A NEW METHOD OF POSITION AND FORCE
CONTROL FOR
ROBOTIC DEBURRING AND GRINDING
By
KUOSHU TING
Diploma
Mingchi Institute of Technology
Taipei, Taiwan
1991
Submitted to the Faculty of the
Graduate College of the
Oklahoma State University
in partial fulflilment of
the requirements for
the Degree of
MASTER OF SCIENCE
December, 1996
A NEW .METHOD OF POSITION AND FORCE
CONTROL FOR
ROBOTIC DEBURRlNG AND GRINDING
Thesis Approved:
Dean of the Graduate College
u
ACKNOWLEDGMENTS
Countless thanks to Dr. Lawrence L. Hoberock, my major adviser, for his
encouragement, advice, and many hours of guidance throughout by graduate program.
Many thanks also go to Dr. Eduardo Misawa and Dr. Gary E. Young for serving on my
committee. Their suggestions and support were very helpful throughout tbis study.
Thanks to Duli Hong for his literature review and earlier work. Financial support
during my graduate study was provided by the Oklahoma Center for Integrated Design
and Manufacturing, together with the Oklahoma Center for the Advancement of Science
and Technology.
This work is dedicated to my mother, JinYueh Lin, for her love, understanding,
and encouragement over the years.
III
TABLE OF CONTENTS
Chapter Page
1. INTRODUCTION. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 1
Automation Using Robotics .. . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . .. 1
Problem Background. ........... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 2
Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 4
Objectives of This Study. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
II. SYSTEM MODELING . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 11
Manipulator Dynamics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 11
UCBerkeley/NSK SCARA Robot .................................. 15
Grinding Modeling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
Ill. CONTROL APPROACHES . ............... ... ............ .. ... . ... 32
Feedback Linealization .. . ........................ . ............... 32
Impedance Control ......... . ................... .. .. . ........... 35
Hybrid Impedance Control. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
Switching Control .......... . .... . .............................. 44
IV. COMPUTER SIMULATIONS ..... .. ..................... . ......... . 51
Simulation Parameters and Motion Plan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
Burr Simulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 54
Simulations for Impedance Control .... . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 60
Simulations for Hybrid Impedance Control .. ... . ..... .. ... . ......... . 83
Simulations for Switching Control ...... .. ...... . ................... 91
V. CONCLUSIONS AND RECOMMENDATIONS ............... . .. . ... . . 128
Summary and Conclusions ... . ................................... 128
Recommendations ............................. ............... .. . 131
REFERENCES ....... .. .......................... . ............... .. 132
iv
APPENDIX ........................................................ 136
APPENDIX ANSK MOTOR SPECIFICATIONS .................... 137
v
LIST OF TABLES
Table Page
1. Robot Links Specifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 25
vi
LIST OF FIGURES
Figure Page
1.1 Grinding an Edge 3
2.1 Schematic Diagram of TwoArm SCARA Robot. . . . . . . . . . . . . . . . . . . . . .. .. 6
2.2 Robotic Grinding Schematic [3] ....... . ... . . . . .. . .. . ... .. .... .. ... . . 19
2.3 Motor and Arm Configuration for Stiffness Calculation . . ...... . .. . .... . . . 26
3.1 Diagram of Control Structure [3] ..... . .... . . . ..... . . . . . . .. . . . . ...... 35
3.2 Hybrid Impedance Control Diagram ........... . .... . ................. 42
4.1 End Point Motion of Robot to Desired Trajectory .. . . . ... . .. . .. . ... .. ... 53
4.2 Motion History of TwoArm SCARA Robot ........................... 54
4.3 Geometry of Sinusoidal Burrs ... . ..... . .................... . .... . ... 56
4.4 Illustration of Large Upset Burrs .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.5 Illustration of Scallop Burrs . ..................... . ..... . .... . ...... 58
4.6 SIMULINK Block Diagram. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.7 Results of Simulation 1 with Impedance Control: Position Errors
Smooth Straight Edge .................. . . . . . . . . . . . . . . . . . . . . . . . .. 63
4.8 Results of Simulation 1 with Impedance Control: External Forces
Smooth Straight Edge . ... . ........ . ........... . ............ . .... 64
4.9 Results of Simulation I with Impedance Control: Motor Torques
Smooth Straight Edge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.10 Results of Simulation 2 with Impedance Control: Position Errors
RandomHeight Sinusoidal Burrs ......... . ......... . ... . ..... . .. .. , 66
Vll
4.11 Results of Simulation 2 with Impedance Control: External Forces
RandomHeight Sinusoidal Burrs ............................... . ... 67
4.12 Results of Simulation 2 with Impedance Control: Motor Torques
RandomHeight Sinusoidal Burrs ....................... . ... . ....... 68
4.13 Results of Simulation 3 with Impedance Control: Position Errors
RandomHeight Sinusoidal Burrs with Desired Force Compensation . .... . .. 69
4.14 Results of Simulation 3 with Impedance Control:
Depth of Cut and Remaining Depth of Cut
RandomHeight Sinusoidal Burrs with Desired Force Compensation . . .. .. . . 70
4.15 Results of Simulation 3 with Impedance Control: External Forces
RandomHeight Sinusoidal Burrs with Desired Force Compensation ........ 71
4.16 Results of Simulation 3 with Impedance Control: Motor Torques
RandomHeight Sinusoidal Burrs with Desired Force Compensation .... .. .. 72
4.17 Results of Simulation 4 with Impedance Control: Position Errors
Large Upset Burrs . ..... . ....................................... 73
4.18 Results of Simulation 4 with Impedance Control:
Depth of Cut and Remaining Depth of Cut
Large Upset Burrs ............................. . ....... . ........ 74
4.19 Results of Simulation 4 with Impedance Control: External Forces
Large Upset Burrs ..... . ................ . ...... . .... ............. 75
4.20 Results of Simulation 4 with Impedance Control: Motor Torques
Large Upset Burrs ....... . .. . .................. .. . . . ............ 76
4.21 Results of Simulation 5 with Impedance Control: Position Errors
Large Upset Burrs with Desired Force Compensation ...... . ...... . ..... 77
4.22 Results of Simulation 5 with Impedance Control:
Depth of Cut and Remaining Depth of Cut
Large Upset. Burrs with Desired Force Compensation .. . . . . . . . . . . . . . . . . . 78
4.23 Results of Simulation 5 with Impedance Control: External Forces
Large Upset Burrs with Desired Force Compensation ... . ........ . ..... , 79
4.24 Results of Simulation 5 with Impedance Control: Motor Torques
Large Upset Burrs with Desired Force Compensation ..... . ...... . ..... , 80
VllJ
4.25 Results of Simulation 6 with Hybrid Impedance Control: Position Errors
Smooth Straight Edge ........................................ . .. 85
4.26 Results of Simulation 6 with Hybrid Impedance Control:
Depth of Cut and Remaining Depth of Cut
Smooth Straight Edge ..... . ..................................... 86
4.27 Results of Simulation 6 with Hybrid Impedance Control: External Forces
Smooth Straight Edge ...................................... . . . .. 87
4.28 Results of Simulation 6 with Hybrid Impedance Control: Position History
Smooth Straight Edge .................... . . . . . . . . . . . . . . . . . . . . . .. 88
4.29 Results of Simulation 6 with Hybrid Impedance Control: Motor Torques
Smooth Straight Edge .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .. 89
4.30 Results of Simulation 7 with Switching Control: Position Errors
RandomHeight Sinusoidal Burrs, ETG Workpiece .............. . ...... 95
4.31 Results of Simulation 7 with Switching Control: External Forces
RandomHeight Sinusoidal Burrs, ETG Workpiece .................. . .. 96
4.32 Results of Simulation 7 with Switching Control: Motor Torques
RandomHeight Sinusoidal Burrs, ETG Workpiece .......... .. ......... 97
4.33 Results of Simulation 8 with Switching Control: Position Errors
RandomHeight Sinusoidal Burrs, DTG Workpiece ............. . ....... 98
4.34 Results of Simulation 8 with Switching Control: External Forces
RandomHeight Sinusoidal Burrs, DTG Workpiece ..................... 99
4.35 Results of Simulation 8 with Switching Control: Motor Torques
RandomHeight Sinusoidal Burrs, DTG Workpiece .... ... .... . ...... . . 100
4.36 Results of Simulation 9 with Switching Control: Position Errors
RandomHeight Sinusoidal Burrs, DTG Workpiece, No Torque Limits ...... 101
4.37 Results of Simulation 9 with Switching Control:
Depth of Cut and Remaining Depth of Cut
RandomHeight Sinusoidal Burrs, DTG Workpiece, No Torque Limits ...... 102
4.38 Results of Simulation 9 with Switching Control: External Forces
RandomHeight Sinusoidal Burrs, DTG Workpiece, No Torque Limits ...... 103
ix
4.39 Results of Simulation 9 with Switching Control: Motor Torques
RandomHeight Sinusoidal Burrs, DTG Workpiece, No Torque Limits ... .. . 104
4.40 Results of Simulation 10 with Switching Control: Position Errors
Large Upset Burrs, ETG Workpiec .... . ......... . .......... . ... . .. . 105
4.41 Results of Simulation 10 with Switching Control:
Depth of Cut and Remaining Depth of Cut
Large Upset Burrs, ETG Workpiec .......... . ...... . .. . .. . ......... 106
4.42 Results of Simulation 10 with Switching Control: External Forces
Large Upset Burrs, ETG Workpiec .................... . ......... . .. 107
4.43 Results of Simulation 10 with Switching Control: Motor Torques
Large Upset Burrs, ETG Workpiec ............. . . . . . ... . .. . . . ... . .. 108
4.44 Results of Simulation 11 with Switching Control: Position Errors
Large Upset Burrs, ETGWorkpiece, No Torque Limits ..... . ...... . .... 109
4.45 Results of Simulation 11 with Switching Control:
Depth of Cut and Remaining Depth of Cut
Large Upset Burrs, ETG Workpiece, No Torque Limits . . . . .. . . . ... . ... . 110
4.46 Results of Simulation 11 with Switching Control: External Forces
Large Upset Burrs, ETG Workpiece, No Torque Limits .. . ............. . 111
4.47 Results of Simulation 11 with Switching Control: Motor Torques
Large Upset Burrs, ETG Workpiece, No Torque Limits ...... . . . ........ 112
4.48 Results of Simulation 12 with Switching Control: Position Errors
Large Upset Burrs, DTG Workpiece, No Torque Limits . . . .. . .... . .. . .. . 113
4.49 Results of Simulation 12 with Switching Control:
Depth of Cut and Remaining Depth of Cut
Large Upset Burrs, DTG Workpiece, No Torque Limits . . .. . . ....... . .. . 114
4.50 Results of Simulation 12 with Switching Control: External Forces
Large Upset Burrs, DTG Workpiece, No Torque Limits ...... ... . .. . . . .. 115
4.51 Results of Simulation 12 with Switching Control: Motor Torques
Large Upset Burrs, DTG Workpiece, No Torque Limits . .. ...... . ... . .. . 116
4.52 Results of Simulation 13 with Switching Control: Position Errors
Scallop Burrs, ETG Workpiece .. ... ... . .... .. .. . ........... . . . ... . 117
x
4.53 Results of Simulation 13 with Switching Control:
Depth of Cut and Remaining Depth of Cut
Scallop Burrs, ETG Workpiece .................................... 118
4.54 Results of Simulation 13 with Switching Control: External Forces
Scallop Burrs, ETG Workpiece .................................... 119
4.55 Results of Simulation 13 with Switching Control: Motor Torques
Scallop Burrs, ETG Workpiece .................................... 120
4.56 Results of Simulation 14 with Switching Control: Position Errors
Scallop Burrs, DTG Workpiece, No Torque Limits ................ ...... 121
4.57 Results of Simulation 14 with Switching Control:
Depth of Cut and Remaining Depth of Cut
Scallop Burrs, DTG Workpiece, No Torque Limits .......... ............ 122
4.58 Results of Simulation 14 with Switching Control: External Forces
Scallop Burrs, DTG Workpiece, No Torque Limits ..................... 123
4.59 Results of Simulation 14 with Switching Control: Motor Torques
Scallop Burrs, DTG Workpiece, No Torque Limits ..................... 124
Xl
B
b
C(q,q)
C(q)
d
D
E
FJq)
F/lq, q)
G(q)
Glq)
I
J(q)
K
Kd
Kf
Kf i
Kp
NOMENCLATURE
desired damping matrix
width of cut
centrifugal and Coriolis matrix in joint space
centrifugal and Coriolis matrix in task space
depth of cut
wheel diameter
modulus of elasticity
force error matrix
position error matrix
force vector arising from actuator torque at the end effector
desired force vector
external force vector acting on the end effector of the manipulator
Coulomb friction force vector in the joints
friction force vector in task space
component of force limit
normal grinding force
tangential grinding force
gravity vector in joint space
gravity vector in task space
moment of inertia
Jacobian matrix
desired stiffness matrix
derivative position gain matrix
proportional force gain matrix
integral force gain matrix
derivative position gain matrix
xii
M(q)
Md
Mlq)
q
If
s
v
V
inertia matrix in joint space
desired inertia matrix
inertial matrix in task space
joint variable vector
the time derivative of q
Laplace operator
Lyapunov function
derivative of Lyapunov function
normal velocity of end effector.
tangential velocity of end effector
workpiece feed rate
position vector in task space
desired position vector in task space
position acceleration
impedance matrix
scalar impedance
material removal rate
metal removal parameter
coefficient of grinding fiction
input torque vector
xiii
CHAPTER I
INTRODUCTION
Automation Using Robotics
Traditional deburring and grinding of metal parts has been considered labor
intensive, monotonous, and tedious work, occurring offline, and often undertaken in dirty
and noisy environments. Such manual cleanup operations usually increase costs and parts
inconsistency. An alternative is to automate such operations, implemented by employing
robots. Robots can be employed to good advantage in deburring and grinding because of
the following properties:
• They can operate three shifts per day.
• They accurately reproduce repetitive motions.
• They can process parts faster than humans.
• They can work in noisy and dirty environments without degradation in
performance.
For many companies, the use of robots for automation has considerably reduced costs and
improved qUality.
Robots have been successfully employed in the foundry industry for grinding gates,
risers, and flash, as well as for various chamfering of internal and external edges [1].
Other industries have successfully employed robotic operations for brushing, polishing,
buffing, and grinding [2]. Automatic parts being deburred and finished with robots include
transmission and steering knuckle housings, connecting rods, and plastic moldings [2].
Robots also have been used for automatic welding and spray painting [3].
Industrial robots have been used for various deburring and finishing operations, but
most applications are dedicated to large quantities of a specific part with simple geometry,
easily adapted for automatic operation. Unlike offline manual operations, the use of
robotic deburring and grinding requires considerable planning to ensure optimum results.
In this thesis, we consider robotic deburring and grinding whereby a robot arm carries a
grinding tool (end effector) to follow a desired trajectory. This involves motion of the end
effector in both free space and in constrained space. Development of an effective and
efficient position and force control strategy is the main focus of this research. In next
section, the problems addressed by this application are described.
