Design optimization of parallel manipulators 
for high-speed precision machining applications 
 
Anatol Pashkevich*, Damien Chablat** 
Philippe Wenger*** 

*Ecole des Mines de Nantes, 
Nantes, France, (e-mail: anatol.pashkevich@emn.fr) 
** Institut de Recherches en Communications et Cybernétique de Nantes, 
Nantes, France, (e-mail: damien.chablat@irccyn.ec-nantes.fr) 
*** Institut de Recherches en Communications et Cybernétique de Nantes, 
Nantes, France, (e-mail: philippe.wenger@irccyn.ec-nantes.fr ) 
Abstract: The paper proposes an integrated approach to the design optimization of parallel manipulators, 
which is based on the concept of the workspace grid and utilizes the goal-attainment formulation for the 
global optimization. To combine the non-homogenous design specification, the developed optimization 
technique transforms all constraints and objectives into similar performance indices related to the 
maximum size of the prescribed shape workspace. This transformation is based on the dedicated dynamic 
programming procedures that satisfy computational requirements of modern CAD. Efficiency of the 
developed technique is demonstrated via two case studies that deal with optimization of the kinematical 
and stiffness performances for parallel manipulators of the Orthoglide family.   
Keywords: Computer-aided design, operation research application, multiobjective optimization, parallel 
robotic manipulators, Orthoglide robot. 

1. INTRODUCTION 
Many modern material processing operations, especially in 
automotive and aerospace industry, require high-accuracy 
positioning and high-speed motion of a work tool that is 
carried by a robotic manipulator (Brogardh, 2007). However, 
at present, classical serial manipulators have already reached 
limits 
of 
their 
performances. 
In 
contrast, 
parallel 
manipulators are claimed to offer better accuracy, lower 
mass/inertia properties, and higher structural rigidity (Merlet, 
2000). These features are induced by their specific kinematic 
structure, which eliminates the cantilever-type loading and 
makes 
them 
attractive 
for 
innovative 
machine-tool 
architectures (Tlusty et al., 1999). But practical utilization of 
the potential benefits requires development of efficient 
optimisation 
techniques, 
which 
should 
satisfy 
the 
computational requirement of relevant CAD systems. 
Generally, 
design 
of 
parallel 
manipulators 
involves 
simultaneous optimization of two types of criteria that 
evaluate 
respectively 
the 
kinematic 
and 
the 
kinetostatic/dynamic properties. Both of these groups include 
a number of performance measures that essentially vary 
through the workspace but should satisfy the prescribed 
bounds at any work-point. Up to now, the existing design 
methods provide separate solutions for the kinematic and 
kinetostatic objectives, while this paper proposes an 
integrated 
approach 
allowing 
to 
achieve 
desired 
performances for any prescribed rectangular workspace and 
to find corresponding parameters taking into account the 
interaction with the manufacturing process. 
In related works, the kinematic design is usually based on the 
concept of ‘critical points’ that allows essentially reduce the 
size of the optimization problem and, in some cases, even 
leads to analytical solutions. In, particular, for the Orthoglide 
family of manipulators, this approach produced several semi-
analytical design strategies that give the manipulators 
geometry (link lengths and joint limits) for any given cubic 
workspace with the desired velocity transmission factors. 
However, the kinetostatic design requires much more 
computing efforts since it operates with more detailed 
description of the links (cross-sections, moment of inertia, 
material properties, etc.). One of a key problem here is 
computing the stiffness matrix that evaluates the effect of the 
applied 
external 
torques/forces 
on 
the 
compliant 
displacements of the end-effector. 
The main contribution of this paper is in the area of CAD 
methodology and application of the operation research 
methods to the integrated design optimization of complex 
mechanical structures, such as parallel robots. In contrast to 
the previous work (Pashkevich et al., 2006), the proposed 
approach does not rely on the ‘critical points’ concept. 
Instead of this, it operates with the ‘workspace grid’ that is 
evaluated using a dedicated dynamic-programming-based 
algorithm allowing, for each particular set of the design 
parameters, to estimate the largest cuboid-shaped workspace 
with the desired properties. Further, the workspace 
parameters are evaluated in the frame of the global 
optimization, using the goal-attainment formulation. Finally, 
it yields a set of Pareto-optimal solutions that satisfies the 
design objectives and are presented the CAD user. 
Pashkevich A., Chablat D., Wenger P., “Design optimization of parallel manipulators for high-speed precision 
machining applications”, 13th IFAC Symposium on Information Control Problems in Manufacturing, 3 - 5 
Juin, 2009, Moscou, Russie. 
 
