arXiv:0705.1282v1  [cs.RO]  9 May 2007
Proceedings of the WORKSHOP on
Fundamental Issues and Future Research Directions for Parallel Mechanisms and Manipulators
October 3–4, 2002, Quebec City, Quebec, Canada
Cl´ement M. Gosselin and Imme Ebert-Uphoff, editors
Design of a Three-Axis Isotropic Parallel Manipulator for Machining Applications:
The Orthoglide
PHILIPPE WENGER, DAMIEN CHABLAT
Institut de Recherche en Communications et Cybern´etique de Nantes (IRCCyN)
1, rue de la No¨e, 44321 Nantes, France
email: Philippe.Wenger@irccyn.ec-nantes.fr
email: Damien.Chablat@irccyn.ec-nantes.fr
Abstract: The orthoglide is a 3-DOF parallel mechanism de-
signed at IRCCyN for machining applications. It features three
fixed parallel linear joints which are mounted orthogonally and a
mobile platform which moves in the Cartesian x-y-z space with
fixed orientation. The orthoglide has been designed as function
of a prescribed Cartesian workspace with prescribed kinetostatic
performances. The interesting features of the orthoglide are a
regular Cartesian workspace shape, uniform performances in all
directions and good compactness. A small-scale prototype of the
orthoglide under development is presented at the end of this pa-
per.
1
Introduction
Parallel kinematic machines (PKM) are interesting alternative
designs for high-speed machining applications and have been
attracting the interest of more and more researchers and com-
panies. Since the first prototype presented in 1994 during the
IMTS in Chicago by Gidding&Lewis (the Variax), many other
prototypes have appeared.
However, the existing PKM suffer from two major draw-
backs, namely, a complex Cartesian workspace and highly non
linear input/output relations. For most PKM, the Jacobian ma-
trix which relates the joint rates to the output velocities is not
constant and not isotropic. Consequently, the performances (e.g.
maximum speeds, forces accuracy and rigidity) vary consider-
ably for different points in the Cartesian workspace and for dif-
ferent directions at one given point. This is a serious drawback
for machining applications (Kim (1997); Treib et al. (1998);
Wenger et al. (1999)). To be of interest for machining applica-
tions, a PKM should preserve good workspace properties, that is,
regular shape and acceptable kinetostatic performances through-
out. In milling applications, the machining conditions must re-
main constant along the whole tool path (Rehsteiner (1999);
Rehsteiner et al. (1999)). In many research papers, this crite-
rion is not taking into account in the algorithmic methods used
for the optimization of the workspace volume (Luh et al. (1996);
Merlet (1999)).
The orthoglide optimization is conducted to define a 3-axis
PKM with the advantages a classical serial PPP machine tool
but not its drawbacks. Most industrial 3-axis machine-tool have
a serial PPP kinematic architecture with orthogonal linear joint
axes along the x, y and z directions. Thus, the motion of the
tool in any of these directions is linearly related to the motion
of one of the three actuated axes. Also, the performances are
constant in the most part of the Cartesian workspace, which is
a parallelepiped. The main drawback is inherent to the serial
arrangement of the links, namely, poor dynamic performances.
The orthoglide is a PKM with three fixed linear joints
mounted orthogonally. The mobile platform is connected to the
linear joints by three articulated parallelograms and moves in the
Cartesian x-y-z space with fixed orientation. Its workspace shape
is close to a cube whose sides are parallel to the planes xy, yz
and xz respectively. The optimization is conducted on the ba-
sis of the size of a prescribed cubic workspace with bounded
velocity and force transmission factors. Two criteria are used
for the architecture optimization of the orthoglide, (i) the condi-
tioning of the Jacobian matrix of the PKM (Golub et al. (1989);
Salisbury et al. (1982); Angeles (1997)) and (ii) the manipula-
bility ellipsoid (Yoshikawa (1985)).
The first criterion leads to an isotropic architecture and to
homogeneous performances in the workspace. The second cri-
terion permits to optimize the actuated joint limits and the link
lengths of the orthoglide with respect to the aforementioned two
criteria.
Next section presents the orthoglide. The kinematic equa-
tions and the singularity analysis is detailed in Section 3. Sec-
tion 4 is devoted to the optimization process of the orthoglide
