arXiv:0708.3723v1  [cs.RO]  28 Aug 2007
Parametric stiffness analysis of the Orthoglide
F´elix Majou1,2, Cl´ement Gosselin2, Philippe Wenger1 and Damien Chablat1
1 IRCCyN ∗, 1 rue de la No¨e, 44321 Nantes, France
2 D´epartement de G´enie M´ecanique, Universit´e Laval, Qu´ebec, Canada, G1K 7P4
1er novembre 2021
Abstract
This paper presents a parametric stiffness analysis of the Orthoglide, a 3-DOF translational Parallel Kine-
matic Machine. First, a compliant modeling of the Orthoglide is conducted based on an existing method. Then
stiffness matrix is symbolically computed. This allows one to easily study the influence of the geometric design
parameters on the matrix elements. Critical links are displayed. Cutting forces are then modeled so that static
displacements of the Orthoglide tool during slot milling are symbolically computed. Influence of the geometric
design parameters on the static displacements is checked as well. Other machining operations can be modeled.
This parametric stiffness analysis can be applied to any parallel manipulator for which stiffness is a critical issue.
Keywords : Parallel Kinematic Machine ; Stiffness ; Parametric analysis;
1
Introduction
Parallel manipulators are claimed to offer good stiffness and accuracy properties, as well as good dynamic
performances. This makes them attractive for innovative machine-tool structures for high speed machining [1],
[2], [3]. When a parallel manipulator is intended to become a Parallel Kinematic Machine (PKM), stiffness
becomes a very important issue in its design [4], [5], [6]. In this paper is presented a parametric stiffness analysis
of the Orthoglide, a 3-axis translational PKM prototype developped at IRCCyN [7].
One of the first stiffness analysis methods for parallel mechanisms is based on a kinetostatic modeling [8].
It proposes to map the stiffness of parallel mechanisms by taking into account the compliance of the actuated
joints. This method is not appropriate for PKM whose legs, unlike hexapods, are subject to bending [9]. In
[10], a stiffness estimation of a tripod-based PKM is proposed that solves this problem. However the stiffness
model presented is not general enough. This gap is filled in [11]. The method is based on a flexible-link lumped
parameter model that replaces the compliance of the links by localized virtual compliant joints and rigid links.
The approach has two differences from that presented in [10] (i), the way the link compliances are modeled
(ii), the equations allowing the computation of the stiffness model, that are more general.
This method is applied to the Orthoglide for a parametric stiffness analysis. The stiffness matrix elements
are symbolically computed. This allows an easy analysis of the influence of the Orthoglide critical design
parameters. No numerical computations are conducted until graphical results are generated. First we present
the Orthoglide, then the compliant model. The results showing the influence of the parameters are presented
next.
2
Compliant modeling of the Orthoglide
2.1
Kinematic architecture of the Orthoglide
The Orthoglide is a translational 3-axis PKM prototype designed for machining applications. The mobile
platform is connected to three orthogonal linear drives through three identical RPaR serial chains (where R
∗Institut de Recherches en Communications et Cybern´etique de Nantes: UMR CNRS 6597, ´Ecole Centrale de Nantes, ´Ecole
des Mines de Nantes, Universit´e de Nantes
1
stands for Revolute and Pa for Parallelogram) (Fig.1), and it moves in the Cartesian workspace while maintai-
ning a fixed orientation. The Orthoglide is optimized for a prescribed workspace with prescribed kinetostatic
performances. Its kinematic analysis, design and optimization are fully described in [12].
B1
i1
P
x
z
y
j1
A1
C1
A2
B2
C2
A3
C3
B3
Fig. 1 – The Orthoglide kinematic architecture
2.2
Parameters for compliant modeling
The parameters used for the compliant modeling of the Orthoglide are presented in Figure 2 and in Table
1. These parameters correspond to a “beam-like” modeling of the Orthoglide legs. The foot has been designed
to prevent each parallelogram from colliding with the corresponding linear motion guide.
λ
Parallelogram bars
Foot
P
fL
d
LB
e
Fig. 2 – Leg geometrical parameters
Parameter
Description
Lf
Foot length, see Fig.2
hf, bf
Foot section sides
If1 = bf.h3
f/12
Foot section moment of inertia 1
If2 = hf.b3
f/12
Foot section moment of inertia 2
d
Distance
between
parallelogram
bars, see Fig.2
LB
Parallelogram bar length, see Fig.2
SB
Parellelogram
bar
cross-section
area
Tab. 1 – Geometric parameters
2.3
Compliant modeling with flexible links
In the lumped model described in [13], the leg links are considered as flexible beams and are replaced by
rigid beams mounted on revolute joints plus torsional springs located at the joints (Fig. 3).
From deriving the relationship between the force F and the deformation y(x), the local torsional stiffness
k can be computed :
EIy′′(x)
=
F(L −x)
2
y(L)
L
E, I
F
x
y
(a) Flexible beam
y(L)
F
θ
k
y
x
(b) Virtual rigid beam
Fig. 3 – General model for flexible link
...
EIy(L)
=
FL3/3
→θ ≃y(L)/L
=
FL2/3EI
k
=
FL/θ
→k
=
3EI/L
If the Orthoglide leg actuator is locked, then one leg can withstand one force F and one torque T (Fig. 4),
that are transmitted along the parallelogram bars and the foot. For a compliant modeling that uses virtual
compliant joints, it is important to understand how external forces are transmitted, and what their effect on
the leg links is.
Actuator
locked
F
T
F
T
F
T
Fig. 4 – Forces transmitted in a leg
Three virtual compliant joints are modeled along the Orthoglide leg. They are described in Table 2. The
determination of the virtual joint stiffnesses is not detailed here to save space. Since the actuator is assumed to
be much stiffer than the virtual compliant joints, it is not included in foot stiffnesses. The leg links compliances
modeled in Table 2 were selected beforehand as the most significant. The joint compliances are not taken into
account in our model.
3
Symbolic computing of the Orthoglide stiffness matrix
The virtual compliant joints stiffnesses depend on the design parameters. In this section, symbolic expres-
sions of stiffness matrix elements are symbolically computed. This part is based on a stiffness model that was
fully described in [11]. Therefore, the description of the model will only be summarized here. Fig. 5 represents
the lumped model of a leg with flexible links.
The Jacobian matrix Ji of the ith leg of the Orthoglide is obtained from the Denavit-Hartenberg parameters
of the ith leg with flexible links. This matrix maps all leg joint rates (including the virtual compliant joints)
into the generalized velocity of the platform, i.e.,
Ji ˙θi = t
where
˙θT
i =
 ˙θi1
˙θi2
˙θi3
˙θi4
˙θi5
˙θi6
˙θi7
˙θi8

