publications in this area are avaiiable. In cornparison, the task space approach is
less understood.
Redundant robots differ from non-redundant robots in t hat they have more degrees
of freedom than are required to obtain the desired kinematic and static properties
of end-effectors. Accordingly, the torque space can be divided into two parts: one
affects the properties of end-effectors in the task space; and the other affects only the
internal motion in the joint space. In other words. whether the joint space or the task
space approach is adopted. the controller of a redundant robot is cornposed of t a o
parts: an external subcontroller for the task domain and an internal subcontroller for
internal motion. The internal motion. which has no efFect on the motion in the task
space. is sued to optimize the performance of the robots based on certain criteria.
The controller in the torque space drives the robot to follow the motion in the joint
space. The joint space controller deals directly with two subspaces in the joint space.
There are different approaches to developing task space controilers. Generally,
the external subcontroller is designed in the task space and transformed into the
torque space. and the internal subcontroller is designed in the internal subspace of
the torque space according to some criteria. In a joint space approach, the dimensional
inconsistency is resolved through the decomposition of joint space which determines
the desired joint space trajectory. In the task space approach, it is resolved through
the decomposition of the torque space.
The task space controller realized in torque space is also cornposed of two parts:
ext ernal and internal subcont rollers. I t is our goal t O est abiish guidelines for designing
tas k space cont rollers for redundant robots which are cornposed of two subcontrollers.
1.4 Contributions of this Research
The objective of this thesis is to provide a theoretical formulation for the design of
task space control of both non-redundant and redundant robots. There are three
fundamental issues involveci:
0 The description of constrained tasks in task space;
0 Guidelines for the design of PIF control:
0 Guidelines for the design of task space control of redundant robots.
The contribution of this thesis is the development of theoretical frameworks for
the design of P /F control and task space control of redundant robots. These are
based on the Principle of Invariance. the Principle of Minimum Interaction. and the
Principle of Yon- Interaction. Al1 developments use fundamentals of the propert ies of
a screw, and subsequent decomposition of the task space. The principle of invariance
is introduced as a guide for the decomposition of the task space. Based on this
principle, a generic met hodology is proposed for the descript ion of const rained tasks.
This principle is also used as a guide for the decomposition of joint and torque spaces
of redundant robots.
The principle of minimum interaction is introduced specifically for P/F control
design. A generic methodology for the design of PIF control is devised based on this
principle. Together they form the theoretical frarnework for P/F control design.
The principle of non-interaction is introduced to deal with control probelms of
redundant robots formulated in task space. A framework is also built for the design
of task space control for redundant robots.
This thesis provides the framework (foundations? guidelines and generic rnethod-
ologies) for the design of task space controllers for both non-redundant and redundant
robots.
1.5 Research Outline
This thesis consists of five chapters. Chapter 1 introduces the introduction. a descrip-
tion of the research motivation and a summary of contributions, and the outline of
the thesis.
The principle of invariance is proposed in Chapter 2. A brief introduction and
the principle of invariance are given in Section 2.1; the concept of task space is clari-
fied, and in Section 2.2 the screw space is introduced to represent the task space. In
Section 2.3, the properties of the screw space are investigated, and in Section 2.4 the
reciprocity property in screw space is defined. A reciprocity-based decomposition of
the task space is developed in Sections 2.5. 2.6 and 2.7. In Section 2.8, the equiv-
alence between the proposed decomposi t ion and the indefinite inner product based
decomposition is proven. The decompositions of both joint and torque spaces are
developed in Section 2.9. In Section 2-10, an overview is presented of the invariant
decomposi tions.
In Chapter 3. a theoretical frarnework is described. After a brief introduction in
Section 3.1. the interactions between two subcontrollers are studied. and the principle
of minimum interaction is proposed in Section 3.2. A dynamic model of constrained
robot ic systems is developed in Section 3.3. Based on the principle of minimum inter-
action, a generic methodology for the design of P/F controllers is developed. Using
the decornposed dynamic model, the established P/F control rnethods are analyzed
based on the principle of minimum interactions as stated in Section 3.5. A perfor-
mance criterion for P/F control is proposed in Section 3.6. To illustrate the effect of
the interactions between the two su bcont rollers. two examples are given in Section
3.7.
Chapter 4 presents a theoretical frarnework for the design of task space control
of redundant robots. The motivation is described in Section 4.1. In Section 4.2, the
principle of non-interaction is proposed and a generic methodology is developed for
the design of task space control of redundant robots. In Section 4.3, both the principle
of minimum interaction and the principle of non-interaction are applied to the design
of P/F control of redundant robots.
Conclusions are presented in Chapter 5 along r i t h some recommendations for
future research.
Chapter 2
Invariance Issues in the
Decomposition of Task and Joint
2.1 Introduction
2.1.1 ThePrincipleof Invariance
In applied sciences, physical laws are studied with the help of mathematical mod-
els. Mat hemat ical models are idealized represent ations of physical systems. It is a
fundamental requirement t hat the mat hematical model reflect the nature of physical
systems regardless of the way in which physical systems are described.
Robotics. like ot her applied sciences, utilizes mathemat ical models to represent
physical systems. The physical models here refer to robotic systems and their in-
teracting environments. For robotic systerns, a mathematical model, here a robotic
model. should describe the kinematics, s tatics and dynarnics of physical systems. The
relationship between the mathematical and physical models is independent of the way
physical models are described. For example, the relationship between motion in joint
space and motion in task space is independent of the reference frame and unit sys-
tems: units and reference frames have nothing to do with the effect of joint motion on
the movernent of end-effectors. This invariance between physical and mat hemat ical
models is a cornerstone of robotics theory.
Task definit ion and descript ion are indispensable in robotics research and ap-
plicat ions. Representat ion of t asks should also reflect the physical nature of tasks
in the real world regardless of the way tasks are described. In task definition and
description. there are many aspects for which this invariance relationship must be
maintained. The focus of this chapter is to describe constrained tasks and tasks for
redundant robots. For constrained tasks, both task and joint spaces are decomposed.
The joint and torque spaces for redundant robots are always divided into external and
interna1 motions. Both task definitions and descriptions are involved in this decom-
position. So the decompositions must maintain the consistency between the physical
system (robotic system) and the mathematical model. This can be summa.rized as
The Ptinczple of Invariance, which states that the decomposition must be kept in-
variant with respect to changes of reference frames and units and to transformations
between task and joint (and torque) spaces.
2.1.2 Chapter Outline
The importance of invariance has recently attracted the attention of robotics re-
searchers. and a relevant body of literature has been published in this area. In this
chapter. based on the description of task space and its decomposition, a systematic
clarification of invariance is conducted, and the principle of invariance is introduced
and applied to the decompositions of task and joint (and torque) spaces, respectively.
The chapter is organized as follows. First, the definition of task space is clarified
and screw space is shown to be a representation of task space. Based on a thorough
study of the properties of screw space, a reciprocity-based decomposition is proposed
which is proven to be equivalent to previous decompositions that use different ap-
proaches in [43]. Xlso a generd methodology is devised for the decompositions of
joint and torque spaces for redundant robots. The criteria for both decompositions
satisfy the invariance requirement. The results proposed in this chapter form the
basis of the research in the remainder of t his thesis.
2.2 Task Spaces
.A robot ic manipulator is designed. programmed and cont rolled to execute certain
tasks. Generally, tasks are defined in terms of motion and statics of end-effectors of
robot ic manipulators.
Task space is a concept closely associated wit h the description of tasks. Usually,
the ta& space is considered as the physical space in which tasks are defined and
described. The physical space in which the motion of end-effectors occurs is regarded
as the task space in most task space related research ( [ 2 ] and [40]). It seems plausible
to equate the 3-D physical space with the task space for planar robots which are
frequently used in the literature as an experimental facility to validate proposed
control schemes. However, a careful examination of physical space reveals that it can
not be a task space . The reasons are:
a a task incorporates both the kinematics and statics of end-effectors; physical
space, however. can not be the space for bot h kinematic and static representa-
t ions.
O to describe kinematics and statics of the end-effectors in spatial motion requires
at least 6 variables. Physical space is of insufficient dimension for description
of spatial kinematics and st at ics of end-effectors.
Physical space is the space in which the motion of end-effectors occurs, not the
space in which the end-effector kinematics and stat ics can be properly described. The
ta& space should meet the following conditions:
O because tasks are depicted in terms of motion and force, task spaces should be
trvofold: a space is required for the motion and i t s dual space for the force.
O task spaces should be well-defined with rnetrics. the physical significance of
which should be associated with tasks.
a task spaces should be endowed with either a motion or statics unit, depending
on which space is referred to.
It will be showo in the next section t hat the screw space m e t s these requirernents.
2.3 Screw Representation of Kinematics and Stat-
ics
The content in first subsection is adopted from (41). In the rest of this section, the
author will illustrate the propertiies of screw space, which were discussed by many
researchers such as Samual [41].
2.3.1 Euclidean Group Representation of Rigid Body Dis-
placement
The displacernent of a rigid body is an isometry of p, which includes both translation
and rotation. The set of al1 position and orientation preserving isometries is the
special Euclidean group in three dimensions. This group is denoted by SE(3). It
should be recalled that the finite motion of a rigid body can not be described in a
six-dimensional space; it can only be represented in terms of a group.
One of the possible representations of an element g E S E ( 3 ) is the well-known
4 x 4 homogeneous transformation matrix:
where R E SO(3) is a 3 x 3 rotation matrix and t is a translation in 3-D physical
space. Besides the homogeneous transformation matrix, Clifford algebra and other
methods can also be used to represent SE(3). In [41] it is shown that SE(3) has the
structure of a smooth six-dimensional manifold.
Small rigid body displacements are represented in velocity space as tangent vectors
to SE(,Y). This vector space is called the Lie algebra of SE(3). and is denoted as s e (3).
Motions of a rigid body occur in se(3). whereas its statics occurs in se(3) ' , which is
dual to se(3) [29].
If we use a screw vector to represent rigid body displacement. velocity and force,
we make the following definitions:
0 velocity screw? a twist t representing an instantaneous velocity of a rigid body.
In mathematical terms. it is the vector tangent to the motion manifold of the
rigid body;
force screw, or wrench W, representing a force on a rigid body. I t is the vector
dual to the tangent vector.
A bilinear operation on se(3) and se(3)' is defined to represent energy:
or in a differential form:
2.3.2 Duality between Motion and Force-Ray and Axis Co-
ordinates of a Screw
When represented by a screw vector, the motion of a rigid body in physical space is
a twist cornprising a rotation around and a translation along the screw axis. When
the a i s passes through the origin of reference frame, the twist is given by:
where w is the angular velocity vector and v the translational velocity in the direction
of W.
When the reference frame is moved along a vector d, the new twist is:
The statics of a rigid body can also be described by a screw, the wrench. It
consists of a moment m around, and a force f along the wis of the screw. When the
axis passes through the origin of the reference frame. the wrench is given by:
When the reference frame is displaced dong a vector d, the new wrench is:
It is easy to see the asymmetry between the twist (2.5) and the wrench (2.7).
These two expressions represent two basic types of screws, which underlie two types
of coordinates: ru y coordinates (2 .5 ) and azis coordinates (2.7).
These two types of coordinates can be interpreted in terms of two screw represen-
tations of a line. As shown in Figure 2.1, for a line 1 passing through a point P. two
vectors, I and p x 1, are sufficient to determine this line. This line can be expressed
in either ray coordinates
or axis coordinates
Obviously, i f in (2.5) is the twist in ray coordinates, and W' in (2.7) is the
Figure 2.1: Line Representation of A Screw
wrench in axis coordinates. Just like the representation of a line, the representation
of any screw can be expressed in either ray or axis coordinates. For the sarne screw,
a e denote by S its ray coordinates and by s its axis coordinates. The relationship
between s and 5 is:
w here
It should be noted that the above relation between s and is a correlation [26].
2.3.3 Properties of a Screw under Transformation
The representation of a screw is not only coordinate-dependent, but also reference
frame-dependent . When the reference frame is changed, the screw configuration
changes accordingly. In Figure 2.2, we show a screw and two reference frames ozyz 1 1 1 1
and o x y r . The distance between the origins of the two frames is represented by
Pl. The relationship between the representation of the screw in the two frames will
reveal the properties of a screw under transformation.
A screw in ray coordinates is represented in reference frame oxyz (Figure 3.2) by:
where i is the vector of direction coordinates of 1 in frame o x y t ; i, j, k are the unit
vectors of the axes qy,: of the frame oxyz; and P is the skew-symmetric matrix l t f 1
associated with the vector p. When represented in reference frame O x y r , the screw
Figure 2.2: A Screw in Different Frames
becomes:
1 / ' 1 - 1 .l
where 7 is the vector of coordinates of 1' in frame O x y r : i . J . k' are the unit coor- 1 1 1 1
dinate vectors of O x y i ; and P; is the skew-symmetric matrix associated with the
translation vector / I l l
The relation between the coordinates in the frames oxyz and O x y 2 is
where Pl is the skew-symmetric matrix associated with the translation vector pi with
respect to frame oz yz. and R is the transformation matrix defined by:
The coordinates of a screw can be represented as vectors in abstract space R6.
narnely
where 14 and ifX' are the 3 x 3 identity matrices of units of 1 and p x 1: respectively,
and At E R 6 . 1 1 1 1
If the screw is represented in irame O x y 2 . its coordinates are:
Thus. in penerai. a screw in ray coordinates can be represented by:
s' = F,~,~~~ h; SPA,,
and the general form of the base of screw space in ray coordinates:
t' = F~~~~~~ hpi I;P
Similarly, we can get the general form of the base of screw space in axis coordinates:
W' = F I ~ Y ~ L ~ iY2 1 g R 6 (2.18)
in which
and
2.4 Screw Spaces and Their Properties: Orthog-
onality or Reciprocity?
In the previous sections, screws are described from a physical perspective. In this
section. we will study them in terms of mathematical concepts. As mentioned. a.
screw c m be expressed into two forms: in ray coordinates or in axis coordinates. It
is the convention of this thesis. unless specified. t hat a twist refers to a screw in ray
coordinates and a wrench refers to a screiv in axis coordinates [Z].
