This document summarizes multi-objective optimization of a Tricept parallel manipulator using an evolutionary algorithm. It discusses using a particle swarm optimization (PSO) technique to optimize the manipulator's conditioning index, workspace volume, and global conditioning index (GCI) simultaneously. The methodology involves calculating inverse kinematics and the Jacobian to determine performance parameters. Single-objective PSO is used to optimize the conditioning index alone. Results show the minimum conditioning index and corresponding design variables. PSO also optimizes the workspace volume alone. A multi-objective PSO with a weighted sum strategy then optimizes all parameters simultaneously, showing their relationships and validating results against single-objective optimization.

1. Multi objective Optimization of a Tricept Parallel Manipulator using Evolutionary Algorithm
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Multi objective Optimization of a Tricept Parallel Manipulator
Using Evolutionary Algorithm
Syed Saad Farooq1*
, Aamer Ahmed Baqai2
, Sajid ullah Butt3
, Wasim Akram Tarar4
, Amjad
Baig5
Mechanical Engineering
National University of Sciences
and Technology
Rawalpindi, 46000, Pakistan
ABSTRACT
Parallel manipulator famous for its rigidity and precision needs optimization in different performance parameters like conditioning
index, workspace volume and the global index. This paper will emphasis on the special type of parallel namely Tricept mechanism.
Tricept is a 3 DOF UPS mechanism with one static base and the moving head platform. Mechanisms previously modeled with its
inverse kinematics and jacobians that lead us towards the conditioning index and the workspace volumes and optimized through
the genetic algorithms. With a view to compare with previous work, Weighted Particle swarm (PSO) technique has been introduced
here for the multi objective optimization and results achieved through this technique was compared and validated with already
published optimization results.
Keywords: Inverse kinematics, Conditioning index, Workspace volume, GCI, Single Objective Optimization, Multi objective
Particle Swarm Optimization (MOPSO), Weighted Sum Strategy
1. INTRODUCTION
Manipulators can be expressed as sub part of robot [1] and are controlled by the motors and drives which have
computer based numerical control [2]. Parallel manipulators are famous for its rapid acceleration and immediate
precise movements as compared to the conventional machining. It has mechanical simplicity in its structure and
requires less installation efforts. It has less moving weight [3]. On the other hand, end effecter is limited to a certain
workspace and have complex inputs and outputs solutions. It is difficult to find high number of singularities in parallel
manipulators. [4].
The workspace is defined as the volume of the region end effectors that can occupy throughout its maximum reach
[5]. Reachable workspace is that volume of space in which the end effecter can reach all its points through at least
from one orientation, whereas the most important term the dexterous workspace is the volume of the space in which
the end effecter can reach its all points from all possible orientations. Basically the dexterous is the subset of the
reachable workspace.
Conditioning number provides the sensitivity ratio for dexterity. Dexterity is actually the measure of sensitivity
between the end effector and the actuator movement [4] [6]. This condition number uses the singular values of jacobian
so it better explains the singularities and links nearness to singularities. Furthermore it also explains the error in the
design and stiffness [4].
Global Conditioning Index is based on the requirement whether the user needs the local conditioning or the global
conditioning. If the user want the results to be with respect to global. He should use the global conditioning index for
its simulation results [7].
Conventional Single objective deals with the optimization of parameters independently. Evolutionary algorithms
(EA’s) will be discussed in this work for the process to get optimized design variables. Here are some points for better
understanding to use the evolutionary algorithms for this task. [8]. EA’s are used when there exists the uncertainty in
the solutions. Secondly when there are multiple design variables involved. Thirdly when there is complex constraints
in their calculations and if there are more numbers of local and global optimum points, evolutionary algorithms are
approached.
* Syed Saad Farooq: Tel.: (0092) 321-6842766; E-mail: syedsaad34@gmail.com

