dynamicsandcontrolofparallelmanipulators ......dynamics and control of parallel manipulators with...

Dynamics and Control of Parallel Manipulators

with Actuation Redundancy

A thesis submitted to

The Hong Kong University of Science and Technology

in partial fulfillment of the requirements for

the Degree of Master of Philosophy in

Electrical and Electronic Engineering

by

Cheng Hui

Department of Electrical and Electronic Engineering

Master of Philosophy in Electronic Engineering

HKUST

May, 2001

Dynamics and Control of Parallel Manipulators with Actuation

Redundancy

by

Cheng Hui

Approved by:

Dr. Zexiang Li

Thesis Supervisor

Dr. Bertram E Shi

Thesis Examination Committee Member (Chairman)

Dr. Pengcheng Shi

Thesis Examination Committee Member

Prof. H.S. Kwok

Acting Head of Department

Department of Electrical and Electronic Engineering

The Hong Kong University of Science and Technology

May, 2001

Contents

List of Figures v

List of Tables vi

Abstract viii

Acknowledgments ix

1 Introduction 1

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Objectives and contribution . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Outline of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Advantages of Redundant Actuation 5

2.1 Remove Direct Kinematic Singularity . . . . . . . . . . . . . . . . . . . 5

2.2 Improve Cartesian Stiffness in the Workspace . . . . . . . . . . . . . . . 7

2.3 Achieve Homogenous and symmetric Force Tranmission . . . . . . . . . 9

2.4 Others . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3 Dynamics of redundantly actuated parallel manipulators 12

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.2 Previous Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.2.1 Nakamura’s method . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.2.2 Ropponen’s method . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.2.3 Lagrange’s Equations with Constraints . . . . . . . . . . . . . . . 18

i

3.3 Lagrange-D’Alembert Formulation . . . . . . . . . . . . . . . . . . . . . 20

3.3.1 Dynamics of Nonredundant Closed-chain Mechanisms . . . . . . 21

3.3.2 Dynamics of Redundant Closed-chain Mechanisms . . . . . . . . 23

4 Control of Parallel Manipulator with Redundant Actuation 27

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4.2 PD Control Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4.3 Augmented PD control . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4.4 Computed torque control . . . . . . . . . . . . . . . . . . . . . . . . . . 33

5 Experimental Devices 36

5.1 Design goals and the Prototype of the Manipulator . . . . . . . . . . . . 36

5.2 Forward kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

5.3 Inverse Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

5.4 Dynamics of the Two DOF Redundant Parallel Manipulator . . . . . . . 41

5.4.1 The measured and computed parameters . . . . . . . . . . . . . 44

5.4.2 System Identification . . . . . . . . . . . . . . . . . . . . . . . . . 45

6 Experiments And Results 50

6.1 Test motions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

6.2 Individual joint torque . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

6.3 Control algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

6.4 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

6.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

7 Conclusions and future work 64

7.1 conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

7.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

Publications 67

A Motor Specifications 68

ii

Bibliography 68

iii

List of Figures

2.1 The schematics of a two DOF non-redundant and a redundant parallel

manipulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2 A comparison of Euclidean two norm of Cartesian stiffness in the non-

redundant and redundant case . . . . . . . . . . . . . . . . . . . . . . . 9

2.3 The ratio between the minimum and maximum singular values in the

non-redundant and redundant case . . . . . . . . . . . . . . . . . . . . . 11

4.1 Diagram of the first type of PD control scheme . . . . . . . . . . . . . . 31

4.2 Diagram of the second type of PD controller in the joint space . . . . . 32

4.3 Diagram of the second type of PD controller in the Cartesian space . . . 32

4.4 Diagram of Augmented PD control scheme . . . . . . . . . . . . . . . . 34

4.5 Diagram of computed torque control scheme . . . . . . . . . . . . . . . . 34

5.1 The Prototype of two DOF redundantly actuated parallel mechanism . . 38

5.2 The motion control system structure . . . . . . . . . . . . . . . . . . . . 39

5.3 Coordinates of the two DOF redundantly actuated parallel mechanism . 40

5.4 Schematics of a two DOF overactuation parallel mechanism and its tree-

structure mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

5.5 The proportion between the physical torque and the commanded torque 46

6.1 The test motion in the workspace . . . . . . . . . . . . . . . . . . . . . . 51

6.2 The profiles of desired acceleration, velocity and position for test motion 1 55

6.3 The profiles of desired acceleration, velocity and position for test motion 2 56

iv

6.4 The torque components of the actuators with an end-effector acceleration

of 7.2m/s2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

6.5 The torque components of the actuators with an end-effector acceleration

of 15.0m/s2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

6.6 Tracking errors of the end-effector for test motion 1 and 2 . . . . . . . . 59

6.7 Actuated joint position error for test motion 1 . . . . . . . . . . . . . . 60

6.8 Actuated joint position error for test motion 2 . . . . . . . . . . . . . . 61

6.9 Actuated joint torque for test motion 1 . . . . . . . . . . . . . . . . . . 62

6.10 Actuated joint torque for test motion 2 . . . . . . . . . . . . . . . . . . 63

v

List of Tables

5.1 Link mass, center of mass and inertia parameters . . . . . . . . . . . . . 45

5.2 The constant parameters αi, βi and γi . . . . . . . . . . . . . . . . . . . 45

5.3 The updated constant parameters αi, βi and γi . . . . . . . . . . . . . . 49

6.1 The performance of PD contorl, augmented PD control and computed

torque control in test motion 1 with acceleration 7.2m/s2 . . . . . . . . 53

6.2 The performance of PD control, augmented PD control and computed

torque control in test motion 2 with acceleration 15.0m/s2 . . . . . . . 53

A.1 Specifications of the servo motor . . . . . . . . . . . . . . . . . . . . . . 68

vi

Dynamics and control of parallel manipulators with redundant

actuation

by

Cheng Hui

for the degree of

Master of Philosophy in Electrical and Electronic Engineering

at The Hong Kong University of Science and Technology

in May, 2001

Abstract

Over the last few years, parallel robots have been under increasing developments from

a theoretical view point as well as for practical applications. They have good features

such as high payload, high stiffness and high speed. However, a great disadvantage

is their limited workspace. Singularities are abundant in parallel robots. When a

mechanism is in a singular configuration, it loses stiffness completely and becomes

uncontrollable. Singularities make the limited workspace of parallel manipulators even

smaller. Redundant actuation has been proposed recently as an efficient approach to

remove singularities over the workspace. It also has advantages of improving Cartesian

stiffness, achieving uniform output forces, optimizing internal and external forces and

so on.

Dynamic modeling and controlling are two fundamental issues of robotics. Dynam-

ics of redundant parallel manipulators is much more complicated than that of serial

vii

robots due to multiple closed chains and redundant actuation. A new scheme to com-

pute the inverse dynamics is proposed using Lagrange-D’Alembert’s principle. The new

strategy is more computationally efficient for real-time control. Independent joint con-

trol and model based control methods represented for redundant parallel manipulators

are formulated, i.e. PD control, augmented PD control and computed torque control.

The stability analysis of the proposed controllers is presented.

A two DOF planar parallel manipulator with redundant actuation was built as a

test bed to investigate the dynamics and control of redundant closed-chain mechanisms.

The dynamic model was formulated using the proposed scheme in the research. The

unknown parameters of the system are obtained by both experimental measures and

identification methods. To verify the estimated dynamic model and to evaluate the

proposed control schemes, the control algorithms were implemented on the mechanism.

The experimental results are analyzed.

There is little research about the dynamics and control of redundantly actuated

parallel manipulators. The work done on the dynamic modeling and controlling will be

valuable attempts in the field.

viii

Acknowledgments

I wish to express my warmest gratitude to Dr Zexiang Li, my thesis supervisor, for his

encouragement and supervision during this work. I would also like to thank him for

providing me with an ideal working environment.

I would also like to thank Prof. Hongrui Wang and Prof. Weigong Song of Dept. of

EE, Yanshan University for their supervision in Yanshan University. I feel much grateful

for their encouragement during my work both in Yanshan University and HKUST.

I am deeply indebted to my group mates S.L. Jiang, G.F. Liu, Z.H. Xiong, Y.K.

Yiu for their valuable suggestions and enlightening discussions, especially Z.H. Xiong

and G.F. Liu for their kindly assistance during the experimental work in programming.

I would like to thank Dr X.Z. Wu for developing the parallel robot. I would also like

to thank Mr Y.L. Wu, Mr Z.Z Lu and Dr J. Li for their cooperation in constructing

the experimental device.

I would like to thank my roomates Fang Huirong, Yin Li and Zhang Chuqian for

their encouragement and the happy times I have shared with them.

I would also like to thank Dr Bertram E Shi and Dr PengCheng Shi for serving on

my thesis committee.

As always I wish to express my hearty gratitude to my family, to my parents, my

sister and my brother for their encouragement and love in all my endeavors.

ix

Chapter 1

Introduction

1.1 Motivation

Currently, most existing manipulators strongly resemble a human arm. They are con-

stituted of a succession of rigid bodies, each of them being linked to its predecessor

and a succession by a one degree-of-freedom joint. This architecture is called a serial

robot or open-chain manipulator. A comparison of the characteristics of several typ-

ical industrial manipulators(mass of the robot and load capacity and repeatability),

has shown that the low transportable load and poor accuracy are both inherent in the

mechanical architecture of existing serial manipulators[1]. Because of serial robot de-

ficiencies, parallel manipulators have been developed by a few researchers. A parallel

manipulator is a closed-loop kinematic chain mechanism whose end-effector is linked

to the base by several independent kinematic chains. A rather exhaustive enumeration

of parallel robots mechanical architectures and their versal applications are described

in [1] [2].

Parallel manipulators have the nice features of high load, high speed, and high accu-

racy. However, compared with the serial counterparts, a great disadvantage of parallel

manipulators is the limited workspace. Singularities are abundant in this type of me-

chanical structure [3] [4] [5]. When a manipulator is in singular configuration, it loses

all stiffness and becomes uncontrollable. Singularities make the limited workspace of

1

1.1. MOTIVATION 2

parallel manipulators even smaller. Redundant actuation has been recently proposed as

an efficient method to remove singularities over the workspace. Parallel manipulators

with redundant actuation are classified as parallel manipulators whose number of actu-

ators is greater than that of the task space coordinates. Besides removing singularities

over the workspace, redundant actuation also has the advantages of making the struc-

ture lighter and faster, optimizing force distribution and improving Cartesian stiffness

[6] [7]. Redundant actuation has been employed successfully on parallel mechanisms

such as multi-fingered hands, walking machines, haptic displays, machining tools and

cooperating robots [8] [9] [7] [10]. Usually direct or semi-direct drives(low gear reduc-

tions) are utilized in these applications. In [11], an actuation redundant parallel robot

with high-reduction gears was constructured.

Dynamics plays an important role in the control of parallel robots in some applica-

tions such as flight simulators or pick-and-place operations involving fast manipulators.

It is therefore necessary to establish the dynamics relations for closed-loop mechanisms

before a control scheme for fast manipulators can be established. This complex subject

has been extensively studied [12] [13] [14]. But these approaches mostly only con-

sider the dynamics of normally actuated closed-chain mechanisms. The dynamics of

redundantly actuated parallel manipulators are more complicated than non-redundant

ones. Recently, a few researchers proposed different approaches to formulate the in-

verse dynamics of redundant closed-chain mechanisms [15] [16] [17]. A new scheme was

proposed my research. It is more computationally efficient than the previous methods.

It has been shown that the dynamics of a redundant parallel manipulator satis-

fies the same structure properties as those of a serial one. This similarity motivates

the application of existing controllers designed for serial manipulators applied to re-

dundant parallel manipulators. The dynamics of redundant parallel manipulators is

quite complicated. Trying to apply the model based control methods to the redundant

closed-chain manipulators, identifying the complete dynamic model creates a bottle-

1.2. OBJECTIVES AND CONTRIBUTION 3

neck. PD plus simple gravity control strategy was experimentally tested by [18] and

[19]. A new robust independent joint control scheme was proposed in [20]. Model based

control methods were employed to a direct drive parallel mechanism with redundant

actuation by [16]. A parallel redundant manipulator PA-R-MA was built [11] with high-

reduction gear boxes. A decoupled Cartesian motion controller was proposed by [21]

and the experimental results showed the feasibility of this high-speed robot concept.