.Any screw can be parameterized in either abstract space or screw space. Let us
take the twist as an example. From the previous section, we can see that a twist can
b e expressed as follows
I f 1 1
It should be noted that, from now on, the primed coordinate frame O x y r will no
longer be used as a general frame, but will be replaced by the frame oxyz.
The twist can also be represented in the twist space spanned by t:
t = t A t .
The relationship between the twist space and the abstract space is clear:
In the same way. we can get the relationship between the wrench space and the
abstract space
The properties of screw spaces (bot h twist and wrench spaces) can be investigated
with the help of relations (2.31) and (2.23).
The abstract space spanned by R6 is a 6-dimensional vector space. It can be
endowed wi t h the met ric which makes orthogonal decompositions meaningful.
The differences between abstract spaces and the screw spaces described in (2.21 )
and (2.22) determine the properties of screw spaces.
The representation of a screw space is coordinate frame dependent, as expressed
a Screw spaces are endowed wit h units by eit her I,' or 1:.
a Screw spaces are not homogeneous because of the introduction of IilI,' (or
Il-* 1;).
a Orthogonality. which is established in the abstract space. is no longer valid in
screw spaces. Instead, it is replaced by reciprocity. The decomposition of a
screw space is based on reciprocity. This issue is so important that the rest of
this subsection will be devoted to it.
Any six-dimensional (abstract) vector space R6 can be decomposed into two ar-
bit rary and mut ually independent subspaces:
which are orthogonal; t hus
Let us examine the images of &-, and R, in a screw space with respect to o x y r .
According to (2.3.3).
Their inner product is defined by:
It is obvious that orthogonality can not be established in this screw space since
the inner product is non-invariant under changes of the origin of the reference frame
and/or unit. Geometrically speaking. the introduction of matrices K1 and 1; distort
the structure of the abstract space in which orthogonality is established.
Similarly, the images of &-, and R, in CI; are:
and the inner product is:
The vanishing of this inner product also varies with translations of the reference
origin and changes of units.
However, the orthogonality of subspaces &, and R, in Rb is inherited by sub-
spaces of different screw spaces. For ts-, in ray coordinates and LVm in axis coordi-
nates' the following holds:
because f ïTK2 = O and 41: = h l s ( h is a scaling parameter of the units). Similarly,
the following relation can be obtained:
The relationship between t6-,and W, (or t , and W6-*) is called reciprocity.
This concept is different from orthogonality in that the related "orthogonaln subspaces
belong to different screw spaces.
2.5 Description of Constrained Tasks
The screw spaces spanned by t and CV are the spaces in which the spatial mo-
tion and force of robotic end-effectors are represented. When robots are required
to execute a task in contact with environrnents, both twist and wrench spaces are
decomposed. In designing P/F controllers. a correct description of constrained tasks
is a prerequisite. It is the purpose of this section to define a framework for the de-
script ion of const rained tasks. Exist ing approaches to decompositions of const rained
task spaces are first reviewed. and their shortcomings are pointed out from both
mathematical and physical viewpoints; then a new approach is proposed.
2.5.1 A Brief Review of the Related Literature
The key elements in P/F control are kinematic and static filters which define motion
and force control variables, respectively. The derivat ion of the existing filters is based
on the decomposition of the task space into two "orthogonalc subspaces: motion
and constraining force subspaces. The orthogonal decompo~ition~ which forms the
basis of hybrid control, seerns so plausible that it has been taken for granted since its
incept ion [39].
The *orthogonal9 P/F decomposition approach is based on the foundation laid
by Mason [33]. The main concept in this work is intuitive and is based on the or-
thogonality of the subspaces of possible displacements and of reaction forces of the
robot manipulator in contact with the environment. To validate these arguments,
hfason in 1331 defines C Surfnces in real 3-D space. which are visible surfaces between
the extremities of the solid and the free space. In an ideal domain in which a ma-
nipulator is considered an ideal end-effector described in terms of position a n d force
variables, it is claimed t hat there are counterparts of the C-surfaces. hypersurfaces in
R6. According to (331, the position control variables lie in the vector space tangent
to hypersurfaces. while the end-effector forces are restricted to be orthogonal to the
tangent space.
However, in the screw space, there is no meaningful positive definite inner product
or metric [39] based upon which a C-surface could be constructed and orthogonality
established. Moreover, the motion of a rigid body can be represented only in terms
of manifolds. Because of the lack of a positive definite inner product. ort hogonality
does not hold in the screw space. and no orthogonality-based decornposition can be
established. In it is shown t hat ort hogonality-based decomposition in the screw
space is non-invariant under changes of origin of the reference frame. This argument
is validated by [Y] and [%]. More recently. (151 introduced an indefinite inner product-
based decomposi t ion t hat maint ains the decomposit ion invariance.
In [33] it is pointed out that position and force controls are a pair of dual con-
cepts. It is also asserted there that, in the ideal domain. the description of the ideal
end-effector is twofold: a point in the motion space corresponding to position and
orientation; and a point in the force space corresponding to force and moment. In
our approach. this duality yields the foundation of a new and invariant decomposition
for hybrid control.
Ort hogonality-based (non-invariant ) decomposition has been accepted by the robotics
comrnunity for a long tirne. This is partly due to &the simplicity and symmetry of
the problem" [8]. In most hybrid control research, the models considered are either
planar robots or robots whose motion is restricted to a plane. In this case, the motion
of the end-effector can be expressed in terms of translations only. rendering it simple
and symmetrical compared with general spatial motion (rotations and translations).
The underlying problems are disguised by this simplicity and symmetry. For spatial
robotics, sirnplicity and symmetry disappear. and orthogonality-based decomposition
is found to be no longer valid.
2.5.2 Previous Approaches to the Decomposition of the
Task Space
Previous approaches to decomposi t ion were based rnainly on the t heoretical founda-
tion given in [33]. To illustrate the main idea, we consider the peg-in-hole task as an
example (Figure 2.3). In this framework (331, the end-ef€ector velocity and force are
represented as vectors in a six-dimensional linear vector space (spanned by B) over
the reals:
Figure 2.3: .A Typical Cons t rained Task: Peg-in-Hole
where B = diag(l.1.1.1.1.1) and A,.,\, E R6.
For the peg-in- hole case ( Figure 2.3).
fz = O Tz = O
It is intuitive and plausible to decompose the space spanned by B (with respect
to the reference frame OZ yz) into two orthogonal subspaces:
P defines the basis of the motion subspace and Q the basis for the force subspace.
Thus, any vector 1, in B cari be decomposed as follows:
1, = lp + iq = P X p + QXQ
We multiply (2.33) by P* and solve to get:
Similady, if we rnultiply (2.33) by QT. we get:
T herefore,
-1 T IQ = Q A Q = Q[Q=Q] Q 1 = S j l r
where the kinematic filter is
and the static filter is
SI = Q [ Q ~ Q I - ' Q ~ (2.37)
The filters project any vector in space B ont0 the motion or constraining force
subspaces. respectively. For the peg-in-hole ta&,
Based on this approach, we observe the following:
0 The kinematics and statics of the rigid body (end-effector) are represented in
one space.
The decomposition and filters are based on the ort hogonalit- established in the
space B.
These two points form the basis of most decompositions and definitions of filters.
Cnfort unately. t hey are fundamentally Bawed. In the following sections. we will
elaborate on t his contention.
2.5.3 Non-Invariance of the Ort hogonality-Based Decom-
position and Filters
In the peg-in-hole task (Figure 2.3)' the motion subspace and its complementary
subspace are:
here. P and Q are orthogonal to each ot her. and this orthogonality is used to construct I l I I
the kinestatic filters in (2.31) and (2.3'7). If we assume that O r y 2 is a translation ' I I 1
of or yr dong x, the decomposition wi t h respect to the reference frame is o r y z :
1 1 1 1
rvhere P' and Q' are the counterparts of P and Q in the frame O x y z and
and
Note t hat II':& = Is only if r = O. So generally
P'*Q' # P ~ Q = 0,
1 1 1 1
and orthogonality does not hold with respect to O x y : . This means that the orthog-
onality between the two subspaces is non-invariant under changes of origin.
The kinestatic filters, which are called "selection matrices' in the literature (e-g,
[39]). define force control and motion control variables. Because they reflect the
nature of the natural const rained system, t heir representat ions in different frames
should satisfy the same relationships in order to keep the mathematical and physical
models consistent.
FVe take the motion selection matrix Su (2.36) as an example. In reference frame
oxyz:
where f is the coordinates of twist of the end-effector i in ray coordinates and Fu is the
coordinates of selected position and velocity controlled components of i. In reference
-f
where t , = A'&, t' = Kl i and S: is the transformed
Clearly Si is a simiiari ty transformation of Su:
I I I 1
filter in the frame O x y t .
(2.44)
This relation must be maintained under frame transformation. However, the
ort hogonality- based filter does not satisfy this requirement. The filter Sv based on
the orthogonali ty decomposit ion (2.34) is:
Sv = P[P=P]- 'P=.
I l I I
In reference frame O x y z , by definition, the filter is
S; = P ' [ p f T p ' l - l p ' T
Referring to (2.39),
However. according to (2.44). S: is defined as:
In order to keep the filter invariant under the transformation defined by K I . it is
necessary that K r & = I6 to keep S: = s:. However. this requirement does not hold
as long as the displacement of the reference frame origin is non-zero. It is obvious
that the orthogonality-based filter is non-invariant under translation of the reference
frame.
2.5.4 Description of the Task: Decomposition of Task Spaces
Control of constrained tasks differs from pure position control in that the former
requires a description of const raints. O ften, the cons traints are represented in tems
of algebraic equations. However, not al1 the constraints can be expressed explicitly
in this way. For example, when both translational and rotational degrees of freedom
are constrained, it is impossible to find algebraic equations to express the constraints
because they form a submanifold of SE(3) rather than a surface or a hypersurface.
In other words, when expressed in SE@), the constraints a.re not coincident with
physical surfaces in &dimensional space. In addition, the description of a task could
inciude information about constraint force, whereas algebraic equations of the physical
const raint can not incorporate statics informat ion.
When a rigid body is subjected to rn independent constraints, it loses m degrees
of freedom. Its motion is then reduced to a submanifold of SE(3). The motion
of the constrained system can be studied through the first-order kinematics of the
constrained system occurring in the space tangent to the manifold. The first-order
kinematics can be represented as a twist (the screw of motion). The statics of the
rigid body can be represented as a wrench, which is the dual of a twist.
In the constrained case, the twist space can be split into two subspaces: one
corresponding to allowable motions, called the jreedom twist subspace I; and the other
corresponding to constrained motions. called the constraint twist subspace c l . Thus.
the twist space is decomposed into two subspaces (Figure 2 - 4 :
t = [I. el.
This description is represented in the frame oxyz as:
f = F , z p 6 - m
According to [33], f corresponds to artificial constraints. and c to natural con-
straints.
The wrench space (in axis coordinates) can also be decomposed into two parts
(Figure 2.4):
W = [ F , Cl,
where C(= F,,It&) represents the constraint wrench subspace , and F(= FlyL
represents the freedom wrench subspace (for example, friction).
The decompositions of both t and CV are not arbitrary. They depend on the
nature of the constrained tasks. When the end-effector is in touch with constraints,
the freedom wrench, which is in the subspace F, does no work dong the direction of
the constraint twist, and only affects the freedom twist. Thus:
Here the constraint twist subspace is the intersection of al1 sets of constrained motion
33
The Space of the Twist The Space of the Wrench
Joint Space Torque Space
Figure 2.4: Decomposition of Task Spaces
Dually. the constraint wrench. which is in subspace C. does no work along the
direction of the freedom twist and affects only the constraint twist. Thus
f -C = O (2.33)
Equat ions (2.5 1)-(2.54) represent the decomposition of the constrained t asks and
describe of the nature of the constrained tasks. They specify the goals of PIF con-
trollers. Failing to satisfy these conditions will result in the violation of constraints
or the failure to attain control goals.
2.6 Kinematic and Static Filters for Twists and
Wrenches
In a P /F control scheme. the basic elements are the filters. These identify position
and force control variables. respectively, in order to guarantee consistency between
the specified task and environmental constraints. The filter dealing with the position
control variables is called the kinematic filter. and the filter dealing with the force
control variables is called the static filter.
Kinernatic Filter
Based on (2.49), a twist of the end-efTector subject to m constraints can be
decomposed into two parts:
where p j E &-, is the vector of coordinates in the freedom twist subspace
and p, E R, is the vector of coordinates in the constraint twist subspace. By
multiplying by on both sides of (2 .55) and solving it for p j and E r , we obtain:
Therefore. the kinematic filter for the freedom twist in ray coordinates is:
P/ = f ( F ~ f)-' F*. (2.56)
Similady, the kinematic filter for the constraint twist in ray coordinates is:
Stat ic Fiiter
In the wrench space in avis coordinates. any wrench can be decomposed as
follo~vs:
where p~ E &-, is t he vector of coordinates in the subspace of the freedom
wrench and pc E R, is the vector of coordinates in the subspace of the con-
stra.int wrench. Multiplying / = on both sides of ('7.58), we get:
The static filter for the freedom wrench is:
36
pF = ~ ( j ~ ~ ) - l j ~
and the static filter for t h e constraint wrench is:
2.7 Invariance of the Reciprocity-Based Decom-
positions and Filters
In the reciprocity- based decomposi t ions. t he twist and wrench are generally ex-
pressed in duai coordinat,es. The bases of the subspaces of the twist space are:
and the bases of the subspaces of the wrench space are:
Because
the following relations are readily established in conjunction with (2.51 )-(3.54):
The above relations form the basis of the decompositions of the ta& spaces with
respect to the new reference frame. It follows t hat reciprocity-based decomposit ions
are invariant under changes of the origin of the reference frame.
The filters are also invariant under changes of reference frame. The kinematic I I l
filter for the freedom twist with respect to O x y z' (Figure 2.3) is:
P; = / ' ( F ' ~ ~ ' ) - ~ F ' ~ = K' ~ ( F * K ; I < , f ) - l ~ * h ; T = K , f ( ~ ~ f ) - I ~ ~ h ' ; l = K , P ~ K ; '
(2.70) I f 1 1
The static filter for the constraint wrench with respect to O I: y 2 is:
T T pi = c ' ( C ~ C ~ ) - ~ ; * = I ~ C ( C Kl l i 2 ~ ) - ' c T ï i ~ = K 2 ~ ( c T ~ ) - ' c T h ; - l = IGPCI\;-'
(2.71)
In the same way, we can get the kinematic filter for the constraint twist and the
static filter for the freedorn wrench:
and
From the above. we can see that reciprocity-based filters are invariant under
changes of the reference frame.
