arXiv:0707.3665v1  [cs.RO]  25 Jul 2007
A Comparative Study of Parallel Kinematic
Architectures for Machining Applications
Philippe Wenger1, Cl´ement Gosselin2 and Damien Chablat1
1Institut de Recherche en Communications et Cybern´etique de Nantes ∗
1, rue de la No¨e, 44321 Nantes, France
2D´epartement de g´enie m´ecanique, Universit´e Laval
Qu´ebec, Qu´ebec, Canada, G1K 7P4
Philippe.Wenger@irccyn.ec-nantes.fr
Abstract: Parallel kinematic mechanisms are interesting alternative designs for machining ap-
plications. Three 2-DOF parallel mechanism architectures dedicated to machining applications
are studied in this paper. The three mechanisms have two constant length struts gliding along
fixed linear actuated joints with different relative orientation. The comparative study is con-
ducted on the basis of a same prescribed Cartesian workspace for the three mechanisms. The
common desired workspace properties are a rectangular shape and given kinetostatic perfor-
mances. The machine size of each resulting design is used as a comparative criterion. The
2-DOF machine mechanisms analyzed in this paper can be extended to 3-axis machines by
adding a third joint.
1
Introduction
Most industrial 3-axis machine tools have a PPP kinematic architecture with orthogonal joint
axes along the x, y, z directions. Thus, the motion of the tool in any of these direction is linearly
related to the motion of one of the three actuated axes. Also, the performances (e.g. maximum
speeds, forces, accuracy and rigidity) are constant in the most part of the Cartesian workspace,
which is a parallelepiped. In contrast, the common features of most existing PKM (Parallel
Kinematic Machine) are a Cartesian workspace shape of complex geometry and highly non lin-
ear input/output relations. For most PKM, the Jacobian matrix which relates the joint rates
to the output velocities is not constant and not isotropic. Consequently, the performances may
vary considerably for different points in the Cartesian workspace and for different directions at
one given point, which is a serious drawback for machining applications [8]. To be of interest
for machining applications, a parallel kinematic architecture should preserve good workspace
properties (regular shape and acceptable kinetostatic performances throughout). It is clear that
some parallel architectures are more appropriate than others, as it has already been shown in
previous studies [6, 7].
The aim of this paper is to compare three parallel kinematic archi-
tectures. To limit the analysis, the study is conducted for 2-DOF mechanisms but the results
can be extrapolated to 3-DOF architectures. The three mechanisms studied have two constant
length struts gliding along fixed linear actuated joints with different relative orientation. Each
∗IRCCyN: UMR n◦6597 CNRS, ´Ecole Centrale de Nantes, Universit´e de Nantes, ´Ecole des Mines de Nantes
1
mechanism is defined by a set of three design variables. Given a prescribed Cartesian rectan-
gular region with given kinetostatic performances, we calculate the link dimensions and joint
ranges of each mechanism for which the prescribed region is included in a t-connected region of
the mechanism and the kinetostatic constraints are satisfied. Then, we compare the size of the
resulting mechanisms. The organisation of this paper is as follows. The next section presents
the mechanism studied. Section 3 is devoted to the comparison of three architectures. The last
section concludes this paper.
2
Kinematic study
2.1
Serial Topology with Three Degrees of Freedom
Most industrial machine tools use a simple PPP serial topology with three orthogonal prismatic
joint axes (Figure 1).
X
Z
Y
Figure 1: Typical industrial 3-axis machine-tool
For a PPP topology, the kinematic equations are :
J ˙ρ = ˙p
with
J = 13×3
where ˙p = [x y z]T is the velocity-vector of the tool center point P and ˙ρ = [ρ1 ρ2 ρ3]T is
the velocity-vector of the prismatic joints. The Jacobian kinematic matrix J being the identity
matrix, the ellipsoid of manipulability of velocity and of force [1] is a unit sphere for all the
configurations in the Cartesian workspace.
The problem of the PPP topology is that the
actuator controlling the Y axis supports at the same time the workpiece and the actuator
controlling the displacement of the X axis, which affects the dynamic performances. To solve
this problem, it is possible to use more suitable kinematic architectures like parallel or hybrid
topologies.
2.2
The Parallel Mechanisms Studied
We focus our study on the use of a 2-DOF parallel mechanism (Figure 2) for the motion of the
table of the machine tool depicted in (Figure 1).
The joint variables are the variables ρ1 and ρ2 associated with the two actuated prismatic
joints and the output variables are the position of the tool center point P = [x y]T .
The
mechanisms can be parameterized by the lengths L0, L1 and L2, the angles α1 and α2 and the
actuated joint ranges ∆ρ1 and ∆ρ2 (Figure 2). To reduce the number of design parameters,
2
P
A
C
B
D
r1
a2
q2
q1
a1
r2
L1
L2
Figure 2: Two degree-of-freedom paral-
lel mechanism
P
A
C
B
D
r1
r2
Figure 3: Two degree-of-freedom paral-
lel mechanism with control of the ori-
entation
we impose L1 = L2 and ∆ρ1 = ∆ρ2. This simplification also provides symmetry and, in turn,
reduces the manufacturing costs.
To control the orientation of the reference frame attached to the tool center point P, two
parallelograms can be used which also increase the rigidity of the structure (Figure 3).
2.3
Kinematics of the Parallel Mechanism Studied
The velocity ˙p of P can be written in two different ways. By traversing the closed-loop (ACP −
BDP) in the two possible directions, we obtain
˙p = ˙c + ˙θ1E(p −c)
(1a)
and
˙p = ˙d + ˙θ2E(p −d)
(1b)
where E is the rotation matrix,
E =
 0
−1
1
0

