Stiffness Analysis of 
Overconstrained Parallel Manipulators 
Anatol Pashkevich1,2, Damien Chablat1*, Philippe Wenger1 
1Institut de Recherches en Communications et Cybernétique de Nantes,  
UMR CNRS 6597, 1 rue de la No, 44321 Nantes, France 
 
2Ecole des Mines de Nantes,    
4 rue Alfred-Kastler, Nantes 44307, France 
 
 
 
Abstract. The paper presents a new stiffness modeling method for overconstrained parallel manipulators 
with flexible links and compliant actuating joints. It is based on a multidimensional lumped-parameter model that 
replaces the link flexibility by localized 6-dof virtual springs that describe both translational/rotational compliance 
and the coupling between them. In contrast to other works, the method involves a FEA-based link stiffness 
evaluation and employs a new solution strategy of the kinetostatic equations for the unloaded manipulator 
configuration, which allows computing the stiffness matrix for the overconstrained architectures, including 
singular manipulator postures. The advantages of the developed technique are confirmed by application 
examples, which deal with comparative stiffness analysis of two translational parallel manipulators of 3-PUU 
and 3-PRPaR architectures. Accuracy of the proposed approach was evaluated for a case study, which focuses 
on stiffness analysis of Orthoglide parallel manipulator. 
Keywords: Parallel mechanisms, Stiffness modeling, Parallelogram-based linkage, Orthoglide robot 
 
 
1. Introduction 
 
Parallel manipulators have become more and more popular in industrial applications, 
including high-accuracy positioning and high-speed machining [1, 2]. This growing attention is 
inspired by their essential advantages over serial manipulators, which have already reached the 
dynamic performance limits (bounded by high masses of the machine components required to 
support sequential joints, links and actuators). In contrast, parallel manipulators are claimed to 
offer better accuracy, lower mass/inertia properties, and higher structural rigidity (i.e. stiffness-
to-mass ratio) [3]. These features are induced by their specific kinematic structure, which resists 
the error accumulation in kinematic chains and allows convenient actuators location close to the 
manipulator base. Besides, the links act in parallel against the external force/torque, eliminating 
the cantilever-type loading and increasing the manipulator stiffness [4]. The latter makes them 
attractive for innovative machine-tool architectures [5-7], but practical utilization of the potential 
benefits requires development of efficient stiffness analysis techniques, which satisfy the 
computational speed and accuracy requirements of relevant design procedures [8]. 
Generally, the stiffness analysis evaluates the effect of the applied external torques and 
forces on the compliant displacements of the end-effector. Numerically, this property is defined 
through the “stiffness matrix” K, which gives the relation between the translational/rotational 
displacement and the static forces/torques causing this transition. The inverse of K is usually 
called the “compliance matrix” and is denoted as k. As follows from mechanics, K is 66 semi-
definite non-negative matrix, where structure may be non-diagonal to represent the coupling 
between the translation and rotation [9]. Besides, this matrix may be not-symmetrical under the 
static load [10], but standard stiffness analysis focuses on the non-loaded structures. 
* Corresponding author: 
Tel: +33 240 37 69 48; Fax:  +33 240 37 69 30; 
E-mail: Damien.Chablat@irccyn.ec-nantes.fr 
A. Pashkevich, D. Chablat, P. Wenger / Stiffness Analysis of Overconstrained Parallel Manipulators 
 
2
Similar to other manipulator properties (kinematical, for instance), the stiffness essentially 
depends on the force/torque direction and on the manipulator configuration. Hence, to provide 
the designer with integrated performance criteria, various scalar indices are usually computed 
(such as the best/worst/average stiffness with respect to the rotation or translation). They are 
typically derived using the singular-value decomposition of K. However, there are still a number 
of open questions here regarding the significance of these indices for a particular manufacturing 
task. Besides, since the matrix K varies through the workspace, corresponding global 
benchmarks must be computed. In some cases, a relevant analysis produces the “stiffness maps”, 
which describe the end-effector compliance as a function of the manipulator configuration [11-
13].  
Several approaches exist for the computation of the stiffness matrix, which differ in the 
modeling assumptions and computational techniques. They are the Finite Element Analysis 
(FEA), the matrix structural analysis (MSA), and the virtual joint method (VJM) that is often 
called the lumped modeling.  
The FEA method is proved to be the most accurate and reliable, since the links/joints are 
modeled with its true dimension and shape [14]. Its accuracy is limited by the discretisation step 
only. However, because of high computational expenses required for the repeated re-meshing, 
this method is usually applied at the final design stage for the verification and component 
dimensioning. For example, in [15], a FEA model was used to evaluate the static rigidity and 
natural frequencies of the T3R1 parallel robot. Also, this method is widely used for validation of 
other stiffness analysis techniques [16-18] and for the comparative study [19].  
The MSA method is a common technique in mechanical engineering [20], it incorporates the 
main ideas of the FEA but operates with rather large flexible elements (beams, arcs, cables, etc.). 
This obviously yields reduction of the computational expenses and, in some cases, allows even 
obtaining an analytical stiffness matrix. For parallel manipulators, the relevant stiffness model is 
a combination of flexible beams and nodes, where each beam is defined by two nodes and 
described by 1212 stiffness matrix derived from the Euler-Bernoulli presentation. Then, these 
matrices are “assembled” in accordance with the superposition principle and the manipulator 
geometry, to produce the desired 66 matrix for the whole mechanism. Sometimes this approach 
is also referred to as the “distributed stiffness” modeling. One of the first examples of MSA 
application for the problem of interest is the stiffness analysis of a Stewart platform [21], which 
was performed under the assumption that the links are not subject to bending. This approach was 
also used in [22, 23] for other manipulators and/or other modeling assumptions. Some resent 
MSA-based results are obtained for the Delta-type mechanisms [24]. This method gives a 
reasonable trade-off between the accuracy and computational time, provided that link 
approximation by the beam elements is realistic. Because it involves rather high-dimensional 
matrix operations, it is not attractive for the parametric stiffness analysis and analytical 
modeling. 
Finally, the VJM method, which is also referred to as the “lumped modeling”, is based on 
the expansion of the traditional rigid model by adding virtual joints (localized springs), which 
describe the elastic deformations of the manipulator components (links, joints and actuators). 
A. Pashkevich, D. Chablat, P. Wenger / Stiffness Analysis of Overconstrained Parallel Manipulators 
 
3
This approach originates from the work of Gosselin [25], who evaluated parallel manipulator 
stiffness taking into account only the actuators compliance and by presenting them as one-
dimensional linear springs (the links were assumed to be rigid, and the passive joints to be 
perfect). Besides, the compliance in all actuated joints was assumed to be equal. The latter 
allowed reducing the stiffness analysis to the analysis of the condition number of the Jacobian 
matrix. Further development of VJM allowed taking into account the links flexibility, which 
were presented as rigid beams supplemented by linear and torsional springs [26]. There are a 
number of variations and simplifications of the VJM method, which differ in modeling 
assumptions and numerical techniques. In particular, it was applied to the CaPAMan, Orthoglide 
and H4 robots, specific variants of Stewart-Gough platform, manipulators with US/UPS legs and 
other kinematic machines [27-32]. Generally, the lumped modeling provides acceptable accuracy 
in short computational time, so it is widely used at the pre-design stage, especially for the 
analytical parametric analysis. However, it is very hypothetic and operates with simplified 
stiffness models that are composed of one-dimensional springs that do not take into account the 
coupling between the rotational and translational deflections. There are also other restrictions, 
which limit its applications to non-overconstrained mechanisms.  
This paper presents a new stiffness modelling method, which combines advantages of the 
above mentioned approaches. It is based on a multidimensional lumped-parameter model that 
replaces the link flexibility by localized 6-dof virtual springs that describe both the 
linear/rotational deflections and the coupling between them. The spring stiffness parameters are 
evaluated using FEA modelling to ensure higher accuracy. In addition, it employs a new solution 
strategy of the kinetostatic equations, which allows computing the stiffness matrix for the 
overconstrained architectures, including the singular manipulator postures. This gives almost the 
same accuracy as FEA but with essentially lower computational effort because it eliminates the 
model re-meshing through the workspace.  
Since the developed technique is targeted to the design optimization, it relies on the 
assumption that the manipulator is located in unloaded equilibrium configuration. This allows 
evaluating the symmetrical part of the general stiffness matrix while neglecting the skew-
symmetrical components, which describe effects caused by the external loading and relevant 
changes in Jacobian [12, 33, 34]. It is obvious that at the preliminary design stage, which focuses 
on the conceptual issues (such as comparison of alternative manipulator architectures, defining 
critical components in the kinematic chains, deciding on the stiffness specifications for the links, 
etc.), this assumption is practical and reasonable. Besides, for a particular case study presented 
below, validity of the assumption is justified numerically. 
The remainder of this paper is organized as follows. Next section introduces a general 
methodology of deriving/computing of the kinematic and stiffness model. Section 3 describes the 
manipulator compliant elements and proposes FEA-based technique for the evaluating their 
parameters. Section 4 includes application examples, which deal with comparative stiffness 
analysis of two translational parallel manipulators and demonstrate advantages of the proposed 
technique. Section 5 summarizes the main contributions of this work and defines future research 
directions. 
A. Pashkevich, D. Chablat, P. Wenger / Stiffness Analysis of Overconstrained Parallel Manipulators 
 
4
2. General methodology 
 
2.1. Manipulator Architecture 
Let us consider a general n-dof parallel manipulator, which consists of a mobile platform 
connected to a fixed base by n identical kinematics chains (Fig. 1). Each chain includes an 
actuated joint “Ac” (prismatic or rotational) followed by a “Foot” and a “Leg” with a number of 
passive joints “Ps” inside. Generally, certain geometrical conditions are assumed to be satisfied 
with respect to the passive joints to eliminate the undesired platform rotations and to achieve 
stability of desired motions. 
 
Base 
Ps
Ps
Ps
Ps
Ps
Ps
Ac
Mobile platform
Ps
Ps
Ps
Ps
Ps
Ps
Ac
L 
L 
F
F
 
Chain # 1
Chain # n
 
Fig. 1. Schematic diagram of a general n-dof parallel manipulator 
(Ac – actuated joint, Ps – passive joints, F – foot, L - Leg) 
Typical examples of such architectures are: 
(a) 3-PUU translational parallel kinematic machine (Fig 2a); where each leg consists of a rod 
ended by two U-joints (with parallel intermediate and exterior axes), and active joint is 
driven by linear actuator [35]; 
(b) Delta parallel robot (Fig 2b) that is based on the 3-RRPaR architecture with parallelogram-
type legs and rotational active joints [36]; 
(c) Orthoglide parallel robot (Fig 2c) that implements the 3-PRPaR architecture with 
parallelogram-type legs and translational active joints [37]. 
Here R, P, U and Pa denote the revolute, prismatic, universal and parallelogram joints, 
respectively. 
It should be noted that examples (b) and (c) illustrate specific cases of the overconstrained 
mechanisms, for which the standard stiffness analysis methods cannot be applied directly. In 
particular, for the Orthoglide mechanism the architectural particularities can be summarized as 
follows: (i) each kinematic chain prevents the platform from rotating about two orthogonal axes; 
(ii) any combination of two kinematic chains suppresses the three platform rotations, (iii) the 
whole set of three kinematic chains also suppresses the three platform rotations. Hence, the 
kinematical architecture of this manipulator includes excessive (redundant) constrains that are in 
certain agreement for the nominal parameter values. However, such a spatial overconstrained1 
arrangement ensures essential increase of the rigidity with respect to the external force. This 
motivates development of dedicated stiffness analysis techniques that are presented below. 
 
1 In the robotic literature, an alternative definition of the “overconstrained manipulator” is also used. Some authors [39] use this 
the term in wide-sense, as the opposite to the “redundant manipulator”, in order to distinguish mechanisms with any additional 
kinematic chains or specific location/orientation of the joints that cause reduction of the workspace dimension (down to planar, 
spherical, etc.). In this paper, the term “overconstrained” is referred to as the specific case, when the number of imposed 
constrains is greater that the resulting loss in the degrees-of-freedom [40]; it is also known as the “repeated” or “common” 
A. Pashkevich, D. Chablat, P. Wenger / Stiffness Analysis of Overconstrained Parallel Manipulators 
 
5
(a) 3-PUU translational PKM [35] 
(b) Delta parallel robot [36] 
(c) Orthoglide parallel robot [37] 
 
Base 
Mobile platform 
R 
R 
R 
R 
R 
R 
R 
R 
R 
R 
R 
R 
F 
F 
F 
L 
L 
L 
P 
P 
P 
 
 
Base 
Mobile platform
R
R
R
R
R
R
R
R
R
R
R
R
R
R
R
R
R
R
R
R
R
F 
F 
F 
L 
L 
L 
 
 
Base 
Mobile platform 
R 
R
R 
R 
R 
R 
P 
R 
R 
R 
R 
R 
R 
P 
R
R 
R
R
R
R 
P
F 
F 
F 
L 
L 
L 
 
 
 
 
 
Fig. 2. Typical 3 dof translational parallel mechanisms 
 
 
 
2.2. Basic Assumptions 
To evaluate the manipulator stiffness, let us apply a modification of the virtual joint method 
(VJM), which is based on the lump modeling approach [25, 26]. According to this approach, the 
original rigid model should be extended by adding virtual joints (localized springs), which 
describe elastic deformations of the links. Besides, virtual springs are included in the actuating 
joints to take into account stiffness of the mechanical transmissions and the control loop. To 
overcome difficulties with parallelogram stiffness modeling, let us first replace the manipulator 
legs (see Fig. 2) by rigid links with configuration-dependent stiffness. (Such transformation will 
be justified further while deriving the stiffness model for the parallelogram-based legs). 
 
 
 
Ac 
Rigid Foot 
Mobile platform 
(rigid) 
Base platform 
(rigid) 
U
Rigid Leg 
U
 6-d.o.f.
spring
 6-d.o.f.
spring
 6-d.o.f.
spring 
 
Fig. 3. Flexible model of a single kinematic chain (Ac – actuating joint, U – universal joint) 
 
This transforms the general architecture into the extended n-xUU case allowing treating all 
the considered manipulators in a similar manner. Under such assumptions, each kinematic chain 
of the manipulator can be described by a serial structure (Fig. 3), which includes sequentially: 
(a) a rigid link between the manipulator base and the ith actuating joint (part of the base 
platform) described by the constant homogenous transformation matrix
i
Base
T
; 
(b) a 1-dof actuating joint with supplementary virtual spring describing the control loop 
stiffness, which is defined by the homogenous matrix function 
0
0
(
)
i
i
a q 
V
 where 
0
iq  is the 
actuated coordinate and 
0
i is the virtual spring coordinate; 
A. Pashkevich, D. Chablat, P. Wenger / Stiffness Analysis of Overconstrained Parallel Manipulators 
 
6
(c) a 6-dof virtual spring describing the actuator mechanical stiffness, which is defined by the 
homogenous matrix function
1
6
(
,...
)
i
i
s 

V
 where 
1
2
3
{
,
,
}
i
i
i



, 
4
5
6
{
,
,
}
i
i
i



 are the virtual 
spring coordinates corresponding to the spring translational and rotational deflections; 
(d) a rigid “Foot” linking the actuating joint and the leg, which is described by the constant 
homogenous transformation matrix 
Foot
T
; 
(e) a 6-dof virtual spring describing the foot stiffness, which are defined by the homogenous 
matrix function 
7
12
(
,
)
i
i
s 

V

, where 
7
8
9
{
,
,
}
i
i
i



 and 
10
11
12
{
,
,
}
i
i
i



 are the spring 
translational/rotational deflections; 
(f) a 2-dof passive U-joint at the beginning of the leg allowing two independent rotations with 
angles 
1
2
{ ,
}
i
i
q q
, which is described by the homogenous matrix function 
1
1
2
(
,
)
i
i
u q q
V
; 
(g) a rigid “Leg” linking the foot to the movable platform, which is described by the constant 
homogenous matrix transformation 
Leg
T
; 
(h) a 6-dof virtual spring describing the leg stiffness, which are defined by the homogenous 
matrix function 
13
18
(
,
)
i
i
s 

V

, where 
13
14
15
{
,
,
}
i
i
i



 and 
16
17
18
{
,
,
}
i
i
i



 correspond to the 
spring translations and rotations respectively; 
(i) a 2-dof passive U-joint at the end of the leg allowing two independent rotations with angles 
3
4
{
,
}
i
i
q
q
, which is described by the homogenous matrix function 
2
3
4
(
,
)
i
i
u
q
q
V
; 
(j) a rigid link from the manipulator leg the end-effector (part of the movable platform) 
described by the constant homogenous matrix transformation 
i
Tool
T
.  
The corresponding mathematical expression defining the end-effector location subject to 
variations of all above defined coordinates of a single kinematic chain i may be written as 
follows 
0
0
1
6
7
12
1
1
2
13
18
2
3
4
(
)
(
,...
)
(
,
)
(
,
)
(
,
)
(
,
)
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
Base
a
s
Foot
s
u
Leg
s
u
Tool
q
q q
q
q

















