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Design and Analysis of a Three Degrees of Freedom Parallel Kinematic Machine
by
Xiaolin Hu
A Thesis Submitted in Partial Fulfillment of the Requirements for the Degree of
Master of Applied Science
in
The Faculty of Engineering and Applied Science
Mechanical Engineering Program
University of Ontario Institute of Technology
August, 2008
© Xiaolin Hu, 2008
CERTIFICATE OF APPROVAL Submitted by: _____________Xiaolin HU_________________ Student #: __100250809____ First Name, Last Name In partial fulfillment of the requirements for the degree of: __________ Master of Applied Science __________ in ____Mechanical Engineering________ Degree Name in full (e.g. Master of Applied Science) Name of Program Date of Defense (if applicable): __________________________________ Thesis Title: Stiffness Analysis of the 3 Degrees of Freedom Parallel Kinematic Machine The undersigned certify that they recommend this thesis to the Office of Graduate Studies for acceptance: ____________________________ ______________________________ _______________ Chair of Examining Committee Signature Date (yyyy/mm/dd) ____________________________ ______________________________ _______________ External Examiner Signature Date (yyyy/mm/dd) ___________________________ ______________________________ _______________ Member of Examining Committee Signature Date (yyyy/mm/dd) ____________________________ ______________________________ _______________ Member of Examining Committee Signature Date (yyyy/mm/dd) As research supervisor for the above student, I certify that I have read the following defended thesis, have approved changes required by the final examiners, and recommend it to the Office of Graduate Studies for acceptance: _________________________ _________________________________ _______________ Name of Research Supervisor Signature of Research Supervisor Date (yyyy/mm/dd)
i
Abstract
Research and development of parallel kinematic machines (hereafter called PKMs) is
being performed more and more actively. The methodologies for PKMs development are
considered as the key for the robot applications in the future.
PKMs feature many advantages over serial robots in terms of accuracy, stiffness,
structural rigidity, dynamic agility, and compactness. However, PKMs have a few of
disadvantages such as treacherous singularities and limited workspace.
The study reported is on design and analysis of a PKM with 3 degrees of freedom
(DOF). The new PKM is designed as a machine tool in various applications in
manufacturing. The PKM is optimized based on the developed stiffness model.
Kinematics and dynamics of the new PKM is also modeled and simulated. The thesis is
organized as follows.
First, the 3-DOF PKM is designed. Its topology is introduced and a CAD model of
the final design is created. The inverse kinematics is analyzed. Jacobian matrix and
velocity equations are derived. The singularities of the PKM structure are studied.
Second, the workspace of the new PKM is studied. The concept of the workspace is
defined. Three methods for the calculation of workspace are compared, and the results are
illustrated.
ii
Third, Static Balancing of the Parallel Kinematic Manipulator is investigated: the
definition and methodology of static balancing are introduced. Two methods called
Adjusting Kinematic Parameters (AKP) and counterweights are applied to the structure
and the counterweights method leads to static balancing of the PKM. The conditions of
static balancing are given.
Fourth, the stiffness of the 3-DOF Parallel Kinematic Machine is analyzed. The
literatures on the methodologies of stiffness analysis are surveyed. The stiffness model of
the proposed PKM is established, the stiffness matrix is utilized, the stiffness mapping is
generated, and the optimization of the global stiffness of the 3-DOF PKM is performed.
Finally, the dynamic model of the new PKM is studied. It describes the relationship
between the driving forces and the motion of the end-effector platform. Two approaches,
the Newton-Euler and the Lagrange methods, are compared and the later one is selected
to build the dynamic model of the 3-DOF Parallel Kinematic Machine.
The new 3-DOF PKM combining the spatial rotational and translational degrees of
freedom has varied advantages and good potential applications of manufacturing. The
novel design leads to the very efficient models in kinematic, workspace, stiffness and
dynamic analyses. For the focus of the system stiffness, the study of this thesis gives the
solution of how to reconfigure a parallel kinematic machine with adjusting kinematic
parameters and path re-planning to achieve a desired stiffness. The optimal design value
parameters are also suggested for the achieved maximum global stiffness of the PKM.
iii
Acknowledgement
In this presentation, my thesis work is described. I have done this work at the Faculty
of Engineering and Applied Science, University of Ontario Institute of Technology,
Oshawa, Ontario, Canada, during 2006 – 2008.
This work is under the supervision of Dr. Dan Zhang, Associate Professor and Director
– Automotive, Manufacturing and Mechanical Engineering Programs, Faculty of
Engineering and Applied Science, UOIT. I would like to thank him for his invaluable
supervision, full support and critical review of the manuscript. Without his ideas and
continuous encouragements, this work would never have been possible.
iv
Abstract i
Acknowledgement iii
Contents iv
List of Figures v
List of Tables viii
List of Acronyms ix
1. Introduction …………………………………………………………………… 1
1.1. The subject of the Study ……………………………………………… 5
1.2 The Objectives and Scope of the Study ………………………………. 7
1.3 The Organization of the Thesis ……………………………………….. 8
2. Design of the New 3-DOF Parallel Kinematic Machine …………………….... 11
2.1 Introduction …………………………………………………………..... 11
2.2 Geometric Structure ………………………………………………….... 13
2.3 Novelty and Applications …………………………………………….... 16
2.4 Inverse Kinematics of the 3-DOF Parallel Kinematic Machine ……….. 17
2.5 Velocity Equations and Jacobian matrix ……………………………….. 20
2.6 Singularity Analysis ………………………………………………….… 21
2.6.1 Type I Singularity …………………………………………….... 22
2.6.2 Type II Singularity …………………………………………...... 22
3. Workspace Modeling of the 3-DOF Parallel Kinematic Machine …………….. 23
3.1 Definition of Workspace ……………………………………………….. 23
3.2 Methods for Determining Workspace ………………………………..... 23
3.3 Workspace Analysis of the 3-DOF Parallel Kinematic Machine............ 25
v
4. Static Balancing of the 3-DOF Parallel Kinematic Machine…………………… 29
4.1 Introduction …………………………………………………………….. 29
4.2 Static Balancing using Adjusting Kinematic Parameters......................... 31
4.3 Static Balancing using counterweight method ...…....................……….. 34
5. Stiffness Analysis of the 3-DOF Parallel Kinematic Machine ………...…….… 38
5.1 Introduction …………………………………………………………….. 38
5.2 General Stiffness Model ……………………………………………….. 40
5.3 Stiffness Mapping of the 3-DOF Parallel Kinematic Machine ………… 42
5.4 Stiffness Optimization of the 3-DOF Parallel Kinematic Machine ……. 58
5.4.1 Genetic Algorithm ……………………………………………... 58
5.4.2 Optimization Result ……………………………………………. 59
6. Dynamic Modeling of the 3-DOF Parallel Kinematic Machine ……………….. 62
6.1 General methodologies of dynamic modeling ………………………..... 63
6.2 Dynamic Modeling of the 3-DOF Parallel Kinematic Machine ……….. 65
6.3 Dynamic Simulation of the 3-DOF Parallel Kinematic Machine ……… 68
7. Conclusion and Future work …………………………………………………… 73
7.1 Conclusion ……………………………………………………………... 73
7.2 Future work …………………………………………………………….. 75
Reference ………………………………………………………………………………. 77
vi
List of Figures 1.1 The Stewart-Gough platform (Bruyninckx and Hallam 2005)................................... 2
1.2 Delta robot application- ABB's FlexPicker robot....................................................... 3
1.3 Hexapod robot application: Hexapod PAROS Source: MICOS, Germany................. 4
2.1 CAD model of the 3-DOF PKM................................................................................. 14
2.2 Schematic representation of the PKM........................................................................ 18
3.1 The workspace of the PKM........................................................................................ 27
3.2 The boundary of y-z plane of the workspace.............................................................. 28
4.1 The schematic representation with position vectors of components.......................... 32
4.2 The 3-DOC PKM with the counterweights............................................................. 35
5.1 Stiffness mesh graphs in X with L= 100 cm to 200 cm, r=10cm............................... 46
5.2 Stiffness contour graphs in X with L= 100 cm to 200 cm, r=10cm............................ 47
5.3 Stiffness mesh graphs in X with L= 100 cm, r=5cm to 20 cm................................... 48
5.4 Stiffness contour graphs in X with L= 100 cm, r=5cm to 20 cm................................ 49
5.5 Stiffness mesh graphs in Y with L= 100 cm to 200 cm, r=10cm............................... 50
5.6 Stiffness contour graphs in Y with L= 100 cm to 200 cm, r=10cm............................ 51
5.7 Stiffness mesh graphs in Y with L= 100 cm, r=5cm to 20 cm................................... 52
5.8 Stiffness contour graphs in Y with L= 100 cm, r=5cm to 20 cm................................ 53
5.9 Stiffness mesh graphs in Z with L= 100 cm to 200 cm, r=10cm................................ 54
vii
5.10 Stiffness contour graphs in Z with L= 100 cm to 200 cm, r=10cm.......................... 55
5.11 Stiffness mesh graphs in Z with L= 100 cm, r=5cm to 20 cm.................................. 56
5.12 Stiffness contour graphs in Z with L= 100 cm, r=5cm to 20 cm.............................. 57
5.13 The best fitness and the best individuals of the global stiffness optimization.......... 60
6.1 The planar structures of the PKM............................................................................... 66
6.2 CAD model of 3-DOF PKM for dynamic simulation................................................ 69
6.3 The angular velocity 1ω of leg 1................................................................................. 69
6.4 The angular acceleration 1α of leg 1...........................................................................70
6.5 The angular velocity 2ω of leg 2................................................................................ 70
6.6 The angular acceleration 2α of leg 2...........................................................................70
6.7 The angular velocity 3ω of leg 3.................................................................................71
6.8 The angular acceleration 3α of leg 3...........................................................................71
viii
List of Tables
Table 1: The possible Degree of Freedom distributed for each leg.................................. 15
Table 2: Possible joint combinations for different degree of freedom............................. 16
Table 3 Stiffness in X vs. length of the leg....................................................................... 43
Table 4 Stiffness in Y vs. lengths of the triangle side (the moving platform).................. 44
ix
List of Acronyms
DOF: Degree of Freedom
SKM: Serial Kinematic Mechanism
PKM: Parallel Kinematic Machine
AKP: Adjusting Kinematic Parameters
GA : Genetic Algorithms
1
Chapter 1
Introduction
A parallel mechanism is a closed-loop mechanism of which the end-effector is
connected to the base by a multitude of independent kinematic chains. Generally it
comprises two platforms which are connected by joints or legs acting in parallel
(Bruyninckx and Hallam, 2005).