Redundant actuation has been recently introduced into parallel manipulators, and

although interesting theoretical problems remain to be solved, the use of redundant

actuation in many robotic tasks will be necessary and indispensable in the near future.

1.2 Objectives and contribution

The main objectives of the research were:

1. To present a more efficient method for computing the inverse dynamics of redun-

dantly actuated parallel manipulators.

2. To set up a redundant manipulator as a test bed to verify the estimated dynamic

model and investigate the dynamic control of such robots.

3. To evaluate three control algorithms in the control of redundant parallel manip-

ulator, i.e. PD control in joint space and in Cartesian space, augmented PD

control and computed torque control.

Because the computation of inverse dynamics of redundant parallel manipulators

is a complicated and time consuming task, it is difficult to implement the advanced

control methods on these mechanisms. A survey of the previous work on inverse dy-

namic computation is given. Compared with existing methods, the dynamics proposed

in the research is more computationally efficient. Although many new concepts can be

tested using computer simulations, experiments with a real robot provide more valuable

and reliable knowledge of the functioning of theoretical methods and their applicability

1.3. OUTLINE OF THE THESIS 4

in practice. With regard to redundant parallel manipulators with gearing tranmis-

sion, no experimental evaluation of the different control methods has been carried out

or reported. Therefore, the results of this study provide valuable new experimental

knowledge of the use of PD control and model based control methods.

1.3 Outline of the Thesis

The thesis is organized as follows. In Chapter 1, a brief introduction of the research

project that included the motivations and objectives was given.

The advantages of redundant actuation are presented in Chapter 2. Then the

dynamics of redundantly actuated closed-chain manipulators is introduced in Chapter

3. A survey of the previous work on inverse dynamic computation is given. The new

scheme to formulate the dynamics of redundantly actuated closed-chain mechanism is

proposed.

In Chapter 4, the formulation of the control schemes utilized in this project are

presented. The stability analysis of these control algorithms is also presented. After

that, in Chapter 5, the experimental setup including the prototype of the redundant

parallel manipulator, the motion controller and etc., is described. Then the forward

kinematics and inverse kinematics for the robot are presented. The dynamics employing

the proposed new inverse dynamic computation scheme is presented.

To verify the estimated dynamic model and to evaluate the performance of the pro-

posed control methods via PD control, augmented PD control and computed torque

control, experiments were implemented on the constructed redundantly actuated par-

allel manipulator. The experimental results of these control strategies are presented

and analyzed in Chapter 6.

The conclusion and further developments are given in Chapter 7.

Chapter 2

Advantages of Redundant

Actuation

Actuation redundancy makes the design of the parallel structure more complicated. It

introduces not only high computational requirements in calculating the dynamic equa-

tions, but also high challenges in controlling the closed-chain system. In principle, it

would be possible to reduce these disadvantages by driving only the minimum necessary

number of actuators. Hence, it should be clear that redundant actuation is desirable

and advantageous.

2.1 Remove Direct Kinematic Singularity

Singular configurations are abundant within the workspace. When a parallel mecha-

nism is in a singular configuration, it loses its stiffness completely and becomes uncon-

trollable. This property has attracted the attention of several researchers [22] [4] [3].

Let the actuated joint variables be denoted by a vector q and the location of end-

effector be described by a vector x. Then the kinematic constraints imposed by the

5

2.1. REMOVE DIRECT KINEMATIC SINGULARITY 6

limbs can be written in the general form

F (x, q) = 0 (2.1)

where F is an n-dimensional implicit function of q and x. Differentiating equation (2.1)

with respect to time, we obtain a relationship between the input joint rates and the

end-effector output velocity as follows:

Jxx = Jq q (2.2)

where Jx = ∂F∂x and Jq = −∂F

∂q . The derivation above leads two separate Jacobian

matrices. An inverse kinematic singularity occurs when the determinant of Jq goes to

zero. A direct kinematic singularity arises when the determinant of Jx is equal to zero.

A combined singularity occurs when the determinants of Jx and Jq are both zero.

Generally, combined singularity can occur only in manipulators with special kine-

matic architecture. This should be avoided when designing the mechanism. An inverse

kinematic singularity arises whenever any limb is in a fully stretched-out or folded-back

configuration, i.e. as the chain reaches either a boundary of its workspace or an internal

boundary. This type of singularity can be evaded by motion planning. However, direct

kinematic singularity lies within the workspace of the chain and should be removed.

When the mechanism is at the direct kinematic singularities, it cannot maintain its

statically equilibrium position upon external forces. The mechanism has extra degrees

of freedom. For a redundant parallel manipulator, direct kinematic singularities are

characterized by the rank-deficiency of n×m matrix Jx, i.e.

rank(Jx) < m (2.3)

or,

det(h) = 0 ∀h ∈ {m×m submatrices of Jx} (2.4)

It is noticed that only n− (m− 1) conditions given by equation (2.4) are independent.Thus, singularities are found at the intersection of n − (m− 1) hypersurfaces forming

2.2. IMPROVE CARTESIAN STIFFNESS IN THE WORKSPACE 7

a lower dimensional manifold. The dimension of the singularity manifold(DOSM) is

given by

DOSM = m− (n−m+ 1) = m− 1− (n−m) = m− 1−DOR ≤ m− 1 (2.5)

where DOR is the degree of redundancy. Thus, it can be seen that actuation redundancy

can be used to reduce the direct singularities to a lower dimensional manifold in the

task space. In particular, making DOR=m-1 or above, the direct singularities in the

workspace can be reduced greatly or even completely eliminated.

Another interesting explanation of this problem comes from [6]. Let us take the

example of the mechanism present in Figure 2.1. For the non-redundant parallel ma-

nipulator, Jx is a 2 × 2 matrix. When the limbs b1 × b2 = 0, det(Jx) = 0 and direct

kinematic singularity occurs. While for the redundant parallel manipulator, Jx is now

a 3 × 2 matrix. In a singular configuration, its rank will be less than two, and this

occurs only when the determinant of all four 2 × 2 minor matrices extracted from Jx

are all zero. These minor matrices are in fact the Jacobian matrices Jx of all the non-

redundant manipulators that can be extracted from the redundant manipulators by

suppressing a leg. Consequently, if the redundant manipulator is in a direct kinematic

singular configuration, then all the extracted robots have to be in a this type of singu-

larity. This means the three vectors bi for i = 1, 2 and 3, are parallel to one another.

In contrast to the parallel of vectors bi for i = 1, 2, the probability of such an event is

very low within the workspace. Therefore, direct kinematic singularities can be greatly

reduced or even eliminated.

2.2 Improve Cartesian Stiffness in the Workspace

L.W. Tsai [2] presents a stiffness analysis of parallel manipulators. Let τ = [τ1, τ2,

. . . , τn]T be the vector of actuated joint torques and ∆q = [∆q1,∆q2, . . . ,∆qn]T be

the corresponding vector of joint deflections. Then we can relate ∆q to τ by an n × n

2.2. IMPROVE CARTESIAN STIFFNESS IN THE WORKSPACE 8

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

�� Passive JointsActive Joints

a1

b

a2

b2

1

a3

b 3

a1

b1

a2

2b

Figure 2.1: The schematics of a two DOF non-redundant and a redundant parallel

manipulator

diagonal matrix, χ = diag [k1, k2, . . . , kn]T , as follows:

τ = χ∆q (2.6)

For a parallel manipulator, the joint displacement, ∆q, is related to the end-effector

deflection, ∆x = [∆x,∆y,∆z,∆φ,∆θ,∆ψ]T , by the Jacobian matrix J:

∆q = J∆x (2.7)

where J = J−1q Jx is the conventional Jacobian matrix defined in Section 1. By virtual

work principle, the end-effector output forces in terms of the actuated joint torques is

given as the following:

F = JT τ (2.8)

Substituting equation (2.6) into 2.7) and the resulting equation into (2.8) yields

F = K∆x (2.9)

where K = JTXJ is called the stiffness matrix of a parallel manipulator. If the limbs

are of the same type and the spring constants associated with all the drive lines are of

the same value(i.e., k1 = k2 = · · · = kn = k), the stiffness matrix reduces to the form

K = kJTJ (2.10)

2.3. ACHIEVE HOMOGENOUS AND SYMMETRIC FORCE TRANMISSION 9

The two DOF parallel manipulators shown in Figure 2.1 are taken as an example to

analysis the Cartesian stiffness of non-redundant and redundant mechanisms. The

simulation results of Figure 2.2 demonstrate that the presence of a third, redundant

actuator obtains a higher Cartesian stiffness.

0.080.090.10.110.120.13 0.040.050.060.070.080.09

300

400

500

600

700

800

900Euclidean two norm of Cartesian stiffness matrix

Redundant

Non−redundant

Figure 2.2: A comparison of Euclidean two norm of Cartesian stiffness in the non-

redundant and redundant case

2.3 Achieve Homogenous and symmetric Force Tranmis-

sion

The force transmission factor is the ratio of the actuator torques to the torques exerted

on the moving platform. Some robotic applications like haptic devices and walking

machines require an even force transmission in the workspace. It will be shown that

within the workspace, the redundant number of actuators permits a more homogenous

force output.

The velocity constraint between the joint velocity and the end-effector velocity is

2.4. OTHERS 10

expressed by

q = Jx (2.11)

Then in the force domain, it has

F = JT τ (2.12)

The force/torque tranmission ratio can be defined as

‖F‖‖τ‖ =

√τTJJT τ√τT τ

(2.13)

The output bounds for ‖F‖ with respect to the input torque ‖τ‖ are given by the

square roots of the two singular values of the matrix JJT :

σmin‖τ‖ ≤ ‖F‖ ≤ σmax‖τ‖ (2.14)

σ2min and σ2

max can be obtained by a singular value decomposition of JJT . Given

‖τ‖ = 1, the ratio σI = σminσmax

is a useful index to value the output force distribution in

the workspace. The two DOF planar parallel manipulators as shown in Figure 2.1 are

considered. One is normally actuated mechanism, the other is an over actuation and

over constrain mechanism. The simulation results of Figure 2.3 demonstrate that the

presence of a third, redundant actuator obtains a force output more homogenous and

symmetric.

2.4 Others

When a system is redundantly actuated, the forces required by the actuators are inde-

terminate corresponding to the generalized forces. The fact that redundantly actuated

systems are statically indeterminate is usually cited as the reason for optimizing the

internal and external force distribution. Usually the psudo-inverse approach is adopted

to minimize the two norm of the input torque. In this approach the energy consump-

tion is considers. As the ratio torque/weight of electrical motors is almost the same

whatever the size of the motor, the weight of the robot does not increase with the

2.4. OTHERS 11

0.080.09

0.10.11

0.120.13

0.02

0.04

0.06

0.08

0.10

0.2

0.4

0.6

0.8

1

For the case of non−redundant manipulator

The

rat

io o

f the

min

imum

and

max

imum

sin

gula

r va

lue

(a)

0.1 0.15 0.2 0.25 0.3 0.350.05

0.1

0.15

0.2

0.25

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

For the case of redundant manipulator

The

rat

io o

f the

min

imum

and

max

imum

sin

gula

r va

lue

(b)

Figure 2.3: The ratio between the minimum and maximum singular values in the non-

redundant and redundant case

number of actuators. But increasing the number of actuators to gain a redundancy in

the actuation enables the minimization for a given task, the power requirements and,

therefore the total energy comsuption of the robot. In addition, redundant actuation

allows a lighter and faster mechanical structure that can carry the same loads to be

built. It also allows greater safety in case of breakdown of individual actuators. If

the mechanism is redundantly actuated, it can still be controlled if one or more of the

actuators breaks down. It is also interesting to note that the human body makes ample

use of redundant actuation. The human arm, which has three joints with a total degree

of 7 to position and orientation, uses 29 muscle groups for control [23].

