arXiv:1012.2199v1  [cs.RO]  10 Dec 2010
Stiffness modelling of parallelogram-based
parallel manipulators
A. Pashkevich, A. Klimchik, S. Caro and D. Chablat
Institut de Recherche en Communications et en Cybernetique de Nantes, France
Ecole des Mines de Nantes, France
e-mail:
anatol.pashkevich@emn.fr,alexandr.klimchik@emn.fr,stephane.caro@irccyn.ec-nantes
Abstract. The paper presents a methodology to enhance the stiffness analysis of parallel manipu-
lators with parallelogram-based linkage. It directly takes into account the influence of the external
loading and allows computing both the non-linear “load-deflection” relation and relevant rank-
deficient stiffness matrix. An equivalent bar-type pseudo-rigid model is also proposed to describe
the parallelogram stiffness by means of five mutually coupled virtual springs. The contributions
of this paper are highlighted with a parallelogram-type linkage used in a manipulator from the
Orthoglide family.
Key words: stiffness modeling, parallel manipulators, parallelogram-based linkage, external load-
ing, linear approximation.
1 Introduction
In the last decades, parallel manipulators have received growing attention in indus-
trial robotics due to their inherent advantages of providing better accuracy, lower
mass/inertia properties, and higher structural rigidity compared to their serial coun-
terparts (Ceccarelli et al., 2002, Company et al., 2002, Merlet, 2000). These features
are induced by the specific kinematic structure, which eliminates the cantilever-type
loading and decreases deflections due to external wrehches. Accordingly, they are
used in innovative robotic systems, but practical utilization of the potential benefits
requires development of efficient stiffness modeling techniques, which satisfy the
computational speed and accuracy requirements of relevant design procedures.
Amongst numerous parallel architectures studied in robotics literature, of special
interest are the parallelogram-based manipulators that employ special type of link-
age constraining undesirable motions of the end-platform. However, relevant stiff-
ness analysis is quite complicated due to the presence of internal closed kinematic
chains that are usually replaced by equivalent limbs (Majou et al., 2007, Pashkevich
et al., 2009a) whose parameters are evaluated rather intuitively. Thus, the problem
of adequate stiffness modelling of parallelogram-type linkages, which is in the focus
of this paper, is still a challenge and requires some developments.
1
2
A. Pashkevich, A. Klimchik, S. Caro and D. Chablat
Another important research issue is related to the influence of the external load-
ing that may change the stiffness properties of the manipulator. In this case, in ad-
dition to the conventional “elastic stiffness” in the joints, it is necessary to take into
account the “geometrical stiffness” due to the change in the manipulator configura-
tion under the load (Kovecses and Anjeles, 2007). Moreover buckling phenomena
may appear (Timoshenko and Goodier, 1970) and produce structural failures, which
must be detected by relevant stiffness models.
This paper is based on our previous work on the stiffness analysis of over-
constrained parallel architectures (Pashkevich et al., 2009a, 2009b) and presents
new results by considering the loading influence on the manipulator configura-
tion and, consequently, on its Jacobian and Hessian. It implements the virtual joint
method (VJM) introduce by Salisbary (Salisbury, 1980) and Gosselin (Gosselin,
1990) that describes the manipulator element compliance with a set of localized
six-dimensional springs separated by rigid links and perfect joints. The main con-
tribution of the paper is the introduction of a non-linear stiffness model of the
parallelogram-type linkage and its linear approximation by 6x6 matrix of the rank
5, for which it is derived an analytical expression.
2 Problem statement
Let us consider a typical parallel manipulator that is composed of several kinematic
chains connecting a fixed base and a moving platform (Figure 1a). It is assumed, that
at least one chain includes a parallelogram-based linkage that may introduce some
redundant constraints improving the mechanism stiffness. Following the VJM con-
cept, the manipulator chains are usually presented as a serial sequence of pseudo-
rigid links separated by rotational or translational joints of one of the following
types: (i) perfect passive joints; (ii) perfect actuated joints, and (iii) virtual flexible
joints that describe compliance of the links, joints and/or actuators.
Fig. 1 Parallel manipulators with parallelogram-based linkages.
To derive a parallel architecture without internal kinematic loops, the parallel-
ogram based linkage must be replaced by an equivalent bar-type element that has
similar elastic properties. Thus, let us consider the VJM model of the kinematic par-
Stiffness modelling of parallelogram-based parallel manipulators
3
allelogram assuming its compliance is mainly due to the compliance of the longest
links, which are oriented in the direction of the linkage (Figure 1b,c). In order to be
precise, let us also assume that the stiffness of the original bar-elements is described
by a 6 × 6 matrix Kb whose elements are identified using FEA-based modeling.