In recent years, parallel kinematic mechanisms have attracted a lot of attention from
the academic and industrial communities due to their potential applications not only as
robot manipulators but also as machine tools. Generally, the criteria used to compare the
performance of traditional serial robots and parallel robots are the workspace, the ratio
between the payload and the robot mass, accuracy, and dynamic behavior. In addition to
the reduced coupling effect between joints, parallel robots bring the benefits of much
higher payload-robot mass ratios, superior accuracy and greater stiffness; qualities which
lead to better dynamic performance. The main drawback with parallel robots is the
relatively small workspace.
2
A great deal of research on parallel robots has been carried out worldwide, and a large
number of parallel mechanism systems have been built for various applications, such as
remote handling, machine tools, medical robots, simulators, micro-robots, and humanoid
robots. The first design for industrial purposes was completed by Gough in the United
Kingdom, and implemented as a tire testing machine in 1955. Some years later, Gough’s
compatriot Stewart published a design for a flight simulator (Bruyninckx and Hallam
2005). The design illustration of the Stewart-Gough platform is shown in Fig. 1.1.
Fig. 1.1 the Stewart-Gough platform (Bruyninckx and Hallam 2005)
Thereafter, many applications of parallel robots can be found in various industries and
fields, such as manufacturing production configurations (Necsulescu 1985), Micro
parallel robot for medical applications (Merlet 2001), assembly robot arms in automotive
applications (Mintchell 2002), deep sea exploration (Saltaren 2007), etc. More recently,
they have been used in the development of high precision machine tools (Zhang 2000).
Machine tools used by industries are the conventional Serial Kinematics Mechanisms
(SKMs). There are two requirements for these machine structures:
3
• Robust design for higher power utilization, and
• Lightweight construction for high feeds and accelerations
These two requirements are rather contradictory in SKMs. The conventional SKMs
appears to have reached their limits both physically (considering the feedrates and
accelerations) and also in terms of purchase cost. This has necessitated a totally new
concept in machine tool construction - the Parallel Kinematics Machine (PKM). There
are several types of PKMs: the triglides, tripods and the chaste hexapods and hex glides.
The Delta robot is the inspiration for this. In 1985, Clavel (EPF-Lausanne) developed
the Delta robot. In 1993, Ingersol presented the first Parallel Kinematic Machine Tool,
the Hexapod, which triggered a worldwide trend in developing a new class of machine
tools (Wu 2008).
Fig. 1.2 The Delta robot application- ABB's FlexPicker robot
4
Fig. 1.3 Hexapod robot application: Hexapod PAROS [Source: MICOS, Germany]
The visible advantages of PKMs over SKMs are their simplicity and light weight
construction. The parallel, closed frame in PKMs results in the improved dynamic
characteristics combined with lower maintenance costs. This is in contrast to SKMs
where sturdy slides are needed in each direction leading to cost and weight escalations
(Parallel Kinematics' Machines for the future 2003). As a result, the Parallel Kinematic
Machines are becoming increasingly popular in the machine tool industry around the
world. Examples of industrial applications are well illustrated in (Weck and Staimer
2002). In the academic field, Chen and You (2000) investigated the effects of the design
parameters on the singularity of a six degrees-of-freedom 6-3-3 parallel link machine tool.
Li and Katz (2005) introduced a new reconfigurable high-speed drilling machine which is
based on a planar parallel kinematic mechanism. They focused on the input motion
planning for ideal drilling operation, and for fast and accurate point-to-point positioning.
Gao et al. (2006) presented a novel 5-DOF fully parallel kinematic machine tool and
5
addressed the relationship between the reachable workspace and the configuration
dimensional parameters of the parallel machine.
Following this trend, the study of this thesis presents a new design of the Parallel
Kinematic Machine with three degrees of freedom and focuses on stiffness analysis of the
PKM.
1.1 The subject of the study
For the purpose of the PKM design and its stiffness analysis, the subject of this thesis
investigates different aspects of the PKM research such as kinematics, jacobian matrix,
singularity, workspace and stiffness.
A basic problem in the study of Manipulator Kinematics is called Forward
Kinematics. This is the static geometrical problem of computing the position and
orientation of the end-effector of the manipulator. Specifically, given a set of joint angles,
the forward kinematics problem is to compute the position and orientation of the tool
frame relative to the base frame.
Contrasting to Forward Kinematics, Inverse Kinematics is about the more difficult
converse problem: Given the position and orientation of the end-effector of the
manipulator, calculate all possible sets of joint angles that could be used to attain this
given position and orientation. This is a fundamental problem in the practical use of
manipulators.
6
In addition to static positioning problems, manipulators in motion are analyzed. The
notions of linear and angular velocity of a rigid body are examined and these concepts are
used to analyze the motion of a manipulator. Also, the forces acting on a rigid body are
considered so that the application of static forces with manipulator is studied. The study
of both velocities and static forces leads to a matrix entity called the Jacobian Matrix of
the manipulator which is essential for the static and dynamic analysis of a PKM.
Singularity configurations are particular poses of the end-effector, for which PKMs
lose their inherent infinite rigidity, and in which the end-effector will have uncontrollable
degrees of freedom. Therefore, it is very important to avoid such situations when
designing a parallel kinematic machine.
The workspace of a robot is an important criterion in evaluating manipulator
geometries. Robot workspace is the ability of a robot to reach a collection of points
(workspace) which depends on the configuration and size of their links and wrist joint.
The workspace may be found mathematically by writing equations that define the robot’s
links and joints including their limitations, such as ranges of motions for each joint.
Generally, larger workspaces give PKMs wider applications.
From the viewpoint of mechanics, the stiffness is the measurement of the ability of a
body or structure to resist deformation due to the action of external forces. The stiffness
of a parallel kinematic machine at a given point of its workspace can be characterized by
its stiffness matrix. This matrix relates the forces and torques applied at the gripper link
in Cartesian space to the corresponding linear and angular Cartesian displacements. In
many applications, stiffness is a very important performance specification for Parallel
7
Kinematic Machines (PKM), because it is strictly related to accurate positioning and high
payload capability as well as dynamic performance.
In the context of mechanical systems, dynamics is the study of how the forces acting
on bodies make these bodies move. For a robot, this is to describe how to model the
dynamic properties of chains of interconnected rigid bodies that form the robot, and how
to calculate the chains’ velocities and accelerations under a given set of internal and
external forces and/or desired motion specifications.
1.2 The Objectives and Scope of the Study
Usually, higher stiffness of parallel kinematic machines offers accuracy properties,
high payload capability, and good dynamic performances. This makes PKMs attractive to
the machine tool industry. Therefore stiffness becomes a very important issue in PKM
design. Particularly the stiffness analysis is not separated from the workspace calculation.
The overall stiffness and workspace of PKMs vary based on configuration of the
structures. In fact, there is always a tradeoff between stiffness and workspace when
designing PKMs. The main issue of this thesis is to design a parallel kinematic machine
to achieve a desired or optimal stiffness at one or more poses in its workspace.
The objective of this thesis is to study how to reconfigure a parallel kinematic
machine with adjusting kinematic parameters and path re-planning to achieve a desired
stiffness at a desired pose in its workspace. To reach this aim, the objectives are set as
follows:
8
1. Development of geometric design model. The geometric design includes selecting
proper links and joints and the type of the base and end-effector to satisfy the
desired motion. It also takes into account the working volume, and mechanical
interferences. The selected design is modeled using Solidworks.
2. Kinematic Analysis and singularity configurations. The inverse kinematic analysis
is studied. Two types of singularities are investigated.
3. Workspace simulation. The geometrical method is applied to determine the
PKM’s workspace. This is also the essential for stiffness analysis.
4. Development of the stiffness model of the 3-DOF PKM and mapping the stiffness
of the 3-DOF PKM.
5. Design optimization of the global stiffness of the 3-DOF PKM.
6. Dynamic modeling of the 3-DOF PKM for the dynamic performance is strictly
related to the stiffness.
The result of the above study is a new design of parallel kinematic machine with three
degrees of freedom which could have potential various applications in the machine tool
industry.
1.3 The Organization of the Thesis
Chapter 2 presents the design of the new 3-DOF Parallel Kinematic Machine. Some
concepts underlying kinematic structure development will be addressed, such as the
Chebychev-Grübler-Kutzbach criterion, a basic theory for kinematic chain degree-of-
freedom distribution and criteria for better and practical kinematic structures of machine
9
tools. Novelties and potential applications of the 3-DOF PKM are indicated. Upon the
established CAD model of the structure, the solution of the inverse kinematic problem,
Jacobian matrices and global velocity equations are developed and the singularity
configurations are investigated.