2.8 An Equivalent Approach: Indefinit e Inner Produc
Based Decomposition
2.8.1 Other Kinestatic Filters and Decompositions
In recent years, it has been recognized that a theoretical basis for the decomposi-
tion of a task space is needed because of the non-invariance of orthogonality-based
decomposition and kinematic and static filters. In (261, Lipkin and Duffy stress the
difference between the screws in ray and axis coordinates. In p i ] , they get the same
decompositions of task space:
based o n the conditions
which results in the following kinernatic filters
P ~ L D = j ( f T ~ f r 1 f T 1 PcLD = C ( C ~ L C ) - ' C ~ A . (2.76)
However, as stated in [-'il, these filters fail when either f or c is an isotropic
subspace because
Lipkin and Duffy investigate the properties of isotropic subspaces and point out
that in the case where isotropic subspaces exist. the screw space can be divided into at
rnost four non-intersecting subspaces. They derive the projection matrices for these
subspaces. However, they fail to define the kinestatic filters based on this novel and
correct decomposi t ion.
In [43], an indefinite inner product-based decomposition is introduced to get an
invariant decomposition in screw space. This approach is equivalent to that of Lipkin
and Duffy [Z]. Invariant kinestatic filters are also derived based on the indefinite
inner produc t - based decomposi tion. In the following subsections. t his decomposit ion
of screw space is reviewed. and its equivalence with the decomposition proposed in
this chapter is proven.
2.8.2 Indefinite Inner Product-Based Decomposition
Instantaneous work can be represented in two ways. One is the bilinear form in terms
of ray and axis coordinates that leads to the approach discussed above. The other is
the indefinite inner product which defines work as follows:
where w(= AW) is in ray coordinates, and w is in axis coordinates. The operator
A. a 6 x 6 indefinite matrix. generates an indefinite inner product.
Let us express bot h twists and wrenches in axis coordinates. denote the freedom
twist subspace as -11 and the constraint wrench subspace as C. To satisfy the condition
that the constraining wrench does no work on the freedom subspace. the following
must hold:
SflTLlc = 0.
In general, LCI n C # O(empty set). The intersection
is defined as a radical 1271 of the subspaces Y and C. which can be decornposed as
40
rvi t h the following relations holding:
.WAR = O G ~ A R = O R*AR = o.
Furthermore. t here exists a subspace D E R6 which satisfies
On the basis of the above definitions. a 6-system of screws is decomposed as
(Figure 2.5):
R6 = .ï.1&3 R S D.
.4ny screw f (in axis coordinates) is then described as:
w here
Ei = PiE. i = -il. C, R, D.
The projection matrices in (2.Y-L) are:
Decomposition of screw space basd on indefinite inner product in axis coordinates
Decomposition of the space of twist in ray coordintaes
Decompositiun of the Space of wrench in axis coordinates
Figure 2.5: Indefinite ber-Product Based Decomposition
Thus. the freedom twist is given by:
and the constraint wrench by:
rvhere P.,] + PR and PC + PR are the kinematic and static filters. respectively. based
on the indefinite inner product decomposi tion.
2.8.3 Equivalence of Indefinite Inner Product-Based and
Reciprocity-Based Decompositions
Based on the indefinite inner product-based decomposition, the screw space can be
decornposed into at most iour non-intersecting subspaces. Any subspace of of twist
and wrench spaces obtained by reciprocity-based decomposition can be defined by
di fferent combinat ions of some of t hese iour subspaces.
In the wrench space, C represents the constraint wrench and (iC.1, D) the freedom
wrench (attributed to motion control).
In the twist space, Ji represents the freedom twist and (c, D) represents the
constraint twist. If the twist is expressed in ray coordinates (Figure 2.5):
J = A(.VI. R) c = l ( C . D).
If the wrench is expressed in axis coordinates. then
F = (.il, D) C = (C. R). (2.88)
According to the definitions of filters(Z.56) and (2.37). the kinematic fiiter is:
which is the filter for the twist in ray coordinates. Its counterpart in axis coordinates
is:
S, = AS,P = . ~ I ( . @ ~ I . ~ I ) - ~ . ~ I ~ A + R(D~AR)-WA. (2-89)
In the same way. we get the static filter based on definitions (2.61) and (2.62):
Because the filters are invariant. Sv and Sc are also invariant. Frorn equations
( 2 3 9 ) and (2.90)
and
Thus, it is easily concluded that the indefinite inner product-based decomposition
is equivalent to the reciprocity-based decomposition because both of them generate
the same kinestatic filters which are invariant under changes of origin of the reference
frame.
2.9 Decomposition in the Joint Domain
Task cont rollers. although designed based on the inforrnat ion in task space. are even-
tuallg realized in torque space. Task space is where tasks are described and control
goals are set. The characteristirs of controlled robots in task space are determined by
the performance of controllers designed in torque space. In t his section. we discuss the
transformation from task space to joint (and torque) space. In the past. the transfor-
mation from task space to joint (and torque) space has been frequently used in inverse
kinemat ics. hy brid control design and robot performance anaiysis wi t hout a deeper
understanding of its characteristics. In t his section different types of transformations
are investigated: this is indispensable for the proper design of ta& space-based robot
cont rollers.
In this section. the transformations from task spaces to joint (and torque) spaces
for both non-redundant and redundant robots are described. We begin with the
transformation for non-redundant robots because it is unique in a non-singular con-
figuration.
2.9.1 Decomposit ion in the Joint Domain: NomRedundant
Robots
Joint and torque spaces are a pair of dual spaces. Both joint and torque spaces
are R" with dimension of either angular dispalcement or torque. They have their
counterparts in the task space: joint variable space corresponds to t h e twist space
and the joint torque space to the wrench space. In the constrained case, both joint
spaces can be decomposed into two parts. These decompositions are based on the
corresponding decompositions of the task space.
It is well knoivn t hat the twist space and the joint velocity space are linked t hrough
the Jacobian:
where f is the twist of motion. q is the vector of joint velocities and J is the Jacobian.
The corresponding freedom and const raint subspaces of the joint variable space
can be derived as follows:
where TI and Tc are the bases of the images of freedom and constraint subspaces of
the task space in the joint space, respectively.
The mapping from the wrench space to the joint torque space is dual to the
rnapping from t h e joint variable space to the twist space:
7 = pw.
where r is the vector of joint torque in the joint torque space? and w is the wrench
in the wrench space.
The subspace of the joint torque space attributed to motion control (corresponding
to the freedom wrench) is spanned by:
and the subspace of the joint torque space attributed t o the contact force (corre-
sponding to the constraint wrench) is spanned by:
Therefore from equation (2.51)-(2.54). the following relationships betw-een joint and
torque spaces can be immediately est ablished. These relations represent the objectives
of the P/F controllers. and they must be satisfied at al1 times during the execution
of a task.
2.9.2 Invariance Issues in the Mappings Between Different
Spaces
In t h e previous section. a decomposition was proposed for t h e description of con-
strained tasks; it highlights certain properties of the ta& space (screw space), mainly
the invariance under transformations of the reierence frame. In this section, we an-
alyze the invariance issue for mappings betneen different spaces. This plays an im-
portant role in the design of P/F controllers.
Three types of spaces were used in the description of constrained tasks: abstract
space R6. screw space (the twist space and wrench space) and joint space (the spaces
containing variables and torque).
The screw space differs from the abstract space in two respects:
rn The screw space has physical rneaning and dimensions(units): the abstract space
is dimensionless ( unit less).
0 The screw space is non-invariant under transformations of the reference frame
and units: the abstract space is origin-independent.
The relations between the abstract and screw spaces are in Section 2.3.3. Be-
cause of the mappings I,', I& h; and &, representing the units and variance with
translations of the reference frame, the orthogonality, which is established in the ab-
stract space. does not hold in the screw space. Also the duality of the mappings (1,'
vs. 15 and Kl vs. K2) brings forward the duality of the screw spaces with their
representations in ray and axis coordinates (Figure 2.6) .
The relations between the screw spaces and joint spaces can be expressed in terms
of the Jacobian. Denoting the base of the space of twist as t and the base of the wrench
space as CV, the bases of the corresponding joint spaces are:
and
;V = ~ C V . (2.101)
From (2.91), (2.95). (2.97) and (2.93). the following relations are obvious:
Because any twist can be thought of as a linear combination of vectors in the
space spanned by the Jacobian, every column of the Jacobian is a base twist in the
same coordinates as the twist t. In the case where the twist is in ray coordinates, the
Jacobian is:
abstract space s 1 The space of twist
Joint space
The space of wrench
(6)
jT T T J ~ ( = J KI Ii)
I ' * 1 v
torque space 1
Figure 2.6: Transformation from Task Domain to Joint Domain
where each column of J is the image in abstract space of the corresponding column
of the Jacobian J . Thus. the base of the joint variable space is:
and the base of the joint torque space is:
Because J is a matrix defined in the abstract space where orthogonality holds.
the ort hogonality is meaningful in the joint spaces as well.
Although the orthogonality is meaningful in both the abstract space and the
joint space. the decomposition of constrained tasks is conducted in the screw space
because tasks are naturally described t here. .As mentioned above, the decomposi tion
is based on the reciprocity between the twist space in ray coordinates and the wrench
s Pace in axis coordinates. As shown in Fig.7. the twist and wrench spaces are
decomposed according to equations (2.51 ) to (2.54). Correspondingly. the abstract
space is decornposed into two subspaces (2.23) which are orthogonal to each other.
It has been shown that the orthogonality between &-, and R, is inherited by the
reciprocity between the freedom twist subspace and the constraint wrench subspace
and between the constraint twist subspace and the freedom wrench subspace. In the
joint variable and torque spaces. the orthogonality holds between the freedom joint
variable subspace and the constraint joint torque subspace and between the constraint
joint variable subspace of and the freedom joint torque subspace as shown in equations
(2.99) to (2.102). It should be noted that orthogonality between t h e images of &-m
and R, in the joint and torque spaces can not be guaranteed because
and
T hese expressions would vanis h only if the following condition holds:
where A is a scalar. The mapping from the abstract space to the joint variable space
is J- ' : and the mapping from the abstract space to the joint torque space is J ~ .
Because J . in general. is not an orthogonal matrix. it distorts the structure of the
abstract space through these mappings so that the images of f&-, and R, in the
joint and torque spaces are no longer orthogonal. The non-orthogonal images of the
mappings from the abstract space to joint spaces have a significant coupling effect on
the dynarnics of the PIF controllers. an issue which will be addressed in the second
part of this chapter.
2.9.3 Decomposition in the Joint Domain: Redundant Robots
Redundant robots differ from non-redundant robots in that the former have more
degrees of freedom t han t h e task description needs. that is, the dimension of the joint
space of redundant robots is larger than that of the twist space and the dimension
of torque space is larger than that of the wrench space. In the case where redundant
robots execute free motion tasks and their task spaces are not partitioned. the joint
space is already decomposed into two subspaces: the external joint subspace which
affects the external motion in the task space: and the interna1 joint subspace which
has no efect on the external motion but provides the flexibility for different kinds of
motion resolutions in the joint space for the defined motion in the twist space. When
the task spaces are decomposed. the external joint space is further partitioned into
two subspaces which correspond to the freedom and constraint twist subspaces. In
the following two subsections. ive will study the decompositions of joint and torque
spaces for undecomposed task spaces and t hen t h e decompositions of joint and torque
spaces for constrained tasks.
Kinernatics in Joint Space and Statics in Torque Space
The link between the task space and the joint space is the Jacobian of robotic manip-
ulators. Any twist (in ray coordinates) of the end-effector is the linear combination
of the Jacobian columns:
and any wench applied by the end-effector is generated by the torque in torque space:
where J is the 6 x n Jacobian. where n(< 6 ) is the number of rnanipulator joints: q
is the velocity vector in the joint space. and r is the vector in the torque space.
Because E is the twist in ray coordinates. each column of the Jacobian is a twist
in ray coordinates representing the contribution of each joint velocity to the twist
motion of the end-effector. So the Jacobian can be expressed as
where J is the image of the Jacobian in R6. which is invariant under the changes in
origin and unit.
Because n < 6 , the general solution to equation (2.109) is typically represented in
the form
w here
is the pseudoinverse of J , which minirnizes the index q T ~ & and A is a positive-definite
n x n matrix. Then (1 - J+L+h) is a projection operator onto the nuil space of J - and
O ( € Rn) is a vector corresponding to the internal motion.
Any twist and wrench can be represented as:
where At and Aiv are the images of the twist t and w in R6, which are invariant
under changes in ongin and unit.
Combining (2.114) and (Till),
The internal space can also be represented in terms of variables in abstract space:
r -pj = r - AJ~(J.AJ~)-~J = I - A J ~ ( J A J ~ ) - ~ J .
The joint velocity vector can be represented as:
where Jf = A J ~ ( J A J ~ ) - ' . It is easy to see that:
In redundant robots. the base of its joint space is divided into two parts (Figure
2.7):
7 = ( J e . J ' ) .
where J e = J+. and J i ( € R " ~ ( ~ - ~ ) ) is defined in such a way that
f?ange(Ji) = Range(l - J C J )
So we have
where Ji is a ( n - 6) x n rnatrix of rank n - 6.
The subspace J e is called the extemal jo int subspace because it defines the motion
of the end-effector. and the subspace J' is called the interna1 joint space since it has
no influence on the motion of the end-effector [35].
In a sarne way, we can get the base of the torque space (Figure 2.7):
where T e = JT. and T' E R*'("-~) is defined in such a way that
Joint Space
Torque Space
Figure 3.7: Decomposition of Joint and Torque Spaces for Redundant Robots
where Tl is an ( n - 6) x n matrix of rank n - 6 .
The ex t emal torque subspace. spanned by J'(T'). corresponds to the motion and
force control of the end-effector and the rest of the torque space. the internal torque
subspace spanned by T'. corresponds to the control of the internal motion of the robot.