Chapter 3

Dynamics of redundantly

actuated parallel manipulators

3.1 Introduction

In this chapter I discuss the dynamics of redundantly actuated parallel manipulators

based on the dynamics of non-redundant ones. While the kinematics of parallel ma-

nipulators have been studied extensively during the last two decades, few papers can

be found on the dynamics of parallel manipulators. The dynamical analysis of paral-

lel manipulators is complicated by the existence of multiple closed-loop chains. Sev-

eral approaches have been proposed, including the Newton-Euler formulation [24] [12]

[25], Lagrangian formulating [26] [27], the principle of virtual work [13] [28] and the

D’Alember’s principle [15] [29].

The traditional Newton-Euler formulation requires the equations of motion to be

written once for each body of a manipulator. This inevitably leads to a large number

of equations and results in poor computational efficiency. The Lagrangian formulation

eliminates all of the unwanted reaction forces and moments at the outset. It is more

efficient than the Newton-Euler formulation. However, because of the numerous con-

straints imposed by closed loops of a parallel manipulator, deriving explicit equations of

motion in terms of a set of independent generalized coordinates becomes a prohibitive

12

3.1. INTRODUCTION 13

task. To simplify the problem, additional coordinates along with a set of Lagrangian

multipliers are often introduced. In this regard, the principle of virtual work appears

to be more efficient. D’Alembert’s principle has been shown to be the most efficient

approach to formulate a transformation of the generalized actuating forces between a

closed-chain mechanism and the corresponding reduced system. The basic procedure

to compute the inverse dynamics of the closed-chain mechanisms involves the following

steps:

1. A closed-chain mechanism is transformed into an open-chain tree-structure mech-

anism by making a virtual cut. The cut is made at a passive joint, and the rest

of the passive joints are considered to have virtual actuators.

2. The joint torques of the open-chain tree-structure mechanism are computed for

the same motion as in the closed-chain mechanism. No force or torque interaction

between the cut joints is assumed.

3. The joint torques of the original closed-chain mechanism are computed from the

open-chain joint torques by taking into account the constraints.

The dynamics of redundantly actuated parallel manipulators is based on the analy-

sis of non-redundant parallel robots, but is much more complicated due to redundant

actuation. Papers on the dynamics of redundant actuated closed-chain mechanisms

are scarce. Nakamura [15] presented the dynamics computation for non-redundant and

redundant closed-chain mechanisms. The scheme is computationally efficient compared

with conventional methods. However, ambiguity in his derivation in the case of actuated

redundancy exists. Ropponen [16] made some improvements to Nakamura’s method.

Lagrangian equations with constraints [17] presenting the dynamics of constrained sys-

tem are also a method to derive the dynamics of redundant parallel manipulators. In

this chapter, a survey is presented on existing methods of inverse dynamics compu-

tation of redundant closed-chain mechanism. Then a new computational scheme for

the inverse dynamics of closed-chain mechanisms with redundant actuation, founded

on the Lagrange-D’Alembert’s formulation, will be presented explicitly and clearly.

3.2. PREVIOUS WORK 14

3.2 Previous Work

3.2.1 Nakamura’s method

Based on the three procedures mentioned above, Nakamura proposed a general and

systematic computational scheme of the inverse dynamics of closed-chain mechanisms.

It is derived using the D’Alembert’s principle. To account for the constraints, only

the Jacobian matrix of the passive joint angles in terms of actuated ones is computed.

It makes the inverse dynamic computation scheme computationally efficient because

compensation for the Lagrange multipliers is not required. The inverse dynamics of

closed-chain mechanisms normally being actuated and redundantly actuated is dis-

cussed in his paper [15].

1. Dynamics of Nonredundant Closed-chain Mechanisms

Joints of a closed-chain mechanism can be classified into actuated and passive joints.

Suppose the ambient joint vector and the joint torque vector of the open-link mechanism

be denoted by

q =

qa

qp

∈ RNa+Nu (3.1)

where qa and qp correspond to the actuated joints and passive joints, Na is the number

of actuated joints, and Np is the number of passive joints excluding the virtually cut

joint. When the tree-structure mechanism makes the same motion as required for

the original closed-chain mechanism, qp is uniquely determined as a function of qa as

follows:

qp = pq(qa) (3.2)

The virtual displacement of the generalized joint coordinates of the tree-structure mech-

anism are given by

δq =

δqa

δqp

=

I

∂qp

∂qa

δqa =Wδqa (3.3)


where

W =

I

∂qp

∂qa

=

I

J

∈ R(Na+Np)×Na (3.4)

where I ∈ RNa×Np is an identity matrix, and J = ∂qp

∂qais the Jacobian matrix of the

passive joints with respect to the actuated ones. Matrix W is the Jacobian matrix

of all the joints with respect to the actuated joints. Assume the Lagrange functions

of the original closed-chain mechanism and the open-link mechanism be Lc and Lt

respectively. That is

Lc = Lc(qa, qa)

Lt = Lt(qa, qp, qa, qp) (3.5)

The dynamics of the open-chain mechanism is represented by

d

dt(∂Lt

∂q)− ∂Lt

∂q= τT (3.6)

Similarly, the dynamics of the closed-chain mechanism is represented by

d

dt(∂Lc

∂qa)− ∂Lc

∂qa= τT

c (3.7)

where τ denotes the joint torque of the open-link mechanism, and τc denotes the joint

torque of the correspondingly closed-link mechanism. Because the energy of the closed-

link mechanism and the correspondingly open-link mechanism is the same, we have

Lc(qa, qa) = Lt(qa, qp, qa, qp). By pure mathematical derivation, it has been proved in

Nakamura’s paper that[d

dt(∂Lc

∂qa)− ∂Lc

∂qa

]δqa − τTWδqa = 0 (3.8)

Since equation(3.8) holds for arbitrary δqa, the following relationship is obtained:

d

dt(∂Lc

∂qa)− ∂Lc

∂qa= τTW (3.9)

Furthermore, the dynamics of the original closed-link mechanism is given by

d

dt(∂Lc

∂qa)− ∂Lc

∂qa= τT

c (3.10)


Hence, it follows

τc =W T τ (3.11)

For the nonredundant actuation of closed-chain mechanisms, qa is independent in de-

termining the motion of the closed-chain mechanism. This means there is only one

way to represent the unactuated joints as a function of actuated joints. Also it can be

claimed the coefficients of the equation(3.8) are zero.

2. Dynamics of Redundant Closed-chain Mechanisms

In the case of redundant actuation, the actuated joint coordinate vector qa is no

longer fully independent since there are more actuators than are needed to perform the

motion. Thus, equation(3.2) can be written as

qp = qpi(qa) i = 1, . . . , Nr+1 (3.12)

where Nr is the number of redundant actuators, and qpi represents Nr+1 ways to write

the passive joint coordinates as a function of the actuated joint coordinates. For a

redundant actuation system that includes Nr redundant actuators, the passive joint

coordinates are represented by Nr + 1 independent ways as a function of the actuated

joints. For example, if we have three actuators of which one is redundant(Nr = 1),

there are two possible ways to represent the passive joints as a function of the actuated

joints. Therefore, Equation(3.11) is no longer a necessary condition for relating τc and

τ in the redundant actuation case. Using Equation(3.12), Nakamura proposed function

q0p as follows:

qp = q0p(qa) =

∑i

αiqip(qa) (3.13)

where α are scalars and satisfy

∑i

αi = 1 i = 1, . . . , Nr+1 (3.14)


Therefore, the joint torque of closed-link mechanism is obtained as follows:

τc = W T τ (3.15)

W =

I

∂q0p

∂qa

∈ R(Nta+Nu)×Nu

=

I

0

+

Nr+1∑i=1

αi

0

J i

(3.16)

J i =∂qi

p

∂qa(3.17)

where none of αi are dependent and subject to Equation(3.14). Eliminating αNr+1

using Equation(3.14) yields

W =

I

JNr+1

+

Nr∑i=1

αi

0

J i − JNr+1

(3.18)

which includes independent αi only.

3.2.2 Ropponen’s method

Ropponen studied the dynamics of redundant actuation based on Nakamura’s dynamics

computation scheme. Nr is the number of redundant actuators. Consequently, there

are Nr + 1 ways to represent the passive joints as a function of the actuated joints. In

Ropponen’s new formulation, to determine the motion in an independent way, just one

representation of qp in Equation(3.12) is used, namely,

W0 =

I

∂qp1

∂qa

=

I

J1

∈ R(Na+Na)×Na (3.19)

Let qin ∈ Rin be the collection of independent actuators among qa ∈ RNa , where

Nin < Na is the number of independent actuators. Accordingly, let the dependent, or

redundant actuators be qr ∈ RNr , where Nr = Na −Nin. Thus, qa can be partitioned

as follows

qa =

qin

qr

(3.20)


Next, qr can be written as a function of qin as

qr = qr(qin) (3.21)

The relation between the infinitesimal motions of qa and qin can be written as

δqa =

δqin

∂qr

∂qinδqin

= Sδqin (3.22)

where S ∈ RNa×Nin means the instantaneous coefficient that relates to qa and qin.

Next, by substituting δqa and W0 into Equation(3.9), also considering Equation(3.10),

it has

τTc Sδqin = τTW0Sδqin (3.23)

Since δqin is an independent variable, the following relationship is obtained

ST τc = STW T0 τ (3.24)

3.2.3 Lagrange’s Equations with Constraints

Only the holonomic constraints are considered. A constraint is said to be holonomic

if it restricts the motion of the system to a smooth hypersurface in the unconstrained

configuration space Q. Holonomic constraints can be represented locally as algebraic

constraints on the configuration space,

hi(q) = 0, i = 1, . . . , k. (3.25)

Each hi is a mapping from Q to � which restricts the motion of the system. Assume

that the constraints are linearly independent and, hence, the matrix ∂h∂q is full row rank.

The constraint forces of a set of holonomic constraints of the form in equation (3.2.3)

are linear combinations of the gradients of the constraint functions hi : Q → �. Lettingh : Q → �k represent the vector-valued constraint function, we can thus write the

constraint force as

Γ =∂hT

∂qλ (3.26)


where λ ∈ �k is the vector of relative magnitudes of the constraint forces. These

constraint forces can be viewed as acting normally to the constraint surface, with the

magnitude of the forces chosen to insure that the system remains on the constraint

surface defined by equation (3.2.3). Note that no work is done by constraints when the

system is moved along feasible trajectories since

Γq = λT ∂h

∂qq = λT d

dt(h(q)) = 0 (3.27)

More generally, for a system with configuration spaceQ, we consider velocity constraints

of the form

A(q)q = 0 (3.28)

where A(q) ∈ �k×n represents a set of k velocity constraints. A constraint of this form

is called a Pfaffian constraint. A Pfaffian constraint is said to be integrable if there

exists a vector-valued function h : Q → �k such that

A(q)q = 0∂h

∂qq = 0 (3.29)

The constraint forces for a set of Pfaffian constraints prevent motion of the system in

directions which would violate the constraints. In order to include these forces in the

dynamics, we must add one additional assumption about the nature of the constraints.

Namely, it is assumed that the forces which are generated by the constraints do no

work on the system. This assumption is often referred to as d’Alembert’s principle.

Let L(q, q) represent the Lagrangian for the unconstrained system and let the con-

straints have the form

A(q)q = 0 A(q) ∈ �k×n (3.30)

The equations of motion are formed by considering the constraint forces as an additional

force which affects the motion of the system. Hence, the dynamics can be written in

vector form as

d

dt

∂L

∂q− ∂L

∂q+AT (q)λ−Υ = 0 (3.31)

3.3. LAGRANGE-D’ALEMBERT FORMULATION 20

where the columns AT form an non-normalized basis for the constraint forces and

λ ∈ �k gives the relative magnitudes of the forces of constraint. As before, Υ represents

nonconservative and externally applied forces.