Chapter 3 focuses on the workspace modeling of the 3-DOF Parallel Kinematic
Machine. First, the definition of workspace and methods for determining workspace are
introduced, and then the workspace analysis of the 3-DOF Parallel Kinematic Machine is
performed using geometrical method.
Chapter 4 talks about the static balancing. The static balancing is a very important
research topic in the theory of machines and mechanisms, and recently being applied to
the design of Parallel Kinematic Machines. A static balanced PKM has better dynamic
characteristics and less vibration caused by motion. For this objective, the static
balancing of the new designed PKM is conducted and the conditions for static balancing
are given.
Chapter 5 describes the main objective of the study- stiffness analysis of the 3-DOF
Parallel Kinematic Machine. First, the methodology of stiffness analysis is investigated.
Three methods - matrix structural analysis, Finite Element Analysis and calculation of the
Jacobian matrix are compared and the latter one has been adopted in this thesis. Second, a
general Stiffness model of PKMs is established and stiffness matrix is derived. Third,
stiffness mapping with adjusting kinematic parameters is presented, and the optimization
of the global stiffness of the 3 DOF PKM is performed.
Chapter 6 discusses the dynamic model of the 3-DOF Parallel Kinematic Machine, as
utilized to find the relationship between the PKM motion and joint forces. Two
10
approaches, the Newton-Euler and the Lagrange methods, are compared and the later one
is used to build the dynamic model of the 3-DOF Parallel Kinematic Machine.
Chapter 7 brings together the most important conclusions and observations of the
study and suggestions of the work to be done in the future.
11
Chapter 2
Design of the New 3-DOF Parallel
Kinematic Machine
2.1 Introduction
Over the past decades, parallel mechanisms have received increasing attention from
researchers and industry. Parallel manipulators have the advantages of high rigidity, high
load capacity, low inertia, no accumulation of position error, and ease of force feedback
control over serial robots. Therefore, they have good application prospect in the fields of
assembly task, manufacturing, underground projects, space technologies, sea projects,
medicine and biology projects, etc. Examples of this type of robots can be found in the
motion platform for the pilot training simulators and the positioning device for high
precision surgical tools because of the high force loading and very fine motion
12
characteristics of the closed-loop mechanism. Recently, researchers are trying to utilize
these advantages to develop parallel-type robot based multi-axis machining tools
and precision assembly tools.
Since most machining tasks require a maximum of five axes, 3-DOF parallel
mechanism design with three axes is becoming popular (Zhang and Wang 2005). For the
family of 3-DOF parallel manipulators, there are much different architectures. The most
famous robot with three translation degrees of freedom is the DELTA, mentioned in
Chapter1, which is proposed by Clavel and marketed by the Demaurex Company and
ABB under the name IRB 340 FlexPicker (Wang and Liu 2003). It consists of three arms
connected to universal joints at the base. The key design feature is the use of
parallelograms in the arms, which maintains the orientation of the end-effector. Another
representative 3-DOF robot is the Tricept that issued from a patent by K.E. Neumann
(Merlet 2006). In that structure, the end-effector has a stem translating along its axis
freely. There is also one spherical 3-DOF parallel manipulator which is worth mentioning:
a camera-orienting device, called “the agile eye” which is presented by Gosselin and St-
Pierre (1997). The architecture has three spherical chains used with rotary actuators and
the desired orientation workspace corresponds to a cone of vision of 140° opening.
Although researchers have presented so many new design concepts of parallel robots,
there are few manipulators that can combine the spatial rotational and translational
degrees of freedom, For example, two spatial translations and one rotation (Liu and Wang
2003). This is the motivation of novel design of the 3-DOF Parallel Kinematics Machine
presented in this thesis.
13
In the new design, the end-effector (the moving platform) of the PKM has three
independent motions: translations about y, z axes and rotation about y axis is
investigated. First, the configuration of the new parallel kinematic machine is
demonstrated. Then the kinematics including inverse kinematics, forward kinematics,
Jacobian matrix and velocity equations are derived. Finally the singularities and
workspace are studied.
2.2 Geometric Structure
The new 3-DOF Parallel Kinematic Machine is composed of a base structure, a
moving platform and 3 legs connecting the base and platform. Of those three legs, two of
them are in the same plane and consist of identical planar four bar parallelograms with
chains connected to the moving platform by revolute joints, while the third leg is one bar
connected to the moving platform by a S-joint. There is one revolute joint on the upper
side of each leg, and the revolute joint is linked to the base by an active prismatic joint.
A CAD model of the 3-DOF parallel kinematic machine is shown in Fig. 2.1.
φ
14
Fig. 2.1 CAD model of 3-DOF PKM
The DOFs for a closed-loop PKM is examined using the Chebychev-Grübler-
Kutzbach’s formula:
∑=
+−−=g
iifgndM
1)1( (2-1)
Where:
M : the system DOFs of the assembly or mechanism
d : the order of the system ( d =3 for planar motion, and d =6 for spatial motion)
n : the number of links including the frames
g : the number of joints,
if : the number of DOFs for the thi joint
In the model showed in Fig. 2.1, the parallelogram has been applied to the structure of
legs. It can act the role of improving the kinematics performance and the leg stiffness can
be increased largely (Liu and Wang 2005). The disadvantage is that parallelogram as a
15
structure has the possibility to accumulate errors which eventually results in error
accumulation of PKMs.
In regard to the types of actuated joints, they can be either revolute or prismatic. Since
the prismatic joints can easily achieve high accuracy and heavy loads, the majority of the
3-DOF parallel mechanisms in reality use actuated prismatic joints. A prismatic joint can
have an extensible length or a fixed length (Zhan, Zhang, and Yang 2005).
There are many options when designing the leg of a PKM. According to those two
table, one can find the structure of PKM presented in this thesis has 5-5-5 DOFs
distributed for each leg, and 1S1R1P for the third and 1S1Parallelogram1P for joint
combinations. By applying Eq.2.1, the Chebychev-Grübler-Kutzbach’s formula, the
degree of freedom of the 3-DOF PKM can be computed:
(2-2)
Table 1: The possible Degree of Freedom distributed for each leg (Zhang, 2000)
Degree of
Freedom
Number
of Legs
Possible architectures
M=3 3
3 6 6 12
4 5 6
5 5 5 56
3)555()198(6 =+++−−=M
1lf 2lf 3lf
16
Table 2: Possible joint combinations for different degree of freedom (Zhang 2000)
2.3 Novelty and Applications
The benefit of the PKM presented in this thesis can be concluded as follows:
1. The parallelogram joints can greatly increase the stiffness of the legs.
2. Two identical chains offer good symmetry.
17
3. The spatial joint linked between the third leg and the moving platform gives the
rotation for the moving platform about y axis.
4. Three actuators drive linear motions of the prismatic joints, and that motivates the
rotary movement of the end-effector. In other words, the input of simple linear motion
results in the output of rotary movement.
5. High dynamic performance due to low moving mass.
The practical applications can be developed to machine tools and other manipulating
devices. The target industries are automotive, aerospace, manufacturing and electronics,
etc.
2.4 The Inverse Kinematics of the Parallel Kinematic
Machine
A kinematics model of the manipulator is developed as shown in Fig. 2.2. Vertices of
the output platform are donated as platform joints )3,2,1( =ipi , and vertices of the base
joints are donated as )3,2,1( =ibi . Fixed on the base frame, a global reference system
xyzOO −: is located at the point of intersection 21bb and 3Ob . Another reference system,
called the moving frame '''':' zyxOO − , is located at the center of 21 pp on the moving
platform.
18
Fig. 2.2 schematic representation of the PKM
The given position and the orientation of the end-effector (the moving platform) are
specified by its three independent motions: y, z translations and φ rotation about y axis.