T
T
V
V
T
V
V
T
V
V
T


 (1)  
where the matrix function 
(.)
a
V
 is either an elementary rotation or translation, the matrix 
functions 
1(.)
u
V
 and 
2(.)
u
V
 are compositions of two successive rotations, the spring matrix 
(.)
s
V
 is composed of six elementary transformations, and 
1,...
i
n

. In the rigid case, the virtual 
joint coordinates 
1
18
,
i
i



 are equal to zero, while the remaining ones (both active 
0
iq  and 
passive 
1
4
,
i
i
q
q

) are obtained through the inverse kinematics, ensuring that all n matrices 
,
1,...
i i
n

T
 are equal to the prescribed one, that characterizes the desired spatial location of the 
moving platform (kinematic loop-closure equations). Particular expressions for all components 
of the product (1) may be easily derived using standard techniques for homogenous 
transformation matrices. 
It should be noted that the kinematic model (1) includes 24 variables (1 for active joint, 4 for 
passive joints, and 19 for virtual springs). However, some of the virtual springs are redundant, 
since they are compensated by corresponding passive joints (with aligning axes) or by 
combination of passive joints. For computational convenience, nevertheless, it is not reasonable 
to detect and analytically eliminate redundant variables at this step, because the technique 
developed below allows easy and efficient computational elimination (without increasing the 
size of the required matrix inverse, which is equal to 66 independent of the virtual joint 
number). 
A. Pashkevich, D. Chablat, P. Wenger / Stiffness Analysis of Overconstrained Parallel Manipulators 
 
7
2.3. Differential kinematic model 
To evaluate the manipulator ability to respond to external forces and torques, let us first 
derive the differential kinematic equation describing relations between the end-effector location 
and small variations in the joint variables. For each ith kinematic chain, this equation can be 
generalized as follows 
 
,
1,...
i
i
i
i
q
i
i
n







t
J
θ
J
q
, 
(2) 
where vector δ
(δ
, δ
, δ
, δ
, δ
, δ
)T
i
xi
yi
zi
xi
yi
zi
p
p
p




t
 describes the end-effector translation 
δ
(δ
, δ
, δ
)T
i
xi
yi
zi
p
p
p

p
 and rotation δ
(δ
, δ
, δ
)T
i
xi
yi
zi





with respect to the Cartesian axes; 
vector 
0
18
(
,
)
i
i
T
i



θ

 collects all virtual joint coordinates, the vector 
1
4
(
,
)
i
i
T
i
q
q



q

 
includes all passive joint coordinates, symbol ' ' stands for variation with respect to the rigid 
case values, and 
i

J , 
i
q
J  are matrices of sizes 6x19 and 6x4 correspondingly. It should be noted 
that the derivative for the actuated coordinate 
0
iq is not included in 
q
J  but it is represented in the 
first column of 


J  through the variable 
0
i. 
The desired matrices 
i

J , 
i
q
J , which are the only parameters of the differential model (2) 
may be computed from (1) analytically using some software support tools such as Maple, 
MathCAD or Mathematica. However, a straightforward differentiation usually yields very 
awkward expressions that are not convenient for further computations. On the other hand, the 
fractionized structure of (1), where all variables are separated, allows applying an efficient semi-
analytical method. To present this technique, let us assume that for the particular virtual joint 
variable 
i
j
 the model (1) is rewritten as 
 
(
)
L
i
R
i
ij
j
j
ij





T
H
V
H , 
(3) 
where the first and the third multipliers are the constant homogenous matrices, and the second 
multiplier is the elementary translation or rotation. Then the partial derivative of the homogenous 
matrix 
iT  with respect to 
i
j
 at the point 
0
i
j

 may be computed from a similar product where 
the internal term is replaced by the matrix 
(.)
j

V
 that admits a very simple analytical 
presentation. In particular, for the elementary translations and rotations about the X-axis these 
derivatives are: 
 
0 0 0 1
0 0 0 0
0 0 0 0
0 0 0 0
x
Tran











V
;   
0
0 0 0
0
0 1 0
0
1 0 0
0
0 0 0
x
Rot











V
. 
(4) 
Furthermore, since the derivative of the homogenous matrix 
(
)
L
i
R
i
ij
j
j
ij







T
H
V
H  may be 
presented as 
 
0
0
0
0
0
0
0
iz
iy
ix
iz
ix
iy
i
iy
ix
iz
p
p
p































T
, 
(5) 
A. Pashkevich, D. Chablat, P. Wenger / Stiffness Analysis of Overconstrained Parallel Manipulators 
 
8
So, the desired jth column of the jacobian 
i

J  can be easily extracted from 
i
T  (using the matrix 
elements 
14
T, 
24
T, 
34
T, 
23
T, 
31
T, 
12
T). 
The jacobians 
q
J  can be computed in a similar manner, but the derivatives are evaluated in 
the neighborhood of the “nominal” values of the passive joint coordinates 
i
jnom
q
 corresponding to 
the rigid case (these values are obtained from the inverse kinematics). However, a simple 
transformation 
i
i
i
j
j
j
nom
q
q
q



 
and 
a 
corresponding 
factoring 
of 
the 
function 
(
)
(
)
(
)
i
i
i
q j
j
q j
j
q j
j
nom
q
q
q



V
V
V
 allow applying the above approach. It is also worth mentioning 
that this technique may be also applied in analytical computations, allowing one to avoid bulky 
transformations required for the straightforward differentiating. 
 
 
2.4. Kinetostatic and Stiffness Models 
For the manipulator kinetostatic model that describes the force-and-motion relation, it is 
necessary to introduce additional equations that define the virtual joint reactions to the 
corresponding spring deformations. In accordance with the adopted stiffness model, the 
following virtual springs are included in each kinematic chain: 
 1-dof virtual spring describing the actuator control loop compliance; 
 6-dof virtual spring describing the actuator mechanics compliance; 
 6-dof virtual spring describing mechanical compliance of the foot;  
 6-dof virtual spring describing mechanical compliance of the leg.  
Assuming that the spring deformations are small enough, the required relations may be 
expressed by linear equations 
0
i
i
ctr
K











;       
1
6
i
i
act
i
i



























K


;         
7
12
i
i
Foot
i
i



























K


;         
13
18
i
i
Leg
i
i



























K


, (6) 
where 
i
j

 is the generalized force for the jth virtual joint of the ith kinematic chain, 
ctr
K
 is the 
actuator control loop stiffness (scalar), and 
act
K
, 
Foot
K
, 
Leg
K
 are 66 stiffness matrices for the 
mechanics of the actuator, foot and leg respectively. It should be stressed that, in contrast to 
other works, these matrices are assumed to be non-diagonal. This allows taking into account 
complicated coupling between rotational and translational deformations, while usual lump-based 
approach does not consider this phenomena [25]. Some examples of such compliance matrices 
will be given in subsequent sections. 
For analytical convenience, expressions (6) may be collected in a single matrix equation 
 
θ
,
1,...
i
i
i
n



τ
K
θ
 
(7) 
where 
0
18
(
,
)
i
i
i
T





τ

 is the aggregated vector of the virtual joint reactions, and 
(
,
,
,
)
ctr
act
Foot
Leg
diag K

θ
K
K
K
K
 is the aggregated spring stiffness matrix of size 1919. 
Similarly, one can define the aggregated vector of the passive joint reactions 
1
4
(
,
)
i
i
i
T
q
q
q


τ

 
but all its components must be equal to zero: 
A. Pashkevich, D. Chablat, P. Wenger / Stiffness Analysis of Overconstrained Parallel Manipulators 
 
9
 
,
1,...
i
q
i
n


τ
0
. 
(8) 
To find the static equations corresponding to the end-effector motion 
i
t , let us apply the 
principle of virtual work assuming that the joints are given small, arbitrary virtual displacements 
(
,
)
i
i


θ
q  in the equilibrium neighborhood. Then the virtual work of the external force 
if  
applied to the end-effector along the corresponding displacement 
i
i
i
i
q
i






t
J
θ
J
q  is equal 
to the sum (
)
(
)
T
i
T
i
i
i
i
q
i



f J
θ
f J
q . For the internal forces, the virtual work is 
θ
Ti
i


τ
θ  since 
the passive joints do not produce the force/torque reactions (the minus sign takes into account the 
adopted directions for the virtual spring forces/torques). Therefore, because in the static 
equilibrium the total virtual work is equal to zero for any virtual displacement, the equilibrium 
conditions may be written as  
 
Ti
i
i




J
f
τ ;        
Ti
q
i


J
f
0 . 
(9) 
This gives additional expressions describing the force/torque propagation from the joints to the 
end-effector.  
Hence, the complete kinetostatic model consists of five matrix equations (2), (7)…(9) where 
either 
if or 
i
t  are treated as known, and the remaining variables are considered as unknowns. 
Obviously, since separate kinematic chains posses some degrees-of-freedom, this system cannot 
be uniquely solved for given if . However, vice versa, for given end-effector displacement 
i
t , it 
is possible to compute both the corresponding external force 
if and the internal variables, 
i
θ , 
i

τ , 
i
q  (i.e. virtual spring reactions and displacements in passive joints, which may also provide 
useful information for the designer). 
Using the above equations, the desired Cartesian stiffness matrix may be derived in a 
straightforward way, by differentiating (9) and relevant eliminating the redundant variables. In 
general case, this produces the stiffness matrix that consists of two components: (i) the 
symmetrical part, which describes the manipulator intrinsic stiffness properties in the 
neighborhood of the “unloaded equilibrium” (i.e. reaction to the changes in the joint 
coordinates); and (ii) the skew-symmetrical part that takes into account changes in the 
manipulator Jacobian (due to the equilibrium shift caused by the externally applied force) [12, 
33, 34]. However, for the preliminary design purposes, the primary interest focuses on the 
symmetrical part that is evaluated below assuming that effect of the external forces is negligible. 
Since matrix 
θ
K is non-singular (it describes the stiffness of the virtual springs), the variable 
i
θ  can be expressed via 
if  using equations 
Ti
i
i




J
f
τ  and 
θ
i
i


τ
K
θ . This yields 
substitution 
1
θ
(
)
Ti
i
i





θ
K
J
f  allowing reducing the kinetostatic model to a system of two 
matrix equations  
 
1
θ
(
)
T
i
i
i
i
q
i
i






J K
J
f
J
q
t ;     
Ti
q
i


J
f
0  
(10) 
with unknowns 
if  and 
i
q . This system can be also rewritten in a matrix form 
 
θ
i
i
q
i
i
T
i
i
q























S
J
f
t
q
0
J
0
 
(11) 
A. Pashkevich, D. Chablat, P. Wenger / Stiffness Analysis of Overconstrained Parallel Manipulators 
 
10
where the sub-matrix 
1
θ
θ
T
i
i
i




S
J K
J  describes the spring compliance relative to the end-
effector, and the sub-matrix 
i
q
J  takes into account the passive joint influence on the end-effector 
motions. Therefore, for a separate kinematic chain, the desired stiffness matrix 
i
K  defining the 
motion-to-force mapping 
 
i
i
i


f
K
t , 
(12) 
can be computed by direct inversion of relevant 1010 matrix in the left-hand side of (11) and 
extracting from it the 66 sub-matrix with indices corresponding to 
θ
iS . It is also worth 
mentioning 
that 
computing 
θ
iS  
requires 
66 
inversions 
only, 
since 
1
1
1
1
1
diag(
,
,
,
)
ctr
act
Foot
Leg
K






K
K
K
K
 and  
 
1
1
1
1
T
T
T
T
i
i
i
i
i
i
i
i
i
ctr
ctr
ctr
act
act
act
Foot
Foot
Foot
Leg
Leg
Leg
K 
























S
J
J
J
K
J
J
K
J
J
K
J
, 
(13) 
where 
,...
i
i
ctr
Leg


J
J
 are the corresponding submatrices of the Jacobian 
i

J . 
Solvability of system (11) in general case, i.e. for any given 
i

J  and 
i
q
J , cannot be proved. 
Moreover, if the matrix 
i
q
J  is singular, the passive joint coordinates 
iq  cannot be found 
uniquely. From a physical point of view, it means that if the kinematic chain is located in a 
singular posture, then certain displacements 
i
t  can be generated by infinite combinations of the 
passive joints. But for the variable 
if  the corresponding solution is unique (since the matrix 
i

J  is 
obviously non-singular if at least one 6 dof spring is included in a serial kinematic chain). On the 
other hand, the singularity may produce an infinite number of stiffness matrices for the same 
spatial location of the end-effector and for different values 
iq  provided by the inverse 
kinematics. A special technique to tackle this case, based on the singular value decomposition, is 
presented in appendix A. 
After the stiffness matrices 
i
K  for all kinematic chains are computed, the stiffness of the 
entire manipulator can be found by simple addition 
 
1
n
m
i
i

K
K  
(14) 
This follows from the superposition principle, because the total external force corresponding to 
the end-effector displacement t  (the same for all kinematic chains) can be expressed as 
1
n
i
i

f
f  where 
i
i


f
K
t . 
It should be stressed that, for a separate kinematic chain, the stiffness matrix 
i
K is not 
invertible, since some motions of the end-effector do not produce the virtual spring reactions 
(because of passive joints influence). However, for the entire manipulator, the stiffness matrix 
m
K  is usually positive definite and invertible for all non-singular postures (with respect to 
iq ). 
For example, for the 3-dof translational manipulators presented in Fig.2, 
(
)
2
i
rank

K
 but 
3
1
(
)
6
i
i
rank



K
, which ensures the manipulator structure resistance to all possible end-
effector displacements. 
 
2.5. Comparison with other results 
The main advantage of the proposed methodology is its applicability to overconstrained 
mechanisms. To describe it in details, let us briefly review an alternative technique [41] that was 
A. Pashkevich, D. Chablat, P. Wenger / Stiffness Analysis of Overconstrained Parallel Manipulators 
 
11
applied to the 3-dof translational manipulator. This technique originates from the same principal 
equations but the solution strategy of the known method begins from straightforward elimination 
of the passive joint variables 
iq  using the differential kinematic equations (2) only. Obviously, 
the feasibility of this step depends on the solvability of the equivalent matrix system  
 
1
1
1
2
2
1
2
2
3
3
3
3
q
q
q





















































t
I
J
J
θ
q
I
J
J
θ
q
θ
J
I
J
q
 
(15) 
where t  and 
i
q  are treated as unknowns. In the non-constrained case (for the 3-PUU 
architecture, for instance) the matrix in the left-hand side of (15) is square, of size 1818. So, it 
can be inverted usually. However, for overconstrained manipulators, this matrix is non-square, 
and system (15) cannot be solved uniquely. For example, for manipulators with parallelogram-
type legs (Orthoglide, Delta, etc.) the matrix size is 1815. So, in [31] three additional (virtual) 
passive joints were introduced to solve the problem. But, obviously, such a modification changes 
the manipulator architecture and also its stiffness matrix, doubting validity of the corresponding 
model. 
Besides, the proposed method allows computing the stiffness matrix even for the singular 
manipulator postures and does not incorporate the least-square pseudo-inversions applied by 
other authors. This is achieved by using another solution strategy, which is applied to for each 
kinematic chain separately and considers the kinematic and static-equilibrium equations 
simultaneously. Formal theoretical proof of this feature is based on the singular-value 
decomposition and is presented in Appendix A, where system (11) is solved for the general case, 
independent of a relevant manipulator posture (singular or non-singular). In particular, for each 
kinematic chain, the derived analytical solution allows detecting the subspace of the dimension   
that defines the motions, which do not cause force/torque reactions of the virtual springs. The 
advantages of the developed method are presented in Sub-section 4.5, where the stiffness matrix 
is computed for the “flat” and “bar” singularities of the Orthoglide manipulator. These 
advantages are also confirmed by the numerical analysis the Orthoglide parallel manipulator 
presented below. 
Some additional conveniences are included in the modeling stage. In particular, the 
kinematic models of the chains may include several redundant springs that are totally 
compensated by relevant passive joints. However, there is no need to eliminate these springs 
from the model manually, since they do not increase the matrix sizes in system (11). This allows 
including in the model 6-dof virtual springs of general type derived directly from FEA-modeling, 
without any modifications. 
Another advantage of the proposed technique is that it can be generalized easily. Within this 
paper, it is applied to the stiffness modelling of n-dof manipulators with actuators located 
between the base and the foot. However, it can be easily modified to cover other actuator 
locations, which may be included in the foot or in the leg. A further generalization is related to 
the similarity of the kinematic chains. This assumption can be easily relaxed here as it influences 
on the Jacobian computing only. After the Jacobians are determined, the stiffness matrices for all 
chains may be computed in the same manner and then aggregated.  
A. Pashkevich, D. Chablat, P. Wenger / Stiffness Analysis of Overconstrained Parallel Manipulators 
 
12
3.  Evaluating model parameters 
 
The adopted stiffness model of each kinematic chain includes four compliant components, 
which are described by one 1-dof spring corresponding to the actuator control loop and three 6-
dof springs corresponding to the actuator transmission and the manipulator links (see Fig. 3). Let 
us present particular techniques for their evaluation. 
 
