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1
Modeling and Design Optimization of a
Robot Gripper Mechanism
Alaa HASSAN1,, Mouhammad ABOMOHARAM2
1University of Lorraine, Equipe de Recherche sur les Processus Innovatifs (ERPI), Nancy, France
2Higher Institute for Applied Sciences and Technology (HIAST), Damascus, Syria
Abstract
Structure modeling and optimizing are important topics for the design and control of robots. In this paper, we
propose a process for modeling robots and optimizing their structure. This process is illustrated via a case study
of a robot gripper mechanism that has a closed-loop and a single degree of freedom (DOF) structure. Our aim is
to conduct a detailed study of the gripper in order to provide an in-depth step-by-step demonstration of the design
process and to illustrate the interactions among its steps. First, geometric model is established to find the
relationship between the operational coordinates giving the location of the end-effector and the joint
coordinates. Then, equivalent Jacobian matrix is derived to find the kinematic model; and the dynamic model is
obtained using Lagrange formulation. Based on these models, a structural multi-objective optimization (MOO)
problem is formalised in the static configuration of the gripper. The objective is to determine the optimum force
extracted by the robot gripper on the surface of a grasped rigid object under geometrical and functional constraints.
The optimization problem of the gripper design is solved using a non-dominated sorting genetic algorithm version
II (NSGA-II). The Pareto-optimal solutions are investigated to establish some meaningful relationships between
the objective functions and variable values. Finally, design sensitivity analysis is carried out to compute the
sensitivity of objective functions with respect to design variables.
Keywords
Robot modeling; Multicriteria design optimization; NSGA-II; Sensitivity analysis
1. Introduction
Robot design is a very complex process involving great modeling and simulation efforts. It has suffered an
important progress in the last decades and many approaches deal with this issue. Major steps in robot manipulator
design are; kinematics design, dynamics design, thermal design, and stiffness design [1]. In particular, robot
modeling and structural analysis are required in all industries. To address these requirements, a design process is
proposed in this paper that combines both; robot modeling and geometrical optimization. The proposed process is
a sub-process of the general robotics design process in which modeling and optimization activities play essential
and complementary roles in the design. As an illustrative case study, we carry out a modeling and an optimal
design of a planar single degree of freedom (DOF) mechanism that is used for robot hands or grippers. These kind
of mechanisms is amply used because of its simplicity and it only needs one actuator to move it, so many robots
use this kind of mechanisms as gripper. However, many researches deal with geometric, kinematic, and dynamic
modeling of the robots using different techniques. Some others work on optimization methods for multicriteria
robot design optimization. A survey of these research works is presented in the following paragraphs.
Corresponding author. Tel: +33372743500. E-mail address: alaa.hassan@univ-lorraine.fr. Postal address: 8 rue
Bastien Lepage, 54000 Nancy, France
2
Modeling is essential for design specifications, simulation, and advanced control of robots. Different techniques
of modeling are available for modeling robots, especially for parallel and closed-loop robots due to their
complexity [2-5]. Ibrahim and Khalil presented kinematic and dynamic modeling of three degrees of freedom 3-
RPS (revolute, prismatic, and spherical) parallel robot [6]. This robot is characterized by a coupling between the
6-DOF of the platform. After presenting a (6×3) kinematic Jacobian matrix, they developed a reduced (3×3)
Jacobian matrix relating the linear velocity of the platform with respect to the three actuated joints. In another
paper, Khalil and Guegan presented closed form solutions for the inverse and direct dynamic models of the Gough-
Stewart parallel robot. The models are obtained in terms of the Cartesian dynamic model elements of the legs and
of the Newton-Euler equation of the platform [7]. Andrzej et al. used forward and inverse kinematic problem as
well as working space and strength analysis issues for the construction of 3-DOF tripod electro-pneumatic parallel
manipulator [8]. Qin et al. proposed analytical modelling of a two-staged parallel mechanism composed by a rigid
platform in a serial connection with a compliant platform [9]. Hassan and Abomoharam performed a study of a
gripper that has two closed loop structure. After finding geometric and kinematic models, they determined the
geometrical solution space and verified it via a CAD model of the gripper [10]. Ha et al. employed Hamilton’s
principle, Lagrange multiplier, geometric constraints, and partitioning method to derive the dynamic equations of
a slider-crank mechanism. They showed that dynamic formulation could give a good interpretation of a slider-
crank mechanism by comparing the numerical simulations with experimental results [11]. Özgür and Mezouar
exploited screw theory expressed via unit dual quaternion representation and its algebra to formulate both the
forward (position and velocity) kinematics and pose control of an n-DOF robot arm [12].