and to the presentation of the prototype.
1
2
Description of the Orthoglide
Most existing PKM can be classified into two main families.
The PKM of the first family have fixed foot points and vari-
able length struts and are generally called “hexapods”. They
have a Stewart-Gought parallel kinematic architecture. Many
prototypes and commercial hexapod PKM already exist like
the Variax-Hexacenter (Gidding&Lewis), the CMW300 (Com-
pagnie M´ecanique des Vosges), the TORNADO 2000 (Hexel),
the MIKROMAT 6X (Mikromat/IWU), the hexapod OKUMA
(Okuma), the hexapod G500 (GEODETIC). In this first family,
we find also hybrid architectures with a 2-axis wrist mounted
in series to a 3-DOF tripod positioning structure (the TRICEPT
from Neos Robotics).
The second family of PKM has been more recently investi-
gated. In this category we find the HEXAGLIDE (ETH Z¨urich)
which features six parallel (also in the geometrical sense) and
coplanar linear joints. The HexaM (Toyoda) is another exam-
ple with non coplanar linear joints. A 3-axis translational ver-
sion of the hexaglide is the TRIGLIDE (Mikron), which has
three coplanar and parallel linear joints. Another 3-axis trans-
lational PKM is proposed by the ISW Uni Stuttgart with the
LINAPOD. This PKM has three vertical (non coplanar) linear
joints. The URANE SX (Renault Automation) and the QUICK-
STEP (Krause & Mauser) are 3-axis PKM with three non copla-
nar horizontal linear joints. The SPRINT Z3 (DS Technology)
is a 3-axis PKM with one degree of translation and two degrees
of rotations. A hybrid parallel/serial PKM with three parallel in-
clined linear joints and a two-axis wrist is the GEORGE V (IFW
Uni Hanover).
PKMs of the second family are more interesting because the
actuators are fixed and thus the moving masses are lower than in
the hexapods and tripods.
The orthoglide presented in this article is a 3-axis transla-
tional parallel kinematic machine with variable foot points and
fixed length struts. Figure 1 shows the general kinematic archi-
tecture of the orthoglide.
The orthoglide has three parallel PRPaR identical chains
(where P, R and Pa stands for Prismatic, Revolute and Parallel-
ogram joint, respectively). The actuated joints are the three or-
thogonal linear joints. These joints can be actuated by means of
linear motors or by conventional rotary motors with ball screws.
The output body is connected to the linear joints through a set
of three parallelograms of equal lengths L
=
BiCi, so that
it can move only in translation.
The first linear joint axis is
parallel to the x-axis, the second one is parallel to the y-axis
and the third one is parallel to the z-axis. In figure 1, the base
points A1, A2 and A3 are fixed on the ith linear axis such that
A1A2 = A1A3 = A2A3, Bi is at the intersection of the first
revolute axis ii and the second revolute axis ji of the ith paral-
lelogram, and Ci is at the intersection of the last two revolute
joints of the ith parallelogram. When each BiCi is aligned with
the linear joint axis AiBi , the orthoglide is in an isotropic con-
figuration (see 4.4) and the tool center point P is located at the
intersection of the three linear joint axes. In this configuration,
the base points A1, A2 and A3 are equally distant from P. The
symmetric design and the simplicity of the kinematic chains (all
joints have only one degree of freedom, Fig. 2) should contribute
to lower the manufacturing cost of the orthoglide.
The orthoglide is free of singularities and self-collisions.
The workspace has a regular, quasi-cubic shape.
The in-
put/output equations are simple and the velocity transmission
factors are equal to one along the x, y and z direction at
the isotropic configuration, like in a serial PPP machine
(Wenger et al. (2000)).
B1
i1
P
x
z
y
j1
A1
C1
A2
B2
C2
A3
C3
B3
Figure 1: Orthoglide kinematic architecture
ii
ji
ri
P
qi
bi
Bi
Ai
Ci
L
Figure 2: Leg kinematics
3
Kinematic Equations and Singularity Analysis
3.1
Static Equations
Let θi and βi denote the joint angles of the parallelogram about
the axes ii and ji, respectively (fig. 2). Let ρ1, ρ2, ρ3 denote the
linear joint variables, ρi = AiBi. In a reference frame (O, x, y,
z) centered at the intersection of the three linear joint axes (note
that the reference frame has been translated in Fig. 1 for more
legibility) , the position vector p of the tool center point P can
be defined in three different ways:
p =