 
3.1. Actuator compliance 
The actuator compliance, described by the scalar parameter 
1
ctr
K  and 66 matrix 
1
act

K
, 
depends on both the servomechanism mechanics and the control algorithm. Since most modern 
actuators implement a digital PID control, the main contribution to the compliance is done by the 
mechanical transmissions. The latter are usually located outside the feedback-control loop and 
consist of screws, gears, shafts, belts, etc., whose flexibility is comparable with the flexibility of 
the manipulator links. Because of the complicated mechanical structure of the servomechanisms, 
these parameters are usually evaluated from static load experiments, by applying the linear 
regression to the experimental data. 
 
3.2. Link Compliance 
Following a general methodology, the compliance of the manipulator links is described by 
66 symmetrical positive definite matrices 
1
1
,
Leg
Foot


K
K
 corresponding to 6-dof springs with 
relevant coupling between translational and rotational deformations. This distinguishes our 
approach from other lumped modeling techniques, where the coupling is neglected and only a 
subset of deformations is taken into account (presented by several 1-dof springs). 
The simplest way to obtain these matrices is to approximate the link by a beam element for 
which the non-zero elements of the compliance matrix may be expressed analytically: 
11
L
k
EA

;  
3
22 3
z
L
k
EI

;  
3
33 3
y
L
k
EI

;  
44
L
k
GJ

;  
55
y
L
k
EI

;  
66
z
L
k
EI

;  
2
35
2
y
L
k
EI

;  
2
26
2
z
L
k
EI

 (16) 
Here L is the link length, A is its cross-section area, Iy, Iz, and J are the quadratic and polar 
moments of inertia of the cross-section, and E and G are the Young’s and Coulomb’s modules 
respectively.  
 
(a) Single-beam model 
(b) Multi-beam model 
(c) FEA-based model 
Link  
Single-beam 
approximation  
6-d.o.f. 
virtual spring  
 
Link  
Multi-beam
approximation 
6-d.o.f. 
virtual springs 
 
Fx 
Link  
Reference 
object 
Mz
 
Fig. 4. Evaluation of the stiffness matrix for the Orthoglide foot 
 
However, for certain link shape, the accuracy of the single-beam approximation can be 
insufficient. In this case, the link can be approximated by a serial chain of the beams, whose 
A. Pashkevich, D. Chablat, P. Wenger / Stiffness Analysis of Overconstrained Parallel Manipulators 
 
13
compliance is evaluated by applying the same method (i.e. considering the kinematic chain with 
6-dof virtual springs, but without passive joints). This leads to the resulting compliance matrix 
1
1
T
Link
b
b
b



K
J K J , where
b
J  and 
1
b

K
 incorporate the Jacobian and the compliance matrices for 
all virtual springs. Examples of the single- and multi-beam approximation of the manipulator 
foot (for Orthoglide robot) are shown in Fig. 4; corresponding numerical values are presented in 
Table 1 where they are compared with the FEA modeling results. 
 
 
TABLE 1.  
Comparison of the link stiffness models for the Orthoglide foot 
 
 
Compliance Matrix Elements 
Method 
k11 
k22 
k33 
k44 
k55 
k66 
 
mm/N 
10-4 
mm/N 
10-4 
mm/N 
10-4 
rad/N mm 
10-7 
rad/N mm 
10-7 
rad/N mm
10-7 
(a) Single-beam approximation 
3.45 
18.1 
3.45 
2.10 
0.91 
2.10 
(b) Four-beam approximation 
2.77 
17.9 
4.34 
2.11 
0.91 
1.95 
(c) FEA-based evaluation 
2.45 
15.9 
3.24 
2.07 
1.71 
2.06 
 
 
 
3.3. FEA-based evaluation of stiffness parameters 
 
For complex link geometries, when the multy-beam approximation is too rough, the most 
reliable results can be obtained from the FEA modeling. To apply this approach, the CAD model 
of each link should be extended by introducing an auxiliary 3D object (Fig. 4c), a “pseudo-rigid” 
body, which is used as a reference for the compliance evaluation. Besides, the link origin must 
be fixed relative to the global coordinate system. Then, sequentially and separately applying the 
forces 
,
,
x
y
z
F
F
F  and torques 
,
,
x
y
z
M
M
M  to the reference object, it is possible to evaluate 
corresponding linear and angular displacements, which allow computing the stiffness matrix 
columns. 
The main difficulty here is to obtain accurate displacement values by using proper FEA-
discretization (“mesh size”). Besides, to increase accuracy, the translational and rotational 
displacements must be evaluated using the redundant data set describing the reference body 
motion. For this reason, it is worth applying a dedicated SVD-based algorithm (Appendix B), 
which allows minimizing the sum of the residual squares. As follows from our study for the 
Orthoglide robot (Table 1), the single-beam approximation of the Orthoglide foot gives accuracy 
of about 50%, and the four-beam approximation improves it up to 30% only.  
It is worth mentioning that the high computational expenses of FEA is not a critical issue 
here, because the proposed technique involves only a single evaluation of the link stiffness (in 
contrast to the straightforward FEA-modeling for the entire manipulator, which requires 
complete re-computing for each manipulator posture). 
 
A. Pashkevich, D. Chablat, P. Wenger / Stiffness Analysis of Overconstrained Parallel Manipulators 
 
14
4. Application examples 
To demonstrate the efficiency of the proposed methodology, let us apply it to the 
comparative stiffness analysis of 3-dof translational mechanisms that employ the Orthoglide 
architecture [37, 38]. This problem was previously studied using other techniques [31, 40], but 
the results were essentially different from those obtained from both the FEA-modeling and from 
the physical experiments. Thus, this section presents several stiffness models for the Orthoglide 
(Fig. 5) and compares their accuracy with the FEA model of the entire manipulator. It is assumed 
that influence of the gravity is negligible, since relevant FEA showed very small deflection of 
the end-platform caused by the weight of the links (less than 0.005 mm). 
 
(a) U-joint based architecture 
(b) Parallelogram based architecture 
(c) Workspace and 
critical points Q1 and Q2 
x
z
y
B1
P
A1
C1
A2
C2
A3
C3
B3
 
B1
i1
P
x
z
y
j1
A1
C1
A2
B2
C2
A3
C3
B3
 
x
z
y
Q2
Q1
 
 
Fig. 5. Kinematics of two 3-dof translational mechanisms employing the Orthoglide architecture 
 
 
4.1. Manipulator geometry 
The Orthoglide is a Delta-type parallel manipulator dedicated to 3-axis rapid machining 
applications that was developed to meet the advantages of both serial and parallel kinematic 
architectures (regular homogeneous workspace with good dynamic performances and stiffness). 
This manipulator consists of three parallel PRPaR identical chains actuated by three mutually 
orthogonal linear drives, which are arranged to ensure almost isotropic workspace kinematic 
properties and to restrict the end-effector motions in translation, with the velocity transmission 
factors close to 1.0 similar to the conventional XYZ-machines. Moreover, to increase the 
manipulator stiffness, the kinematic chains impose redundant constraints on the mobile platform 
(because only two parallelograms would be sufficient to restrict the motion in translation). 
Hence, this is an over-constrained structure that cannot be evaluated using standard lump 
modeling techniques. 
 
P
             
 
 
Fig. 6. CAD model of and Orthoglide and its prototype 
A. Pashkevich, D. Chablat, P. Wenger / Stiffness Analysis of Overconstrained Parallel Manipulators 
 
15
This architecture was implemented in the Orthoglide prototype (Fig. 6), which was built in 
Institut de Recherche en Communications et Cybernetique de Nantes (IRCCyN) and satisfies the 
following design objectives: cubic Cartesian workspace of size 200200200 mm, Cartesian 
velocity and acceleration in the isotropic point 1.2 m/s and 14 m/s2; payload 4 kg; transmission 
factor range 0.5–2.0. The manipulator kinematics, including the direct and inverse 
transformations, is described in details in our previous paper [42]. Here we propose the 
manipulator stiffness model that, in contrast to previous works, does not ignore the over-
constraining feature. Also, we compare two alternative architectures, which are kinematically 
equivalent but differ in the stiffness capabilities. 
 
 
4.2. Stiffness of U-Joint Based Manipulator 
First, let us derive the stiffness model for the simplified Orthoglide mechanics, where the 
legs are comprised of equivalent limbs with U-joints at the ends (Fig 5a). Accordingly, to retain 
major compliance properties, the limb geometry corresponds to the parallelogram bars with 
doubled cross-section area. 
Let us assume that the world coordinate system is located at the end-effector reference point 
corresponding to the isotropic manipulator posture (when the legs are mutually perpendicular 
and parallel to relevant actuator axes). For this assumption, the geometrical models of separate 
kinematic chains can be described by the expression (1), where 
{ , , }
i
x y z

 and the product 
components are defined via the standard translational/rotational operators 
(.),
(.),
(.)
x
y
z
T
T
R

 as 
follows: 
 
1
0
0
0
1
0
0
;
0
0
1
0
0
0
0
1
x
base
L
r













T
   
0
0
1
0
1
0
0
;
0
1
0
0
0
0
0
1
y
base
L
r













T
   
0
1
0
0
0
0
1
0
;
1
0
0
0
0
0
1
z
base
L
r













T
 
(17) 
 
1
0
0
0
1
0
0 ;
0
0
1
0
0
0
0
1
x
tool
r












T
           
0
1
0
0
0
1
0 ;
1
0
0
0
0
0
0
1
y
tool
r












T
              
0
0
1
1
0
0
0 ;
0
1
0
0
0
0
0
1
z
base
r












T
 
(18) 
 
0
0
0
0
(
)
(
)
a
x
q
q





V
T
;      
Foot 
T
I ;       
( )
Leg
x L

T
T
 
(19) 
 
1
6
1
2
3
4
5
6
(
,
)
(
)
(
)
(
)
(
)
(
)
(
)
s
x
y
z
x
y
z














V
T
T
T
R
R
R

 
(20) 
 
1
1
2
1
2
(
,
)
(
)
(
)
u
z
y
q
q
q
q


V
R
R
;       
2
3
4
3
4
(
,
)
(
)
(
)
u
y
z
q
q
q
q


V
R
R
; 
(21) 
Here L, r are the manipulator geometrical parameters (the leg length and end-effector offset 
respectively), and the remaining variables are the same as in equation (1). Because the end-
effector of the rigid manipulator is restricted to the translational motions, the nominal values of 
the passive joint coordinates are subject to the specific constrains, 
2
3
0
q
q


 and 
1
4
0
q
q


, 
which are implicitly incorporated in the direct/inverse kinematics. However, the flexible model 
A. Pashkevich, D. Chablat, P. Wenger / Stiffness Analysis of Overconstrained Parallel Manipulators 
 
16
allows variations all passive joint coordinates around the nominal values, so the jacobians 
q
J  
must be computed for four variables 
1
4
,
q
q

. 
Using the link stiffness parameters obtained by the FEA-modeling (see Appendix C) and 
applying the proposed methodology, we computed the compliance matrices for three typical 
manipulator postures, the principal components of which are presented in Table 2. Below, they 
are compared with the compliance of the parallelogram-based manipulator. It should be noticed 
that the equivalent model of the UU-leg includes two parallelogram bars with corresponding 
stiffness matrix 
/2
Bar
k
. 
 
4.3. Stiffness of Parallelogram Based Manipulator 
Further, let us considerer the 3-PRPaR architecture where the manipulator legs are composed 
of the kinematic parallelograms (see Fig. 5b), which corresponds to the final design of the 
Orthoglide prototype. This obviously imposes some additional kinematic constrains compared to 
the 3-PUU case and should increase the stiffness, but quantitative comparison requires relevant 
modeling. 
Before evaluating the compliance of the entire manipulator, let us derive the stiffness matrix 
of the parallelogram. Using the adopted notations, the parallelogram equivalent model may be 
written as 
 
2
2
7
8
(
)
( )
(
)
(
,
,
)
Plg
y
x
y
s
q
L
q






T
R
T
R
V
 
(22) 
where, compared to the 3-PUU case, the third passive joint is eliminated (it is implicitly assumed 
that 
3
2
q
q

). On the other hand, the original parallelogram may be split into two serial 
kinematic chains (the “upper” and “lower” ones) that yields the same multi-serial-chain parallel 
architecture as considered above. Hence, the parallelogram compliance matrix may be derived 
using the same stiffness modeling technique proposed in this paper. 
Assuming that the main contribution to parallelogram compliance is caused by the bar links of 
the length L, the geometry and flexibility of these kinematic chains can be described by the 
expressions  
 
1
1
6
2
(
/2)
(
)
( )
(
,
) 
(
)
( /2)
up
up
up
up
up
z
y
x
s
y
z
d
q
q
L
q
q
d











T
T
R
T
V
R
T

 
(23) 
 
1
1
6
2
( /2)
(
)
( )
(
,
) 
(
)
(
/2)
dn
dn
dn
dn
dn
z
y
x
s
y
z
d
q
q
L
q
q
d











T
T
R
T
V
R
T

 
(24) 
where L, d are the parallelogram geometrical parameters), 
1
2
,
,
{
,
}
i
i
q
q
i
up dn



 are the 
variations of the passive joint coordinates, q  is the parallelogram state coordinate defining its 
“rigid” posture, 
(.)
s
V
 describes displacements in the virtual springs, and the sub/superscripts 
“up” and “dn” correspond to the upper and lower chain respectively (Fig. 7). 
 
(a) 
(b) 
 
x 
L 
q2 
z 
x
z 
6-d.o.f.  
spring 
6-d.o.f.  
spring 
d 
 
x
L 
z
x 
z 
5 d.o.f.  
spring 
q2 
 
A. Pashkevich, D. Chablat, P. Wenger / Stiffness Analysis of Overconstrained Parallel Manipulators 
 
17
Fig. 7. Lump stiffness model of the parallelogram (a) and its equivalent presentation (b). 
Computing Jacobians for the upper chain with respect to 
iq

 and 
i yields 
 
2
2
0
0
0
0
0
1
1
0
0
q
up
q
q
d
d
LS
LC























J
;      
0
0
0
2
0
1
0
0
2
2
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
q
q
q
q
up
q
q
q
q
q
q
d
C
S
dC
dS
S
C
C
S
S
C

























J
 
(25) 
where 
sin( )
q
S
q

 and 
sin( )
q
C
q

. For the lower part, the expressions are similar and differ in 
signs of d  only. Then, defining the bar-link stiffness in the general form as 
Bar
ij
K




K
 and 
performing relevant matrix transformations for both kinematic chains, the parallelogram stiffness 
matrix is presented analytically as 
 
11
22
26
2
2
2
22
2
22
44
2
2
11
2
2
2
2
22
22
26
66
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
( )
2
4
8
0
0
0
0
0
4
0
0
0
8
4
q
q
Plg
q
q
q
K
K
K
d C K
d S K
K
q
d C K
d S K
d S K
K
K



























K
 
(26) 
where it is assumed that the x-axis is directed along to the L-links (i.e., similar to the 3-PUU 
case). It is also worth mentioning that the stiffness parameters 
33
K  and 
55
K
(which depend on 
yI , 
see equations (16)) are completely compensated by the passive joints, and there is no 
translational stiffness in the z-direction. Moreover, the rotational stiffness around z-axis is 
defined by the parameter 
11
K (describing the bar compression/tension).  
More detailed analysis of the matrix 
(0)
Plg
K
 at the isotropic point using expressions (16) 
shows that most of the elements are doubled compared to the stiffness of a single beam. But the 
rotation about the z-axis (matrix elements 
55
K
, where the term 
/
y
L EI  is replaced by 
2
/2
d EA
L ) 
demonstrates essential increase of the stiffness. Numerically, it can be evaluated as the 
ratio
2
/(8
)
y
d A
I
, which for the rectangular cross-section of size b h

 is reduced to 
2
1.5 ( / )
d h
 
where the dimension h corresponds to the axis y. Besides, there is some stiffness increase in the 
x-axis rotation (element 
44
K
) but it is not so high. For instance, for the considered case study 
(Orthoglide prototype), the increase in 
55
K
 is about 25 times, while the element 
44
K
 increases 
roughly by 10%. However, for the entire manipulator, three parallelograms yield essential 
stiffness increase for all rotational axes. 
It worth mentioning that, for such description of the parallelogram stiffness, the kinematic 
chains of 3-PRPaR manipulator are described by the same equations as in the 3-PUU case, but 
the joint variable 
3
iq  is not treated as an independent one, since 
2
3
0
i
i
q
q


. The latter must be 
taken account while computing the jacobians 
i
q
J  which size is reduced to 6x3 and corresponds to 
three passive joints 
1
2
4
,
,
i
i
i
q q
q .  
A. Pashkevich, D. Chablat, P. Wenger / Stiffness Analysis of Overconstrained Parallel Manipulators 
 
18
To avoid the rank-deficiency of matrix (26) that is produced by the third raw and column, it 
is necessary to eliminate the third translational spring. This allows the matrix inversion while 
computing 
i

S  (see equations (13)) and reduces the parallelogram stiffness model to a 5-dof 
virtual spring that includes two translational and three rotational components with relevant 
coupling between them. Another way to avoid the above singularity is to introduce an arbitrary 
“fictitious” stiffness at the intersection of the zero raw and column of (26), since its influence 
will be totally compensated by the corresponding passive joint presented in the Jacobian 
i
q
J . The 
latter approach allows minimizing preliminary analytical derivation and replacing them by 
numerical computations. 
Using this model and applying the proposed technique, we computed the compliance 
matrices for three typical non-singular manipulator postures (Table 2) and also the stiffness 
matrixes for two singular postures (Table 3). As follows from the comparison with the U-joint 
case, within the dexterous workspace, the parallelograms allow multiplying the rotational 
stiffness roughly by 10. This justifies application of this architecture in the Orthoglide prototype 
design [37]. 
 