The position is given by the position vectors oO )'( , and the orientation is given by the
rotation matrix Q as followings:
[ ]Tzyx=O)(O' (2-3)
where 0=x (rotating about y axis), and
[ ]⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−=
φφ
φφ
cos0sin010
sin0cosQ (2-4)
19
where the angle φ is the rotation DOF of the output platform around y axis. The
coordinates of the point ip in reference system 'O can be described by the vector
)3,2,1( =ipi
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡−=
0
0)(
00)(
00)( ''' r
rr
OOO 221 ppp (2-5)
the vectors )3,2,1( =ibi in frame xyzOO −: will be defined as position vectors of
actuating joints:
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡−=
0
0)(
00)(
00)( 33
21
ρρρ
bbb 21 (2-6)
the vector )3,2,1( =ipi in frame xyzOO −: can be written as
[ ] OOO )()()( ' O'pQp ii += (2-7)
That is
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
+
−=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡+
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡−
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−=
zry
r
zy
r
O
φ
φ
φφ
φφ
sin
cos0
00
cos0sin010
sin0cos)( 1p (2-8)
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
+=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡+
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−=
zry
r
zy
r
O
φ
φ
φφ
φφ
sin
cos0
00
cos0sin010
sin0cos)( 2p (2-9)
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡+=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡+
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−=
zyr
zyrO
00
0
0
cos0sin010
sin0cos)( 3
φφ
φφp (2-10)
20
The inverse kinematics of the manipulator can be solved by writing the following
constraint equation:
L=− ii bp (2-11)
Hence, one can obtain the required actuator inputs from Eq. (2-9):
(2-12)
(2-13)
(2-14)
2.5 Velocity Equations and Jacobian Matrix
Equations (2-12), (2-13) and (2-14) can be differentiated with respect to time to
obtain the velocity equations, which leads to
(2-15)
(2-16)
(2-17)
Rearranging Eqs. (2-13), (2-14) and (2-15) leads to an equation of the form
pBρA && = (2-18)
where ρ& is the vector of input velocities defined as
[ ]Tρρρ 321 &&&& =ρ (2-19)
φφρ cos)sin( 2221 rrzyL ++−−=
φφρ cos)sin( 2222 rrzyL +−−−=
ryzL ++−= 223ρ
0)cossin()sin()cos( 111 =+++++− φφφρφρφρ &&&& zrzrzyyr
0)cossin()sin()cos( 222 =−+−++− φφφρφρφρ &&&& zrzrzyyr
0)]([)]([ 33 =++−++− zzyryry &&& ρρρ
21
and p& is the vector of output velocities defined as
[ ]Tzy φ&&&& =p (2-20)
Matrices A and B can be expressed as
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−+−
−=
3
2
1
000cos000cos
ρρφ
ρφ
ryr
rA (2-21)
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−+−−++
=0
)cossin(sin)cossin(sin
3
2
1
zryzrrzyzrrzy
ρφφρφφφρφ
B (2-22)
The Jacobian matrix of the manipulator can be written as
BAJ 1−= or ABJK 11 −− == (2-23)
2.6 Singularity Analysis
Singularity configurations are particular poses of the end-effector, for which
manipulators lose their inherent infinite rigidity, and in which the end-effector will have
uncontrollable degrees of freedom. Most manipulators have singularities at the boundary
of their workspace, and some have singularities inside their workspace. When a
manipulator is in a singular configuration, it becomes failed at the moment. It is very
important to avoid such situations when designing a manipulator.
In the parallel manipulator, singularities occur in configuration where either Jacobian
matrix A or B becomes singular.
22
2.6.1 Type I Singularity
For given ByAx = , the first type of singularity, Type I, occurs when the following
condition is applied:
0)det( =A (2-24)
From Eq. (2-21) one can obtain:
(2-25)
(2-26)
(2-27)
That is: when those three legs are normal to the O-xy plane, Type I singularity occurs.
2.6.2 Type II Singularity
The second type of singularity-Type II occurs when the following condition is applied:
0)det( =B (2-28)
One can observe those three legs in the O-xy plane when 321 ,, ρρρ reach the
maximum. That are z=0 and 0=φ leading to 0)det( =B . So Type II singularity occurs at
this position.
0cos 1 =− ρφr
0cos 2 =− ρφr
0)( 3 =−+ ρry
23
Chapter 3
Workspace Analysis of a 3-DOF
Parallel Kinematic Machine
3.1 Definition of the workspace
In this thesis, the maximal workspace or reachable workspace is being considered.
The definition is: all the locations of operation point (or moving platform) that may be
reached with at least one orientation of the platform.
3.2 Methods for determining workspace
Various approaches may be used to determine the workspace of a PKM, such as
geometrical method, discretisation method and numerical method.
24
The common one is geometrical method. The purpose of this approach is to determine
geometrically the boundary of the robot workspace. The principle is to deduce from the
constraints on each leg of a geometrical object WL that describes all the possible locations
of the operation points (or moving platform), that satisfy the leg constraints. One such
object is obtained for each leg and the PKM workspace is constituted of the intersection
of all WL.
From Eqs. (2-12), (2-13) and (2-14), one can obtain
(3-1)
(3-1)
(3-3)
Applying this principle to the PKM in this thesis, the WL for each leg can be
represented by Eqs. (3-1), (3-2) and (3-3).
Then the workspace of the manipulator is the intersection of the three enveloping
faces. The main interest of the geometrical approach is that it is usually very fast and
accurate but it requires a good computational geometry library to perform the calculations.
In this case, there is no solution obtained when solving Eqs. (3-1), (3-2) and (3-3) by
Matlab. However, from the principle of this approach we can still obtain the geometrical
relationship between the coordinate system on the moving platform and the one on the
base. Hence we get the formulas that are x, y and z coordinates of the moving platform
represented by the constraints: rotation angles of the joints.
By investigating the structure, we find that the constraints come from the mechanical
limit on the S-joint (maximal rotation angle range from -45 to 45 degree) and the
21
222 )cos()sin( φρφ rLrzy −−=++
22
222 )cos()sin( φρφ rLrzy −−=−+
2223 )]([ Lzry =+−− ρ
25
mechanical interference on the revolute joints, and parallelogram mechanisms. Among
those, the limitation of the rotation angle of the spatial joint is in the dominate position.
That means the rotation angle has reached its limit before the mechanical interferences
happen. As a result, we can obtain the formulas that are x, y and z coordinates of the
moving platform represented by the rotation angle of the spatial joint.
3.3 Workspace analysis of the 3-DOF Parallel
Kinematic Machine
In Fig.2.2, fixed on the base frame, a global reference system xyzOO −: is located at
the point of intersection 21bb and 3Ob . Another reference system, called the moving frame
'''':' zyxOO − , is located at the center of 21 pp on the moving platform. We may consider
the coordinate transformation from xyzOO −: to '''':' zyxOO − as Homogeneous
Transformation passing Ob3, b3P3 and P3O’. Therefore, we have
[ ] [ ]TT zyxzyx 1'''1 A= (3-4)
[ ]zyx ,, and [ ]zyx ′′′ ,, are coordinates in the system O and O’ respectively. A is
Homogeneous Transformation Matrix. Then we can composite Homogeneous
Transformation Matrix A by
'3333 OppbOb AAAA = (3-5)
where
26
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡−
=
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
=
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
=
10000100
0100001
1000cos100sin0100001
10000100
0100001
'3333
rllP
OPPbOb AAAθθ
That is
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−++−
=
1000cos100
sin0100001
22
θθ
LLyzL
A (3-6)
substituting (3-6) into (3-4):
2 2
1 0 0 0
0 1 0 sin0 0 1 cos
1 10 0 0 1
x xy yL z y Lz zL
θθ
⎡ ⎤′⎡ ⎤ ⎡ ⎤⎢ ⎥⎢ ⎥ ⎢ ⎥′ − + +⎢ ⎥⎢ ⎥ ⎢ ⎥= ⎢ ⎥′⎢ ⎥ ⎢ ⎥−⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦⎣ ⎦
(3-7)
we have
2 22 sincos
x x
y y L z Lz z L
θθ
′ =
′ = + − +′ = −
(3-8)
where l is the length of the leg, r is the length of the moving platform’s side shown in
Fig.2.2, θ is the rotation angle of the S-joint from negative z axial to b3P3.
Eq. (3-8) is the geometrical relationship between the coordinate system on the moving
platform and the one on the base. In other words, [ ]zyx ′′′ ,, , the coordinates of the
operation point on the moving frame '''':' zyxOO − , can be represented by [ ]zyx ,, , the
coordinates of the operation point on the global reference system xyzOO −: .
27
Geometrically we know that x’ and z’ have the following relationship in the moving
frame '''':' zyxOO − ,
(3-9)
As a result, we can combine Eqs. (3-8) and (3-9) to obtain the workspace of the PKM
shown in Figure 3.1 (In the graph, x ∈ [-10, 10], y ∈ [-100, 200] and z ∈[-150, 0], unit:
cm).
Fig.3.1 the workspace of the PKM
(Initial input: L=100cm, r=10cm, oo 45 to45−=θ )
The boundary on y-z plan can be shown as Fig. 3.2. The path in the left side is
representing the situation when 0min3 =ρ and °= 45θ , while the path in the right is
showing the situation when 100max3 =ρ and °= 45θ . From the graph, we can see the
222 rzx =′+′
28
boundary of y from the beginning of the left line (y minimum) to the end of the right line
(y maximum) and also the boundary of z from the top or maximum to the bottom or
minimum. We also can conclude that the boundary of x is from –r to r from the
geometrical shape of the moving platform because the structure is designed so that there
is no movement along x axis. In the graph, x ∈ [-10, 10], y ∈ [-100, 200], z ∈[-150, 0],
unit: cm.
Fig. 3.2 The boundary of y-z plane of the workspace
29
Chapter 4
Static Balancing of the Parallel
Kinematic Machine with 3 Degree of
freedom
4.1 Introduction
The static balancing is a very important research topic in the theory of machines and
mechanisms and recently has been applied to the design of Parallel Kinematic Machines
(PKMs). A static balanced PKM has better dynamic characteristics and less vibration
caused by motion. Static balancing is defined as the set of conditions under which the
weight of the links of the mechanism does not produce any torque (or force) at the
actuators under static conditions, for any configuration of the manipulator or mechanism
(Wang and Gosselin 2000). For example, consider a planar parallel robot in a vertical
30
plane. The masses of the links induce torques or force to the actuators and so does the
weight of end-effector. This also happens in the spatial parallel manipulator and the
problem is much more complex. Particularly the problem becomes serious for the parallel
manipulator with a heavy moving platform/end-effector. The aim of the static balancing
is to reduce or ideally to cancel those torques or forces. There are two main methods of
static balancing developed by researchers over decades: (i) making the total mass center
of a mechanism stationary using counterweights and (ii) making the total potential energy
of a mechanism constant using the elastic elements (Ouyang and Zhang 2005).