 
 
    
 
2. DESIGN PROBLEM AND METHODOLOGY 
Manipulator design traditionally begins with the selection of 
a kinematic framework and achieving certain geometric goals 
such as workspace size and dexterity. Besides, for particular 
manufacturing tasks, the manipulator geometry is optimized 
with respect to the desired velocity transmission factors. This 
yields a wire-frame CAD model of a relevant mechanism that 
defines basic dimensions of the links and spatial locations of 
all active and passive joints, as well as the joint limits. 
Further, when the geometry is established, the design focuses 
on locating the motion actuators and on the assigning the 
mass properties (i.e. the links shape, cross-sections, material 
type, etc.). The primary goal here is the implementation of 
the 
above 
geometric/kinematic 
objectives 
while 
simultaneously 
satisfying 
the 
desired 
performance 
requirements 
(payload, 
speed, 
acceleration, 
accuracy, 
deflection, etc.). This leads to a complete solid 3D model of 
the manipulator that is used in the global optimization loop.  
Because of very nonlinear and highly coupled relations 
between the design objectives and the design parameters, the 
global optimization is still a challenge for the CAD-based 
design. Existing approaches usually ignore the detailed and 
physically-reasonable description available in the CAD and 
operate with the secondary design parameters (masses, 
moment of inertia, etc.) that are treated as independent ones. 
The latter may produce a meaningless result that can not be 
implemented in practice (an evident example can be found in 
(Asada and Slotine, 1986), where the optimization produced 
incompatible values of link length, mass and inertia). On the 
other hand, a straightforward optimization is also non-
applicable here because of the very high computing expenses 
for the CAD- embedded routines that are extensively invoked 
for the evaluation of the performance measures. These 
motivate a problem-specific formalization of the manipulator 
design procedure presented below. 
To formulate the design problem, let us define the 
manipulator geometry by the mapping 
:
W
g 
, where 
1
n




Φ

 and 
1
n
p
p


W

 denote respectively the 
configuration space and the workspace; 
i are the joint 
coordinates, 
ip  are the coordinates of the end-effector; and n 
is the number of degrees of freedom. Besides, for each 
workspace point 

p
W , let us define the matrices 
( , )
v
K
p , 
( , )
s
K
p , 
( , )
m
K
p , that describe various  mechanical 
properties of  the manipulator (velocity and force  
transmission, stiffness, mass distribution, etc.) for any given 
set of the design parameters . Let us also assume that for 
each type of the matrices 
,
{ , , ,...}
v s m

K
, there are 
defined 
physically 
consistent 
scalar 
measures 
(
),
{ , , ,...}
i t c




K
 that may be directly included in the 
design objectives or constraints. Some examples of such 
measures 
(isotropy, 
transmission 
factors, 
compliance 
coefficients, etc.) are presented in the following section. 
Similarly, for the global evaluation of the manipulator, let us 
introduce the performance measures 
( , ),
{ , , , ...}
m l w



g π
 
that depend both on the adopted geometrical structure g and 
the physical parameters of the links π . Examples of the 
global measures include the total mass of the manipulator, the 
length of the principal links, the workspace size, etc. 
Then, following the general methodology adopted for the 
considered application area (high-speed machining), the 
design optimization problem can be stated as achieving the 
best values of the performance indices 
( , )
min ,
g