TABLE 2 
Comparison of translational and rotational compliance for 3-PUU and 3-PRPaR manipulators 
MANIPULATOR 
ARCHITECTURE 
Point Q0 
, ,
0.00
x y z
mm

 
 
 
Point Q1 
, ,
73.65
x y z
mm

 
 
 
 
Point Q2 
, ,
126.35
x y z
mm

 
 
 
tran
k
 
[mm/N] 
rot
k
 
[rad/Nmm] 
tran
k
 
[mm/N] 
rot
k
 
[rad/Nmm] 
 
tran
k
 
[mm/N] 
rot
k
 
[rad/Nmm] 
3-PUU manipulator 
2.7810-4 
20.910-7 
10.910-4 
24.110-7 
 
71.310-4 
25.810-7 
3-PRPaR manipulator 
2.7810-4
 
1.9410-7
 
9.8610-4
 
2.0610-7
 
 
21.210-4
 
2.6510-7
 
 
 
 
4.4. Accuracy of the proposed model 
To validate the developed stiffness modeling technique, the 3-PRPaR Orthoglide prototype 
was evaluated using several different lump models and the FEA-based model. The obtained 
results are presented in Table 4. As follows from them, accuracy of the previous methods [31] is 
about 25…30%. In contrast, the proposed model that uses 6-dof virtual springs with FEA-based 
evaluation of the link stiffness parameters, gives accuracy of about 5% for almost all workspace 
points. The only exceptions are the manipulator configurations that are close to the “flat 
singularity” (point Q2), when the error for the rotational stiffness rises up to 20%.  
This motivated development of the extended stiffness model for the parallelogram legs, 
which takes into account flexibility of the “axes” corresponding to the links d (see Fig. 7). 
However, while it yielded essential improvement in accuracy for the translational stiffness, the 
accuracy of the rotational components in the neighborhood of Q2 was about 15% only. The latter 
A. Pashkevich, D. Chablat, P. Wenger / Stiffness Analysis of Overconstrained Parallel Manipulators 
 
19
motivates further research that takes into account the joint stiffness (in addition to the links and 
actuators). However, in general, these results confirm advantages of the developed technique and 
justify its application for the pre-design stage. 
 
 
TABLE 3 
Stiffness matrices Ktran of 3-PUU and 3-PRPaR manipulators in singular configurations 
MANIPULATOR 
ARCHITECTURE 
Flat” singularity 
, ,
/
6
x y z
L

 
 
 
 
“Flat” singularity 
, ,
/
3
x y z
L

 
 
 
tran
K
,   [N/mm]  
 
tran
K
,   [N/mm] 
3-PUU manipulator 
3
1.48
0.74
0.74
0.74
1.48
0.74
10
0.74
0.74
1.48
tran

















K
 
(
)
2
tran
rank

K
 
 
3
1.78
1.78
1.78
1.78
1.78
1.78
10
1.78
1.78
1.78
tran












K
 
(
)
1
tran
rank

K
 
3-PRPaR manipulator 
3
1.54
0.77
0.77
0.77
1.54
0.77
10
0.77
0.77
1.54
tran

















K
 
(
)
2
tran
rank

K
 
 
3
4.65
4.65
4.65
4.65
4.65
4.65
10
4.65
4.65
4.65
tran












K
 
(
)
1
tran
rank

K
 
 
 
 
 
TABLE 4 
Comparison of compliance modelling results for 3-PRPaR Orthoglide 
STIFFNESS MODEL 
Point Q0 
, ,
0.00
x y z
mm

 
Point Q1 
, ,
73.65
x y z
mm

 
 
Point Q2 
, ,
126.35
x y z
mm

 
tran
k
 
[mm/N] 
rot
k
 
[rad/Nmm] 
tran
k
 
[mm/N] 
rot
k
 
[rad/Nmm] 
 
tran
k
 
[mm/N] 
rot
k
 
[rad/Nmm] 
Lump model of F.Majou et al. (2007) 
with additional passive joints 
3.6810
-4 
2.7710
-7 
13.810
-4 
2.7710
-7 
 
34.310
-4 
2.7810
-7 
Modified model of F.Majou et al. (2007) 
without additional passive joints 
3.6810
-4 
1.2610
-7 
12.510
-4 
1.2610
-7 
 
24.710
-4 
1.2610
-7 
Over-constrained lump model 
with 6-dof springs 
2.7810
-4
 
1.9410
-7
 
9.8610
-4
 
2.0610
-7
 
 
21.210
-4
 
2.6510
-7
 
Extended over-constrained model 
with 6-dof springs  
2.9310
-4 
2.0210
-7
 
10.210
-4
 
2.1510
-7
 
 
21.910
-4
 
2.7610
-7
 
FEA-based model 
3.0510
-4 
2.0510
-7 
10.910
-4 
2.1710
-7 
 
26.810
-4 
2.6710
-7 
A. Pashkevich, D. Chablat, P. Wenger / Stiffness Analysis of Overconstrained Parallel Manipulators 
 
20
5. Conclusions 
 
The paper proposes a new systematic method for computing the stiffness matrix of 
overconstrained parallel manipulators. It is based on a multidimensional lumped model of the 
flexible links, whose parameters are evaluated via the FEA-modeling and describe both the 
translational/rotational compliances and the coupling between them. In contrast to previous 
works, the method employs a new solution strategy of the kinetostatic equations, which 
considers simultaneously the kinematic and static relations for each separate kinematic chain and 
then aggregates the partial solutions in a total one. This allows computing the stiffness matrices 
for overconstrained mechanisms for any given manipulator posture, including singular 
configurations and their neighborhood. Another advantage is the computational simplicity that 
requires low-dimensional matrix inversion compared to other techniques. Besides, the method 
does not require manual elimination of the redundant spring corresponding to the passive joints, 
since this operation is inherently included in the numerical algorithm. Using the proposed 
methodology, we also derived the analytical 5-dof stiffness model of the parallelogram-based 
link. 
The efficiency and accuracy of the proposed method was demonstrated through application 
examples that compare the stiffness of two parallel manipulators of the Orthoglide family (with 
U-joint based and parallelogram based links). Relevant simulation results have confirmed 
essential advantages of the parallelogram based architecture and validated adopted design of the 
Orthoglide prototype. Accuracy of the proposed model was evaluated via comparison with FEA 
modeling. 
While applied to mechanisms with similar kinematic chains and actuators located between 
the base and foot, the method can be extended to other parallel architectures to cover different 
actuator locations and dissimilar chain geometry. So, future work will focus on the stiffness 
modeling of more complicated parallel mechanisms (such as the Verne machine) and also on the 
experimental verification of the stiffness models for the Orthoglide robot. Another prospective 
research direction is the stiffness analysis of heavy manipulators, for which the influence of 
gravity is essential and cannot be neglected. 
 
A. Pashkevich, D. Chablat, P. Wenger / Stiffness Analysis of Overconstrained Parallel Manipulators 
 
21
Acknowledgments 
This work has been partially funded by the European projects NEXT, acronyms for “Next 
Generation of Productions Systems”, Project no° IP 011815. 
 
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A. Pashkevich, D. Chablat, P. Wenger / Stiffness Analysis of Overconstrained Parallel Manipulators 
 
24
Appendices 
 
 
Appendix A. SVD-based computing of stiffness matrix for a kinematic chain with passive joints 
 
Let us consider the kinetostatic model of a separate kinematic chain 
q


S f
J
q
t ;   
T
q 
J
f
0  
which defines portion of the external force 
6
R

f
 and the variations of the passive joint 
coordinates 
m
R

q
 corresponding to the end-effector motion 
6
R

t
(for convenience, the 
chain-number indices i are omitted). Here m is the passive joint number, 
q
J  is the passive joint 
jacobian of size 6xm, and 

S  is the 6x6 positive-definite symmetric matrix of the virtual springs 
compliance relative to the end-effector (see Section 2 for details). 
To find the desired mapping 


f
K
t , let us apply the SVD decomposition to the jacobian 
q
J  that yields to the following factorisation 
T
q
q
q
q



J
U
Σ
V  
where 
q
U  and 
q
V  are the orthogonal matrices of the size 66 and mm respectively (i.e., 
T
q
q 
U U
I , 
T
q
q 
V V
I ), and 
q
Σ  is the 6m quasi-diagonal matrix with non-negative elements 
1,
m



 (singular values). Then, after substitution 
q
J  and left-multiplication of the first 
equation by 
T
q
U
 and the second one by 
T
q
V , the original system may be rewritten as 
(
)
(
)
T
T
T
T
q
q
q
q
q
q







U S U
U f
Σ
V
q
U
t ;    
(
)
T
T
q
q


Σ
U f
0 . 
The latter may be treated as the orthogonal linear transformations of the variables  
T
q




t
U
t  ;   
T
q

f
U f  ;      
T
q




q
V
q  
with respect to which the considered system is simplified down to:  
T
q
q
q








U S U f
Σ
q
t ;    
T
q



Σ
f
0 . 
Further, because the matrix 
q
Σ  is quasi-diagonal, the second equation may be re-written in a 
scalar from as  
0,
1,
k
kf
k
m





, 
which gives the trivial solutions for the first r components of the variable f  
0,
1,
kf
k
r

 
corresponding to 
0
k

 where 
1,
k
r
 and 
(
)
q
r
rank

J
 (usually, r
m

 within the dexterous 
workspace, and only for some singular configurations r
m

.). The remaining (n-r) components 
of f  corresponding to 
0
k

 can be found from the portion of the first matrix equation which 
in the scalar form is presented as 
A. Pashkevich, D. Chablat, P. Wenger / Stiffness Analysis of Overconstrained Parallel Manipulators 
 
25
6
1
;
1,
6
T
k
k
q
q
l
k
k
l r
f
t
k
r








u S u
 
where 
6,
1,...6
k
q
R
k


u
 are the orthogonal vector-columns comprising the matrix 
q
U . Hence, 
in the matrix form, the expression for the vector variable f  can be written as 
(6
)
1
(6
)
(
)
r r
r
r
T
d
d
r
r
q
q



















0
0
f
t
0
U
S U
 
where 
1
6
[
,
]
d
r
q
q
q


U
u
u

 is a “rank-deficient” part of the matrix 
q
U  obtained by its partitioning 
[
,
]
r
d
q
q
q

U
U
U
 into two submatrices of the size 6 r
 and 6 (6
)r


 respectively:  
Thus, after restoring the original variables f , q  and left-multiplication by 
q
U , the above 
relation is transformed into  
1
(
)
T
r
q
r
d
T
q
q
T
d
d
d
q
q
q
























0
0
U
f
U
U
t
0
U
S U
U
 
which after simplification yields the final expression for the stiffness of a kinematic chain 
1
(
)
T
T
d
d
d
d
q
q
q
q



K
U
U
S U
U
 
where the matrix K  is obviously conservative. 
In this expression, the matrix 
1
T






S
J K J  describes the spatial location and compliance of 
the virtual springs, and the matrix 
d
q
U  characterizes impact of the passive joints, which slacken 
the springs effect by accepting certain motions without the force/torque reactions. The rank of 
the obtained matrix 
(
)
6
rank
r


K
 depends on the number of passive joints m and the 
kinematic chain posture ( r
m

). Apparently, for the case without passive joints, the expression 
for 
1


k
K
 is reduced to the known one, i.e. 
1
T





k
J K J . 
 
 
Appendix B. CAD-based computing of the link compliance matrix 
 
Let us assume that the FEA-modelling provided six data sets describing the displacement of 
the reference object caused by successive applications of the forces 
,
,
x
y
z
F
F
F  and the torques 
,
,
x
y
z
M
M
M  along the axes of the virtual spring coordinate system. Each such data set may be 
formally described as {
,
=1,2,
}
k
k k
m
p
d

 where 
k
p  and 
k
d  are respectively the Cartesian 
position and the Cartesian displacement of the kth node in the link-base coordinate system. 
To evaluate the reference object translation (
,
,
)
x
y
z
p
p
p



 and rotation (
,
,
)
x
y
z






 
relative to the virtual spring centre 
0
p , let us fit the data by the model 
0
0
(
)
(
)
k
k
k




d
R p
p
t
p
p
, 
which includes, as the parameters, the translation vector t  and the orthogonal rotation matrix R  
of sizes 3x1 and 3x3 respectively. After defining 
0
k
k


g
p
p  and 
0
k
k
k



g
p
p
d , the model may 
be rewritten in the form  
A. Pashkevich, D. Chablat, P. Wenger / Stiffness Analysis of Overconstrained Parallel Manipulators 
 
26
;
1,
k
k
k
m




g
R g
t

, 
which is known in the matrix analysis as the “Procrustes problem” and admits the minimum least 
square solution [43] 
1
1
1
1
;
m
m
T
k
k
k
k
R
m
m










V U
t
g
R
g  
via the SVD factorization of the following 3x3 matrix 
1
m
T
T
k
k
g
g
g
k



g g
U
Σ
V  
where 
g
U  and 
g
V  are the orthogonal matrices of the size 33, and 
g
Σ  is the 3x3 diagonal 
matrix (positive definite, if m  3). For small displacements 
k
d , the rotation matrix may be re-
written in the differential from, which gives explicit expressions for the rotation angles: 
23
x
r



;   
13
y
r



;   
12
z
r



. 
The translational displacements are extracted from the vector t : 
1
xp
t


;   
2
y
p
t


;   
3
zp
t


. 
Then the obtained values 
,
,
x
z
p




 are scaled by dividing by the corresponding force/torque 
amplitude. And, after applying this algorithm to all six data sets (corresponding to 
,
,
x
y
z
F
F
M

), 
the desired compliance matrix of size 6x6 is constructed as  
(
)/
(
)/
(
)/
(
)/
(
)/
(
)/
(
)/
(
)/
(
)/
x
x
x
x
y
y
x
z
z
y
x
x
y
y
y
y
z
z
CAD
z
x
x
z
y
y
z
z
z
p F
F
p F
F
p M
M
p
F
F
p
F
F
p
M
M
F
F
F
F
M
M


























k







 
where 
(
)
x
y
p F

denotes the displacement 
xp

 caused by the applied force 
y
F

, etc. Finally, to 
compensate some small computational errors, the obtained matrix is symmetrized 
(
)/2
T
link
CAD
CAD


k
k
k
 
by averaging the non-diagonal symmetrical elements. 
 
A. Pashkevich, D. Chablat, P. Wenger / Stiffness Analysis of Overconstrained Parallel Manipulators 
 
27
 
Appendix C. Compliance parameters of the Orthoglide links 
 
For the Orthoglide manipulator, the actuator compliance was evaluated as 
5
10
/
ctr
k
mm N


 
while the links compliance matrices were computed via the FEA-based simulation using 
technique presented in Appendix 2, which yielded: 
 
 
4
4
6
4
4
6
3
6
5
6
7
5
7
6
6
7
2.45 10
2.73 10
0
0
0
5.48 10
2.73 10
3.24 10
0
0
0
7.04 10
0
0
1.59 10
9.90 10
1.27 10
0
0
0
9.90 10
2.07 10
0
0
0
0
1.27 10
0
2.06 10
0
5.48 10
7.04 10
0
0
0
1.71 10
Foot


























































k
 
 
 
5
2
4
2
4
6
4
6
4
6
4.50 10
0
0
0
0
0
0
8.01 10
0
0
0
3.98 10
0
0
3.64 10
0
1.71 10
0
0
0
0
3.76 10
0
0
0
0
1.71 10
0
1.09 10
0
0
3.98 10
0
0
0
2.65 10
Bar










































k
 
 
 
6
5
7
5
7
8
7
8
7
8
1.99 10
0
0
0
0
0
1.29 10
0
0
0
2.61 10
0
0
1.50 10
0
7.64 10
0
0
0
0
6.81 10
0
0
0
0
7.64 10
0
8.23 10
0
0
2.61 10
0
0
0
2.67 10
Axis










































k
; 
 
 
6
7
6
7
7
7
8
7
10
10
1.88 10
0
0
0
0
0
0
3.83 10
0
0
0
0
0
0
9.99 10
2.90 10
0.45 10
0
0
0
2.90 10
1.55 10
0
0
0
0
0.45 10
0
5.19 10
0
0
0
0
0
0
4.86 10
Act










































k
 
 
In these matrices, the units are [mm], [rad], [N] and [Nmm] for the length, angle, force and 
torque respectively. As following from these results, the most compliant manipulator 
components are the foots and the parallelogram bars, while the remaining elements have rigidity 
of 5 – 10 times higher. 
 