Problem Background
In many robot applications, manipulators are commanded in more or less
unconstrained environments. An unconstrained or "free" environment is a 3 dimensional
work space in which there is no contact between the moving robot arms and any other
objects, and no external force, other than gravity. acts on the end effector or other robot
moving parts. Control of the position of the end effector in such environments is relatively
straightforward. More advanced robotic applications involve interaction between the
robot end effector, or other moving robot links, and the environment. Robotic deburring
2
and grinding requires the end effector to follow a desired trajectory in both constrained
and unconstrained space as illustrated in Figure 1.1. An important issue here is to design a
controller to achieve stable contact transition and external force regulation with minimum
impact and bouncing. Such a control strategy usually may be divided into three operation
modes: free motion mode, transition or impact mode, and constrained motion mode [4].
In the transition and constrained motion mode, if a large burr is encountered, a sharp
surface change may cause the end effector to leave the workpiece. Limit cycle response
or instability may be excited. Therefore, appropriate control is important for efficient tool
utilization and accurate production of desired finished profiles. A further concern in
grinding is the potential of burning the workpiece or destruction of the tool if the grinding
forces are excessive. This can be avoided by controlling the normal .and tangential
grinding forces to lie below the burning or damage limits for the given cutting conditions.
wor kp iece
nomul
f orce
deSir ed
surface
unfinished
surP o.ce
grinding wheel center
r obot ic end
effector
Figure 1. 1 Grinding an Edge
3
Various investigators [5,6, 7, 8] have proposed switching control strategies to
handle the transition or impact modes. For example Marth, et al [8], employed position
control in the unconstrained direction and force control in the constrained direction for a
simple end effector probe contacting a smooth edge. However, such control approaches
may not suitable for robotic deburring and grinding, because force control in a constrained
direction requires tracking a desired force trajectory, which means a precise force model
and known surface geometry are required. In robotic deburring and grinding, the end
effector will encounter constrains in both tangential and normal directions, and we assume
the geometry of the workpiece is not precisely known. Moreover force control in a
constrained direction does not guarantee accurate production of a desired finished contour
on the workpiece.
Literature Review
In this section, we briefly review some control strategies, proposed by previous
investigators, including impedance control, hybrid position/force control, and hybrid
impedance control. These control strategies deal with the control of a constrained
manipulator, which may be suitable for robotic deburring and grinding. A more complete
review is given by Hong [3]. We note, however, that these investigations typically
. consider only contact by an end effector probe with a relatively smooth workpiece, and do
not consider material removal of the workpiece by the end effector.
4
Impedance Control
Impedance control was fust proposed by Hogan in 1985 [9]. His central idea was
to assume a relationship between the position of the end effector and the contact force
exerted by the constraining environment. This relationship can be modeled by a
generalized linear impedance consisting of inertial, damping, and stiffness characteristics.
Impedance control regulates the relationship between the end effector position and the
contact force, called the mechanical impedance [10]. The fundamental relationship is
given by
Z(s) = Fe(s)
Xes)
( 1.1)
where s is the Laplace operator and Fe (s), X (s), and Z(s) are the Laplace representations
of the external force, position, and impedance, respectively. Typically, a generalized
expression for the impedance is given by
(1.2)
where M d' B, and K represent desired inertia, damping, and stiffness, respectively.
Impedance control has attracted a significant number of investigators, [11, 12, 13, 14],
because it provides a stable and unified control structure for the three different regions of
operation, namely, free motion, transition or impact, and constrained motion modes. On
the other hand, unless the exact environment model is known and is integrated into the
motion plan, the external force can not be independently regulated with impedance control
after contact, such that, it is difficult to handle both position and force regulation in a
5
constrained environment [4]. However, proper design of an impedance controller can
guarantee the stability of manipulators in contact with environments. Details of impedance
control will be addressed more completely in Chapter ill, including stability analysis and
controller design for robotic deburring and grinding.
Hybrid .PositionIForce Control
Hybrid position/force control, flrst proposed by Raibert and Craig [15], is a control
strategy dealing with tasks requiring force control in some di~ections and position control
in others. A hybrid position/force controller has the following three characteristics [16]:
• Position control is employed in directions for which a natural force constraint
exists. J •
• Force control is employed in directions for which a natural position con traint
exists.
• Appropriate combinations of force and position control modes are employed
along the coordinates of an arbitrary reference frame.
Typically, a hybrid position/force controller is unable to regulate the relation between the
end effector position and contact force because it neglects the manipulator'S impedance.
Moreover, the position of the end effector and contact force along one degree of freedom
(DOF) can not be controlled independently, such that for complex tasks like robotic
deburring and grinding, such a controller is unsuitable.
6
Hybrid Impedance Control
Hybrid impedance control combines impedance control and hybrid position/force
control into one strategy [17]. It treats the contact environment as a linear impedance and
assumes the manipulator can be effectively decoupled into singleDOF linear subsystems.
Then, a duality principle is employed to decide which control should be used in each
subsystem. In short, an inertial environment requires a positioncontrolled manipulator, a
capacitive environment requires a forcecontrolled manipulator, and a resistive
environment allows either force or position control [17]. Once the type of control method
has been decided, the impedance of the end effector is chosen accordingly. Such a
controller provides more flexibility than those mentioned earlier, and may be applied to
robotic deburring and grinding. In Chapter III, we will further investigate and implement
this control algorithm, and simulation results will presented in Chapter IV.
Impact Control
In robotic deburring and grinding, an impact force may occur when the end
effector contacts the workpiece or encounters a large burr. This impulsive force may
deviate the end effector off the workpiece. It could induce unstable dynamics and damage
the end effector and workpiece. Strategies for impact control, or contact transition
control,. to solve this problem have been studied by several investigators. Such
investigations may be broadly classified into two categories, namely, impedance control
and switching control [4]. Impedance control is appealing because it provides a stable and
7
unified control strategy for both free and constrained environments without the need for
switching control algorithms on impact. Several switching (discontinuous) controllers
have been investigated, mainly during the last five years. A common result is that while
force can be regulated if contact is continuous, instability can arise if bouncing occurs after
impact. Because of this, the overall contact stability problem has not been completely
addressed for realistic deburring and grinding problems. Recently, Tam, et al [4],
proposed a new control strategy using "positive acceleration feedback to control the
transient force response to reduce the peak impulsive force and bouncing". The new
method employs a position control to eliminate the unexpected bouncing and reestablish
contact. Tarn showed that the number of bounces is finite and that the last bounce always
corresponds to the transition from free space to constrained space. Stable contact is
guaranteed. Tam's work may have potential for developing an improved control method
for robotic deburring and grinding. In [18], Pagilla uses another approach for impact
control. He assumes the end effector and environment are rigid, and there is no
penetration. By employing a simple rigid body collision and coefficient of restitution to
model impact, Pagilla experimentally and numerically shows that bouncing can be
eliminated in finite time.
While the work reviewed above may have relevance to our problem herein, we
note a significant difference. For robotic deburring and grinding, the robot arm carries a
grinding wheel or deburring tool rotating at high speed. When the workpiece is contacted,
such tools will immediately cut into workpiece such that the "hard" surface assumed by
8
previous investigators immediately disappears. Accordingly, the impact force is likely
much smaller in our operation, such that the approaches by Pagilla, Tarn, and others may
not be suitable for robotic deburring and grinding.
Objectives of This Study
From our literature review, most proposed control algorithms for manipulators
operating in constrained and unconstrained environments employ some type of force
control for stable contact while tracking a desired trajectory. In the work herein, we seek
high accuracy in the finished workpiece profile using robotic finishing. That is, we are
interested in employing force control only when the normal or tangential forces exerted by
the workpiece on the end effector exceed some prespecified limits, at which point we are
prepared to compromise on position accuracy, otherwise, we desire highly accurate
position control. We assume that the actual geometry of the workpiece is unknown and
we wish to finish workpieces of different materials. Employing a forcetracking strategy
under these assumptions will be very difficult. On the other hand, impulsive forces may
need to be regulated when the end effector contacts the workpiece or encounters a large
burr. Such demands increase the difficulty in implementing robotic deburring and
grinding.
This research investigates position control and force regulation of a simple twoarm
SCARA robot carrying a powered tool at its end effector used for deburring and
grinding. Based on Hong's work [3], we extend the grinding models to encompass easy
to grind (ETG) materials and difficult to grind(DTG) materials. A new switched control
9
method is developed for this operation, and other control algorithms are investigated to
compare performance for robotic deburring and grinding. The remainder of this thesis will
describe system modeling, the new control approach for robotic deburring and grinding,
and computer simulation results. Chapter II describes the dynamics of the robot and the
force model for ETGIDTG materials. In Chapter III, a new switched controller is
presented for position control and force regulation in robotic deburring and grinding. We
also investigate impedance control and hybrid impedance control in this application.
Chapter IV discusses surface characteristics for various surface irregularities and burrs.
Simulation results for various controllers are presented, together with analysis and
discussion. Chapter V follows with conclusions and recommendations.
10
CHAPTER II
SYSTEM MODELING
In this chapter, we first address the equations describing the dynamics of a
manipulator having n links. We have elected to use as our simulation test bed, a model of
a SCARA robot developed at UCBerkeley [19] using NSK drives. The grinding forces
for our study are derived from conventional grinding models. The stiffness of the robotic
arm will also be discussed. Control strategies and simulations are based on the models
developed here.
Manipulator Dynamics
A robotic manipulator can be considered as a set of n rigid bodies connected in a
serial chain with friction acting at the joints. The equation describing the dynamics of such
a device in free space can be expressed in "joint space" as [10]:
M(q)q+ C(q,q)q+ Ff(q,q) + G(q) ='t (2.1)
where q is an n x 1 joint variable vector, q is the time derivative of q, M(q) is an n x n
inertia matrix, C( q, q) represents an n X n matrix that describes the centrifugal and
Coriolis terms in the dynamics of the manipulator, G( q) is an n x 1 vector containing terms
11
arising from forces due to gravity, F J q) is an n x 1 vector that specifies the effects of
Coulomb friction force in the joints, and 't is an n x 1 vector that defmes input torques
from the actuators of the manipulator.
Since the natural description of a desired trajectory and interaction force are given
in "task space", it is desired to express the dynamics of a manipulator in task space as
[3, 16]:
(2.2)
where F is a n x 1 force vector arising from actuator torque at the end effector, Ml q) and
el q) are n x n matrices corresponding to the inertial matrix and centrifugal/Coriolis matrix
in task space, and Glq) and Ffrlq, q) are n x 1 vectors of gravity and friction force terms in
task space. For simplicity, we consider the task space to be the Cartesian (reference)
space in this study.
Note that the force term, F, arising from actuator torques at the end effector is
applied by the actuators at the joints, using the relationship
(2.3)
where J(q) is the nXn manipulator Jacobian matrix written in the same frame as F and X.
The Jacobian matrix is defined by [3]
J(q) = iJL(q)
iJq
12
(2.4)
where L( q) is a continuous function of the joint space vector found from manipulator
kinematics and geometric relationships. It relates the n x 1 task space vector X to
generalized joint coordinates q by
X = L(q) (2.5)
When the end effector contacts an object, such as a workpiece, a force term arises
on the right of (2.1) due to the environment, such that (2.1) becomes
M(q)q + C(q,q)q + Ff (q,q) + G(q) = 't  JT Fe (2.6)
where Fe is the n x 1 vector that defines the task space force or torque acting on the end
effector of the manipulator. Similarly, for such contact, (2.2) becomes
M,(q)X + C,(q,q)X + Ff,(q,q)+ G,(q) = F  Fe (2.7)
We can derive the relationship between the terms of (2.6) and those of (2.7). First,
premultiply (2.6) by the inverse of the Jacobian to obtain
or from (2.3),
r
Now differentiate (2.5) twice with respect to time to obtain
x = J(q)q
X = J(q)q+ hq)q
13
(2.10)
(2.11)
/ 
Eq. (2.10) is assumed to be nonsingular. Solving for q and q gives
(2.12)
(2.13)
Substituting from (2.12) and (2.13) into (2.9) yields
(2.14)
from which we derive the expressions for the terms in the task space dynamics in (2.7) as
Mr = ]T(q)M(q)]I(q)
C{ = ] T (q)[C(q,q)]1  M(q)]' j(q)] I(q)]
F ft = ] T (q)Ff(q,q)
G{ = ] T (q)G(q)
(2.15)
In practical applications, the control input, torques 't , are commanded in joint
space, and encoders and tachometers are usually placed on the motor shafts, such that
positions and velocities are measured in joint space. To obtain motion of the end effector
in task space, "forward kinematics" is employed for transformation. Accordingly, we can
derive a more convenient and useful expression for the manipulator dynamics by
substituting from (2.13) into (2.6), which yields
These new manipulator dynamics will be used to design the control laws for deburring and
grinding in Chapter III.
14
UCBerkeleylNSK SCARA Robot
Detailed information on typical robot installations for deburring and grinding can
not be readily obtained because of a certain degree of proprietary information surrounding
many of these installations. This is because robot manufacturers desire to withhold
information about their robots from their actual and potential competitors. Thus, detailed
modeling information is typically not reported in trade publications, nor is such
information provided by robot manufacturers to customers. In this study, we employ a
UCBerkeleylNSK SCARA robot, which consists of only four major mechanical parts,
two direct drive motors from Nippon Seiko K.K. (NSK) and two aluminum links, as a
benchmark for our simulations because the technical data for this robot have been
published [19], and its configuration as a two axis robotic arm provides a planar
workspace, appropriate for our study.
In this work, we are concerned with end effector motion and force acting only in a
horizontal plane, parallel to the planes of motion of the SCARA planar robot. We employ
a model with only two degrees of freedom, namely rotations of the two main arms of the
SCARA robot about their vertical axes, as shown in Figure 2.1. Because gravity has no
effect in the horizontal plane, the gravity term in (2.6) vanishes, and the dynamic equation
in joint space for this simple model reduces to a secondorder nonlinear differential
equation given by [19]
M(q)ij + C(q,q)q + Ff (q,q) = 1:  JT Fe (2.17)
where q, 1: and Fare 2 x 1 vectors as defined previously, and
15
M=[~I
~I
~2]
~2
c=
 "
[CII
C21
" q2 " " \
\
\
\
\
\ ql
\
\
I
1 __ _
Figure 2.1 Schematic Diagram of TwoArm SCARA Robot
with
~I = PI + 2P2 COS(q2)
~2 = ~l = P3 + P2 COS(q2)
m22 = P3
Cll = 2P2 sin(q2)42
CI2 =  P2 sin(q2 )42
C21 = P2 sin(q2 )ql
C22 = 0
where pI, p2, and p3 are constant terms dependent on the manipulator's geometric
dimensions and masses of components, given by [19]
16
(2.l7a)
(2.17b)
/~
P2 = II i2cml + lJ /2mp
P3 = I) + 14 + I" + U;cm4 + li mp) (2.17c)
P1 = II + 12 +I3c +11~m2 +L~(m3 +m4 +mp)+ P3
In (2.17c) I, and 13 are rotor inertias of Motors 1 and 2; 12 and 14 are inertias of Links
1 and 2 about their own gravity centers; 13c is the stator inertia of Motor 2; Ip is the
payload inertia; Tn, and ~ are the masses of Motors 1 and 2; mz and m4 are the masses
of Links 1 and 2; mp is the mass of the payload; IJ and l2 are the lengths of Links 1 and
2; and l,c and L2c are the radii of gyration for Links 1 and 2. The Coulomb friction matrix
is described by:
if ki;1 > 0
if Ittl = 0 and Iqil> th; i = 1,2
if Ittl = 0 and Iq;1 ~ thl
where thi is the magnitude (unitless) of the friction torque and i = 1,2. Note that thjs
number th; is also used as the switch limit for Iq;l.