π
π
 
 (1) 
subject to the constraints 


0
( , )
,
,
&
S








K
p
p
W

 
 (2) 
that must be satisfied for all point of the cuboid workspace 
0
W  of size a b c
, which includes the manufacturing task. 
It should be noted that the latter assumption (concerning the 
workspace shape) is essential here and allows considerably 
speed-up the optimization routines. Since in practice this 
problem can not be solved by the direct search methods, in 
the following subsections there will be presented the 
discretisation scheme and relevant optimization algorithms 
allowing to obtain desired solutions in reasonable time.  
3. PERFORMANCE MEASURES 
Let us present the most essential performance measures that 
are used in mechanical design of manipulators. Traditionally 
they are directly included in the design constraints/objectives 
to be satisfied or optimized throughout the prescribed 
workspace. However, in this paper each performance 
measure is preliminary converted in an alternative form that 
defines the workspace subset where the relevant criterion is 
higher/lower of the desired value.  
3.1  Geometric and Kinematic Performances 
In robot design, the manipulator architecture is usually 
defined outside of the main optimization loop and highly 
depends on an application target. Within the CAD system, 
this architecture is described as an assembly of the 
links/joints that is parameterized by the link lengths and the 
joint limits. This set of the parameters is sufficient for 
evaluating the basic geometric and kinematic specifications 
such as the workspace size, dexterity, velocity transmission, 
reachability, etc. Using this model, the workspace W  may be 
generated in a straightforward way, using the direct kinematic 
equations 
( )
p = g  and the joint limits . It worth 
mentioning that, for parallel robots, the direct kinematics is 
usually non-trivial and an analytical solution do not exist in 
the general case (Merlet, 2006).   
Since for the considered application (high-speed machining) 
the desired workspace is the cuboid 
0
W  of size 

a b c

, 
the relevant performance measure may be defined by the 
largest similar object 

a
b
c





 inscribed in 
0
W , i.e.  


0
0
,
(
);
( , )
argmax
(
)
abc








T
W
T
W
T
T
W
W   (3) 
 
 
 
    
 
where , T are respectively the scalar scaling factor and the 
coordinate transformation operator in the Cartesian space. 
This notion is the fundamental issue of this paper and is 
discussed in details in the following sections. 
  For the kinematic performances, complete information is 
contained in the Jacobian matrix 
( )




J
g
 that is 
commonly evaluated using the condition number 
( )
cond J , 
or the largest/smallest singular values 
min
max
,


 (the latter 
are also referred to as the “velocity transmission factors”). 
However, because the Jacobian varies throughout the 
workspace, it should be defined a global metric that is usually 
computed by averaging or by detecting the worst case. 
Hence, in this paper, we suggest to redefine the global 
kinematic metric by using the above notion of the “largest 
inscribed cuboid”. For instance, if the condition number 

k J is used as the local metric, the manipulator global 
performances is evaluated by the size of the cuboid 
workspace 
that 
satisfies 
the 
design 
specification 
max
( ( ))
k
k

J p
 : 
0
(
);
abc
k


W
T
W
 
 (4) 




max
0
,
( , )
arg max
( , )
;
(
)
k
k








T
T
J p π
p
T
W
W  
This allows operate in a similar way with both geometric and 
kinematic performance measures. 
3.2  Elastic Performances 
For parallel manipulators, elasticity is an essential 
performance measure since it is directly related to the 
positioning 
accuracy 
and 
the 
payload 
capability. 
Mathematically, this benchmark is defined by the stiffness 
matrix, 
which 
describes 
the 
relation 
between 
the 
linear/angular displacements of the end-effector (wrench) and 
the external forces/torques applied to the tool. It is obvious 
that the elasticity is highly dependent upon geometry, 
materials and link shapes that are completely defined within 
the CAD model. 
The stiffness matrix may be computed using three different 
methods: the Finite Element Analysis (FEA), the matrix 
structural analysis (SMA), and the virtual joint method 
(VJM) (Alici & Shirinzadeh, 2005). The first of them, FEA, 
is proved to be the most accurate and reliable but requires 
very high computational efforts for repeated  3D remeshing 
over the whole workspace. The second method, SMA, also 
incorporates the main ideas of the FEA, but operates with 
rather large structural elements that allow some reduction of 
the computational expenses. And finally, the VJM method is 
based on the expansion of the traditional rigid model by 
adding the virtual joints (localized springs), which describe 
the elastic deformations of the links. It is the most efficient 
technique for the design optimization, which was recently 
enhanced by the authors to handle a case of the 
overconstrained manipulators (Pashkevich et al., 2008). 
Within this approach, each ith kinematic chain of the 
manipulator is described by the kinetostatic model that 
defines the differential kinematics and elasticity taking into 
account the active, passive and virtual joints:  
i
i
i
i
q
i