c and d represent the position vector of the points C and D, respectively.
Moreover, the velocity ˙c and ˙d of the points C and D are given by,
˙c =
c −a
||c −a|| ˙ρ1 =
 cos(α1)
sin(α1)

˙ρ1
,
˙d =
d −b
||d −b|| ˙ρ2 =
 cos(α2)
sin(α2)

˙ρ2
The two unactuated joint rates ˙θ1 and ˙θ2 can be eliminated from equations (1a) and (1b) by
dot-multiplying the former by p −c and the latter by p −d, thus obtaining
(p −c)T ˙p = (p −c)T
c −a
||c −a|| ˙ρ1
(2a)
(p −d)T ˙p = (p −d)T
d −b
||d −b|| ˙ρ2
(2b)
Equations (2a) and (2b) can now be cast in vector form, namely,
A ˙p = B ˙ρ
3
where A and B are, respectively, the parallel and serial Jacobian matrices, defined as
A =
 (p −c)T
(p −d)T

,
B =
 (p −c)T ((c −a)/||c −a||)
0
0
(p −d)T ((d −b)/||d −b||)

and with ˙ρ defined as the vector of actuated joint rates and ˙p defined as the vector of velocity
of point P:
˙ρ =

˙ρ1
˙ρ2

,
˙p =

˙x
˙y

When A and B are not singular, we can study the Jacobian kinematic matrix J [2],
˙p = J ˙ρ
with
J = A−1B
(3a)
or the inverse Jacobian kinematic matrix J−1, such that
˙ρ = J−1 ˙p
with
J−1 = B−1A
(3b)
2.4
Parallel Singularities
The parallel singularities occur when the determinant of the matrix A vanishes [3, 4], i.e. when
det(A) = 0. In this configuration, it is possible to move locally the tool center point whereas
the actuated joints are locked.
These singularities are particularly undesirable, because the
structure cannot resist any force and control is lost. To avoid any deterioration, it is necessary
to eliminate the parallel singularities from the workspace.
For the mechanism studied, the parallel singularities occur whenever the points C, D, and P
are aligned (Figure 4), i.e. when θ1 −θ2 = kπ, for k = 1, 2, ....
P
A
C
B
D
r1
r2
Figure 4: Parallel singularity
P
A
C
B
D
r1
r2
Figure 5: Structural singularity
They are located inside the Cartesian workspace and form the boundaries of the joint workspace.
Moreover, structural singularities can occur when L1 is equal to L2 (Figure 5). In these config-
urations, the control of the point P is lost.
2.5
Serial Singularities
The serial singularities occur when the determinant of the matrice B vanishes, i.e.
when
det(B) = 0. When the manipulator is in such configurations, there is a direction along which no
Cartesian velocity can be produced. The serial singularities define the boundary of the Cartesian
workspace [Merlet 97].
For the topology studied, the serial singularities occur whenever θ1 −α1 = π/2 + kπ, or
θ2 −α2 = π/2 + kπ, for k = 1, 2, ... (Figure 6), i.e whenever AC is orthogonal to CP or BD is
orthogonal to DP.
4
P
A
C
B
D
r1
r2
Figure 6: Serial singularity
2.6
Application to Machining
For a machine tool with three axes as in (Figure 1), the motion of the table is performed
along two perpendicular axes.
The joint limits of each actuator give the dimension of the
Cartesian workspace. For the parallel mechanisms studied, this transformation is not direct. The
resulting Cartesian workspace is more complex and its size smaller. We want to have a Cartesian
workspace which will be close to the Cartesian workspace of an industrial serial machine tool.
For our 2-DOF mechanisms, we will prescribe a rectangular shape Cartesian workspace. In
addition, the workspace must be reduced to a t-connected region, i.e. a region free of serial and
parallel singularities [9]. Finally, we want to prescribe relatively stable kinetostatic properties
in the workspace.
2.7
Velocity Amplification Study
In order to keep reasonable and homogeneous kinetostatic properties in the Cartesian workspace,
we study the manipulability ellipsoids of velocity defined by the inverse Jacobian matrix J−1
[1]. For the mechanisms at hand, the inverse Kinematic Jacobian matrix J−1 given in equation
(3b) is simple. In this case, the matrices B and J−1 are written simply,
B = 1
L1