The first method, counterweight, is also called mass redistribution. This problem was
addressed early by Dunlop and Jones (1996) who suggested the use of counterweights to
balance a 2-DOF parallel robot used for antenna aiming, and by Jean and Gosselin (1996)
who analyzed static balancing of planar parallel manipulator using counterweights and
gave some examples. The second method, using spring, was used by Mahalingam and
Sharan (1986) who used both methods on a serial industrial robot. After that, researchers
began to pay more attention to static balancing of spatial parallel robots. Wang and
Gosselin (2000) investigated four types of spatial four-degree-of-freedom parallel
manipulators who also applied both method to bring the mechanism to static equilibrium
in any configuration of their workspace with zero actuator forces or torques. Ebert-
Uphoff, Gosselin and Laliberte (2000) performed the research on spatial parallel platform
mechanism and discussed the static balancing problem using the adding spring method.
This research was motivated by the use of parallel platform manipulators as motion base
in commercial flight simulators, which was a great example to show how important the
static balancing is. Russo, Sinatra and Xi (2005) focused more on the first method, using
31
counterweights, to study the static balancing of the six-degree-of-freedom platform type
parallel manipulator and derived the conditions for static balancing. However, both
methods have their disadvantages. The drawback of counterweights is that it increases the
joint forces or torques and also adding masses leads to increase the inertia, which has a
negative effect on the dynamic characteristics and energy efficiency. On the other hand,
the method using springs introduces more unknowns and may be difficult for some
situations. Furthermore, it is limited to perform static balancing only on gravity direction.
Recently an interesting idea of static balancing or force balancing method called
Adjusting Kinematic Parameters (AKP) is proposed by Ouyang and Zhang (2005). As
discussed, applying the counterweights method to a mechanism or manipulator is actually
to change the mechanical design of the objective. In fact, the method of Adjusting
Kinematic Parameters follows this principle but in a different way. The counterweights
method changes the mechanical design of the objective by introducing additional masses
or mass redistribution, while the method of AKP modifies the design parameters to
achieve static equilibrium. In this thesis, the parallel Kinematic machine showed in Fig.
2.1 is studied. The methods of Adjusting Kinematic Parameters and counterweights are
both selected to study the static balancing of the PKM.
4.2 Static Balancing using Adjusting Kinematic
Parameters
32
AKP method shares the same principle with Counterweights method as stated above,
so first the total mass center of the PKM is calculated. The total mass of the structure M
can be expressed as:
∑=
+=3
3iiP mmM (4-1)
where Pm is the mass of the moving platform, )3,2,1( =imi is the masses of the thi legs.
So vector of the total mass center of the structure can be found from:
∑=
+=3
1iiiPP mmM rrr (4-2)
where r is the position vector of the total mass center, Pr is the position vector of the
moving platform and ir is the position vector of the thi leg.
Fig.4.1 schematic representation with position vectors of components
From the geographic property of the structure one can see that y axial is parallel to y’
axial. So in the quadrilateral Ob3P3O’, showed in Fig. 4.1, the coordinate of point G,
33
which is the mass center of the moving platform, can be obtained according to the
coordinate of P3. That is G ),3
,0( zyr+ . Then the position vector of the mass center of the
moving platform OG is:
TP zyr ),
3,0( +=r (4-3)
And also in Fig. 4.1, the position vectors of the mass centers of the three legs can also be
calculated. In triangle ii gOP , ig is the mass center of the ith leg.
ii
ii
i LLL
LL bPr −
+= (4-4)
where iii gbL =
That is,
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
+
−−−=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡−−
+⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
+
−=
zLrLyL
LLrL
LLLL
zry
r
LL
11
1
111111
1
sin
)(cos1
00
sin
cos
φ
ρφρ
φ
φr (4-5)
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
+−
−−=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡−
+⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
+−=
zLrLyL
LLrL
LLLL
zry
r
LL
22
2
222222
2
sin
)(cos1
00
sin
cos
φ
ρφρ
φ
φr (4-6)
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡−++=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡−
+⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡+=
zLLLyLrL
LLLL
zyr
LL
3
3333333
3 )(0
1
0
00ρρr (4-7)
Substituting Eqs. (4-5) (4-6) and (4-7) into Eq. (4-2), and let ),,( zyx rrr=r , one can writes
[ ]221112 )()(cos)( ρρφ LLLLrLLMLmrx −+−−−= (4-8)
34
⎥⎦⎤
⎢⎣⎡ −++++++= 33333321 )()(
31 ρLLrLmyLmmLmLLmrLm
MLr PPy (4-9)
[ ]ϕsin)()(1213321 rLLzLmmLmLm
MLr Pz −++++= (4-10)
According to the principle of the Counterweights or Adjusting Kinematic Parameters,
a sufficient condition for the total mass center of the manipulator to be fixed is that the
coefficients in Eqs. (4-8), (4-9) and (4-10) have to be ZERO. Therefore one can conclude,
for example in Eq. (4-8),
02112 =−=−=− LLLLLL (4-11)
That is
LLL == 21 (4-12)
Eq. (4-12) means that the mass centers of leg 1 and leg 2 are both located at vertices
1P and 2P in order to make the x coordinate of the total mass center of the manipulator
stationary. In the real application, that is impossible. So, the method of Adjusting
Kinematic Parameters does not apply in this case and the counterweights method needs to
be used for the PKM.
4.3 Static Balancing using counterweight method
For balancing the 3-DOF PKM, the approach proposed by Russo, Sinatra and Xi
(2005) is adopted. The method to add a pantograph connecting the end-effector to the
base (the pantograph is fixed to the moving platform on the point by a spherical joint
35
and fixed to the point by an universal joint), one counterweight on the end-effector as
well as one counterweight on each leg of the PKM as shown in Figure 4.2.
Fig.4.2 the 3-DOC PKM with the counterweights
In the situation, the total mass of the structure M becomes
∑∑==
+++++=3
1
*3
1
**
ii
iiaaPP mmmmmmM (4-13)
where is the mass of the platform counterweight, is the mass of the legs
counterweights, am and are the mass of the pantograph and the pantograph
counterweight. In this case, the global center of the mass of the manipulator is written as
∑∑==
+++++=3
1
**3
1
****
iii
iiiaaaaPPPP mmmmmmM rrrrrrr (4-14)
where *Pr is the platform counterweight position, *
ir is the legs counterweight
position, ar and *ar is the pantograph and the pantograph counterweight position
vectors. All the position vectors can be derived from Figure 4.2 and then
substituted in to Eq. (4-14), one can write
36
∑∑==
⎥⎦
⎤⎢⎣
⎡+′⋅+⎟⎟
⎠
⎞⎜⎜⎝
⎛−+⎥⎦
⎤⎢⎣⎡ +′⋅++
⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟
⎟⎠
⎞⎜⎜⎝
⎛+⋅++⋅+=
3
1
***
3
1
****
)(121)(
21
)()(
iiii
iiii
aa
aaPP
Ll
Llmm
lmlmmmM
bpQhbpQh
hh
hhgQhgQhr
(4-15)
where g is the vector center of mass of the moving platform with respect to the frame
'''':' zyxOO − , h is the position of 'O with respect to the fixed frame xyzOO −: , p′ is
the position of the moving platform with respect to the moving frame '''':' zyxOO − , ib is
the position vector of the ith actuating joint. Eq. (4-15) can be rewritten as
iCAM bBQhr ++= (4-16)
where
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛−+++++= ∑∑
==
3
1
**3
1
*** 1
21
ii
ii
aa
aapp m
LlmlmlmmmA
hh (4-17)
⎥⎦
⎤⎢⎣
⎡′⎟⎟
⎠
⎞⎜⎜⎝
⎛−+′++= ∑∑
==
3
1
**3
1
** 121
iii
iiipp m
Llmmm ppggB (4-18)
⎥⎦
⎤⎢⎣
⎡+= ∑∑
==
3
1
**3
1 21
ii
ii m
LlmC (4-19)
The conditions for static balancing can be given as
0 ,0 ,0 === CA B (4-20)
From condition 0=C , one can obtain
3,2,1 , 2
3
1*
* =−= ∑=
iLmml
i i
i (4-21)
Negative sign indicates the counterweights should be placed opposite side of the L.
Form condition 0=A , one can obtain
37
⎥⎥⎦
⎤
⎢⎢⎣
⎡++++−= ∑
= hh a
ai
iippa
almmmmm
ml
3
1
***
* )( (4-22)
Negative sign indicates the counterweights should be placed opposite side of the al .
Finally from condition 0=B , one can obtain
*
3
1
*
*)(
p
iiiip
m
mmm ∑=
′++=
pgg (4-23)
In Eq. (4-23), ip′ is constant while g , the position vector of the mass center of the
moving platform, is not. However, one can fix the mass center of the moving platform at
point O′ so as to make *g stable. Thus all three conditions for static balancing are
satisfied and the static balancing of the 3-DOF PKM is achieved.
For the design improvement, the next step should be optimization of the
counterweights, ** , , aaP mmm and *im . As stated at the beginning, the drawback of
counterweights is that adding masses leads to increase the inertia, which has a negative
effect on the dynamic characteristics and energy efficiency. So, for example, the
optimization objective can be set up to reduce the total mass of the counterweights by
optimum design of ** , , aaP mmm and *im .
38
Chapter 5
Stiffness Analysis of the 3-DOF
Parallel Kinematic Machine
5.1 Introduction
In many applications, stiffness is a very important performance specification for
Parallel Kinematic Machines (PKM), because it is strictly related to accurate positioning
and high payload capability.
Three main methods have been used to generate and analyze the mechanism stiffness
models.