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Some types of Parallel Manipulators like Gough Stewart Platform is a 6DOF basic architecture and has spherical
prismatic spherical architecture explained in [9]. 3 DOF RRR architecture has been well explained by [10] and [4].
This architecture has all joints revolute and does not possess translations. Orthoglide mechanism has been illustrated
by [11]. This structure moves in the x, y, z directions having fixed orientation and heavily used for the machining
purposes. Tricept Manipulators which is a center of discussion in this work has three legs with prismatic actuated and
center leg has UPS architecture which is connected from base to the moving platform above.it has prismatic actuators.
This type of structures has been well explained by [12] and [13].
2. PROBLEM FORMULATION
Figure 1: Tricept Mechanism [13]
This mechanism shown in figure 1 has 3 DOF and combination of joints include the two rotations and one
translation. Actuated joint is prismatic and it has SPS configuration but later on one spherical has been replaced by
the Universal joint so then it became the UPS structure. The center link connects the base to the moving platform.
When the structure is static the line passing through the universal joint of the moving platform is parallel with the x
and y axis of base. When the prismatic joint is activated other universal and spherical joints are passive with that
prismatic joint movement [13].
In the study of tricept, [13] has explained the design of a Tricept parallel manipulator with his weighing techniques,
kinematic equations, jacobian and parameters used to control the regularity is inverse condition number and workspace
optimization. He has used the genetic algorithm technique for its optimization. In this work it is ought to make sure
the same design through the particle swarm optimization (PSO) technique and this will bring here the optimum values
of the performance parameters using PSO. Conditioning index and workspace volume are major performance
parameters that will be optimized together with the corresponding optimized design variables as multi objective.
Finally analysis will be given that will correlate the results of PSO with the genetic algorithm (GA) previously done.
3. METHADOLOGY
In this portion the proposed methodology, the optimization by evaluating the multiple performance parameters
will be discussed. Starting from the kinematic solutions, division of four sections will lead towards multi objective
weighted particle swarm optimization which is as follows.
3.1 KINEMATIC SOLUTIONS AND JACOBIAN
In order to calculate the performance parameters like conditioning index and global index, first find the inverse
kinematics of the whole structure. Here are some steps to calculate the inverse kinematics and then to conditioning
index which will finally to the global index.
1. Formulate position vectors of limbs with respect to frame ‘O’ which is base frame i.e. OB1, OB2, OB3
2. Formulate position vectors of limbs with respect to ‘P’ frame which is the moving frame i.e. PA1, PA2,
PA3
3. Considering center link and make the rotation and translational matrices.
4. Then from closed loop procedure, find the position vector indicating from base to moving platform.

3. Multi objective Optimization of a Tricept Parallel Manipulator using Evolutionary Algorithm
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Ai= 𝑄 𝑂
𝑃
*PAi+ OP (1)
Where
Ai=Transformation from base point 'O’ to the moving ‘P’, i ranges from 1 to 3
𝑄 𝑂
𝑃
=Rotation matrix from point of base to moving platform
OP= Position vector from base to moving platform.
5. Then from the constraint equations, proceed towards the inverse kinematics.
|| (Ai - Bi) ||=qi (2)
Where i approaches from 1 to 3,
{q1, q2, q3} denotes the actuated lengths of joints configuration and { 𝜑, 𝜃, c } is the Cartesian coordinates.
Where 𝜑 denotes the rotation angle along x axis and 𝜃denotes the rotation angle along the y axis whereas c
is the translation along z axis.
6. Now when 5th
step is done, jacobian has been formulated. In this step,take differentials on both side of
equation 2, then rearrange the above 2 equation into Equation 3 form where this will separate the inverse and
forward kinematic matrices.
Jx 𝑥̇ = Jq 𝑞̇ (3)
7. From this step onwards, path towards the conditioning index is clear which the first performance index is.
So by using the equation 4, by taking the inverse of K gives ‘k’ which is the conditioning index.
K= ||J||*||J-1
|| (4)
k=1/K (5)
8. Further check results globally by using the global indexing performance index GCI. Simply in other words
global index is the mean of the conditioning index in a prescribed volume around its workspace.
3.2 WORKSPACE VOLUME
There are many methods adopted by many researchers for the calculation of workspace volumes. Analytical and
numerical approaches have been introduced previously in [14]. Same concept has been used here. Firstly It takes the
whole of the workspace as a cube which have three axis x, y and z respectively then it takes the subspace, a cylinder
in particular for the workspace calculation. It restricts the legs and the platforms of the manipulator around a cylinder
and from the inverse kinematic solutions of the parallel manipulator. By keeping in view the constraints, this searches
each q’s in that subspace which forms the closed cylinder. After each z increasing, this will try to find out the solutions
which are trapped inside or onto the surface of that subspace. [14]
3.3 CONVENTIONAL SINGLE OBJECTIVE OPTIMIZATION
Optimization process of one parameter irrespective of other performance parameters will be of major concern of
this section. These evolutionary algorithms as discussed previously in section 1 perform swiftly for the findings of
local and global minima and maximal points. Other traditional methods like bracketing and elimination optimization
techniques does not guarantee findings of optimum points. They can skip their local and global points.
So in order to have that EA’s in our parallel manipulator calculations, some types are given below.
1. Ant colony Optimization
2. Genetic algorithms optimization.(GA)
3. Particle swarm optimization (PSO)
PSO usually takes the advantage of lesser iterations and its higher convergence rate in the start than Genetic algorithm
[15]. Hence in this work Particle swarm optimization (PSO) technique to optimize the design variables is used.
3.31 PARTICLE SWARM OPTIMIZATION
This work will have an emphasis on particle swarm optimization and work will deal with this technique throughout
our optimization. It is the social behavior of birds which this algorithm follows. When the birds move in search of
food and all don’t know the exact location of food. Finally the food is located by one bird and it is found to be nearest
so now all the birds will follow that food which has been found by one of their bird [16]. That bird can be name as a

