Dynamic Modeling and Control of a 6

DOF Parallel

Kinematic

Mechanism

Based Reconfigurable meso

Milling
Machine Tool
by
Adam
Yi Le
A thesis submitted in conformity with the requirements
for
the degree of Master of Applied Science
Department of Mechanical and Industrial Engineering
University of Toronto
©
Copyright by
Adam
Yi Le 2012
ii
Dyna
mic Modeling and Control of a 6

DOF Parallel

Kinematic

Mechanism

Based Reconfigurable meso

Milling Ma
chine Tool
Adam
Yi Le
Master of Applied Science
Department of Mechanical and Industrial Engineering
University of Toronto
2012
Abstract
In this thesis, a methodology for rigid body dynamic modeling and control design is
presented for a 6
degree

of

freedom
(DOF)
parallel

kinematic

mechanism

based reconfigurable
meso

milling machine tool
(RmMT)
with submicron tracking accuracy requirement
. The
dynamic modeling of the parallel kinematic mechanism
(PKM)
is
formulated using the
Lagrangian method with the applic
ation of
principle of energy equivalence and coordinate
transformations to separate the mechanism into serial sub

systems. The rigid body gyroscopic
force is also modeled using this approach and its effect as a disturbance is analyzed and
compensated.
The
contour errors for both position and orientation are formulated t
o increase
machin
ing accuracy. The
dynamic model of the system
is linearized through feedback
linearization and the
contour error
based
feedback
control law
is formulated using the convex
combination design approach to satisfy a set of design specifications simultaneously. The
dynamic model and its control
methodology are simulated and verified
within the
MATLAB
Simulink environment.
iii
Acknowledgments
I would
like to express my sincere appreciation to my supervisors, Professor J. K. Mills
and Professor B. Benhabib, for giving me the opportunity to participate in this research project as
well as providing continuous
guidance,
encouragement,
and
support
.
I wou
ld like to thank my colleagues and friends from the Laboratory for Nonlinear
Systems Control and the Computer Integrated Manufacturing Lab for their assistance
:
Mr. Masih
Mahmoodi, Mr. Hay Azulay,
Dr. Henry Chu, Mr. Stephen Haley,
Mr. Shael Markin,
Mr. Lu
Cong,
Mr. David Schacter,
Mr. Ashish Macwan, and
Dr. Matthew Mackay
.
I have learned
greatly from them and t
hey are a
continuous
source o
f inspiration and encouragement.
I would also like to thank Prof. Steven Waslander
at the University of Waterloo for his
enthusiasm and support.
I greatly appreciate the encouragement and motivation that he provided
me
to pursue the field of mechatronics
and controls.
I would also like to acknowledge NSERC for providing the funding to
make the
CANRIMT
project and my research possible.
Finally, I would like to express my deepest gratitude to my family for their
never ending
support, encouragement and patience.
iv
Table of Contents
Ac
knowledgments
................................
................................
................................
..........................
iii
Table of Contents
................................
................................
................................
...........................
iv
List of Tables
................................
................................
................................
................................
vii
List of Figures
................................
................................
................................
..............................
viii
List of Appendices
................................
................................
................................
..........................
x
List of Nomenclatures
................................
................................
................................
....................
xi
Chapter 1
Introduction
................................
................................
................................
...............
1
1.1
Thesis Scope and Background
................................
................................
.........................
1
1.2
Thesis Objectives and
Contributions
................................
................................
................
4
1.3
Thesis Outline
................................
................................
................................
..................
5
Chapter 2
Kinematic Behaviour of Parallel Kinematic Mechanism
................................
.........
6
2.1
System Description
................................
................................
................................
..........
6
2.2
Kinematic Equations
................................
................................
................................
........
7
2.3
Kinematic Behaviour of the PKM
................................
................................
..................
1
0
2.3.1
Work

space Definition
................................
................................
............................
10
2.3.2
Kinematic Singularity
................................
................................
.............................
12
2.4
Summary
................................
................................
................................
........................
20
Chapter 3
Rigid Body
Dynamic Modeling
................................
................................
..............
21
3.1
Literature Review
................................
................................
................................
...........
22
3.2
Underlying Principles for Dynamic Modeling
................................
...............................
24
3.2.1
Principle of Energy Equivalence
................................
................................
............
25
3.2.2
Coordinate Transformation
................................
................................
.....................
26
3.3
Dynamic Modeling Methodology
................................
................................
..................
28
3.3.1
Division of the PKM
................................
................................
...............................
28
v
3.3.2
Kinetic and Potential Energies of Serial Chains
................................
.....................
31
3.3.3
Dynamic Equation in the Standard Form
................................
................................
36
3.4
Simulation Development
................................
................................
................................
39
3.5
Summary
................................
................................
................................
........................
42
Chapter 4
Dynamic Mo
deling of Rigid Spindle Gyroscopic Forces
................................
.......
43
4.1
Introduction
................................
................................
................................
....................
43
4.2
Spindle Dynamic Modeling Methodology
................................
................................
.....
44
4.3
Spindle Simulation
................................
................................
................................
.........
48
4.3.1
Rotor Assembly Design
................................
................................
..........................
48
4.3.2
Simulation Trajectory
................................
................................
.............................
51
4.3.
3
Simulation Results
................................
................................
................................
..
52
4.4
Summary
................................
................................
................................
........................
56
Chapter 5
Contour Error Formulation
................................
................................
.....................
57
5.1
Literature Review
................................
................................
................................
...........
58
5.2
Translational Contour Error Formulation
................................
................................
......
61
5.2.1
Tangential Approximation Approach
................................
................................
.....
61
5.2.2
Circular Approximation Approach
................................
................................
.........
63
5.2.3
Implementation and Simulation
................................
................................
..............
66
5.3
Orientation Contour Error Formulation
................................
................................
.........
71
5.3.1
Tangential Approach
................................
................................
...............................
71
5.3.2
Implementation Issues
................................
................................
............................
75
5.4
Summary
................................
................................
................................
........................
76
Chapter 6
Rigid Body Control Design
................................
................................
....................
77
6.1
Control Strategy Outline
................................
................................
................................
77
6.2
Feedback Linearization
................................
................................
................................
..
78
6.3
Design Specification
................................
................................
................................
......
80
vi
6.3.1
Definitions
................................
................................
................................
...............
80
6.3.2
Robust S
pecification Formulation
................................
................................
..........
82
6.4
Convex Combination Control Design
................................
................................
............
85
6.4.1
System Framework
................................
................................
................................
.
85
6.4.2
Desi
gn Procedure
................................
................................
................................
....
87
6.5
Simulations
................................
................................
................................
.....................
89
6.6
Summary
................................
................................
................................
........................
91
Chapter 7
Conclusions and Recommendations
................................
................................
.......
93
7.1
Conclusions
................................
................................
................................
....................
93
7.2
Recommendations
................................
................................
................................
..........
94
References
................................
................................
................................
................................
.....
96
Appendix A
................................
................................
................................
................................
.
101
vii
List of Tables
Table 4