Similarly, the dynamic equation in task space for this simple model can reduce to a
secondorder nonlinear differential equation from (2.7) as
(2.18)
with coefficient matrices defined by (2.15) and the appropriate matrices and vectors
defined as for (2.17). For simplicity to implement dynamic analysis and control based on
(2.18), we define the xy horizontal reference system plane as identical with the task space
17
tn, defined by two orthogonal axes normal and tangential to the surface at an idealized
point of contact of the grinding tool. Thus, we assume a straightline nominal surface
edge aligned with the x axis. Considering the implications of following a curved edge, the
task coordinate system changes as the tool contact point moves along an arbitrary curved
edge in the reference system. At each time step in simulation, this requires two steps of
transformation namely, from robot joint space to task space and from task space to
reference space. A very fast and efficient computation will be an important issue in the
design of a manipulator system used for such applications.
Grinding Modeling
In this section we model the grinding forces based on conventional grinding
operations. The grinding conditions will be specified for calculating the grinding forces.
Force limits to prevent damage to the workpiece and breakdown forces for a selected
grinding wheel will also be discussed. Finally, we investigate the effects of different
workpiece materials.
Force Modeling
In steady grinding operations, the grinding forces can usually be treated as two
orthogonal forces: Fn normal to the contact surface and F/ tangential to the contact
surface, as shown in Figure 2.2. Hahn and Lindsay [20] have experimentally investigated
the grinding process and developed an empirical equation for the normal grinding force as
18
where
F" = normal grinding force
Zw = material removal rate
A In = metal removal parameter
By definition [3],
F = Zw
n A
m
(2.19)
(2.20)
where Vw is the workpiece feed rate, d is the depth of cut, and b is the width of cut. Hahn
and Lindsay have also proposed an empirical equation to predict Am based on
experimental data, which yields errors of +/ 20 % for easytogrind (ETG) materials [20].
n
+44 ____ ~t burrs
wheel
y
Figure 2.2 Robotic Grinding Schematic [3]
19
Robotic deburring and grinding differs from conventional grinding because of the
compliant structure and mobility of the robot. Material feeding is accomplished by
moving the robot relative to the workpiece rather than feedin of the workpiece to a
stationary grinding wheel as in conventional grinding. Hong [3] developed normal and
tangential grinding force equations for robotic deburring and grinding by employing Hahn
and Lindsay's experimental equations for conventional grinding forces, together with
geometry and kinematics. His results yielded
where
FlI = [V, + (0.285D / d + l)v" ]db / Am
F, = JlFlI ,
D = the wheel diameter
( .
d = depth of cut
FlI = normal grinding force
Ft = tangential grinding force
Am = metal removal parameter
b = width of cut
~ = coefficient of grinding fiction, (0 ~ ~ ~ 1 )
VII = normal velocity of end effector
VI = tangential velocity of end effector
20
(2.21)
In our simulation work in Chapter IV, we will employ (2.21) to calculate grinding forces.
In actual implementation, normal and tangential grinding forces would be measured
directly by a force sensor at the end effector.
ETG and DTG Materials
The grinding wheel speed, workpiece hardness, dressing lead, and depth of dress
are the four most important parameters affecting the metal removal parameter, Am. [20] .
The workpiece hardness can usually be classified into two categories: easytogrind (ETG)
and difficulttogrind (DTG) materials [20]. Materials classified as ETG are chrome, cast
iron, aluminum, and soft steel. DTG materials are many steels in the M and T categories
of tool steels, titanium alloys, and highnickel steels. In this section, we calculate the
metal removal parameter, Am' for an ETG and a DTG material.
For ETG materials, Am' the metal removal parameter can be predicted within
20 % by a semiempirical equation given by [20]
Am = (0.021 (2.22)
where
Vw = workspeed, fpm
Vs = wheel speed, fpm l = inch per wheel revolution
De = conformity, or the equivalent diameter, inch
21
d = grain size in wheel, inch
vol = approx. volume percent of bonding material in the wheel
C = diametric depth of dress, inch
Rc = value of Rockwell Hardness
In (2.22),
in 515/304
. is used to cancel the power of units for A . The parameter vol can be
lb m
estimated from the empirical relationship given by
vol = 1.33 Hd + 2.2 S  8.0 (2.23)
where
Hd = wheel hardness, denoted by H, I, J, K, L, M, etc. with H = 0, 1= 1, J = 2,
K=3,etc.
S = wheel structure number, 4, 5, 6, etc.
De, conformity, or equivalent diameter, is the degree to which the wheel surface fits or
conforms to the workpiece surface. For surface grinding, De := D.(, grinding wheel
diameter. For the workspeed, Vw, we will use the relative speed of workpiece to the end
effector, which is chosen as 0.012 mls for our simulation later. As an example, we
calculate the metal removal parameter, Am ' for an ETG material as follows [20, 21].
Material: R 60, AISI 52100 steel, width 10 mm.
Grinding wheel: Ds = 2 inch, Vs = 18000 fpm
type: 80K5V, grain size (d): 0.01 inch
22
dress lead (I): 0.004 ipr
dressing compensation (C): 0.001 inch
Vw = 2.362 fpm (0.012 mls)
(2.362 )3/19 (1 + 2.0.001). (.004)"119 .18000
Am = (0.021) ~43::":1::8/3=0040~0 ::""""':""3:" ".: 0"".:0:"0"4': ":::::::::::::
2 . (1.33 ·3 + 2.2·5  8)°.47 .0.01 5/38 .6027119
= 0.00871 in3/(min, lb) = 5.3484x 1010 m3/(sec, N)
For DTG materials, we have been unable to locate a suitable equation to calculate
Am' Thus we use values from experimental data and assume that grinding conditions,
wheel dressing, and rotary speed are the same as described for experiments in [20]. For
an Rc 64, M4 material, the value for Am ranged from 0.000035 to 0.0028 in3/(min, lb)
[20]. As a reasonable example, we choose Am = 0.002 in3/(min, lb) (1.228 X 1010
m3/(secN)) to simulate the grinding force for a DTG material.
Eq. (2.21) and the value for Am developed here for an ETG and a DTG material
will be used for computer simulations in Chapter IV.
Grinding Force Limits
In considering possible limits to applied grinding forces, we consider potential
thermal damage to the workpiece and breakdown forces of the grinding wheel. Thermal
damage to a workpiece may be caused by excessive grinding temperature and can be
classified into three common types: workpiece bum, workpiece tempering, and induced
23
residual stresses in the workpiece [22]. Workpiece burn and tempering have apparently
not been studied extensively in grinding operations, perhaps because induced residual
stresses are more commonly encountered. Accordingly, we focus on induced residual
stress as the major concern for thermal damage in this work.
The grinding process invariably produces residual stresses in the vicinity of the
finished surface. When residual tensile stresses exist, a workpiece surface is thermally
damaged because such stresses lead to reduced fatigue strength and cracking. Although
residual compressive stresses can also be generated, their magnitudes are much smaller
than residual tensile stresses. Usually, residual compressive stresses are induced after
grinding by cold working operations. There are three principal means to reduce thermal
damage: decrease contact time by increase grinding speed, decrease force intensity and
wheelspeed, and maintain wheel sharpness.
The normal grinding force to cause thermal cracking for an ETG material with
Am= 0.0064 in3/(min, lb) and work surface speed = 1200 fpm is about 320 lblin (56.04
N/mm) [20]. The normal grinding force to cause thermal cracking for our ETG materials
will be larger than 320 lb/in because our value for Am is greater than 0.0064 in3/(rnin, Ib).
However, the normal grinding force to cause thermal cracking for our DTG material is
unavailable from the literature. Our workpiece width is 10 mm, such that the normal
grinding force to cause thermal cracking is at least 560.4 N for our ETG material.
Following [21], we select a grinding wheel designated 80KSV for our ETG and DTG
materials, for which the breakdown force is 483 N (48.3 N/mm x 10 mm). Since this
wheel breakdown force is lower than our workpiece thermal damage limiting force, we
24
will use the breakdown force as a force limit in our simulation in Chapter IV.
Motor and Robot Ann Stiffness
The stiffness of the UCBerkeley NSK SCARA robot is not given in the available
literature. In this section, we estimate the stiffness of the NSK motors and our robot links
for worstcase conditions, which will allow us to determine if our robot is sufficiently rigid
to justify i.gnoring robot arm flexibility. The robot links are made from aluminum, and the
specifications are given in Table 1 [19].
Table 1 Robot Links Specifications
,~
~ '>. ....... 1,. t Inertia Length
Link 1 0.360 kg m2 0.36m
Link 2 0.051 kg m2 0.24m
. "
The two robot joint motors used are made by NSK, Model 1410 for the first
(lower) axis and Model 608 for the second (upper) axis. The moment rigidities of these
two motors are: Motor 1 (first axis), Mid = 3.27 X 106 N  m / rad, and Motor 2 (second
axis), Mk2 = 2.80 X 105 N  m / rad [23]. Motor specifications from [23] are given in
Appendix A. The most compliant configuration for the motors and robot arms is that for
which both links lie along a straight line in the fully extended position, illustrated in
25
Figure 2.3(a).
0,36 M
l 1
link 1 2 t6X
(a)
F F
0,24 [') 0,36 [')
l2 II ) T=FI'
link 2 link ]
*~llXx~ee *AllX~el
(b)
Figure 2.3 Motor and Arm Configuration for Stiffness Calculation
We assume there is no reduction gear, such that the robot arm joints are directly coupled
to the rotors of the motors. By considering Figure 2.3(b), it can be seen that including the
deflections of Motor 1, Link 1, Motor 2, and Link 2, the total deflection Mis
(2.24)
26
where &bl and Lllb2 are the deflections due to bending of Links 1 and 2, respectively,
and &91' and &92 are the deflections due to motor rotation for Links 1 and 2,
respectively. Dividing (2.24) by P, we obtain the overall stiffness Ke as
(2.25)
where KI and K2 are the stiffnesses of Links 1 and 2, respectively, and Kml and Km2 are the
stiffnesses of Motors 1 and 2, respectively, as seen from the far ends of their respective
links. From (2.25), we can see that overall stiffness Ke of the assembly may be modeled as
four springs in series. With the moment rigidity Mk given, the stiffness for a motor can be
calculated by referring to Figure 2.3, as shown below. Assume a point force F acts at the
end of Link 2, and that Link 1, Motor 1, and Link 2 fonn a rigid assembly. Then the
stiffness (as seen by F) of Motor 2, Km2 , is defined by assuming the rigid assembly rotates
through small angle ~8 , displacing the end of Link 2 by distance Lll02. Then we have for
Now assuming the small angle approximation
together with the definition of motor rigidity [23] for Motor 2, we obtain
M = P'[2
k2 ~e
27
(2.26)
(2.27)
(2.28)
Employing (2.28) and (2.27) in (2.26) yields
K = Mu
m2 1 2
2
(2.29)
Using a similar development for the stiffness of Motor 1 (as seen by F and T in Figure
2.3b), we obtain
(2.30)
Using the rigidity values above for the two motors and the lengths of the two robot arms
given in Table 1, we obtain KmJ = 1.51 x107 N I m and Km2 = 4.86 X 106 N 1m.
To determine KJ and K2, we assume that each link is a cantilever beam fixed at its
left end with a point load applied at the free end on the right of Link 1, and a point load
and moment load applied at the free end on the right of Link 2. We consider two cases of
area section for each link: a circular ring and a square tubular section. The end deflection
of a cantilever beam with a point load at the free end is given by [24]
FL3
Ax =
b 3EI
(2.31)
where Axb is the free end deflection, F is end load, L is length, E is modulus of elasticity,
and I is moment of inertia. The stiffness K can then be defined as
(2.32)
Therefore K2 is given by
28
K = 3EI2
2 I 3
2
(2.33)
For Link 1, there is a moment, T, applied at the free end, in addition to the point load F.
The deflection fum caused by this moment is given by [25]
fu = T·l/
m 2EI 1
(2.34)
where T = F X l2' Then, the stiffness of Link 1 is
KI = F = 6EI1
Lhm + Llxb 211
3 + 3l21/
(2.35)
The mass density, p, of aluminum is 2800 kg/m3 and the link inertia, I, is 0.36 kg m2 for
Link: 1 and 0.051 kg m2 for Link 2. The equation for ann inertia is given by
(2.36)
where m is the link mass and r is the link radius of gyration, which from Figure 2.3 i 0.18
m for Link 1 and 0.12 m for Link 2. The mass can be calculated by
m=pAL (2.37)
where A is the link section area and L is the link length. From (2.32), (2.33), and given
parameters, we obtain A, = 0.011 m2 and A2 = 5.27 X 10.3 m2 for Link 1 and Link 2,
respectively. With these values for Ai, the stiffness for different cross sections of each link
can be calculated as shown below.
29
. """"'"
Case I: Circular Ring Cross Section
Assume d; = 0.8D;
A = 1C(D/ d/)
I 4
DJ = 0.1972 m, d/ = 0.1578 m
D2 = 0.1365 m, d2 = 0.1092 m
The modulus of elasticity, E, for aluminum is 70 x 109 N/m2 and the moment of inertia, I,
for a circular ring is defined as [24]
(2.38)
Then (2.38) and (2.35), (2.33) yield K/= 9.8564 x 107 N/m, K2= 1.5284 x 108 N/m.
Case II: Square Tubular Section
Assume dl = 0.8D,
I
Ai = D/  d/ = 0.36 D/
r /"71~  + 
I
D/ = 0.1748 m, d/ = 0.1398 m
D2 = 0.1210 m, d2 = 0.0968 m
The moment of inertia, I, for a square tubular section is [24]
30
(D4 _d4) /. = I I
I 12 (2.39)
Combining (2.39) and (2.35), (2.33) yields K/ = 1.0346 X 108 N/m, Kz = 1.602J X J08
N/m.