In the framework of the VJM approach, the geometry of the parallelogram-based
linkage can be descried by the following homogeneous transformations
Ti = Tz(η d/2) Ry(qi1) Tx(L) Tx(θi1) Ty(θi2) Tz(θi3) Rx(θi4)
Ry(θi5) Rz(θi6) Ry(qi2) Tz(−η d/2) Ry(−qi2)
(1)
where η = (−1)i, Ti is a 4 × 4 homogenous transformation matrix, Tx, ..., Rz are
the matrices of elementary translation and rotation, L, d are the parallelogram ge-
ometrical parameters, qij are the passive joint coordinates, θij are the virtual joint
coordinates, i = 1,2 defines the number of the kinematic chain, j identifies the co-
ordinate number within the chain. It should be noted that using this notation, the
closed loop equation can be expressed as T1 = T2 .
For further computational convenience, the homogenous matrix equations (1)
may be also rewritten in the vector form as
t = gi(qi,θi),
(2)
where vector t = (p,ϕ) is the output frame pose, that includes its position p =
(x,y,z)T and orientation ϕ = (ϕx,ϕy,ϕz), vector qi = (qi1,qi2) contains the passive
joint coordinates, and vector θi = (θi1,...,θi6) collects all virtual joint coordinates of
the corresponding chain. Using the above assumptions and definitions, let us derive
the stiffness model of the parallelogram-based linkage.
3 Static equilibrium
Let us derive first the general static equilibrium equations of the parallelogram as-
suming that it is virtually divided into two symmetrical serial kinematic chains with
the geometrical and elastostatic models t = gi(qi,θi), τi = Ki θi, i = 1,2 where the
variables θi correspond to the deformations in the virtual springs, Ki is the spring
stiffness matrix, and τi incorporates corresponding forces and torques. Taking into
account that the considered mechanism is under-constrained, it is prudent to assume
that the end-points of both chains are located at a given position t and to compute
partial forces Fi corresponding to the static equilibrium.
To derive equations, let us express the potential energy of the mechanism as
E(θ1,θ2) = 1
2
2
∑
i=1
θ T
i Kθi θi
(3)
4
A. Pashkevich, A. Klimchik, S. Caro and D. Chablat
In the equilibrium configuration, this energy must be minimised subject to the geo-
metrical constraints t = gi(qi,θi), i = 1,2. Hence, the Lagrange function is
L(θ1,θ2,q1,q2) = 1
2
2
∑
i=1
θ T
i Kθi θi +
2
∑
i=1
λ T
i (t−gi(θi,qi))
(4)
where the multipliers λi may be interpreted as the external forces Fi applied at the
end-points of the chains. Further, after computing the partial derivatives of L(...)
with respect to θi,qi,λi and setting the derivatives to zero, equations of the static
equilibrium can be presented as
JT
θiλi = Ki θi;
JT
qi λi = 0;
t = gi(qi,θi);
i = 1,2
(5)
where Jθi = ∂gi(...)/∂θi and Jqi = ∂gi(...)/∂qi are the kinematic Jacobians. Here,
the configuration variables θi, qi and the multipliers λi are treated as unknowns.
Since the derived system is highly nonlinear, the desired solution can be obtained
only numerically. In this paper, it is proposed to use the following iterative procedure
"
λ ′
i
q′
i
#
=
"
Jθi K−1
θi JT
θi
Jqi
JT
qi
0
# "
t−gi + Jqi qi + Jθi θi
0
#
;
θ ′
i = K−1
θi JT
θi λ ′i (6)
where the symbol “ ′ ” corresponds to the next iteration. Using this iterative pro-
cedure, for any given location of the end-point t, one can compute both the partial
forces Fi and the total external force, which allows to obtain the desired “force-
deflection” relation F = f(t) for any initial unloaded configuration.
4 Stiffness matrix
To compute the stiffness matrix of the considered parallelogram-based mechanism,
the obtained “force-deflection” relations must be linearized in the neighborhood of
the static equilibrium. Let assume that the external forces Fi, i = 1,2 and the end-
point location t of both kinematic chains are incremented by some small values
δFi, δt and consider simultaneously two equilibriums corresponding to the state
variables (Fi,θi,qi,t) and (Fi +δFi,θi +δθi,qi +δqi,t+δt). Under these assump-
tions, the original system (5) should be supplemented by the equations
(Jθi + δJθi)T (λi + δλi) = Kθi (θi + δθi);
(Jqi + δJqi) (λi + δλi) = 0;
t+ δt = gi(qi,θi)+ Jθi δθi + Jqi δqi;
(7)
where δJθi and δJqi are the differentials of the Jacobians due to changes in (θi,qi).