The scalars λi, . . . , λk are called Lagrange multipliers. In the case that the Lagrangian

has the form L(q, q) = 12 q

TM(q)q − V (q)(kinetic minus potential), an explicit formula

for the Lagrange multipliers can be derived. The equations of the motion can be written

as

M(q)q + C(q, q)q +N(q, q) +AT (q)λ = F (3.32)

where F corresponds to the vector of external forces and N(q, q) includes nonconser-

vative forces as well as potential forces. Differentiating the constraint equation (3.30)

yields

A(q)q + A(q)q = 0 (3.33)

and, solving q from equation (3.32), we obtain

(AM−1AT )λ = AM−1(F − Cq −N) + Aq (3.34)

The configuration dependent matrix AM−1A is full rank if the constraints are inde-

pendent and, hence, the Lagrange multipliers are

λ = (AM−1AT )−1(AM−1(F − Cq −N) + Aq) (3.35)

Using the equation, the Lagrange multipliers can be computed as a function of the

current state, q and q, and the applied forces, F .

3.3 Lagrange-D’Alembert Formulation

The schemes proposed by Nakamura and Ropponen are more computationally efficient

than conventional methods because they avoid of computing the Lagrangian multipliers.

However, in Nakamura’s proposition, there is ambiguity when considering the redun-

dant actuation. The scheme of Ropponen’s eliminates the ambiguity, but it is complex


with respect to the dynamic equation. Lagrange’s equations with constraints are theo-

retical clear while the computation of the Lagrange multipliers is very time-consuming.

In the following, Lagrange-D’Alembert formulation will be employed to derive the dy-

namics of redundantly actuated closed-chain mechanisms. Lagrange-D’Alembert for-

mulation is convenient and useful of derive the equations of motion without explicitly

solving for the instantaneous constraint forces present in the system. In essence, this

proceeds by projecting the motion of the system onto the feasible directions and ig-

noring the forces in the constrained directions. In doing so, we will be able to get a

more concise description of the dynamics which is in a form well suited for closed-loop

control.

3.3.1 Dynamics of Nonredundant Closed-chain Mechanisms

At a given configuration q ∈ Rn, the instantaneous set of directions in which the system

is allowed to move is given by the null space of the constraint matrix, A(q). We adopt

the classical notation and call a vector δq ∈ Rn which satisfies A(q)δq = 0 a virtual

displacement. D’Alembert’s principle states that the forces of constraint do no virtual

work. Hence,

(AT (q)λ)δq = 0 forA(q)δq = 0 (3.36)

The Lagrange-D’Alembert equations are as follows:

(d

dt

∂L

∂q− ∂L

∂q− τ)δq = 0 (3.37)

where δq ∈ Rn satisfies

A(q)δq = 0 (3.38)

Let the joint vector of the open-link system be qt =

qa

qp

, where qa is the actuator

joint vector, qp is the passive joint vector in the open-link system. Let the structure

equations on the tree system be

h(qt) = h(qa, qp) = 0 (3.39)


Then, by differentiating equation (3.39), we have

∂h

∂qaδqa +

∂h

∂qpδqp = 0 (3.40)

Thus, from the Lagrange-D’Alembert formulation, it follows

(d

dt(∂Lt

∂qt)− ∂Lt

∂qt− τT

t )δqt = 0 (3.41)

where Lt is the Lagrangian for the open-chain system. Suppose normally actuated

and away from actuator singularity, the matrix ∂h∂qp

is square and invertible, and the

configuration space can be smoothly parameterize by the actuator joint vector qa.

Consequently,

(d

dt

∂Lt

∂qt− ∂Lt

∂qt− τT

t )δqt (3.42)

=[d

dt

∂Lt

∂qa− ∂Lt

∂qa− τT

a ,d

dt

∂Lt

∂qp− ∂Lt

∂qp− τT

p

] δqa

δqp

= 0 (3.43)

Then,

(d

dt

∂Lt

∂qa− ∂Lt

∂qa− τT

a )δqa+

(d

dt

∂Lt

∂qp− ∂Lt

∂qp− τT

p )∂qp

∂qaδqa = 0 (3.44)

where

∂qp

∂qa= − ∂h

∂qp

−1 ∂h

∂qa(3.45)

As δqa is free(arbitrary), the coefficients of the equation (3.45) is zero. After rearrange-

ment of the terms, we have

[d

dt

∂Lt

∂qt− ∂Lt

∂qt

] I

∂qp

∂qa

= τT

a + τTp

∂qp

∂qa(3.46)

The applied torque τp of the passive joints is zero if the friction in the unactuated

joints is ignored. Thus, the torque τa is the just the required torque for the closed-

chain mechanism. Also consider

(d

dt

∂Lt

∂qt− ∂Lt

∂qt)T = τt (3.47)


Take transpose, equation (3.46) becomes

τa =W T τt (3.48)

where

W =

I

∂qp

∂qa

(3.49)

Equations (3.48) and (3.49)have the same form as those in [15]. However, it can be seen

clearly that the derivation of the dynamics of closed-chain mechanism is more simple

and straightforward.

3.3.2 Dynamics of Redundant Closed-chain Mechanisms

Proposition 1

The joint torque τa of a redundantly actuated closed-link mechanism, for a given mo-

tion, is computed by (3.50) from the joint torque τt of the corresponding open-chain

mechanism required to make the same motion.

W T τt = ST τa (3.50)

where W = ∂q∂qin

, S = ∂qa

∂qin, q is the ambient space of the open-chain mechanism, qa is

the vector of actuated joint angles and qin is the independent generalized coordinates

of the system.

Proof of proposition 1:

In the case of redundant actuation, δqa is not arbitrary, equation (3.46) does not

hold any more. The dynamics of closed-chain mechanisms will be derived as follows.

Let qin be the independent generalized coordinates of the closed-chain system. Un-

der the kinematic constraints, the actuated joints qa and the passive joints qp can be

parameterized by qin:

qa = qa(qin) (3.51)

qp = qp(qin) (3.52)


Differentiate the two equations, we have

δqa =∂qa

∂qinδqin (3.53)

δqp =∂qp

∂qinδqin (3.54)

Making use of the same notation in the Lagrange-D’Alembert equations in the nonre-

dundant case yields

(d

dt(∂Lt

∂qt)− ∂Lt

∂qt− τT

t )δqt

= (d

dt(∂Lt

∂qa)− ∂Lt

∂qa− τT

a )δqa + (d

dt(∂Lt

∂qp)− ∂Lt

∂qp− τT

p )δqp

=[(d

dt(∂Lt

∂qa)− ∂Lt

∂qa− τT

a )∂qa

∂qin+ (

d

dt(∂Lt

∂qp)− ∂Lt

∂qp− τT

p )∂qp

∂qin

]δqin = 0(3.55)

Since qin is free, the equations become

[d

dt(∂Lt

∂qa)− ∂Lt

∂qa,

d

dt(∂Lt

∂qp)− ∂Lt

∂qp

] ∂qa

∂qin

∂qp

∂qin

= τT

a

∂qa

∂qin+ τT

p

∂qp

∂qin(3.56)

where τp is the torque of the passive joints, and τa is the torque of the actuated joints.

Ignore the friction of the passive joints, τp is zero. Also consider

(d

dt

∂Lt

∂qt− ∂Lt

∂qt)T = τt (3.57)

Hence, we have

W T τt = ST τa (3.58)

where

W =

∂qa

∂qin

∂qp

∂qin

(3.59)

S =∂qa

∂qin(3.60)

The dynamics of the open-chain system has been studied extensively, and has the

following form:

M(q)q + C(q, q)q +N(q) = τt (3.61)


where M(q) and C(q, q) is the inertial and Coriolis matrix respectively, N(q) is the

vector of actuator torque. Substitute equation (3.61) into equation (3.58), we have

W T (M(q)q + C(q, q)q +N(q)) = ST τa (3.62)

Let q =

qa

qp

, we have

q =∂q

∂qinqin =

∂qa

∂qin

∂qp

∂qin

qin =Wqin (3.63)

q = W ˙qin +Wqin (3.64)

Thus, we can write the dynamics of the redundantly actuated closed-chain mechanism

as

M qin + Cqin + N = ST τa (3.65)

where

M = W TMW

C = W TMW +W TCW

N = W TN

Proposition 2

The inertial matrix M(q) and the Coriolis matrix C(q, q) satisfy the following struc-

tural properties:

1. M (q) is symmetric and positive definite matrix.

2. ˙M − 2C is skew-symmetric matrix.

Proof of proposition 2:

Since M is symmetric and positive definite, M is obviously symmetric and positive

definite.

˙M − 2C = W TMW +W TMW +W TMW − 2W TMW − 2W TCW

= W T (M − 2C)W + W TMW −W TMW


While,

W TMW −W TMW + (W TMW −W TMW )T = 0 (3.66)

Hence W TMW −W TMW is skew-symmetric. For the open-chain system, the matrix

M − 2C is skew-symmetric, hence W T (M − 2C)W is skew-symmetric. Consequently˙M − 2C is skew-symmetric since it is the sum of two skew symmetric matrices.

Actually, the independent coordinates qin can be chosen as any well-behaved pa-

rameterization such as the actuated joint position or the end-effector coordinates in

Cartesian frame. If the independent coordinates are chosen from the actuated joint an-

gles, the control strategy is joint control law. While if the end-effector coordinates are

chosen to parameterize the configuration space, the dynamics given in equation (4.1)

represents the equations of motion relative to the workspace coordinates. In the case of

redundant actuation, there are multiple choices with respect to the independent coor-

dinates. The most commonly used forces distribution algorithm is the Moore-Penrose

weighted Pseudo-inverse of the inverse kinematic rate relationship. This method results

in a vector of joint torques that has a minimum weighted Euclidian norm. Employing

the Moore-Penrose Pseudo-inverse and allowing the weighted matrix be the identity

matrix, the dynamics (4.1) of a redundant parallel manipulator can be written as:

τa = (ST )+(M qin + Cqin + N) (3.67)

where (ST )+ = S(STS)−1, which satisfies S(ST )+ = I, and (ST )+ST = I.

Chapter 4

Control of Parallel Manipulator

with Redundant Actuation

4.1 Introduction

In Chapter 3, the dynamics of redundantly actuated closed-chain mechanisms has been

obtained as

M qin + Cqin + N = ST τa (4.1)

where

M = W TMW

C = W TMW +W TCW

N = W TN

Also it has been proved that the inertial matrix M(q) and the Coriolis matrix C(q, q)

satisfy the following structural properties:

1. M (q) is symmetric and positive definite matrix.

2. ˙M − 2C is skew-symmetric matrix.

Therefore, the structure of the equations of motion of closed-chain manipulators is

similar to that of open-chain dynamics robot models. This similarity motivates the

27


application of existing trajectory controllers designed for serial manipulators applied to

closed-chain manipulators, i.e., PD control, augmented PD control, computed torque

control and adaptive control methods.

Independent joint controllers (of PD or PID type) are usually employed in indus-

trial robot manipulators because they are very computationally efficient, and are easy

to implement. However, independent joint controllers cannot achieve a satisfactory

performance due to their inherent low rejection of disturbances and parameter varia-

tions. Because of such limitation, model-based control algorithms were proposed [30]

that have the potential to perform better than independent joint controllers that do not

account for manipulator dynamics. In spite of all the foregoing good features, model-

based control relies its potential on the correctness and completeness of the dynamic

models that are just idealizations of the physical components of the robots, i.e., the

manipulator, the actuators, the joint transmission, the transducers, etc. For open-chain

mechanisms, a lot of research effort was placed on model-based control so as to demon-

strate the effectiveness of these approaches. These control schemes were also intensively

experimentally tested. Experimental evaluation of feed forward and computed torque

control was presented in [31] and [32]. Moreover, to improve the performance of in-

dependent joint controllers, some robust design of independent joint control schemes

[20] [33] were proposed as alternatives to model-based control. The dynamic model

of a robot manipulator is described by a set of nonlinear, highly coupled, differential

equations.