a + ρ1 + cos(θ1) cos(β1)L + e
sin(θ1) cos(β1)L
−sin(β1)L


(1a)
2
p =


−sin(β2)L
a + ρ2 + cos(θ2) cos(β2)L + e
sin(θ2) cos(β2)L


(1b)
p =


sin(θ3) cos(β3)L
−sin(β3)L
a + ρ3 + cos(θ3) cos(β3)L + e


(1c)
where a = OAi, e = CiP and we recall that L = BiCi, ρi =
AiBi.
3.2
Kinematic Equations
Let ˙ρ be referred to as the vector of actuated joint rates and ˙p as
the velocity vector of point P:
˙ρ = [ ˙ρ1 ˙ρ2 ˙ρ3]T ,
˙p = [ ˙x ˙y ˙z]T
˙p can be written in three different ways by traversing the three
chains AiBiCiP:
˙p = n1 ˙ρ1 + ( ˙θ1i1 + ˙β1j1) × (c1 −b1)
(2a)
˙p = n2 ˙ρ1 + ( ˙θ2i2 + ˙β2j2) × (c2 −b2)
(2b)
˙p = n3 ˙ρ3 + ( ˙θ3i3 + ˙β3j3) × (c3 −b3)
(2c)
where bi and ci are the position vectors of the points Bi and Ci,
respectively, and ni is the direction vector of the linear joints, for
i = 1, 2,3.
3.3
Singular configurations
We want to eliminate the two idle joint rates ˙θi and ˙βi from
Eqs. (2a–c), which we do upon dot-multiplying Eqs. (2a–c) by
ci −bi:
(c1 −b1)T ˙p
=
(c1 −b1)T n1 ˙ρ1
(3a)
(c2 −b2)T ˙p
=
(c2 −b2)T n2 ˙ρ2
(3b)
(c3 −b3)T ˙p
=
(c3 −b3)T n3 ˙ρ3
(3c)
Equations (3a–c) can now be cast in vector form, namely
A ˙p = B ˙ρ
where A and B are the parallel and serial Jacobian matrices, re-
spectively:
A =


(c1 −b1)T
(c2 −b2)T
(c3 −b3)T


(4a)
B =


η1
0
0
0
η2
0
0
0
η3


(4b)
with ηi = (ci −bi)T ni for i = 1, 2, 3.
The parallel singularities (Chablat et al. (1998)) occur when
the determinant of the matrix A vanishes, i.e. when det(A) = 0.
In such configurations, it is possible to move locally the mobile
platform whereas the actuated joints are locked. These singu-
larities are particularly undesirable because the structure cannot
resist any force. Eq. (4a) shows that the parallel singularities oc-
cur when:
(c1 −b1) = α(c2 −b2) + λ(c3 −b3)
that is when the points B1, C1, B2, C2, B3 and C3 are copla-
nar (Fig. 3). A particular case occurs when the links BiCi are
parallel (Fig. 4):
(c1 −b1)
//
(c2 −b2)
and
(c2 −b2)
//
(c3 −b3)
and
(c3 −b3)
//
(c1 −b1)
x
z
y
Figure 3: Parallel singular configuration when BiCi are coplanar
x
z
y
Figure 4: Parallel singular configuration when BiCi are parallel
Serial singularities arise when the serial Jacobian matrix B is
no longer invertible i.e. when det(B) = 0. At a serial singularity
a direction exists along which any cartesian velocity cannot be
3
produced. Eq. (4b) shows that det(B) = 0 when for one leg i,
(bi −ai) ⊥(ci −bi).
The optimization of the orthoglide will put the serial and
parallel singularities far away from the workspace (see 4.4).
4
Design and Performance Analysis of the Orthoglide
For usual machine tools, the Cartesian workspace is generally
given as a function of the size of a right-angled parallelepiped.
Due to the symmetrical architecture of the orthoglide, the Carte-
sian workspace has a fairly regular shape in which it is possible
to include a cube whose sides are parallel to the planes xy, yz
and xz respectively (Fig. 5).
The aim of this section is to define the dimensions of the or-
thoglide as a function of the size LWorkspace of a prescribed cu-
bic workspace with bounded transmission factors. We first show
that the orthogonal arrangement of the linear joints is justified by
the condition on the isotropy and manipulability: we want the
orthoglide to have an isotropic configuration with velocity and
force transmission factors equal to one. Then, we impose that the
transmission factors remain under prescribed bounds throughout
the prescribed workspace and we deduce the link dimensions and
the joint limits.
4.1
Condition Number and Isotropic Configuration
The Jacobian matrix is said to be isotropic when its condition
number attains its minimum value of one (Angeles (1997)). The
condition number of the Jacobian matrix is an interesting perfor-
mance index which characterises the distortion of a unit ball un-
der the transformation represented by the Jacobian matrix. The
Jacobian matrix of a manipulator is used to relate (i) the joint
rates and the Cartesian velocities, and (ii) the static load on the
output link and the joint torques or forces. Thus, the condition
number of the Jacobian matrix can be used to measure the uni-
formity of the distribution of the tool velocities and forces in the
Cartesian workspace.
4.2
Isotropic Configuration of the Orthoglide
For parallel manipulators, it is more convenient to study the con-
ditioning of the Jacobian matrix that is related to the inverse
transformation, J−1. When B is not singular, J−1 is defined
by:
˙ρ = J−1 ˙p with J−1 = B−1A
Thus:
J−1 =