Comparing the two types of beams, we see that the square tubular section has the
largest stiffness. We therefore use the values for KJ and Kz for a square tubular section in
(2.25), together with the previously determined values for Kml and Km2, to obtain the
equivalent stiffness Ke for the most compliant robot arm orientation as
Ke = 3.4735 X 106 N/m = 19836.58 lb/in
For a maximum normal force of 450 N, which we will employ in Chapter IV, this
corresponds to a "worsecase" end deflection of 0.1296 mm. For a more reasonable robot
configuration than worst case, we assume that 112 of this value is more representative,
namely 0.0648 mm. As we shall see, such deflection is small compared to most position
errors in our simulations. We conclude that this value of Ke is sufficiently large to ignore
robot flexibility. However, this prediction may not truly represent overall robot
compliance because the drive train may introduce more flexibility than the links. To
predict a more accurate stiffness of more realistic structural elements is beyond the scope
of this study.
We have developed equations describing the dynamics of a twolink SCARA robot
and modeled the grinding forces for robotic deburring and grinding. In the next chapter,
we will employ these models to investigate and design control methods for these
operations.
31
CHAPTER III
CONTROL APPROACHES
In this chapter, several control approaches are investigated. A well known
feedback lineaJization method is used to linealize manipulator dynamics. Based on
feedback linealization, several control laws are developed for robotic deburring and
grinding. A new switched control method is proposed for this operation to improve
position accuracy and force regulation.
Feedback Linealization
Feedback linealization is an approach used to control nonlinear systems, which has
attracted considerable study recently. The basic idea of feedback linealization is to
transform a nonlinear dynamic system into a linear one, in order that linear control theory
can be applied to the transformed dynamic system. It is achieved by "exact state
transformations and feedback, rather than by linear approximations of the dynamics" [27J.
The nonlinear control used to produce the transformation is constructed by feedback
linealization and is called inner loop control [10]. The designer can then design an outer
loop control using c1assicallinear control approaches by specifying performance such as
32
tracking, disturbance rejection, and robustness.
The dynamic equations of a two degree of freedom SCARA robot are nonlinear
and coupled. Feedback linealization is employed to linealize the manipulator dynamics by
inner loop control before considering the design of position and force controllers by outer
loop control. Computedtorque control [16] is a special application of feedback
linearization for manipulator dynamic nonlinearities, which has been widely applied in
robot control. This approach amounts to canceling the nonlinearities of a nonlinear system
so that the closedloop dynamics become linear. In this project, we employ this method to
construct an inner loop control structure before designing the outer loop controllers. The
manipulator dynamic equation given by (2.16) is repeated here as
(3.1)
The problem of controlling a complicated system such as described by (3.1) can be
handled by a partitioned controller [16], with torque 't given by
t = at' + p (3.2)
where 't is the n x 1 vector of joint torques, 't I is the "servo" portion of the control law
and is based on outer loop considerations, and a and ~ are functions chosen to decouple
and cancel the nonlinear terms in the complete dynamic system. The control law given by
(3.2) is the modelbased portion of the controller [16], which establishes an inner control
loop as shown in Figure 3.1. Following Craig [16], we choose
a = M(q)Fl (q)
~ = M(q)r1 (q)j(q)q + C(q,q)q + F,(q,q) + G(q) + JT F:
(3.3)
33
In Figure 3.1, the inner loop feedback term N(q, q) is given by
N(q, q) = M(q)JJ (q)j(q)q + C(q, q)q + F/q,q) + G(q) (3.3a)
Substituting (3.3) into (3.2), the modelbased portion of the control law becomes
Now employing the right of (3.4) for 't in (3.1) yields
X=t' (3.5)
Eq. (3.5) shows that the acceleration of the end effector is equal to the servo portion of
the control law , which can be designed to achieve design specifications, such as minimum
tracking error and (desired) disturbance rejection. If we design the servo controller for t'
properly, the desired motion of the manipulator can be achieved from the computed
torque control law (3.4), assuming available motor torque does not saturate.
In order to employ (3.4), it must be assumed that the manipulator dynamics are
known exactly with perfect sensors for the measurement of forces, positions, and
velocities. However, in practice there exist modeling and measurement errors, which may
cause inexact cancellation of dynamics of the nonlinearities in (3.4). It is possible that a
lack of robustness could arise from inexact cancellation of dynamics of the nonlinearities,
but treatment of this problem is beyond the scope of this research. We are concerned here
mainly with outer loop design in the absence of inner loop uncertainty. In the following
sections, we design and analyze some outer loop controllers, based on the control
structure developed above.
34
'  lil
I I
il'1put
IOuter Loop II
I II
I
I
I
I
I
I
I
]nner Loop
1'1
F" orword
I
I
I
I
I
I
I
I
I
I
I
I I
L _____ JL __ _ ____ ____ __________ J
Figure 3.1 Diagram of Control Structure [3]
Impedance Control
Impedance control regulates the relation of position to force and changes the
dynamic behavior of the system. It may be suitable for robotic deburring and grinding to
track a desired trajectory while accommodating the cutting forces produced by the cutting
process. To implement impedance control, the first step is to specify the desired behavior
of the target impedance. Hogan [9] points out that the target impedance consists of some
inertial, damping, and stiffness characteristics that describe the relation between the
position of the end effector and the force exerted by the environment. Typically, this
impedance can be expressed as
(3.6)
35
where s is the Laplace operator, Z( s) is the 2 x 1 impedance matrix, and M d' B, and K
represent 2 x 2 desired inertia, damping, and stiffness matrices, respectively. According to
Ho gan [11], the effect of the target impedance approach can be represented in the time
domain by
MdX + B(X  Xd )+ K(X Xd ) =~ (3.7)
where X is the position vector in task space, Xd is the desired position vector in task
space, and Fe is the external force acting on the end effector. Because the external force
produced by the constrained environment opposes the motion of the end effector, we use
a minus sign for Fe in (3.7). Solving for X gives
Substituting the right side of (3.8) for X in (3.1) yields the control law torque as
where we have defined errors tlX and LlX by
tlX = Xd  X
LlX = Xd  X
and nonlinear "gains" GI' G2 , G3 , and G4 by
GJ =JT(q)G2
G2 = M(q)rlM~J
G3 = G2Md
G4 = C(q, q)q + Ft(q, q) + G(q)
36
(3.8)
(3.9)
~/' ...........
Eg. (3.9) has been developed containing both joint and task space terms to
facilitate implementation, instead of developed solely in task space [3]. This is because
robot positions and velocities are measured in joint space, while desired positions and
velocities are given in task space. Essentially, the impedance control law amounts to a
proportional plus derivative (PD) position controller, augmented by external force
feedback. Note that if the manipulator moves in free space with no external force acting
on the end effector, the impedance becomes zero. Conversely if a manipulator is
motionless in constrained space for any applied torque, the impedance is infinite.
Therefore, pure position and pure force control are considered as special cases of
impedance control.
Hong [3] and McCormick and Schwartz [12] discuss an alternate impedance
control strategy described by
(3.10)
where Fd is a desired force vector (required for material removal during grinding jn our
case). Solving for X and substituting in (3.1) yields a control law torque given by
(3.11)
This alternate impedance control strategy commands desired forces along with desired
positions and velocities for robotic deburring and grinding operations. If the needed
grinding force Fd is modeled well and surface geometry is known, desired forces may be
COITlmanded to increase the performance of impedance control. We will evaluate these
two types of impedance controllers in Chapter IV.
37
System stability using impedance control is dependent on the target impedance
parameter matrices, manipulator dynamics, and the constrained environment. If the target
impedance matrices Md , B, and K are selected as symmetric, positive definite matrices,
Kazerooni, et al [14], show the linear impedance control is stable in contact with any
directly coupled, stable, linear environment. Colgate and Hogan [13] use the Nyquist
criterion to show the stability of the feedback linearlized impedance controller (differents
from local linear approximation), which is employed here, in contact with a linear, passive
environment. The drawback of these analyses is the modeling of contact interactions as a
directly coupled linear system. Such a model of interactions is extremely restrictive.
McCormick and Schwartz [12] observed that contact dynamics cause instability when the
level of force feedback is sufficiently increased. Based on the small gain theorem,
Kazerooni, et al [26], presented an input/output stability proposition for bounded force
feedback gain, but two difficulties arise. First, an accurate model of the force environment
must be known in order to insure certain necessary conditions, and second, a design based
on given sufficient conditions may result in an overly conservative control law [12]. In
general, achieving a guarantee of global stability of an impedance control law is very
diffi.cult in practice.
Hybrid Impedance Control
Hybrid impedance control (HIC) was proposed by Anderson and Spong (17),
combining impedance control and hybrid position/force control. It treats the contact
environment as a linear impedance and assumes manipulator dynamics can be decoupled
38
into singleDOF linear subsystems in task space, which in our deburring and grinding
operation is described by directions tangential with and normal to the surface. The main
idea of HIC is to employ a duality principle to decide which control should be used for
different environments in each subsystem. Before using this "duality principal", the
environment must be modeled. The scalar impedance Ze defined here is the ratio of the
Laplace transforms of scalar force F and scalar velocity V. It can be represented by a
complex number with real part R(w) and imaginary part X(w) for any given frequency was
Ze(w) = R(w) + jX(w) (3.12)
According to Anderson and Spong [17], the impedance of the environment can be
classified into three categories: inertial, resistive, and capacitive impedances given by
Inertial impedance
Resistive impedance
Capacitive impedance
where 0 < C < 00. In Laplace notation, Ze is given by
MdS
Ze(s) = Mds + B
K
MdS + B+S
Inertial impedance
Resistive impedance
Capacitive impedance
(3.13)
(3.14)
where Md, B, and K represent desired scalar inertia, damping, and stiffness, respectively.
By the duality principle, if the environment is capacitive, a forcecontrolled manipulator
with noncapacitive impedance is required; if the environment is inertial, a positioncontrolled
manipulator with noninertial impedance is applied; and if the environment is
39
resistive, either a forcecontrolled manipulator or a positioncontrolled manipulator with
nonresistive impedance may be applied.
In our deburring and grinding task, the environment is inertial when the end
effector moves in free space before contact. According to the duality principle, the inertial
environment requires a positioncontrolled manipulator. Thus, we choose a capacitive
manipulator impedance as
(3.15)
The corresponding differential equation is
( Md(X  Xd )+ R(X  Xd )+ K(X  Xd ) =F (3.16)
which is an impedance control identical to that in (3.7), except that the external force, F, is
zero in free space. As for impedance control, we obtain our outer loop control from
(3.16) with F= 0 as
(3.17)
for the manipulator with unconstrained motion. After contact, we consider the
environment to be capacitive in the normal direction (assuming the material to be deburred
acts like a spring in the normal direction), and resistive in the tangential direction [3].
Based on the duality principle, we use a forcecontrolled manipulator with noncapacitive
impedance in the normal direction. For the resistive environment in the tangential
direction, either position control or force control should be applied. Considering the
nature of the deburring and grinding task, we prefer a position control with a capacitive
40
manipulator impedance in tangential direction [3]. Based on (3.14), we can select our
target manipulator impedances as
normal direction
tangential direction Z, = Md1S + B, + K, (3.18)
s
A force control in the normal direction needs to command a desired normal force Fdll and
a position control in the tangential direction needs to command desired tangential position
Xdr, velocity Vdh and acceleration adt. Then, in the time domain, the corresponding
differential equations are
normal direction
(3.19)
where a, v, and x represent scalar acceleration, velocity, and position, respectively,
subscripts n and t denote normal and tangential directions, respectively, subscript d shows
desired quantities, M, B, and K are positive scalars of desired mass, damping, and stiffness,
respectively, and F is external force. Now rearrange (3.19) to fit the servo portion of the
control law in (3.2), which yields
.. [a,] , X= ='t
all
(3 .20)
where
41
Eq. (3.20) is the outer loop control in Figure 3.1. Combining (3.20) and (3.4), we obtain
the control law for hybrid impedance control, as illustrated in Figure 3.2, for the
manipulator with constrained motion, which fits the general structure given in Figure 3.1.
q F or warcl x
L__ l q Kinel'1Q. tiC x
lin
'___ ...L___ ~ _ ___=____ _ ___L.:._=_____l En vir on r'l e nt ~~I
Figure 3.2 Hybrid Impedance Control Diagram
In robotic deburring and grinding, it is intuitive to design the manipulator with a
large impedance (small compliance) in the normal direction and small impedance (large
compliance) in the tangential direction. A large impedance in the normal direction can
42
cause the end effector to remain insensitive to the grinding forces and remain very close to
the commanded (desired) trajectory. A large impedance implies that a position control
should be applied in the normal direction. This contrasts with force control used in the
hybrid impedance control law developed above. Although using force control in the
normal direction may provide stable contact with the workpiece, since displacement in this
direction is adjusted indirectly by force control, large position errors may occur with this
approach. However, the force environment for the deburring and grinding task is more
complicated than mere contact or loss of contact, and can not be represented by a simple
linear impedance. For simple tasks such as edgefollowing or "peginhole" operations,
where nonzero contact force exists in the normal direction and zero contact force is
assumed in the tangential direction, a hybrid impedance control strategy may be suitable.
We will evaluate this type of control by simulation in the next chapter.
Stability analysis of hybrid impedance control has not been properly addressed in
the literature, probably because the design strategy is so intuitive. If the target
(manipulator) impedance matrices are real, symmetric, and positive definite, the target
dynamics are stable. However, this does not guarantee stability of the complete system.
Moreover, hybrid impedance control developed for deburring and grinding operations
involves switching control after contact because of the change of environment. Contact
stability is a difficult problem, which we discuss in the stability analysis of the next section.
43
Switching Control
In this study, we seek high accuracy in the finished workpiece profile using robotic
deburring and grinding. We assume that the actual geometry of the workpiece is
unknown, and we wish to finish workpieces of different materials. From our literature
review, most proposed control methods for manipulators operating in constrained and
unconstrained environments employ some type of force control to obtain stable contact
while tracking a desired trajectory. To employ a forcetracking strategy under our
assumptions would be very difficult because the surface geometry is unknown and a
precise force generation model is required. Moreover, impulsive forces need to be
considered when the end effector contacts the workpiece or encounters a large burr. Such
demands increase the difficulty in implementing robotic deburring and grinding.
We have reviewed in Chapter I previous work on impact controL Pagilla, et al
[18], employed a simple rigid body collision and coefficient of restitution to model impact
to demonstrate that bouncing can be eliminated in finite time. However, this approach i
not suitable when penetration of the workpiece occurs, as in our deburring problem.
Tarn's work [4] may have potential for developing an improved control method for
robotic deburring and grinding. In the problem at hand, we consider position control and
force regulation of a simple twoarm SCARA robot carrying at its end effector a powered
tool used for deburring and grinding. This tool rotates at high speed, while the end
effector moves at low speeds in directions tangential with and normal to the nominal
surface of the workpiece. When the workpiece is contacted, such tools immediately cut
44
into workpiece, such that the "hard" surface assumed by previous investigators, [4, 8, 18],
immediately disappears. Accordingly, impact forces are likely much smaller in our
operation, assuming the grinding and robot motor torques can accommodate such forces
and provide stable contact. Consider a grinding or deburring tool in contact with a large
burr, which suddenly ends, such that the tool momentarily looses contact with workpiece
material. We assume that the normal distance from this point to the next point of surface
contact is sufficiently smaIl and that the normal distance to the desired trajectory is also
small such that the normal velocity of the end effector, under position control in free
space, does not become large. This implies that the tool approaches the next surface
contact with a relatively low normal velocity. A low approach velocity, coupled with the
material removal capacity by the tool, is expected to eliminate bouncing of the tool. A
further concern in grinding and deburring is the potential of burning the workpiece or
.
damage to the grinding or deburring tool if the material removal forces are excessive.