Different researches of the optimum design of robot manipulators are available in the works of [13–16]. Xie et al.
proposed a decoupled 3-DOF parallel tool head without parasitic motion. Using the atlases of the tool architecture
as bases, the optimal kinematic design of the tool head is carried out [17]. Jiang et al. presented a dynamic modeling
and redundant force optimization of a 2-DOF parallel kinematic machine with kinematic redundancy in order to
minimize the position errors of the manipulated platform [18]. Nevertheless, in real robot design problems, the
number of design parameters can be very large, and their influence on the value to be optimized (the objective
function) can be very complicated, having a strongly non-linear character. In these complex cases, stochastic
optimization techniques including evolutionary algorithms such as genetic algorithms (GA) may offer solutions to
the problem [19]. Coello et al. proposed GA-based multiobjective optimization hybrid technique to optimize the
counterweight balancing of a robot arm [20]. Jamwal et al. used a modified genetic algorithm to optimize the
kinematic design of a parallel ankle rehabilitation robot [21]. Osyczka and Krenich discussed some new methods
for multicriteria design optimization using evolutionary algorithms. The main aims of these methods is to reduce
the computing time and to facilitate the decision making process. Examples of a robot gripper mechanism and a
clutch break design are presented in this paper showing that these methods can be used to solve different design
optimization problems [22]. Gao et al. described the implementation of genetic algorithms and artificial neural
networks as an intelligent optimization tool for the dimensional synthesis of the spatial 6-DOF parallel
manipulator. The multi-objective optimization (MOO) problem was consisted of two functions: system stiffness
and dexterity, which are derived according to kinematic analysis of the parallel mechanism [23].
The rest of the paper is organized as follows. The proposed modeling and optimal design process of the robots and
its advantages are described in Section 2. In Section 3, our case study of a robot gripper mechanism is described
and its geometric modeling is recalled. Section 4 reviews the kinematic modeling of the gripper, then, the dynamic
model is derived in Section 5. After describing and modeling the gripper, the corresponding multi-objective
optimization problem is formalized in Section 6. Section 7 describes the solution algorithm of the optimization
problem, and the non-dominated sorting genetic algorithm version II (NSGA-II), it discusses the results. Section
8 presents the sensitivity analysis of the gripper mechanism design. Finally, Section 9 summarizes the contributions
and results made in this paper and gives some perspectives.
2. Modeling and Optimal Design Process
The design of robots is a complex engineering task, in which certain mathematical models are required. This task
can often be seen as an optimization problem in which the robot parameters or structure describing the best quality
design is sought. In this paper, an integrated modeling-optimizing robot design process is proposed where the
modeling steps are combined with the optimal structural design process, Fig. 1 illustrates this proposed process.
During this procedure, the geometric information is transferred from one step to the next step. The modeling stage
information is captured as input by the optimization stage, while the optimal design information feeds back the
modeling stage. These interactions give the designer the advantage to better define the design parameters and to
take into account both the modeling and the optimization issues in one integrated process. These two issues are
3
often handled separately as presented in the literature survey above, but in our presented process, they are
combined together in order to benefit of their complementarity.
The process starts by defining the problem that must be solved. Depending on the objectives of the study, the
applied steps may vary; it could be finding the geometric, kinetic, or dynamic models of the robot using different
techniques. These models are important to apply high performance control algorithms, to improve stiffness, to
increase payload, to improve force/torque capacity, etc. The objective could be also finding the optimal design that
aims at enhancing the performance indexes by adjusting the structural parameters, such as the geometrical lengths.
In the optimal design, several performance indices are involved, such as stiffness, transmission ratio, and accuracy.
The modeling stage starts by the geometric modeling, which represents the relations between the location vector
of the end-effector X and the joint coordinate vector q (Eq. (1)). Several methods and notations have been proposed
to find the geometric model; the most widely used one is that of Denavit-Hartenberg [24]. However, this method
is developed for simple serial-structured robot. Khalil and Kleinfinger have proposed a unified description of
parallel and tree-structured robots [25].
𝑋 = 𝑓(𝑞) (1)
Kinematic model is to find the relation between the end-effector velocity and the joint velocities. Kinematic model
could be written using the Jacobian matrix J. This matrix appears in calculating the derivation of the geometric
model. It gives the differential variations of the operational coordinates 𝑋̇ in terms of the differential variations of
the joint coordinates 𝑞̇ (Eq. (2)). For parallel manipulator, the key concept is to “break” the parallel manipulator
into “simple” serial chains. Derived from the loop-closure or constraint equations, the equivalent Jacobian matrix
could be found in terms of active and passive joint variables.
𝑋̇= 𝐽(𝑞). 𝑞̇ (2)
The Jacobian matrix has multiple applications in robotics. It facilitates the calculation of singularities and of the
dimension of accessible operational space of the robot [26]. In static force model, we use the Jacobian matrix in
order to calculate the forces and torques of the actuators in terms of the forces and moments exerted on the
environment by the end-effector. The static model is essential in structural analysis as well as in formulating the
optimal design problem.
Fig. 1. Robot modeling and optimal design process.
Robot design problem
Geometric modeling
Kinematic modeling
Dynamic modeling
Modeling stage
Design parameterization
Optimization problem
formulation
(objective functions,
constraints)
Optimal design solutions
(optimization algorithm)
Design sensitivity
analysis
Optimized?
