FACTA UNIVERSITATIS
Series: Mechanical Engineering Vol. 18, N

o
1, 2020, pp. 79 - 89

https://doi.org/10.22190/FUME190428006A

Original scientific paper


WORKSPACE ANALYSIS AND OPTIMIZATION

OF THE PARALLEL ROBOTS BASED ON COMPUTER-AIDED

DESIGN APPROACH

Badreddine Aboulissane, Larbi El Bakkali, Jalal El Bahaoui

Team Modeling and Simulation of Mechanical Systems, Faculty of Sciences,

Abdelmalek Essaadi University, Tetouan, Morocco

Abstract. This paper provides workspace determination and analysis based on the

graphical technique of both spatial and planar parallel manipulators. The computation

and analysis of workspaces will be carried out using the parameterization and three-

dimensional representation of the workspace. This technique is implemented in CAD

(Computer Aided Design) Software CATIA workbenches. In order to determine the

workspace of the proposed manipulators, the reachable region by each kinematic chain is

created as a volume/area; afterwards, the full reachable workspace is obtained by the

application of a Boolean intersection function on the previously generated volumes/areas.

Finally, the relations between the total workspace and the design parameters are

simulated, and the Product Engineering Optimizer workbench is used to optimize the

design variables in order to obtain a maximized workspace volume. Simulated annealing

(SA) and Conjugate Gradient (CG) are considered in this study as optimization tools.

Key Words: CAD Software, Design Parameters, Optimization, Parallel Robots,

Workspace Analysis

1. INTRODUCTION

A parallel manipulator is a mechanism in which there are two or more closed kinematic
chains attaching the base to the mobile platform. Nowadays, most of the manipulators are
serial architecture; parallel robots exhibit many advantages, such as high speeds and
accelerations, low mobile masses, high stiffness, and great accuracy. The most notable
disadvantage of the parallel manipulators is their relatively small workspaces. One can use
the workspace volume or surface as an objective function for optimization. In this sense, the
researchers focused on workspace determination as a performance index in order to design
robots for specific industrial applications. The calculation of the parallel robots’ workspace

Received April 28, 2019 / Accepted January 25, 2020

Corresponding author: Badreddine Aboulissane

Team Modeling and Simulation of Mechanical Systems, Faculty of Sciences, Abdelmalek Essaadi University,

BP. 2121 M'Hannech II, Tetouan, Morocco

E-mail: b.aboulissane@gmail.com

80 B. ABOULISSANE, L. EL BAKKALI, J. EL BAHAOUI

is a complex problem due to the kinematic modeling difficulty. However, the problem of
workspace optimization of the parallel manipulators to obtain a prescribed workspace has
been investigated in few articles. The concept of the prescribed workspace is a significant
issue to optimize and to synthesize a robot. The actuated joint variables, the range of joints
motion and the mechanical interferences between the links essentially influence the parallel
manipulators’ workspace. In this paper, we focus on some areas of the space that surrounds
the manipulator, and limiting its workspace to the prescribed area. Several papers studied
this problem based on geometrical techniques, and using optimization algorithms to
synthesize the design parameters of the parallel manipulators.