This can be avoided by controlling the robot such that norma] and tangential forces lie
below the burning or damage limits, which were addressed in Chapter II. Based on these
considerations, we examine a new switching control to implement deburring and grinding.
First, we divide our deburring and grinding operations into two phases, namely free space
motion and constrained space motion. In free space, a position controller is used to
follow a desired trajectory. After contact, when the material removal forces are below the
force limits developed in Chapter II, we employ the same position controller to guarantee
high accuracy of workpiece edge position. When the grinding forces approach the force
limits, force control will be employed to maintain material removal forces below the
45
grinding force limit. This control strategy is different from Hong's approach [3], which
under simultaneous position (PD) and force (PI) control will degrade the position
accuracy of the workpiece edge and require the command of a desired (but difficult to
determine) force.
A position controller can be easily implemented using proportional and derivative
control [4]:
(3.21)
or
(3.22)
where Kd and Kp represent derivative and proportional position gain matrices, respectively,
and the position and velocity error matrices e p and e p are defined by
ep = Xd  X
ep = Xd  X
(3.22a)
This position controller will be employed for position control in free space and constrained
space if the material removal forces are below the force limits.
Based on Tarn's work [4], force control using measured position acceleration, X ,
can be developed as
(3.23)
46
where Kf and Kfi are proportional and integral force gain matrices, respectively, and ef is a
force error matrix defined by
(3.24)
where subscriptj indicates a vector component and Fjlim is a force component limit. We
avoid a force time derivative in (3.23) because it is difficult to obtain a noiseforce time
derivative from a force sensor, which typically contains high frequency components in its
measurements. Implementing position acceleration feedback can be difficult, and it adds
an acceleration sensor, which will increase hardware cost and typically would provide a
very noisy signal. As a tradeoff, we propose eliminating the position acceleration
feedback in (3.23), such tbat the force controller becomes
(3.25)
Eq. (3.4) combined with (3.22) and (3.24) establishes our proposed new switching
controller. To implement such a control strategy requires measurements of joint position,
velocity, and force acting on the end effector by the environment. We assume encoders
and tachometers exist on the shafts of the actuators to measure tbe position and velocity
of each joint. A 2axis force sensor mounted at the end effector on the second link is
assumed for force measurements.
Stability Analysis
Stability of Position Control. For the position tracking we use a PD controller, as
given by (3.21), which after introducing (3.22a) yields the equation for error dynamics as
47
(3.26)
From (3.26), it can be seen that epj = 0 is an asymptotically stable eqUilibrium point for
the closedloop system when Kd and K p are positive diagonal matrices.
Stability of Force Control. Once the external forces Fj equal FjLim , we switch to PI
force control to insure normal and tangential forces remain below their limits to avoid
damage to the workpiece and tool. If we employ Tarn's [4] force control in (3.23), where
position acceleration feedback is introduced to cancel the effect of acceleration in c1osedloop
dynamics, we substitute the right side of (3.23) in (3.5) to obtain
(3.27)
for each degree of freedom because the components en of vector ef are decoupled. Kjj and
Kfij are the nonzero elements of diagonal matrices Kfand Kfi. respectively. Obviously, the
equilibrium point is e D = O. We assume (i) the trajectory remains in the constrained space
and (ii) the gains are positive. Because of decoupling, choose Lyapunov functions .\tj a
(3.28)
Since we know e fj < 0 except at the equilibrium point, VJ is positi ve and Vj 7 00 as
lieD 117 00 • Differentiating the right side of (3.28) with respect to time and employing
(3.27) gives
(3.29)
Thus, we see that ~ is negative definite and the system is asymptotically stable [27].
48
Now suppose we eliminate X in (3.23) to obtain a force control in (3.25) that is
easier to implement. The error dynamic equation then becomes
(3.31)
While we have not been able to prove stability for (3.31), extensive simulations show that
the results of this force controller are very close to those of Tam's controller with
acceleration feedback. In what follows, we consider switching between position and force
control by partitioning the problem into the two areas of concern, namely, switching
stability at contact and switching stability in constrained space. )
Stability at Contact. If the material removal forces are below the force limits after
contact, our pure position controller is employed for both free space and constrained
space. The grinding forces are treated as undesired disturbances and there is no controller
switch. The nominal stability of the position control has been established, above, by
assuming the grinding and robot motor torques can accommodate such forces and provide
a stable contact. If the torques exceed saturation limits, which means robot nonlinearities
cannot be properly canceled, multiple deburring passes will be needed to insure torques
remain under the limits. However we have been unable to prove stabiJity under torque
saturation.
If at contact the material removal forces exceed the force limits during contact,
switching occurs from position control to force control. This is similar to Tarn's problem
[4], if acceleration feedback is used. His and our switching control strategy employ a
position control in free space to eliminate unexpected bouncing and reestablish contact.
49
Our employment of material removal at contact is expected to soften any bouncing
tendency. Tarn has showed that the number of switches is finite and that the last switch
always corresponds to the transition from free space to constrained space [4, 8]. In
implementation, this requires that the sampling rate of measurement be higher than the
bouncing frequency . Without acceleration feedback as proposed by Tam, we have no
contact stability guarantee.
Switching Stability in Constrained Space. First consider that a single switch from
position control in steady state to force control occurs in constrained space and that the
trajectory remains in constrained space, meaning no loss of contact or "bounceoff'. A
nonoscillatory force transient response can be achieved, if the integral gain is small
enough. Even for relatively large integral gain, a desired nonoscillatory transient
response can still be obtained by an appropriate choice of K f and K ft' such that no loss
of contact occurs after switching [4]. Therefore, a single switch from a steady state of
position control to force control may remain stable. A single switch from steady state
force control to position control is stable if gains are properly chosen. Now, considering
frequent switching between the two controllers around Flimir , while our simulations
indicate stability and good dynamic behavior with suitable gains choices, we have been
unable to prove stability of the complete system. While such proof is important, it is
beyond the scope of this work.
In this chapter, we have discussed and proposed several controllers which may be
suitable for robotic debuITing and grinding. In next chapter, we will use computer
simulation to test and evaluate the performance of these approaches.
50
CHAPTER IV
COMPUTER SIMULATIONS
In this chapter, we numerically evaluate several control approaches for robotic
deburring and grinding using the Berkeley twoarm SCARA robot described in Chapter II.
Several types of burrs have been generated numerically to simulate rough edges, and a
motion plan has been designed for computer simulations. The simulation results are
presented for different controllers for an "easytogrind" (ETG) and a "difficulttogrind"
(DTG) material. We assume that the computations can be performed quickly enough that
the continuous time assumption is valid.
Simulation Parameters and Motion Plan
A UCBerkeley/NSK SCARA robot [19] has been employed as a benchmark for
our simulations because its configuration as a two axis robotic arm provides a planar
workspace, and because the technical data for this robot are available. The manipulator
parameters used in (2.17) for this robot are given as [3]:
II = 0.2675 kg m2 , 12 = 0.36 kg m2 , 13 = 0.0077 kg m2 , 14 = 0.051 kg m2 ,
13e = 0.04 kg m2 , II' = 0.046 kg m2 ;
m1 = 73 kg, m2 = 10.6 kg, m3 = 12 kg, and m4 = 4.85 kg, mp = 6.81kg;
L1 = 0.36 m, [2 = 0.24 m, Lle = 0.139 m, and 12c = 0.099 m;
51
J
thl = 5.5 N  m, and th2 = 0.9 N m.
The maximum torques for motors 1 and 2 are 245.0 Nm and 39.2 Nm, respectively.
These torque limits are used in a saturation function in computer simulations to avoid
overloads of the robot actuators.
The ETG and DTG workpieces to be deburred, or ground, in our examples are
metal plates of Rc 60, AISI 52100 steel and Rc 64, M4 alloy, respectively, with a thickness
of 10 nun. We propose to grind the edges of these plates, such that we take this thickness
as the active width of cut b. The diameter of grinding wheel is 50.8 mm (2 in.). We
assume the grinding wheel diameter is large compared to the peak heights of burrs, and
that the thickness of the grinding wheel is greater than the thickness of workpiece. The
grinding wheel and grinding conditions are described in Chapter n. The values of the
metal removal parameter Am are repeated here as 0.00871 in3/(min,lb) (5.3484 x 1010
m3/(sec, N)) for the ETG material and 0.002 in3/(rnin, lb) (1.228 x 1010 m3/(sec, N)) for
the DTG material. For simplicity, in the simulations of this study, we set the grinding
friction coefficient at Il = 0.7. Note from (2.21) that this reasonably high friction
coefficient means that the tangential grinding forces will be relatively large, although this
could be reduced by employing lubricating coolant.
The desired position and velocity of the grinding trajectory are given in Cartesian
space. The total simulation time is set at 10 seconds, and the workpiece edge to be
ground is aligned in the x direction of Cartesian space and is designated by
0.1 m ~ x ~ 0.2 m. The desired motion plan for simulations is as follows:
52
• Desired velocity X.d = [0.0°12 ] rnls
• [
0.08 + 0.012 t] Desired position Xd = m
0.4
• Desired acceleration X, ~ [~] mis'
where the matrix notation [;] indicates components in the x and y directions, and t is the
current simulation time. The starting point of the end effector is [ 0.08 ] ffi. Figure 4.1
0.399
illustrates a sample of end point motion of the robot reaching the desired trajectory in
Cartesian space.
Position History
400.2
400.0 r
399.8
E 399.6
g
> 399.4
399.2
399.0
398.8
80 100 120 140 160 180 200
X (mm)
Figure 4.1 End Point Motion of Robot to Desired Trajectory
53
A scaled line drawing of the robot link. positions at the beginning and end of the desired
trajectory is given in Figure 4.2. For force control, we employ an absolute force limit of
483 N, as developed in Chapter II. In our simulations, we employ a somewhat smaller
value of 450 N to provide a small margin for error.
Motion History of TwoAnn SCARA Robot
500rr~r_._.~~_.~
450
400
350
E'300
S
.~ 250
~ :> 200
150
100
50
50 100
end effector
150 200 250
XAxis (mm)
300 350
Figure 4.2 Motion History of TwoArm SCARA Robot
Burr Simulation
400
BUITS are unwanted irregUlarities on the edge surface of a workpiece. In practice,
we assume they are unpredictable and unmeasurable, causing variations in the cutting
force. In this section, we numerically generate three different types of burrs, namely
54 J
randomheight sinusoidal, large upset, and scallop to simulate rough surfaces of
workpieces.
A typical burr is highly variable. Kazerooni, et al [28], generated a geometric
model of a burred workpiece edge from statistical data based on the burr height and root
thickness measurements made on aircraft engine parts. In this work, average burr height
ranged from 0.25 to 0.75 mm (0.01 to 0.03 in.) and the thickness varied from 0.025 to
0.075 rnrn (0.001 to 0.003 in.). In our study herein, the thickness of all burrs is taken to
be the thickness of the workpiece b = 10 mm. For a smooth edge, we assume a desired
depth of cut he of 0.5 rnm. For sinusoidal burrs, we used burrs with height hb varying
from a to 0.1 mrn with an average height ha of 0.05 mm and a nominal desired depth of
cut he of 0.5 mrn. The spatial frequency ofburrsjb was chosen as 2 burrs/mm. We use a
sinusoidal function to generate an individual burr, with the magnitude generated by a
uniform random number [3]
Yburr = hd rand) x sin(21t jb x,) 0::; X, ::; 0.5 mm (4.1)
where
Ylmrr = ycoordinate of burr edge
hb( rand) = burr height randomly generated every 0.5 mrn
Examples of burr geometry and the sinusoidal burr edge are illustrated in Figure 4.3a and
b, respectively.
55 J
y ""'t ~~       ~f des;r ed contour
..£
'3> X
(a)
399.55 .r~__=____r__.
399.54
399.53
~399.52
E
.s399.51
c: g 399.50
'ij;
o
0.. 399.49
>
399.48
399.47
399.46
399.45 '____ '_____. ....1...____ "'____ '
100.0 100.5 101.0 101.5 102.0
X Position (mm)
(b)
Figure 4.3 Geometry of Sinusoidal Burrs
For large upset burrs, we begin with a rough surface modeled by randomheight sinusoidal
burrs with average burr height ha = 0.04 mm and burr frequency /b = 2 burrs/mm. On this
surface, we superimpose 3 stepup, step down pulses of beight 0.6 mm and width 20 mm,
separated by 10 mm, as illustrated in Figure 4.4. This was handled in the simulations by
step changes in the nominal desired depths of cut from he = 0.4 to he = 1.0 mm, and back.
56
399.8 r,rr,,
399.7
399. 6 ~.u+fI"Ml
399.5
EE 399.4 c
~ 399.3
·00
o
~ 399.2
399.1
399.0
398.9
workpiece
free space
398.8 '___ '___ L ____ ...L..___ '___ ~
100 120 140 160 180 200
XPosition (mm)
Figure 4.4 Illustration of Large Upset Burrs
Some automobile parts are produced by a ballshaped end mill tool [3], which can leave a
surface with a regular "scallopshaped" contour. The size and frequency of scallops are
dependent on the tool dimension and the number of passes per unit width of surface. In
this study, we examine scallopshaped burrs by assuming the diameter of the ballshaped
mill is 30 mm, with a 5 m.m span of tool passes. This can produce a 0.21 nun scallop
height hb and a frequency fb 200 scaHops per meter, illustrated in Figure 4.5.
57
399.60 ,,.,"r,
399.55
workpiece
399.50 f\ f\ A f\ f\ f\ f\ f\ f\ f\ f\ A f\ A fI A f\ A A f\
I 399.45
'""
c:::
~ 399.40
·00
o a>.. 399.35
399.30
399.25
free space
399.20l...1.1....L....Ll
100 120 140 160 180 200
XPosition (mm)
Figure 4.5 Illustration of Scallop Burrs
Several artificial surfaces have been developed to investigate robotic deburring
and grinding here. In next section, we use computer simulation to employ different
controllers in grinding these artificial surfaces. In our simulations, we will investigate
three issues: (i) dynamic behavior of contact between the end effector and the workpiece,
(ii) achievable perfonnance with and without motor torque limits, and (iii) ability to
accommodate large upset burrs. All source code is written in MA TLAB 4.2c [29], and
simulations are completed by SIMULINK 1.3c [30J using the automatic step size, Runge
Kutta 45 algorithm. Figure 4.6 presents a block diagram of the SIMULINK code.