It has been pointed out by [29] that, because the dynamics satisfies the same two

properties as those of serial robots, the extension of existing control schemes designed

for serial robots to closed-chain mechanism was obvious. However, due to the com-

plication of the dynamics, the model-based controllers are difficult to implement on

redundant closed-chain robots. Research work on the control of closed-chain manipu-

lators is scarce. F. Ghorbel and R. Gunawardana built the Rice Planar Delta Robot as

a test bed to perform control experiments for closed-chain mechanisms. PD plus sim-


ple gravity control strategy was experimentally tested by [18] and [19]. A new robust

independent joint control scheme was proposed in [20]. Direct drives greatly reduce

joint friction and backlash effects. Hence direct-drive robots have been preferred as

good experimental devices to test advanced control strategies. A redundantly actu-

ated direct-drive closed-chain mechanism was built to perform the model-based control

strategies. The experimental evaluation of the three control schemes was presented in

[16]. The parallel redundant manipulator PA-R-MA was built to implement control

strategies for fast parallel robots with high-reduction gear boxes [11]. The dynamics of

the closed-chain system was simplified by assuming the mass of links were insignificant.

A decoupled Cartesian motion controller was proposed by [21] and the experimental

results showed the feasibility of this high-speed robot concept.

Parallel manipulators tend to introduce additional dynamics because of coupling

due to the kinematic constraints between the cooperating mechanisms. The redundant

actuation offers an unlimited variety of control inputs that will effect any given motion.

These two issues are the major complications in the control of cooperating systems. Due

to the redundancy of actuation, the set of forces needed to move the system along a path

corresponding to a particular payload trajectory is not unique. The most commonly

used forces distribution algorithm is the Moore-Penrose weighted Pseudo-inverse of the

inverse kinematic rate relationship. This method results in a vector of joint torques

that has a minimum weighted Euclidian norm.

Position control and trajectory tracking are the most common tasks for robot ma-

nipulators; given a desired trajectory, the joint torques be chosen so that the manip-

ulator follows that trajectory. The control strategy should be robust with respect to

initial condition errors, sensor noise, and modeling errors. In the following sections,

the control schemes will be presented for trajectory tracking, and their stability will be

discussed.

4.2. PD CONTROL SCHEME 30

4.2 PD Control Scheme

PD control is the simplest type of control strategy and is often used to compute the

control forces in industrial manipulators. In this type of control, each axis of the manip-

ulator is controlled as a single-input/single-output(SISO) system. The error between

the actual and desired joint position is used as feedback to control the actuator asso-

ciated with each joint. It treats each actuator in the system individually and ignores

dynamics. PD control is popular in industrial field because it is easy to implement,

requires little computation time during real-time operation, and offers proven stability

with a specified range of feedback parameters. [19] and [18] proposed PD plus full

gravity compensation control law for closed-chain mechanisms. To investigate the ap-

plicability of the proposed control law, experiments were performed successfully. The

proposed control law is confined to nonredundant closed-chain mechanisms. In this

section two different PD control algorithms will be proposed for redundantly actuated

parallel manipulators.

1. The first PD control law is based on the position and velocity error of each

actuator in the joint space. To implement the joint control architecture, the

values for the joint position error and the joint rate error of the closed-chain

system are used to compute the joint torque.

τc = kpe+ kv e (4.2)

where e = θd − θ, which is the vector of position error of the individual actuated

joints. The joint position error and the joint rate error are multiplied by the

individual proportional and derivative control gains of the open-chain components

to obtain the joint control forces τc. The first control algorithm considers neither

the dynamics, nor the actuated redundancy. It is the most simplest control law

for a closed-chain mechanism with redundant actuation. The diagram of this

control law is shown in Figure 4.1.

2. The second PD control strategy is based on the position and velocity error of the

4.2. PD CONTROL SCHEME 31

q

q .

a(t)

a(t)K

pK

v

Manipulator

Parallel

q da(t)

q .

da(t)

e

e.

+-

+

-

Figure 4.1: Diagram of the first type of PD control scheme

proposed independent coordinates of the system. The control law is

ST τc = kpe+ kv e (4.3)

where S is the Jacobian matrix between the actuated joint position rate qa and the

position rate of the independent coordinates qin, e = qdin−qin, and e = qdin− qin.

Due to the redundant actuation, the distribution of the torque in the actuated

joints is undetermined. In the research the strategy of minizing the two Euclidian

norm of the input actuator torque is employed, which tends to minimize the

following cost function of τc :

min τTc τc (4.4)

It can be shown that the solution is

τc = S(STS)−1(kpe+ kv e) (4.5)

Figure 4.2 is the diagram of the second type of PD controller when the inde-

pendent generalized coordinates are chosen from the actuated joints. If the end-

effector coordinates are chosen as the independent generalized coordinates of the

system, the control strategy is the PD controller in the cartesian space. The

control diagram is shown in Figure 4.3.

4.3. AUGMENTED PD CONTROL 32

+)T

S(Parallel Manipulator

q din

Kp

Kv

q in

inq (t)

(t)

.. .e

e-q din

-

+

+

Figure 4.2: Diagram of the second type of PD controller in the joint space

+)T

S(Parallel Manipulator

Xd

d

X

Jacobian Matirx J

Forward Kinematics

X

X.

.

qa

.q

(t)Kp

Kv

e

e.

(t)a

+ -

+

-

Figure 4.3: Diagram of the second type of PD controller in the Cartesian space

4.3 Augmented PD control

Augmented PD control augments the basic PD controller by compensating for the ma-

nipulator dynamics in the feed forward way. The feed forward terms can be computed

off-line. The augmented PD control law(also called nonlinear feedforward PD control)

is:

τc = (ST )+( ˆM(q)qdin + C(q, q)qdin + N(q, q)−Kv e−Kpe) (4.6)

where e = qin − qdin, Kv and Kp are constant gain matrices satisfying Kv,Kp ≥ 0.

Stability Proof:

Combining the dynamics (3.67) with the proposed control law (4.6), the closed loop

4.4. COMPUTED TORQUE CONTROL 33

system can be written as

(ST )+(M(q)e+ C(q, q)e+Kv e+Kpe) = 0 (4.7)

Premultiplying equation (4.7) by ST to obtain

M(q)e+ C(q, q)e+Kv e+Kpe = 0 (4.8)

since ST (ST )+ = I when ST has full rank. Choose the Lyapunov function candidate

V (e, e, t) =12eT M e+

12eTKpe+ εeT M e (4.9)

which is positive definite for ε sufficiently small since M ≥ 0 and Kp ≥ 0. Evaluating

V along trajectories of (4.8):

V = eT M e+12eT M e+ eTKpe+ εeT M e+ εeT (M e+ ˙

M)

= −eT (Kv − εM)e+12eT ( ˙M − 2C)e+ εeT (−Kpe−Kv e− Ce+ ˙

Me

= −eT (Kv − εM)e− εeTKpe+ εeT (−Kv +12˙M)e

Choosing ε ≥ 0 sufficiently small insures that V is negative definite and, hence, the

system is exponentially stable. In practice, if appropriate Kv and Kp is chosen to

track the desired trajectory with certain degree of accuracy, the matrices M(q) and

C(q, q) can be substituted by M(qd) and C(qd, qd). Hence, the feed forward part of

the control law (4.6), i.e. ˆM(q) ¨qdin + C(q, q) ˙qdin can be calculated off-line to improve

the computational efficiency. Assume parameterize the redundant closed-chain system

by the end-effector coordinates, the diagram of the augmented PD control is shown in

Figure 4.4.

4.4 Computed torque control

The scheme of the computed torque control is shown in Figure 4.5. This scheme utilizes

nonlinear feedback to decouple the manipulator. The control law is given as

τc = (ST )+(M(q)(qdin −Kv e−Kpe) + C(q, q)qin + N(q, q)) (4.10)


(q)

Feedforward Compensation

M..Xd + (q,q)

.Xd

.C^^

X d

X.

d

..

+)T S( ManipulatorParallel

.q.

-

+

-

e

e

+Xd

d

X

X

X

.

.

K

K

p

v

.

++

+

qa(t)

Forward Kinematics

Jacobian Matirx J

a(t)

Figure 4.4: Diagram of Augmented PD control scheme

- Kpe- Kv ) +)T

S(+ C(q,q)

^ .XM(q)^

d e.

X

Xd-

+

-

.

e

e.

+X

..

.

X

+ N^

(q)

X d..

Xd.

Forward Kinematics

Jacobian Matirx J

Manipulator

Parallel

.a (t)q

qa (t)

Figure 4.5: Diagram of computed torque control scheme

where e = qin − qdin is the tracking error, Kv and Kp are constant feedback gain

matrices.

Stability Proof:

Combining the dynamics (3.67) with the proposed control law (4.6), the closed loop

system can be written as

(ST )+M(q)(e+Kv e+Kpe) = 0 (4.11)

Premultiplying equation (4.11) by ST to obtain

M(q)(e+Kv e+Kpe) = 0 (4.12)


Since ST (ST )+ = I when ST has full rank. Moreover, M is positive definite, (4.12)

simplifies to

e+Kv e+Kpe = 0 (4.13)

Since the error equation (4.13) is linear, it is easy to chooseKv andKp so that the overall

system is stable and e → 0 exponentially as t → ∞. Usually, if let Kv = kvI,Kp = kpI

with s2+kvs+kp Hurwitz polynomial, then the control law (4.10) applied to the system

(3.67) results in exponentially trajectory tracking. In the control scheme depicted in

(4.5), the system is parameterized by the end-effector coordinates. Hence, the computed

torque controller is workspace control.

Chapter 5

Experimental Devices

5.1 Design goals and the Prototype of the Manipulator

The increasing demand for a higher level of productivity in the semiconductor manufac-

turing industry has imposed very stringent requirements on the accuracy and speed of

manufacturing equipment. Traditional x-y tables commonly used as motion platforms

are designed on the basis of LM guides, ball(lead) screws and driving motors. The

distinguished feature of this construction is that one is placed on the top of the other.

Such a design has the advantage of a large workspace but suffers from the following

drawbacks:

1. Since the guide of the top axis is an additional and useless load for the underlying

axis, the maximum acceleration of the system is limited.

2. The serial structure implies that the positioning error of one of the axes will

impact on the positioning accuracy of the next axis. Thus, the positioning error

of the whole system is an accumulated sum of the error of the constituting axes.

3. In order to maintain a high level of system accuracy, high precision elements(LM

guides and ball screws) will be chosen since their accuracy directly affects the

accuracy of the whole system. Hence, associated costs rise.

36

5.1. DESIGN GOALS AND THE PROTOTYPE OF THE MANIPULATOR 37

In contrast to a serial manipulator, a parallel manipulator has the potential to improve

the acceleration, the accuracy, and reduce the cost:

1. A parallel manipulator can be made much lighter since its links are mainly sub-

jected to tension and pull load. A light system makes it easier to achieve higher

values of accelerations.

2. The positioning error of a parallel manipulator is the mean error of the individual

chains in contrast to the sum in the case of a serial manipulator.

3. Links cost less than that of LM guides and ball(lead) screws.

Parallel manipulators present many advantages. However, there are certain problems

inherited by its parallel structure. Singularities are commonly encountered within the

workspace when the manipulator gains one or more degrees of freedom. When the

manipulator falls into the neighborhood of the singularity, it losses stiffness completely.

Redundant actuation is introduced to eliminate singularities within the workspace.

Because of these considerations, a two DOF parallel manipulator with redundant ac-

tuation was developed for epoxy writing applications. The redundant mechanism is

shown in Figure 5.1. In addition, it was built as a test-bed to investigate the funda-

mental problems of dynamic modeling and controlling closed-chain mechanisms with

redundant actuation. The parallel redundant manipulator is an RRR-type mechanism.