Gallant and Boudreau [1] used a Genetic Algorithm in order to optimize a 3-DOF planar
parallel manipulator to obtain a workspace as close as possible to a prescribed one.
Singularities and workspace of planar 3-RPR parallel mechanism for maximal singularity-
free workspace by optimizing the geometric parameters are investigated by Jiang and
Gosselin [2, 3] and Yang and O’Brien [4]. Di Gregorio and Zanforlin [5] studied the
workspace of the 3-RUU and the DELTA robot. They concluded that these robots could
have the same closure equations and workspace when some geometric conditions are
satisfied. Chablat et al. [6, 7] compared 3-DOF parallel kinematic machines using two
design criteria: regular workspace shape and a kinetostatic performance index that needs to
be as homogenous as possible throughout the workspace ; this technique is based on the
interval analysis method. In [8] the workspace optimization of translational 3-UPU parallel
robot is performed using its parameterization by two design variables, which are the
prismatic joint stroke and the distance between the base and the mobile platform. Zhao,
Chu, and Feng [9] discussed the analogous symmetry properties between the workspace
and the mechanism structure. Gao, Liu, and Chen [10] analyzed the relationship between
the shapes of the workspaces and the link lengths for 3-DOF planar parallel manipulators;
his results are useful for the designers to optimize the robots regarding the workspace index.
Hay and Snyman [11, 12] focused on the numerical multi-level optimization for the
synthesis of the 3-DOF parallel manipulators for a desired workspace. In [13] the
workspace of Gough-Stewart platform was optimized using the Genetic Algorithm. The
idea is to minimize the areas, which do not belong to the intersection between two areas: the
workspace of the robot and the prescribed workspace. A genetic algorithm based method is
used also in [14] to deal with the optimal dimensional synthesis of the DELTA robot for a
prescribed workspace. The geometrical approaches have been used to represent the
workspace of the parallel manipulators, by Assad Arrouk et al. [15], Aboulissane et al. [21],
Bonevet al. [16], Gosselin [17], and Merlet [18]. The principle of these methods is to
deduce, from the constraints on each limb, a geometrical entity (sub-workspace) which
describes all the possible poses of the tool center point that satisfy the leg constraints. Then,
the robotic manipulator workspace is generated by the intersection of all the sub-
workspaces. Tsirogiannis et al. [22] presented an overall structural design optimization
approach for a robot arm link seeking mass reduction and satisfaction of manufacturability
with SLS AM technique.

In this paper, a graphical based technique is addressed for workspace’s determination,
analysis and optimization of two parallel robots, which are the 3-RPR planar manipulator,
and the DELTA robot.

The first section is dedicated to the description of the proposed manipulators. In the next
section, we present the steps to determine the workspace of both robots. The last section is
about the comparison of the two optimization methods, applied to the workspace of the
DELTA robot.

Workspace Analysis and Optimization of the Parallel Robots Based on Computer-Aided Design Approach 81

2. KINEMATIC SCHEME AND DESCRIPTION OF THE 3-RPR AND THE DELTA ROBOTS

2.1. The 3-RPR Planar Parallel Robot

Fig. 1 shows the Kinematic scheme of the 3-RPR planar robot. This mechanism is a

parallel robot with closed loop chains. Three actuated prismatic joints are linked to passive

joints , , and fixed to the base, and , , fixed to the mobile platform. The
actuated prismatic joints coordinates are given by the length of the legs, named , , and
. The orientation of the mobile platform is given by angle . The components of points
and are respectively and . Each limb generates an annular region
bounded by two concentric circles with radii of and , the centers of the circles
are defined by Eq. (1), and Eq. (2):

cos( ) sin( )
ci ai bi bi

x x x y    (1)

sin( ) cos( )
ci ai bi bi

y y x y    (2)

Fig. 1 Kinematic diagram of the 3-RPR planar parallel robot

The vector describing the 3-RPR parallel manipulator parameters is defined as follows:

1 min max 1 1 2 2 3 3[ ]
T

c c c c c c
x y x y x y   (3)

2.2. The DELTA Parallel Robot

The robot under study in this section is the DELTA parallel robot depicted in Fig.

2(a); it is composed of a triangular moving platform linked to a triangular fixed base with

three closed parallel chains. Each one consists of an actuated rotational joint mounted

near to the fixed base; the parallelograms and spherical joints transmit the movement of

the mobile platform. In this work, all the three arms of the manipulator are identical in

terms of geometrical parameters (Fig. 2(b)).