S8
simulation time
Muxl Demux Mux3
u2
x
xadot
xdot
adot y
ydot
Figure 4.6 SIMULINK Block Diagram
59
Simulations for Impedance Control
In this section, we investigate five simulations of impedance control for robotic
deburring and grinding:
Simulation 1: Smooth Straight Edge
• The desired workpiece trajectory was a straight edge, from (0.1,0.4) m to (0.2, 0.4) m
in reference space.
• The "rough" surface was modeled by a smooth edge with a desired depth of cut he =
0.5 mm, and the workpiece material was ETG.
• The target impedance control matrices for the impedance control law in (3.7) were
selected as:
[10 0] [7746 0] [1500000 0] Md = kg, B = N  s I m, K = N I m o 10 0 8000 0 1600000
These values were chosen after a number of trials because position accuracy could be
improved by increasing parameter values in matrices K and B.
• Results showing position errors, external forces, and motor torques are given in
Figures 4.74.9. The external forces applied to the end effector are positive, such that
the torques applied to the workpiece are negative.
Simulation 2: RandomHeight Sinusoidal Burrs
• The same simulation conditions as in Simulation 1 were used, except the smooth edge
was replaced by a rough surface described by sinusoidal burrs with average burr height
60
ha = 0.05 rom and burr frequency /b = 2 burrs/rom. The nominal desired depth of cut
he is 0.5 mm and the workpiece material was ETG.
• Results showing position errors, external forces, and motor torques are given in
Figures 4.104.12.
Simulation 3: RandomHeight Sinusoidal Burrs with Desired Force Compensation
• In this simulation, we used the same simulation conditions as in Simulation 2, except
we employed an alternate impedance controller, described by (3.10), by including
desired contact forces at the desired trajectory. The desired contact forces were
obtained from Eq. (2.21) for desired velocity and desired depth of cut.
• Results showing position errors, depth of cut and remaining depth of cut, external
forces, and motor torques are given in Figures 4.134.16. Remaining depth of cut is
defined by the end point position of the end effector after grinding minus desired
position.
Simulation 4: Large Upset Burrs
• The same simulation conditions as in Simulation 1 were used, except the rougb surface
was modeled as large upset burrs, as in Figure 4.4. The material was ETG.
• Results showing position errors, depth of cut and remaining depth of cut, external
forces, and motor torques are given in Figures 4.17 4.20.
61
Simulation 5: Large Upset Burrs with Desired Force Compensation
• In this simulation, we used the same simulation conditions as in Simulation 4, except
we employed the alternate impedance controller, described by (3.10), by including
desired contact forces at the desired trajectory.
• Results showing position errors, depth of cut and remaining depth of cut, external
forces, and motor torques are given in Figures 4.214.24.
62
nr±rt
Tangential Position Error
0.1
0.08
E E 0.06  ~
0
l:: 0.04
Ql
0.02
0
0 2 3 4 5 6 7 8 9 10
time (sec)
Normal Position Error
0.1
0.08
.Es 0.06
~
0
l:: 0.04
Q)
0.02
0
0 2 3 4 5 6 7 8 9 10
time (sec)
Figure 4.7 Results of Simulation 1 with Impedance Control: Position Errors
Smooth Straight Edge
63
Tangential External Force
80~~~~~~~~~. ~
60
z (l) 40 e o 20
O~~~~~~~~J~~ o 2 3 456
time (sec)
7
Normal Extemal Force :(Limit 450 N)
8 9 10
150.,r,...r,.
_100
z CD
U
' o
 50
O~~~~~~~LL~ o 1 2 3 4 5 6 7 8 9 10
time (sec)
Figure 4.8 Results of Simulation I with Impedance Control: External Forces
Smooth Straight Edge
64
J
Torque 1 (Limit: 245 Nm)
300 I
200
E
zI 100
CD
::J 0 .0...
E
100
200
0 2 3 4 5 6 7 8 9 10
time (sec)
Torque 2 (Limit: 39.2 Nm)
40
1
 20 1 E
I
Z
Q) 0 I ::J
.0...
0 20
40
0 2 3 4 5 6 7 8 9 10
time (sec)
Figure 4.9 Results of Simulation 1 with Impedance Control: Motor Torques
Smooth Straight Edge
65
Tangential Position Error
0.1
0.08
.Es 0.06
....
0
~ 0.04
Q)
0.02
a a 2 3 4 5 6 7 B 9 10
time (sec)
Normal Position Error
0.1
0.08
.Es 0.06
....
0
~ 0.04
Q) i
0.02 ...I
0 a 2 3 4 5 6 7 8 9 10
time (sec)
Figure 4.10 Results of Simulation 2 with Impedance Control: Position Errors
RandomHeight Sinusoidal Burrs
66
Tangential External Force
80~~~~''~~~~
60
z  Q) 40 e o 20
O~~~~~~~~~~~ o 2 3 456
time (sec)
7
Normal Extemal Force :(Limit 450 N)
8 9 10
150.~.,.,......~
..100
z  Q) e
.E 50
o~~~~~~~~~~~~~ o 2 3 456
time (sec)
7 8 9
Figure 4.11 Results of Simulation 2 with Impedance Control: External Forces
RandomHeight Sinusoidal Burrs
67
10
Torque 1 (Limit: 245 Nm)
300
200
E
zI 100
(J) :0::J 0 ""\ .....
E
100
200
0 2 3 4 5 6 7 8 9 10
time (sec)
Torque 2 (Limit: 39.2 Nm)
40
 20
E
I
Z
(J) 0
:::J
.0....
0
 20
•• n .• u ..... lu ... ... ..... . oL . ...... ,. ·n '" ..... "T' ..... ''''
40 I
0 2 3 4 5 6 7 8 9 10
time (sec)
Figure 4.12 Results of Simulation 2 with Impedance Control: Motor Torques
RandomHeight Sinusoidal BUITS
68
E
E
"'" .....
.o.... .....
(])
Tangential Position Error
0.0 1 .r:r,rr,,......~___,_,_ ____.
0.005
o Jl..... IJ"VP
0.005
E
E .....
.o.... .....
(])
0.01~~~~~~~~~~ o 1 2 3 456
time (sec)
Normal Position Error
7 8 9
O.05.~r...r~~,......~
0.05~~~L~L~L~~~ o 1 2 3 456
time (sec)
7 8 9
Figure 4.13 Results of Simulation 3 with Impedance Control: Position Errors
RandomHeight Sinusoidal Burrs with Desired Force Compensation
69
10
Depth of Cut
0.5 ~..ftJrA/MM_~M~\'IHHtrWM_.H
EOA
E
~0.3
a.
~ 0.2
0.1
O~~~~LL~~~~
E
E
100 110 120 130
x 103
140 150 160
XCoordinate (mm)
Remaining Depth of Cut
170 180 190 200
5r~r'~~r~~~~__
~ 0
a. Q)
"C
_5~~~L~ ____ ~ __ ~ ____ ~ ____ L ____ ~ _ ~
100 110 120 130 140 150 160
XCoordinate (mm)
170 180 190
Figure 4.14 Results of Simulation 3 with Impedance Control: Depth of Cut and
Remaining Depth of Cut
RandomHeight Sinusoidal Burrs with Desired Force Compensation
70
200
.. J
100
80
z 60 '"
Q) .u... .E 40
20
0
0 2 3
Tangential External Force
456
time (sec)
7
Normal External Force :(Limit 450 N)
8 9 10
150~~~~~~~~~~~
_100 z
Q)
~ o
 50
O~~~~L~~~~L__ ~ _ ~ o 2 3 456
time (sec)
7 8 9
Figure 4.15 Results of Simulation 3 with Impedance Control: External Forces
RandomHeight Sinusoidal Burrs with Desired Force Compensation
71
10
300
 200 E
zI 100
Q.l
::J 0 .0.."."
,0
100
200
40
_ 20
E
I
Z
Q.l 0
::J e o
 20
0
40 o
1 2
.......
2
3
_." ...... 7'""
3
Torque 1 (Limit: 245 Nm)
456 7
time (sec)
Torque 2 (Limit: 39.2 Nm)
." ."
~ '''' ...
456
time (sec)
"
7
B 9
,
B 9
Figure 4.16 Results of Simulation 3 with Impedance Control: Motor Torq ues
RandomHeight Sinusoidal Burrs with Desired Force Compensation
72
10
....
10
_0.4
E
eE .... 0.2
Q)
Of'
o
0.8
0.6 E
E
';:"0.4
..o......
Q)
0.2
o o
1
r
2 3
2 3
Tangential Position Error
456
time (sec)
Normal Position Error
,
456
time (sec)
7 8 9
7 8 9
Figure 4.17 Results of Simulation 4 with Impedance Control: Position Errors
Large Upset Burrs
73
10
10
'E0.8
E
;;0.6
i5..
~ 0.4 J..WM1r'Jr<..NNI
Depth of Cut
i ,
0.2
?0~0~~1~10~1~2~0~1~30~1~4015LO1~6017LO1~8019~0~200
XCoordinate (mm)
0.6
E
SO.4
~
i5..
Q)
'0 0.2
o
100 110 120 130
Remaining Depth of Cut
I
, ,
140 150 160
XCoordinate (mm)
 , I
..!
I
I
I 
1 I
I
;]
\ I LJ
170 180 190 200
Figure 4.18 Results of Simulation 4 with Impedance Control: Depth of Cut and
Remaining Depth of Cut
Large Upset Burrs
74
150
z_ 100 Q)
2
.2 50
o o
Jo.l. "'" ~,
2 3
Tangential External Force
"II ''''
456
time (sec)
7
Normal External Force :(Limit 450 N)
8
,
•
I I
"~
9 10
250.r..r~~...
200
~ 150
Q)
() .E 100
50
O~J~~~~~_J __ ~_L ___ _L ____ ~ _ ~
o 2 3 456
time (sec)
7 8 9
Figure 4.19 Results of Simulation 4 with Impedance Control: External Forces
Large Upset Burrs
75
10
300
200
E
zI 100
Q)
:::l 0 .0.. 0 100
200
40
E 20
I
Z
Q) 0
:::l e o
 20
0
40 o
2
2
3
3
Torque 1 (Limit: 245 Nm)
456 7
time (sec)
Torque 2 (Limit: 39.2 Nm)
. r
4
, ..
~
5
time (sec)
6 7
8 9
8 9
Figure 4.20 Results of Simulation 4 with Impedance Control: Motor Torques
Large Upset Burrs
76
J
10
1
J
1
10
_0.4
E
eE 0.2
~
Q)
0/1
0.5
E°.4
E
:: 0.3 e
05 0.2
o
0.1
° °
2
1 2
3
(
3
Tangential Position Error
456
time (sec)
Normal Position Error
,
456
time (sec)
7 8 9
.
7 8 9
Figure 4.21 Results of Simulation 5 with Impedance Control: Position Errors
Large Upset Burrs with Desired Force Compensation
77
10
.J
10
E .s
..c a.
1.5
~ 0.5
o
100
0.5
EOA
E
:;; 0.3
a.
..g 0,2
0.1
o
100
\ ... m ..", 'I"'
110 120 130
(
110 120 130
Depth of Cut
~
140 150 160
XCoordinate (mm)
Remaining Depth of Cut
r
140 150 160
XCoordinate (mm)
0
I\,
170
170
~
0
180 190
0
180 190
Figure 4.22 Results of Simulation 5 with Impedance Control: Depth of Cut and
Remaining Depth of Cut
Large Upset Burrs with Desired Force Compensation
78
I
200
i
.J
.,
200
200
150
.z. . Q) 100
f2
.E
50 ro
o
300
_200 z Q)
.u...
o
 100
o o
2
....
2
....
3
Tangential External Force
456
time (sec)
I
7
Normal External Force :(Limit 450 N)
3 456
time (sec)
7
8 9
.
8 9
Figure 4.23 Results of Simulation 5 with Impedance Control: External Forces
Large Upset Burrs with Desired Force Compensation
79

10
10
Torque 1 (Limit: 245 Nm)
300
200
E
zI 100 1 (]J
::l 0 t:r
"
.8
100
200
0 1 2 3 4 5 6 7 8 9 10
time (sec)
Torque 2 (Limit: 39.2 Nm)
40
E 20
I
Z 0 (]J
::l
t:r
"
.8 20
40
0 1 2 3 4 5 6 7 8 9 10
time (sec)
Figure 4.24 Results of Simulation 5 with Impedance Control: Motor Torques
Large Upset Burrs with Desired Force Compensation
80
Discussion and Analysis for Impedance Control
The results of Simulations 15 indicate that impedance control is able to achieve
stable performance if the impedance parameter matrices are positive definite. Steady state
can be reached quickly if impedance parameters are selected properly. Large steady state
position errors exist in Simulation 1,2, and 4 because impedance control does not control
the position directly, but instead regulates the relationship between the end effector
position and the contact force. Steady state position errors can be improved by an
alternate impedance controller (3.10), with a commanded desired force term.
In Simulation l, we investigated grinding performance on a smooth straight edge,
with desired depth of cut he = 0.5 rnm, to test the performance of impedance control. The
results show that steady state is reached quickly and steady state position errors occur in
both tangential and normal directions after contact with the workpiece. By trial and error,
a set of target parameter matrices were selected. High values for the position gains in the
K matrix were chosen to increase position accuracy in both normal and tangential
directions .. Position accuracy could be improved by increasing parameter values in
matrices K and B, with Md fixed, but this requires more time for simulation with only slight
increases in the performance. Practical limits in implementation exist for these values, and
attaining zero steady state error is not possible with finite values.
In Simulation 2, we investigated impedance control with randomheight sinusoidal
burrs on a straight edge. The results are similar to these in Simulation 1, except small
irregular variations occur,. caused by the contact force variations from randomheight
sinusoidal burrs. Our simulation results show that approximately 86 % of unwanted
81
paz
materials are removed by grinding, which is different from Hong's results [3] that show
the end effector barely contacting the workpiece. We believe Hong's work is in error. In
Simulation 3, an alternate impedance control (3.10) is employed to deburr the same
surface as in Simulation 2. Simulation results, Figure 4.134.16, show that in contrast to
results from Simulation 2, position errors remain close to zero, and external contact forces
in steady state remain close to the commanded forces, namely those required to remove
materials to reach the desired edge trajectory. These results indicate that if surface
geometry is known and the needed grinding force Fd is modeled well, desired forces may
be commanded to increase the performance of impedance control. However, in real
operations, burrs are highly irregular, and it is difficult to model the desired grinding
forces precisely.
In Simulations 4 and 5, we simulate impedance control in deburring an edge with
large upset burrs illustrated in Figure 4.4. Simulation results show that the maximum
depth of cut that can be reached by this robot for the ETG workpiece material is about 0.6
mm because the torque of motor 2 saturates. This torque limit also causes large position
errors for large upset burrs. To improve position accuracy, either a larger torque motor or
multiple passes of cut should be employed. An alternate impedance control can improve
the performance of robotic deburring and grinding only when the motor torques do not
saturate.