t
J
θ
J
q  
θ
i
i


τ
K
θ ,   
i
q 
τ
0 ,   
1,...
i
n

 
(5) 
where vector δ it  describes the end-effector translation and 
rotation (wrench) with respect to the Cartesian axes; vector 
i
θ  collects all virtual and active joint coordinates, vector 
i
q  includes all passive joint coordinates, 
i

J  and 
i
q
J  are the 
kinematic Jacobians with respect to the virtual/actuated and 
passive joints, 
i

τ , 
i
q
τ  are the aggregated vectors of the 
virtual/active and passive joint reactions, and 
θ
K  is the 
aggregated spring stiffness matrix of the links (composed of 
6x6 stiffness matrices of all virtual springs and the actuator 
stiffness coefficients). As it proved in (Pashkevich et al., 
2008), the desired force-wrench relation 
i
i
i


f
K
t  can be 
computed from the system 
1
θ
T
i
i
i
q
i
i
T
i
i
q


























J K
J
J
f
t
q
0
J
0
, 
 (6) 
by the straightforward inversion of the relevant matrix and 
extracting from it the upper-left sub-matrix of size 66. Then, 
for the entire manipulator, the stiffness is computed by 
aggregation all kinematic chains: 
1
n
m
i
i

K
K  
Using the stiffness matrix, one can evaluate the elastic 
deflection at the tool-point caused by a particular machining 
operation. However, following the above proposed approach, 
it is prudent to compute a relevant workspace-based metric, 
i.e.  
0
(
);
abc
s


W
T
W
 
 (7) 




max
0
,
( , )
argmax
,
;
(
)
m
m
K










T
T
p π
f
p
T
W
W
where 
,
m
m

f
 are respectively the eternal force and the upper 
limit of the elastic deflection defined by the design 
specifications. 
3.3  Dynamic Performances 
The manipulators dynamics is determined by the link inertial 
characteristics that bound the highest reachable accelerations 
and define capability to execute a given manufacturing task. 
Relevant evaluation techniques are generally based on two 
concepts: 
the 
dynamic 
isotropy 
and 
the 
dynamic 
manipulability. The first of them (Asada, 1986) deals with 
generalized inertia ellipsoid (GIE) that geometrically 
represents 
the 
generalized 
inertia 
matrix 
-T
-1

G
J DJ referring to the end effector, where 
( , )
D q π  is 
the manipulator inertia matrix with respect to the joint space 
and 
( , )
J q π  is the kinematic Jacobian. Using the GIE, the 
design is aimed at transforming the ellipsoid to a sphere 
(without respect to its radius), in order to ensure that the 
 
 
 
 
    
 
attitude to produce end-effector accelerations does not 
depend on the direction. Hence, from practical point view, 
this classical approach seeking for 


( )
1
cond

D p
 should be 
reformulated taking into account the GEI size. The latter can 
be done by bounding the norm 
0
( )
m

D p
 , for instance the 
spectral norm allows to bound the GEI axis length. Further, 
similarly to the above, it is necessary to compute a relevant 
workspace-inscribed cuboid, i.e.  
0
(
);
abc
m


W
T
W
 
 (8) 




0
0
,
( , )
arg max
,
;
(
)
m








T
T
G p π
p
T
W
W  
Similar 
approach 
can 
be 
applied 
to 
the 
dynamic 
manipulability metric (Yoshikawa, 1985) that evaluates 
ability of the manipulator to transform the inputs 
forces/torques 
into 
the 
output 
accelerations. 
This 
transformation 
is 
defined 
by 
the 
matrix 
product 
1
( , )
( , )