.\lthough orthogonality is meaningful in both joint and torque spaces. the sub-
spaces in (Z.11Y) and (2.121) are not orthogonal to each other: the external joint
subspace J e and the internal subspace J' are not orthogonal: neither are T e and T'.
However, the following relations are satisfied:
These relations mean that. mithout considering dynamic effects. the torque in the
external torque subspace has no static influence on the motion in the internal joint
space. and the torque in the internal torque subspace has no static influence on the
motion in the external joint space. These relations form the b a i s for the design of
task space controllers for redundant robots.
Decompositions of the Joint Space and the Torque Space
The freedom and constraint twist subspaces f and c have their images in the external
subspace of the joint space as follons:
and the images of the freedom and constraint wrench subspaces in the external sub-
space of the torque space are:
Te = J ~ ( F . C ) = (T;. T;). (2.127)
The joint space decomposition (Figure 2 3 ) . following (2.113) and ('1.126). is
7 = (3;. J t . J ' ) .
and the torque space decomposition (Figure ?.Y), following (2.131) and (2.127), is
O bviously, the following relations hold true based on Equat ions (2.128), (2.125).
(2-51) - (2.54):
The decomposition of the task space. expressed in Equations (2.49) and (2.50).
and that of the joint space, expressed in Equations (2.136) and (2.127) are primitive
goals for P/F controllers to realize. This is because the P/F controller should first
control the robot to comply with the constraining environment, and then regulate the
position and force following the desired trajectories in the freedom twist subspace and
the constraint wrench subspace. The ideal controllers, whose output is the wrench in
the task space and the torque in torque space in the static sense. must be designed
Joint Space
extemal joint subspace
Torque Space
external torque subspace
Tt
intemal joint subspace
internai torque subspace
r
Figure ?.Y: The Decomposed Joint and Torque Spaces for Redundant Robots Subject to Constrained Taslis
so that they satisfy conditions (2.51)-(2.54) in the task space and conditions ('2.130),
(2.131) and (2.132) in the torque space.
2.10 Concluding Remarks
In this chapter. the consistency requirement between physical and mathematical mod-
els is applied to the modeling of constrained tasks and tasks for redundant robots.
The principle of invariance is introduced as a guide for the decompositions of both
task and joint (and torque) spaces. This principle requires that the criteria for the
decompositions b e kept invariant with respect to changes of reference frames and
units.
The decompositions of task spaces of constrained tasks are clarified according
to the principle of invariance. X general method is also proposed for the invariant
decornpositions of task spaces of constrained tasks. Invariant kinematic and static
filters are also derived based on the decomposi tions. Furthermore? the relationships
among abstract. screw and joint (and torque) spaces are thoroughly studied. and the
transformations arnong these spaces are now well understood. The decompositions of
task spaces of constrained tasks are rnapped into the joint domain and both the joint
and torque spaces are also invariantly decomposed. This work lays a solid foundation
for P /F control design for robots executing constrained tasks.
The principle of invariance is also applied to the decornpositions of joint and torque
spaces of redundant robots. In these decompositions. two nul1 spaces (interna1 joint
and internal torque subspaces) are defined. The results of these decompositions will
be used in Chapter 4.
Overall, this chapter deals with the fundamentals of decompositions of task and
joint (and torque) spaces and provides theoretical preliminaries for the research pre-
sented in the subsequent chapters.
Chapter 3
The Principle of Minimum
Interaction and It s Application t O
P/F Control Design
Introduction
In this chapter, a framework is presented for the analysis and design of P/F control.
In general. for constrained tasks under P/F control. in the initial state the end-
effector may or may not be in contact with the environment. while in the desired
state. the end-effector is not only in contact \vit h the environment. but also it executes
simultaneously commanded trajectories of motion and force. Therefore, the controller
should help by attaining two goals:
0 keeping the robot manipulator's motion in cornpliance with constraints; and
a moving the robot rnanipulator along desired trajectories without violating the
environmental constraints.
Because PIF controllers are combinations of both position and force subcon-
t rollers. the knowledge developed for design of pure position and pure force cont rollers
can be exploited. The second goal can be easily realized using direct extensions of pure
position and force control to position and force subcontrollers. Hoivever. achieving
the first goal is an issue specific to P/F controller research.
Because P/F control is a combination of position and force control. in order to
control robots in cornpliance with constraints. tivo fundamental design issues must be
considered:
a position and force subcontrollers must control a robot in cornpliance with con-
st raints; and
a both subcontrollers must cooperate. rather than oppose with each other.
The solution to these problems leads to a theoretical framework for the design
of PIF control that consists of a set of guidelines and a generic methodology. The
guidelines are summarized as the Principle of hfinimum Interaction.
T h e chapter is organized as follows. First, interactions between position and force
subcontrollers are investigated. Based on the interaction analysis. the prînciple of
minimum interaction is proposed. A decomposed dynamic mode1 of a constrained
robotic system is proposed for use in the analysis and design of PIF control; Then
a generic methodology is devised of the design of P/F control. The proposed P/F
controller. along wit h ot her generic P/F controllers. is analyzed in terms of kinematic
stability and the interaction between the two subcontrollers. Finally, two illustrative
simulations are conducted to show the effects of the interaction on the performance
of PIF controllers.
3.2 Principle of Minimum Interaction
The primary requirement for a P/F controller is to control the motion and the force
of the robot so that they comply with the constraining environment and with the
force constraints.
As mentioned above. the P/F controllers proposed in the last decade are primitive.
Often t hey are based on arbi trary combinat ions of force and position subcontrollers:
there is a lack of specific design criteria. As any controller, a P/F controller should
be stable. However. to design a high performance P/F controller. specific criteria
must be defined because it is a non-trivial combination of position and force controls.
Furthermore. the nature of the interaction between the subcontrollers must be fully
understood.
X system can be properly cont rolled oaly if sufficient informat ion about the system
is available. Moreover. inelevant information incorporated in the feedback loop may
generate an undesired dist urbance. Good performance can be obtained when enough
information is available and irrelevant information is eliminated.
Specific criteria for the design of a P/F controller can be derived from performance
specifications and control objectives. These criteria could be defined in the form of
design rules:
0 The position subcontrolIer, which uses information about the ireedom twist,
should control only the freedom twist, and not affect the constraint twist;
a The force subcontroller, which uses information about the constraint wrench,
should control only the wrench of constraint. and not affect the freedom twist.
In conclusion, the rules state t hat :
The position subcontroller should not affect the constraint motion because there
is no information about it in the position feedback loop.
O The force subcontroller should not affect the freedom motion because there is
no information about it in the force feedback loop.
These rules can be illustrated by selected cases:
a When the end-effector is required to exert a pre-defined force on a constraining
surface while moving along a trajec tory on the surface. any influence the position
subcontroller has on the contact force is not controllable because no information
about the freedom twist is available to the force subcontroller. In principle, it
is impossible to control the contact force accurately when there is an influence
from the position subcontroller.
a When there are errors in the geometric mode1 of the constraints. it is possible
for the end-effector to lose contact with the constraining surface. A good per-
formance P/F subcontroller should be able to compensate for modeling errors.
In other words. the force subcontroller of the PIF controller should be able to
return the end-effector so that contact with the constraining surface is main-
tained. However. if the force subcontroller affects the freedom twist. the motion
of the end-effector will deviate from the normal to the constraining surface. and
some motion will appear in the freedom subspace. This motion can not be
controlled by the force controller because of the lack of information about the
freedom twist in the force control loop.
It can be easily observed that, if the term position subcontroiler is replaced by the
freedom wrench and the term force subcontroller by the constraint wrench, the above
rules are equivalent to the decomposition of the constrained task space given in the
Chapter 2 [(Ml)-(2.54)]. This implies that, if a P/F controller is designed based on
the above rules. the decomposition of the task space and the control system objectives
are consistent with the constraints. The relationship of the interactions between the
su bcontrollers and compliance wi th const raints is now clear: if the interactions can
be reduced to a minimum. the compliance of robot motions with constraints can be
mauimized. Therefore, in order to design a high performance P/F contrcller. the
following guideline must be observed: the interactions between the position and force
subcontrollers must be reduced to a minimum. We cal1 this rule the Principle O/
Jlinimum Interaction.
3.3 Dynamics of Constrained Robotic Systems
3.3.1 A Brief Review of Dynamics of Constrained Robotic
Systems
In both position and PIF control designs. dynarnic models play the sarne role:
tbey provide a platform for control design and analysis. In position control: the
dynamic model is a representation of the relationship between motion and act uating
force. In P/F cont rol. the dynamics must represent the relat ionships between motion.
constraining force and driving force. When robots execute const rained tasks. t heir
dynamics is decomposed into two parts: motion dynamics and force dynamics. Wit h
such a decomposed dynarnic model. position and force subcontrollers can be designed
separatel- and previous knowledge of pure position and pure force control can be
fully utilized. Also, the interaction between the subcontrollers can be investigated in-
depth. .A correct decomposed dynamic model is indispensable for P/F control design
and analysis.
There have been many publications about the dynamic modeling of constrained
robotic systems. In [31], the constrained systern is modeled as a singular system,
and in [32] a decomposition of the dynamic model is obtained through a nonlinear
transformation. Based on t his decomposition. a cornputer torque-based met hod for
PIF control is developed. In [35], the dynamics of a constrained robotic system is
given using a descriptor system formulation - a class of singular system. A singu-
lar system. alt hough t heoretically suitable for constrained dynamics analysis. is not
adequate for the synt hesis and design of P/F controllers.
Such model. as well as most other formulations of dynamics for PIF control, is
based on the following descript ion of the robot ic manipulator-environment system:
where g E R6 is the vector of joint variables: M ( ~ ) E Px6 is the inertia matrix: h
denotes the nonlinear terms, and r E R~ denotes the vector of the input torque. Here.
f represents the force exerted on the environment, and
where J is the Jacobian of the robot and W, is the actual contact wrench: o E R"
is the bilateral representat ion of the constraining environment.
A careful examination of constraint representation would indicate that Equation
(3.2) is not correct. In most cases. the constraints are unilateral. The end-effector
can still move along the normal direction away from the constraining surface, for
instance. when the robotic manipulator executes profile tracking task. Even for tasks
in which the bilateral constraint assumption holds t heoretically (e.g. peg-in-hole).
there is still a clearance between the rnating parts. In fact, Equation (3.2) is only one
of the control goals of the P/F cootroller. requiring that motion control of the end-
effector be performed so that it complies with the environment. In the next section,
a decomposed dynamic mode1 of constrained robotic systems is given. Based o n it, a
generic methodology for P/F control design is derived.
3.3.2 Mode1 of Robot Systems for P/F Control
Equation (3.1) can be rewritten as:
According to the decomposition of r he constrained task space given in Chapter 2,
the joint motion of the robotic system can also be decomposed into two parts: the
freedom motion and the constraint motion. as follows:
where is the twist of the end-effector, and if and i, are the projections of t in the
freedom and constraint twist subspaces. Xote that
and
where t j and t , are images of t and t, in two subspaces of abstract space, respect ively.
Then. the joint space motion from Equation (3.5) becomes:
where Tf and Tc are the bases of freedom and constraint joint variable subspaces. as
defined in Equations (2.94) and (3.95) of Chapter 2. The term Tctc represents the
constraint motion that is to be suppressed by the controller. This term can not be
eliminated from the kinematic mode1 because, in most cases, the constraints are not
bilateral. It is the task of the PIF controller to control the robot so that it complies
with the environment, in other words. to annihilate the effect of this terrn.
The acceleration can be also expressed as:
The constraint wrench is also defined:
and the actual wrench consists of two parts:
where C is the base of the constraint wrench subspace and, F is the base of the
freedom wrench subspace. Here, CV; is the image of the constraint wrench in the
abstract space, and CV;t is the image of the freedom rvrench in the abstract space. an
example of which is the friction wrench. In this case. ive assume that the constraining
environment is smoot h; heace the friction wrench is zero.
Yote that the static filter is given by:
From the above three equations. we get the constraint wrench
Thus f in equation (3.4) is
where Xc is the base of the joint torque subspace attributed to the constraint defined
in Equation (2.98) of Chapter 2.
The dynamic equations can be now written in terms of new variables as follows:
rvhere hi = M( i j t f + SctC) + h.
Suppose that gd is the desired twist of the end-effector, and its image in the
abstract space is td. To simplify the dynamic equation, we use the computed torque
technique, assuming that the exact mode1 is available:
where TI is an additional input to the system. The dynamic Equation (3.10) of the
system becomes:
i~[T~(i, - id) + r&] = 7,.
The velocity error of the end-effector is defined as:
where ë is the twist error of the end-effector; and ëf and ë, are the projections of ë
onto the freedom twist and constraint subspaces. Furthermore:
where f and c are the bases of the freedom twist and constraint subspaces; and e l
and e, are the coordinatcs of ëf and ë, with respect to the above bases, respectively.
From the above equation, we get
where e, and ec are the images in the abstract space of the twist errors ël and Sc.
The equation of the dynamic system becomes:
This equation will be used to analyze the performance of various PIF controllers
in Sections 4 and S.
3.4 Design of P/F Controllers
In order to ensure that the execution of the constrained tasks is consistent with
the constraints, the principle of minimum interaction must be satisfied. When the
dynamics is taken into consideration. the position and force subcontrollers appear to
interact with each other as follows:
The freedom and constraint motions influence each other through the linear
inertia force. Because of the freedorn motion the position controller affects the
constraint motions and the contact forces.
The freedom and constraint motions influence each other through the nonlinear
inertia force.
Modified Mode1
To eliminate the influence of dynamics on the consistency, in the case where accu-
rate knowledge of the dynamic model is available, a computed torque technique is
suggested:
r = MT,&+ hi + f + T *
which yields the closed loop:
where r1 is specific to the controller proposed in this Chapter, whereas TI is for general
use, as described in the previous sections.
The motion of this model must comply with the constraints and be in zero-force
contact wit h the environment. The model is universal because the specific features of
constrained tasks are eliminated via the computed torque technique. The influence
of the nonlinear inertia force on the consistency is also eliminated.