1 Mass Property of Representative Spindle
................................
................................
.....
50
Table 5

1 Sample Contour/Lag Error Controllers with Weighting Adjustment
...........................
70
Table 6

1 List of Design Specifications
................................
................................
........................
81
Table 6

2 Sample
Controller Design
................................
................................
............................
89
Table 6

3 Design Specifications Results
................................
................................
......................
90
Table 6

4 Performance of Convex Combination Controller
................................
.........................
91
viii
List o
f Figures
Figure 2

1 3 PPRS PKM Structure
................................
................................
............................
6
Figure 2

2 Nominal
Positions and Sign Conventions of Joint Variables
................................
.
8
Figure 2

3 Platform and Tool
................................
................................
................................
...
9
Figure 2

4
Extended Workspace
................................
................................
.............................
11
Figure 2

5
Limited Workspace
................................
................................
...............................
11
Figure 2

6 Singularity Scenario 1
................................
................................
...........................
13
Figure 2

7 Singularity Scenario 2
................................
................................
...........................
14
Figure 2

8 Rotation of Tool Platform from 0° to

90° in Euler Angle
................................
15
Figure 2

9 Singular Pose from Figure 2

10
................................
................................
............
16
Figure 2

10 Rotation of Tool Platform from 0° to 90° in Euler Angle
β
...............................
17
Figure 2

11 Front and Top View of Singular Pose in Limited Workspace
............................
18
Figure 2

12 Isometric View of Singular Pose in Limited Workspace
................................
....
18
Figure 2

13
Singularities in Limited Workspace with Inclination Angle from 0° to 90°
......
19
Figure 3

1
Stewart Gough Platform
................................
................................
........................
21
Figure 3

2
Division Method 1: Intuitive division of the PKM into 4 subsystems
..................
29
Figure 3

3 Coordinate Transformation Strategy for Link Division Method 1
.......................
30
Figure 3

4 Division Method 2: Isolate Passive Links
................................
.............................
30
Figure 3

5
Coordinate Transfor
mation Strategy for Link Division Method 2
.......................
31
Figure 3

6 Tool Spindle Casing and Platform
................................
................................
........
33
Figure 3

7 Dynamic Motion of the Passive Link
................................
................................
....
34
Figure 3

8
Block Diagram of Real RmMT System
................................
................................
39
Figure 3

9
Block Diagram of Simulink RmMT System
................................
.........................
40
Figure 3

10
MATLAB Simulink Block Diagram
................................
................................
...
42
ix
Figure 4

1
Spindle and Tool Assembly
................................
................................
..................
43
Figure 4

2
NSK Dental Drill [26]
................................
................................
...........................
49
Figure 4

3
Representative Spindle Dimensions
................................
................................
......
50
Figure 4

4
Simulation Trajectory
................................
................................
............................
51
Figure 4

5 Tool Velocity
................................
................................
................................
........
51
Figure 4

6
Position Error Induced by Gyroscopic Forces
................................
......................
52
Figure 4

7 Position Error
................................
................................
................................
........
53
Figure 4

8
Gyroscopic Forces vs. Controller Torque Output in Magnitude
...........................
53
Figure 4

9
Effect of Spindle Speed on Position Error
................................
............................
54
Figure 4

10
Effect o
f Spindle Mass on Position Error
................................
...........................
55
Figure 4

11
Effect of Tool Cutting Speed on Position Error
................................
..................
56
Figure 5

1 Contour Error Definition
................................
................................
.......................
57
Figure 5

2
Contour Error vs. Cartesian Tracking Error
................................
..........................
58
Figure 5

3
Tangential Estimate of Contour Error
................................
................................
...
59
Figure 5

4
Contour Error Control Structure [39]
................................
................................
....
60
Figure 5

5 Spatial Tangential Approximation of Contour Error
................................
............
62
Figure 5

6
The Circular Approximation and Computation of Vector
.............................
64
Figure 5

7
T
angential and Circular Contour Error Performance Comparison
.......................
68
Figure 5

8
Comparison between Axial Tracking Control and Contour/Lag Control
.............
69
Figure 5

9 Contour/Lag Error Control Performance
................................
..............................
71
Figure 5

10
Tangential Approach for Orientation Contour Error
................................
..........
73
Figure 5

11
Input Output Relationship between
and
................................
................
75
Figure 6

1
Uniform Framework
................................
................................
..............................
86
Figure 6