J p π
D p π
, so the corresponding cuboid is defined as  
0
(
);
abc
a


W
T
W
 
 (9) 
1
min
max
,
0
( , )
( , )
;
( ,
)
arg max
(
)
;
i
i
a
























T
J p π
D p π
τ
T
p
T
W
W
 
where 
min
a
, 
max
i
 are the desired acceleration and the 
maximum force/torque in the ith actuated joint respectively. 
Thus, independent of the physical meaning, all performance 
measures are presented in a similar way allowing essentially 
simplify the design optimization process. 
4. WORKSPACE-BASED METRICS 
The above presented technique for the global evaluation of 
the manipulator performances is based on an auxiliary 
computational problem, i.e. estimation of the largest cuboid-
shaped sub-workspace where the relevant criterion is higher 
or lower of the desired value. Let us attack this problem 
numerically, using the workspace discretisation and applying 
the dynamic programming.  
To satisfy the desired cuboid proportions

a b c

, let us 
define 
the 
workspace 
grid 

ijk
G
that 
includes 
the 
manipulator workspace 
0
W  and possesses uniform but 
different 
steps 
along 
the 
Cartesian 
axes, 
namely 


0
0
0
/
;
/
;
/
a N
b N
c N
, where 
0
N  defines the discretisation 
precision. Besides, for each node of the grid, let us compute 
relevant local performance measure and define a 3D binary 
matrix 


0,1
ijk


, where 
1
ijk

 if the corresponding 
design constraint/objective is satisfied, and 
0
ijk


 
otherwise. For computation conveniences, let us also set 
0
ijk


 if 
0
ijk
G
W

. 
Thus, the original problem is reduced to searching for the 
largest cubic submatrix inside of 

ijk

 containing non-zero 
values only. The latter can be efficiently solved applying the 
following algorithm that operates with additional integer 
matrix 

ijk

 that define sizes of the candidate solutions with 
the vertexes ( , , )
i j k : 
Step 0. Set 
0,
, ,
ijl
i j k



  
Step 1. Set 
ijk
ijk


 for 
 






1&
,
1&
,
1&
,
i
j k
j
i k
k
i j








 
Step 2. for 
max
2 :
i
i

 do 
for 
max
2 :
j
j

 do 
for 
max
2 :
k
k

 do 
if 
1
ijk

  then 
1, ,
,
1,
, ,
1
1,
1,
1, ,
1
,
1,
1
1,
1,
1
,
,
,
1
min
,
,
,
i
j k
i j
k
i j k
ijk
i
j
k
i
j k
i j
k
i
j
k































 
Step 3. Find 
max(
) 1
ijk
d 

 ; 
0
0
0
( ,
,
)
arg max(
)
ijk
i
j k


 
Step 4. Retrieve from the grid 

ijk
G
 the desired cuboid 
bounded by the indices 
0
0
0
(
,
,
)
i
d j
d k
d



 and 
0
0
0
( ,
,
)
i
j k
. 
 
Validity of this routine and correctness of the relevant 
recurrent expression can be proved using the standard ideas 
of the dynamic programming, similar to finding the largest 
square block in two-dimensional binary matrix. 
Hence, for each performance measure and each set of the 
design parameters, it can be computed a workspace-based 
metrics composed of the coordinate ranges of the largest 
cuboid-shaped sub-workspace with the desired proportions.  
5. DESIGN PROCEDURE 
Following the common engineering concepts adopted in 
robotic practice, let us divide the design procedure into two 
basic steps: (i) geometric/kinematic design that provides the 
manipulator geometry, including the link lengths, joint 
locations 
and 
joint 
limits, 
and 
(ii) 
integrated 
kinetostatic/dynamic design that deals with assigning the 
shape and mass properties for all joints and links. This 
separation 
allows 
essentially 
simplify 
optimization 
algorithms and improve their convergence. 
In the frame of the proposed methodology, both design 
problems can be reformulated in a similar way: 
( )
min ,
if
i


π
π
 
 (10) 
subject to the workspace-based constraints 


0
0
0
(
( ))
abc
j
size W
a
b
c
j




π
 
 (11) 
where indices ,  define particular design objectives and 
constraints. For instance, for the geometric and kinematic 
design, the objectives are to minimize the manipulator 
dimensions (link lengths), while the constraints define the 
desired workspace size and the range of the velocity 
transmission factors. For the kinetostatic and dynamic design, 
the objectives deal with minimizing the component masses, 
and the constraints identify the prescribed stiffness and the 
accelerations transmission characteristics (for the fixed 
geometry).  
 