(1/η1)(c1 −b1)T
(1/η2)(c2 −b2)T
(1/η3)(c3 −b3)T


(5)
with ηi = (ci −bi)T ni for i = 1, 2, 3.
The matrix J−1 is isotropic when J−1J−T
= σ213×3,
where 13×3 is the 3 × 3 identity matrix. Thus, we must have,
1
η1
||c1 −b1|| = 1
η2
||c2 −b2|| = 1
η3
||c3 −b3||
(6a)
(c1 −b1)T (c2 −b2) = 0
(6b)
(c2 −b2)T (c3 −b3) = 0
(6c)
(c3 −b3)T (c1 −b1) = 0
(6d)
Equation (6a) states that the orientation between the axis of the
linear joint and the link BiCi must be the same for each leg i.
Equations (6b–d) mean that the links BiCi must be orthogonal
to each other. Figure 6 shows the isotropic configuration of the
orthoglide. Note that the orthogonal arrangement of the linear
joints is not a consequence of the isotropy condition, but it stems
from the condition on the transmission factors at the isotropic
configuration (see next section).
4.3
Manipulability Analysis
For a serial PPP machine tool, Fig. 7, a motion of an actuated
joint yields the same motion of the tool (the transmission factors
are equal to one). In the purpose on our study, this factor is
calculated from linear joint to the end-effector.
For a parallel machine, these motions are generally not
equivalent. When the mechanism is close to a parallel singu-
larity, a small joint rate can generate a large velocity of the tool.
This means that the positioning accuracy of the tool is lower in
some directions for some configurations close to parallel singu-
larities because the encoder resolution is amplified. In addition,
a velocity amplification in one direction is equivalent to a loss of
rigidity in this direction.
The manipulability ellipsoids of the Jacobian matrix
of
robotic
manipulators
was
defined
several
years
ago
(Salisbury et al. (1982)). This concept has then been applied as a
performance index to parallel manipulators (Kim (1997)). Note
that, although the concept of manipulability is close to the con-
cept of condition number, these two concepts do not provide the
same information. The condition number quantifies the proxim-
ity to an isotropic configuration, i.e. where the manipulability
ellipsoid is a sphere, or, in other words, where the transmission
factors are the same in all the directions, but it does not inform
about the value of the transmission factor.
The manipulability ellipsoid of J−1 is used here for (i) justi-
fying the orthogonal orientation of the linear joints and (ii) defin-
ing the joint limits of the orthoglide such that the transmission
factors are bounded in the prescribed workspace.
We want the transmission factors to be equal to one at the
isotropic configuration like for a PPP machine tool. This con-
dition implies that the three terms of Eq. (6) must be equal to
one:
1
η1
||c1 −b1|| = 1
η2
||c2 −b2|| = 1
η3
||c3 −b3|| = 1
(7)
which implies that (bi −ai) and (ci −bi) must be collinear for
each i.
4
x
z
y
LWorkspace
Figure 5: Cartesian workspace
x
z
y
Figure 6: Isotropic configuration of the Orthoglide mecanism
Since, at this isotropic configuration, links BiCi are orthog-
onal, Eq. (7) implies that the links AiBi are orthogonal, i.e. the
linear joints are orthogonal. For joint rates belonging to a unit
ball, namely, || ˙ρ|| ≤1, the Cartesian velocities belong to an el-
lipsoid such that:
˙pT (JJT ) ˙p ≤1
The eigenvectors of matrix (JJT )−1 define the direction of its