Stop
Yes
Design update
Optimization stage
No
4
The dynamic model is the relation between the torques (and/or forces) applied to the actuators and the joint
positions, velocities and accelerations. The relation in Eq. (3) represents the dynamic model.
= 𝑓(𝑞, 𝑞̇, 𝑞̈ , 𝜏𝑒𝑥𝑡) (3)
where
is the joint torques and forces. 𝑞, 𝑞̇, 𝑞̈ are the vectors of joint positions, velocities, and accelerations,
respectively. 𝜏𝑒𝑥𝑡 is the vector representing the external forces and moments that the robot exerts on its
environment. The dynamic model is typically used in actuator dimensioning and in robot simulation and control.
Several formalisms are used to obtain the dynamic model; the Lagrange multipliers, the principle of virtual work,
and the Newton-Euler formulation.
The optimization stage is a process consists of design parameterization, optimization problem formulation, optimal
design solutions, and design sensitivity analysis. The principal role of design parameterization is to define the
geometric parameters that characterize the structural model of the robot and to collect a subset of the geometric
parameters as design variables. Modeling stage supports design parameterization task by generating mathematical
models that describe geometric structure, kinetic, and dynamic behaviours of the robot. Hence, geometric
parameters and the structural design problem can be derived from the modeling stage. Only proper design
parameterization will yield a good optimum design, since the optimization algorithm will search within a design
space that is defined for the optimal design problem.
In optimization problem formulization step, stiffness, transmission ration, accuracy, cost, etc. can be defined by as
objective functions with appropriate constraint bounds. This involves the selection of objective functions,
expressed in terms of the design variables, which we seek to minimize or maximize. Beside the objective functions,
it involves the selection of a set of variables to describe the design alternatives. Constraint functions are the criteria
that the robot variables have to satisfy for each feasible design. In real world, it is common that a given structural
design problem has multiple and often conflicting objectives. This will result in a multi-objective optimization
problem from this step.
After defining the multi-objective design optimization problem, the next step is to find the optimal design
solutions. The presence of multiple objectives in a problem, in principle, gives rise to a set of optimal solutions
(largely known as Pareto-optimal solutions), instead of a single optimal solution. In the absence of any further
information, one of these Pareto-optimal solutions cannot be said to be better than the other. This demands a user
to find as many Pareto-optimal solutions as possible. Since evolutionary algorithms (EAs) work with a population
of solutions, a simple EA can be extended to maintain a diverse set of solutions. With an emphasis for moving
toward the true Pareto-optimal region, an EA can be used to find multiple Pareto-optimal solutions in one single
simulation run. The NSGA-II proposed by Deb et al. [27], is one of the EAs widely used to solve MOO problems.
Design sensitivity analysis is used to compute the sensitivity of objective functions with respect to design variables.
Based on the design sensitivity results, a design engineer can decide on the direction and amount of design change
needed to improve the objective functions. In addition, design sensitivity information can provide answers to “what
if” questions by predicting objective function perturbations when the perturbations of design variables are
provided. Regarding the geometric robot design, sensitivity analysis is of a great value to designers if a realistic
and economical allocation of tolerances on design variables is to be achieved. There are different approaches to
performing a sensitivity analysis like scatter plots, regression analysis, and partial derivative methods. Depending
on the design problem, the design engineer could test some of the optimal solutions to improve the robot design at
each iterative step. As a result, new designs could be obtained from optimization and sensitivity analysis steps.
Thus, the geometric model, in the modeling stage, has to be updated for the optimal set of design variables supplied
by the optimization stage.
In the rest of this paper, the proposed modeling and optimal design process is applied to a robot gripper mechanism.
The motivation of this case study is to model this closed-loop structure, and then to design it optimally.
3. Description of the gripper mechanism and geometric
modeling
The gripper is planner closed-loop mechanism with a single DOF. The gripping force F, applied on the object, is
generated by the actuating force P. Due to the symmetry; we can perform the study on a half of the mechanism
that is composed of three links and four joints (one prismatic and three revolute joints), as shown in Fig. 2..
5
The notations of Khalil and Kleinfinger [25], are used to describe the geometry of the closed-loop structure of the
gripper. The definition of the local link frames are given in Fig. 2, while the geometric parameters are given in
Table 1.
Fig. 2. Link frames of gripper mechanism.