The first method is based on matrix structural analysis for the calculation of the
stiffness matrix of PKM. Li, Wang, and Wang (2002) used this approach to establish the
stiffness model of a Stewart platform-based PKM, considering the deformation of the
39
frame. Yoon, Suehiro, and Tsumaki (2004) applied this method on a compact modified
Delta Parallel Mechanism they developed to derive the compliance matrix by considering
elastic deformations of both parts and bearings in the structure. Deblaise, Hernot and
Maurine (2006) also adopted matrix structural analysis on a more general case to derive
the stiffness matrix for each element of a Stewart platform and then assemble individual
ones into a system stiffness matrix. This approach fits well with machines that can be
modeled with beam elements, and it easily takes into account all mechanical effects
acting on all elements of their structure.
The second methods are those based on the Finite Element Analysis. Corradini and
Krut (2003) used Finite Element Analysis to evaluate the stiffness on a multi-beam
articulated model of a PKM where all the joints are translated into displacement
relaxations. Bouzgarrou, Fauroux, and Gogu (2004) built a global meshed FEM model of
a PKM and performed stiffness study.
The third method relies on the calculation of the parallel mechanism’s Jacobian
Matrix. One of the first stiffness analyses was conducted by Gosselin (1990). In that work,
the stiffness of a PKM is mapped onto its workspace whereas links are supposed as
perfectly rigid. The approach has been widely adopted in most of the works on stiffness
analysis. El-Khasawneh and Ferreira (1999) addressed the problem of finding the
minimum and maximum stiffnesses and the directions in which they occur for a
manipulator in a given posture and also discussed the computation of stiffness in an
arbitrary direction. Sanger, Chen, and Zhang (2000) concerned the displacement of the
end-effector of a manipulator when subjected to an externally applied force system, and
used Jacobian Matrix to derive the end-effector stiffness and compliance for serial and
40
parallel manipulators. Zhang and Liang (2004) applied this method into the research of
reconfigurable manufacturing system and introduced lumped models for joints and link
compliances to establish a general stiffness model for the family of reconfigurable PKMs.
All of the above stiffness mappings using the calculation of the parallel mechanism’s
Jacobian Matrix are under the assumption of no pre-loading in PKMs. On the other hand,
some researchers extended the study to the situation considering the stiffness effect
introduced by pre-loading. For example, Svinin, Hosoe and Uchiyama (2002) as well as
Simaan and Shoham (2003) studied the stiffness matrix expression which contained the
effect coursed by pre-loading (self-weight of the moving platform or the antagonistically
acting driving forces). However, they also stated that pre-loading effect could be
neglected for manipulators with high joint stiffness and depended on the specific
applications.
5.2 General stiffness Model
The stiffness of a parallel mechanism is dependent on the joint's stiffness, the leg's
structure and material, the platform and base stiffness, the geometry of the structure, the
topology of the structure and the end-effector position and orientation. Since stiffness is
the force corresponding to coordinate i required to produce a unit displacement of
coordinate j , the stiffness of a parallel mechanism at a given point of its workspace can
be characterized by its stiffness matrix. This matrix relates the forces and torques applied
at the gripper link in Cartesian space to the corresponding linear and angular Cartesian
displacements. It can be obtained using kinematic and static equations (Zhang 2000).
41
The stiffness of a PKM may be evaluated by using an elastic model for the variations
of the joint variables as functions of the forces that are applied to the link. In this model,
the change θδ in the joint variables θ when a joint force f is applied on the link is
expressed as
θf δk= (5-11)
where k is the elastic stiffness of the link, supposed to identical for all legs.
From the velocity equation
XJθ && = (5-12)
where θ& is the vector of joint rates, and X& is the vector of Cartesian rates. Matrix J is
Jacobian matrix, one can conclude that
XJθ δδ = (5-13)
where θδ and Xδ represent joint and Cartesian infinitesimal displacements, respectively.
The forces and moments applied at the gripper under static conditions are related to the
forces or moments required at the actuators to maintain the equilibrium by the transpose
of the Jacobian matrix J. That is
fJF T= (5-14)
where f is the vector of actuator forces or torques, and F is the generalized vector of
Cartesian forces and torques at the gripper link.
Then substituting Eqs. (5-11), (5-13) into Eq. (5-14), yields
XJJF δTk= (5-15)
Hence, K, the stiffness matrix of the mechanism in the Cartesian space is then given by
the following expression
JJK Tk= (5-16)
42
which is the equation given in (Gosselin 1990).
5.3 Stiffness mapping of 3-DOF Parallel Kinematic
Machine
From Eqs. (2-21), (2-22) and (2-23) in Section 2.4, the Jacobian matrix can be
expressed as
BAJ 1−=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−+−
−=
3
2
1
000cos000cos
ρρφ
ρφ
ryr
rA
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−+−−++
=0
)cossin(sin)cossin(sin
3
2
1
zryzrrzyzrrzy
ρφφρφφφρφ
B
Substituting Eqs. (2-21) (2-22) into (2-23), that is
⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
−+
−−
−−
−
−+
−+
−
=
01
cos)cossin(
cossin
cos
cos)cossin(
cossin
cos
3
2
2
22
1
1
11
ρ
ρφφφρ
ρφφ
ρφ
ρφφφρ
ρφφ
ρφ
ryz
rzr
rrz
ry
rzr
rrz
ry
J (5-17)
Now the Jacobian matrix in Eq. (5-17) can be substituted into the stiffness model in
Eq. (5-16) in order to obtain the stiffness maps for the PKM. A program has been written
with the software Matlab. Given the values of the length of the leg, L (L = 100cm, 120cm,
43
150cm and 200cm), the length of the triangle side (the moving platform), r (r=5cm, 10cm,
15cm and 20cm), and the stiffness of the actuators k (k = 1000 N/m) the stiffness mesh
graphs and contour maps in X, Y and Z with adjusting kinematic parameters are shown in
Figures 5.1-5.12.
From the stiffness mesh graphs in X (Figs. 5.1 and 5.3) and Y (Figs. 5.5 and 5.7), one
can conclude that the stiffnesses in X and Y axis increase when the magnitude of y
coordinate increases, while they have little changes with varied rotation angles. That
means the position of the end-effector is the main factor to determine the stiffness in the
situations. On the contrary, from the stiffness mesh graphs in Z (Figs. 5.9 and 5.11), one
can conclude that the stiffness in Z axis increases when the magnitude of rotation angle φ
increases, while it has comparatively small changes with varied y coordinates. In this
situation, the rotation angle φ takes the major role of determining the stiffness. The
conclusions also give the reason of essential difference between Figs. 5.3 and 5.11. The
curve is smooth In Fig.5.3 because the rotation angle φ is slightly modified the stiffness
in X axis; while, in Fig. 5.11, the curve has sharp changes at some points which
demonstrates discontinuity in the slop. This is because the rotation angle φ is the major
factor which greatly changes the stiffness in Z axis.
By comparing the stiffness mesh graphs and contour maps in X (Figs.5.1 and 5.2), Y
(Figs.5.5 and 5.6) and Z (Figs.5.9 and 5.10) with the different length of the legs, one can
conclude that the stiffness is increasing when the length of the legs is decreasing and vice
versa. This trend applies in the X, Y, and Z directions. For example, in Fig. 5.2 Stiffness
contour graphs in X direction, different stiffnesses of the structure with different leg
lengths are shown in the following table.
44
L (m) K-minimum read in graph (N/m) K-maximum read in graph (N/m)
1.00 1200 3000
1.20 1100 2000
1.50 1050 1500
2.00 1050 1250
Table 3 Stiffness in X vs. length of the leg
On the other hand, by comparing the stiffness mesh graphs and contour maps in X
(Figs.5.3 and 5.4), Y (Figs.5.7 and 5.8) and Z (Figs.5.11 and 5.12) with the different
length of the triangle side (the moving platform), one can conclude that the stiffness in X
is keeping unchanged when the length of the triangle side is changing. But the stiffness in
Y and Z are increasing when the length of the triangle side is increasing. For example, in
Fig. 5.8 Stiffness contour graphs in Y direction, different stiffnesses of the structure with
different lengths of the triangle side (the moving platform) are shown in the following
table.
r (m) K-minimum read in graph (N/m) K-maximum read in graph (N/m)
0.05 550 950
0.10 550 1000
0.15 600 1100
0.20 600 1200
Table 4 Stiffness in Y vs. lengths of the triangle side (the moving platform)
45
As a result, the desired stiffness on X, Y and Z directions can be achieved by
adjusting kinematic parameters. Moreover, when using this 3-DOF PKM as a machine
tool, one can judge if the PKM is good enough to perform the machining job by
considering the workloads in X, Y and Z directions. From the stiffness contour graphs in
the three directions, one can always choose a suitable path to finish the machining task
under the specific stiffness. For example, showing in the stiffness contour graphs in X
when L = 1.00m, if the workload in X direction requires the stiffness under 1200 N/m, all
the working paths located in the region under 1200 N/m could be chosen. However, if the
workload increases, the path could be moved up to the region between 1400 N/m and up.
At the same time, upon the desired stiffness, there are PKMs with different designed
lengths of legs to serve the different situation, such as; the task requires the PKM to reach
the objective in a further distance.
On the other hand, in the situations when the lengths of legs are required to be fixed,
the desired stiffness still can be achieved by adjusting the length of the triangle side of the
moving platform.