Stiffness Analysis of 
Overconstrained Parallel Manipulators 
Anatol Pashkevich1,2, Damien Chablat1*, Philippe Wenger1 
1Institut de Recherches en Communications et Cybernétique de Nantes,  
UMR CNRS 6597, 1 rue de la No, 44321 Nantes, France 
 
2Ecole des Mines de Nantes,    
4 rue Alfred-Kastler, Nantes 44307, France 
 
 
 
Abstract. The paper presents a new stiffness modeling method for overconstrained parallel manipulators 
with flexible links and compliant actuating joints. It is based on a multidimensional lumped-parameter model that 
replaces the link flexibility by localized 6-dof virtual springs that describe both translational/rotational compliance 
and the coupling between them. In contrast to other works, the method involves a FEA-based link stiffness 
evaluation and employs a new solution strategy of the kinetostatic equations for the unloaded manipulator 
configuration, which allows computing the stiffness matrix for the overconstrained architectures, including 
singular manipulator postures. The advantages of the developed technique are confirmed by application 
examples, which deal with comparative stiffness analysis of two translational parallel manipulators of 3-PUU 
and 3-PRPaR architectures. Accuracy of the proposed approach was evaluated for a case study, which focuses 
on stiffness analysis of Orthoglide parallel manipulator. 
Keywords: Parallel mechanisms, Stiffness modeling, Parallelogram-based linkage, Orthoglide robot 
 
 
1. Introduction 
 
Parallel manipulators have become more and more popular in industrial applications, 
including high-accuracy positioning and high-speed machining [1, 2]. This growing attention is 
inspired by their essential advantages over serial manipulators, which have already reached the 
dynamic performance limits (bounded by high masses of the machine components required to 
support sequential joints, links and actuators). In contrast, parallel manipulators are claimed to 
offer better accuracy, lower mass/inertia properties, and higher structural rigidity (i.e. stiffness-
to-mass ratio) [3]. These features are induced by their specific kinematic structure, which resists 
the error accumulation in kinematic chains and allows convenient actuators location close to the 
manipulator base. Besides, the links act in parallel against the external force/torque, eliminating 
the cantilever-type loading and increasing the manipulator stiffness [4]. The latter makes them 
attractive for innovative machine-tool architectures [5-7], but practical utilization of the potential 
benefits requires development of efficient stiffness analysis techniques, which satisfy the 
computational speed and accuracy requirements of relevant design procedures [8]. 
Generally, the stiffness analysis evaluates the effect of the applied external torques and 
forces on the compliant displacements of the end-effector. Numerically, this property is defined 
through the “stiffness matrix” K, which gives the relation between the translational/rotational 
displacement and the static forces/torques causing this transition. The inverse of K is usually 
called the “compliance matrix” and is denoted as k. As follows from mechanics, K is 66 semi-
definite non-negative matrix, where structure may be non-diagonal to represent the coupling 
between the translation and rotation [9]. Besides, this matrix may be not-symmetrical under the 
static load [10], but standard stiffness analysis focuses on the non-loaded structures. 
* Corresponding author: 
Tel: +33 240 37 69 48; Fax:  +33 240 37 69 30; 
E-mail: Damien.Chablat@irccyn.ec-nantes.fr 
A. Pashkevich, D. Chablat, P. Wenger / Stiffness Analysis of Overconstrained Parallel Manipulators 
 
2
Similar to other manipulator properties (kinematical, for instance), the stiffness essentially 
depends on the force/torque direction and on the manipulator configuration. Hence, to provide 
the designer with integrated performance criteria, various scalar indices are usually computed 
(such as the best/worst/average stiffness with respect to the rotation or translation). They are 
typically derived using the singular-value decomposition of K. However, there are still a number 
of open questions here regarding the significance of these indices for a particular manufacturing 
task. Besides, since the matrix K varies through the workspace, corresponding global 
benchmarks must be computed. In some cases, a relevant analysis produces the “stiffness maps”, 
which describe the end-effector compliance as a function of the manipulator configuration [11-
13].  
Several approaches exist for the computation of the stiffness matrix, which differ in the 
modeling assumptions and computational techniques. They are the Finite Element Analysis 
(FEA), the matrix structural analysis (MSA), and the virtual joint method (VJM) that is often 
called the lumped modeling.  
The FEA method is proved to be the most accurate and reliable, since the links/joints are 
modeled with its true dimension and shape [14]. Its accuracy is limited by the discretisation step 
only. However, because of high computational expenses required for the repeated re-meshing, 
this method is usually applied at the final design stage for the verification and component 
dimensioning. For example, in [15], a FEA model was used to evaluate the static rigidity and 
natural frequencies of the T3R1 parallel robot. Also, this method is widely used for validation of 
other stiffness analysis techniques [16-18] and for the comparative study [19].  
The MSA method is a common technique in mechanical engineering [20], it incorporates the 
main ideas of the FEA but operates with rather large flexible elements (beams, arcs, cables, etc.). 
This obviously yields reduction of the computational expenses and, in some cases, allows even 
obtaining an analytical stiffness matrix. For parallel manipulators, the relevant stiffness model is 
a combination of flexible beams and nodes, where each beam is defined by two nodes and 
described by 1212 stiffness matrix derived from the Euler-Bernoulli presentation. Then, these 
matrices are “assembled” in accordance with the superposition principle and the manipulator 
geometry, to produce the desired 66 matrix for the whole mechanism. Sometimes this approach 
is also referred to as the “distributed stiffness” modeling. One of the first examples of MSA 
application for the problem of interest is the stiffness analysis of a Stewart platform [21], which 
was performed under the assumption that the links are not subject to bending. This approach was 
also used in [22, 23] for other manipulators and/or other modeling assumptions. Some resent 
MSA-based results are obtained for the Delta-type mechanisms [24]. This method gives a 
reasonable trade-off between the accuracy and computational time, provided that link 
approximation by the beam elements is realistic. Because it involves rather high-dimensional 
matrix operations, it is not attractive for the parametric stiffness analysis and analytical 
modeling. 
Finally, the VJM method, which is also referred to as the “lumped modeling”, is based on 
the expansion of the traditional rigid model by adding virtual joints (localized springs), which 
describe the elastic deformations of the manipulator components (links, joints and actuators). 
A. Pashkevich, D. Chablat, P. Wenger / Stiffness Analysis of Overconstrained Parallel Manipulators 
 
3
This approach originates from the work of Gosselin [25], who evaluated parallel manipulator 
stiffness taking into account only the actuators compliance and by presenting them as one-
dimensional linear springs (the links were assumed to be rigid, and the passive joints to be 
perfect). Besides, the compliance in all actuated joints was assumed to be equal. The latter 
allowed reducing the stiffness analysis to the analysis of the condition number of the Jacobian 
matrix. Further development of VJM allowed taking into account the links flexibility, which 
were presented as rigid beams supplemented by linear and torsional springs [26]. There are a 
number of variations and simplifications of the VJM method, which differ in modeling 
assumptions and numerical techniques. In particular, it was applied to the CaPAMan, Orthoglide 
and H4 robots, specific variants of Stewart-Gough platform, manipulators with US/UPS legs and 
other kinematic machines [27-32]. Generally, the lumped modeling provides acceptable accuracy 
in short computational time, so it is widely used at the pre-design stage, especially for the 
analytical parametric analysis. However, it is very hypothetic and operates with simplified 
stiffness models that are composed of one-dimensional springs that do not take into account the 
coupling between the rotational and translational deflections. There are also other restrictions, 
which limit its applications to non-overconstrained mechanisms.  
This paper presents a new stiffness modelling method, which combines advantages of the 
above mentioned approaches. It is based on a multidimensional lumped-parameter model that 
replaces the link flexibility by localized 6-dof virtual springs that describe both the 
linear/rotational deflections and the coupling between them. The spring stiffness parameters are 
evaluated using FEA modelling to ensure higher accuracy. In addition, it employs a new solution 
strategy of the kinetostatic equations, which allows computing the stiffness matrix for the 
overconstrained architectures, including the singular manipulator postures. This gives almost the 
same accuracy as FEA but with essentially lower computational effort because it eliminates the 
model re-meshing through the workspace.  
Since the developed technique is targeted to the design optimization, it relies on the 
assumption that the manipulator is located in unloaded equilibrium configuration. This allows 
evaluating the symmetrical part of the general stiffness matrix while neglecting the skew-
symmetrical components, which describe effects caused by the external loading and relevant 
changes in Jacobian [12, 33, 34]. It is obvious that at the preliminary design stage, which focuses 
on the conceptual issues (such as comparison of alternative manipulator architectures, defining 
critical components in the kinematic chains, deciding on the stiffness specifications for the links, 
etc.), this assumption is practical and reasonable. Besides, for a particular case study presented 
below, validity of the assumption is justified numerically. 
The remainder of this paper is organized as follows. Next section introduces a general 
methodology of deriving/computing of the kinematic and stiffness model. Section 3 describes the 
manipulator compliant elements and proposes FEA-based technique for the evaluating their 
parameters. Section 4 includes application examples, which deal with comparative stiffness 
analysis of two translational parallel manipulators and demonstrate advantages of the proposed 
technique. Section 5 summarizes the main contributions of this work and defines future research 
directions. 
A. Pashkevich, D. Chablat, P. Wenger / Stiffness Analysis of Overconstrained Parallel Manipulators 
 
4
2. General methodology 
 
2.1. Manipulator Architecture 
Let us consider a general n-dof parallel manipulator, which consists of a mobile platform 
connected to a fixed base by n identical kinematics chains (Fig. 1). Each chain includes an 
actuated joint “Ac” (prismatic or rotational) followed by a “Foot” and a “Leg” with a number of 
passive joints “Ps” inside. Generally, certain geometrical conditions are assumed to be satisfied 
with respect to the passive joints to eliminate the undesired platform rotations and to achieve 
stability of desired motions. 
 
Base 
Ps
Ps
Ps
Ps
Ps
Ps
Ac
Mobile platform
Ps
Ps
Ps
Ps
Ps
Ps
Ac
L 
L 
F
F
 
Chain # 1
Chain # n
 
Fig. 1. Schematic diagram of a general n-dof parallel manipulator 
(Ac – actuated joint, Ps – passive joints, F – foot, L - Leg) 
Typical examples of such architectures are: 
(a) 3-PUU translational parallel kinematic machine (Fig 2a); where each leg consists of a rod 
ended by two U-joints (with parallel intermediate and exterior axes), and active joint is 
driven by linear actuator [35]; 
(b) Delta parallel robot (Fig 2b) that is based on the 3-RRPaR architecture with parallelogram-
type legs and rotational active joints [36]; 
(c) Orthoglide parallel robot (Fig 2c) that implements the 3-PRPaR architecture with 
parallelogram-type legs and translational active joints [37]. 
Here R, P, U and Pa denote the revolute, prismatic, universal and parallelogram joints, 
respectively. 
It should be noted that examples (b) and (c) illustrate specific cases of the overconstrained 
mechanisms, for which the standard stiffness analysis methods cannot be applied directly. In 
particular, for the Orthoglide mechanism the architectural particularities can be summarized as 
follows: (i) each kinematic chain prevents the platform from rotating about two orthogonal axes; 
(ii) any combination of two kinematic chains suppresses the three platform rotations, (iii) the 
whole set of three kinematic chains also suppresses the three platform rotations. Hence, the 
kinematical architecture of this manipulator includes excessive (redundant) constrains that are in 
certain agreement for the nominal parameter values. However, such a spatial overconstrained1 
arrangement ensures essential increase of the rigidity with respect to the external force. This 
motivates development of dedicated stiffness analysis techniques that are presented below. 
 
1 In the robotic literature, an alternative definition of the “overconstrained manipulator” is also used. Some authors [39] use this 
the term in wide-sense, as the opposite to the “redundant manipulator”, in order to distinguish mechanisms with any additional 
kinematic chains or specific location/orientation of the joints that cause reduction of the workspace dimension (down to planar, 
spherical, etc.). In this paper, the term “overconstrained” is referred to as the specific case, when the number of imposed 
constrains is greater that the resulting loss in the degrees-of-freedom [40]; it is also known as the “repeated” or “common” 
A. Pashkevich, D. Chablat, P. Wenger / Stiffness Analysis of Overconstrained Parallel Manipulators 
 
5
(a) 3-PUU translational PKM [35] 
(b) Delta parallel robot [36] 
(c) Orthoglide parallel robot [37] 
 
Base 
Mobile platform 
R 
R 
R 
R 
R 
R 
R 
R 
R 
R 
R 
R 
F 
F 
F 
L 
L 
L 
P 
P 
P 
 
 
Base 
Mobile platform
R
R
R
R
R
R
R
R
R
R
R
R
R
R
R
R
R
R
R
R
R
F 
F 
F 
L 
L 
L 
 
 
Base 
Mobile platform 
R 
R
R 
R 
R 
R 
P 
R 
R 
R 
R 
R 
R 
P 
R
R 
R
R
R
R 
P
F 
F 
F 
L 
L 
L 
 
 
 
 
 
Fig. 2. Typical 3 dof translational parallel mechanisms 
 
 
 
2.2. Basic Assumptions 
To evaluate the manipulator stiffness, let us apply a modification of the virtual joint method 
(VJM), which is based on the lump modeling approach [25, 26]. According to this approach, the 
original rigid model should be extended by adding virtual joints (localized springs), which 
describe elastic deformations of the links. Besides, virtual springs are included in the actuating 
joints to take into account stiffness of the mechanical transmissions and the control loop. To 
overcome difficulties with parallelogram stiffness modeling, let us first replace the manipulator 
legs (see Fig. 2) by rigid links with configuration-dependent stiffness. (Such transformation will 
be justified further while deriving the stiffness model for the parallelogram-based legs). 
 
 
 
Ac 
Rigid Foot 
Mobile platform 
(rigid) 
Base platform 
(rigid) 
U
Rigid Leg 
U
 6-d.o.f.
spring
 6-d.o.f.
spring
 6-d.o.f.
spring 
 
Fig. 3. Flexible model of a single kinematic chain (Ac – actuating joint, U – universal joint) 
 
This transforms the general architecture into the extended n-xUU case allowing treating all 
the considered manipulators in a similar manner. Under such assumptions, each kinematic chain 
of the manipulator can be described by a serial structure (Fig. 3), which includes sequentially: 
(a) a rigid link between the manipulator base and the ith actuating joint (part of the base 
platform) described by the constant homogenous transformation matrix
i
Base
T
; 
(b) a 1-dof actuating joint with supplementary virtual spring describing the control loop 
stiffness, which is defined by the homogenous matrix function 
0
0
(
)
i
i
a q 
V
 where 
0
iq  is the 
actuated coordinate and 
0
i is the virtual spring coordinate; 
A. Pashkevich, D. Chablat, P. Wenger / Stiffness Analysis of Overconstrained Parallel Manipulators 
 
6
(c) a 6-dof virtual spring describing the actuator mechanical stiffness, which is defined by the 
homogenous matrix function
1
6
(
,...
)
i
i
s 

V
 where 
1
2
3
{
,
,
}
i
i
i



, 
4
5
6
{
,
,
}
i
i
i



 are the virtual 
spring coordinates corresponding to the spring translational and rotational deflections; 
(d) a rigid “Foot” linking the actuating joint and the leg, which is described by the constant 
homogenous transformation matrix 
Foot
T
; 
(e) a 6-dof virtual spring describing the foot stiffness, which are defined by the homogenous 
matrix function 
7
12
(
,
)
i
i
s 

V

, where 
7
8
9
{
,
,
}
i
i
i



 and 
10
11
12
{
,
,
}
i
i
i



 are the spring 
translational/rotational deflections; 
(f) a 2-dof passive U-joint at the beginning of the leg allowing two independent rotations with 
angles 
1
2
{ ,
}
i
i
q q
, which is described by the homogenous matrix function 
1
1
2
(
,
)
i
i
u q q
V
; 
(g) a rigid “Leg” linking the foot to the movable platform, which is described by the constant 
homogenous matrix transformation 
Leg
T
; 
(h) a 6-dof virtual spring describing the leg stiffness, which are defined by the homogenous 
matrix function 
13
18
(
,
)
i
i
s 

V

, where 
13
14
15
{
,
,
}
i
i
i



 and 
16
17
18
{
,
,
}
i
i
i



 correspond to the 
spring translations and rotations respectively; 
(i) a 2-dof passive U-joint at the end of the leg allowing two independent rotations with angles 
3
4
{
,
}
i
i
q
q
, which is described by the homogenous matrix function 
2
3
4
(
,
)
i
i
u
q
q
V
; 
(j) a rigid link from the manipulator leg the end-effector (part of the movable platform) 
described by the constant homogenous matrix transformation 
i
Tool
T
.  
The corresponding mathematical expression defining the end-effector location subject to 
variations of all above defined coordinates of a single kinematic chain i may be written as 
follows 
0
0
1
6
7
12
1
1
2
13
18
2
3
4
(
)
(
,...
)
(
,
)
(
,
)
(
,
)
(
,
)
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
Base
a
s
Foot
s
u
Leg
s
u
Tool
q
q q
q
q

