principal axes of this ellipsoid and the square roots ξ1, ξ2 and
ξ3 of the eigenvalues of (JJT )−1 are the lengths of the afore-
mentioned principal axes. The velocity transmission factors in
X
Z
Y
Figure 7: Typical industrial 3-axis PPP machine-tool
the directions of the principal axes are defined by ψ1 = 1/ξ1,
ψ2 = 1/ξ2 and ψ3 = 1/ξ3. To limit the variations of this factor
in the Cartesian workspace, we impose
ψmin ≤ψi ≤ψmax
(8)
throughout the workspace. This condition determines the link
lengths and the linear joint limits. To simplify the problem, we
set ψmin = 1/ψmax.
4.4
Design of the Orthoglide for a Prescribed Workspace
The aim of this section is to define the position of the fixed
point Ai, the link lengths L and the linear actuator range ∆ρ
with respect to the limits on the transmission factors defined in
Eq. (8) and as a function of the size of the prescribed workspace
LWorkspace.
Our process of optimization is divided into three steps.
1. First, we determine two points Q1 and Q2 in the prescribed
cubic workspace such that if the transmission factor bounds
are satisfied at these points, they are satisfied in all the pre-
scribed workspace.
2. The points Q1 and Q2 are used to define the leg length L as
function of the size of the prescribed cubic workspace.
3. Finally, the positions of the base points Ai and the linear ac-
tuator range ∆ρ are calculated such that the prescribed cu-
bic workspace is fully included in the Cartesian workspace
of the orthoglide.
Step 1: The transmission factors are equal to one at the
isotropic configuration. These factors increase or decrease when
the tool center point moves away from the isotropic configura-
tion and they tend towards zero or infinity in the vicinity of the
singularity surfaces. It turns out that the points Q1 and Q2 de-
fined at the intersection of the workspace boundary with the axis
x = y = z (figure 8) are the closest ones to the singularity sur-
faces, as illustrated in figure 9 which shows on the same top view
the orthoglide in the two parallel singular configurations of fig-
ures 3 and 4. Thus, we may postulate the intuitive result that if
the prescribed bounds on the transmission factors are satisfied at
5
Q1 and Q2, then these bounds are satisfied throughout the pre-
scribed cubic workspace. Although we could not derive a simple
formal proof, we have verified numerically that this result holds.
x
z
y
Q2
Q1
Figure 8: Points Q1 and Q2
x
y
B1
B2
A1
A2
C1
C2
C’1
C’2
B’1
B’2
P
P’
Parallel singularities
Parallel
singularities
Workspace
Q1
Q2
Serial  singularities
Figure 9: Points Q1 and Q2 and the singular configurations (top
view)
Step 2: At the isotropic configuration, the angles θi and βi
are equal to zero by definition. When the tool center point P
is at Q1, ρ1 = ρ2 = ρ3 = ρmin (Fig. 10). When P is at Q2,
ρ1 = ρ2 = ρ3 = ρmax (Fig. 11).
We pose ρmin = 0 for more simplicity.
On the axis (Q1Q2), β1 = β2 = β3 and θ1 = θ2 = θ3. We
note,
β1 = β2 = β3 = β
and
θ1 = θ2 = θ3 = θ
(9)
Upon substitution of Eq. (9) into Eqs. (1a–c), the angle β can be
C1
C2
x
y
Q2
B1
B2
A1
A2
Q1
Figure 10: Q1 configuration
x
y
Q2
B1
C1
B2
A2
C2
Q1
A1
Dr
Dr
Figure 11: Q2 configuration
written as a function of θ,
β = −arctan(sin(θ))
(10)
Finally, by substituting Eq. (10) into Eq. (5), the inverse Jacobian
matrix J−1 can be simplified as follows
J−1 =