is the vector containing the 8 actuated, passive and compliant joint rates of leg i, t is the twist of the platform.
The Pa joint parameterization imposes ˙θi5 = −˙θi5bis, which makes ˙θi5 and ˙θi5bis dependent. ˙θi5 is chosen to
3
Stiffness and description
Figure
kact : translational stiff-
ness of the prismatic ac-
tuator
F
2EIf2/Lf : foot bending
due to torque T
T
fL
3EIf1/Lf : foot bending
due to force F
F
fL
ESBd2/2LB
: differen-
tial tension of parallelo-
gram bars due to torque
T
LB
T
Tab. 2 – Virtual compliant joints modeling
Z 1
Z2
Z 3
Z5bis
Z6
Z 7
Z4
Z 8
Z5
P
Fig. 5 – Flexible leg
model the circular translational motion, and finally Ji is written as
Ji =

0
ei2
ei3
ei4
ei1
ei2 × ri2
ei3 × ri3
ei4 × ri4
0
ei6
ei7
ei8
ei5 × ri5 −ei5bis × ri5bis
ei6 × ri6
ei7 × ri7
ei8 × ri8

in which eij is the unit vector along joint j of leg i and rij is the vector connecting joint j of leg i to the platform
reference point. Therefore the Jacobian matrix of the Orthoglide can be written as :
J =