It introduces an additional linkage with one redundant actuator. It is equipped with

three permanent magnet synchronous servo drives with harmonic gear boxes. Mounted

on a base plate, they form an isosceles triangle. Each leg of the mechanism has two

links, one of which is actuated and the other is passive. The three legs meet at a center

joint, where the epoxy writing tool in die bonding or general dispensing machine would

be mounted. All the links are of the same length 70mm. Harmonic gears are utilized

as speed reducers with a reduction ratio of 25. The harmonic gears possess benefits

like small size, low inertia, and small backlash. The servo-system element is a Sanyo

Denki p2 series AC servo-pack. The servos run in current-controlled mode. Motor-side

incremental encoders are used to feed back the joint position of the actuators.

5.2. FORWARD KINEMATICS 38

Figure 5.1: The Prototype of two DOF redundantly actuated parallel mechanism

The structure of a motion control system is shown in Figure 5.2. The control

system is based on a four-axis DSP motion control board developed by the Automation

Technology Center of HKUST. It provides exceptional utilities for: (1) High speed

encoders inputs, (2) various type I/O and (3) continuous contouring. The superior

performance 32-bit digital signal processor SHARC ADSP-21061 along with the 16-bit

resolution servo output, and fastest servo loop update rate and 8MHZ encoder inputs

provide exceptionally precise position control.

5.2 Forward kinematics

The forward kinematic problem is to compute the position of the end point C in the

Cartesian frame with respect to the joint angles θ1, θ2 and θ3 in Figure 5.3. Ai denotes

the ith actuated joint vector with coordinates XAi =

xAi

yAi

, Bi represents the ith

passive joint vector with coordinates XBi =

xBi

yBi

, and C represents the end-effector

5.3. INVERSE KINEMATICS 39

Figure 5.2: The motion control system structure

position with XC =

xC

yC

. From figure(5.3), we have

|XC −XBi|2 = L2, i = 1, 2, 3 (5.1)

where

XBi =

xBi

yBi

=

xAi + L cos θi

yAi + L sin θi

i = 1, 2, 3 (5.2)

Solving for XC and YC yields forward kinematics equations

xC =‖XB1‖2(yB2 − yB3) + ‖XB2‖2(yB3 − yB1)+ ‖ XB3‖2(yB1 − yB2)

2[xB1(yB2 − yB3) + xB2(yB3 − yB2) + xB3(yB1 − yB2)](5.3)

yC =‖XB1‖2(xB3 − xB2) + ‖XB2‖2(xB1 − xB3) + ‖XB3‖2(xB2 − xB1)

2[xB1(yB2 − yB3) + xB2(yB3 − yB2) + xB3(yB1 − yB2)](5.4)

Note that without the third leg, any two-leg planar manipulator would have two so-

lutions for the forward kinematics. Whereas, over constraint makes the forward kine-

matics have the unique solution.

5.3. INVERSE KINEMATICS 40

2

2

1

3

3

2

2

A1

B

b

B

a

A

L=70mm

B

A

L

L

ϕ1θ1

La3

θ2

ϕ2

3

3

b3LC

L1b

L1a

ϕθ

Figure 5.3: Coordinates of the two DOF redundantly actuated parallel mechanism

5.3 Inverse Kinematics

The inverse kinematics problem is to find the joint angles θ1, θ2, θ3 of actuators given

the position of C in the Cartesian frame. For any triangle AiBiC in Figure 5.3, we

have

‖ BiC‖2 = ‖AiBi‖2 + ‖AiC‖2 − 2‖AiBi‖‖AiC‖ cos (θi − ϕi) (5.5)

where ϕi = arctan yc−yAixc−xAi

. Hence θi can be solved as

θi = arccos‖BiC‖2 + ‖AiC‖2 − det ‖AiBi‖2

2‖AiBi‖‖AiC‖ + ϕi (5.6)

Two solutions are obtained from the equation (5.6) due to the uncertainty of the arccos

function. Two solutions for each leg amount to eight solutions for the manipulator. The

actuated joints of the manipulator are located on the corners of an isosceles triangle.

In the Cartesian frame, the coordinates of actuated joints are A1(0, 62), A2(184, 0) and

A3(184, 124) respectively, and the home position of the end-effector C is chosen as

(92, 62). By inverse kinematics, the joint angles of the actuators are θ1 = 48.918, θ2 =

251.563 and θ3 = 183.609 with respect to the home position of the end-effector C in

the Cartesian frame.

5.4. DYNAMICS OF THE TWO DOF REDUNDANT PARALLELMANIPULATOR 41

5.4 Dynamics of the Two DOF Redundant Parallel Ma-

nipulator

First, the dynamic equation of the equivalent open-chain mechanism should be ob-

tained. The open-train mechanism consists of three two-link planar manipulators. The

schematic of the mechanism is shown in Figure 5.4.

According to [17], the dynamics of each two-link planar manipulator is:

Mi

θi

ϕi

+ Ci

θi

ϕi

=

τai

τpi

(5.7)

where

Mi =

Mi11 Mi12

Mi21 Mi22

=

αi + 2βi cosϕi γi + βi cosϕi

γi + βi cosϕi γi

Ci =

Ci11 Ci12

Ci21 Ci22

βi sinϕiθi −βi sinϕi(θi + ϕi)

βi sinϕiθi 0

and

2

2

2

2

1

3

3

1

1

3

31

2

2

3

3

A1

a

θ

B

b

B

ϕ

aθ

A

L=70mm

Bϕ

b

aθ

A

ϕ

LC

L

L

L

L1b

(a)

τa1

p1τ

τ

τ

τ

τ

a2

p2

a3

p3

(b)

Figure 5.4: Schematics of a two DOF overactuation parallel mechanism and its tree-

structure mechanism


αi = Izi1 + Izi2 +mi1r2i1 +mi2(L2 + r2

i2)

βi = mi2ri2L

γi = Izi2 +mi2r2i2 (5.8)

Izi1 and Izi2 represent the inertia of the links ai and bi respectively for i = 1, 2, 3.

Similarly, mi1 and mi2 are the mass of the outer and inner links, ri1 and ri2 denote the

center of the mass of the links. Combining the three open chains’ equations together,

the dynamic model of the tree structure mechanism is obtained as:

Mq +Nq = τ (5.9)

where

M111 0 0 M112 0 0

0 M211 0 0 M2120

0 0 M311 0 0 M312

M112 0 0 M122 0 0

0 M212 0 0 M2220

0 0 M312 0 0 M322

θ1

θ2

θ3

ϕ1

ϕ2

ϕ3

+

C111 0 0 C112 0 0

0 C211 0 0 C2120

0 0 C311 0 0 C312

C112 0 0 C122 0 0

0 C212 0 0 C2220

0 0 C312 0 0 C322

θ1

θ2

θ3

ϕ1

ϕ2

ϕ3

=

τa1

τa2

τa3

τp1

τp2

τp3

(5.10)

Next, Jacobian matrix should be obtained to represent the constraints. In the ex-

periment, the end-effector Cartesian space coordinates u = [x y]T are chosen as the

independent coordinates. As shown in Figure 5.4, ai and bi denote the vectors of the

first and second moving links of the ith limb, θi denotes the orientation angles with

respect to the x-axis, and ϕi denotes the relative orientation angles between ai and bi.


Its kinematics satisfies:

e1 0 0

0 e2 0

0 0 e3

θ1

θ2

θ3

=

b1x b1y

b2x b2y

b3x b3y

x

y

e1 0 0

0 e2 0

0 0 e3

ϕ1

ϕ2

ϕ3

= −

d1x d1y

d2x d2y

d3x d3y

x

y

(5.11)

Where ei = dixbiy − diybix, which represents the magnitude of ai × bi, bix and biy are

the x and y components of ai and bi respectively, and dix and diy denote the x and y

components of the vectors ACi correspondingly. Let x and y denote the position of the

end-effector(point C), Aix and Aiy represent the task space coordinates of the three

motors mounted in the base, and L is the length of the link(all the links have the same

length). By the forward kinematics, x, y and ϕi can be calculated from the measured

joints angle θi. Hence bix, biy, dix and diy can be calculated according to

bix = L cos(θi + ϕi)

b1y = L sin(θi + ϕi)

dix = L cos(θi) + L cos(θi + ϕi) = x−Aix

diy = L sin(θi) + L sin(θi + ϕi) = y −Aiy

From the above kinematics equations, Jacobian matrix representing the relationship

between velocity in the task space and that in the joint space is obtained. Let ri = 1ei.


Consequently, the Jacobian matrix representing the constraints is as follows:

W =∂(θ1, θ2, θ3, ϕ1, ϕ2, ϕ3)

∂(x, y)=

b1xr1 b1yr1

b2xr2 b2yr2

b3xr3 b3yr3

−d1xr1 −d1yr1

−d2xr2 −d2yr2

−d3xr3 −d3yr3

S =[∂(θ1, θ2, θ3)

∂(x, y)

]=

b1xr1 b1yr1

b2xr2 b2yr2

b3xr3 b3yr3

(5.12)

Therefore, the dynamics of the closed-link mechanism are:

ST τc = Mu+ Cu (5.13)

where, M =W TMW, N =W TMW +W TCW.

5.4.1 The measured and computed parameters

As mentioned above, the link parameters required to calculate the elements of αi, βi

and γi in equation (5.8) are mass, the center of mass and the terms of inertia. The links

were detached in order to measure these parameters. The mass of each component was

determined with a balance. The center of mass was located by balancing each link on a

knife edge. Each link can be simplified as two homogenous rectangular bars. Since the

center of mass has been obtained experimentally, the inertia of each link was computed.

The link masses, center of mass and inertia are reported in Table 5.1. Consequently,

the constant αi, βi and γi can be obtained. The values are shown in Table 5.2.

In modeling the system, the mass, length and joint angles all united into the stan-

dard units. The corresponding torque has the unit N.m. Since the commanded torque

for the GM-400 motion control board is the digital value, the proportion should be

obtained between the torque of the unit N.m and the commanded digital value of the


Table 5.1: Link mass, center of mass and inertia parameters

Links Mass(kg) Center of mass(m) Inertia(Kg.m2)

Link11 0.55 0.045 6.214e-4

Link12 0.504 0.0508 3.08e-4

Link21 0.55 0.045 6.214e-4

Link22 0.376 0.0512 3.065e-4

Link31 0.55 0.045 6.214e-4

Link32 0.125 0.035 1.001e-4

Table 5.2: The constant parameters αi, βi and γi

Links α β γ

Link1 0.0058 0.0018 0.0016

Link2 0.0049 0.0013 0.0013

Link3 0.0026 0.0003 0.0003

torque. Driving the motor using the sets of commanded digital values, and recording

the actual torque of the motor correspondingly, a set of points was obtained. Using

the least square method, the points are approximated as a line as shown in Figure 5.5.

The proportion, i.e. the slope of the two lines, is 19528 and 20670 respectively. Their

average value 20100 is utilized as the proportion representing the relationship between

the actual torque and the commanded torque.

5.4.2 System Identification

To determine the total rotational inertia and the friction of the parallel manipulator

with geared electric drives, the parameter identification method was used. The inertia

includes the effective motor and gear inertia. The accurate modeling of friction is

generally considered as fairly difficult. In the experiment, the approximated friction


0.14 0.16 0.18 0.2 0.22 0.24 0.26 0.28 0.3 0.32 0.343500

4000

4500

5000

5500

6000

6500

7000

7500

8000

the measured torque(N.m)

The

dig

ital v

alue

of t

orqu

e

K=20670

(a)

−0.4 −0.35 −0.3 −0.25 −0.2 −0.15−8000

−7500

−7000

−6500

−6000

−5500

−5000

−4500

−4000

−3500

K=19528

(b)

Figure 5.5: The proportion between the physical torque and the commanded torque

was modeled as

Fric = Kf θ (5.14)

The basic strategy behind the estimation procedures is to estimate the unknown pa-

rameters by driving the manipulator with commanded torque. To present the general

dynamic model of the manipulator, it is assumed that (1) all links are rigid bodies and

(2) the gears are ideal in terms of elasticity and backlash, i.e., the motors are rigidly

coupled to the links, so that there are no extra degree of freedom.