46
Fig. 5.1 Stiffness mesh graphs in X with L= 100 cm to 200 cm, r=10cm
47
Fig. 5.2 Stiffness contour graphs in X with L= 100 cm to 200 cm, r=10cm
48
Fig. 5.3 Stiffness mesh graphs in X with L= 100 cm, r=5cm to 20 cm
49
Fig. 5.4 Stiffness contour graphs in X with L= 100 cm, r=5cm to 20 cm
50
Fig. 5.5 Stiffness mesh graphs in Y with L=100cm to 200cm, r=10cm
51
Fig. 5.6 Stiffness contour graphs in Y with L=100cm to 200cm, r=10cm
52
Fig. 5.7 Stiffness mesh graphs in Y with L=100cm, r=5cm to 20 cm
53
Fig. 5.8 Stiffness contour graphs in Y with L=100cm, r=5cm to 20cm
54
Fig. 5.9 Stiffness mesh graphs in Z with L=100cm to 200cm, r=10cm
55
Fig. 5.10 Stiffness contour graphs in Z with L=100cm to 200cm, r=10cm
56
Fig. 5.11 Stiffness mesh graphs in Z with L=100cm, r=5cm to 20cm
57
Fig. 5.12 Stiffness contour graphs in Z with L=100cm, r=5cm to 20cm
58
5.4 The Stiffness Optimization of the 3-DOF Parallel
Kinematic Machine
5.4.1 Genetic Algorithms
The genetic algorithms (GA) is a very good method for solving both constrained and
unconstrained optimization problems. It is based on the natural selection. The genetic
algorithm repeatedly modifies a population of individual solutions. At each step, the
genetic algorithm selects individuals at random from the current population to be parents
and uses them produce the children for the next generation. From generations to
generations, the population is going toward an optimal solution. The genetic algorithm
could be applied to solve a variety of optimization problems, such as including problems
in which the objective function is liner or nonlinear, continuous or discontinuous and
differentiable or non-differentiable, etc.
In the thesis, GA is applied to optimize the global stiffness globalK of the 3-DOF PKM.
Stated in Eq. (5-16), the stiffness of the PKM is expressed by a 3 × 3 matrix. The
diagonal elements of the matrix are the manipulator’s pure stiffness in each direction
(Zhang 2000). To obtain the maximum stiffness in each direction, one can write the
following objective function, or called the fitness function in GA.
332211 kkkKglobal ++= (5-18)
59
where )3 ,2 ,1( =ikii represents the diagonal elements of the PKM’s stiffness matrix.
Then the objective is to maximize globalK in GA.
Before running GA, one has to set up some parameters for it. First, the fitness function
should be created. It is the function of the objective being optimized. In this case, the
fitness function is the Eq. (5-18). The optimization functions in the GA minimize the
objective or fitness function, for example, f(x). If the objective is to maximize the f(x), it
can be done by minimizing -f(x), because the point at which the minimum of -f(x) occurs
is the same as the point at which the maximum of f(x) occurs.
Second, the number of variables is to be determined. In the PKM structure, there are
two design parameters considered to be optimization variables. They are the length of the
leg “L” and the length of the end-effector’s side “r”. It is noticed that values of
coordinates y and φ in Eq. (5-18) also contribute to the val . So the vector of optimization
variables is therefore
],,,[ Lry φ=s (5-19)
and their bound are
y∈[-100, 100] cm, φ ∈[- 2/ ,2/ ππ ] rad, L∈[100, 200] cm, r∈[5, 20] cm.
Finally, the population size and the generation number have to be selected. The
generation number is the maximum number of iterations the GA performs and the
population size specifies how many individuals there are in each generation. In this case,
the population size is set to 20, and the maximum generation number is 100.
5.4.2 Optimization Result
60
The following figure displays a plot of the best and mean values of the fitness function
at each generation. The points at the bottom of the plot denote the best fitness values,
while the points above them denote the averages of the fitness values in each generation.
The plot also displays the best and mean values in the current generation numerically at
the top of the figure.
Fig. 5.13 The best fitness and the best individuals of the global stiffness optimization
The optimal parameters are obtained after 59 generations as follows:
],,,[ Lry φ=s = [-70.2469 0.7197 19.9878 100.0273]
which suggest the optimal design value for the length of the leg and the length of the end-
effector’s side are
L = 100 cm, r = 20 cm
61
and the maximum global stiffness of the PKM is
332211 kkkKglobal ++= =9527.3442 (N/m)
The following result of the global stiffness optimization is output from GA.
The result suggests that the leg length of the PKM should be 100cm and the length of
the end-effector’s side should be 20cm in order to make the global stiffness of the PKM
reaches maximum. That also corresponds with the conclusions in section 5.3.
62
Chapter 6
Dynamic Modeling of the 3-DOF
Parallel Kinematic Machine
Dynamics of PKMs is the science of studying the forces required to cause motion. To
accelerate a PKM from rest to a desired speed, then decelerate and finally back to the rest
again, a complex set of forces or torques must be applied by joint actuators. Therefore,
finding the relationships between the accelerations, velocities and positions of the end-
effector and the joint forces is the main task for dynamic analysis of PKMs. Those
relationships can be obtained by dynamic modeling, or in other words, finding the
dynamic equations of motion. The dynamic equations of motion generally serve two
purposes on PKMs: control and simulation. When controlling a PKM in a desired motion,
one needs to calculate these actuator torques using the dynamic equations of motion. On
the other hand, by rearranging the dynamic equations so that accelerations and velocities
are computed as the function of actuator torques, it is possible to simulate how a PKM
would move under the application of actuator torques. An understanding of the
63
manipulator dynamics is important from several different perspectives. First, it is
necessary to properly define the size of the actuators and other manipulator components.
Without a model of the manipulator dynamics, it becomes difficult to predict the actuator
force requirements and in turn equally difficult to properly select the actuators. Second, a
dynamics model is useful for developing a control scheme. With an understanding of the
manipulator dynamics, it is possible to design a controller with better performance
characteristics than would typically be found using heuristic methods after the
manipulator has been constructed. Moreover, some control schemes such as the computed
torque controller rely directly on the dynamics model to predict the desired actuator force
to be used in a feed forward manner. Third, a dynamical model can be used for computer
simulation of a robotic system (Wu 2008).
6.1 General methodologies of dynamic modeling
As stated in the previous chapters, parallel robots have various practical advantages
over serial robots and have been widely used in industries. However, the dynamic
modeling of parallel robots presents an inherent complexity, due to their closed-loop
structure and kinematic constraints (Khalil and Guegan 2004). Moreover, the model
algorithms cannot be generalized. When used in a real-time control framework the
resulting models must be simplified as they usually demand a very high computational
effort (Wu 2008). The dynamic model of a parallel manipulator is usually developed
following one of two approaches (Callegari 2006): the Newton-Euler or the Lagrange
methods. The Newton-Euler approach uses the free body diagrams of the rigid bodies.
64
The Newton-Euler equation is applied to each single body and all forces and torques
acting on it are obtained. Harib and Srinivasan (2002) used the Newton-Euler formulation
to derive the rigid body dynamic equations when doing dynamic analysis of Stewart
platform-based machine tool structures. The Lagrange method describes the dynamics of
a mechanical system from the concepts of work and energy. This method enables a
systematic approach to the motion equations of mechanical systems. Li (1988) presented
a computational method to derive the complete Lagrange-Euler equations of motion for
robot manipulators, which reduced the number of computations when using the
Lagrange-Euler method. By comparison of the two methodologies, the Lagrange-Euler
method which is built based on the viewpoint of energy is a mature one (Zhu et al. 2007).
Although solving Lagrange equation is very complex and there are large amounts of
calculations, modern computing software like Matlab could give the solution. Therefore,
Lagrange-Euler method is used in this thesis to build the dynamic model of the 3-DOF
PKM. Because Lagrange-Euler method is an "energy-based" approach to dynamics, one
should start by developing the expressions of kinematic energy and potential energy of
the manipulator. The very familiar expressions of kinematic energy and potential energy
of a point mass are well known as
mghumvk == ,21 2 (6-1)
while from (Craig 1989) the kinetic energy of a manipulator is given by
ΘΘΘΘΘ, &&& )(21)( Mk T= (6-2)
where )(ΘM is the nn× manipulator matrix, and Θ is the vector of joint angles of the
manipulator. It is noticed that the kinetic energy of a manipulator can be described by a
65
scalar formula as a function of joint position and velocity, )( ΘΘ, &k . Similarly the
potential energy of a manipulator can be written as
∑∑==
−==n
iC
Ti
n
ii i
muu11
)()( ΘPgΘ (6-3)
where g is 13× gravity vector and )(ΘPiC is the position vector of the mass center of
the thi link. The potential energy of a manipulator can also be described by a scalar
formula as a function of joint position, )(Θu .
The Lagrange-Euler method provides a means of deriving the equations of motion from a
scalar function called the Lagrangian, which is defined as the difference between the
kinetic and potential energy of a mechanical system. The Lagrangian of a manipulator is
)()()( ΘΘΘ,ΘΘ,L uk −= && (6-4)
The equations of motion for the manipulator are then given by
τΘΘΘ
=∂∂
+∂∂
−∂∂ ukk
dtd
& (6-5)
where τ is the 1×n vector of actuator torques.
6.2 Dynamic Modeling of the 3-DOF Parallel
Kinematic Machine
Using the Lagrange Method described above, one can establish the dynamic model of
the 3-DOF PKM by deriving the equation of motion. Giving the condition that the legs of
the PKM have the same mass and moment of inertias on all directions and the mass of the
66
end-effector is also as same as that of the leg, one can write the kinematic energy and
potential energy expressions of the legs as follows
222
21
21
221
iiii mILmk ρθθ &&& ++= (6-6)
where )3,2,1( =iki are kinematic energies of the legs, and )3,2,1( =iiρ are travelling
distances of the prismatic joints which are obtained by investigating the geometry
structures shown in Fig.6.1.