J1
0
0
0
J2
0
0
0
J3


One then has :
J ˙θ = Rt
(1)
4
with
R =


I6
I6
I6


T
and
t =
 Ω
V

˙θ being the vector of the 24 joint rates, that is ˙θ =
 ˙θT
1
˙θT
2
˙θT
3

. Unactuated joints are then eliminated
by writing the geometric conditions that constrain the two independent closed-loop kinematic chains of the
Orthoglide kinematic structure :
J1 ˙θ1 = J2 ˙θ2
(2)
J1 ˙θ1 = J3 ˙θ3
(3)
From (2) and (3) one can obtain (see [11] for details) :
A ˙θ′ = B ˙θ′′
where ˙θ′ is the vector of joint rates without passive joints and ˙θ′′ is the vector of joint rates with only passive
joints. Hence :
˙θ′′ = B−1A ˙θ′
Then a matrix V is obtained (see [11] for details) such that :
˙θ = V ˙θ′
(4)
From (1) and (4) one can obtain :
JV ˙θ′ = Rt
(5)
As matrix R represents a system of 18 compatible linear equations in 6 unknowns, one can use the least-square
solution to obtain an exact solution from (5) :
t = (RT R)−1RT JV ˙θ′
Now let J′ be represented as :
J′ = (RT R)−1RT JV
Then one has :
t = J′ ˙θ′
(6)
According to the principle of virtual work, one has :
τ T ˙θ′ = wT t
(7)
where τ is the vector of forces and torques applied at each actuated or virtual compliant joint and w is the
external wrench applied at the end effector, point P. Gravitational forces are neglected. By substuting (6) in
(7), one can obtain :
τ = J′T w
(8)
The forces and displacements of each actuated or virtual compliant joint can be related by Hooke’s law, that
is for the whole structure one has :
τ = KJ∆θ′
(9)
with :
KJ =


A
0
0
0
A
0
0
0
A


5
and :
A =


kact
0
0
0
0
3EIf1
Lf
0
0
0
0
2EIf2
Lf
0
0
0
0
ESBd2
LB


(10)
∆θ′ only includes the actuated and virtual compliant joints, that is by equating (8) with (9) :
KJ∆θ′ = J′T w
hence :
∆θ′ = K−1
J J′T w
Pre-multiplying both sides by J′ one obtains :
J′∆θ′ = J′K−1
J J′T w
(11)
Substituting (6) into (11), one obtains :
t = J′K−1
J J′T w
Finally the compliance matrix κ is obtained as follows :
κ = J′KJ
−1J′T
In the Orthoglide case we obtain a simplified compliance matrix :
κ =








κ11
0
0
0
κ15
κ15
0
κ11
0
κ24
0
κ26
0
0
κ11
0
κ35
0
0
κ24
0
κ44
0
κ46
κ15
0
κ35
0
κ55
0
κ15
κ26
0
κ46
0
κ66








(12)
And the Cartesian stiffness matrix is :
K = J′KJ
−1J′T −1
Remarks :
From the expression of matrix A (see eq. 10) we deduce that all elements of stiffness matrix K can be
factored either in factors of E or in factors of kact. As kact is assumed infinite compared to other stiffnesses,
the elements of K which are factors of kact must be eliminated. Also obviously when E increases, K elements
increase too, which is in accordance with intuition. Young’s modulus E should then be eliminated from K
elements expressions because its influence needs no further study.
In eq. (12) one should notice that κ11 = κ22 = κ33, and κ15 = κ16. Also, we noticed that when X=Y=Z we
have κ44 = κ55 = κ66.
4
Parametric stiffness analysis
In this part, we study the in uence of the geometric parameters on the elements of the Orthoglide stiffness
matrix. For a realistic inspection of the symbolic expressions of these elements, it is necessary to replace some
parameters by numerical values, once the symbolic expressions have been obtained. Numerical values of the
design parameters come from the Orthoglide prototype [14].
6
4.1
Qualitative analysis at the isotropic configuration
There is an isotropic configuration in the Orthoglide workspace [7]. This configuration gives a good insight
of the Orthoglide general performances. It happens when the tool point P is at the intersection of the three
actuated joint axes (see Fig. 1). In this configuration, the stiffness matrix K is diagonal and the symbolic
expressions of its elements are simplified :
K = E/2








Ka
1
1
1
1
1
1
Ka
1
1
1
1
1
1
Ka
1
1
1
1
1
1
Kb
1
1
1
1
1
1
Kb
1
1
1
1
1
1
Kb








E = 2Ka being the torsional stiffness with
Ka =
 
LB
d2SB
+ 3Lfcos2λ
hfb3
f
!−1
(13)
and E = 2Kb being the translational stiffness with
Kb =
 