After relevant transformation, neglecting high-order small terms and expanding the
Stiffness modelling of parallelogram-based parallel manipulators
5
differentials via the Hessians of the function Ψi = gi(qi,θi)T λi : H(i)
qq = ∂2Ψi/∂q2
i ,
H(i)
qθ = ∂2Ψi/∂q∂
i θi, H(i)
θθ = ∂2Ψi/∂θ 2
i , the system of equations may be rewritten as
Jθi δλi + H(i)
qθ δqi + H(i)
θθ δθi = Kθi δθi;
Jqi δλi + H(i)
qq δqi + H(i)
qθ δθi = 0;
Jθi δθi + Jqi δqi = δt
(8)
After analytical elimination of the variable δθi and defining k(i)
θ = (Kθ −H(i)
θθ)−1,
one can obtain a matrix equation
"
δλi
δqi
#
=
"
Jθi ·k(i)
θ ·JT
θi
Jqi + Jθi ·k(i)
θ ·H(i)
θq
JT
qi + H(i)
qθ ·k(i)
θ ·JT
θi H(i)
qq + H(i)
qθ ·k(i)
θ ·H(i)
θq
#−1
·
"
δt
0
#
(9)
which yields the linear relation δλ = Kci δt defining the Cartesian stiffness matrix
Kci for each kinematic chain. Taking into account the architecture of the consid-
ered mechanism, the total stiffness matrix may be found as Kc = Kc1 + Kc2. These
expressions allow numerical computation of the Cartesian stiffness matrix for a gen-
eral case, both for loaded and unloaded equilibrium configuration.
However, in the case of unloaded equilibrium, the above presented equations may
be essentially simplified:
Kp(q) = 2


K11
0
0
0
0
0
0
K22
0
0
0
K26
0
0
0
0
0
0
0
0
0
K44 + d2 C2
q K22
4
0
d2 S2q K22
8
0
0
0
0
d2 C2
q K11
4
0
0
K26
0
d2 S2q K22
8
0
K66 + d2 S2
q K22
4


,
(10)
where Cq = cosq, Sq = sinq, S2q = sin2q, q = q1i (Figure 1b,c), Kij are the elements
of the 6 × 6 stiffness matrix of the parallelogram bars Kb which are assumed here
to be the only elements of the linkage that posses non-negligible stiffness. In the
following example, this expression is evaluated from point of view of accuracy and
applicability to stiffness analysis of the parallelogram-based manipulators.
5 Illustrative example
To demonstrate validity of the proposed model and to evaluate its applicability
range, let us apply it to the stiffness analysis of the Orthoglide manipulator (Fig-
ure 1a). It includes three parallelogram-type linkage where the main flexibility
6
A. Pashkevich, A. Klimchik, S. Caro and D. Chablat
source is concentrated in the bar elements of length 310 mm. Using the FEA-based
software tools and dedicated identification technique (Pashkevich et al., 2009c), the
stiffness matrix of the bar element was evaluated as
Kb =


2.20·104
0
0
0
0
0
0
1.81·101
0
0
0
−2.84·103
0
0
7.86·101
0
1.25·104
0
0
0
0
3.48·104
0
0
0
0
1.25·104
0
2.66·106
0
0
−2.84·103
0
0
0
5.85·105


,
(11)
where for linear/angular displacements and for the force/torque are used the fol-
lowing units: [mm], [rad], [N], [Nmm]. This bar-element was also evaluated with
respect to the structural stability under compression, and the FEA-modelling pro-
duced three possible buckling configurations presented in Figure 2. It is obvious
that these configurations are potentially dangerous for the compressed parallelo-
gram. However, the parallelogram may also produce some other types of buckling
because of presents of the passive joints.
Fig. 2 Critical forces and buckling configurations of a bar element
employed in the parallelograms of Orthoglide.
For the parallelogram-base linkage incorporating the above bar elements, expres-
sion (11) yields the following stiffness matrix
Kp = 2


2.20·104
0
0
0
0
0
0
1.81·101
0
0
0
−2.84·103
0
0
0
0
0
0
.0
0
0
5.64·104
0
1.25·104
0
0
0
0
2.64·107
0
0
−2.84·103
0
1.25·104
0
5.92·105


,
(12)
corresponding to the straight configuration of the linkage (i.e. to q = 0). Being in
good agreement with physical sense, this matrix is rank deficient and incorporates
exactly one zero row/column corresponding to the z-translation, where the parallelo-
gram is completely non-resistant due to specific arrangement of passive joints. Also,
as follows from comparison with doubled stiffness matrix (11) that may be used as a
reference, the parallelogram demonstrates essentially higher rotational stiffness that
mainly depends on the translational stiffness parameters of the bar (moreover, the
Stiffness modelling of parallelogram-based parallel manipulators
7
rotational stiffness parameter K55 of the bar is completely eliminated by the pas-
sive joints). The most significant here is the parallelogram width d that explicitly
presented in the rotational sub-block of Kp.