T
T
V
V
T
V
V
T
V
V
T


 (1)  
where the matrix function 
(.)
a
V
 is either an elementary rotation or translation, the matrix 
functions 
1(.)
u
V
 and 
2(.)
u
V
 are compositions of two successive rotations, the spring matrix 
(.)
s
V
 is composed of six elementary transformations, and 
1,...
i
n

. In the rigid case, the virtual 
joint coordinates 
1
18
,
i
i



 are equal to zero, while the remaining ones (both active 
0
iq  and 
passive 
1
4
,
i
i
q
q

) are obtained through the inverse kinematics, ensuring that all n matrices 
,
1,...
i i
n

T
 are equal to the prescribed one, that characterizes the desired spatial location of the 
moving platform (kinematic loop-closure equations). Particular expressions for all components 
of the product (1) may be easily derived using standard techniques for homogenous 
transformation matrices. 
It should be noted that the kinematic model (1) includes 24 variables (1 for active joint, 4 for 
passive joints, and 19 for virtual springs). However, some of the virtual springs are redundant, 
since they are compensated by corresponding passive joints (with aligning axes) or by 
combination of passive joints. For computational convenience, nevertheless, it is not reasonable 
to detect and analytically eliminate redundant variables at this step, because the technique 
developed below allows easy and efficient computational elimination (without increasing the 
size of the required matrix inverse, which is equal to 66 independent of the virtual joint 
number). 
A. Pashkevich, D. Chablat, P. Wenger / Stiffness Analysis of Overconstrained Parallel Manipulators 
 
7
2.3. Differential kinematic model 
To evaluate the manipulator ability to respond to external forces and torques, let us first 
derive the differential kinematic equation describing relations between the end-effector location 
and small variations in the joint variables. For each ith kinematic chain, this equation can be 
generalized as follows 
 
,
1,...
i
i
i
i
q
i
i
n







t
J
θ
J
q
, 
(2) 
where vector δ
(δ
, δ
, δ
, δ
, δ
, δ
)T
i
xi
yi
zi
xi
yi
zi
p
p
p




t
 describes the end-effector translation 
δ
(δ
, δ
, δ
)T
i
xi
yi
zi
p
p
p

p
 and rotation δ
(δ
, δ
, δ
)T
i
xi
yi
zi





with respect to the Cartesian axes; 
vector 
0
18
(
,
)
i
i
T
i



θ

 collects all virtual joint coordinates, the vector 
1
4
(
,
)
i
i
T
i
q
q



q

 
includes all passive joint coordinates, symbol ' ' stands for variation with respect to the rigid 
case values, and 
i

J , 
i
q
J  are matrices of sizes 6x19 and 6x4 correspondingly. It should be noted 
that the derivative for the actuated coordinate 
0
iq is not included in 
q
J  but it is represented in the 
first column of 


J  through the variable 
0
i. 
The desired matrices 
i

J , 
i
q
J , which are the only parameters of the differential model (2) 
may be computed from (1) analytically using some software support tools such as Maple, 
MathCAD or Mathematica. However, a straightforward differentiation usually yields very 
awkward expressions that are not convenient for further computations. On the other hand, the 
fractionized structure of (1), where all variables are separated, allows applying an efficient semi-
analytical method. To present this technique, let us assume that for the particular virtual joint 
variable 
i
j
 the model (1) is rewritten as 
 
(
)
L
i
R
i
ij
j
j
ij





T
H
V
H , 
(3) 
where the first and the third multipliers are the constant homogenous matrices, and the second 
multiplier is the elementary translation or rotation. Then the partial derivative of the homogenous 
matrix 
iT  with respect to 
i
j
 at the point 
0
i
j

 may be computed from a similar product where 
the internal term is replaced by the matrix 
(.)
j

V
 that admits a very simple analytical 
presentation. In particular, for the elementary translations and rotations about the X-axis these 
derivatives are: 
 
0 0 0 1
0 0 0 0
0 0 0 0
0 0 0 0
x
Tran











V
;   
0
0 0 0
0
0 1 0
0
1 0 0
0
0 0 0
x
Rot











V
. 
(4) 
Furthermore, since the derivative of the homogenous matrix 
(
)
L
i
R
i
ij
j
j
ij







T
H
V
H  may be 
presented as 
 
0
0
0
0
0
0
0
iz
iy
ix
iz
ix
iy
i
iy
ix
iz
p
p
p































T
, 
(5) 
A. Pashkevich, D. Chablat, P. Wenger / Stiffness Analysis of Overconstrained Parallel Manipulators 
 
8
So, the desired jth column of the jacobian 
i

J  can be easily extracted from 
i
T  (using the matrix 
elements 
14
T, 
24
T, 
34
T, 
23
T, 
31
T, 
12
T). 
The jacobians 
q
J  can be computed in a similar manner, but the derivatives are evaluated in 
the neighborhood of the “nominal” values of the passive joint coordinates 
i
jnom
q
 corresponding to 
the rigid case (these values are obtained from the inverse kinematics). However, a simple 
transformation 
i
i
i
j
j
j
nom
q
q
q



 
and 
a 
corresponding 
factoring 
of 
the 
function 
(
)
(
)
(
)
i
i
i
q j
j
q j
j
q j
j
nom
q
q
q



V
V
V
 allow applying the above approach. It is also worth mentioning 
that this technique may be also applied in analytical computations, allowing one to avoid bulky 
transformations required for the straightforward differentiating. 
 
 
2.4. Kinetostatic and Stiffness Models 
For the manipulator kinetostatic model that describes the force-and-motion relation, it is 
necessary to introduce additional equations that define the virtual joint reactions to the 
corresponding spring deformations. In accordance with the adopted stiffness model, the 
following virtual springs are included in each kinematic chain: 
 1-dof virtual spring describing the actuator control loop compliance; 
 6-dof virtual spring describing the actuator mechanics compliance; 
 6-dof virtual spring describing mechanical compliance of the foot;  
 6-dof virtual spring describing mechanical compliance of the leg.  
Assuming that the spring deformations are small enough, the required relations may be 
expressed by linear equations 
0
i
i
ctr
K











;       
1
6
i
i
act
i
i



























K


;         
7
12
i
i
Foot
i
i



























K


;         
13
18
i
i
Leg
i
i



























K


, (6) 
where 
i
j

 is the generalized force for the jth virtual joint of the ith kinematic chain, 
ctr
K
 is the 
actuator control loop stiffness (scalar), and 
act
K
, 
Foot
K
, 
Leg
K
 are 66 stiffness matrices for the 
mechanics of the actuator, foot and leg respectively. It should be stressed that, in contrast to 
other works, these matrices are assumed to be non-diagonal. This allows taking into account 
complicated coupling between rotational and translational deformations, while usual lump-based 
approach does not consider this phenomena [25]. Some examples of such compliance matrices 
will be given in subsequent sections. 
For analytical convenience, expressions (6) may be collected in a single matrix equation 
 
θ
,
1,...
i
i
i
n



τ
K
θ
 
(7) 
where 
0
18
(
,
)
i
i
i
T





τ

 is the aggregated vector of the virtual joint reactions, and 
(
,
,
,
)
ctr
act
Foot
Leg
diag K

θ
K
K
K
K
 is the aggregated spring stiffness matrix of size 1919. 
Similarly, one can define the aggregated vector of the passive joint reactions 
1
4
(
,
)
i
i
i
T
q
q
q


τ

 
but all its components must be equal to zero: 
A. Pashkevich, D. Chablat, P. Wenger / Stiffness Analysis of Overconstrained Parallel Manipulators 
 
9
 
,
1,...
i
q
i
n


τ
0
. 
(8) 
To find the static equations corresponding to the end-effector motion 
i
t , let us apply the 
principle of virtual work assuming that the joints are given small, arbitrary virtual displacements 
(
,
)
i
i


θ
q  in the equilibrium neighborhood. Then the virtual work of the external force 
if  
applied to the end-effector along the corresponding displacement 
i
i
i
i
q
i






t
J
θ
J
q  is equal 
to the sum (
)
(
)
T
i
T
i
i
i
i
q
i



f J
θ
f J
q . For the internal forces, the virtual work is 
θ
Ti
i


τ
θ  since 
the passive joints do not produce the force/torque reactions (the minus sign takes into account the 
adopted directions for the virtual spring forces/torques). Therefore, because in the static 
equilibrium the total virtual work is equal to zero for any virtual displacement, the equilibrium 
conditions may be written as  
 
Ti
i
i




J
f
τ ;        
Ti
q
i


J
f
0 . 
(9) 
This gives additional expressions describing the force/torque propagation from the joints to the 
end-effector.  
Hence, the complete kinetostatic model consists of five matrix equations (2), (7)…(9) where 
either 
if or 
i
t  are treated as known, and the remaining variables are considered as unknowns. 
Obviously, since separate kinematic chains posses some degrees-of-freedom, this system cannot 
be uniquely solved for given if . However, vice versa, for given end-effector displacement 
i
t , it 
is possible to compute both the corresponding external force 
if and the internal variables, 
i
θ , 
i

τ , 
i
q  (i.e. virtual spring reactions and displacements in passive joints, which may also provide 
useful information for the designer). 
Using the above equations, the desired Cartesian stiffness matrix may be derived in a 
straightforward way, by differentiating (9) and relevant eliminating the redundant variables. In 
general case, this produces the stiffness matrix that consists of two components: (i) the 
symmetrical part, which describes the manipulator intrinsic stiffness properties in the 
neighborhood of the “unloaded equilibrium” (i.e. reaction to the changes in the joint 
coordinates); and (ii) the skew-symmetrical part that takes into account changes in the 
manipulator Jacobian (due to the equilibrium shift caused by the externally applied force) [12, 
33, 34]. However, for the preliminary design purposes, the primary interest focuses on the 
symmetrical part that is evaluated below assuming that effect of the external forces is negligible. 
Since matrix 
θ
K is non-singular (it describes the stiffness of the virtual springs), the variable 
i
θ  can be expressed via 
if  using equations 
Ti
i
i




J
f
τ  and 
θ
i
i


τ
K
θ . This yields 
substitution 
1
θ
(
)
Ti
i
i





θ
K
J
f  allowing reducing the kinetostatic model to a system of two 
matrix equations  
 
1
θ
(
)
T
i
i
i
i
q
i
i






J K
J
f
J
q
t ;     
Ti
q
i


J
f
0  
(10) 
with unknowns 
if  and 
i
q . This system can be also rewritten in a matrix form 
 
θ
i
i
q
i
i
T
i
i
q























S
J
f
t
q
0
J
0
 
(11) 
A. Pashkevich, D. Chablat, P. Wenger / Stiffness Analysis of Overconstrained Parallel Manipulators 
 
10
where the sub-matrix 
1
θ
θ
T
i
i
i




S
J K
J  describes the spring compliance relative to the end-
effector, and the sub-matrix 
i
q
J  takes into account the passive joint influence on the end-effector 
motions. Therefore, for a separate kinematic chain, the desired stiffness matrix 
i
K  defining the 
motion-to-force mapping 
 
i
i
i


f
K
t , 
(12) 
can be computed by direct inversion of relevant 1010 matrix in the left-hand side of (11) and 
extracting from it the 66 sub-matrix with indices corresponding to 
θ
iS . It is also worth 
mentioning 
that 
computing 
θ
iS  
requires 
66 
inversions 
only, 
since 
1
1
1
1
1
diag(
,
,
,
)
ctr
act
Foot
Leg
K






K
K
K
K
 and  
 
1
1
1
1
T
T
T
T
i
i
i
i
i
i
i
i
i
ctr
ctr
ctr
act
act
act
Foot
Foot
Foot
Leg
Leg
Leg
K 
























S
J
J
J
K
J
J
K
J
J
K
J
, 
(13) 
where 
,...
i
i
ctr
Leg


J
J
 are the corresponding submatrices of the Jacobian 
i

J . 
Solvability of system (11) in general case, i.e. for any given 
i

J  and 
i
q
J , cannot be proved. 
Moreover, if the matrix 
i
q
J  is singular, the passive joint coordinates 
iq  cannot be found 
uniquely. From a physical point of view, it means that if the kinematic chain is located in a 
singular posture, then certain displacements 
i
t  can be generated by infinite combinations of the 
passive joints. But for the variable 
if  the corresponding solution is unique (since the matrix 
i

J  is 
obviously non-singular if at least one 6 dof spring is included in a serial kinematic chain). On the 
other hand, the singularity may produce an infinite number of stiffness matrices for the same 
spatial location of the end-effector and for different values 
iq  provided by the inverse 
kinematics. A special technique to tackle this case, based on the singular value decomposition, is 
presented in appendix A. 
After the stiffness matrices 
i
K  for all kinematic chains are computed, the stiffness of the 
entire manipulator can be found by simple addition 
 
1
n
m
i
i

K
K  
(14) 
This follows from the superposition principle, because the total external force corresponding to 
the end-effector displacement t  (the same for all kinematic chains) can be expressed as 
1
n
i
i

f
f  where 
i
i


f
K
t . 
It should be stressed that, for a separate kinematic chain, the stiffness matrix 
i
K is not 
invertible, since some motions of the end-effector do not produce the virtual spring reactions 
(because of passive joints influence). However, for the entire manipulator, the stiffness matrix 
m
K  is usually positive definite and invertible for all non-singular postures (with respect to 
iq ). 
For example, for the 3-dof translational manipulators presented in Fig.2, 
(
)
2
i
rank

K
 but 
3
1
(
)
6
i
i
rank



K
, which ensures the manipulator structure resistance to all possible end-
effector displacements. 
 
2.5. Comparison with other results 
The main advantage of the proposed methodology is its applicability to overconstrained 
mechanisms. To describe it in details, let us briefly review an alternative technique [41] that was 
A. Pashkevich, D. Chablat, P. Wenger / Stiffness Analysis of Overconstrained Parallel Manipulators 
 
11
applied to the 3-dof translational manipulator. This technique originates from the same principal 
equations but the solution strategy of the known method begins from straightforward elimination 
of the passive joint variables 
iq  using the differential kinematic equations (2) only. Obviously, 
the feasibility of this step depends on the solvability of the equivalent matrix system  
 
1
1
1
2
2
1
2
2
3
3
3
3
q
q
q





















































t
I
J
J
θ
q
I
J
J
θ
q
θ
J
I
J
q
 
(15) 
where t  and 
i
q  are treated as unknowns. In the non-constrained case (for the 3-PUU 
architecture, for instance) the matrix in the left-hand side of (15) is square, of size 1818. So, it 
can be inverted usually. However, for overconstrained manipulators, this matrix is non-square, 
and system (15) cannot be solved uniquely. For example, for manipulators with parallelogram-
type legs (Orthoglide, Delta, etc.) the matrix size is 1815. So, in [31] three additional (virtual) 
passive joints were introduced to solve the problem. But, obviously, such a modification changes 
the manipulator architecture and also its stiffness matrix, doubting validity of the corresponding 
model. 
Besides, the proposed method allows computing the stiffness matrix even for the singular 
manipulator postures and does not incorporate the least-square pseudo-inversions applied by 
other authors. This is achieved by using another solution strategy, which is applied to for each 
kinematic chain separately and considers the kinematic and static-equilibrium equations 
simultaneously. Formal theoretical proof of this feature is based on the singular-value 
decomposition and is presented in Appendix A, where system (11) is solved for the general case, 
independent of a relevant manipulator posture (singular or non-singular). In particular, for each 
kinematic chain, the derived analytical solution allows detecting the subspace of the dimension   
that defines the motions, which do not cause force/torque reactions of the virtual springs. The 
advantages of the developed method are presented in Sub-section 4.5, where the stiffness matrix 
is computed for the “flat” and “bar” singularities of the Orthoglide manipulator. These 
advantages are also confirmed by the numerical analysis the Orthoglide parallel manipulator 
presented below. 
Some additional conveniences are included in the modeling stage. In particular, the 
kinematic models of the chains may include several redundant springs that are totally 
compensated by relevant passive joints. However, there is no need to eliminate these springs 
from the model manually, since they do not increase the matrix sizes in system (11). This allows 
including in the model 6-dof virtual springs of general type derived directly from FEA-modeling, 
without any modifications. 
Another advantage of the proposed technique is that it can be generalized easily. Within this 
paper, it is applied to the stiffness modelling of n-dof manipulators with actuators located 
between the base and the foot. However, it can be easily modified to cover other actuator 
locations, which may be included in the foot or in the leg. A further generalization is related to 
the similarity of the kinematic chains. This assumption can be easily relaxed here as it influences 
on the Jacobian computing only. After the Jacobians are determined, the stiffness matrices for all 
chains may be computed in the same manner and then aggregated.  
A. Pashkevich, D. Chablat, P. Wenger / Stiffness Analysis of Overconstrained Parallel Manipulators 
 
12
3.  Evaluating model parameters 
 
The adopted stiffness model of each kinematic chain includes four compliant components, 
which are described by one 1-dof spring corresponding to the actuator control loop and three 6-
dof springs corresponding to the actuator transmission and the manipulator links (see Fig. 3). Let 
us present particular techniques for their evaluation. 
 