82 B. ABOULISSANE, L. EL BAKKALI, J. EL BAHAOUI

(a)

(b)

Fig. 2 (a) Geometric scheme of the Delta robot; (b) the DELTA robot parameterization

The independent design variables for the DELTA robot are:

2 1 2[ ]
T

L L R r  (4)

3. WORKSPACE DETERMINATION OF THE PROPOSED ROBOTS

First, we need to determine workspace of the proposed robots. We can define this

region as the reachable positions and rotations by the end-effector center point, generally

located on the platform of the robots. In this work, we used a geometrical technique for

the representation of the workspace through the CATIA software, and there are no limits

for all the revolute joints used for each manipulator.

3.1. Workspace of the 3-RPR Planar Robot

For this robot, we obtain the workspace by the intersection of three circular areas, which

correspond to the areas accessible by the end-effector point center when each leg is taken as

a serial manipulator. Fig. 3 shows the steps we followed to generate the workspace of the 3-

RPR manipulator in the CATIA.

(a)

(b)

(c)

Fig. 3 Steps of the workspace determination of the 3-RPR robot[15]

Workspace Analysis and Optimization of the Parallel Robots Based on Computer-Aided Design Approach 83

Fig. 3(a) depicts the first step; it consists of creating three annular regions in the

Sketcher workbench; then the Pad and Pocket commands are applied to the drawing with

a finite thickness in the Part Design workbench. The second step is performed under the

Part Design workbench; it consists of applying the first intersection Boolean operation.

Fig. 3(b) above shows the obtained result.

The final step to determine the workspace of the 3-RPR is to apply a second

intersection Boolean operation on the two remaining regions in the second step. The

obtained shape corresponds to the theoretical workspace of the mechanism shown in Fig.

3(c). This step is also done in the Design Part workbench.

The area of the 3D model presented in Fig. 3(c) is calculated by using a smart area

parameter. Since the thickness of this 3D model is neglected, the workspace area of the 3-

RPR robot is obtained, dividing by two, the area previously calculated.

The design parameters used to obtain this workspace are tabulated in Table 1:

Table 1 Design parameters of the 3-RPR manipulator

Design parameters Limb 1 Limb 2 Limb 3

xci (mm) -188,632 132,159 56,473

yci (mm) -43,697 -141,512 185,209

min (mm) 120

max (mm) 270

 (degrees) 102
Area (cm²) 197,02

For each orientation , the workspace of the robot has different shape and area. Table
2 illustrates the workspace of the robot for few orientations of the mobile platform.

Table 2 Variation of the workspace with respect to the orientation of the mobile platform

A=70,611 cm²

A=156,077 cm²

A=197,668 cm²

A=144,380 cm²

A=29,600 cm²

A=22,411 cm²

3.2. Workspace of the DELTA robot

The workspace of the DELTA Parallel robot is defined as a three dimensional volume

in the Cartesian space; this volume is reached by a point on the mobile platform. The

84 B. ABOULISSANE, L. EL BAKKALI, J. EL BAHAOUI

equations used to determine the workspace of a parallel robot are generally complex to

solve by using the traditional approaches. Hence, the CAD-based approach is used in this

work to determine geometrically the workspace of the DELTA parallel robot.

The parallel robot workspace robot can be quickly generated as an area or a volume

using the CATIA, then the complex technique such the numerical method. Discretization

based techniques produce an approximate form of a low quality workspace. To improve it,

it is necessary to use other graphical methods. By implanting the problem of workspace

determination in a CAD software, these techniques will become more reachable to industry

and more precise. As the first step, the proposed method for workspace determination of the

DELTA robot consists in assuming all legs to be independent serial arms having the mobile

platform as tool center point. Then, the region swept by the tool center point of each arm is

determined for a given orientation of this point.

In [14] the workspace of the DELTA robot is presented by following Eq. (5):

2 2 2 2 2 2 2 2 2

1 2 1
[( ) ] 4 [( ) ]

t t t t t
x r y z L L L x r z        (5)

with r R r   and:

cos sin

sin cos

t i i

x x y

y x y

z z

 

 


  
 

(6)

(a)

(b)

(c)

(d)

Fig. 4 Generation of the workspace for a Delta robot based on the CATIA V5. (a) The

three torus intersect; (b) First intersection Boolean operation; (c) Second Intersection;

(d) The workspace of the Delta robot (z < 0)

As shown in Fig. 4, the workspace of the

Delta robot is based on three tori. The first step

consists of drawing a circle with a radius , and
another circle with a radius passing by the
center of the first circle (Fig. 5).