Table 1
Geometric parameters of the gripper.
j
r
j
j
d
j
j
Pj
j
0
1
0
0
0
0
1
2
r
0
0
90
1
0
2
0
3
3
d
0
0
1
3
0
4
4
d
90
0
2
4
0
0
5
d
0
0
3
5
P(j) denotes the frame antecedent to frame j, and σj = 1 if joint j is prismatic and σj = 0 if it is revolute. The
homogeneous transformation matrix iTj, which defines the frame Rj relative to frame Ri, is obtained as a function
of four geometric parameters (αj, dj, θj, rj). Thus iTj, are obtained as:
11
11
01
cos sin 0 0
sin cos 0 0
0 0 1 0
0 0 0 1
T
,
3 3 3
33
13
cos sin 0
sin cos 0 0
0 0 1 0
0 0 0 1
d
T
,
5
35
1 0 0
0 1 0 0
0 0 1 0
0 0 0 1
d
T
(4)
3
1 0 0
0 0 1
0 0 1 0
0 0 0 1
tool
f
e
T
,
2
02
1 0 0 0
0 0 1
0 1 0 0
0 0 0 1
r
T
,
4 4 4
2444
cos sin 0
0 0 1 0
sin cos 0 0
0 0 0 1
d
T
The geometric closed-loop constraint can be expressed as
0 1 3 0 2
5
1 3 2 4
T T T T T
(5)
Thus
4 1 3
3 1 5 1 3 4
3 1 5 1 3 2
cos cos( )
sin sin( )
d d d
d d r
(6)
6
4. Kinematic modeling
Let
1,..., , 0
n
qq
be the constraint equations (Eq. (6)) previously found. Taking the time derivative
and rearranging these equations, Eq. 7 can be obtained as
*0K q K q
(7)
where
2
r
is the actuated variable (joint) and
1 3 4
,, T
are the passive variables (joints). n is the total
number of the gripper joints, and m be the number of the actuated joints. Columns of the
n m m
matrix [
Kq
] are the partial derivatives of
q
with respect to the actuated variables
i
, i = 1,…, m. Columns of the
n m n m
matrix [
*
Kq
] are the partial derivatives of
q
with respect to the passive variables
i
, i
= 1,…, n-m.
If
*
det 0Kq
, we can determine the linear and angular velocities of the passive joints (
) in terms of only
( the linear and angular velocities of the actuated joints)
1
*
KK
(8)
Hence, one can write the linear and angular velocities of the end-effector, denoted by tool in Fig. 2, with respect
to the fixed coordinate frame R0 as
0*
0*
tool v v
tool
v J J
JJ
(9)
which leads to
,
,
1
0 * *
1
0 * *
v eq
eq
tool v v
J
tool
J
v J J K K
J J K K
(10)
Define the equivalent Jacobian matrix Jeq as
,
,
v eq
eq eq
J
JJ
(11)
Eq. (9) can be written as
0
0tool
tool eq
tool
v
VJ
(12)
Thus, kinematic model of the gripper can be found by the calculation of Jeq. The derivative of constraint equations
Eq. (6) with respect to time gives
3 1 5 1 3 5 1 3
*3 1 5 1 3 5 1 3
0 sin sin sin 0
1 , cos cos cos 0
0 1 1 1
d d d
K K d d d
(13)
The transformation matrix
0tool
T
=
0 1 3
13
..
tool
T T T
is combined of rotation matrix
0tool
R
and translation vector
0tool
P
with respect to R0
1 3 1 3 1 1 3 1 3
1 3 1 3 1 3 1
0
00
13
cos sin 0 3cos cos sin
sin cos 0 cos 3sin sin
0 0 1 0
0 0 0 1
0 0 0 1
tool
tool tool
d f e
ed
T
RP
f
(14)
The linear velocity of the tool
0tool
v
is calculated by the derivation of position vector
0tool
P
with respect to time
7
1
3 1 1 3 1 3 1 3 1 3
3 1 1 3 1 3 1 3
*
33
4
02 1
d sin cos sin cos sin 0
d cos cos sin cos s
0
0 in 0
0 0 00
tool
vv
e f e f
ev er
JJ
ff
(15)
While the angular velocity vector of the tool
0tool
is obtained from the angular velocity matrix
.
0 0 0
tool tool
0
. 0 ,
0
x
zy
T
z x tool y
yx z
RR
(16)
hence
4
*
1
032
0 0 0 0
0 0 0 0
0 1 1 0
tool
J
r
J
(17)
Substituting Eqs. (13), (15), and (17) into Eq. (10) gives
,v eq
J
and
,eq
J
, then, the equivalent Jacobian matrix Jeq
is obtained
1 1 3 5 1 3
53
1 1 3 5 1 1 1 3
53
1
53
sin cos (d )sin
d sin
sin cos d cos sin sin
d sin
0
0
0
sin
d sin
eq
ef
fe
J
(18)
Based on the obtained kinematic model, we can establish the relationship between the external applied
forces/torques and the joint forces/torques. To find this relationship, in our case study, the principle of virtual work
is applied
00
. X .
TT
tool tool
F
(19)
where
0tool
F
is the vector of the external forces/torques applied on the tool, assuming that there is no other external
forces/torques.
0Xtool
is the virtual displacement vector of the tool.
is the vector of forces/torques applied on
the actuated joints.
is the virtual displacement vector of the actuated joints.
Substituting Eq. (12) into Eq. (19) gives
0.
TT
tool eq
FJ
(20)
Therefore
0
.