The P/F cootroller for the modified dynarnic system (3.15) consists of two parts:
a position subcontroller with output r p . and a force subcontroller with output r ~ :
Then. (3.153 is given as follows:
T,E, + T,ë, = M - ~ ( T ~ + .rF) = rif + +TFf,
in which
and
are t ransformed position and force su bcont rollers for the plant (3.15).
Xccording to the principle of minimum interaction, the transformed position sub-
controller rSf should not influence the constraint motion Tcëc, and the transformed
force subcontroller TT should not influence the freedom motion Tf ë ,. So the following
inner products must vanish:
and
In order to satisfy these requirements, 7;' must be spanned by NF and T;' by
.Vc. Because the position and force subcontrollers are first designed in task space and
then transformed into torque space. they can expressed be in the following forms:
and
where wF and wC are the position and force subcontrollers in the wrench space. In
order for r:f and TF to satisfy conditions (3.18) and (3.19), the position and force
su bcont rollers must be properly designed .
Design of P/F Controllers in Task Space
Design of Position Subcont rollers
The position subcontroller is designed based on information about the freedom
twist. so we have the following primitive motion subcontroller:
where Kp and Ku(€ Rx6) are the position and velocity feedback gain matrices,
respect i vely.
The key point in the design of the position subcontroller is the selection of
feedback gain matrices, which can not be arbitrary but must be done such that
the requirements (3.18) and (3.19) are satisfied. In order that TF not affect the
constraint motion? it should be in the freedom joint subspace spanned by NF,
and therefore, the domain of operation of position subcontroller W F should
be contained in the freedorn wrench subspace, and the following relation must
hold:
To sat isfy t his condition, the position and velocity feedback gain matrices m u t
be:
where K; and K(E R ( ~ - ~ ) ~ ( ~ - ~ ) ) are negative definite symrnetrical matrices.
We ob t ain the following position subcontroiler:
Design of Force Subcontrollers
The force error between the actual (measured) wrench W, and the commanded
wrench wd can be expressed in axis coordinates as:
The force controI is based on information about the constraint wrench. The
primitive form of a force subcontroller is:
where fi E Px6 is the force feedback gain matrix.
The design of a force subcontroller must also follow the rules mentioned above.
They require that the force subcontroller should not affect the freedom twist,
that is. the transformed force subcontroller T$* in the toque space should not
influence the motion in the constraint joint subspace. Dually, the position
subcontroller in the wrench space (task space ) should not affect the constraint
twist subspace:
Thus, the force subcontroller's domain of operation should be in the constraint
wrench subspace which is spanned by C. Therefore, condition (3.28) holds if
and only if:
where & is an rn x rn symmetncal negative definite rnatrix.
Thus? we obtain the following force subcontroller:
P/F Controllers
From the above subsections, we can get the transformed position and force subcon-
trollers
and
$* = ~ ~ ~ ~ k c ~ ë j
The PIF controller (3.16) for the plant (3.15) is:
In conclusion. where there is exact knowledge of the dynamic model, the overall
generic P/F controller is obtained as follows:
The P/F controller designed here is cornposed of two parts: a model-based feed-
forward controller and the feedback controller. The feedforward eliminates the inter-
action due to the inertia force, and the feedback controller is designed so that the
position and force subcontrollers do not interact with each other directly.
3.5 Analysis of Known P/F Controllers
Several basic controllers have been proposed in the Literature of the last decade
whose purpose is to control the force and position simultaneously. These controllers
are based on somewhat arbitrary combinations of position and force subcontrollers.
In this section, the performance of these controllers, dong with that of the controller
proposed here, will be andyzed and compared using the proposed decomposition of
the task and the dynamic model.
The performance of a controller is a multifold issue. In general, it depends on
the ad hoc choice of gain matrices- the specific requirements of the task and even
the hardware setup. Thus, a consistent cornparison between various controllers is
not straightforward. However, based on the formulation presented in the previous
sections, al1 PIF controllers must satisfy the principle of minimum interaction. The
interaction between the position and force subcontrollers is a specific criterion for
judging the performance of P/F controllers. There is an additional fundamental
requirement for al1 controllers: st abili ty.
3.5.1 Decomposed Dynamic Model
A Decomposed Dynamic Model for Stability Analysis
Our aim is to study the kinematic stability issue in P/F control as it applies to the
constrained robot. In order to filter out the dynamic instability, we assume that there
exists a zero-force contact between the end-effector and a stiff environment. In ot her
words, the end-effector is subject to virtual constraints in free space. In this case. the
desired and measured forces are zero.
In general, the controller TI in (3.13) consists of the position and force subcon-
t rollers. Thus:
Observe that the above notations are different from those in (3.16): the former
refers to existing P/F controllers, whereas the latter represents the proposed P/F
cont roller,
However, in the case of the zero-force contact requirement. the effect of the force
subcontroller can be neglected, which implies that:
Thus. the dynamic equation of the robotic mônipulator is (3.13):
To study the stability of the freedom and constraint motions separateiy, the dy-
namic model of the constrained robotic system is decomposed. The model of the
freedom motion of (3.33) can be obtained by filtering out the constraint motion using
premultiplication by NF because of Ti -NF = 16-m and NF **Tc = 0:
Equat ion (3.34) provides a plat form for the stability analysis of PIF controllers.
A Decomposed Dynamic iModel for Consistency Analysis of P/F Con-
t roIlers
According to t he Principle of 4Linimum Interaction. the position subcontrollers in
P / F controller should not affect the constraint motion and the contact force. and the
force subcontroller should not affect the freedom motion and the freedom wrench.
This means that the position and force subcontrollers should not interact with each
other. However. in practice. the position and force subcontrollers interact directly
and indirect ly.
a Direct interaction occurs when the position subcontroller, which must control
the freedom motion only, affects the constraint motion and the contact force.
and the force subcontroller, which must control the contact force, affects the
freedom motion.
O Indirect interaction occurs when the inertia force due to the freedom motion.
which is cont rolled by the position subcontroller. affects the constraint motion,
which is supposed to be suppressed by the force subcontroller. Also, it occurs
when the inertia force due to the constraint motion, which is controlled by
the force subcontroller, affects the freedom motion, rvhich is supposed to be
cont rolled by the position subcontroller only.
The dynamic mode1 for the analysis of the interaction betrveen the position and
the force subcontrollers can be derived from Equation (3.33)
Tfëf + Teec = M-' (T~ + T ~ ) .
Pre-multiplying the above equation by Tf yields:
TTT,~, + T:T,Z, = T;M-'T, + T;M-%,. (3.36)
From the above equation, we can see that the freedom motion is influenced by
16
the position subcontroller represented by TTM-' r,: . the force subcontroller represented by T T M - ' ~ ' rvhich is the direct interaction
between the position and the force subcontrollers:
a the constraint motion represented by TTT,~,, rvhich is the indirect interaction
betrveen the position and the force subcont rollers.
Multiplying Equation(S.35) by Tc pields:
From the above equation, we can see that the constraint motion is influenced by
O the force subcontroller represented by TTM-' rt;
the position subcontroller represented by T ~ M - ' T,, which is the direct interac-
t ion between the position and the force su bcontrollers:
O the freedorn motion represented by T:TIë,, which is the indirect interaction
bet ween the position and the force subcont rollers.
Equations (3.36) and (3.37) can be used to study the interaction issue for various
P /F controllers.
In order for the PIF controllers to be consistent, there should be no direct or indi-
rect interaction between the force and the position subcont rollers. Direct interaction
vanishes only when the following conditions are satisfied:
and
T ~ M - ~ T , , = 0.
Indirect interaction vanishes only when the following condition is satisfied:
Conditions (3.3Y) and (3.39) could be satisfied by the appropriate design of the
P / F controller. However. this condition (3.40) depends on the structure of the robot
and the task and can not. in general. be satisfied by the controller design. It will not
be used as a criterion for judging PIF controllers because it does not depend on the
cont roller design.
To satisfy conditions (3.38) and (3.39), the force subcontroller should be designed
in the subspace MNc, and the position subcontroller in the subspace M N F .
In general. the consistency is of two kinds: static and dynamic. The static consis-
tency is guaranteed by (3.23) and ( U s ) , and the dynamic consistency is guaranteed
by (3.38) and (3.39). Statically. in order that the position subcontroller not affect the
constraint motion, the position subcontroller should be designed in the wrench sub-
space F. To avoid affecting the freedom motion. the force subcontroller should be de-
signed in the wrench subspace C. To maintain consistency, the position subcont roller
should be in the subspace NF of the joint torque space and the force subcontroller in
the subspace .Vc of the joint torque subspace. If these conditions are satisfied, the
consistency is guaranteed in the static sense.
Hoivever. the dynamic consistency is further affected by the inertia matrix M,
that acts as a transformation which distorts the structure of the abstract space from
which the joint and torque variables are derived. This causes the orthogonality be-
t w e n the joint variable and joint torque subspaces to vanish. The dynamics of the
robotic system endows the joint space with an M-'-based inner product. To main-
tain consistency in the dynamic se nse . the joint torque space should also be art ificially
endowed with M-based inner product. Therefore, to compensate for the influence of
dynamics. the force subcontroller should be in the subspace MNc and the position
subcontroller in the subspace M-VF.
Even if the static consistency requirement is satisfied. the position and force sub-
controllers in a PIF controller will still interact with each other directly because the
dparnics is not considered. This will be discussed in the following subsections.
3.5.2 Hybrid Control
The original hybrid control proposed in [:39] is based on an abstract space decom-
position. In general. this orthogonal decomposition holds true only in the case of a
planar robot for which the motion of the end-effector is specified by ttvo translation
coordinates. In this subsection. ive will apply the reciprocal decomposition to hybrid
control and examine its kinematic stability.
Formulated in terms of a reciprocity-based decomposition, the hy brid controller
is:
where the selection matrices S and I - S in the original formulation ivere replaced
by the kinematic and static filters PI and Pc; e f is the image in abstract space of the
wrench error ef.
Kinematic Stability Analysis
In the case of virtual constraints (3.33):
r, = rl = I < : T ~ ~ ~ + h ; " ~ ~ é ~ .
According to (3.34). the motion in the freedom subspace is:
In order for (3.43) to be stable, i\';M-' !i: TI and I ~ M - ' I < ~ T ~ must be negative
definite. This can be accomplished if Ii," and ~i: are suitably chosen. However. the
terrns .vT. Tl and M are configuration-dependent. Thus ~i,h and Iit must be updated
on-line in order to maintain stability.
Consistency Analysis
The position subcontroller(3.4'1) and the force subcontroller
might not satisfy consistency conditions (3.38) and (3.39) with an arbitrary choice of
feedback gain matrices q. and h;". Therefore. the force subcontroller may affect
the freedom motion. and the position subcontroller may affect the constraint motion
and contact force directly.
In hybrid control (3.41), the twist error in the freedorn twist subspace f is first
transformed into the error in the joint variable subspace T j , and the wrench er-
ror in the constraint wrench subspace C is transformed into the torque error in the
torque subspace .Vc. To satisfy the consistency requirement. the position subcon-
troller should be designed in :b to achieve static consistency and in MiVF to achieve
dynamic consistency. However. arbitrary choices of feedback matrices might not guar-
antee bot h stabili ty and st at ic consistency. let alooe dynamic consistency
3.5.3 Resolved Acceleration Force Control (RAFC)
X resolved acceleration force subcontroller is a model-based P/F cont roller t hat takes
several forms: joint space formulation [-El, operational s p c e formulation (211 and
iinpedance control (201. In this subsection, we study RAFC in joint space.
Formulat ion of ResoIved Acceleration Force C o n t r o l
Reformulated in terrns of reciprocity, the RAFC takes a new form:
in which
because
Comparing the above expression with (3.7). we get
~ - l & = - ~ ~ t , - rctc. -
The RAFC controller ( 3 . 44 is:
T = M J - ~ ptid + hl + jT pC Wd
Substituting the RAFC into Equation (3.10) yields:
in w hich the position subcontroiler is
Tp = M j-I(pfhpRë + P , K , R ~ ) ,
and the force subcontroller is
Consistency Analysis
According to the principle of minimum interaction. t h e position subcontroller, which
is supposed to control the freedom motion, should be designed based solely on the
information about it. Inclusion of other information such as the constraint motion
in the feedback loop might result in undesired disturbances. This requires that the
motion information be filtered before being used by the controller. However. in the
R-AFC. the position (or velocity) feedback matrix I c ( o r Ii:) is between the kinematic
filter P; and the position error @(or e ) . The kinematic filter Pf in the RXFC. which
is supposed to be applied directly to the twist space in which ê and e are represented,
is applied to the distorted twist spaces by K p and K f . So the position subcontroller
is based on incomplete information about the motion in the freedom twist subspace
and unnecessary information about the motion in the constraint twist subspace. This
means that the position can not be fully controlled due to lack of information; and the
position subcontroller is still affected by the force subcontroller through the constraint
twist which is not filtered out because of the distortion of the twist space.
The RXFC violates the principle of minimum interaction in yet another way. The
position subcontroller (3.46) of the RAFC is in the subspace MTr (TI is the base of the
freedom joint variable subspace, as defined in Equation (2.94) of Chapter 2) instead
of in the subspace M*VF to satisfy condition (3.39). The force subcontroller. which is
in the joint torque subspace spanned by JTC. can not satisfy condition (3.3s) either.
Therefore. the position and force subcontrollers in the RAFC are directly interacting.
Stability Analysis of the RAFC
In order for the position subcontroller to operate based on exclusive information
about the motion in the freedom twist suhspace, the error twist space should be
filtered before the controller is applied. so the following conditions rnust be satisfied:
Shen. the kinematic filter would operate properly and the PIF controller would be
based on the decomposition of the ta& space. In this case. the position subcontroller
of the RAFC is:
Equation (l3.3-1) for the freedom motion becomes:
ëI = .V;M-'~, = F ~ K : f e + F ~ K ! f ë f. (3.50)
In order to keep F ~ K ; f and F'K; f invariant with respect to changes of the
reference frame. h',R and h;R must be:
so that
and
where G, and G, E R ( ~ - ~ ) ~ ( ~ - ~ ) are negative definite mat rices.
From Equation (2.51) in Chapter 2. we know that FT f is invariant with respect
to changes of the reference frame: and F=I\VR f are also invariant.