 
3.1. Actuator compliance 
The actuator compliance, described by the scalar parameter 
1
ctr
K  and 66 matrix 
1
act

K
, 
depends on both the servomechanism mechanics and the control algorithm. Since most modern 
actuators implement a digital PID control, the main contribution to the compliance is done by the 
mechanical transmissions. The latter are usually located outside the feedback-control loop and 
consist of screws, gears, shafts, belts, etc., whose flexibility is comparable with the flexibility of 
the manipulator links. Because of the complicated mechanical structure of the servomechanisms, 
these parameters are usually evaluated from static load experiments, by applying the linear 
regression to the experimental data. 
 
3.2. Link Compliance 
Following a general methodology, the compliance of the manipulator links is described by 
66 symmetrical positive definite matrices 
1
1
,
Leg
Foot


K
K
 corresponding to 6-dof springs with 
relevant coupling between translational and rotational deformations. This distinguishes our 
approach from other lumped modeling techniques, where the coupling is neglected and only a 
subset of deformations is taken into account (presented by several 1-dof springs). 
The simplest way to obtain these matrices is to approximate the link by a beam element for 
which the non-zero elements of the compliance matrix may be expressed analytically: 
11
L
k
EA

;  
3
22 3
z
L
k
EI

;  
3
33 3
y
L
k
EI

;  
44
L
k
GJ

;  
55
y
L
k
EI

;  
66
z
L
k
EI

;  
2
35
2
y
L
k
EI

;  
2
26
2
z
L
k
EI

 (16) 
Here L is the link length, A is its cross-section area, Iy, Iz, and J are the quadratic and polar 
moments of inertia of the cross-section, and E and G are the Young’s and Coulomb’s modules 
respectively.  
 
(a) Single-beam model 
(b) Multi-beam model 
(c) FEA-based model 
Link  
Single-beam 
approximation  
6-d.o.f. 
virtual spring  
 
Link  
Multi-beam
approximation 
6-d.o.f. 
virtual springs 
 
Fx 
Link  
Reference 
object 
Mz
 
Fig. 4. Evaluation of the stiffness matrix for the Orthoglide foot 
 
However, for certain link shape, the accuracy of the single-beam approximation can be 
insufficient. In this case, the link can be approximated by a serial chain of the beams, whose 
A. Pashkevich, D. Chablat, P. Wenger / Stiffness Analysis of Overconstrained Parallel Manipulators 
 
13
compliance is evaluated by applying the same method (i.e. considering the kinematic chain with 
6-dof virtual springs, but without passive joints). This leads to the resulting compliance matrix 
1
1
T
Link
b
b
b



K
J K J , where
b
J  and 
1
b

K
 incorporate the Jacobian and the compliance matrices for 
all virtual springs. Examples of the single- and multi-beam approximation of the manipulator 
foot (for Orthoglide robot) are shown in Fig. 4; corresponding numerical values are presented in 
Table 1 where they are compared with the FEA modeling results. 
 
 
TABLE 1.  
Comparison of the link stiffness models for the Orthoglide foot 
 
 
Compliance Matrix Elements 
Method 
k11 
k22 
k33 
k44 
k55 
k66 
 
mm/N 
10-4 
mm/N 
10-4 
mm/N 
10-4 
rad/N mm 
10-7 
rad/N mm 
10-7 
rad/N mm
10-7 
(a) Single-beam approximation 
3.45 
18.1 
3.45 
2.10 
0.91 
2.10 
(b) Four-beam approximation 
2.77 
17.9 
4.34 
2.11 
0.91 
1.95 
(c) FEA-based evaluation 
2.45 
15.9 
3.24 
2.07 
1.71 
2.06 
 
 
 
3.3. FEA-based evaluation of stiffness parameters 
 
For complex link geometries, when the multy-beam approximation is too rough, the most 
reliable results can be obtained from the FEA modeling. To apply this approach, the CAD model 
of each link should be extended by introducing an auxiliary 3D object (Fig. 4c), a “pseudo-rigid” 
body, which is used as a reference for the compliance evaluation. Besides, the link origin must 
be fixed relative to the global coordinate system. Then, sequentially and separately applying the 
forces 
,
,
x
y
z
F
F
F  and torques 
,
,
x
y
z
M
M
M  to the reference object, it is possible to evaluate 
corresponding linear and angular displacements, which allow computing the stiffness matrix 
columns. 
The main difficulty here is to obtain accurate displacement values by using proper FEA-
discretization (“mesh size”). Besides, to increase accuracy, the translational and rotational 
displacements must be evaluated using the redundant data set describing the reference body 
motion. For this reason, it is worth applying a dedicated SVD-based algorithm (Appendix B), 
which allows minimizing the sum of the residual squares. As follows from our study for the 
Orthoglide robot (Table 1), the single-beam approximation of the Orthoglide foot gives accuracy 
of about 50%, and the four-beam approximation improves it up to 30% only.  
It is worth mentioning that the high computational expenses of FEA is not a critical issue 
here, because the proposed technique involves only a single evaluation of the link stiffness (in 
contrast to the straightforward FEA-modeling for the entire manipulator, which requires 
complete re-computing for each manipulator posture). 
 
A. Pashkevich, D. Chablat, P. Wenger / Stiffness Analysis of Overconstrained Parallel Manipulators 
 
14
4. Application examples 
To demonstrate the efficiency of the proposed methodology, let us apply it to the 
comparative stiffness analysis of 3-dof translational mechanisms that employ the Orthoglide 
architecture [37, 38]. This problem was previously studied using other techniques [31, 40], but 
the results were essentially different from those obtained from both the FEA-modeling and from 
the physical experiments. Thus, this section presents several stiffness models for the Orthoglide 
(Fig. 5) and compares their accuracy with the FEA model of the entire manipulator. It is assumed 
that influence of the gravity is negligible, since relevant FEA showed very small deflection of 
the end-platform caused by the weight of the links (less than 0.005 mm). 
 
(a) U-joint based architecture 
(b) Parallelogram based architecture 
(c) Workspace and 
critical points Q1 and Q2 
x
z
y
B1
P
A1
C1
A2
C2
A3
C3
B3
 
B1
i1
P
x
z
y
j1
A1
C1
A2
B2
C2
A3
C3
B3
 
x
z
y
Q2
Q1
 
 
Fig. 5. Kinematics of two 3-dof translational mechanisms employing the Orthoglide architecture 
 
 
4.1. Manipulator geometry 
The Orthoglide is a Delta-type parallel manipulator dedicated to 3-axis rapid machining 
applications that was developed to meet the advantages of both serial and parallel kinematic 
architectures (regular homogeneous workspace with good dynamic performances and stiffness). 
This manipulator consists of three parallel PRPaR identical chains actuated by three mutually 
orthogonal linear drives, which are arranged to ensure almost isotropic workspace kinematic 
properties and to restrict the end-effector motions in translation, with the velocity transmission 
factors close to 1.0 similar to the conventional XYZ-machines. Moreover, to increase the 
manipulator stiffness, the kinematic chains impose redundant constraints on the mobile platform 
(because only two parallelograms would be sufficient to restrict the motion in translation). 
Hence, this is an over-constrained structure that cannot be evaluated using standard lump 
modeling techniques. 
 
P
             
 
 
Fig. 6. CAD model of and Orthoglide and its prototype 
A. Pashkevich, D. Chablat, P. Wenger / Stiffness Analysis of Overconstrained Parallel Manipulators 
 
15
This architecture was implemented in the Orthoglide prototype (Fig. 6), which was built in 
Institut de Recherche en Communications et Cybernetique de Nantes (IRCCyN) and satisfies the 
following design objectives: cubic Cartesian workspace of size 200200200 mm, Cartesian 
velocity and acceleration in the isotropic point 1.2 m/s and 14 m/s2; payload 4 kg; transmission 
factor range 0.5–2.0. The manipulator kinematics, including the direct and inverse 
transformations, is described in details in our previous paper [42]. Here we propose the 
manipulator stiffness model that, in contrast to previous works, does not ignore the over-
constraining feature. Also, we compare two alternative architectures, which are kinematically 
equivalent but differ in the stiffness capabilities. 
 
 
4.2. Stiffness of U-Joint Based Manipulator 
First, let us derive the stiffness model for the simplified Orthoglide mechanics, where the 
legs are comprised of equivalent limbs with U-joints at the ends (Fig 5a). Accordingly, to retain 
major compliance properties, the limb geometry corresponds to the parallelogram bars with 
doubled cross-section area. 
Let us assume that the world coordinate system is located at the end-effector reference point 
corresponding to the isotropic manipulator posture (when the legs are mutually perpendicular 
and parallel to relevant actuator axes). For this assumption, the geometrical models of separate 
kinematic chains can be described by the expression (1), where 
{ , , }
i
x y z

 and the product 
components are defined via the standard translational/rotational operators 
(.),
(.),
(.)
x
y
z
T
T
R

 as 
follows: 
 
1
0
0
0
1
0
0
;
0
0
1
0
0
0
0
1
x
base
L
r













T
   
0
0
1
0
1
0
0
;
0
1
0
0
0
0
0
1
y
base
L
r













T
   
0
1
0
0
0
0
1
0
;
1
0
0
0
0
0
1
z
base
L
r













T
 
(17) 
 
1
0
0
0
1
0
0 ;
0
0
1
0
0
0
0
1
x
tool
r












T
           
0
1
0
0
0
1
0 ;
1
0
0
0
0
0
0
1
y
tool
r












T
              
0
0
1
1
0
0
0 ;
0
1
0
0
0
0
0
1
z
base
r












T
 
(18) 
 
0
0
0
0
(
)
(
)
a
x
q
q





V
T
;      
Foot 
T
I ;       
( )
Leg
x L

T
T
 
(19) 
 
1
6
1
2
3
4
5
6
(
,
)
(
)
(
)
(
)
(
)
(
)
(
)
s
x
y
z
x
y
z














V
T
T
T
R
R
R

 
(20) 
 
1
1
2
1
2
(
,
)
(
)
(
)
u
z
y
q
q
q
q


V
R
R
;       
2
3
4
3
4
(
,
)
(
)
(
)
u
y
z
q
q
q
q


V
R
R
; 
(21) 
Here L, r are the manipulator geometrical parameters (the leg length and end-effector offset 
respectively), and the remaining variables are the same as in equation (1). Because the end-
effector of the rigid manipulator is restricted to the translational motions, the nominal values of 
the passive joint coordinates are subject to the specific constrains, 
2
3
0
q
q


 and 
1
4
0
q
q


, 
which are implicitly incorporated in the direct/inverse kinematics. However, the flexible model 
A. Pashkevich, D. Chablat, P. Wenger / Stiffness Analysis of Overconstrained Parallel Manipulators 
 
16
allows variations all passive joint coordinates around the nominal values, so the jacobians 
q
J  
must be computed for four variables 
1
4
,
q
q

. 
Using the link stiffness parameters obtained by the FEA-modeling (see Appendix C) and 
applying the proposed methodology, we computed the compliance matrices for three typical 
manipulator postures, the principal components of which are presented in Table 2. Below, they 
are compared with the compliance of the parallelogram-based manipulator. It should be noticed 
that the equivalent model of the UU-leg includes two parallelogram bars with corresponding 
stiffness matrix 
/2
Bar
k
. 
 
4.3. Stiffness of Parallelogram Based Manipulator 
Further, let us considerer the 3-PRPaR architecture where the manipulator legs are composed 
of the kinematic parallelograms (see Fig. 5b), which corresponds to the final design of the 
Orthoglide prototype. This obviously imposes some additional kinematic constrains compared to 
the 3-PUU case and should increase the stiffness, but quantitative comparison requires relevant 
modeling. 
Before evaluating the compliance of the entire manipulator, let us derive the stiffness matrix 
of the parallelogram. Using the adopted notations, the parallelogram equivalent model may be 
written as 
 
2
2
7
8
(
)
( )
(
)
(
,
,
)
Plg
y
x
y
s
q
L
q






T
R
T
R
V
 
(22) 
where, compared to the 3-PUU case, the third passive joint is eliminated (it is implicitly assumed 
that 
3
2
q
q

). On the other hand, the original parallelogram may be split into two serial 
kinematic chains (the “upper” and “lower” ones) that yields the same multi-serial-chain parallel 
architecture as considered above. Hence, the parallelogram compliance matrix may be derived 
using the same stiffness modeling technique proposed in this paper. 
Assuming that the main contribution to parallelogram compliance is caused by the bar links of 
the length L, the geometry and flexibility of these kinematic chains can be described by the 
expressions  
 
1
1
6
2
(
/2)
(
)
( )
(
,
) 
(
)
( /2)
up
up
up
up
up
z
y
x
s
y
z
d
q
q
L
q
q
d











T
T
R
T
V
R
T

 
(23) 
 
1
1
6
2
( /2)
(
)
( )
(
,
) 
(
)
(
/2)
dn
dn
dn
dn
dn
z
y
x
s
y
z
d
q
q
L
q
q
d











T
T
R
T
V
R
T

 
(24) 
where L, d are the parallelogram geometrical parameters), 
1
2
,
,
{
,
}
i
i
q
q
i
up dn



 are the 
variations of the passive joint coordinates, q  is the parallelogram state coordinate defining its 
“rigid” posture, 
(.)
s
V
 describes displacements in the virtual springs, and the sub/superscripts 
“up” and “dn” correspond to the upper and lower chain respectively (Fig. 7). 
 
(a) 
(b) 
 
x 
L 
q2 
z 
x
z 
6-d.o.f.  
spring 
6-d.o.f.  
spring 
d 
 
x
L 
z
x 
z 
5 d.o.f.  
spring 
q2 
 
A. Pashkevich, D. Chablat, P. Wenger / Stiffness Analysis of Overconstrained Parallel Manipulators 
 
17
Fig. 7. Lump stiffness model of the parallelogram (a) and its equivalent presentation (b). 
Computing Jacobians for the upper chain with respect to 
iq

 and 
i yields 
 
2
2
0
0
0
0
0
1
1
0
0
q
up
q
q
d
d
LS
LC























J
;      
0
0
0
2
0
1
0
0
2
2
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
q
q
q
q
up
q
q
q
q
q
q
d
C
S
dC
dS
S
C
C
S
S
C

























J
 
(25) 
where 
sin( )
q
S
q

 and 
sin( )
q
C
q

. For the lower part, the expressions are similar and differ in 
signs of d  only. Then, defining the bar-link stiffness in the general form as 
Bar
ij
K




K
 and 
performing relevant matrix transformations for both kinematic chains, the parallelogram stiffness 
matrix is presented analytically as 
 
11
22
26
2
2
2
22
2
22
44
2
2
11
2
2
2
2
22
22
26
66
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
( )
2
4
8
0
0
0
0
0
4
0
0
0
8
4
q
q
Plg
q
q
q
K
K
K
d C K
d S K
K
q
d C K
d S K
d S K
K
K



























K
 
(26) 
where it is assumed that the x-axis is directed along to the L-links (i.e., similar to the 3-PUU 
case). It is also worth mentioning that the stiffness parameters 
33
K  and 
55
K
(which depend on 
yI , 
see equations (16)) are completely compensated by the passive joints, and there is no 
translational stiffness in the z-direction. Moreover, the rotational stiffness around z-axis is 
defined by the parameter 
11
K (describing the bar compression/tension).  
More detailed analysis of the matrix 
(0)
Plg
K
 at the isotropic point using expressions (16) 
shows that most of the elements are doubled compared to the stiffness of a single beam. But the 
rotation about the z-axis (matrix elements 
55
K
, where the term 
/
y
L EI  is replaced by 
2
/2
d EA
L ) 
demonstrates essential increase of the stiffness. Numerically, it can be evaluated as the 
ratio
2
/(8
)
y
d A
I
, which for the rectangular cross-section of size b h

 is reduced to 
2
1.5 ( / )
d h
 
where the dimension h corresponds to the axis y. Besides, there is some stiffness increase in the 
x-axis rotation (element 
44
K
) but it is not so high. For instance, for the considered case study 
(Orthoglide prototype), the increase in 
55
K
 is about 25 times, while the element 
44
K
 increases 
roughly by 10%. However, for the entire manipulator, three parallelograms yield essential 
stiffness increase for all rotational axes. 
It worth mentioning that, for such description of the parallelogram stiffness, the kinematic 
chains of 3-PRPaR manipulator are described by the same equations as in the 3-PUU case, but 
the joint variable 
3
iq  is not treated as an independent one, since 
2
3
0
i
i
q
q


. The latter must be 
taken account while computing the jacobians 
i
q
J  which size is reduced to 6x3 and corresponds to 
three passive joints 
1
2
4
,
,
i
i
i
q q
q .  
A. Pashkevich, D. Chablat, P. Wenger / Stiffness Analysis of Overconstrained Parallel Manipulators 
 
18
To avoid the rank-deficiency of matrix (26) that is produced by the third raw and column, it 
is necessary to eliminate the third translational spring. This allows the matrix inversion while 
computing 
i

S  (see equations (13)) and reduces the parallelogram stiffness model to a 5-dof 
virtual spring that includes two translational and three rotational components with relevant 
coupling between them. Another way to avoid the above singularity is to introduce an arbitrary 
“fictitious” stiffness at the intersection of the zero raw and column of (26), since its influence 
will be totally compensated by the corresponding passive joint presented in the Jacobian 
i
q
J . The 
latter approach allows minimizing preliminary analytical derivation and replacing them by 
numerical computations. 
Using this model and applying the proposed technique, we computed the compliance 
matrices for three typical non-singular manipulator postures (Table 2) and also the stiffness 
matrixes for two singular postures (Table 3). As follows from the comparison with the U-joint 
case, within the dexterous workspace, the parallelograms allow multiplying the rotational 
stiffness roughly by 10. This justifies application of this architecture in the Orthoglide prototype 
design [37]. 
 