T
eq tool
JF
(21)
Substituting
0
0
0
0
0
0
0
tool
tool
RF
F
(22)
/2P
, and
eq
J
(Eq. (18)) into Eq. (21), the relationship between the actuator force and the force applied on
the object can be found
8
53
5 1 5 1 3
d sin
(d 2 )sin d sin 2
P
Ff
(23)
Eq. (23) is resulted from kinematic modeling, and it will be transmitted to design parametrization. This relationship
is the basis of the optimization problem formulization, as shown in the next paragraphs.
5. Dynamic modeling
To obtain dynamic model of the gripper, Lagrangian formulation is used. As above, n is the total number of joints,
and m is the number of the actuated joints. The Lagrangian formulation for a closed-loop mechanism is
1, 1,....,
nm j
ij
j
i i i
d L L Q i n
dt q q q
(24)
where the scalar Lagrangian is defined from the total kinetic and potential energy
1
( , ) ( )
n
ii
i
L q q KE PE
(25)
i
KE
kinetic energy of the link i
i
PE
potential energy of the link i
i
Q
the externally generalised forces applied on link i, it includes the actuator force/torque (if i is actuated) and the
external forces/torques
1 3 2
T
qr
joint coordinate vector
j
()nm
loop-closure constraint equations
j
()nm
Lagrange multipliers
Eq. (24) can be written in matrix form
,T
M q q C q q q G q Q
(26)
()Mq
nn
mass matrix
,C q q
nn
Coriolis/centripetal matrix, where
1
1
2
nij kj
ik
ij k
kk j i
MM
M
Cq
q q q
Gq
1n
vector of gravity terms, where
()
ii
PE
Gq
()n m n
constraint matrix that is obtained from the partial derivatives of
()nm
constraint equations with
respect to
j
q
,
i
ij j
q
For the gripper mechanism, let
( , , , ), 1,..., ,
i i i i
m l r I i n
denote mass, length, center of gravity location, and
component of inertia matrix of link i, respectively. For planar case, only
ii
I Iz
is relevant, the gravity is along (-
x0) axis (Fig. 2). Thus, Eq. (26) matrices are written as
2 2 2
1 1 1 3 3 1 3 3 1 3 3 3 3 1 3 3
2
3 3 3 1 3 3 3 3 3
2
2cos cos 0
cos 0
00
m r Iz Iz m l r l r Iz r l m r
M q Iz r l m r m r Iz
m
(27)
9
132 2 2 1 2
21
3 1 3 3
1 3 3
sin sin 0
sin 0 0
00
,
0
l m r l
C q q
mr
l m r
(28)
1 3 3 3 1 1 3 1 1
1 3 3 3
sin sin
sin
0
g m r g l m m r
g m rGq
(29)
1 3 5 1 3 1 3 5
1 3 1 3 5 1 3 5
sin sin sin 0
cos cos cos 1
dd
dd
qd
d
(30)
Q
1n
vector of joint torque/force (
) and externally applied torques/forces (
ext
), this vector is written as
ext
Q
(31)
The actuating force vector is
0 / 2 0 T
P
with respect to the fixed coordinate frame R0.
The externally applied force is due to the gripping force F, and it is already expressed in Eq. (22). The externally
applied force term in Lagrangian equation is
T
ext F
(32)
Where
is the matrix of the relationship between the linear velocity of F application point
0tool
V
and the joint
velocity
1 3 2
T
qr
, i.e.
1
03
2
.
tool
V
r
(33)
To find
, as in Eq. (15) and Eq. (17), we can obtain that
3 1 1 3 1 3 1 3 1 3 1
03
2
3 1 1 3 1 3 1 3 1 3
d sin cos sin cos sin 0
d cos cos sin cos sin 0
0 0 0
v
tool
e f e
r
feV
f
fe
(34)
and
1
03
2
000
000
1 1 0
tool r
(35)
which leads to
10
1 3 3 1 1 3 1 3 1 3
3 1 1 3 1 3 1 3 1 3
cos d sin sin cos sin 0
d cos cos sin cos sin 0
0 0 0
0 0 0
0 0 0
1 1 0
v
e f e f
f e f e
(36)
Substituting the right side of Lagrangian equation Eq. (26) gives
3 3 1 1 1 2 5 1 3 1 1 3 2
5 1 3 1 1 3 2
2
cos sin cos sin cos
sin cos
2
T
fF d F d
fd
P
Q F
(37)
After finding the dynamic model of the gripper, the relationship between the actuator force and the force applied
on the object can be found. By setting the joint velocities, accelerations, and gravity terms to zero, we can find
3 3 1 1 1 2 5 1 3 1 1 3 2
5 1 3 1 1 3 2
2
cos sin cos sin cos
0
0 sin cos
0
2
fF d F d
fF d
P
(38)
Solving this system of equations gives the relationship of the gripping force
53
5 1 5 1 3
sin
sin n2 si 2
dP
Fd f d
(39)
Eq. (39), which is resulted from the dynamic model, is the same one we found in Eq. (23) that is resulted from
kinematic model. Depending on the study objectives, finding kinematic and/or dynamic model could be carried
out, and both lead to the same static equation. This equation is transmitted to optimization stage, particularly to
formulize the gripper optimization problem.