Once conditions (3.45) and (3.51) are satisfied. the Equation (3.50) becomes
ëf = Gpef + Guéf.
This plant is stable with negative definite matrices Gp and G,.
3.5.4 Stiffness Control
The st iffness controller can be reformulated in terms of a reciprocity-based decompo-
si t ion
The feedback gain matrices are redefined according to the invariance requirements
as:
and
i n which f,':, [<:(E ~ ( 6 - m ) x ( 6 - m ) ). and [if (E RmXm ) are al1 negative definite matri-
ces.
The stiffness controller can be also defined as:
The motion subcontroller of the stiffness control is:
rp = X F ( K , S e + K z è f ) .
and the force subcontrol1er is:
Stability Analysis of Stiffness Control
Frorn Equation (3.34) ive have
Let
Then, the equation of freedom motion is:
The plant (3.61) is always stable because 1'; and KU are negative definite and H
is positive definite [?].
Consistency Analysis
For st iffness cont rol, the position subcont roller is designed in the freedom torque
subspace ;VF. and the force subcontroller is designed in the constraint torque subspace
.Vc. so the static consistency requirements (3.23) and (3.38) are sat isfied.
Hoivever. the stiffness control can sat isfy only stat ic consistency requirements and
fails to meet dynamic consistency requirements because the subcontroller (3.57) does
not include the effect of the inertia matrix M and nonlinear inertia forces.
The Relation between Kinematic Stability and Static Consistency
The position and force subcontrollers in a PIF controller are first designed in ta&
space. If the static consistency is to be satisfied. the position subcontroller should be
designed in the freedom wrench subspace F and force subcontroller in the constraint
wrench subspace C, and then transformed into the freedom torque subspace XF and
constraint torque subspace &. The general form of a P/F controller designed ac-
cording to the stat ic consistency requirement is (3.57). which can guarantee kinematic
stability. as discussed above. However. if the static consistency requirement is not
sat isfied. the position subcontroller r d be designed in a torque subspace other t han
.VF. let us sa? .Y;. In this case. the position subcontroller (3.58) will become:
Substituting it into Equation (3.34). we get:
ë, = NzbI-'.~>(h',Se + K V ~ ~ ) .
In t bis case, the matrix
is not symmetric and positive definite. So the stability of the following equation can
not be guaranteed even if h',S and h'; are negative definite:
From the above analysis. we conciude that static consistency is a necessary con-
dit ion for kinematic stability.
3.5.5 Analysis of the Proposed PIF Controller
The stability oi the proposed P/F controller can be examined in the same way as the
stability of Hy brid Control. RAFC and S tiffness Control.
The proposed P/F controller is (3.32):
Noting that e = f ef and ëf = Cef the position subcontroller is:
86
- 1 - r p = M - \ - ~ ( I < ~ , + I\,éf)- T~ -
and the force su bcont roller is:
Stability Analysis
Equation (3.34):
is used as a platform for stability analysis. Expressed in another form. it becomes:
where Hi = ( . v F ~ V ~ ) - ' is a positive definite symmetric rnatrix.
The above plant is always stable because h", and I ' ' , are negative definite [-Il.
Consist ency Analysis
It can easily be seen that the position subcontroller (3.65) in the proposed P/F con-
troller satisfies condit ion (3.391, and the force subcont roller (3.66) satisfies condition
(3.38). Therefore. in the proposed controller, there is no direct interaction between
the force and position subcontrollers.
3.5.6 Cornparisons of Different P /F Controllers
The above analysis reveals the differences in performance of various P/ F controllers
from the perspective of stability and interaction. We summarize the resuits as follows:
0 Hybrid Control: this controller is designed based on the decomposition of the
task space without considering robot dynamics. The kinematic stability can
not be guaranteed. and the position and force subcontrollers interact wit h each
ot her direct ly and indirect ly.
0 Resolved Acceleration Force Control: although the robot dynarnics is taken into
considerat ion. the position subcontroller might be designed using incomplete in-
formation about freedom twist because of the improper location of the kinematic
fiiter which distorts the twist space. The consistency requirement can not be
satisfied. and the position and force subcontrollers interact with each other di-
rectly and indirectly. Stability can be guaranteed only if conditions (3.4s) and
(3.51) are satisfied.
Stiffness Control: it is designed based on the decomposition of the task space.
and it can guarantee the kinematic stability of the position subcontroller. How-
ever. the position and force subcontrollers can still interact with each other
directly and indirectly because the robot dynamics is not considered.
0 The proposed P/F controiler: it is designed based on the principle of mini-
mum interaction. Kinematic stability can be guaranteed and direct interaction
is elirninated. However. indirect interaction may still exist. and it can not be
eliminated by altering the design of the controller. This problem ivill be elabo-
rated in the next section.
3.6 Indirect Interaction Between the Position and
Force Controllers
.As discussed in the above section. the existence of the two terms TfT& and
T F T ~ ~ , violates the principle of minimum interaction. To eliminate the interaction
between the force and position subcontrollers. it is necessary that the following con-
dition be satisfied:
because
Then Equation (3.69) hoids only when:
where X is a scalar. The above equation requires an orthogonal mapping from the
abstract space to the joint space.
This condition is similar to conditions (A.?) in Appendix A and (3.108) in Chapter
2. which are equivalent to maintaining the invariance of mappings [rom the abstract
space to the joint variables and the joint torque spaces.
Surprisingly. unless J=J = AIs. it is impossible to realize the PIF controiler
in such a way that the position subcontroller is responsible only for the freedom
motion. and the force subcontroller is responsible only for the contact force and the
constraint motion. However. this condition provides u s ivith an index for planning
the task optimally that could serve as a guide in the design of robots for force control
purposes.
There have been several other criteria based on the operation of the Jacobian.
Arnong them. the most often cited is the rnanipulability proposed by Yoshikaiva (471:
Based on (3.70), u = h implies that the performance is omnidirectional. The
measure of manipulability reflects a quantitative property and the Equation (3.70)
reflects the invariance of mappings between joint and task spaces and the degree of
interaction between position and force subcont rollers. T hese two design cri teria may
complement each other.
Examples
In this section. two examples are given to show the effect of indirect interactions
on the performance of P/F controllers.
The performance of a PIF controller deteriorates when the position and force
subcontrolIers interact with each other. -4s rnentioned before. t hese interactions de-
pend on many factors: the accuracy of the dynamic model. the methodology based
on which the position and force subcontrollers are designed, and the configuration of
the robotic manipulators. Here. we assume that the perfect dynamic model can be
obtained and a methodology is adopted which e lhinates direct interactions. so the
effect of indirect interactions c m be singled out.
Two kinds of robots wi t h different configurations are used to illust rate:
0 the dependency of the indirect interaction of the position and force subcon-
t rollers on the robot configuration;
0 the effect of indirect interaction on the performance of PIF controllers
The interactions between the subcontrollers are mutual: if a position subcontroller
affects the force subcontroller, the force subcontroller in t urn affects the position sub-
controlier. and vice versa because both interactions will not vanish as long as condition
(3.70) is not satisfied. Due to the reciprocity between the effect of the position sub-
controllers on the force subcontroller and the effect of the force subcontroller on the
position subcontroller, we study only the effect of the position subcontroller on the
force subcont roller.
For each robot, we assume that the exact dynamic model is available. The P/F
controller is designed according to the methodology proposed in t his thesis. If there
is no interaction between the position and force subcontrollers, the position subcon-
troiler should not affect the contact force even in the case where the force subcontroller
does not function at ail. So the interaction can be examined based on the behavior
of the robot when the force subcontroller is turned off.
The task in the following examples is defined as follows: the end-ef€ector is required
to follow a straight line from one point to another on a surface (x=const.), while
executing zero contact force.
This task will be realized by ttr-O robots wit h different configurations: a joint robot
and a Cartesian robot. To examine the effect of the position subcontroller on the force
subcontroller. we turn off the force subcontroller. If the position subcontroller has no
effect on the force subcontroller. the robots should be in zero force contact with the
environment. O t herwise. the end-effectors will ei t her deviate from the const raining
surface or push the surface with a greater-than-zero force.
For this task. the kinematic filter is
and the static filter is:
First. ive explore the joint robot which is schematically shown in Figure 3.1 witli
the constraining surface. The length of each link of the robot is 1 meter. The end-
effector is required to move from (1.366, -0.5) to (1.866, 0.5) in 10 seconds along the
t rajectory:
xd = 1.S66
The dynamic equation is:
where M is the 2 x 2 inertia matrix. and h E R2 is the nonlinear and gravitational
91
Figure 3.1: The Joint Robot with 2 DOFs
term. The joint space variable vector is:
and the task space variable is:
.\ccording to the proposed methodology, the PIF controller is designed as folloas:
where qd is the desired joint acceleration. The position feedback matrix is li, =
diag[ - 100.0, -100.01, and the velocity feedback matrix is Kk = diag[-70.0 . -70.01.
The Jacobian of this robot.
does not sat isfy condit ion (3.70). Though the direct interactions between the position
and force subcontrollers are elirninated. indirect interactions remain due to the non-
orthogonal transformation frorn abstract space to joint (or torque) space. Their effect
can be seen in Figure 3.2. in which the end-effector deviates from the constraining
surface. Because the force subcontroller is turned off, the deviation from the virtual
constraining surface is due solely to t h e effect of the position subcontroller on the
const raint motion.
The trajectory for the Cartesian robot (Figure 3.4 ) is:
Figure 3.2: The Simulated Trajectory for the Joint Robot
Figure 3.3: The Cartesian Robot with 2 DOFs
The kinematic and static fiiters for this task are the same as the ptevious ones.
The Jacobian for the Cartesian robot is as follows:
The generaiized coordinates for the Cartesian robot are shown in Figure 3.3. The
generalized forces are the forces dong each coordinate. The dynamic equation of this
rabot
IV here
According to the proposed met hodolog- the P /F controller is designed as follows:
ivhere q d is the desired joint acceleration. The position feedback matrix is h; =
diag[lO.O.lO.O], and the velocity feedback matrix is Iik = diag[.iO'i. .707]. The result
of the simulation is shown in Figure 3.4.
From the above examples. we can see that the performance of P/F controllers
for robots. t hough designed according the principle of minimum interaction, depends
largely on the robot configuration. For the Cartesian robot, whose Jacobian satisfies
the condition (3.70), the position subcontroller induces no deviation of motion. which
would require a force subcontroller to suppress. However, for the joint robot which
generaily does not satisfy the condition (3.70). the position subcontroller in the P / F
controller results in unexpected motion in the direction of constraint motion. This
deviation can be suppressed only by the force subcontroller, which is supposed to
control the contact force because its design is based solely on the information about
the contact force. From this point of view. the deviation of motion can be a constant
dist urbance to the force subcontroller. In t his case, the performance of P/F controllers
det eriorates.
dosind tnjrdov
Figure 3.4: The Simulated Trajectory for the Cartesian Rabot
Concluding Remarks
This chapter presents a viable t heoretical framework for the design of P/F control
systems built on a thorough understanding of the effects of interactions b e t i w n the
subcontrollers on the performance of the P/F controller.
Because there is no information about the constraining force in the feedback loop
of position subcontroller and no information about the motion in the feedback loop
of the force subcontroller. any interaction between the subcontrollers can only be a
disturbance and deteriorates the performance of the P/F controller. Based on this
finding. the principle of minimum interaction is proposed as a guide for the design of
P/ F cont rollers.
Based on the kinematic and static decompositions of task spaces presented in
Chapter 2 . a decomposed dynamic model of the robots executing constrained tasks
is derived for interaction analysis that can also serve as a platform for the analysis
and design of P/F controllers. Using this dynamic model, we found that the sub-
controllers interact wit h each other both directly and indirectly. Direct interactions
can be eliminated using an the elaborate design for both subcontrollers. but indirect
interactions are robot configuration-dependent and can not be eliminated in any \ v a .
In this chapter. a generic methodology for the design of P/F controllers is pro-
posed based on the principle of minimum interaction. Csing this methodology. the
minimum interaction design for P /F control can be achieved. The resulting PIF con-
t roller is able to eliminate direct interaction between the subcont rollers. but indirect
interactions still exist, causing the performance of the P/F control to deteriorate.
However, the relationship between the indirect interactions and the robot configu-
ration has been established. so it is possible to find a way to reduce them. In this
chapter. a quantitative measure is devised to evaluate the indirect interactions and
the performance of P /F controllers. Two examples are given of the implementation of
the proposed P / F control scheme using robots with different configurations to show
the effect of indirect interactions.
Csing the decomposed dynamic model. the existing generic PIF control schemes
are analyzed in terms of the stability of the position subcontroller and the interac-
tions between the subcontrollers. This dynamic mode1 and the principle of minimum
interaction can aiso be used to assess the impact of future research results in this
area.
Chapter 4
The Principle of Non-Interaction
and Its Application in Task Space
Control Design of Redundant
Robots
4.1 Introduction
Redundant robots have some advantages over non-redundant robots because their
redundancy provides the flexibility to optimize the motion of the whole robotic sys-
tem. In recent decades, many technical papers have been published that investigate
the inverse kinematics and control of redundant robots. The most frequently used
strategy in redundant robot position control is to derive the joint space trajectories
using inverse kinematic techniques. and then control the robots by following these
trajectories. Redundancy (internai motion) has also been used to optimize the mo-
tion of robots with respect to some criteria, including the minimization of actuator
torque [19]. the avoidance of singularities and obstacles [36], and the minimization of
the kinetic energy [?l]. However. this strategy fails in some cases for the following
sections:
In pseudoinverse cont rol of kinematically redundant robots. the existence of
drift makes the trajectories in the joint space unrepeatable. The repetitive
trajectory in the task space is therefore unpredictable in the joint space. and to
store the joint space trajectory information requires huge memory [Zj].
0 When redundant robots are required to execute the P/F control task. irnple-
menting the inverse kinematics is inappropriate because, wit h predefined t ra-
jectory in the joint space, these robots lack the ability to compensate for the
modeling error of the constraints. In this case, the force controller will be in
confiict wit h the position subcootroller.
To make the P/ F controller sensitive to const raining environments and to avoid
the huge memory requirement, i t is advantageous to design controllers for redundant
robots in the task space.