The configuration space was parameterized by the end-effector position in the Carte-

sian space. The actuators joint position can be obtained by encoders. The joint position

rate was obtained by numerical differentiation. The position rate of the end-effector in

the Cartesian space was computed by Jacobian matrix. Also by numerical differentia-

tion of the position rate, the acceleration of the end-effector in the Cartesian space was

obtained. The dynamic equation (5.13) only considers the dynamics of the rigid body.

When the dynamics of the actuators is involved, the dynamic model becomes

ST τm =W T (τ + Iaq +Kmq)

n(5.15)

where n = 25 is the gear ratio, the vector τm represent the torque of the motor, τ is the

torque of the open-train mechanism, and q denotes the ambient configuration space.


The 6× 6 diagonal matrix Ia denotes the sum of mass moment of inertia of the motor

and the gear. Km is also a 6× 6 diagonal matrix revealing the sum of all friction terms

in the actuators(in the bearings and gears, inner damping of the parts, etc.), and

W =∂(θ1, θ2, θ3, ϕ1, ϕ2, ϕ3)

∂(x, y), S =

[∂(θ1, θ2, θ3)

∂(x, y)

]

Since the three motors are the same, the first three diagonal elements of Ia denoting the

motor and gear inertia of the actuators were assumed to be equal, while the other diag-

onal elements denoting the inertia of the passive joints are zero. Similarly, the diagonal

elements of Km represented the friction coefficients of the actuated joints and passive

joints respectively. The friction coefficients of the three actuators were assumed the

same, the same truth for the three passive joints. Therefore, the inertia matrix has the

form as Ia = diag(I1, I2, I3, 0, 0, 0) and the matrix Km = diag(K1,K2,K3,K4,K5,K6).

Since the velocity and acceleration in the Cartesian space were calculated by nu-

merical differentiation. Therefore, the velocity and acceleration was noisy, especially

for the acceleration. A one dimension window filter of a length nine was utilized to

filter the velocity signal. Parameterize the configuration space with the end-effector

coordinates in the Cartesian space, equation (5.15) can be written as

ST τm =W T (M + Ia)WX + (W T (M + Ia)M +W T (C +Km)M)X

n(5.16)

As mentioned above, the mass, center of mass and inertia of the rigid bodies were

known experimentally. Equation (5.16) becomes

nST τm −W TMWX + (W TMW +W TCW )X

=W T IaWX + (W T IaM +W TKmM)x (5.17)

In order to apply the least square estimation, equation (5.17) should be revised as a

linearized model in terms of the unknown parameters. The linearized equation is as


follows

nST τm −W TMWX + (W TMM +W TCM)X =

v1 v2 v3

v4 v5 v6

I1

K1

K2

(5.18)

where

X =

x

y

Let Wd denote W , we have

v1 = (W 211 +W 2

21 +W 231)x+ (W11W12 +W21W22 +W31W32)y

+(W11Wd11 +W21Wd21 +W31Wd31)x+ (W11Wd12 +W21Wd22 +W31Wd32)y

v2 = (W 211 +W 2

21 +W 231)x+ (W11W12 +W21W22 +W31W32)y

v3 = (W 241 +W 2

51 +W 261)x+ (W41W42 +W51W52 +W61W62)y

v4 = (W 212 +W 2

22 +W 232)y + (W11W12 +W21W22 +W31W32)x

+(W12Wd11 +W22Wd21 +W32Wd31)x+ (W12Wd12 +W22Wd22 +W32Wd32)y

v5 = (W 212 +W 2

22 +W 232)y + (W11W12 +W21W22 +W31W32)x

v6 = (W 242 +W 2

52 +W 262)y + (W41W42 +W51W52 +W61W62)x

With the measured parameters, the left part of equation (5.18) is known. The dynamic

model is now of the form

Y = Φχ

where χ denotes the unknown parameters. By least square estimation, the solution for

the above equation is

χ = (ΦTΦ)−1ΦTY

In the experiment, the estimation procedures used 700 samples, one every 5ms. The

estimated parameters I1,K1 and K2 are equal to 0.0289, 0.8315 and 0.0938 respectively.

I1, i.e. the inertia of the motor and the gear can be involved into the inertia matrices

of the rigid bodies. The updated parameters αi, βi and γi are shown in Table 5.3.

Table 5.3: The updated constant parameters αi, βi and γi

Links α β γ

Link1 0.0347 0.0018 0.0016

Link2 0.0338 0.0013 0.0013

Link3 0.0315 0.0003 0.0003

Chapter 6

Experiments And Results

In the research, PD control, augmented control and computed torque control methods

were implemented on the redundantly actuated parallel manipulator developed in the

robotic laboratory of Automation Technology Center, HKUST. The control algorithms

and the stability of the control laws were presented in Chapter 4. Despite voluminous

publications on the control of serial manipulators, little research has been done on

the control of parallel manipulators. Research on parallel manipulator with redundant

actuation is very scarce. Using the estimated dynamic model of the system, the different

control methods were employed not only to verify the proposed dynamic model in

Chapter 3 but also to evaluate the performance of the control algorithms.

6.1 Test motions

Two test trajectories were generated off-line to evaluate the performance of differ-

ent control algorithms. The workspace of the mechanism and the test motion in the

workspace are shown in Figure 6.1. Trajectory 1 spaned the workspace diagonally with

the displacement of 20.36mm. Trajectory 2 was along the same direction with the dis-

placement of 17.11mm. The maximum acceleration of the end-effector in the two test

motions was 7.2m/s2 and 15m/s2 respectively. Accordingly, the maximum velocity of

the end-effector was 0.216m/s and 0.3m/s. The profiles of the desired acceleration,

velocity and position of the end-effector are shown in Figure 6.2 and 6.3.

50

6.2. INDIVIDUAL JOINT TORQUE 51

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

A3

A2

1A(9.2, 6.2)(0, 6.2)

(18.4, 0)

(18.4, 12.4)

(5.9, 6.2)

(12.5, -0.2)

(12.5, 12.7)

Figure 6.1: The test motion in the workspace

6.2 Individual joint torque

For the planar manipulator, the total torque of a revolute robot joint consists of iner-

tial torque, Coriolis and centrifugal torque and friction torque. To find out how the

torque components of the redundant robot are distributed, the torque command was

partitioned into inertia, Coriolis and centrifugal and friction terms. The torque distri-

butions for the test motions with acceleration values 7.2m/s2 and 15.0m/s2 are given

in Figure 6.4 and 6.5. It can be seen that the terms of Coriolis and centrifugal torque

is insignificant. When the velocity values of the actuated joints are constant, the joint

torque is dominant by the friction torque. In the test motion 1, the joint velocity of the

actuated joint 1 is small. The stiction was compensated by 0.02N.m for joint 1. Also,

in the test motion 2, the stiction was compensated by 0.02N.m for joint 1. The value

of static is obtained experimentally, which is the least torque to drive the motor.

6.3 Control algorithms

The following four control methods were employed to tracking the two test trajectories.

6.4. EXPERIMENTAL RESULTS 52

1. PD controller with joint position and velocity reference of the actuators:

τc = kp(qda − qa) + kv(qda − qa)

2. PD controller with position and velocity reference of the end-effector:

τc = (ST )+(kp(Xd −X) + kv(Xd − X))

3. Augmented PD controller, where PD controller is with position and velocity ref-

erence of the end-effector:

τc = (ST )+( ˆM(q)qdin + C(q, q)qdin + kp(Xd −X) + kv(Xd − X))

4. Based on the analysis in the former section, the terms of Coriolis and centrifugal

torque was insignificant. The computed torque control law can be simplified as:

τc = (ST )+(M (q)(qdin + kp(Xd −X) + kv(Xd − X))

6.4 Experimental Results

In this section, some preliminary results are presented for the four control algorithms

implemented on the redundantly actuated parallel manipulator, i.e. PD controllers,

augmented PD controller and computed torque controller. The program was written in

C language. The sampling time was 4ms for the controllers. The trajectories presented

in the previous section were used for this experiment. The tracking errors of the four

controllers in Cartesian space for test motion 1 and 2 are shown in Figure 6.6. The

tracking errors in the joint space for test motion 1 and 2 are shown in Figure 6.7 and

6.8. The torques of the actuated joints in test motions 1 and 2 are shown in Figure 6.9

and 6.10 respectively.

6.5 Discussion

The experimental results of the four types of control algorithms were presented above.

To discuss the two kinds of PD controllers conveniently, the PD control with position

6.5. DISCUSSION 53

and velocity reference of the actuators is called as decoupled PD control, and the other

PD control with the position and velocity reference of the end-effector is called as

coupled PD control. The performance of the four control methods will be evaluated by

comparing the tracking accuracy of the end-effector. By analyzing the torques of the

actuators of each control strategy, the accuracy of the estimated dynamic model and its

effect on the model based control will be discussed. The tracking accuracy was evaluated

by the root square mean error(RSME). RSME=√

Err2

n . The experimental results for

test motion 1 and test motion 2 are presented in Table 6.1 and 6.2 respectively.

Table 6.1: The performance of PD contorl, augmented PD control and computed torque

control in test motion 1 with acceleration 7.2m/s2

Controller RSME(Pos./Vel.) Kp/Kv Running time(ms)

Decoupled PD 1.2285/0.0753 18/0.16 2.71

Coupled PD control 1.5021/0.0565 28000/700 2.81

Augmented PD 1.0447/0.0471 8500/400 2.81

Computed torque 0.9592/0.0436 3000/120 3.18

Table 6.2: The performance of PD control, augmented PD control and computed torque

control in test motion 2 with acceleration 15.0m/s2

Controller RSME(Pos./Vel.) Kp/Kv Running time(ms)

Decoupled PD 1.9785/0.1323 32/0.22 2.71

Coupled PD 1.7220/0.1024 38000/1200 2.81

Augmented PD 0.7143/0.0606 14000/700 2.81

Computed torque 0.9778/0.0547 6000/300 3.18

The experimental results and the analysis described above showed that computed

torque control and augmented PD control performed similarly in regard to the tracking

6.5. DISCUSSION 54

accuracy. Decoupled PD control and coupled PD control presented nearly the same

performance. The model based control, i.e. augmented PD control and computed

torque control improved the tracking accuracy compared with the simple PD control

method. Especially, the improvement was greater in the high speed motion.

One goal of these experiments was to test the estimated dynamical model of the

system for control purposes. The input torques recorded in the real-time control of

each control method were compared with the desired actuated joint torques. The

results showed that the model was accurate when the joint velocity is not constant.

While when the joint velocity of the actuators is constant, there is much difference

between the real-time computed torques and the desired torques. It is known that

the joint torque is dominanted by the friction torque, during the joint velocity of the

actuators is constant. Therefore, it may be claimed that the simplified friction model

is insufficient. The estimated inertia parameters of the motors and gears had certain

degree of accuracy.

Another result is that the torques required by the actuators in the decoupled PD

control was larger than those of coupled PD control. The reason for this is because

the coupled PD control employs actuation redundancy. The strategy to minimize two

norm of the input torque reduces the value of the input torques. Decoupled PD control

is only dependent on the position and velocity errors of the actuators. Hence, coupled

PD control is more energy efficient than decoupled PD control.