2
Bode Plot for Response to 1N Magnitude Exogenous Disturbance
.....................
92
x
List of Appendices
Appendix A
Parameters of the RmMT used in MATLAB
Simulink…………………………..101
xi
List of Nomenclatures
Latin Letters
State matrix in state space formulation
Partial derivative of kinematics w.r.t.
Acceleration
Limiting constant in a design specification
Lie bracket of
Input matrix in state space formulation
Partial derivative of kinematics w.r.t.
Binormal unit vector in the Frenet frame
Binormal unit vector in the orientation Frenet frame
Coriolis and centrifugal force matrix
Coriolis and
centrifugal force matrix with uncertainty terms
Gyroscopic forces of spindle rotor assembly
Compact notation for expression
Center of curvature
Elements of the Coriolis and
centrifugal force matrix
Design specification
Compact notation for expression
Vertical prismatic joint variable
Moving reference frame at tool tip
Cartesian tracking error
Orientation error vector
Euler angular tracking error
Frenet frame
Applied Force
Orientation Frenet frame
xii
New Frenet frame rotated by
about
General state equations in state model
Gravitational
force vector
Gravitational force vector with uncertainty terms
General input equations in state model
Transformation matrix between angular velocity and Euler
angle derivatives
Closed loop transfer matrix
Convex combined closed loop
transfer matrix
Set of all closed loop transfer matrices
Moment of inertia of link
Moment of inertia of platform
Inertia tensor about y and z axis in the local frame
Identity matrix
Jacobian matrix between
,
Pseudo

Jacobian matrix between
,
Orientation Jacobian matrix
Reference orientation Jacobian matrix
Sample controller
Convex combined controller
Diagonal derivative gain matrix
Diagonal proportional
gain matrix
Lie derivative of
w.r.t.
Length of passive link
Length of spindle rotor assembly
Distance between tool tip and tool platform
The Lagrangian
Mass and inertia matrix
xiii
Mass and inertia
matrix with uncertainty terms
Mass of circular prismatic actuator
Mass of passive link
Mass matrix of passive link
Mass of platform
Mass of spindle rotor assembly
Mass of vertical prismatic actuator
Mass of vertical guide rail
Skew symmetric matrix
Normal unit vector in the Frenet frame
Normal unit vector in the orientation Frenet frame
Stationary Cartesian frame
Orientation vector of the tool
Reference
orientation of the tool
Plant transfer function matrix
A function to satisfy condition 2 for the existence of solution
to
Moving reference frame at centre of tool platform
Moving reference frame at
centre of mass of passive link
Generalized force
Active joint space generalized coordinates
7 DOF joint space generalized coordinates with spindle
rotation
Passive and active joint variables for sub

chains
Passive joint variables
Compact notation for the expression
Convex combined expression
Radius of PKM circular base
xiv
ZYZ Euler angle rotation
matrix
Element ro
tation matrix about
Radius of tool platform
Radius of workspace
Radius of a circular trajectory
Kinetic energy
Total kinetic energy of actuator sub

chain
Kinetic energy of circular prismatic actuator
Kinetic energy of passive link
Kinetic energy of tool platform
Kinetic energy of spindle rotor assembly
Kinetic energy of vertical prismatic actuator
Kinetic
energy of vertical guide rail
Current time
A time instant in the past
Tangential unit vector of the Frenet frame
Tangential unit vector of the orientation Frenet frame
Difference between
and
Linear control law
Controller input vector in the uniform framework
Potential energy
Total potential energy of actuator sub

chain
Potential energy of passive link
Potential energy of platform
Potential energy of spindle rotor assembly
Potential energy of vertical prismatic actuator
Velocity
Velocity of passive link
xv
Greek Letters
Magnitude of tool tip velocity
Exogenous input vector in the uniform framework
Position of circular prismatic joint
Position of passive spherical joint
Position of vertical prismatic joint
Task space generalized coordinates of the tool tip
Position of the centre of tool platform in
Task space reference trajectory
Position of the center of mass of spindle rotor assembly
Position of the tool tip
Euler angle
orientation of the tool tip
7 DOF task space generalized coordinates with spindle
rotation
6 DOF task space generalized coordinates with spindle
rotation
Tas
k space generalized coordinates with
牥灬pc楮i
Sensed output vector in the uniform framework
Feedback linearized state variables
Regulated output vector in the uniform framework
Azimuth angle of workspace
Inclination angle of workspace
Compact notation for expression
Compact notation for expression
Virtual displacement
Bounded maximum limit for
xvi
Bounded minimum limit for
Bounded maximum limit for
Bounded maximum limit for
Bounded maximum limit for
Experimental Constants in forming a bounded limit for
Position contour error
Position contour error
estimate
Position lag error
Position lag error estimate
Orientation error in orientation Frenet frame
Orientation contour error
Orientation contour error estimate
Orientation lag error estimate
Position error in the Frenet frame
Sum of rotation of spindle variable
and Euler angle
Elements of the mass and inertia matrix
Angle between
and
Circular prismatic joint variable
Angle of reference trajectory w.r.t. horizontal axis
Example generalized coordinates
Vector of coefficients for convex combination
Real number
Rotation of the spindle rotor assembly
Radius of curvature
Generalized dynamic forces
xvii
Dynamic forces of serial sub

chains
Nonlinear compensation terms from feedback linearization
Dynamic forces of the platform
Active joint space generalized forces
Task space
generalized forces
General disturbance forces
Unit vector in the direction between
and
Square matrix containing closed loop system performances
corresponding to each design specification for each sample
controller
Functional on a transfer matrix
Specification: Euler angle tracking
Specification: Euler angle derivative tracking
Passive revolute joint variable
Specification: robust
Specification: task space actuation
effort
Specification:
Specification: contour velocity tracking
Specification: lag position tracking
Specification: lag velocity tracking
Vector of design specification constants
Angular velocity of passive link
Angular velocity of platform
Angular velocity of spindle rotor assembly
1
Chapter 1
Introduction
1.1
Thesis Scope and
Background
In the past decade,
meso

scale components (0.5 to 5.0 mm in size) has flourished in a
wide range of applications such as medical, electronics, automotive, optics and avionics.
However, the research efforts of the machine
tool
design industry have remained focused on the
manuf
acturing of conventional macro sized components and nano/micro sized
micro

electromechanical systems (
MEMS
). Of the manufacturing methods currently employed in
machining meso

scaled workpieces, some are processes intended for MEMS and others involve
downsi
zing, to a certain extent, traditional machine tools. However, the capabilities of these
processes are limited in terms of geometry and material of the meso

scale workpiece.
The
primary motivation for this project is to design a Reconfigurable meso