1/c1
0
0
1/c2

and J−1 = 1
L1

(1/c1)(p −c)T
(1/c2)(p −d)T

with ci = cos(θi −αi) i = 1, 2
The square roots γ1 and γ2 of the eigenvalues of (JJT )−1 are the values of the semi-axes
of the ellipse which define the two factors of velocity amplification (from the joint rates to the
output velocities), λ1 = 1/γ1 and λ2 = 1/γ2, according to these principal axes. To limit the
variations of this factor in the Cartesian workspace, we pose the following constraints,
1/3 < λi < 3
(4)
This means that for a given joint velocity, the output velocity is either at most three times
larger or, at least, three times smaller. This constraint also permits to limit the loss of rigidity
(velocity amplification lowers rigidity) and of accuracy (velocity amplification also amplifies the
encoder resolution). The values in equation (4) were chosen as an example and should be defined
precisely as a function of the type of machining tasks.
5
3
Comparative Study
3.1
The Three Parallel Mechanism Architectures Studied
The three parallel mechanism architectures studied are the following:
1. The biglide1 mechanism with α1 = 0 and α2 = π (Fig. 7)
2. The biglide2 mechanism with α1 = π/2 and α2 = π/2 (Fig. 8)
3. The orthoglide mechanism with α1 = π/4 and α2 = 3π/4 (Fig. 9)
The biglide1 mechanism studied (Fig. 7) has been used for example in the hexaglide and in
the triglide [10].
P
A
C
B
D
r1
r2
L
L0
L
Figure 7: The biglide1 mechanism
The biglide2 mechanism (Fig. 8) has been used in the Linapod and in [10, 12].
P
A
C
B
D
r1
r2
L
L0
L
Figure 8: The biglide2 mechanism
L0
r1
r2
C
A
B
D
P
L
L
Figure 9: The orthoglide mechanism
The third mechanism (Fig. 9) was introduced in [11] and extended to 3-DOF in [10]. The
main constraint of this design is AC ⊥BD, which makes it isotropic, i.e. the Jacobian matrix
of this mechanism is isotropic in some configurations.
3.2
Determination of the Mechanism Dimensions
To determine the mechanism dimensions, we proceed in several steps as follows. Let L = L1 = L2
be the common link lengths, let L0 be the distance between the attachment points A and B of
the prismatic joints and let ∆ρ be the range of the actuated joints. The length L0 and the joint
6
range ∆ρ are determined consecutively as function of L by using the condition on the velocity
amplification factor (see section 2.7). For all mechanisms, we can show that the maximal (resp.
minimal) velocity amplification factor is reached at the configuration for which the distance
between C and D is maximal (resp. minimal). For the biglide1 and for the orthoglide, the
maximal (resp. minimal) velocity amplification factor is reached at the configuration where C
is on A and D is on B (resp. where C is on C’ and D is on D’) (figures 10 and 12).
A
r1
L
L0
P
B
r2
L
C’
D’
Figure 10: The biglide1 mechanism
P
A
C’
B
r1
r2
L
L
L0
D’
Figure 11: The biglide2 mechanism
L0
r1
r2
A
B
D’
P
L
L
C’
Figure 12: The orthoglide mechanism
By first writing that the maximal factor must be smaller than 3 in the first configuration, we
can calculate L0. Then ∆ρ is calculated by writing that in the opposite configuration the velocity
amplification factor must be larger than 1/3. For the biglide2, the maximal (resp. minimal)
amplification factor is reached at the configuration where C is on A and D is on D’(resp. where
C and D lie on an horizontal line) (figure 11). In this case, we first calculate L0 at the minimal
factor configuration and ∆ρ is then calculated at the maximal factor configuration. The values of
L0 and ∆ρ obtained for all mechanisms are given in the first two rows of table 1. All derivations
and computations have been obtained with MAPLE.
Then, for each mechanism, we determine the maximum rectangular surface S which can be
included in the Cartesian workspace (figures 13 to 15). We have used the parametric sketcher
of a CAD system to perform this task. The area of the surfaces S obtained are given in the last
row of table 1. The last step is the scaling of the mechanism link dimensions and joint ranges in
7
Mechanism
L0
∆ρ
S
Biglide1
1.946L
0.547L
0.107L
Biglide2
0.458L
0.529L
0.249L
Orthoglide
1.961L
1.109L
0.885L
Table 1: Dimensions and rectangular Cartesian workspace surface as function of L