Compared to Hong's results [3], our results show significant improvement in
steady state position error in the normal direction. Hong's results show large steady state
errors in the nonna! direction that are approximately equal to the deviation of the average
82
rough edge position from the desired normal position. This is probably caused by
programming mistakes. Our simulation results show that 86 % or more of desired depth
of cut can be reached by an impedance controller.
Based on these results, we conclude that an impedance controller provides a wellbehaved
controller for both free space and the constrained environment. It may be
suitable for "rough" deburring and grinding operations or edge following tasks. If the
surface geometry is known and desired grinding force Fd is modeled well, desired forces
may be commanded to increase the performance of impedance control.
Simulations for Hybrid Impedance Control
The hybrid impedance control law of (3.17), (3.20), and (3.4), with position
control in unconstrained space and in the tangential direction after contact, and force
control in the normal direction after contact, is investigated in this section. The following
simulation were employed:
Simulation 6: Smooth Straight Edge
• The desired workpiece trajectory was a straight edge starting from (0.1,0.4) m to
(0.2,0.4) m in reference space.
• The "rough" surface was modeled by a smooth straight edge with a desired depth of
cut he = 0.5 mm and the workpiece material was ETG.
83
• After some initial trials, impedance parameters for constrained space were chosen for
Eq. (3.20) as fit = 5kg, bt = 200 Ns/m, kt = 30000 N/m, mn = 100 kg, bn = 50000 Ns/
m. For unconstrained space, the impedance matrices for Eq. (3.17) were selected as
M d = [ 5 0] kg. B = [2000 0] N  s I m, K = [30000 0]N I m o 5 0 2000 0 30000
• Simulation results showing position errors, depth of cut and remaining depth of cut,
external forces, position history, and motor torques are given in Figures 4.254.29.
84
E
E ~
0... .
~
Q)
4
3 E
E
';:'2
2 ~
Q)
1
0
0 1 2 3
Tangential Position Error
456
time (sec)
Normal Position Error
7 8 9 10
1.5r,.....~~~~
0.5
0 \
0.5
0 1 2 3 456
time (sec)
7 8 9
Figure 4.25 Results of Simulation 6 with Hybrid Impedance Control: Position Errors
Smooth Straight Edge
85
10
Depth of Cut
0.8 '( I
 0.6 E
E 
;;0.4
0.
Q)
'0
0.2
0
80 100 120 140 160 180 200
XCoordinate (mm)
Remaining Depth of Cut
0.1
I
0 J
E E J
;; 0.1
0.
Q)
'0
0.2
0.3
V
80 100 120 140 160 180 200
XC':nmrlin::ltp. fmm\
Figure 4.26 Results of Simulation 6 with Hybrid Impedance Control:
Depth of Cut and Remaining Depth of Cut
Smooth Straight Edge
86
100
80
z 60 Q)
~ 40 0
20
0
0 2 3
Tangential External Force
456
time (sec)
7
Normal External Force :(Limit 450 N)
I I
8 9 10
150r,.r.r..,~~
_100
z
<ll
U....
.E 50
O~~~J~LL_ __ ~ ___ ~ ___ L__ ~ _ ~
o 2 3 456
time (sec)
7 8 9
Figure 4.27 Results of Simulation 6 with Hybrid Impedance Control: External Forces
Smooth Straight Edge
87
10
Position History
400.4,.~,.~~~~
400.J
400
399.8
E
.s399.6
>
399.4
399.2
399
398.8 L____ L___ L ____ ..!.___ ~ ______ ___.J
80 100 120 140 160 1 80 200
X (mm)
Figure 4.28 Results of Simulation 6 with Hybrid Impedance Control: Position History
Smooth Straight Edge
88
d
Torque 1 (Limit: 245 Nm)
150
_100
E
I
Z
CD 50
~
0""
~
.8 0
50
0 2 3 4 5 6 7 8 9 10
time (sec)
Torque 2 (limit: 39.2 Nm)
0
_10
E
I
Z
; 20
~
0""
'
0
 30
40
0 2 3 4 5 6 7 8 9 10
time (sec)
Figure 4.29 Results of Simulation 6 with Hybrid Impedance Control: Motor Torques
Smooth Straight Edge
89
Discussion and Analysis for Hybrid Impedance Control
After some trial simulations, we were able to obtain results showing small
oscillation and steady state position errors, as illustrated in Figures 4.254.29. From
simulation results, we found that zero steady state could be reached before contact, but
after contact, large transient force and position oscillations occur. This is caused by
switching from position control to force control in the normal direction at contact.
Because force control attempts to achieve desired forces, the dynamic effect of impact
causes fluctuations in position, external forces, and depth of cut.
From (3.19), Hong [3] shows that
where
V n = normal velocity Fn = normal grinding force
Bn = damping parameter Fdn = desired normal grinding force
Thus the impact velocity can be reduced by increasing the value of Bn. At steady state,
FdnFn is close to zero, such that the normal velocity should also approach zero jf Bn is not
small. Increasing the values of ml> b" and k/ improves motion tracking in the tangential
direction, which is why large values were selected for the position control in the tangential
direction. However, position control in the tangential direction will be degraded by the
force control in the normal direction, because the force control adjusts position in the
90
normal direction to control the normal force.
To implement hybrid impedance control in robotic deburring and grinding is
difficult, because including desired forces requires a known surface geometry and accurate
grinding force modeling. Displacement in the normal direction is adjusted indirectly by
force control, such that large position errors may occur with this approach. Moreover the
force environment for the deburring and grinding task is more complicated than mere
contact or loss of contact, which cannot be represented by a simple linear impedance. For
simple tasks such as edgefollowing or "peginhole" operations, where nonzero contact
force exists in the normal direction and zero contact force is assumed in the tangential
direction, a hybrid impedance control strategy may be suitable. We conclude, however,
that it is unsuitable for deburring and grinding.
Simulations for Switching Control
In this section, we employ simula60ns using different materials and rough edges to
investigate the performance of our proposed new switching control. Results for this
control, as described by (3.4), (3.21), and (3.25), are presented for the following
simulations:
Simulation 7: RandomHeight Sinusoidal Burrs, ETG Workpiece
• The desired workpiece trajectory was a straight edge, from (0.1, 0.4) m to (0.2, 0.4) m
in reference space.
91
• The rough surface is modeled by sinusoidal burrs with average burr height ha = 0.05
mm and burr frequency jb = 2 burrs/mm. The nominal desired depth of cut he is 0.5
mm, and the material is ETG.
• After some initial trials to tune the controller, the proportional and derivative gain
matrices were chosen for position control (3.21) as:
[900 0] I
Kp = ° 9OO~'
Similarly, the proportional and integral gain matrices for force control (3.25) were
chosen as:
:[0.005 0]1 m
Kf = ° 0.005 52 N'
[0.00006 0] 1 m
Kfi = . ° 0.00006~· N
• Simulation results showing position errors, external forces, and motor torques are
given in Figures 4.304.32.
Simulation 8: RandomHeight Sinusoidal Burrs. DTG Workpiece
• The same simulation conditions as in Simulation 7 were used, except the workpiece
material was changed to DTG.
• Simulation results showing position errors, external forces, and motor torques are
given in Figures 4.334.35.
92
Simulation 9: RandomHeight Sinusoidal Burrs. DTG Workpiece. No Torgue Limits
• The same simulation conditions as in Simulation 8 were used, except torque limits on
the motors were removed.
• Simulation results showing position errors, depth of cut and remaining depth of cut,
external forces, and motor torques are given in Figures 4.364.39.
Simulation 10: Large Upset Burrs. ETG Workpiece
• The same simulation conditions as in Simulation 7 were used, except the rough surface
was modeled as large upset burrs, as in Figure 4.4.
• Simulation results showing position errors, depth of cut and remaining depth of cut,
external forces, and motor torques are given in Figures 4.404.43.
Simulation 11: Large Upset Burrs. ETG Workpiece, No Torgue Limits
• The same simulation conditions as in Simulation 10 were used, except torque limits on
the motors were removed.
• Simulation results showing position errors, depth of cut and remaining depth of cut,
external forces, and motor torques are given in Figures 4.444.47.
Simulation 12: Large Upset Burrs, DTG Workpiece. No Torque Limits
• The same simulation conditions as in Simulation 11 were used, except the workpiece
was changed to DTG material.
• Simulation results showing position errors, depth of cut and remaining depth of cut,
external forces, and motor torques are given in Figures 4.484.51.
93
Simulation 13: Scallop Burrs, ETG Workpiece
• The same simulation conditions as in Simulation 7 were used, except the rough surface
was modeled by scallop burrs, as illustrated in Figure 4.5.
• Simulation results showing position errors, depth of cut and remaining depth of cut,
external forces, and motor torques are given in Figures 4.524.55.
Simulation 14: Scallop Burrs, DTG Workpiece. No Torque Limits
• The same simulation conditions as in Simulation 13 were used, except the workpiece
was changed to DTG material and torque limits on the motors were removed.
• Simulation results showing position errors, depth of cut and remaining depth of cut,
external forces, position history, and motor torques are given in Figures 4.564.59.
94
Tangential Position Error
0.15
0 .1 E I
E J .... 0.05 ..!
~ 0 i ........
Cl)
0
0.05
0 2 3 4 5 6 7 8 9 10
time (sec)
Normal Position Error
1.5
E .s .... 0.5
..o......
Cl)
o \~
0.5 L...__ L ____ ..l....__ L ____ ...l...__ L ____ L__ l ____ L..____ L..:....~
o 2 3 4 5 6 7 8 9 10
time (sec)
Figure 4.30 Results of Simulation 7 with Switching Control: Position Errors
RandomHeight Sinusoidal Burrs, ETG Workpiece
95
I S
100
80
z 60
Q)
.t..>.. 0 40
20
0
0
150
_100 z
Q)
~ o
 50
2
Tangential External Force
,
I
I
3 4 5 6 7 8 9 10
time (sec)
Normal External Force :(Lirnit 450 N)
O~~~~LL~~L~~ o 2 3 456
time (sec)
7 8 9
Figure 4.31 Results of Simulation 7 with Switching Control: External Forces
RandomHeight Sinusoidal Burrs, ETG Workpiece
96
10
Figure 4.32 Results of Simulation 7 with Switching Control: Motor Torques
RandomHeight Sinusoidal BUITs, ETG Workpiece
97
Tangential Position Error
0.6
0.4
E
E ... 0.2 ..:
.0. . "
Q)
0
0.2
0 2 3 4 5 6 7 8 9 10
time (sec)
Normal Position Error
1.5
!
... E 1
E 0.5 I ... 1 .0..
I
... Q)
0 \
0.5
0 1 2 3 4 5 6 7 8 9 10
time (sec)
Figure 4.33 Results of Simulation 8 with Switching Control: Position Errors
RandomHeight Sinusoidal BUITs, DTG Workpiece
98 
Tangential External Force
250
200
~ 150
Ql
u
(5100
50
0
0 2 3 4 5 6 7 8 9 10
time (sec)
Normal External Force :(Limit 450 N)
400
300
z Ql 200 u
'
.2
100
0
0 2 3 4 5 6 7 8 9 10
time (sec)
Figure 4.34 Results of Simulation 8 with Switching Control: External Forces
RandomHeight Sinusoidal Burrs, DTG Workpiece
99
Torque 1 (limit: 245 Nm)
20
0 E
~ 20
Q.l
540 .....
.8
60
80
0 1 2 3 4 5 6 7 8 9 10
time (sec)
Torque 2 (Limit: 39.2 Nm)
0
_10
E
I
Z
; 20
~ .c....r
0 30
40
0 2 3 4 5 6 7 8 9 10
time (sec)
Figure 4.35 Results of Simulation 8 with Switching Control: Motor Torques
RandomHeight Sinusoidal BUffs, DTG Workpiece
100
Tangential Position Error
0.15
0.1
E
E.... 0.05
..0.... .. ~Mf'w,.\t"'<.,.J./\ ....r ... .\'\.A'\,"P.
OJ
0
0.05
0 2 3 4 5 6 7 8 9 10
time (sec)
Normal Position Error
1.5
1
~
E
E .... 0.5 "'j
0 ,
.'... I
OJ I
0
I 0.5
0 2 3 4 5 6 7 8 9 10
time (sec)
Figure 4.36 Results of Simulation 9 with Switching Control: Position Error
RandomHeight Sinusoidal BUffS, DTG Workpiece, No Torque Limits
101
Depth of Cut
0 . 6.,.r=~~~
EO.4
E ~
i5..
~0.2
OLL~LL~~
80 100 120 140 160 180 200
XCoordinate (mm)
Remaining Depth of Cut
0 . 08r,~.....~
0.06 E
E
:;; 0.04
i5..
Q)
"0
0.02
OL~~~L~~
80 100 120 140 160 180 200
XCoordinate (mm)
Figure 4.37 Results of Simulation 9 with Switching Control: Depth of Cut and Remaining
Depth of Cut
RandomHeight Sinusoidal BUITs, DTG Workpiece. No Torque Limits
102
, ,
400
300
z Q.l 200
2
0 100
0
0
500
400
~300
Q.l u .E 200
100
0
0
1 2
1 2
3
Tangential Extemal Force
456
time (sec)
7
Normal Extemal Force :(Limit 450 N)
3 456
time (sec)
7
8 9
8 9
Figure 4.38 Results of Simulation 9 with Switching Control: External Forces
RandomHeight Sinusoidal Burrs, DTG Workpiece, No Torque Limits
103
10
10
Torque 1 (No Limit)
50 r.~_,_,,_..~
E 0 I
Z
Q)
::::l
.0... 50
.8
100
0 2 3 4 5 6 7 8 9 10
time (sec)
Torque 2 (No Limit)
0 ,
I
E 50
I
I
i
Z I
Q) ::::l I
0 E 100
,
~~~~..w~1
150
0 2 3 4 5 6 7 8 9 10
time (sec)
Figure 4.39 Results of Simulation 9 with Switching Control: Motor Torques
RandomHeight Sinusoidal Burrs, DTG Workpiece, No Torque Limits
104
E .s
~
0
~
~
Q)
E
E ~
0
~
~
Q)
0.6
0.4
0.2
0
0.2
0 1 2 3
Tangential Position Error
456
time (sec)
Normal Position Error
7 8 9 10
1.5r..,r..~~r_~~
0.5
0
0.5
0 1 2 3 456
time (sec)
7 8 9
Figure 4.40 Results of Simulation 10 with Switching Control: Position Errors
Large Upset Burrs, ETG Workpiece
105
10
Depth of Cut
E'1 .s
~
0.
~ 0.5
°8~O~1~00~o ~1~2~~~L~~~ 140 160 180 200
E .s
~ a.