Milling
Machine Tool
System (RmMT) aimed specifically at meso

scale workpieces and fills the gap between
manufacturing of nano/micro and macro components. The RmMT in development will seek to
have a 5

axis configuration with a parallel

kinematic

mechanism
(PKM) b
ased
structure capable
of producing 3D sculptured meso

scale workpieces with positioning accuracies of less than
0.1
µ
m
[1]
.
The synthesis of the RmMT
is
modularized
into smaller design projects focusing
several
key aspect of the design:
mechanical structure, sensors, actuators, controllers and workholding
tools.
The
objective
of this thesis is the
development
of
rigid body dynamic model of the PKM
structure and
the f
ormulation of
closed

loop
control based on the dynamic model.
The primary challenges of the project are to construct the dynamic model in either task
space or joint space suitable for the purpose of control and also to meet the high accuracy
requirement o
f the RmMT from the rigid body control perspective. Two further topics relevant to
machine tool control are analyzed within this thesis in order to improve the accuracy
performance.
First the contour errors
for both position and orientation are formulated
and the
control algorithm is constructed based on these errors as oppose to Cartesian tracking errors. The
2
spindle gyroscopic forces induced by the spindle rotor assembly as a disturbance force is also
modeled and analyzed.
Since the design
and construction
of the RmMT machine
prototype
is a
separate
and
concurrent project, there is no available test

bed for conducting physical experiments
. As a
result, there are no means to obtain information such as the variation of parameters between the
d
ynamic model and the physical system
. Thus
,
the thesis emphasizes
the general methodology of
dynamic modeling and control
development. This design methodology, which is applied to a
design of a preliminary representative PKM design
and simulated within MAT
LAB Simulink
,
will also be applicable to a larger family of PKM
configurations
, including the final
design for
the RmMT
.
The PKM structure is rarely seen in
the machine tool industry
. The most popular form of
PKM is the
Stewart

G
ough mechanism and most
pa
pers
discussing PKMs inevitably
refer
to it
in
order to demonstrate and
to verify their analysis. However, PKM configurations can vary greatly.
Their
closed looped chains
provide
better structure stiffness and accuracy
when compared to
serial kinematic mec
hanisms (SKM)
. However, the cross coupled nature of these closed looped
chains makes dynamic modeling of PKMs much
more complicated
than
SKMs
[1]
.
The primary
source
of this complexity is the existence of passive joint variables that cannot be described
explicitly in terms of active joint variables
[2]
.
In this thesis, the Principle of Energy Equivalence
is introduced to divide the PKM into serial elements, where upon the Lagrangian dynamic
formulation can be directly applied similar to SKM
s. The dynamics of the serial elements are
summed together to form the total PKM dynamic model. This methodology proves to be
straightforward and procedural and can be applied to a wide range of PKM configurations.
The PKM, when applied in robotic applicat
ions such as gauging, experiences little
external dynamic forces. The same cannot be said for machining operations. The primary
external force experienced by the PKM is the cutting force.
While this meso

milling cutting
force
must be compensated within the
machine control algorithm,
it is an ongoing research
subject of its own and
will not be discussed in
this thesis. Another external force
that
is
often
neglected in conventional machines is the rigid body gyroscopic forces from the spindle
rotor
.
3
This is b
ecause in conventional machines, the spindle speed is relatively slow and the force
output from its linear stages is
large enough that
the gyroscopic forces are
relatively
insignificant. As a result,
there is
no
apparent
literature discussing the mode
ling
and analysis of
rigid body gyroscopic forces for machining applications
.
Due to a combination of high accuracy
requirement and high
spindle speed
(in the range of
150
,000
to 350
,000
rpm
for the RmMT),
the
effect of
gyroscopic forces
may be a significant en
ough disturbance to warrant direct
compensation.
For a machining operation, it has been recognized that
minimizing
tracking errors along
the orthogonal Cartesian coordinates may not deliver the
most accurate part finish
. Koren
[3]
proposes that minimizing the contour error, defined as the normal error between the position of
the tool and the reference is of higher importance.
While there have been efforts at determ
ining
the exact contour error, it is
computationally expensive. Subsequently, most research effort has
been finding better methods in estimating contour errors
[4]
. In this thesis, a
novel position
contour error
approach is developed
based on the circular approximation of a tool trajectory
.
On
the other hand, little research can be found on the developm
ent of orientation contour error for 5