6. Optimization problem formulation
The goal of the optimization problem is to find the dimensions of the gripper elements and to optimize objective
functions simultaneously by satisfying the geometric and force constraints. The vector of six design variables is
3 4 5
, , , , ,x d d d l e f
, where
3 4 5
, , , , ,d d d l e f
are the gripper link variables used in geometrical modeling. The
structure of geometrical dependencies of the mechanism is described in Fig. 3. The angles
1
2
and
3
can be written in terms of the design variables as
C
y/2
Fig. 3. Force distribution and geometrical variables of the gripper mechanism.
11
2 2 2
53
5
2 2 2
35
3
arccos 2
arccos 2
g d d
gd
g d d
gd
(40)
where
2
2
4
4
arctan
g d l z
d
lz
6.1 Objective functions
Based on the relationship between the gripping force and the actuator force in Eq. (23) or Eq. (39), we can
rewrite this relationship in terms of the design variables,
and
5
5
sin( )
,cos( )cos( ) cos( ) 2
dP
F x z df
(41)
The objective functions of the gripper are borrowed from [22] to perform a bi-objective study to understand the
trade-off between chosen objectives. These two objective functions can be formulated as follows:
1. The first objective function (Eq. (42)) can be written as the difference between the maximum and
minimum gripping forces for the assumed range of gripper ends displacement. The minimization of this
objective ensures that there is not much variation in the gripping force during the entire range of operation
of the gripper. Thus, it ensures the minimization of stress variation in the gripper links.
1max ( , ) min ( , )
z
z
f x F x z F x z
(42)
2. The second objective function (Eq. (43)) is the force transmission ratio, the ratio between the applied
actuating force
P
and the resulting minimum gripping force at the tip of the gripper end. The
minimization of this objective will ensure that the gripping force experienced at the tip of link f has the
largest possible value. Thus, it ensure the maximization of force transmission ratio.
2min ( , )
z
P
fx F x z
(43)
In the previously mentioned multi-objective optimization problem, both objective functions depend on the vector
of decision variables and on the displacement
z
. The parameter
z
is the displacement parameter of the gripper
actuator, which takes its minimum value at 0 and its maximum value at
max
Z
. Checking for a number of different
solution vectors (
x
), it is observed that
F
1max ( , ) min ( , )
z
z
f x F x z F x z
is a decreasing function as shown in
Fig. 4. So that, the maximum value of
F
takes place at
0z
and the minimum value takes place at
max
zZ
.
Thus, the objective functions may be written as:
1 max
2max
( ,0) ( ,Z )
( , )
f x F x F x
P
fx F x Z
(44)
6.2 Constraints
Before presenting the gripper constraints, we need to conclude the relationship between the displacement of the
gripper end
y
and the displacement of the gripper actuator
z
(Fig. 3).
45
, 2 sin siny x z d d c
(45)
where
22
c f e
arctan e
f
12
Fig. 4. Variation of force
F
with the displacement
z
for a typical design vector
x
.
From the geometry of the gripper, a number of non-linear constraints can be derived:
1. The minimum displacement between the ends of gripper (corresponding to the maximum displacement
of actuator value at
max
Z
) should be less than the minimum dimension of the gripping object:
1 min max
: , 0g x Y y x Z
(46)
2. The distance between the gripper ends corresponding to
max
Z
should be greater than zero:
2 max
: , 0g x y x Z
(47)
3. The maximum distance between the gripping ends corresponding to no displacement of actuator (
0z
)
should be greater than the maximum dimension of gripping object:
3 max
: ,0 0g x y x Y
(48)
4. Maximal range of the gripper ends displacement (
G
Y
) should be greater than or equal to the distance
between the gripping ends corresponding to no displacement of actuator:
4: ,0 0
G
g x Y y x
(49)
5. Maximal displacement of actuator should be greater than
l
, the actuator stroke should not reach the point
O
(Fig. 3):
5 max
:0g x l Z
(50)
6. The gripper ends displacement (
y
) decreases when the actuator displacement (
z
) increases. To ensure
this condition and to not reverse the gripper ends displacement direction, the angle
should be less than
2
:
6:0
2
g x z
(51)
7. The angle
is the gripper declination refers to the horizontal axis, a stable gripping requires a limitation
of this angle:
7:5gx
(52)
8. Geometrical properties are preserved by two constraints on
to maintain the triangle
OAB
. From Eq.
(40), the arccos function input should be less than 1:
2
2 2 2 2 2 2
5 3 5 3
55
11
22
g d d g d d
gd gd
After simplification and substitution of
g
, we get:
2 2 2 2
22
3 5 4 3 5 4
12
0d d d l z d d d l z
aa
So that,
1
a
and
2
a
should be positives for all
max
0,zZ
.