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leader. PSO is searching algorithm. PSO start with the same process of initialization [17] .Steps for this algorithm is
given below.
The equation below is an update velocity equation for step 5 in the above flow chart.
vnew = vg(j)+c1*r1*(pbest-x(i,j))+c2*r2*(gbest(j)-x(i,j)) (6)
Whereas
vnew = New velocity after update
vg = Global velocity of the particle
pbest = Particle best is same at start as x(i,j).
x(i,j) = It is the value of particle taken from ith
row and jth
column from the start to size of
the swarm ‘n’
gbest = global best is the global best of the swarm corresponding to the fitness value of the
objective function.
c1 = first constant
c2 = second constant
r1 = first random value
r2 = second random value
These c1 and c2 are the constants and weights assigned to each particle during its updating and
usually these constants should both sum up to 4 in simulations whereas r 1 and r2 both are random
values taken 0 to 1.
Similarly the position update of the particle takes place in accordance with the velocity update equation
which represent in this form normally [18].
xnew = x(i,j)+ vnew (7)
Where xnew is the new position of the particle.
In following all the steps stated above, the maximum swarm values will optimize and finally declare the final constant
value as maxima optimum point.

5. Multi objective Optimization of a Tricept Parallel Manipulator using Evolutionary Algorithm
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3.4 MULTIOBJECTIVE PARTICLE SWARM OPTIMIZATION (MOPSO)
MOPSO Uses the technique of PSO stated previously in section 3.31. Here in multi objective make one more
function which will have the all the performance parameters to be in function so that it can have a variation of all the
parameters at the same time through one function. New function will be treated as objective function for the job and
the performance parameters act as function variables. After that all the process for the optimization remains the same
[19].There are many methods to form that new function. Proceed towards weighted sum strategy here for further
calculations.
In this multi objective technique the function is formed by assigning the weights. Each variable is assigned weight
which can be changeable according to the user demands. And the equation will be like the following [20] Maximize
y now and fitness value is examined using this function.
y=w1*z (i,1)+w2*z (i,2)+ w3*z (i,3) (8)
w1, w2 and w3 are the two weights assigned to the conditioning index, workspace volume and global Conditioning
index (GCI). This method is good for continuous and convex problems however local optima usually achieve to
discontinuous functions as well. [20]
4. APPLICATION AND RESULTS
Table 1 Geometric Constraints
Actuator
lengths(mm)
Angle
(rad)
d (mm) b(mm) a(mm)
400-750 -1 to +1 20-200 300-500 200-300
Where d is the length of the joint from C to P point respectively and also ‘b’ is the length of the static platform from
point O to B1 whereas ‘a’ is the length from point P to A1 of the moving platform as shown in table 1.
Here from the section 3.1 the main objective is to go to the inverse kinematics equation, finding a relation between
the Cartesian and joint coordinate system. Keeping in mind the actuator lengths (q’s) joint coordinates and the passive
x which is the Cartesian coordinates, derivation of the jacobian equation separating these two terms is as under
equation (9).
(9)
Equation (9) when simplifies give the inverse kinematic solutions. By doing small calculations, finally found the
jacobian matrix as
,
J= [P Q R] (10)