axis machine
tools. In this thesis, an
orientation contour error formulation method developed
based on the tangential approximation of the orientation trajectory
[5]
.
Feedback linearization is used to linearize the PKM dynamics through direct
compensation of the nonlinear terms such as the spindle gyroscopic forces.
While contour error
tracking is most important design specifica
tion for the RmMT, there are a number of other
specifications that must be met, such as steady state contour error, velocity tracking, actuation
effort and robust performance. All of these design specifications exhibit the convex property
such that t
he con
vex combination control design method developed by Liu and Mills
[6]
can be
app
lied to obtain a set of linear controllers that can satisfy all of the design specifi
cations
simultaneously.
4
1.2
Thesis
Objectives and
Contributions
Often in
the
literature, the different topics relevant to the RmMT dynamic modeling and
control are analyzed separately. The goal of
the thesis is to combine the
se
topics to form a
coherent and pr
ocedural design methodology for the development of rigid body dynamic model
and control of PKM based machine tools.
The primary objectives of this thesis are:
1.
To develop a straightforward
and implement
able
methodology for deriv
ation of
rigid body
dynamics of a
general group of PKMs
configurations
.
2.
To apply feedback linearization and convex combination control design methodology to meet
a set of multiple control specifications
simultaneously
for a PKM
based machine tool.
T
here are also s
econdary objectives
with the purpose of better meeting the tracking error
design requirement of the PKM:
1.
Applying the dynamic modeling methodology in deriving the rigid body dynamics of spindle
rotor gyroscopic forces.
2.
Formulating contour errors for both translation and orientation motions in a PKM structure
and evaluating these formulations for its effectiveness in simulation implementations.
The main contribution of this thesis
is the development of the rigid body dyn
amic model
and control design methodology for PKM based applications, which in this case is the RmMT.
The methodology combines a number of research topics which are often analyzed separately to
produce a rather straightforward design procedure. A further c
ontribution is within this
procedure
, principles such as contour error and convex combination control design
are applied
to
a 6DOF spatial PKM, which has not been done
prior in literature.
Another contribution is the analysis of rigid body s
pindle gyrosco
pic forces, which is not
found in literature
for machine tools. This research provides a better understanding of effects of
5
the gyroscopic forces as a
disturbance
, which is
very
relevant in high spindle speed high
accuracy
meso

milling environment.
Finally
, a novel position contour error formulation is developed using circular
approximation. The method is entirely geometrical, however it takes into account the time delay
between the reference point and the tool point without explicitly referencing time vari
ables.
1.3
Thesis O
utline
The remainder of the thesis is organized as follows. Chapter 2 discusses the kinematics of
a representative PKM structure. The inverse kinematics and the system Jacobian
are
derived. The
chapter also examines singular cases of the PK
M, and establishes a work

space for the RmMT.
Chapter 3 introduces the two concepts central to the development of rigid body dynamic models
for PKMs: the Principle of Energy Equivalence and Transformation of Generalized Coordinates.
These concepts are appl
ied in the Lagrangian formulation to derive the 2
nd
order dynamic model
with respect to the active joint variables or the task space generalized coordinates. The chapter
then describes the implementation of the dynamic model within the MATLAB Simulink
envi
ronment. Chapter 4 analyzes the effect of rigid body gyroscopic forces for a high speed
spindle rotor. For conventional machine tools, the gyroscopic forces typically neglected
.
However, for the RmMT with high accuracy requirement, the gyroscopic forces ar
e modeled and
compensated. Chapter 5 discusses the formulation of contour errors for both translation and
rotational motion in order to further improve the accuracy of the control algorithm. Chapter 6
discusses the rigid body control strategy for the RmMT,
applying feedback linearization and
convex combination linear controller design to satisfy a series of design specifications
simultaneously. Finally, Chapter 7 summarizes the findings of the thesis and offers concluding
remarks as well as recommendations
for future work.
6
Chapter 2
Kinematic Behaviour
of Parallel Kinematic Mechanism
A p
arallel kinematic mechanism (PKM) can be considered as a suitable structure for
meso

milling machine tools due to its lower inertia and higher velocity than
its
conventional
multi

axis counterparts. PKMs have
also
been proven to exhibit greater structural stiffness and
higher accuracy than
do
serial mechanisms
[1]
.
In this
chapter, an explicit active joint dynamic
model is developed for a
3
×
PP
RS
PKM
(where P, R, and S denote prismatic, revolute, and
spherical joints
,
respectively
,
and the underlined prismatic joints are actuated)
that will serve as a
candidate structure for
a
Reconfigurable meso

Milling Machine Tool (RmMT). The kinematics
of this PKM is briefly introduced, followed by a discussion of the two key concepts used in the
derivation of the dynamic model. The methodology of the derivation is then discussed in detail
.
2.1
System Description
The PKM structure
,
selected as a candidate structure for the RmMT
,
was first presented
by Kim
et al.
in
[7]
.
It is a
3
×
PP
RS
mechanism. The system is shown in
Figu
re
2

1
.
There
are
other possible
PKM configurations under consideration for the RmMT
currently
being
considered
in
the
overall
research project; however, this particular configuration is analyzed in
detail, with emphasis on the general methodology of developing the dynamic model.
Figu
re
2

1
3 PPRS PKM Structure
7
The PKM in
Figu
re
2

1
consists of a circular base of radius
on which three circular
prismatic joints
,
described by the angular joint variables
,
are mounted at points
. Three
vertical columns are mounted to the circular prismatic joints. The vertical prismatic joints
described by the linear joint variables
are situated on these three columns respectively at
points
. The tool spindle platform is connected to the three columns through three links of
length
. The
se
links
, referred to thereafter as passive links,
are connected to the three
columns through revolute joints
. The links
are connected to the tool spindle platform through
spherical joints at points
. The prismatic joints
and
are actuated joints
while
the revolute
joints
and
the spherical joint at
are passive joints.
The radius of the tool platform is
denoted as
and the length between the tool platform and the tip of the tool, or the tool centre
point (TCP) at point
is denoted as
.
A stationary coordinate reference f
rame
is
defined at the center of the circular base of the system. A moving reference frame
is defined
at the
TCP
.
2.2
Kinematic Equations
The kinematic models of the PKM consist of forward and inverse kinematics. The
objective of the forward

kinematic problem is to determine the position and the orientation of
TCP in terms of active joint variables, whereas the objective of the
inverse