 
 
    
 
For computational conveniences, let us transform the design 
constraints and present them in the scalar form as 
0
( )
k
k
h
h

π
. 
Then, to generate a specific Pareto-optimal solution, let us 
apply the goal attainment technique that yields the following 
nonlinear programming formulation: 
,
min


π  
 (12) 
subject to 
0
0
( )
;
( )
;
i
i
i
i
f
w
f
h
h
i





π
π
 
 (13) 
Here  is an unrestricted scalar variable, 
0
iw 
 are 
designer-selected weighting coefficients, and 
0
if  are the 
goals to be realized for each design objective (usually, the 
designer can extract them from the design specifications). In 
this formulation, minimization of , tends to force the 
specifications to meet their goal. If, at the solution point,  is 
negative, the goals have been over-attained; if  is positive, 
then the goals have been under-attained. The method is 
appealing since it is possible for the designer to specify 
unrealizable objectives and still obtain a solution which 
represents a compromise. Advantages of this approach will 
be demonstrated in the following section that focuses on the 
application examples. 
6. APPLICATION EXAMPLES 
6.1  Optimization of Orthoglide Geometry 
Let us consider first the problem of the geometric synthesis 
of the Orthoglide parallel manipulators that was addressed in 
our previous paper (Pashkevich et al., 2006). However, the 
previous solution is valid only for the “cubic type” design 
specifications (i.e. for the case 
0
0
0
a
b
c


). Hence, it is 
prudent to generalize the former results for a general case. 
The Orthoglide architecture includes three identical parallel 
chains that are actuated by linear drives with mutually 
orthogonal axes. Each kinematic chain is formally described 
as PRPaR, where P, R and Pa denote the prismatic, revolute, 
and parallelogram joints respectively. The output machinery 
is connected to the legs in such a manner that the tool moves 
in the Cartesian space with fixed orientation (i.e. restricted to 
translational motions). Assuming that the manipulator legs 
are not necessarily equal, the manipulator Jacobian may be 
described by the following equations  
1
1
1
1
y
z
x
x
x
x
x
z
y
y
y
y
y
x
z
z
z
z
p
p
p
ρ
p
ρ
p
p
p
ρ
p
ρ
p
p
p
ρ
p
ρ





























J
 
 (14) 
where p = (px, py, pz)T is the output vector of the TCP 
position,  = (x, y, z)T is the input vector of the prismatic 
joints variables, i{x, y, z}. It allows computing the 
condition number and the velocity transmission factors for 
each workspace point 
(
,
,
)
x
y
z
p
p
p
. Then, using the 
workspace grid, each set of the geometrical parameters 
{
,
,
}
x
y
z
L L
L
 can be evaluated by the metric 

a b c

. 
The optimization results were normalized with respect to the 
desired workspace size and tabulated. One of the case studies 
corresponding 
to 
the 
design 
specifications 


0
1.0 1.0 0.8
W 


 is presented in Fig 1. It contains 
solutions constrained by the equality 
1
2
L
L

 incited by the 
shape of 
0
W . As follows from them, the developed technique 
perfectly accounts non-symmetry of the design specifications 
and produces relevant non-symmetrical solutions for the 
manipulator geometry. For example, compared to the known 
solution corresponding to the symmetrical case, the obtained 
Pareto-optimal set contains solutions, which allow reduce the 
manipulator dimensions by 10…20 %.  
1.45
1.5
1.55 
1.6
1.3
1.4
1.5
1.6
L3 
W0={1.01.01.0} 
W0={1.01.00.8} 
L1=L2 
 
Fig. 1.  Optimal geometrical parameters of Orthoglide. 
 