Q)
"C
0.6
0.4
0.2
0
0.2
80
r
100
XCoordinate (mm)
Remaining Depth of Cut
r ( """\
".J 'J
120 140 160 180
XCoordinate (mm)
Figure 4.41 Results of Simulation 10 with Switching Control: Depth of Cut and
Remaining Depth of Cut
Large Upset Burrs, ETG Workpiece
106
'
200
Tangential External Force
150r,...,~~~~===~ I
z_ 100 CD
f:
.E 50
~
M I
O~~~~LJ~~~~L~~L~~~ o 1 2 3 4 5 6 7 8 9 10
250
200
Z150 CD
tJ
(5100
50
o o
..... r ...... ""l
1 2
time (sec)
Normal External Force :(Limit 450 N)
3
.~ ,
,
456
time (sec)
~
I
I
7

I
8 9
Figure 4.42 Results of Simulation 10 with Switching Control: External Forces
Large Upset Burrs, ETG Workpiece
107
.........
"
10
Torque 1 (Limit: 245 Nm)
20
 0
E
I
Z
;20
:::I
.t..T.
0
 40
60
0 1 2 3 4 5 6 7 8 9 10
time (sec)
Torque 2 (Limit: 39.2 Nm)
0
_10
E
I
Z
; 20
:::I
!:
0 30
40
0 1 2 3 4 5 6 7 8 9 10
time (sec)
Figure 4.43 Results of Simulation 10 with Switching Control: Motor Torques
Large Upset Burrs, ETG Workpiece.
108
Tangential Position Error
0.15
0.1
E
E 0.05 '
.0. . '
Q)
0
0.05
0 1 2 3 4 5 6 7 8 9 10
time (sec)
Normal Position Error
1.5
1
E
E 0.5 '
.0... .. Q) .
0 ~ .
0.5
0 1 2 3 4 5 6 7 8 9 10
time (sec)
Figure 4.44 Results of Simulation 11 with Switching Control: Position Errors
Large Upset Burrs, .t;:TG Workpiece, No Torque Limits
109
1.5
'E1
E .c
Q.
~0.5
o
80
0.5
.'Es 1
.c
g.1.5
"0
2
~
100
Depth of Cut
~
 
I,
I
120 140 160 180 200
XCoordinate (mm)
Remaining Depth of Cut
2.5 ~:..L...L...L.="'"""''...L....l__
80 1 00 120 140 160 180 200
XCoordinate (mm)
Figure 4.45 Results of Simulation 11 with Switching Control: Depth of Cut and
Remaining Depth of Cut
Large Upset Burrs, ETG Workpiece, No Torque Limits
110
200
150 z Q) 100
~ o 50
o o
250
200
2150 Q)
(J
0100
50
o
o
"'
1 2
w
ny ""f
1 2
.~L ...
.". ........
3
Tangential Extemal Force
_..u. ....
" I ~
'w
1m
456
time (sec)
7
Normal Extemal Force :(Limit 450 N)
ill. ,. Itb ~ .... .~ .. ~ ,.
'III ,.,"" ... .... .,..
, ...
'III
3 4 5 6 7
time (sec)
....
8 9
"r""" T '"
.
8 9
Figure 4.46 Results of Simulation II with Switching Control: External Forces
Large Upset Burrs, ETG Workpiece, No Torque Limits
111
....
10
..... .. "
10
20
E 0
I
Z
; 20
:::l
E" o
 40
60 o
o
_20
E
I
Z
; 40
:::l
t:T
~ o
 60
80 o
I

1 2 3
... ......
...... . " ~I
1 2 3
Torque 1 (No Limit)
~
 
456
time (sec)
Torque 2 (No Limit)
Iw ....
. .. I.~.
.... .....
.
456
time (sec)
,
.J l J
7 8 9
,
.... w
7 8 9
Figure 4.47 Results of Simulation 11 with Switching Control: Motor Torques
Large Upset Burrs. ETG Workpiece, No Torque Limits
112
I
I 1
'" I
j
I
10
~
10
0.4
0.3
E
E
::0.2 e ~
Q)
E .s
~
..0.... ..
Q)
0.1
a o
1.5
1
0.5
0
0.5
0
r
\
1 2
1 2
.d
I
3
3
Tangential Position Error
~
..
\
456
time (sec)
Normal Position Error
4 5 6
time (sec)
\ I ~
\
7 8 9
7 8 9
Figure 4.48 Results of Simulation 12 with Switching Control: Position Errors
Large Upset Burrs, DTG Workpiece, No Torque Limits
113
I
I
\
10
10
E
E ~ a
~ 0.5
Depth of Cut
O~~~~~~~~L~
E .s
~ a
Q)
"0
80
0.6
0.4
0.2
0
0.2
80
100
100
120 140 160 180
XCoordinate (mm)
Remaining Depth of Cut
I r
" L...J
120 140 160 180
XCoordinate (mm)
Figure 4.49 Results of Simulation 12 with Switching Control: Depth of Cut and
Remaining Depth of Cut
Large Upset Burrs, DTG Workpiece, No Torque Limits
114
200
'
200
I
Tangential External Force
800
600 z Q) 400
~
0 200
0
0 1 2 3 4 5 6 7 8 9 10
time (sec)
Normal External Force :(Limit 450 N)
1000
800
z 600 Q)
.C..J. 0 400
200
0
0 1 2 3 4 5 6 7 8 9 10
time (sec)
Figure 4.50 Results of Simulation 12 with Switching Control: External Forces
Large Upset Burrs, DTG Workpiece, No Torque Limits
115
Torque 1 (No Limit)
50
0 J E
zI 50 Q)
.6... 100
0 150
200
0 1 2 3 4 5 6 7 B 9 10
time (sec)
Torque 2 (No Limit)
0
E 100 I z Q)
:::J c
0200 l
300 a 1 2 3 4 5 6 7 8 9 10
time (sec)
Figure 4.51 Results of Simulation 12 with Switching Control: Motor Torques
Large Upset Burrs, DTG Workpiece, No Torque Limits
116
Tangential Position Error
0.15
0.1
E
E .... 0.05
..0.... ..
Q)
0
0.05
0 1 2 3 4 5 6 7 B 9 10
time (sec)
Normal Position Error
1.5
 1 E
E .... 0.5
.0..... ..
Q)
0
0.5
0 2 3 4 5 6 7 B 9 10
time (sec)
Figure 4.52 Results of Simulation 13 with Switching Control: Position Errors
Scallop Burrs, ETG Workpiece
117
Depth of Cut
0.8 r,,y.~:~
0.6
E
E
::; 0.4
a. Q)
1:)
0.2
OL~L~~L~~~
00 100 1~ 1~ 1W 100 ~
0.1
0.08 E .§. 0.06
.s=
g.0.04
1:)
0.02
o
80
\ \ \
100
XCoordinate (mm)
Remaining Oepth of Cut
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \
120 140 160 180
XCoordinate (mm)
Figure 4.53 Results of Simulation 13 with Switching Control: Depth of Cut and
Remaining Depth of Cut
Scallop Burrs, ETG Workpiece
118

\ 1\
200
Tangential External Force
150,~.r~.r~~c~~
_100
z III
f::
.E 50
r
\ ;
O~~~~~~~~~~L~~
_100
z (I)
u
~ o
 50
o 1 2 3 456
time (sec)
7
Nonnal External Force :(Limit 450 N)
8 9 10
oo~ ~~~~~~~~~~~ 1 2 3 456
time (sec)
7 8 9 10
Figure 4.54 Results of Simulation 13 with Switching Control: External Forces
Scallop Burrs, ETG Workpiece
119
Torque 1 (Limit: 245 Nm)
10
0 E i1O Q)
520
~
0 30
40
0 2 3 4 5 6 7 8 9 '0
time (sec)
Torque 2 (Limit: 39.2 Nm)
0
_10
E
I
Z
; 20
::l c::r ~
.B 30
40
0 1 3 4 5 6 7 B 9 10
time (sec)
Figure 4.55 Results of Simulation 13 with Switching Control: Motor Torques
Scallop Burrs, ETG Workpiece
120
E
E
Tangential Position Error
0.15r,~rT~I~r~,~r~~'~~~~~~~~~f~~p., ~
0.1
":" 0.05 v o VV V VV V V VV V V V V VV V VV V ""
CD
o
0 . 05~~~~~~~~~~~
E
E
o 1 2 3 456
time (sec)
Normal Position Error
7 8 9 10
1.5r,r..r~r_~~~
1
= 0.5 e "
CD
o
0.5~~~~~~~~L~~~=~ o 1 2 3 4 5 6 7 8 9 10
time (sec)
Figure' 4.56 Results of Simulation 14 with Switching Control: Position Errors
Scallop Burrs, DTG Workpiece. No Torque Limits
121
Depth of Cut
0.8r~,~~r~===~
 0.6 E
E
;;0.4
Q.
Q)
"0
0.2
O~~~~~~L~L~~~
80 100 120 140 160 180 200
XCoordinate (mm)
Remaining Depth of Cut
0.25 r..,.T:.r'~~r''=.,
0.2 .Es 0.15
~ g. 0.1
"0 0.05 VVVvVVvvvVVVV VVVVVV v
°8·~01~0~0~~~~~~LL~
120 140 160 180 200
XCoordinate (mm)
Figure 4.57 Results of Simulation 14 with Switching Control: Depth of Cut and
Remaining Depth of Cut
Scallop Burrs, DTG Workpiece, No Torque Limits
122
Tangential External Force
500
400
2300 II I Q.l
0 I
(5200
1

100
0
0 1 2 3 4 5 6 7 8 9 10
time (sec)
Normal Extemal Force :(Limit 450 N)
800
600 z Q) 400 0.. . 0 200
0
0 2 3 4 5 6 7 8 9 10
time (sec)
Figure 4.58 Results of Simulation 14 with Switching Control: External Forces
Scallop Burrs, DTG Workpiece, No Torque Limits
123
Torque 1 (No Limit)
50
E 0
I
Z 50 Q)
::::l e o  100
150
0 1 2 3 4 5 6 7 8 9 1'0
time (sec)
Torque 2 (No Limit)
0
E 50
I z
; 100
::::l
.C.."
0 150
200
0 1 2 3 4 5 6 7 8 9 10
time (sec)
Figure 4.59 Results of Simulation 14 with Switching Control: Motor Torques
Scallop Burrs, DTG Workpiece. No Torque Limits
124
Discussion and Analysis for Switching Control
The position control parameters are chosen by considering the scalar characteristic
equation of a position controller,
ms2 +bs+k = 0 (4.2)
where b, m, and k are scalar gains quantities representing friction, mass, and stiffness,
respectively, and s is Laplace operator. By selecting b2 = 4mk , we obtain critically
damped response [16], which yields the fastest possible nonoscillatory response. The
proportional and integral gains for force control are chosen very small in order to obtain
nonoscillatory force transient response. Simulation result show that steady state can be
reached quickly without oscillatory response for position and force control.
Simulation 7 has the same conditions as Simulation 2 for impedance control in
order to compare the results of two controllers. Simulation results show that our new
controller provides significant improvement, and the tangential and normal position errors
are eliminated by this new control. Figure 4.30 illustrates very accurate position tracking
using this control approach. In Simulation 8, we examined deburring a DTG material
under the same conditions as in Simulation 7. From Figure 4.35, we see that Motor 2
reached its torque limit at numerous times throughout the simulation, which is the cause of
large position errors in Figure 4.33 in both the tangential and normal directions. No
controller can overcome this torque saturation situation, and in order to solve this
problem, either smaller depth of cut should be commanded, or a highertorque motor
should be employed. We assume there are no motor torque limits in Simulation 9, and the
125 I
/
results show a large improvement in Figure 4.36. However nonzero errors continue to
occur, because the normal force limit was reached at numerous times, dictating switches
to force control and giving up position accuracy, as shown in Figure 4.38. The force
control regulated the normal grinding fome reasonably well to the limit of 450 N as shown
in Figure 4.38. The highfrequency force variations were caused by the irregularity of
burrs, which were randomheight sinusoidal in this simulation. In this case, smaller desired
depth of cut in multiple passes of deburring and grinding should be employed in order to
avoid the potential burning of workpiece or tool damage. Note from Figure 4.38 that
switches between position and force control occur with high frequency in this case.
Although, our simulation results do not indicate a potential stability problem with this
frequent switching between two control modes, we have not developed a proof to
guarantee stability for all deburring situations and all choices of controller gains.
In Simulations 10, 11, and 12, we employed large upset burrs to test the
performance of our controller. Simulation results in Figure 4.40 for Simulation 10 with an
ETG workpiece show that the grinding process did not reach the desired contour for large
upset burrs because the torque of Motor 2 saturated at its limit for each upset, indicated in
Figure 4.43. Again, we could employ smaller desired depths of cut or a highertorque
motor for Joint 2 to improve. This is demonstrated in Simulation 11, where we removed
the torque limits on both motors. Figure 4.44 shows that a precise contour is achieved for
the finished workpiece. In simulation 12, with torque limits removed, the grinding process
reached the normal force limit when large upset burrs were encountered for a DTG
material, requiring switching to force control, as seen in Figure 4.50. We have assumed in
126
our modeling that cutting is instantaneous upon contact with a surface, such that impulsive
forces and torques appear in our simulation results when the grinding wheel encounters
large burrs. When the grinding wheel "jumps off' large burrs, it rereaches the workpiece
surface quickly, as seen in Figures 4.48 and 4.49.
Finally, we simulate our controller with scallop burrs for ETG and DTG
workpieces in Simulations 13 and 14, respectively. Figure 4.52 shows periodic nonzero
position errors for the ETG material, caused by torque saturation of Motor 2, shown in
Figure 4.55. For the DTG workpiece with torque limits removed, Figure 4.56 shows
larger position errors than for the ETG material. This is caused by reaching the normal
force limit immediately, with force control in place throughout the simulation, as shown in
Figure 4.58.
From our simulation results, we conclude that our controller can achieve an
accurate finished workpiece edge for robotic deburring and grinding, but also provide the
ability to control grinding forces to avoid potential damage to the workpiece and grinding
tool. This controller appears to be more effective than controllers using impedance and
hybrid impedance control. Like all controllers, however, the physics of the system prevent
achieving accurate finishing on a single pass when force and torque limits are encountered.
In the next chapter, we present conclusions of this work and recommendations for further
study.
127
CHAPTER V
CONCLUSIONS AND RECOMMENDATIONS
Summary and Conclusions
In this study, we have focused on the control for position accuracy under force
limits for a SCARA robot used for deburring and grinding. The following is a summary
and relevant conclusions:
1. Based on traditional grinding mechanics, a grinding model previously developed for
robotic deburring and grinding was employed. The grinding conditions were specified
to calculate grinding forces for an easytogrind (ETG) and difficulttogrind (DTG)
workpiece material. Realistic force limits to prevent heat damage to the workpiece
and tool breakdown for a selected grinding wheel were also determined and employed
in simulations.
2. Stiffness calculations were conducted for the prototype robot employed in this study,
the DCBerkeley NSK SCARA robot, because this infonnation was not available from
the literature. The stiffness of the joint motors and robot links were estimated for
worstcase conditions, which were then used to determine that the robot was
sufficiently rigid to justify ignoring robot arm flexibility.
128 )
3. Two common control approaches for manipulators operating in constrained motion
were investigated for robotic deburring and grinding. Simulation results showed that
impedance control, which provides a stable and unified control structure for both free
and constrained mo