0 5 10 15 20 25 30 35 40 45 50−8

−6

−4

−2

0

2

4

6

8

Sampletime(4ms)

Acc

lera

tion

(m/s

2 )

Desired End−effector Acceleration along X,Y Axis

(a)

0 5 10 15 20 25 30 35 40 45 50−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

Sampletime(4ms)

velo

city

(m

/s)

Desired End−effector Velocity along X,Y Axis

(b)

0 5 10 15 20 25 30 35 40 45 500.092

0.094

0.096

0.098

0.1

0.102

0.104

0.106

Sampletime(4ms)

Pos

ition

(m

)

Desired End−effector Position along X Axis

(c)

0 5 10 15 20 25 30 35 40 45 500.062

0.064

0.066

0.068

0.07

0.072

0.074

0.076

Sampletime(4ms)

Pos

ition

(m

)

Desired End−effector Position along Y Axis

(d)

0 5 10 15 20 25 30 35 40 45 50−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

Sampletime(4ms)

Vel

ocity

of A

ctua

ted

Join

ts (

radi

us/s

)

Velocity of Actuated Joints

Joint 1Joint 2Joint 3

(e)

0 5 10 15 20 25 30 35 40 45 50−100

−80

−60

−40

−20

0

20

40

60

80

100

Sampletime(4ms)

Acc

eler

atio

n of

Act

uate

d Jo

ints

(ra

dius

/s2 )

Acceleration of Actuated Joints


(f)

Figure 6.2: The profiles of desired acceleration, velocity and position for test motion 1

0 5 10 15 20 25 30 35 40−15

−10

−5

0

5

10

15

Sampletime(4ms)

Acc

lera

tion

(m/s

2 )

Desired End−effector Acceleration along X,Y Axis

(a)

0 5 10 15 20 25 30 35 40−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

Sampletime(4ms)

velo

city

(m

/s)

Desired End−effector Velocity along X,Y Axis

(b)

0 5 10 15 20 25 30 35 400.092

0.094

0.096

0.098

0.1

0.102

0.104

0.106

0.108

Sampletime(4ms)

Pos

ition

(m

)

Desired End−effector Position along X Axis

(c)

0 5 10 15 20 25 30 35 400.062

0.064

0.066

0.068

0.07

0.072

0.074

0.076

0.078

Sampletime(4ms)

Pos

ition

(m

)

Desired End−effector Position along Y Axis

(d)

0 5 10 15 20 25 30 35 40−4

−3

−2

−1

0

1

2

3

4

Sampletime(4ms)

Vel

ocity

of A

ctua

ted

Join

ts (

radi

us/s

)

Velocity of Actuated Joints


(e)

0 5 10 15 20 25 30 35 40−200

−150

−100

−50

0

50

100

150

200

250

Sampletime(4ms)

Acc

eler

atio

n of

Act

uate

d Jo

ints

(ra

dius

/s2 )

Acceleration of Actuated Joints


(f)

Figure 6.3: The profiles of desired acceleration, velocity and position for test motion 2

0 5 10 15 20 25 30 35 40 45 50−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

Sampletime(4ms)

Tor

que(

Nm

)

TotalInertiaCoriolis & centrifugalFrictionStiction

(a)

0 5 10 15 20 25 30 35 40 45 50−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

Sampletime(4ms)

Tor

que(

Nm

)

TotalInertiaCoriolis & centrifugalFriction

(b)

0 5 10 15 20 25 30 35 40 45 50−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

Sampletime(4ms)

Tor

que(

Nm

)


(c)

Figure 6.4: The torque components of the actuators with an end-effector acceleration

of 7.2m/s2

0 5 10 15 20 25 30 35 40−0.1

−0.05

0

0.05

0.1

0.15

Sampletime(4ms)

Tor

que(

Nm

)

TotalInertiaCoriolis & centrifugalFrictionStiction

(a)

0 5 10 15 20 25 30 35 40−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

Sampletime(4ms)

Tor

que(

Nm

)


0 5 10 15 20 25 30 35 40−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

Sampletime(4ms)

Tor

que(

Nm

)


(b)

0 5 10 15 20 25 30 35 40−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

Sampletime(4ms)

Tor

que(

Nm

)


(c)

Figure 6.5: The torque components of the actuators with an end-effector acceleration

of 15.0m/s2

0 5 10 15 20 25 30 35 40 45 500

0.5

1

1.5

2

2.5

time(ms)

mm

Position error of end effector

Coupled PDAugmented PDCompute torqueUncoupled PD

(a)

0 5 10 15 20 25 30 35 40 45 500

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

time(ms)

m/s

Velocity error of end effector


(b)

0 5 10 15 20 25 30 35 400

0.5

1

1.5

2

2.5

3

3.5

4

time(ms)

mm

Position error of end effector


(c)

0 5 10 15 20 25 30 35 400

0.05

0.1

0.15

0.2

0.25

time(ms)

m/s

Velocity error of end effector


(d)

Figure 6.6: Tracking errors of the end-effector for test motion 1 and 2

0 5 10 15 20 25 30 35 40 45 50−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

time(ms)

Pos

ition

Err

or o

f Joi

nt 1

(Deg

ree)

Position Error of Joint 1


(a)

0 5 10 15 20 25 30 35 40 45 50−1

−0.5

0

0.5

1

1.5Position Error of Joint 2

time(ms)

Pos

ition

err

or o

f Joi

nt 2

(Deg

ree)


(b)

0 5 10 15 20 25 30 35 40 45 50−2

−1.5

−1

−0.5

0

0.5

1

1.5

2Position Error of Joint 3

time(ms)

Pos

itoin

Err

or o

f Joi

nt 3

(Deg

ree)


(c)

Figure 6.7: Actuated joint position error for test motion 1

0 5 10 15 20 25 30 35 40−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

time(ms)

Pos

ition

Err

or o

f Joi

nt 1

(Deg

ree)

Position Error of Joint 1


(a)

0 5 10 15 20 25 30 35 40−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5Position Error of Joint 2

time(ms)

Pos

ition

err

or o

f Joi

nt 2

(Deg

ree)


(b)

0 5 10 15 20 25 30 35 40−2

−1.5

−1

−0.5

0

0.5

1

1.5

2Position Error of Joint 3

time(ms)

Pos

itoin

Err

or o

f Joi

nt 3

(Deg

ree)


(c)

Figure 6.8: Actuated joint position error for test motion 2

0 5 10 15 20 25 30 35 40 45 50−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25Torque of Joint1

time(ms)

N.m

Desired TorqueCoupled PDAugmented PDCompute torqueUncoupled PD

(a)

0 5 10 15 20 25 30 35 40 45 50−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3Torque of Joint2

time(ms)

N.m


(b)

0 5 10 15 20 25 30 35 40 45 50−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3Torque of Joint3

time(ms)

N.m


(c)

Figure 6.9: Actuated joint torque for test motion 1

0 5 10 15 20 25 30 35 40−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15Torque of Joint1

time(ms)

N.m


(a)

0 5 10 15 20 25 30 35 40−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4Torque of Joint2

time(ms)

N.m


(b)

0 5 10 15 20 25 30 35 40−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5Torque of Joint3

time(ms)

N.m


(c)

Figure 6.10: Actuated joint torque for test motion 2

Chapter 7

Conclusions and future work

7.1 conclusions

In my research I investigated the dynamics and control of redundantly actuated parallel

manipulators. Redundant actuation has been employed successfully in serial robots

to achieve some tasks. It has been introduced recently into parallel manipulators to

improve their performance. Unlike the work done on serial robots, the research on the

dynamics and control of parallel manipulators is much less. Therefore, few results have

been presented in regard to the dynamics and control of redundantly actuated parallel

manipulators. In the thesis, the advantages of redundant actuation was presented to

show its desirability. Then a new method to compute the inverse dynamics of redundant

manipulators was proposed using Lagrange-D’Alembert’s principle. Several control

schemes designed for serial robots were utilized in redundant parallel manipulators.

The stability of these control schemes was also discussed. As an important part of

the research, a planar two DOF parallel manipulator with redundant actuation was

developed as a test bed. The dynamic equations of the mechanism were formulated

using the proposed dynamic model presented in Chapter 3. To verify the estimated

dynamic model and to investigate the control of the redundant parallel manipulators,

the proposed control schemes discussed in Chapter 4 were implemented experimentally

on the planar parallel manipulator.

64

7.1. CONCLUSIONS 65

Redundant actuation has been accepted as an efficient approach to remove singular-

ities of parallel manipulators. But the rigorous investigation is little. Two explanations

were presented to discuss how redundant actuation can remove singularities within the

workspace. The simulation results showed that redundant actuation had the advantages

of improving Cartesian stiffness and achieving uniform output forces.

The dynamics of redundantly actuated closed-chain mechanisms is based on the

previous work on non-redundant closed-chain mechanisms. However, the methods can

not be applied directly to redundant ones. Three main method computing inverse

dynamics of redundantly actuated closed-chain mechanisms were presented. The new

scheme proposed in the thesis is computationally efficient than the previous schemes.

It was shown that the dynamics of redundantly actuated closed-chain mechanisms has

the similar structure to that of serial ones.

The complete dynamics of the rigid links was included in modeling the redundant

closed-chian system. The unknown parameters of the system were obtained both by

experimental measurement, and through the identification method. By analyzing the

input torques recorded in real-time control, it was shown that the estimated models

were accurate except that the joint velocity was constant. When the joint velocity was

constant, the simulation showed that the total torque was dominated by the friction

torque. Consequently, the error of the estimated model mainly was mainly because: (1)

The simplified model of the friction torque was insufficient, (2) The nonlinear factors

introduced by the motor and the gear boxes were neglected. But it was shown the

estimated inertia of the motors and the gears reduction presented certain degree of

accuracy.

The model-based control schemes performed better than simple PD control. The

improvement was more obvious as the speed increased. It was showed that the advanced

control schemes should be employed in high speed applications. Decoupled PD control

and coupled PD control had the similar performance. However, coupled PD control

7.2. FUTURE WORK 66

was more energy efficient than decoupled PD control.

7.2 Future work

Dynamic modeling and controlling are two fundamental and critical issues related to

the research of parallel manipulators with redundant actuation. The work done on this

thesis was a valuable attempt. The future work involves

1. Further investigation into why redundant actuation can remove the singularities

within the workspace should be done. Modern mathematical tools should be

employed to reveal the nature of the problem.

2. Identifying the dynamic model of a system is a highly difficult and challenging

topic, even for serial robots. It is necessary to develop more efficient methods to

identify the dynamic model of redundantly actuated parallel manipulators.

3. The performance of model based control relies on an accurate model of a sys-

tem. However, identifying the accurate dynamic model of a system is very diffi-

cult. Therefore, effective controllers for the versatile application of parallel robots

should be developed. Adaptive control has the potential to improve the track-

ing accuracy, for it updates the unknown parameters online. Adaptive control

algorithm is too complicated to be utilized in high-speed applications. In such

applications, robust independent joint control is a prospective method to improve

the performance of simple PD control.

4. The redundant closed-chain mechanism developed in the project is a planar ma-

nipulator. Spatial mechanisms are expected to be constructed to meet the prac-

tical requirements.

Publications

1. Y.K. Yiu, H. Cheng, Z.H. Xiong, G.F. Liu and Z.X. Li, ”On the Dynamics

of Parallel Manipulators”, IEEE Int. Conf. Robotics and Automation (ICRA),

Shoul, Korea, May 2001.

2. H. Cheng, Y.L. Wu, Z.H. Xiong , Y.K. Yiu and Z.X. Li, ”Analysis and Control

of Parallel Manipulators with Redundant Actuation”, SCI2001, Orlando, U.S.A.,

July 2001.

3. H. Cheng, G.F. Liu, Y.K. Yiu, Z.H. Xiong and Z.X. Li, ”Advantage and Dynam-

ics of Redundanly Actuated Parallel Manipulators”, IROS, U.S.A., Oct. 2001.

(submitted)

4. G.F. Liu, H. Cheng, X.Z. Wu and Z.X. Li, ”Distribution of Singularity and

Optimal Control of Redundant Parallel Manipulators”, IROS, U.S.A., Oct. 2001.

(submitted)

67

Appendix A

Motor Specifications

Table A.1: Specifications of the servo motor

Motor Sanyo Denki P50B05010D

Rated output 100 W

Maximum speed 4500 RPM

Rated torque 0.319 N.m

Continous stall torque 0.353 N.m

Instantaneous maximum stall torque 0.98

Incremental encoder revolution 2000 P/R

Absolute encoder revolution 8192 P/R

Inertia(including ABS R II) 0.098 × 10−4Kg.m2

Power supply 200 VAC

68

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