5
10
15
20
25
z
20
40
60
80
100
F
z (mm)
F (N)
(mm)
13
22
2
8 4 max 3 4
:0g x d l Z d d
(53)
222
9 3 4 4
:0g x d d d l
(54)
9. The geometric bounds of link lengths, or design variables, (in mm), are:
3
4
5
10 50
10 50
10 60
10 50
5 15
50 100
d
d
d
l
e
f
10. The geometric and force parameters are assumed to be as:
min max max
30mm, 70mm, 100mm,Z 25mm
G
Y Y Y
and
95NP
Thus, the gripper optimization problem can be formulated as follows:
Find
* * * * * * *
3 4 5
, , , , ,x d d d l e f
which will satisfy the 9 inequality constraints
1,...,9
k
g x k
(55)
and minimize the two objective functions
*12
min[ ( ), ( )]f x f x f x
(56)
7. Solution algorithm and result discussion
After formulating the optimization problem, the next step is to find an optimal design solution using an appropriate
algorithm. Our gripper case study is a MOO problem; and all of the above constraints must be taken into account.
In order to solve it and find an optimal solution, there are several methods which are proposed in the literature.
Due to the complexity, the size of problem and the importance of reducing the solving time, NSGA-II algorithm
is used. This algorithm results in Pareto front that consists of a set of solutions, which are not dominated by each
other. The Pareto-optimal solutions are thoroughly investigated to establish some meaningful relationships
between the objective functions and variable values.
The NSGA-II procedure is illustrated in Fig. 5. In this procedure, an initial combined population (Rt) is randomly
created and this population must be sort based on the non-domination strategy into each front. The fronts (F1, F2,
F3 ...) are compared with each other and a rank is assigned to each individual according to the fitness value. In
addition, a new parameter, named crowding distance is calculated in order to sort the solutions of F3. It is the
distance of an individual to its neighbors, and the large value of this parameter shows the better diversity in
population. The binary tournament selection is used to select the parents due to the rank and crowding distance.
The mutation and crossover operators are utilized for the creation of a new offspring (Qt+1) and so on for the next
generations.
NSGA-II parameters are as follows: population size = 200, number of generations = 2000, probability of SBX
(simulated binary crossover) recombination = 0.9, probability of polynomial mutation = 0.1, distribution index for
real-variable SBX crossover = 20, and distribution index for real-variable polynomial mutation = 100.
Pt
Qt
Rt
F1
Crowding
distance
sorting
F2
F3
Rejected
Pt+1
Nondominated
sorting Qt+1
Selection
+
Crossover
+
Mutation
14
Fig. 5. NSGA-II procedure [28].
Fig 6 shows the obtained Pareto front resulting from NSGA-II optimization procedure. From this set of solutions,
three are selected to investigate the changes in the gripper configuration. We have taken two extreme solutions (A
and C) and one intermediate solution (B) as shown in Fig. 7. These three solutions are presented in Table 2, the
design variable values are cited with their corresponding objective function values.
To investigate how one optimal solution varies from another, we can analyze the values of six design variables as
a function of one of the objectives (the force transmission ratio for example). It is clear that variables e and f are
fixed at their allowable lower limit value. We notice also that the force transmission ratio (f2) is proportional to
3 4 5
, , ,d d d l
design variables. On the other hand, the difference between the maximum and minimum gripping
forces (f1) is inversely proportional to these design variables. Therefore, the designer task is to select one of the
optimal solutions compromising the two objective functions in terms of the design requirements.
8. Design sensitivity analysis
After selecting an optimal design solution, the last step is to analyse the design sensitivity to design variable
changes. Design sensitivity analysis computes the rate of objective function change with respect to design variable
changes. With the robot structural analysis, the design sensitivity analysis generates a critical information, gradient,
for design optimization. Obviously, the objective function is presumed to be a differentiable function of the design,
at least in the neighbourhood of the optimal solution point, which is found, and selected, in the previous step.
Fig. 6. Set of optimal solutions obtained using NSGA-II.
Fig. 7. Three gripper configurations for solutions A, B, and C.
Table 2
Objective function and design variable values for solutions A, B, and C.
Solution
f1 (N)
f2
d3 (mm)
d4 (mm)
d5 (mm)
l (mm)
e (mm)
f (mm)
A
93.81
6.64
32.92
19.08
54.73
49.85
5.00
50.00
B
138.90
6.25
30.89
18.88
52.31
49.74
5.00
50.00
C
184.2
6.09
29.69
18.80
51.28
49.71
5.00
50.00
f2
100
120
140
160
180
6.1
6.2
6.3
6.4
6.5
6.6
f1 (N)
A
B
C
15
A sensitivity analysis of the mechanism is of a great value to designers if a realistic and economical allocation of
tolerances on link-lengths is to be achieved. Such analysis enables the designer to notice important trends, to
identify most critical link of a given mechanism, and to allocate the tolerances optimally [29]. In the gripper case
study, the design variables are all link-lengths, so the attention is drawn to the effect of practical manufacturing
tolerances on the objective functions. The local gripper design sensitivity to link-length tolerances is studied by
using deterministic approach, based on the worst-case analysis of the individual tolerances (maximum output
tolerance). Selecting an optimal solution (x*) on point B, illustrated in Table 3, the gripper objective functions
sensitivity can be conducted as follows.