kinematic problem is
to determine the active joint variables from given position and orientation
(pose)
of th
e TCP
[8]
.
The solution to the forward kinematic problem in PKMs is typically difficult to obtain because of
the presence of passive joints that cannot be describe
d explicitly in terms of the active joints.
This means the position and orientation of the TCP, described by the task space variables, are
functions of both active and passive joint variables. These variables are not linearly independent
due to their coupl
ed nature. In contrast, the inverse kinematic problem for PKM is
m
ore
straightforward and the actuated joint variables can be computed explicitly in terms of task space
variables. The purpose of solving the inverse kinematic problem is to use the relation
ships in
subsequent analyses, more specifically, to determine the system inverse Jacobian, which is used
frequently in the dynamics formulation and analysis. The inverse kinematic of each sub

chain of
8
the PKM can be solved directly and the derivation of th
e inverse kinematic problem with respect
to the spherical joints is presented in detail in
[7]
.
Let us define a
vector
to represent the task space variables at the
TCP described in the fixed frame
, where
describes the position of the
TCP and
is the
ZYZ
Euler angles describing the orientation of the tool.
Let us
also define a
vector
for
to represent the position of the spherical
joints in the fixed frame
. The inverse kinematic relationships for each sub

chain of the PKM
are
⠲

ㄩ
⠲

㈩
⠲

㌩
Figure
2

2
Nominal Positions and Sign Conventions of Joint Variables
The nominal positions and sign conventions of the joint variables are shown in
Figure
2

2
.
The kinematic relationships between the spherical joints
and
can
be formulated by taking
into account the geometry of the tool platform and the tool length. Each spherical joint are
apart as illustrated in
Figure
2

3
. For si
mplicity of modeling, two moving coordinate frames are
to be used: the frame
at the TCP and the frame
at the centre of the platform (
Figure
2

3
).
9
With respect to the moving frame
, the point
. The rotation matrix
between the frame
and frames
,
is
, where
,
, and
are elementary rotation matrices rotating about current axes as specified by the
ZYZ
Euler
angles convention
. Then
,
with respect to frame
:
⠲

㐩
周T
灯獩瑩潮猠潦⁴桥灨 物ra氠l潩湴s
Ⱐ
睩瑨w獰散琠t漠瑨攠o牡浥m
,
can be described as
⠲

㔩
⠲

㘩
⠲

㜩
周T
灯獩瑩潮oac栠獰he物ra氠
橯j湴Ⱐ
睩瑨w獰sc琠t漠瑨攠o牡浥m
,
is
⠲

㠩
啳楮U 瑨t猠 se琠 潦 牥la瑩潮獨o灳p a汬潷猠 瑨攠 橯j湴 癡物r扬e猠 楮i each 獵s

c桡楮i 瑯t be de獣物扥d
c潭灬e瑥ty渠 e牭猠潦⁴桥⁴ 獫灡se⁶ 物r扬e猠
.
Figure
2

3
Platform and Tool
Let us define a
vector
as the vector of active joint
variables,
where
the
inverse Jacobian is defined as:
10
⠲

㤩
can be
computed analytically within MATLAB using the Symbolic Toolbox. The forward
Jacobian matrix
can be computed numerically within MATLAB by inverting
.
Let us define a
vector
as the vector of passive
and active
joint
variables
of
associated with each of the three sub

chains connected to the tool platform
,
where
a
pseudo

Jacobian matrix
can be defined as:
⠲

1
F楮慬iyⰠle琠畳u摥晩湥 a vec瑯爠
as the vector of passive joint variables.
Th
e
above
Jacobian matrices
can be
computed analytically within MATLAB.
2.3
Kinematic Behaviour of the PKM
2.3.1
Work

s
pace Definition
A
n
extended
Cartesian
work
space of the RmMT, which demonstrates the full
capability
of the PKM structure,
can be visualized as a hemispherical shell. For
each
TCP position
described
by
in this shell, the tool
and the tool
platform can also rotate in
different orientations as described by Euler angles
.
Figure
2

4
shows the extent of
feasible poses in this extended work space.
The work space can also be described in terms of
spherical coordinates
,
where
is the radius of the workspace from
workspace origin, defined at the point
in the fixed coordinate
frame
;
is the azimuth
angle expressed on the
reference plane that passes through the workspace
origin;
i
s the inclination angle measured from the
axis in the frame
. Note that
and
11
is different from the Euler angles
and
, as the former, along with
describes a point
on the workspace and the latter describes the orientation of the tool at a point on the workspace.
Figure
2

4
Extended Workspace
The extended workspace presents a challenge in simul
ation because of the large number
of DOFs available for manipulation. A more
limited
workspace is defined here
and all
subsequent simulation reference trajectories used will be on this workspace unless specified
otherwise. This workspace is shown in
Figure
2

5
.
The dimensions are merely selected for the
purpose of implementing simulation and do not reflect the values in the final design of the
RmMT.
The radius of the w
orkspace is fixed at
. The workspace inclination angle
is between
to
. Furthermore, the orientation of the tool, specified by Euler angles
is to be normal to the workspace surface such that
,
and
.
From Section
2.3.2
Kinematic Singularity
, it is found when the orientati
on of the tool is set to be
normal to the workspace, singular poses occur
in the region
. Thus to avoid these
singular poses, this region is removed from workspace.
Figure
2

5
Limited Workspace
12
2.3.2
Kinematic Singularity
Kinematic singularity occurs when a mechanism loses one or more degrees of freedom
(DOF)
when in certain configurations. For serial mechanisms, singular configurations can be
identified when the Jacobian matri
x
has a determinant of zero, that is, not full rank
[8]
.
For
PKMs, a more general case between the input and output generalized coordinates has to be
considered
[9]
.
Let us suppose
the relationship between the input coordinates or the active joint variables
and the output coordinates or the task space variables
c
an be expressed as follows
,
⠲