TABLE 2 
Comparison of translational and rotational compliance for 3-PUU and 3-PRPaR manipulators 
MANIPULATOR 
ARCHITECTURE 
Point Q0 
, ,
0.00
x y z
mm

 
 
 
Point Q1 
, ,
73.65
x y z
mm

 
 
 
 
Point Q2 
, ,
126.35
x y z
mm

 
 
 
tran
k
 
[mm/N] 
rot
k
 
[rad/Nmm] 
tran
k
 
[mm/N] 
rot
k
 
[rad/Nmm] 
 
tran
k
 
[mm/N] 
rot
k
 
[rad/Nmm] 
3-PUU manipulator 
2.7810-4 
20.910-7 
10.910-4 
24.110-7 
 
71.310-4 
25.810-7 
3-PRPaR manipulator 
2.7810-4
 
1.9410-7
 
9.8610-4
 
2.0610-7
 
 
21.210-4
 
2.6510-7
 
 
 
 
4.4. Accuracy of the proposed model 
To validate the developed stiffness modeling technique, the 3-PRPaR Orthoglide prototype 
was evaluated using several different lump models and the FEA-based model. The obtained 
results are presented in Table 4. As follows from them, accuracy of the previous methods [31] is 
about 25…30%. In contrast, the proposed model that uses 6-dof virtual springs with FEA-based 
evaluation of the link stiffness parameters, gives accuracy of about 5% for almost all workspace 
points. The only exceptions are the manipulator configurations that are close to the “flat 
singularity” (point Q2), when the error for the rotational stiffness rises up to 20%.  
This motivated development of the extended stiffness model for the parallelogram legs, 
which takes into account flexibility of the “axes” corresponding to the links d (see Fig. 7). 
However, while it yielded essential improvement in accuracy for the translational stiffness, the 
accuracy of the rotational components in the neighborhood of Q2 was about 15% only. The latter 
A. Pashkevich, D. Chablat, P. Wenger / Stiffness Analysis of Overconstrained Parallel Manipulators 
 
19
motivates further research that takes into account the joint stiffness (in addition to the links and 
actuators). However, in general, these results confirm advantages of the developed technique and 
justify its application for the pre-design stage. 
 
 
TABLE 3 
Stiffness matrices Ktran of 3-PUU and 3-PRPaR manipulators in singular configurations 
MANIPULATOR 
ARCHITECTURE 
Flat” singularity 
, ,
/
6
x y z
L

 
 
 
 
“Flat” singularity 
, ,
/
3
x y z
L

 
 
 
tran
K
,   [N/mm]  
 
tran
K
,   [N/mm] 
3-PUU manipulator 
3
1.48
0.74
0.74
0.74
1.48
0.74
10
0.74
0.74
1.48
tran

















K
 
(
)
2
tran
rank

K
 
 
3
1.78
1.78
1.78
1.78
1.78
1.78
10
1.78
1.78
1.78
tran












K
 
(
)
1
tran
rank

K
 
3-PRPaR manipulator 
3
1.54
0.77
0.77
0.77
1.54
0.77
10
0.77
0.77
1.54
tran

















K
 
(
)
2
tran
rank

K
 
 
3
4.65
4.65
4.65
4.65
4.65
4.65
10
4.65
4.65
4.65
tran












K
 
(
)
1
tran
rank

K
 
 
 
 
 
TABLE 4 
Comparison of compliance modelling results for 3-PRPaR Orthoglide 
STIFFNESS MODEL 
Point Q0 
, ,
0.00
x y z
mm

 
Point Q1 
, ,
73.65
x y z
mm

 
 
Point Q2 
, ,
126.35
x y z
mm

 
tran
k
 
[mm/N] 
rot
k
 
[rad/Nmm] 
tran
k
 
[mm/N] 
rot
k
 
[rad/Nmm] 
 
tran
k
 
[mm/N] 
rot
k
 
[rad/Nmm] 
Lump model of F.Majou et al. (2007) 
with additional passive joints 
3.6810
-4 
2.7710
-7 
13.810
-4 
2.7710
-7 
 
34.310
-4 
2.7810
-7 
Modified model of F.Majou et al. (2007) 
without additional passive joints 
3.6810
-4 
1.2610
-7 
12.510
-4 
1.2610
-7 
 
24.710
-4 
1.2610
-7 
Over-constrained lump model 
with 6-dof springs 
2.7810
-4
 
1.9410
-7
 
9.8610
-4
 
2.0610
-7
 
 
21.210
-4
 
2.6510
-7
 
Extended over-constrained model 
with 6-dof springs  
2.9310
-4 
2.0210
-7
 
10.210
-4
 
2.1510
-7
 
 
21.910
-4
 
2.7610
-7
 
FEA-based model 
3.0510
-4 
2.0510
-7 
10.910
-4 
2.1710
-7 
 
26.810
-4 
2.6710
-7 
A. Pashkevich, D. Chablat, P. Wenger / Stiffness Analysis of Overconstrained Parallel Manipulators 
 
20
5. Conclusions 
 
The paper proposes a new systematic method for computing the stiffness matrix of 
overconstrained parallel manipulators. It is based on a multidimensional lumped model of the 
flexible links, whose parameters are evaluated via the FEA-modeling and describe both the 
translational/rotational compliances and the coupling between them. In contrast to previous 
works, the method employs a new solution strategy of the kinetostatic equations, which 
considers simultaneously the kinematic and static relations for each separate kinematic chain and 
then aggregates the partial solutions in a total one. This allows computing the stiffness matrices 
for overconstrained mechanisms for any given manipulator posture, including singular 
configurations and their neighborhood. Another advantage is the computational simplicity that 
requires low-dimensional matrix inversion compared to other techniques. Besides, the method 
does not require manual elimination of the redundant spring corresponding to the passive joints, 
since this operation is inherently included in the numerical algorithm. Using the proposed 
methodology, we also derived the analytical 5-dof stiffness model of the parallelogram-based 
link. 
The efficiency and accuracy of the proposed method was demonstrated through application 
examples that compare the stiffness of two parallel manipulators of the Orthoglide family (with 
U-joint based and parallelogram based links). Relevant simulation results have confirmed 
essential advantages of the parallelogram based architecture and validated adopted design of the 
Orthoglide prototype. Accuracy of the proposed model was evaluated via comparison with FEA 
modeling. 
While applied to mechanisms with similar kinematic chains and actuators located between 
the base and foot, the method can be extended to other parallel architectures to cover different 
actuator locations and dissimilar chain geometry. So, future work will focus on the stiffness 
modeling of more complicated parallel mechanisms (such as the Verne machine) and also on the 
experimental verification of the stiffness models for the Orthoglide robot. Another prospective 
research direction is the stiffness analysis of heavy manipulators, for which the influence of 
gravity is essential and cannot be neglected. 
 
A. Pashkevich, D. Chablat, P. Wenger / Stiffness Analysis of Overconstrained Parallel Manipulators 
 
21
Acknowledgments 
This work has been partially funded by the European projects NEXT, acronyms for “Next 
Generation of Productions Systems”, Project no° IP 011815. 
 
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A. Pashkevich, D. Chablat, P. Wenger / Stiffness Analysis of Overconstrained Parallel Manipulators 
 
24
Appendices 
 
 
Appendix A. SVD-based computing of stiffness matrix for a kinematic chain with passive joints 
 
Let us consider the kinetostatic model of a separate kinematic chain 
q


S f
J
q
t ;   
T
q 
J
f
0  
which defines portion of the external force 
6
R

f
 and the variations of the passive joint 
coordinates 
m
R

q
 corresponding to the end-effector motion 
6
R

t
(for convenience, the 
chain-number indices i are omitted). Here m is the passive joint number, 
q
J  is the passive joint 
jacobian of size 6xm, and 

S  is the 6x6 positive-definite symmetric matrix of the virtual springs 
compliance relative to the end-effector (see Section 2 for details). 
To find the desired mapping 


f
K
t , let us apply the SVD decomposition to the jacobian 
q
J  that yields to the following factorisation 
T
q
q
q
q



J
U
Σ
V  
where 
q
U  and 
q
V  are the orthogonal matrices of the size 66 and mm respectively (i.e., 
T
q
q 
U U
I , 
T
q
q 
V V
I ), and 
q
Σ  is the 6m quasi-diagonal matrix with non-negative elements 
1,
m



 (singular values). Then, after substitution 
q
J  and left-multiplication of the first 
equation by 
T
q
U
 and the second one by 
T
q
V , the original system may be rewritten as 
(
)
(
)
T
T
T
T
q
q
q
q
q
q







U S U
U f
Σ
V
q
U
t ;    
(
)
T
T
q
q


Σ
U f
0 . 
The latter may be treated as the orthogonal linear transformations of the variables  
T
q




t
U
t  ;   
T
q

f
U f  ;      
T
q




q
V
q  
with respect to which the considered system is simplified down to:  
T
q
q
q








U S U f
Σ
q
t ;    
T
q



Σ
f
0 . 
Further, because the matrix 
q
Σ  is quasi-diagonal, the second equation may be re-written in a 
scalar from as  
0,
1,
k
kf
k
m





, 
which gives the trivial solutions for the first r components of the variable f  
0,
1,
kf
k
r

 
corresponding to 
0
k

 where 
1,
k
r
 and 
(
)
q
r
rank

J
 (usually, r
m

 within the dexterous 
workspace, and only for some singular configurations r
m

.). The remaining (n-r) components 
of f  corresponding to 
0
k

 can be found from the portion of the first matrix equation which 
in the scalar form is presented as 
A. Pashkevich, D. Chablat, P. Wenger / Stiffness Analysis of Overconstrained Parallel Manipulators 
 
25
6
1
;
1,
6
T
k
k
q
q
l
k
k
l r
f
t
k
r








u S u
 
where 
6,
1,...6
k
q
R
k


u
 are the orthogonal vector-columns comprising the matrix 
q
U . Hence, 
in the matrix form, the expression for the vector variable f  can be written as 
(6
)
1
(6
)
(
)
r r
r
r
T
d
d
r
r
q
q



















0
0
f
t
0
U
S U
 
where 
1
6
[
,
]
d
r
q
q
q


U
u
u

 is a “rank-deficient” part of the matrix 
q
U  obtained by its partitioning 
[
,
]
r
d
q
q
q

U
U
U
 into two submatrices of the size 6 r
 and 6 (6
)r


 respectively:  
Thus, after restoring the original variables f , q  and left-multiplication by 
q
U , the above 
relation is transformed into  
1
(
)
T
r
q
r
d
T
q
q
T
d
d
d
q
q
q
























0
0
U
f
U
U
t
0
U
S U
U
 
which after simplification yields the final expression for the stiffness of a kinematic chain 
1
(
)
T
T
d
d
d
d
q
q
q
q



K
U
U
S U
U
 
where the matrix K  is obviously conservative. 
In this expression, the matrix 
1
T






S
J K J  describes the spatial location and compliance of 
the virtual springs, and the matrix 
d
q
U  characterizes impact of the passive joints, which slacken 
the springs effect by accepting certain motions without the force/torque reactions. The rank of 
the obtained matrix 
(
)
6
rank
r


K
 depends on the number of passive joints m and the 
kinematic chain posture ( r
m

). Apparently, for the case without passive joints, the expression 
for 
1


k
K
 is reduced to the known one, i.e. 
1
T





k
J K J . 
 
 
Appendix B. CAD-based computing of the link compliance matrix 
 
Let us assume that the FEA-modelling provided six data sets describing the displacement of 
the reference object caused by successive applications of the forces 
,
,
x
y
z
F
F
F  and the torques 
,
,
x
y
z
M
M
M  along the axes of the virtual spring coordinate system. Each such data set may be 
formally described as {
,
=1,2,
}
k
k k
m
p
d

 where 
k
p  and 
k
d  are respectively the Cartesian 
position and the Cartesian displacement of the kth node in the link-base coordinate system. 
To evaluate the reference object translation (
,
,
)
x
y
z
p
p
p



 and rotation (
,
,
)
x
y
z






 
relative to the virtual spring centre 
0
p , let us fit the data by the model 
0
0
(
)
(
)
k
k
k




d
R p
p
t
p
p
, 
which includes, as the parameters, the translation vector t  and the orthogonal rotation matrix R  
of sizes 3x1 and 3x3 respectively. After defining 
0
k
k


g
p
p  and 
0
k
k
k



g
p
p
d , the model may 
be rewritten in the form  
A. Pashkevich, D. Chablat, P. Wenger / Stiffness Analysis of Overconstrained Parallel Manipulators 
 
26
;
1,
k
k
k
m




g
R g
t

, 
which is known in the matrix analysis as the “Procrustes problem” and admits the minimum least 
square solution [43] 
1
1
1
1
;
m
m
T
k
k
k
k
R
m
m










V U
t
g
R
g  
via the SVD factorization of the following 3x3 matrix 
1
m
T
T
k
k
g
g
g
k



g g
U
Σ
V  
where 
g
U  and 
g
V  are the orthogonal matrices of the size 33, and 
g
Σ  is the 3x3 diagonal 
matrix (positive definite, if m  3). For small displacements 
k
d , the rotation matrix may be re-
written in the differential from, which gives explicit expressions for the rotation angles: 
23
x
r



;   
13
y
r



;   
12
z
r



. 
The translational displacements are extracted from the vector t : 
1
xp
t


;   
2
y
p
t


;   
3
zp
t


. 
Then the obtained values 
,
,
x
z
p




 are scaled by dividing by the corresponding force/torque 
amplitude. And, after applying this algorithm to all six data sets (corresponding to 
,
,
x
y
z
F
F
M

), 
the desired compliance matrix of size 6x6 is constructed as  
(
)/
(
)/
(
)/
(
)/
(
)/
(
)/
(
)/
(
)/
(
)/
x
x
x
x
y
y
x
z
z
y
x
x
y
y
y
y
z
z
CAD
z
x
x
z
y
y
z
z
z
p F
F
p F
F
p M
M
p
F
F
p
F
F
p
M
M
F
F
F
F
M
M


























k







 
where 
(
)
x
y
p F

denotes the displacement 
xp

 caused by the applied force 
y
F

, etc. Finally, to 
compensate some small computational errors, the obtained matrix is symmetrized 
(
)/2
T
link
CAD
CAD


k
k
k
 
by averaging the non-diagonal symmetrical elements. 
 
A. Pashkevich, D. Chablat, P. Wenger / Stiffness Analysis of Overconstrained Parallel Manipulators 
 
27
 
Appendix C. Compliance parameters of the Orthoglide links 
 
For the Orthoglide manipulator, the actuator compliance was evaluated as 
5
10
/
ctr
k
mm N


 
while the links compliance matrices were computed via the FEA-based simulation using 
technique presented in Appendix 2, which yielded: 
 
 
4
4
6
4
4
6
3
6
5
6
7
5
7
6
6
7
2.45 10
2.73 10
0
0
0
5.48 10
2.73 10
3.24 10
0
0
0
7.04 10
0
0
1.59 10
9.90 10
1.27 10
0
0
0
9.90 10
2.07 10
0
0
0
0
1.27 10
0
2.06 10
0
5.48 10
7.04 10
0
0
0
1.71 10
Foot


























































k
 
 
 
5
2
4
2
4
6
4
6
4
6
4.50 10
0
0
0
0
0
0
8.01 10
0
0
0
3.98 10
0
0
3.64 10
0
1.71 10
0
0
0
0
3.76 10
0
0
0
0
1.71 10
0
1.09 10
0
0
3.98 10
0
0
0
2.65 10
Bar










































k
 
 
 
6
5
7
5
7
8
7
8
7
8
1.99 10
0
0
0
0
0
1.29 10
0
0
0
2.61 10
0
0
1.50 10
0
7.64 10
0
0
0
0
6.81 10
0
0
0
0
7.64 10
0
8.23 10
0
0
2.61 10
0
0
0
2.67 10
Axis










































k
; 
 
 
6
7
6
7
7
7
8
7
10
10
1.88 10
0
0
0
0
0
0
3.83 10
0
0
0
0
0
0
9.99 10
2.90 10
0.45 10
0
0
0
2.90 10
1.55 10
0
0
0
0
0.45 10
0
5.19 10
0
0
0
0
0
0
4.86 10
Act










































k
 
 
In these matrices, the units are [mm], [rad], [N] and [Nmm] for the length, angle, force and 
torque respectively. As following from these results, the most compliant manipulator 
components are the foots and the parallelogram bars, while the remaining elements have rigidity 
of 5 – 10 times higher.