Table 3
Objective functions and selected optimal solution for point B.
Solution
f1* (N)
f2*
d3* (mm)
d4* (mm)
d5* (mm)
l* (mm)
e* (mm)
f* (mm)
B
138.90
6.25
30.89
18.88
52.31
49.74
5.00
50.00
Considering tolerances on link-lengths, the actual lengths of the links deviate from the nominal:
iact i nom i
x x x
. If the tolerances are much smaller than the link-lengths ti << xi, the variation in the objective function f1 may be
written in terms of Taylor series as:
**
1 1 1
1
. . .
2
TT
f x f x x H x x
(57)
where
3 4 5
, , , , ,x d d d l e f
1
1i
f
fx
is the first partial derivative of the objective function f1 with respect to the ith design variable (xi)
1
H
is the second partial derivative matrix called the Hessian matrix,
21
1,ij ij
f
Hxx
In our case, the terms in Taylor series expansion having order three and above could be neglected without
appreciable loss of accuracy. We also assume that the constraints are still satisfied when
i
x
occurs. The magnitude
of the sensitivity coefficients for a design variable indicates the relative importance and influence of that variable
on the variation of the objective function. Fig. 8 shows the values of the first order sensitivity coefficients of the
objective function f1 on the optimal solution point (x*). It shows that the objective function f1 is most sensitive to
the variation of design variables d5 and l, and less sensitive to the other variables.
The same formula in Eq. (57) is applied for the second objective function ( f2). Fig. 9 shows the values of the first
order sensitivity coefficients of the objective function f2 on the optimal solution point (x*). Like f1, the objective
function f2 is most sensitive to the variation of design variables d5 and l. It is less (approximately equal) sensitive
to the variables d3 and d4, and much less sensitive to e and f.
The variation of each objective function can now be evaluated when a rough manufacturing tolerances (variations)
of the design variables are assigned, as illustrated in Table 4. The nominal values of the design variables are equal
to the optimal solution values on point B, while the tolerances are derived from the International Tolerance Grades
table in ISO 286. Based on the worst-case analysis, the total variation of an objective function is therefore the sum
of the individual variations due to each of the design variables considered separately. From Eq. 57, the calculated
maximum variation for each objective function is
122.4Nf
, and
20.414f
Since d5 and l have the major influences on the gripper objective functions, the tolerance intervals of these two
variables can be restricted in order to get the objective function variations within acceptable limits. On the other
hand, it is desirable to give as much of tolerance as possible to keep the manufacturing costs low. The designer
has to select the optimal solution and the assigned tolerance intervals that compromise multiple conflicting design
16
objectives. Once the optimal solution is selected, the geometric model must be updated and the optimization stage
is completed.
Fig. 8. First order sensitivity coefficients of f1 to design variable variations on point B.
Fig. 9. First order sensitivity coefficients of f2 to design variable variations on point B.
Table 4
Manufacturing tolerances (in mm) assigned to the gripper design variables.
3
d
4
d
5
d
l
e
f
0.3
0.2
0.3
0.3
0.1
0.3
9. Conclusion
This paper has proposed a two-stage process for modeling and optimising robot structures. In the modeling stage,
an analytical step-by-step study is introduced to find geometric, kinematic and dynamic models. The optimization
stage shows the procedure to formulize and to solve a design optimisation problem and then to analyse the design
sensitivity. The data flow and interactions between these steps and stages are highlighted when a robot modeling
and optimal design are needed. To illustrate the proposed process, a case study of a robot gripper is carried out.
The process starts by the geometric modeling step to find the geometric closed-loop constraint equations of the
gripper. Based on these equations, the kinematic model is derived in terms of equivalent Jacobian matrix and the
velocity of the actuated joint, then, Lagrangian formulation is employed to find the dynamic model.
The modeling stage data is used in the optimization stage to formalize the optimization problem. The relationship
between the gripper force and the actuator is resulting from the kinematic and dynamic models whereas some of
geometrical constraints are derived from the geometrical model. Two objective functions are expressed: the
minimization of the difference between the maximum and minimum gripping force and the maximization of the
17
force transmission ratio. NSGA-II is applied to solve the problem resulting the Pareto-optimal solutions. Each one
of these solutions represents a configuration of the gripper with a set the link variable values. A local sensitivity
analysis of a trade-off solution has showed that the two objective functions are most sensitive to the variation of
two design variables d5 and l, and less sensitive to the other variables. The manufacturing tolerance intervals of
these two variables can be restricted in order to maintain the objective function variations within acceptable limits.
Further improvements in this design process will focus on other MOO algorithms that may be used to find the
optimal solutions. It is also vital to integrate multi-objective robust optimization principle in order to find multi-
objectively robust Pareto optimum solutions.
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