0.5h3
fbf
L3
f(1 −cos2λ +
E
2kact
!−1
(14)
– From the inspection of Ka we note that increasing the cross-section areas SB and bfhf increases the
torsional stiffness, which is in accordance with intuition. Increasing the distance d between the paralle-
logram bars increases the torsional stiffness. Also, it is no surprise that increasing Lf or LB decreases
the torsional stiffness. It is no surprise that Ka and Kb increase when E increases, and that Kb increases
when kact increases, while Ka does not depend on kact, the prismatic actuated joint stiffness. Ka and
Kb do not depend on e at the isotropic configuration.
– From the inspection of Kb we note that increasing the cross-section area bfhf increases the translational
stiffness, as increasing the foot length Lf does. We also note that if λ (angle between the foot and the
actuated joint axis) decreases toward zero, then the translational stiffness tends toward infinite. This is
in accordance with intuition since at the isotropic configuration, if λ = 0 then the virtual compliant joint
“foot bending due to force F” (see Tab. 2) is not requested because the applied force axis crosses the
virtual compliant joint axis.
Furthermore, if λ = 0, the cantilever disappears. As this cantilever is aimed at preventing each paral-
lelogram from colliding with the corresponding linear motion guide, from a designer point of view the
conclusion could be : “λ should be lowered to increase the translational stiffness, but there should remain
enough cantilever for the workspace volume to be large enough”.
4.2
Quantitative analysis at the isotropic configuration
The qualitative analysis conducted above provides an interesting insight of the influence of the geometrical
parameters on rotational and translational stiffnesses. Quantitative information is now obtained by studying
the influence on Ka and Kb of a +/ −50% variation of the numerical value of each design parameter. The
results are shown on Fig.6 and on Fig.7. From Fig. 6 analysis, we can go further than the qualitative analysis
and see that d, SB and LB have very little influence on the rotational stiffness while Lf , hf bf and λ have
much more influence. From Fig. 7 we note, as deduced from the qualitative analysis, that decreasing Lf or λ
increases the translational stiffness, and that increasing hf can improve the translational stiffness. Increasing hf
does not reduce the workspace volume like reducing Lf does. However increasing hf increases the foot weight,
which could be prejudicial to the dynamic performances. The proof of this forecast needs further investigations.
Quantitative analysis provides another point of view and helps in a better understanding of the influence of
the geometric design parameters on the machine stiffness.
5
Conclusions
A parametric stiffness analysis of a 3-axis PKM prototype, the Orthoglide, was conducted. First, a compliant
model of the Orthoglide was obtained, then an existing stiffness analysis method for parallel manipulators
7
−0.5
0
0.5
0
0.5
1
1.5
2
2.5
3
3.5
4 x 10
7
Parameter variation (%)
Rotational stiffness (Nmm/rad)
h 
L 
b 
f 
f 
f 
(a) Influence of Lf , bf , hf and λ
−0.5
0
0.5
0
0.5
1
1.5
2
2.5
3
3.5
4 x 10
7
Parameter variation (%)
Rotational stiffness (Nmm/rad)
d 
S B 
L B 
(b) Influence of d, LB and SB
Fig. 6 – Quantitative influence of the geometric parameters on the rotational stiffness E = 2Ka
−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
0
0.5
1
1.5
2
2.5 x 10
4
Parameter variation (%)
Translational stiffness (N/mm)
h f 
b f 
L f 
Fig. 7 – Quantitative influence of the geometric parameters hf , bf , Lf and d on the translational stiffness
E = 2Kb
was applied to the Orthoglide. The stiffness matrix elements were computed symbolically. In the isotropic
configuration, the influence of the geometric design parameters on the rotational and translational stiffnesses
was studied through qualitative and quantitative analyses. These provided relevant information for stiffness-
oriented design or optimization of the Orthoglide.
This example of a parametric stiffness analysis shows that simple symbolic expressions carefully built and
interpreted provide broad information on parallel manipulators stiffness features. Comparison of the stiffness
model with a finite elements model and with stiffness tests on the prototype are currently being conducted.
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9