6.2  Optimization of Orthoglide Stiffness 
The stiffness-based design assumes that the manipulator 
geometry is constrained and satisfies the kinematic 
objectives. Hence, before considering a related optimization 
problem, it is prudent to investigate the relationship between 
the kinematic and stiffness specifications for a particular 
manipulator, 
such 
as 
Orthoglide 
prototype 
with 
a 
symmetrical architecture (Chablat & Wenger, 2003). The 
kinematic specifications for this manipulator were defined via 
the velocity transmission factors that should be in the range 
of 0.5…2.0 for the workspace size 200200200 mm3. 
The manipulator stiffness model was derived using the 
“virtual-joint” technique, and the model parameters were 
evaluated via the FEA. Then, the workspace was meshed 
with a step of 1.0 mm and the stiffness matrices were 
computed for each node. Using these data and applying the 
proposed algorithm, it was generated a set of nested cubic 
subspaces with the upper-bounded translational/rotational 
stiffness. Corresponding results demonstrate good agreement 
between the shapes of the velocity and stiffness maps. In 
particular, the location of the cubic workspaces derived via 
constraining the velocity transmission factors and the 
stiffness are almost the same (difference is less than 5%). 
Also, the obtained results justify the velocity transmission 
constraint [0.5; 2.0] used in our previous study, since its 
violation leads to essential decrease of the stiffness. 
 
 
 
    
 
0.5
1
1.5
2
2.5
2.75
2.8
2.85
2.9
2.95
3
3.05
x 10
-4
ktran: Q0

0.5
1
1.5
2
2.5
1.96
1.98
2
2.02
2.04
2.06
2.08
x 10
-7
krot: Q0

0.5
1
1.5
2
2.5
1
1.01
1.02
1.03
1.04
x 10
-3
ktran: Q1

0.5
1
1.5
2
2.5
2.08
2.1
2.12
2.14
2.16
2.18
2.2
x 10
-7
krot: Q1

0.5
1
1.5
2
2.5
2.16
2.18
2.2
2.22
2.24
x 10
-3
ktran: Q2

0.5
1
1.5
2
2.5
2.72
2.74
2.76
2.78
2.8
2.82
x 10
-7
krot: Q2

 
Fig. 2. Optimisation of the Orthoglide parallelogram links 
At the second stage, the developed technique was applied to 
parametrical optimization of the Orthoglide components with 
respect to the stiffness objectives. Here, it was assumed that 
the link lengths are constrained while the cross-sectional 
dimensions were scaled using the factors  and 1/ . The 
latter allowed maintaining the same cross section areas and, 
respectively, the link masses.  
An extraction from the optimization results concerning the 
parallelogram links are presented in Fig. 2. For the sake of 
clarity, the stiffness is evaluated in three characteristic points 
Q0, Q1 and Q2, where Q0 corresponds to the isotropic posture 
and Q1, Q2 define the workspace boundary. As follows from 
them, the manipulator stiffness can be essentially improved 
without increasing the link masses, just by changing the 
cross-section shape according to the scaling factor 
1.8.

 
These results have been also used for the design of a new 
Orthoglide prototype, which is currently under development 
in our laboratory. 
7. CONCLUSIONS 
Parallel robotic manipulators present attractive solutions for 
innovative machine-tool architectures, which should insure 
high-speed and precision machining of large spectrum of 
materials. However, practical utilization of the potential 
benefits requires development of efficient optimisation 
techniques, which should satisfy the computational speed and 
accuracy requirements of relevant CAD procedures. To 
response this challenge, the paper proposes an integrated 
approach to the design optimization of parallel manipulators. 
In contrast to previous works, which are based on the concept 
of “critical points”, the proposed technique is compatible 
with current CAD methodology and allows achieving desired 
performances for any prescribed workspace and to take into 
account an interaction with the manufacturing process. 
The developed approach has been validated by a number of 
case studies, which focus on the design optimization of 
translational parallel manipulators of the Orthoglide family. 
Further work will deal with extending these results for more 
general 
case, 
including 
Orthoglide 
with 
additional 
orientational axes that is currently under development. 
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