To investigate applicability range of the linear model based on the stiffness matrix
(10), it was computed a non-linear “force-deflection” relation corresponding to the
parallelogram compression in the x-direction. This computation was performed us-
ing an iterative algorithm presented in Section 3. As follows from obtained results,
the matrix (10) ensures rather accurate description of the parallelogram stiffness
in small-deflection area. However, for large deflections, corresponding VJM-model
detects a geometrical buckling that limits applicability of the matrix (10).
Similar analysis was performed using the FEA-technique, which yielded almost
the same “force-deflection” plot for the small deflections but detected several types
of the buckling, with the critical forces may be both lower and higher then in the
VJM-modelling. Corresponding numerical values and parallelogram configurations
are presented in Figure 3.
The developed VJM-model of parallelogram was verified in the frame of the stiff-
ness modelling of the entire manipulator. Relevant results are presented in Table 1.
They confirm advantages of the parallelogram-based architectures with respect to
the translational stiffness and perfectly match to the values obtained from FEA-
method. However, releasing some assumptions concerning the stiffness properties
of the remaining parallelogram elements (other than bars) modifies the values of
Fig. 3 Force-deflection relations and buckling configurations for parallelogram compression
(modelling methods: VJM, FEA, and LIN - linear model with proposed stiffness matrix).
Table 1 Stiffness parameters of the Orthoglide manipulator for different assumptions concerning
parallelogram linkage.
Model of linkage
VJM-model
FEA-model
Ktran
Krot
Ktran
Krot
[mm/N]
[rad/N]
[mm/N]
[rad/N]
2x-bar linkage
3.35·103
0.13·106
—
—
Parallelogram linkage
(solid axis)
3.23·103
4.33·106
3.33·103
4.10·106
Parallelogram linkage
(flexible axis)
3.08·103
4.06·106
3.31·103
4.07·106
8
A. Pashkevich, A. Klimchik, S. Caro and D. Chablat
the translational and rotational stiffness. The latter gives a new prospective research
direction that targeted at more general stiffness model of the parallelogram.
6 Conclusions
The paper presents new results in the area of enhanced stiffness modeling of par-
allel manipulators with parallelogram-based linkage. In contrast to other works, it
explicitly takes into account influence of the loading and allows both computing
the non-linear load-deflection relation and detecting the buckling phenomena that
may lead to manipulator structural failure. There is also derived an analytical ex-
pression for relevant rank-deficient stiffness matrix and proposed equivalent bar-
type pseudo-rigid model that describes the parallelogram stiffness by five localized
mutually coupled virtual springs. These results are validated for a case study that
deals with stiffness modeling of a parallel manipulator of the Orthoglide family, for
which the parallelogram-type linkage was evaluated using both proposed technique
and straightforward FEA-modeling.
Acknowledgements The work presented in this paper was partially funded by the Region “Pays
de la Loire” (project RoboComposite).
References
Ceccarelli M., Carbone G., 2002, A stiffness analysis for CaPaMan (Cassino Parallel Manipu-
lator), Mechanism and Machine Theory 37(5):427-439.
Company O., Krut S., Pierrot F., 2002, Modelling and preliminary design issues of a 4-axis
parallel machine for heavy parts handling. Journal of Multibody Dynamics 216:1-11.
Gosselin C., 1990, Stiffness mapping for parallel manipulators, IEEE Transactions on Robotics
and Automation 6(3):377-382.
Kovecses J., Angeles J., 2007, The stiffness matrix in elastically articulated rigid-body systems,
Multibody System Dynamics 18(2):169-184.
Majou F., Gosselin C., Wenger P., Chablat D., 2007, Parametric stiffness analysis of the Or-
thoglide, Mechanism and Machine Theory 42:296-311.
Merlet, J.-P., 2000, Parallel Robots,, Kluwer Academic Publishers, Dordrecht.
Pashkevich A., Chablat D., Wenger Ph., 2009a, Stiffness analysis of overconstrained parallel
manipulators, Mechanism and Machine Theory 44:966-982.
Pashkevich A., Klimchik A., Chablat D., Wenger Ph., 2009b, Stiffness analysis of multichain
parallel robotic systems with loading, Journal of Automation, Mobile Robotics & Intelligent Sys-
tems 3(3):75-82.
Pashkevich A., Klimchik A., Chablat D., Wenger Ph., 2009c, Accuracy improvement for stiff-
ness modeling of parallel manipulators, in Proceedings of 42nd CIRP Conference on Manufactur-
ing Systems, Grenoble, France.
Salisbury J., 1980, Active stiffness control of a manipulator in Cartesian coordinates, in 19th
IEEE Conference on Decision and Control 87-97.
Timoshenko S., Goodier J. N, 1970, Theory of Elasticity, 3d ed., McGraw-Hill, New York.