Each limb of the DELTA robot generates a

torus, Fig.4(a) depicts the intersection of those

three volumes, and the rest of the figures shows

the intersection Boolean operations with the final

shape of the workspace of the robot presented by

Fig. 4(d).

Fig. 5 Torus in Sketcher workbench

Workspace Analysis and Optimization of the Parallel Robots Based on Computer-Aided Design Approach 85

3.3. Workspace analysis of the DELTA robot

Before starting the optimization problem, an analysis between the reachable workspace

and the design parameters is required. We presented examples for each variable and ,
as well as the resulting volume. First we fixed R = 55 mm, r =30 mm, and = 100 mm,
then length is varied from 100 mm to 250 mm. Table 3 shows the boundaries of the
reachable workspace for length of 100 mm, 150 mm, 200 mm, and 250 mm.

Table 3 Workspace shape versus

L2 = 100 mm
W = 4,765 dm

L2 = 150 mm
W = 4,688 dm

L2 = 200 mm
W = 4,661 dm

L2 = 250 mm
W = 4,647 dm

It can be seen from Table 3 that the workspace volume of the DELTA robot increases

with reducing length of the forearm.
Now, to demonstrate the effect of length , forearm is fixed at 250 mm, R = 55 mm,

r = 30 mm, and is varied from 100 mm to 250 mm.

Table 4 Workspace shape versus

L1 = 100 mm
W = 4,647 dm

3
L1 = 150 mm

W = 15,753 dm
3

L1 = 200 mm
W = 37,637 dm

3
L1 = 250 mm

W = 75,040 dm
3

As shown in Table 4, the workspace volume increases considerably by increasing the

length . The volume starts to take a cup-shape.

4. OPTIMIZATION PROBLEM

In robotics, the designer uses numerous indices to evaluate the performance of a
manipulator; among these indices, we can mention the workspace that describes the
potential robot utilization. In this work, we are using the reachable workspace as a
performance index in order to optimize the design parameters of the DELTA robot. The
optimization problem is formulated as follows:

2, ,min 2, 2, ,max

( )

i i i

maximize W

subject to



   
(7)

86 B. ABOULISSANE, L. EL BAKKALI, J. EL BAHAOUI

where W is the workspace volume, and is the vector defined by Eq. (4). The main
purpose of the maximization of the workspace is to expand the capabilities of the DELTA
robot. The parameters that have an effect on the volume and the shape of the reachable

workspace of the manipulator are: and with (j=1,2,3). The parameterization
used in the CATIA software is shown in Fig. 6:

Fig. 6 Parameters of the DELTA robot on the CATIA

For this optimization problem, we used the CATIA “Product Engineering Optimizer”

workbench in which we can use different algorithms such as: Conjugate Gradient method

(CG) a local algorithm and the simulated annealing (SA) a global algorithm. Both the

methods are employed in our present study. The simulated annealing is listed as the

oldest algorithm among the metaheuristics that had an explicit strategy to avoid local

minima; it can be applied to the majority of optimization problems. The behavior of this

algorithm is strongly dependent on the problem addressed [19]. The other algorithm,

which is the Conjugate Gradient, is a mathematical approach used on both linear and

non-linear systems; this approach can be used as an iterative algorithm and a direct

method [20]. To realize this optimization, we choose the last five parameters that are shown

in Fig. 6; we excluded the angles representing the angular offset between different

kinematic chains. The optimization parameters are usually provided with an upper and

lower limit. The main goal of this optimization is to maximize the objective function

represented by the workspace volume of the DELTA robot, based on the parameters

presented in Table 5, which can describe Eq. (7).