.As mentioned in Chapter 1, though the rnethodology for the design of task space
controllers for non-redundant robots has been well established. a design methodology
for task space control does not exist. The task space controller for redundant robots is
different from t hat for non-redundant robots in t hat it consists of two subcontrollers.
an external and an internal. The external controller affects the kinematic and static
properties of the end-effectors, and the internal controller affects t h e interna1 motion.
Therefore. the design the subcontrollers so t hat they cooperate is a problem specific to
the control design of redundant robots. Unfortunately, the existing literature provides
no solution.
In this chapter, a design methodology is devised f ~ r task space control of re-
dundant robots. The principle of non-interaction is first introduced and a generic
methodology is developed on it. The principle of non-interaction and the proposed
methodology form a framework for the design of task space controllers without inter-
action between the subcontrollers. Finally. both the principle of minimum interaction
and the principle of non-interaction are proposed as design guidelines. The sirnulta-
neous implementation of the two principles leads to a methodology for the design of
P / F cont rollers of redundant robots.
4.2 Task Space Controller Design For Redundant
Robots: Position Controller
4.2.1 Previous Work
Previous work on the planning and control of kinematically redundant robots has
focused on the use of internal motion (or self-motion). In the following solution of
inverse kinematics for redundant robots [28]
o is used to achieve the desired performance function. Given a performance function
H. 6 is taken to be the gradient vector with respect to the vector of joint angle
q E P .
o = VH,
in order to minimize the performance function H.
Examples of the proposed performance functions include the joint range avail-
ability [ZS], the manipulability rneasure (481, the condition number [24] and the the
compatibiiity index [j].
The internal motion space spanned by I - J+J is used not only to plan the joint
motion, but also to design the so-called nul1 space controller ([l'il (61). In these
approaches, the following nul1 space controller is proposed:
where Ii is the n x n feedback gain matrix. It is supposed to affect the internal motion
and so improve certain properties such as stability without infiuencing the motion of
the end-effector; bowever. it still affects the motion of the end-effector directly. a
problem which will be addressed in the following subsection.
4.2.2 Task Space Position Controller Design
Principle of Non-Interaction
In previous chapters, the principle of minimum interaction was proposed as a guide
for P /F controller design for non-redundant robots. In this chapter. guidelines for
position controller design for redundant robots will be introduced.
As discussed in Chapter 3: the joint space of a redundant robot is divided into
the external and internal subspaces:
The torque space is also decomposed into the external and internal subspaces:
.V = ( T e , T i ) (4.3)
The above decomposi t ions are based on the foliowing conditions:
which state in the static sense that the torque in the external torque subspace Te will
not affect the internal motion of the robot, and the torque in the internal subspace
Ti will not affect the external joint motion, and subsequently the motion of the end-
effector.
The design of the esternal controller is based on the information about the external
motion and there is no information of internal motion in its feedback loops. Any
influence the external controller exerts on the internal motion can be a constant
disturbance to the internai controller. Also there is no information about the external
motion in the feedback loops of the internal controller because it is designed using
solely the information about the internal motion. The internal controller can be a
source of disturbance if it affects the external behavior of redundant robots. From
the perspective of controllability, we can derive the properties of both subcontrollers:
a The ezternal subcontroller can control only the motion of the end-efictor;
O The internal subcontroller can control only the selJmotion of the robots.
The external subcontroller is expected to affect kinematic and static properties
and the internal controller is supposed to control the internal motion only. Through
the interaction between the subcontrollers, each can cause constant disturbances to
the other. This is because the external subcontroller does not make use of the infor-
mation concerning the internal motion, and there is no information with respect to
the external motion in the feedback loop of the internal subcontroller.
From both the definition and controllability of the external and internal con-
t rollers. ive obt ain the following requirements for bot h cont rollers:
0 The external controller should control only the motion of the end-eflectors;
0 The interna1 controller should control only the self-motion of the robots.
The above guidelines can be summarized as the Pn'nciple of Non-Interaction for
the design of position controllers for redundant robots: the ezternal and internal
subcontrollers should not interact llith each other.
According to (4.6) and ( 4 4 , the principle of non-interaction can be realized in
cont rol design with consideration of dynamic effects as follows:
The erternal controller must be designed in the eztemal torque subspace (spanned
b y T c ) :
0 The in temal controller must be designed in the internal torque subspace (spanned
by Tt).
A Simplified Mode1 for Task Space Controller Design
The dynamic mode1 of a robotic system can be expressed as:
where q E R" is t he joint vector. M(q) E RnX" is the inertia matrix. h denotes the
nonlinear and gravity terms. and r E R" denotes the vector of input torque.
The velocity in the joint space can be decomposed into the external and internal
mot ions:
and the joint acceleration is given as:
It should be noted that in the above expressions. I - PJ = 1 - J+J is used.
Substituting (4.10) into (4.8) results in:
where hl = h + ~ ' 6 + M(J+E + ( 1 - >+ J ) Q ) .
Suppose id is the commanded motion of the end-effector. The computed torque
method is used here to eliminate dynamic effects:
where 7,. is the controller for rnodified model.
Equation (4.11) is thus changed:
where ëe gives the corresponding coordinates of task space error O,(= f - id). Here.
e; is defined as the coordinates of the joint velocity in internal space:
e; = J1O.
The dynamic Equation (1.13) can be written as:
Je& + J:Z; = ru,
where the interaction due to the nonlinear inertia forces is eliminated if t he exact
model is known.
Then. Ï, in (4.11) can be decomposed into two parts:
where T: is the controller for the external motion and ri the controller for the internal
106
mot ion.
Design of a Task Space Controller According to the Principle of Non-
Interaction
.As ive mentioned above. the joint space can be divided into two parts:
where J e is the base of the subspace of the motion which affects the motion of t h e
end-effector. and J' the base of the self-motion (internal) subspace.
According to the principle of non-interaction? the external controller T: should not
affect the internal motion. Therefore it must satisfy:
and the internal controller r: should not influence the external motion. Thus:
Furthermore, we can see that the external controller r: should be designed in the
subspce (of the torque space) spanned by T e . and the internal controller r: should
be designed in the subspace (of the torque space) spanned by T t .
The external controller should first be designed in the task space ( the wrench
space) as follows
where h',l and are 6 x 6 feedback matrices and ë, is the orientation and position
error of the end-effector.
Because Ë', is t h e arench and ë, is the twist. to maintain invariance [-MI. the
feedback matrices KPl and litel should have the following forrns:
where h& and Ki, are matrices in PX".
The external controller can then be expressed as:
and the external torque is
r: = PF, = (F,
The internal controller is designed in torque space to rninimize the scalar index
H .
.lccording to the principle of non-interact ion, t h e internal controller should be
designed in torque subspace Ti, that is, the subspace spanned by I - JTJTC or TiTi:
Finally, from (LE), when the dynamics of the robotic system is taken into con-
sideration and it is known exactly, the ta& space controller is:
Interaction Analysis
In the above subsect ions, we have designed the task space cont roller which eliminates
the direct interactions between the external and internai cont rollers; t hus the external
controller has no direct influence on the internal motion. and the internal controller
does not affect the external motion directly.
However, interaction still exists between the mot ion of the end-effector and the
self-motion. It acts through the inertia force terms because the subspaces J e and Ji
are generally not orthogonal. We cal1 t his kind of interaction indirect in teraction.
Premultiplication of Equation (4.14) by ( J e ) = yields the equation of motion of
end-effector:
and premultiplicat ion of (4.14) by (Ji)= yields the equation of self-motion:
The above equations are derived based on the relations:
Equations (4.23) and (4.24) can be combined into a compact form:
From the above equat ion, the interactions between the mot ion of the end-effector
and the self-motion are obvious because the mat rix
is not block-diagonal. The interactions disappear only when the following relation is
est ablished:
which means that the interna1 and external motion subspaces should be orthogonal to
each other. Recalling that the internal motion subspace is orthogonal to the external
torque subspace, and the external motion subspace is orthogonal to internal torque
subspace. to satisfy condition (4.26). it is necessary t hat the internal motion subspace
be the same as the internal torque subspace and the external motion subspace be the
same as the external torque subspace.
The external motion subspace is spanned bu:
and the externai torque subspace is spanned by:
For both subspaces to be the same. t he rveighting matrix A should be in the
following form:
.4 = XI.
where I is the n x n identity matrix and ,\ is a scalar.
In this case, the internal motion subspace and the torque subspace will also be
the same:
and
In previous approaches (for example. [KI and [6] ), the internal motion controllers
are designed in the internal motion subspace. If the weighting rnatrix is not of the
form (-L.27), the internal motion subspace and torque subspace (spanned by J' and T i ,
respectively ) are not the same, so the internal controller affects the external motion
directly. In Our approach, the internal motion controller is designed in the internal
torque subspace, which is always orthogonal to the external motion subspace in spite
of the form of the weighting matrix. This eliminates the direct interaction between
the external and internal motions.
The weighting matrix A in the pseudoinverse JC plays an important role in the
performance of task space controllers for redundant robots. If it is not of the form
of (4.27), it changes the metrics in both the joint and torque spaces and makes the
internal and external joint (torque) subspaces non-ort hogonal. The external controller
affects the internal motion, and the internal controller affects the external motion
through the inertia forces. In the following sections. A is set to I in order to eliminate
the interaction.
4.3 Design of P/F Control for Redundant Robots
When the end-effector of a redundant robot is required to move in cornpliance
with certain constraints, only the external properties of the end-effector are changed.
In task space, both twist and wrench spaces are decomposed according to the nature
of the constraints. so the position and force subcontrollen form the external subcon-
trollers. Since the task space controllers for redundant robots are already divided into
externd joint and torque spaces. there are two kinds of interactions: the interaction
between the external and interna1 subcontrollers and that between the position and
force subcontrollers in the external controller. The design of PIF control for redun-
dant robots is a combined irnplementation of the principle of minimum interaction
and the principle of non-interaction. This leads to a generic methodology for PIF
cont rol design for redundant robots. The principle of non-interaction is implemented
in the same way as it was in the last section. Howvever, the principle of minimum
interaction is implemented differently: the redundancy is now used to minimize the
indirect interaction between the position and force subcontrollers.
4.3.1 Design of PIF Controllers in the Task Space
Decomposition of Joint and Torque Spaces
In this subsection, we will review the decomposition of the joint and torque spaces
for P/F controller design. As mentioned previously, the spaces of constrained task
are decomposed in the following:
The joint space:
The torque space:
The definitions of f, c, F, C are given in Chapter 2.
The decompositions are defined according to the following conditions:
The external subspace of the joint space is also divided into two parts for the
description of the task:
The external subspace of the torque space is divided into two subspaces:
If ive take the interna1 subspaces into consideration, the decomposition of the joint
space is:
and the decomposition of torque space is:
.u = (TF. TC, T ' )
The decomposition of the joint and torque spaces must satisfy the following con-
dit ions:
These decompositions do not represent a mode1 but set the control targets (pri-
ma- tasks) and guidelines for the position and force controller in task space. The
wrench of the controller must comply with conditions (4.30) and (1.31). In other
words. to achieue the control goals. the position subcontroller in the task space must
be i n the freedom wrench subspace. and the force controller be in the constraint wrench
subspace.
The guidelines for the design of hybrid controllers can also be expressed from
the viewpoint of cont rollabili t y. The position subcont roller. which depends on the
information about the freedom twist subspace. can only control the twist of freedom.
and the force controller. which depends on the force information of the constraint
wrench subspace, can only control the cvrench of constraint properly.
The P/F controller is designed in two stages. It is first designed in the task space.
and then transformed into the torque space.
The P/F controller in task space F, consist s the position and force subcont rollers:
where wF is the position subcontroller and the force subcontroller.
Decomposition of Joint and Torque Spaces
In the following context, a dynamic mode1 of redundant robots subjected to con-
straints is proposed as a platiorm for control design and anaiysis.
The dynamic equation of a redundant robot can be expressed as:
where f represents the force exerted on the environment and
f = J~w,,
where J is the Jacobian of the robot and W, is the actual contact wrench: q5 E Rm
is the bilateral representation of the constraining environment.
Substituting (4.10) into (4.33) results in the following task space formulation:
where hi = h + 3'6 + M ( L + ~ + ( I ->+J)o).
.\ny twist t can be decomposed into two parts:
t = i, + t,.
The dynamic model (4.35) can be represented in terms of the decomposed vari-
ables:
From the above equation. we can see the following
0 The position and force subcontrollers interact with each other through the in-
ertia rnatrix M. It is thus obvious t hat the freedorn twist ~ ; i ~ and constraint
twist Jzà, affect each other through the inertia matrix M.
The position and force subcontrollers interact \vit h each other through the non-
Iinear inert ia force terms. The nonlinear inert ia force hl is a funct ion of external
freedom and constraint motions and of the self-rnotion. through which the force
and position subcontrollers interact wit h each other.
a The external and interna1 subcontrollers interact with each other through the
inertia matrix M.
To eliminate the interactions due to dynamic effects? a computed torque method
is employed:
The dynamic model (4.35) is simplified as:
Ils
In the above equation.
where r:, = J+F, and ri, = TiTIKVH, F, and H will be deterrnined in the foilowing
context.
Design of Position Controllers in Hybrid Control
The position subcontroller is designed based on the information about the freedorn
twist and must be active only in the freedorn wrench subspace. The primitive form.
of the position subcontroller is:
where wF is the position subcontroller in the static sense. Because the position
should influence only the freedom twist: the following condition must be satisfied:
which requires that the position subcontroller should be designed in the freedom
wrench subspace spanned by F. It also requires that &,PI and &PI be invariant.
Since Pf = f ( F = f )-' FT, hTP f and & f should be invariant. To meet the above
requirements, the feedback mat rices s hould be designed as follows:
where I< and < (E R(6-")x (6-m) ) are negat ive definite symmetric mat rices.
The position/orientation error of the end-effector can be decomposed into two
parts:
where ë, is the vector of the position/orientation error in the freedom twist subspace,
and ë, is the vector of the position error in the constraint twist subspace. Here, e,
and e, are the images of ë and ê, in the space R.