ㄱ1
a湤n
摩晦ere湴na瑩湧 䕱畡t楯渠⠱ㄩ yie汤l
,
⠲

ㄲ1
睨w牥
⠲

1
3
)
I映瑨e 摥瑥t浩湡湴n潦 e楴桥爠浡瑲楸
,
or
,
is
zero
for a specific
configuration, then
,
such
a
configuration is singular
[8]
.
Applying this concept to the PKM structure under consideration, Equations (2

1)
–
(2

3)
can be reformul
ated into the form in Equation (2

1
2
) where the passive joint variable in Equation
(2

3), explicit in task

space variables, becomes embedded in Equation (2

2). Differentiating this
would yield the following
⠲

1
4
)
睨w牥
is the identity matrix, and
13
⠲

1
5
)
In
[㝝
Ⱐ 䭩洠 e琠 a氮Ⱐ 睨潭w 潲楧楮a汬y p牥獥湴敤n 瑨t猠 灡牴楣畬r爠
PKM
configuration,
discussed
only
two singular cases.
After a more thorough analysis using
simulation
s
, a total of
four singular
scenarios, including the two c
ases presented by
[7]
,
are
noted.
In
the first case,
it can be
found
that
when
the
Euler angle
.
The rank
of
is reduced to 5.
In the
ZYZ
Euler angle formulation of orientation, if angle
, then
,
angle
and
rotate about the same
Z
axis.
This means that the 6 actuated joint variables are
dependent only on 5 task space variables and the output at the task s
pace gains one degree of
freedom.
Figure
2

6
illustrates this singular scenario. This can also be seen in both
Figure
2

8
and
Figure
2

10
where the condition number for
approaches infinity
at
. Since the
simulations are done numerically, it is not possible to pin point the exact value of angle
such
that
. By evaluating the condition number, an approximate value of angle
can be
determined as the condition number increases rapidl
y near the singularity.
It should be noted that this scenario is not analogous to that of wrist singularity which
exists in serial mechanisms. Wrist singularity occurs when the Jacobian
loses rank and the two
input revolute joints gains arbitrary freed
om while the output task space pose remains fixed.
Figure
2

6
Singularity Scenario 1
In the second case, singularity can occurs when the tool platform and the adjacent passive
link lies on the same p
lane, illustrated in
Figure
2

7
. The tilt angle of the tool platform is
14
described by the Euler angle
, this singularity occurs when
or
is equal to the passive
joint
variables
and that the orientation of the passive link intersects the tool platform centre, which
also describes the angle of the passive link. The simulation shown in
Figure
2

8
, where
,
demonstrates an example of this singularity.
Figure
2

7
Singularity Sce
nario 2
In the third case, from
Figure
2

8
, the second peak in condition number, where
,
is illustrating the singularity which occurs when the mechanism is at its maximum boundary of
reach. This happens when
is tilted
, beyond which there is an abrupt change in
configuration where passive link 1 is shifted
in order for the pose to be structurally feasible.
Mathematically, this implies that infinite forces and acceleration is required to instantane
ously
change the position of a link for an infinitesimal motion in the task space coordinate.
15
Figure
2

8
Rotation of
T
ool
P
latform from 0° to

90° in Euler Angle
16
Finally, there are
a series
of
singular
poses
where
, but the nature of the
singularity
is
not evident by simply examining the geometry of the poses.
The behaviour of th
is
type of
singularity is similar to that of
C
ase
1
, where the task space Euler angles
and
are free
t
o move arbitrarily when all the ac
tuators are fixed. The occurrence of these
singular poses are
dependent on the position of the TCP in the work space, as well as the Euler
angles
,
and the
length of the tool
Given the number of variables which contribute to the
occurrences
, it is
very
difficult to perform a thorough scan across the
extended
work space to map out the
locations of all the singular points.
One example of this singular pose
is shown in
Figure
2

9
and
the associated peak in
condition number can be seen in
Figure
2

10
.
Figure
2

9
Singular Pose from Figure 2

10
17
Figure
2

10
Rotation of
T
ool
P
latform from 0° to 90° in Euler Angle
β
18
A
search
for this type of singularity is
performed in the
limited
workspace
specified
in
Section
2.3.1
,
where
workspace radius
,
Eule
r angle
and angle
is normal
to the workspace. The r
esult shows that singular point
occurs when
and
, is
at
(
Figu
re
2

1
)
.
The pose is shown in
Figure
2

11
and
Figure
2

12
. The region between inclination
angle of
and
yields aside from the two singular points still yield condition numbers much
larger than the rest of the workspace, thus this region is
denoted as near singular and
for
all
subsequent simulations,
this singular region
is avoided.
Figure
2

11
Front and Top View of Singular Pose in Limited Workspace
Figure
2

12
Isometric View of Singular Pose in Limited Workspace
19
Figure
2

13
Singularities in Limited Workspace with Inclination Angle from 0° to 90°
20
2.4
Summary
In this chapter,
the notations to be used for the representative PKM configuration are
presented. The inverse kinematics and the inverse Jacobian of the PKM are also derived. The
chapter then described the full range of the work space of the PKM, but presented a case for
a
limited work space for the purpose of simulation and avoidance of singularities. Finally, four
cases of singular configurations are presented with simulation results.
21
Chapter 3
Rigid Body Dynamic Modeling
The dynamic modeling methods for serial mechanisms are we
ll developed and
straightforward using the Lagrangian formulation. However, dynamic modeling for PKMs is less
developed in comparison due to the increased level of complexity. Furthermore, the bulk of
existing research has been focused on the Stewart
Gough
platform (
Figure
3

1
) while studies for
other PKM configurations are more limited due to the shear degree of variation and limited
industrial applications. The goal i
n dynamic modeling is to derive an explicit dynamic equation
in terms of active joints only in the following form, also referred to as the standard form
⠳