12
21
22
1 cos)sin(sin4
θθθρ LLr +−−= (6-7)
22
21
22
2 cos)sin(sin4
θθθρ LLr +−−= (6-8)
rL+−−+−= 4)sinsin3()sinsin3(
2 212
213 θθθθρ (6-9)
Fig. 6.1 The planar structures of the PKM
The kinematic energy of the end-effector is written as
23
2234 2
121
21 ραθ α &mImLk ++= (6-10)
67
where is the moment of inertia of the end-effector, is the rotation angle of the end-
effector and
rL
2)sin(sinsin 211 θθα −
= − (6-11)
So the total kinematic energy of the PKM is
PImLImLImLkki
i +⎟⎠⎞
⎜⎝⎛ ++⎟
⎠⎞
⎜⎝⎛ ++⎟
⎠⎞
⎜⎝⎛ +==∑
=
23
22
21
4
1 21
41
21
41
21
41 θθθ &&& (6-12)
where
⎟⎠⎞
⎜⎝⎛ +++= αρρρ &&&&
mImP P2
322
212
1 (6-13)
The total potential energy of the PKM is obtained as
)sin3sin(sin2
sin)sinsin(sin2
321
3321
θθθ
θθθθ
++−=
−++−=
Lmg
mgLLmgu (6-14)
From Eqs. (6-12) and (6-14), one can write
( )333
33
2222
2
1111
1
,
,21
,21
θθθθ
θ
θθθθ
θ
θθθθ
θ
∂∂
=∂∂
∂∂
++=∂∂
∂∂
=∂∂
∂∂
+⎟⎠⎞
⎜⎝⎛ +=
∂∂
∂∂
=∂∂
∂∂
+⎟⎠⎞
⎜⎝⎛ +=
∂∂
PkPdtdImLk
dtd
PkPdtdImLk
dtd
PkPdtdImLk
dtd
&&&
&
&&&
&
&&&
&
(6-15)
and
333
222
111
cos23 ,cos
21 ,cos
21 θθ
θθθ
θθθ
θ&&& mgLumgLumgLu
−=∂∂
−=∂∂
−=∂∂ (6-16)
68
Substituting Eqs. (6-15) and (6-16) into Eq. (6-5), one can obtain the equation of motion
of the 3-DOF PKM as
( )⎪⎪⎪⎪⎪
⎩
⎪⎪⎪⎪⎪
⎨
⎧
=−∂∂
−∂∂
++
=−∂∂
−∂∂
+⎟⎠⎞
⎜⎝⎛ +
=−∂∂
−∂∂
+⎟⎠⎞
⎜⎝⎛ +
33333
3
22222
2
11111
1
cos23
cos21
21
cos21
21
τθθθθ
θ
τθθθθ
θ
τθθθθ
θ
&&
&&
&&
&&
&&
&&
mgLPPdtdImL
mgLPPdtdImL
mgLPPdtdImL
(6-17)
where )3,2,1( =iiτ are actuator forces at three prismatic joints of the PKM.
Based on the dynamic model expressed in Eq. (6-17), the control and simulation of the 3-
DOF PKM can be conducted.
6.3 Dynamic Simulation of the 3-DOF Parallel
Kinematic Machine
Dynamic simulation investigates how the mechanism will move under application of a
set of actuator torques. That is, given a set of actuator torques, iτ , calculate the resulting
motion of the manipulator, θ& and θ&& . In section 5.2, the dynamic model of the 3-DOF
PKM has been established, so dynamic simulation of the motion can be conducted based
on that model.
69
Fig. 6.2 CAD model of 3-DOF PKM for dynamic simulation
In this simulation, the following parameters are used
2321321 /400 ,2 cmkgIIIIkgmmmm PP ========
The actuator forces are given as
N16321 === τττ
The three legs named leg 1, leg 2 and leg 3 are shown in Figure 6.2. The angular velocity
)3,2,1( =iiω and angular acceleration )3,2,1( =iiα of three legs are computed in Figures
6.3-6.8.
Fig. 6.3 The angular velocity 1ω of leg 1
70
Fig. 6.4 The angular acceleration 1α of leg 1
Fig. 6.5 The angular velocity 2ω of leg 2
Fig. 6.6 The angular acceleration 2α of leg 2
71
Fig. 6.7 The angular velocity 3ω of leg 3
Fig. 6.8 angular acceleration 3α of leg 3
Figures 6.3 to 6.4 indicate that angular velocities and accelerations of the leg 1 and leg
2 have same magnitude but opposite direction. This is because those two legs are
symmetrical and made of identical chains, and the motion on X axis is constrained, so leg
1 and leg 2 cannot move towards each other. When given the same actuator forces, the
leg 1 and leg 2 have same motion in opposite direction, which makes the end-effector
translate in Z axis. On the other hand, when the leg 3 is given the actuator force, it is
moving at the angular velocity and acceleration shown in Figures 6.7 and 6.8, driving the
72
end-effector translating in Y axis. The curve in Figure 6.8 is not smooth because of the
coupling effect from leg 1 and leg 2.
73
Chapter 7
Conclusions and Future work
7.1 Conclusions
The research described in this thesis is aimed at design of a new PKM with the focus
on system stiffness. The PKM possesses 3 DOF, and is potentially used as a machine tool
in various applications of manufacturing. The CAD model of the design is created; the
inverse kinematics is analyzed; the Jacobian matrix and velocity equations are derived;
the two types of singularities of the structure are studied; and the applied conditions are
given. Upon the new designed PKM, the workspace simulation is presented. The static
balancing is investigated and the conditions for static balancing are given. More
importantly, the study of this thesis gives the solution of how to reconfigure a parallel
kinematic machine with adjusting kinematic parameters and path re-planning to achieve a
desired stiffness. The optimization of the global stiffness of the PKM is performed and
the optimal design parameters are suggested. Equations of motion of the 3-DOF PKM are
74
derived by using the Lagrange Method for dynamic modeling and the dynamic simulation
of the 3-DOF PKM is conducted. In the proceeding chapters, a number of detailed results
are already provided on the respective topics. They are now highlighted again as the
conclusions.
1. Novel design of 3-DOF PKM. This PKM has the advantage of combining the
spatial rotational and translational degrees of freedom, which performs an
important function when used as a machine tool. Moreover, the spatial rotational
DOF results from the two simple linear motion inputs. The application of
parallelogram joints on the two identical legs donates the high stiffness and good
symmetry to the PKM. The simplicity in structure design not only achieves the
objective functions but also leads to the easy inverse kinematics and Jacobian
matrix, which reduces the computational operations in the later analyses of the
PKM.
2. Workspace simulation of the PKM. The geometrical method is selected to
determine the workspace of the PKM. The workspace simulation graphically
describes all the locations of operation points which the end-effector can reach.
That is helpful to define the reach ability of the PKM.
3. The general stiffness model and stiffness mapping of the PKM. The general
stiffness model for PKMs is established. The reliability of this model is verified in
application to the new 3-DOF PKM. As a result, the stiffness mapping of the
PKM is presented. By comparing the stiffness mesh graphs and contour maps, one
can conclude that the stiffness is increasing when the length of the legs is
decreasing and vice versa. In addition, the stiffness mapping offers a means of
75
achieving the desired stiffness of the PKM by adjusting kinematic parameters or
path re-planning.
4. Optimization of the global stiffness. The Genetic Algorithm (GA) is applied to
optimize the global stiffness of the 3-DOF PKM. The optimization result is
obtained after 63 generations. It suggests the optimal design value for the length
of the leg and the length of the end-effector’s side are L = 100 cm, r = 19.99 cm
and the maximum global stiffness of the PKM is 9527.3442 (N/m). From the
results, it can be seen that the optimal parameters should be selected when using
adjusting kinematic parameters method to achieve the maximum global stiffness
of the PKM.
5. Dynamic modeling. Dynamics is the science of studying the forces required to
cause motion and plays an important role in the trajectory generation and control
of parallel robots. In this thesis, the analysis focuses on dynamic modeling of the
PKM. By considering the energy of the whole structure, the expressions of
kinematic and potential energy of the PKM are developed. Then Equations of
motion of the 3-DOF PKM are derived by using the Lagrange Method for
dynamic modeling, which can be used for the simulation and control purposes.
The objective of the research in this thesis is successfully achieved with the focus on
system stiffness as well as analyses on other issues. All the approaches developed
regarding the new 3-DOF PKM could be well extended to other parallel mechanisms.
7.2 Future Work
76
The stiffness model built in this thesis doesn’t consider the material properties of the
structure. For improvement, the lumped models for links and joints should be introduced.
This will involve in the “E - elastic modulus” and “I - section moment of inertia” which
reflect the material properties of the structure. As mentioned in the literature review in
Chapter 4, there are three kinds of research methods on stiffness analysis. These will help
to verify the result given in this thesis if the study is repeated by using the other two
approaches.
The dynamic modeling is presented with the assumption of the followings: 1. legs of
the PKM have the same mass and moment of inertias on all directions; 2. the mass of the
end-effector is as same as that of the leg. For more general cases, different masses and
moment of inertias of the rigid bodies which form the PKM should be considered.
Moreover, the optimization on the mass distribution and moment of inertia is encouraged
to be done for the best dynamic performance of the PKM.
On the other hand, no consideration was given to the PKM’s control issue. This issue
is worth further research before applying this 3-DOF PKM in the industry.
77
Reference
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