Because Pj is the kinematic filter which projects the position/orientation error
onto the freedom twist subspace. we get:
P,e, = 6, = fer
The position subcontroller is obtained from (4.40), (4.44) and (4.42):
Design of Force Controllers in Hybrid Controllers
The force error between the actual (measured) wrench W* and the commanded
wrench wd con be expressed in mis coordinates as:
The force subcontroller is designed based on information about the constraint
wrench. The primitive form of a force subcontroller is:
where & E Px6 is the force feedback gain matrix. Because the force subcontroller
should not affect the freedorn twist,
which requires that the force subcontroller should be designed in the constraint wrench
subspace spanned by C because fTC = O. It also requires that & Pc be invariant.
Considering that Pc = ~(?C)-lc? , we can derive that
where K; is an rn x m symmetric and negative definite matrix.
T h e contact wrench error is:
where ef is the image of the ë j in abstract space.
Thus
pcéJ = c e f (4.51)
The force subcontroller is derived by combining (4.47),(4.49) and (4.51):
Combining the position and force subcontrollers (4.45) and (4.52), the hybrid
controller in the task space is:
4.3.2 Model-Based Hybrid Controllers for Redundant Robots
Wit h F, well-defined. we can write the controller for the modified dynamic model
4 4 . 3 8 )
The controller for the original model (4.35) can be obtained by substituting (4.54)
into (4.37):.
r = M [ J ' ( F K ~ ~ ~ + FA$^/ + ch>ef) + (1 - J 'J+)KVH] + hl + M J + ~ ~ + -wd. (4.55)
This is the controller for a redundant robot executing a contacting task, which is
comparable to ( 4 . 2 ) , the pure position task space controller for redundant robots.
4.3.3 Interaction Analysis of P/F Controllers for Redun-
dant Robots
The dynamic mode1 (4.35) provides a platform for the performance analysis of the
proposed controller. In this section, the emphasis is placed on analysis of the interac-
tions among the subcont rollers. Substit uting (4.55) into (4.35) leads to the following
simplified dynamic model:
because
J+& = ~ + ( 8 , + P,) = J+( fëf +ce , ) . -
lié also have the dynamics of the controlled plant with H to be determined:
The interaction in the controlled systern can be analyzed using t h e decomposed
model. The equation of the freedom motion is obtained by premultiplying Equation
(4.57) by ( J ; ) ~ .
and the equation of the constraint motion is obtained by premultiplying Equation
(4.57) by (JL:)=.
If the weighting matrix for the pseudoinverse of the Jacobian is of the form (4 .Z) ,
the external and internai joint subspaces should be orthogonal to each other. In t his
case.
Then Equations (4.58) and (4.59) are changed as follows:
c T e - ( ~ j ) ~ ~ j ë l + ( J , ) J,e, = K;ef + h':èf,
and
From Equation (4.60). Ive can see that the constraint motion and the interna1
mot ion affect the mot ion of freedom t hrough the coupling terms ( J;)* J: and ( J : ) ~ J i .
Because the constraint motion is affected mainiy by the force subcontroller. the force
subcontroller affects the freedom mot ion indirectly t hrough the term ( J;)' ~ f ë , term .
From Equation (4.61). Ive c m also reach the conclusion t hat the position subcontroller
still affects the constraint motion t hrough the coupling term (J:)= Jjë,.
These interactions between the position and force subcontrollers are inherent and
can not be eliminated by altering the design of the PIF controller. The interactions
will cease to exist only when the coupling term vanishes:
which requires that
where In is the n x n identity rnatrix and A' is a scalar. Because
(J+)*J+ = ( J J * ) - ~ J J * ( J J * ) - ~ = ( ~ ~ ~ 1 - l .
The above relation holds only when the following condition is sat isfied:
J J ~ = X'I,
where A' is a scalar.
This condition, which is configuration dependent. can not generally be met. For-
tunatel- redundant robots have the additional degree of freedom to realize some
indices to improve performance. In this case. it is reasonable for the performance
functions to be:
where a,,, and amin are the maximum and minimum eigenvalues of the rnatrix J J'.
When H reaches O (minimum). al1 the eigenvalues are equal to A': the condition (4.64)
is t hen realized.
The decoupling condition (4.64) for position and force subcontrollers requires the
invariance of orthogonality from abstract space to joint (or torque) space under the
transformation J-' (or J'). This relation is determined by the robot structure and
configuration of the robots, and can not be altered by any kind of controller.
4.4 Concluding Remarks
This chapter presents a theoretical framework for the design of task space control
of redundant robots. The core of this frarnework is the interactions between the
external and internal subcontrollers. Based on an investigation of these interactions
and their effects on the performance of task space controllers, the principle of non-
interaction is proposed as a guide for the design of task space control of redundant
robots. This principle states that there should be no interaction between the external
and internal subcontrollers.
To design the task space control of redundant robots? the dynamics of redundant
robots is formulated in task space. From this model. a generic methodology is de-
veloped for the design of task space control of redundant robots that is based on
the principle of non-interaction. The principle of non-interaction and the proposed
methodology form a theoretical framework for the design of task space control of
redundant robots.
It is found t hat the interactions bet ween the subcont rollers can also be eit her direct
or indirect. Like the interactions discussed in Chapter 3. the direct interactions can
be eliminated by altering the design of the subcontrollers . while and the indirect
interactions are configuration dependent. Hoivever. redundant robots can affect the
indirect interactions with t heir changeable configurations. A met hod is devised for
designing task space controllers for redundant robots wit h zero-interact ion bet ween
the subcontrollers.
The methodologies proposed in t his Chapter are generic in the sense that only
the PD controller is employed to illustrate how they work: moreover. they are based
on the esact knowledge of robot dynamics. Where parameter uncertainty exists. a
robust cont rolling scheme should be applied.
Chapter 5
Conclusions and Suggestions for
Future Research
5.1 Conclusions
In t his research, some fundamental problems related to the design of task space cont rol
have been addressed. Theoretical frarneworks for the design of PIF control and task
space control of redundant robots have been proposed.
The theoretical framework for the design of PIF control is built on the principle
of invariance and the principle of minimum interaction. When applied to P/F con-
trol, the principle of invariance states that the criteria for the decomposition of task
or joint (and torque) spaces should be invariant with respect to changes of reference
frames, unit systems and transformations between task and joint (and torque) spaces.
According to t his principle. a reciproci ty-based decomposition of a screw- based task
space of constrained tasks has been proposed. The principle of invariance is also ap-
plied to the transformation between task and joint (and torque) spaces. Furthermore,
the joint and torque spaces can be invariantly decomposed. The invariant decompo-
sitions provide a clear description of constrained tasks in both task and joint (and
torque) spaces and a sound foundation for the definition of constrained tasks and the
selection of the positiou and force control variables for P/F control.
Because the interactions between the position and force subcontrollers are specific
to P/ F control. t hey were st udied t horoughly The interactions between the subcon-
trollers were classified in terms of the ma? they function. as direct or the indirect.
The principle of minimum interaction was proposed as a guide for the design of P/F
cont rol since the interactions between the subcont rollers cause dist urbances to each
ot her. -4ccording to this principle. a generic methodology was devised for the design
of P/F control that eliminates the direct interactions. It was also proved that the
indirect interactions can not be elirninated by the controller because they depend
on the robot configuration, and any altering the control design can only affect the
dynamic properties of the robot. However. a criterion kvas introduced to measure
the degree of the indirect interactions; it suggests that the elimination of the indirect
interaction requires ort hogonali ty between the decomposed subspaces of the joint and
torque spaces.
The cornerstone of the theoretical framework for the design of task space control
of redundant robots is the principle of non-interaction. The task space controller for
redundant robots consists of two subcontrollers: the e'rternal and the internal. From
the viewpoint of controllabili ty. the interactions between the subcontrollers resul ts
in disturbances to each other that deteriorate the performance of the task space
controller. Accordingly. the principle of non-interaction states that the the external
and internal subcontrollers should not affect each other. A s in the P/F control case.
the interactions between the two subcontrollers can be both direct and indirect, and
the direct interactions can be eliminated by altering the design of the subcontrollers.
The indirect interactions can also be eliminated using kinematic redundancy. .A
methodology was proposed for the design of task space control of redundant robots
based on the principle of non-interaction. In addition, an example was given to
show how to apply both the principle of minimum interaction and the principle of
non-interaction to the design of P/F control of redundant robots.
Overall. the contribution of this research is the development of a methodology for
the design of task space controller, based on which a complete framework is established
for the task space approach to the control design of robotic systems.
5.2 Suggested Future Research Topics
In this thesis. we have mainl- dealt with theoretical issues in task space control.
The irnplementation of the proposed generic met hodologies and t heir experimental
validation are issues for further exploration. Based on the research results of this
t hesis. O t her research endeavors can be pursued:
0 The incorporation of uncertainties in dynamics into task space controller de-
sign. One of the assumptions in this dissertation is the availability of a perfect
dynamic model. Methodologies need to be developed that take into account
dynamic uncertainties for task space controller design.
0 The use of robot configuration-related criteria, which enhances the performance
of task space control. to optirnize the design of robots. !ilethodologies needed to
be developed for such design approaches. This issue is called Integrated Design
of Configuration and Control of robots.
Appendix A
Position Control of
Non-Redundant Robots
The defining feature of a non-redundant robot is t hat the dimension of the task space
is equal t o that of the joint space (and the torque space). If the joint space is divided
into several subspaces by singular surfaces. the mapping from each subspace to the
task space is one-to-one. and can be represented by the Jacobian
where x is the position vector of the end-effector of a robotic manipulator, q E R6 is
the vector of joint variables and J E Px6 is the Jacobian.
Like joint space control, task space control for position control can be divided into
two categories: classical joint control and model-based control. We first investigate
the differences between the design philosophies for classical joint space controllers and
its counterparts in the task space domain.
A.1 Position Controllers wit hout Consideration
of Dynamic Effects
The goveming equation of a robotic system expressed in terrns of joint variables is
given as follows:
where M is an n x n inertia rnatrix of a robotic system with n being the number of
degrees of freedom. For non-redundant robots, n equds the dimension of the task
space, r is the vector of the output torque from the joint actuators and the input
torque to robots, and q is the vector of joint angles. In classical joint control, r is
designed wi t hout considerat ion of robot dynamics:
rvhere lq = q - qd is the error vector between the actual and commanded joint posi-
tions and Aq is the error vector between the actual and commanded joint velocities.
The negative-definite feedback matrices for both joint position and velocity , h', and
rii. map the variables in joint space into the output in torque space. The subscript
j in Tj stands for the controller designed using the joint space approach.
The task space controller F is designed first in the task space without considering
robot dynamics:
where Kr and I\'i are the feedback matrices in the task space.
Then F is transformed into the torque space through the transposed Jacobian:
where r, is T designed using task space approach (the subscript t stands for task
space).
If the following relations are taken into account
then the task space controller can be expressed in terms of the joint variables:
Then T, and r, are equivalent in that. for given h', and Ki in the joint space
approach, we can always find feedback matrices for the task space approach
to achieve the same performance as long as the Jacobian is not singular.
From the above analysis. it can be seen that though their design philosophies
are different. position controllers designed either in joint space or in task space are
equivalent when dynamic effects are not considered.
A.2 Model-Based Position Controllers
The generic methodologv for the design of model-based joint space controllers was
proposed by [30]. It consists of two steps: the computed torque technique is first used
to eliminate dynamic effects and to simplify the model; then a traditional control
scheme such as PID control is applied to the simplified model.
Compared wi t h non-model-based cont roller design schemes. model- based con-
t roller design is more complicated. The computed torque is determined by two factors:
the linear inertia term: and nonlinear inertia terms such as Coriolis and centrifuga1
forces. In classical joint control. nonlinear terms act as disturbances. The inertia
term. which is determined by the inertia matrix, plays an important role in deterio-
rat ing performance. If ive conduct eigenvalue decomposi t ion upon the inert ia matrix
where -4 is the eigenvalue matrix and Z the eigenvector matrix. the dynamic equation
(if nonlinear terms are eliminated ) is the following:
The effects of the inertia term on the performance of controllers are obvious: the
controller torque, which is supposed to influence the joint vector q directly, is first
distorted by the orthogonal transformation 2. After the magnitudes are changed by
A-'. its direction in torque space is altered by ZT. CVithout a n accurate dynamic
model. the effect of the transformations incurred by the inertia matrix is unpre-
dictable.
The model-based approach is adopted to design joint space controllers that have
better and more predictable performance. The design procedure can be described
as follows: the motion is first transformed from task space into joint space through
inverse kinemat ics. and the classical joint cont roller rj is prirnitively designed t hrough
the mappings of feedback matrices from joint space to torque space: then the model-
based control torque rmj is derived in torque space by incorporating the inertia matri':
and the non-linear terms.
where r,, is the model-based joint position controller and id is the commanded joint
accelerat ion vector.
The cont rolled plant wit h a model- based cont roller is:
A comprehensive and systematic study of model-based task space control was
conducted in (211. Operational space is used for the task space and a unified approach
for task space controller design is proposed . In ['l]. robot dynamics ivas formulated
in the task space.
The end-effector equations of motion in task space can be written in the form
where Ft is the wrench in the ta& space and
A = J - ~ M J - ~ .
The model-based task space controller is first designed in task space
where jid is the desired acceleration of the end-efkctor. Then dynamic effects are
t aken into consideration
The model- based position cont roller designed in task space is realized in torque
space [21]:
rmt = J ~ F ~ .
If ive expand rmt in joint variables.
Tm, = J~A(~,+F)+L*TX(X.X) = M J - ~ ( & + F ) + ~ ( ~ , *)-MJ-'& = M(&+ J - ' F ) + ~ ( ~ , q).
The controlled plant with T,, is:
Atj = J-' K ~ A X + L-' KiAx
which is always stable as long as fi, andlc are negative definite.
Comparing r,, in A.1 and rmt in A.2. we find they are equivalent only when
Because rj is designed in the torque space which is spanned by dT. the following
condition must be met in order for Tmj and rmt to be equivalent.
This criterion. which is frequently used in the following context. was first under-
lined by this author. and it emphasizes the importance of kinematic configuration in
determining controller performance.
Though both the task space and the joint space approaches function, their per-
formances are not easily compared due to a basic incomparability of methodology.
However. both the ta& space and the joint space approaches have solid underpin-
nings. This appendix concentrates on the differences between t heir met hodologies.
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