The initial parameters correspond to the workspace shown in Table 6(a) with a volume

W = 4,765 dm
3
. The first optimization is done using the (SA) algorithm, optimized values

are tabulated in Table 5 corresponding to the workspace summarized in Table 6(b) with a

volume W = 215,712 dm
3
. Secondly, we applied the (CG) method. Table 6(c) shows the

shape of the workspace with a volume W = 110,265 dm
3
. This workspace is associated with

the values of the design parameters presented also in Table 5.

Workspace Analysis and Optimization of the Parallel Robots Based on Computer-Aided Design Approach 87

Table 5 Optimization results

Design parameters Initial values Simulated annealing Conjugate gradient

r (mm) 30 193,77 54,538

R (mm) 55 344,198 120,308

L1 (mm) 100 350 283,757

L2 (mm) 100 112,255 250

Volume (dm
3
) 4,765 215,712 110,265

Table 6 Workspace optimization

Without optimization (a)

Simulated Annealing algorithm (b)

Conjugate Gradient algorithm (c)

The number of iterations made to reach the objective is 523 for the (SA) algorithm,

and 602 for the (CG) method. The time needed to achieve these two optimizations is

about 10 minutes. The simulations were performed on a computer that has the following

characteristics: CPU @2.10Ghz, 8.0 GB RAM.

(a)

(b)

Fig. 7 Evolution of the design parameters for (a) SA algorithm; (b) CG method

88 B. ABOULISSANE, L. EL BAKKALI, J. EL BAHAOUI

Fig. 7 presents the evolution of the design variables for both algorithms. From the

parametric analysis previously presented in Table 4 and the design variables evolution

shown in Figs. 7 (a) and (b), we can conclude that L1 have a significant impact on the

volume of the workspace. On the other hand, the two algorithms applied in this study

have set the L1 variable to a maximum value, while other parameters R, r, and L2 are

showing a variation in a large range searching for a maximum volume of the robot's

workspace. Fig. 8 provides a comparison of the convergence rates of the results; it can be

seen that the performance of the (SA) is more superior to that of (CG) method due to the

good speed of convergence with few generations, also, the optimal value reached by the

(SA) algorithm is greater than that obtained by the (GC) algorithm.

Fig. 8 Comparison of convergence cost for SA and GC algorithms

5. CONCLUSION

In this paper, we have focused on the CAD based technique to determine the

workspace of planar and spatial parallel robots. This study is performed on the 3-RPR

planar parallel robot and the DELTA robot. We considered the determination and the

characterization of the workspace of the two manipulators. For this purpose, we have

applied a geometrical approach that has been implemented in CATIA workbenches. We put

in evidence the effectiveness of this technique for the workspace optimization of the

DELTA robot, taking into consideration the joint limits. Two algorithms were applied to

maximize the workspace of the DELTA robot, the simulated annealing and the Conjugate

Gradient algorithms. The best result is related to the (SA) algorithm in terms of

convergence speed and the best optimal value of the workspace volume. The manipulators

studied herein for the workspace analysis and optimization illustrate the efficiency and the

capability of the graphical methodology for the designers, to avoid complex mathematical

equations.

Workspace Analysis and Optimization of the Parallel Robots Based on Computer-Aided Design Approach 89

REFERENCES

1. Gallant, M., Boudreau, R., 2002, The synthesis of planar parallel manipulators with prismatic joints for an
optimal, singularity‐free workspace, Journal of Robotic Systems, 19(1), pp. 13-24.

2. Jiang, Q., Gosselin, C.M., 2006, The maximal singularity-free workspace of planar 3-RPR parallel
mechanisms, International Conference on Mechatronics and Automation, pp. 142-146.

3. Jiang, Q., Gosselin, C.M., 2007, Geometric optimization of planar 3-RPR parallel mechanisms, Transactions
of the Canadian Society for Mechanical Engineering, 31(4), pp. 457-468.