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According to proposed methodology next section demands the single objective optimization. The conditioning
index is being optimized for the set of design variables a, b and d and PSO algorithm is launched. Aim to find minimum
point for this performance index was accomplished and the corresponding design variables saved against that best
minimum point. The execution of the MATLAB code reveals the results in the figure 2 as below
Figure 2 Condition number versus the iterations
The figure 2 result is demonstrated against the iterations of 20 with a step size of 0.1 rad of angles in the MATLAB
code. These values are for the one elevation of ‘c’ for 500 mm elongation. For 20 intervals between the design variables
the iteration started until a smooth constant line comes as it is a sign that the algorithm has found it’s most probably
the optimum point. Iterations are being used here as a stopping criterion. As graph showing the nearby optimum
condition index CI point is .002642 . Table 2 shows the corresponding optimum Design variables.
Table 2 gbest versus design variables
Now go for check of the maximum workspace values. Firstly run the algorithm and then check for the maximum
values of the workspace that it computes.
Figure 3: Maximum workspace values versus the iterations
20 iterations for the 20 intervals has been taken between the design variables and with a step size of 0.1 rad of
angles in the MATLAB code. The curve in figure 3 shown has made its own threshold at 8 iterations. It is because the
curve has reached to its maximum height in 8 iterations and shown too constant therefore it stopped to go till the end
of the iterations and this optimum point is regarded as global maxima. gbest value for the maximum optimized volume
is found to be 950.0733mm3
as shown from the figure 3 and table 2 shows its corresponding optimum design variables
within their geometric constraints.

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Table 3: Maximum volume versus design variables
At this point optimization of single objective calculations has been done. There remains the need to check the
relationship of one performance parameter with the other. One function having these parameter acting as objective
variables are evolved. Here only maximization weighted multi objective PSO has been run as shown in section 3.4
and preference has been allocated to the workspace volume in the following graph such that set 1 for Workspace
volume and 0 for the other two parameters. The preferences can also change. From running the Matlab code, the
following response of figure 4 is achieved.
Figure 4: Multi objective Maxima with Workspace Volume given preference of 1
Table 4: Workspace volume against the set of GCI, Conditioning Index and the Design Variables for the multi objective
maximum Optimization.
Results of table 4 concludes the relationship that workspace volume is inversely proportional to the conditioning
index and conditioning index is directly proportional to global index. Hence Results have been compared with single
objective optimization and validated.
5: CONCLUSION AND DISCUSSION
In our proposed methodology multi objective optimization of Tricept manipulator is an addition to the previous
research work. Further it has been done through the particle swarm technique here with an ingredient of weighted
Strategy. This work has given validation to the results shown in [13] for single objective optimization for volume and
condition numbers optimization respectively. This work concludes that PSO usually takes lesser iterations than
previously used genetic algorithm GA in [13] and declare as faster than GA for this case. GA exerts more computation
on the processor then PSO.PSO has a higher convergence rate then GA for this task. But sometimes PSO can treat the

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local maxima or minima as global ones so some tries are needed to be done to declare the point as nearby global
optimum [18]. It is random. It can try different iterations at start to see the variance of optimum points.
This work has claimed to cover this Tricept mechanism through the PSO algorithm. Research can be extended
through many other evolutionary algorithm techniques like ANT colony Optimization Technique and others. Secondly
it can go for more performance parameters like stiffness index. More constraints can be added. More parallel structures
can be altered by small variation in the design keeping in mind the actual concept of the tricept mechanism.
ACKNOWLEDGEMENT
Moral support of Research mates, guidance by the supervisor and research environment of NUST is acknowledged.
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