1
−tan(θ)
−tan(θ)
−tan(θ)
1
−tan(θ)
−tan(θ)
−tan(θ)
1


Thus, the square roots of the eigenvalues of (JJT )−1 are,
ξ1 = |2 tan(θ) −1|
and
ξ2 = ξ3 = | tan(θ) + 1|
6
And the three velocity transmission factors are,
ψ1 =
1
|2 tan(θ) −1|
and
ψ2 = ψ3 =
1
| tan(θ) + 1|
(11)
Figure 12 depicts ψ1, ψ2 and ψ3 as function of θ along the axis
(Q1Q2).
0
1
2
3
4
5
-180°
y1y2
q
0°
180°
Q2 Q1
Isotropic configuration
Figure 12: The three velocity transmission factors as function of
θ along the axis (Q1Q2)
The joint limits on θ are located on both sides of the isotropic
configuration. To calculate the joint limits, we solve the follow-
ing inequations,
1
ψmax
≤
1
|2 tan(θ) −1| ≤ψmax
(12a)
1
ψmax
≤
1
| tan(θ) + 1| ≤ψmax
(12b)
where the value of ψmax depends on the performance require-
ments. Two sets of joint limits ([θQ1 βQ1] and [θQ2 βQ2]) are
found. The detail of this calculation is given in the Appendix.
The position vectors q1 and q2 of the points Q1 and Q2,
respectively, can be easily defined as a function of L (Figs. 10
and 11),
q1 = [q1 q1 q1]T
and
q2 = [q2 q2 q2]T
(13a)
with
q1 = −sin(βQ1)L
and
q2 = −sin(βQ2)L
(13b)
The size of the Cartesian workspace is,
LWorkspace = |q2 −q1|
Thus, L can be defined as a function of LWorkspace.
L =
LWorkspace
| sin(βQ2) −sin(βQ1)|
Step 3: We want to determine the positions of the base
points, namely, a. When the tool center point P is at Q′
1 defined
as the projection onto the y-axis of Q1, ρ = 0 and, (Fig. 13)
OA2 = OQ′
1 + Q′
1C2 + C2A2
with OA2 = a, OQ′
1 = q1, Q′
1C2 = PC2 = −e and since
ρ = 0, C2A2 = C2B2 −L. Thus,
a = q1 −e −L
C1
x
y
Q2
B1
B2
A1
A2
Q’1
a
C2
Q1
e
L
Figure 13: The point Q′
1 used for the determination of a
Since q1 is known from Eqs. (13a) and (19b), a can be cal-
culated as function of e, L and ψmax.
Now, we have to calculate the linear joint range ∆ρ = ρmax
(we have posed ρmin=0).
When the tool center point P is at Q2, ρ = ρmax. The
equation of the direct kinematics (Eq. (1b)) written at Q2 yields,
ρmax = q2 −a −cos(θQ2) cos(βQ2)L −e
4.5
Prototype
Using the aforementioned two kinetostatic criteria, a small-scale
prototype is under development in our laboratory. The mechani-
cal structure is now finished, (Fig. 14). The actuated joints used
for this prototype are rotative motors with ball screws. The pre-
scribed performances of the orthoglide prototype are a Carte-
sian velocity of 1.2m/s and an acceleration of 14m/s2 at the
isotropic point. The desired payload is 4kg. The size of its pre-
scribed cubic workspace is 200 × 200 × 200 mm. We limit the
variations of the velocity transmission factors as,
1/2 ≤ψi ≤2
(14)
The resulting length of the three parallelograms is L = 310 mm
and the resulting range of the linear joints is ∆ρ = 257 mm.
Thus, the ratio of the range of the actuated joints to the size of the
prescribed Cartesian workspace is r = 200/257 = 0.78. This
ratio is high compared to other mechanisms. The three velocity
transmission factors are depicted in Fig. 15. These factors are
given in a z-cross section of the Cartesian workspace passing
through Q1.
7
Figure 14: The orthoglide prototype
0,0
-1,0
1,0
2,0
0,0
-1,0
1,0
2,0
-1,0
1,0
2,0
0,0
1,0
1,0
1,1
1,2
1,1
1,2
1,1
1,2
x
y
0,0
-1,0
1,0
2,0
-1,0
1,0
2,0
0,0
1,2
1,2
1,3
1,3
1,4
1,4
1,5
1,5
1,6
x
y
Q1
Q1
y2
y3
-1,0
1,0
2,0
0,0
0,8
0,8
0,7
0,7
0,6
0,6
0,7
0,8
x
y
Q1
y1
Figure 15: The three velocity transmission factors in a z-cross
section of the Cartesian workspace passing through Q1
5
Conclusions
Presented in this paper is a new kinematic structure of a PKM
dedicated to machining applications: the Orthoglide. The main
feature of this PKM design is its trade-off between the popular
serial PPP architecture with homogeneous performances and the
parallel kinematic architecture with good dynamic performances.
The workspace is simple, regular and free of singularities
and self-collisions. The Jacobian matrix is isotropic at a point
close to the center point of the workspace. Unlike most existing
PKMs, the workspace is fairly regular and the performances are
homogeneous in it. Thus, the entire workspace is really available
for tool paths. In addition, the orthoglide is rather compact com-
pared to most existing PKMs. A small-scale prototype of this
mechanism is under construction at IRCCyN. First experiments
with plastic parts will be conducted. The dynamic analysis has
not been reported in this article. A rigid dynamic model has been
proposed in (Guegan et al. (2002) and an elastic dynamic model
is now being developed with the software package Meccano.
Acknowledgments
The authors would like to acknowledge the financial support of
R´egion Pays-de-Loire, Agence Nationale pour la Valorisation de
la Recherche, and ´Ecole des Mines de Nantes.
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6
Appendix
To calculate the joint limits on θ and β, we solve the followings
inequations, from the Eqs. 12,
|2 tan(θ) −1| ≤ψmax
(15a)
1
|2 tan(θ) −1| ≤ψmax
(15b)
Thus, we note,
f1 = |2 tan(θ) −1|
f2 = 1/|2 tan(θ) −1|
(16a)
Figure (16) shows f1 and f2 as function of θ along (Q1Q2). The
four roots of f1 = f2 in [−π π] are,
s1
=
−arctan