4. Yang, Y., O’Brien, J.F., 2007, A case study of planar 3-RPR parallel robot singularity free workspace
design, International Conference on Mechatronics and Automation, pp. 1834–1838.

5. Di Gregorio, R., Zanforlin, R., 2003, Workspace analytic determination of two similar translational parallel
manipulators, Robotica, 21(5), pp. 555-566.

6. Chablat, D., Wenger, P., Majou, F., Merlet, J.P., 2004, An interval analysis based study for the design and
the comparison of three-degrees-of-freedom parallel kinematic machines, The International Journal of

Robotics Research, 23(6), pp. 615-624.

7. Chablat, D., Wenger, P., Merlet, J.P., 2007, A comparative study between two three-dof parallel kinematic
machines using kinetostatic criteria and interval analysis, 11th World Congress on Theory of Machines

andMechanisms, Tianjin, April, pp. 1209–1213.

8. Badescu, M., Morman, J.,Mavroidis, C., 2002, Workspace optimization of 3-UPU parallel platforms with
joint constraints, In Proceedings 2002 IEEE International Conference on Robotics and Automation, 4, pp.

3678-3683.

9. Zhao, J.S., Chu, F., Feng, Z.J., 2008, Symmetrical characteristics of the workspace for spatial parallel
mechanisms with symmetric structure, Mechanism and Machine Theory, 43(4), pp. 427-444.

10. Gao, F., Liu, X.J., Chen, X., 2001, The relationships between the shapes of the workspaces and the link
lengths of 3-DOF symmetrical planar parallel manipulators, Mechanism and Machine Theory, 36(2), pp.

205-220.

11. Hay, A.M., Snyman, J.A., 2005, A multi-level optimization methodology for determining the dextrous
workspaces of planar parallel manipulators, Structural and Multidisciplinary Optimization, 30(6),

pp. 422-427.

12. Hay, A.M., Snyman, J.A., 2006, Optimal synthesis for a continuous prescribed dexterity interval of a 3‐dof
parallel planar manipulator for different prescribed output workspaces, International journal for numerical

methods in engineering, 68(1), pp. 1-12.

13. Boudreau, R., Gosselin, C.M., 1999, The synthesis of planar parallel manipulators with a genetic algorithm,
Journal of mechanical design, 121(4), pp. 533-537.

14. Laribi, M.A., Romdhane, L., Zeghloul, S., 2008, Advanced synthesis of the DELTA parallel robot for a
specified workspace, In Parallel Manipulators, Towards New Applications, Intech Open.

15. Assad, K.A., Bouzgarrou, B.C., Stan, S.D., Gogu, G., 2010, CAD based design optimization of planar
parallel manipulators, Diffusion and defect data, Solid state data, Part B, Solid state phenomena, 166, pp.

33–38.

16. Bonev, I.A., Ryu, J., 2001, A geometrical method for computing the constant-orientation workspace of 6-
PRRS parallel manipulators, Mechanism and machine theory, 36(1), pp. 1-13.

17. Gosselin, C., 1990, Determination of the workspace of 6-DOF parallel manipulators, Journal of mechanical
design, 112(3), pp. 331-336.

18. Merlet, J.P., 1995, Determination of the orientation workspace of parallel manipulators, Journal of
intelligent and robotic systems, 13(2), pp. 143-160.

19. Kirkpatrick, S., Gelatt, C.D., Vecchi, M.P., 1983, Optimization by simulated annealing, Science, 220(4598),
pp. 671-680.

20. Shewchuk, J.R., 1994, An introduction to the conjugate gradient method without the agonizing pain.
21. Aboulissane, B., EL Haiek, D., EL Bakkali, L., EL Bahaoui, J., 2019, On the workspace optimization of

parallel robots based on CAD approach, Procedia Manufacturing, 32, pp. 1085-1092.

22. Tsirogiannis, E., Vosniakos, G.C., 2019, Redesign and topology optimization of an industrial robot link for
additive manufacturing, Facta Universitatis-Series Mechanical Engineering, 17(3), pp. 415-424.