ㄩ
睨w牥
is the
mass and inertia
matrix,
is the Coriolis and centrifugal force
matrix and
is gravitational force vector.
The challenge however lies with the passive joints, which cannot be written explicitly in
terms of the active joints in a straightforward manner. The following section pres
ents a brief
discussion on the three main methods proposed thus far for the dynamic modeling of PKMs.
Figure
3

1
Stewart
Gough
Platform
22
3.1
Literature Review
The methods used in modeling serial manipulato
rs, which are standardized, have been
carried over for the modeling of PKMs. The following three methods have received the most
attention and are
also the most well established:
the Newton

Euler formulation, the Lagrangian
formulation
,
and the Principle of
Dynamic Virtual Work formulation.
The Newton

Euler formulation is based on the translational and rotational motions of a
rigid body. In this method, free body diagrams of individual
structural
links are considered, and
the constraint forces between the l
inks are taken into consideration when formulation the
equations and must then be eliminated as part of the process
[8]
. The main
disadvantages for this
method are
that the joint
torques is
not included explicitly when formulating the equations. In
order to derive a closed

form dynamic equation, further arithm
etic operations must be carried out
to present the equations in the desired form of Equation (3

1)
[10]
[11]
.
The research works of
Dasgupta in
1998 on Newton

Euler formulation has been highly
cited
[12]
[13]
. It has been known that Newton

Euler approach is computationally
more
efficient
,
while the Lagrange formulation is more capable at deriving closed

form equations. At the time
,
in the 1990
s, joint

space formulation using the Lagrangian
method
with both active and passive
joint variables as generalized coordinates required the use of Lagrange multipliers and with
increased complexity of the manipulator, Dasgupta believed that the Newton

Eule
r formulation
is a viable alternative in terms of computation cost.
The Principle of Dynamic Virtual Work formulation is an energy method which takes
advantage of the D’Alembert’s Principle stated below
[14]
:
⠳

㈩
This principle comes directly from Newton’s
Second Law
. A new force, called the force of
inertia, defined by
is introduced
in
to the system such that combined with the applied forces
23
on the system, the resultant virtual work for a virtual displacement
is 0 and equilibrium is
achieved. The generalized forces
associated with a generalized coordinate
can be def
ined.
⠳

㌩
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楳
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瑨攠
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晲潭⁴桥⁰潴敮瑩慬湥rgy映瑨 y獴e洮
周T Lagrang楡渠景f浵m
at楯渠楳ia汳漠a渠e湥牧y me瑨潤t瑨慴t楳ic潭灬e瑥t睩瑨 c潮瑲楢畴楯湳i
from kinetic and potential energy. The generalized forces are solved from Lagrange’s Equations,
expressed as
⠳

㐩
睨w牥
is the Lagrang
ian
function, the difference between the kinetic energy and the potential
energy of the system. The Lagrange’s equations can be derived directly from the Principle of
Dynamic Virtual Work
[14]
. If the kinetic energy of the system is
⠳

㔩
I琠景汬潷猠t桡t
⠳

㘩
a湤
⠳

㜩
⠳

㠩
24
From Equation (3

3)
⠳

㤩
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⠳

1
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[ㄵ1
[ㄶ]
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楳
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a猠ㄹ㤲1 Leb牥琬tL極ia湤nLe睩猠
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ex灬慮p瑩潮猠瑯tt
桥楲i灲oce摵牥. 周T c潮瑲楢畴楯湳i潦
[㉝
潮o瑨攠潴桥o 桡n搠c汥l牬y 楬汵獴牡瑥t 瑨t
業灯牴p湣e映瑨 se⁴睯摥 献
3.2
Underlying Principles for Dynamic Modeling
Aside f
rom the Lagrangian based energy method for dynamic modeling, the methodology
for the dynamic modeling of the representative PKM applies two major concepts. These concepts
are the Principle of Energy Equivalence and the Transformation of Generalized Coordin
ates.
The principle of energy equivalence is applied to the PKM in order to divide the parallel
mechanism into a set of serial mechanisms, thus enabling the application of the Lagrange
25
formulation. The transformation of generalized coordinates is required
to obtain the same set of
generalized coordinates for all the component serial mechanisms of the PKM in order to
recombine the dynamics into the complete dynamics of the PKM.
3.2.1
Principle of Energy Equivalence
The principle of energy equivalence
was
original
ly developed by Liu
et al
.
[20]
.
The
principle states that for two systems
–
described by two different but mathematically related sets
of generali
zed coordinates
–
possessing equal properties, occupying the same space and
experience the same trajectory, the virtual work done by the two systems are equal. The
main
contribution
of this principle is that it allows the dynamics of the PKM to be treated
in easily
manageable serially connected pieces, where standardized Lagrangian based modeling
procedures can be systematically applied.
Let us suppose
the two systems described above have the generalized coordinates
and
, and they are related as
⠳

ㄱ1
⠳

ㄲ1
周Tn
,
by ex灥物rnc楮g 瑨攠 獡浥mdyna浩c猬s 瑨t楲i Lagrang楡渠
. The
virtual work done by system
is
⠳

ㄳ1
睨w牥
⠳

ㄴ1
a湤
⠳

ㄵ1
Le琠畳畢獴楴畴攠
䕱畡瑩o湳″

ㄴ湤″

ㄵ湴漠o

1㌬湤na牲an来
2
6
⠳

ㄶ1
w
桩捨猠h汳l
⠳

ㄷ1
潲
⠳

ㄸ1
䅳A摩獣畳獥搠灲e癩潵vlyⰠ景f 瑨攠睨潬w P䭍Ⱐt桥 景fwa牤r 歩湥浡瑩c ex灲e獳s潮o
, where the passive joint variables are written explicitly as
, does not exist.
Fortunately, the inverse kinematic expression
does exist, and this expression
satisfies the condition of the principle of energy equivalence
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