(1 +
√
17)/4

(17a)
s2
=
−arctan(1/2)
(17b)
s3
=
0
(17c)
s4
=
arctan

(−1 +
√
17)/4

(17d)
with
f1(s1) = (−3 +
√
17)/4
f1(s2) = 2
(17e)
f1(s3) = 1
f1(s4) = (3 +
√
17)/4
(17f)
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
0
1
2
3
4
5
q
f1
f2
isotropic configuration
s1
s2
s3
s4
Figure 16: f1 and f2 as function of θ along (Q1Q2)
and
f1(θ) = 0
when
θ = arctan(1/2) −π
(18a)
f2(θ) = 0
when
θ = arctan(1/2))
(18b)
The isotropic configuration is located at the configuration where
θ = β = 0. The limits on θ and β are in the vicinity of this
configuration. Along the axis (Q1Q2), the angle θ is lower than
0 when it is close to Q2, and greater than 0 when it is close to
Q1.
To find θQ1, we study the functions f1 and f2 which are both
decreasing on [0 arctan(1/2)]. Thus, we have,
θQ1
=
arctan
ψmax −1
2ψmax

(19a)
βQ1
=
−arctan
 
ψmax −1
p
5ψ2max −2ψmax + 1
!
(19b)
In the same way, to find θQ2, we study the functions f1 and f2
on [s1 0]. The three roots s1, s2 and s3 define two intervals. If
ψmax ∈[f1(s1) f1(s2)], we have,
θQ2
=
−arctan
ψmax −1
ψmax

(20a)
βQ2
=
arctan
 
ψmax −1
p
2ψ2max −2ψmax + 1
!
(20b)
otherwise, if ψmax ∈[f1(s2) f1(s3)],
θQ2
=
−arctan
ψmax −1
2

(20c)
βQ2
=
arctan
 
ψmax −1
p
ψmax
2 −2ψmax + 5
!
(20d)
9