order to have a same Cartesian workspace rectangular for all mechanisms. The first three rows
of table 2 give the resulting link dimensions and joint ranges.
y
x
Figure 13:
The Cartesian
workspace of the Biglide1
x
y
Figure 14:
The Cartesian
workspace of the Biglide2
x
y
Figure 15:
The Cartesian
workspace of the Orthoglide
3.3
Comparison of the Mechanism Size Envelopes
Table 2 provides the mechanism dimensions and envelope sizes for the three parallel mechanisms
studied, for a prescribed rectangular Cartesian workspace surface of 1 m2.
Mechanism
L0
L
∆ρ
Mechanism envelope surface
Biglide1
5.95
3.05
1.67
16.45
Biglide2
0.92
2.00
1.06
8.50
Orthoglide
2.08
1.06
1.18
3.91
Table 2: Mechanism dimensions and envelope sizes for a same rectangular Cartesian workspace
Figs. 16, 17 and 18 show the three mechanisms along with their Cartesian workspace and the
same rectangular surface in it. We can notice that the orthoglide mechanism has smaller lengths
struts i.e. smaller mass in motion and thus higher dynamic performances than the other two
mechanisms. The biglide2 and the orthoglide mechanisms have similar values of ∆ρ. It should
be noticed, also, that the Cartesian workspace of the biglide2 includes a rectangulle which is far
from a square, whereas it is an exact square for the other two mechanisms. We have calculated
the dimensions of the biglide2 for a square of 1 m2 in its workspace and we have obtained
L0 = 1.075, L = 2.348 and ∆ρ = 1.242.
4
Conclusions
Three 2-DOF parallel mechanisms dedicated to machining applications have been compared in
this paper. The link dimensions and the actuated joint ranges have been calculated for a same
8
y
x
Figure 16: Workspace of the biglide1 mechanism
x
y
Figure 17: Workspace of the biglide2 mech-
anism
x
y
Figure 18:
Workspace of the orthoglide
mechanism
prescribed rectangular Cartesian workspace with identical kinetostatic constraints. The machine
size of each resulting design was used as a comparative criterion. One of the mechanisms, the
orthoglide, was shown to have lower dimensions than the other two mechanisms. This result
shows that the isotropic property of the orthoglide induces interesting additional features like
better compactness and lower inertia. In the future, the comparative study will be continued
using dynamic performance indices.
References
[1] Yoshikawa T., “ Manipulability and redundant control of mechanisms”, Proc. IEEE, Int.
Conf. Rob. And Aut., pp. 1004–1009, 1985.
[2] Merlet J.P., Les robots parall`eles, 2nd ´edition, Herm`es, Paris, 1997.
[3] Chablat D., “Domaines d’unicit´e et parcourabilit´e pour les manipulateurs pleinement par-
all`eles”, PhD Thesis, Nantes, Novembre, 1998.
[4] Gosselin, C. and Angeles, J.
“Singularity Analysis of Closed-Loop Kinematic Chains”,
IEEE Trans. on Robotics and Automation, Vol. 6, No. 3, pp. 281–290, 1990.
9
[5] Golub G.H. and Van Loan C.F., Matrix Computations, The John Hopkins University Press,
Baltimore, 1989.
[6] Wenger, P. Gosselin, C. and Maille. B.
“A Comparative Study of Serial and Parallel
Mechanism Topologies for Machine Tools”, Proc. PKM’99, Milano, pp. 23–32, 1999.
[7] Kim, J. Park, C. Kim, J. and Park, F.C. 1997, “Performance Analysis of Parallel Manip-
ulator Architectures for CNC Machining Applications”, Proc. IMECE Symp. On Machine
Tools, Dallas.
[8] Treib, T. and Zirn, O.
“Similarity laws of serial and parallel manipulators for machine
tools”, Proc. Int. Seminar on Improving Machine Tool Performance, pp. 125–131, Vol. 1,
1998.
[9] Chablat, D. and Wenger, Ph. “On the Characterization of the Regions of Feasible Trajec-
tories in the Workspace of Parallel Manipulators”, in Proc. Tenth World Congress on the
Theory of Machines and Mechanisms, Vol. 3, pp. 1109–1114, Oulu, June, 1999.
[10] Wenger, P. and Chablat, D.
“Kinematic Analysis of a new Parallel Machine Tool: the
Orthoglide”, in Lenarˇciˇc, J. and Staniˇsi´c, M.M. (editors), Advances in Robot Kinematic,
Kluwer Academic Publishers, pp. 305–314, June, 2000.
[11] Chablat D. Wenger P. and Angeles J., “Conception Isotropique d’une morphologie par-
all`ele: Application l’usinage”, 3th Int. Conf. On Integrated Design and Manufacturing in
Mechanical Engineering, Montreal, May 2000.
[12] Horn, W. and Konold, T. “Parallelkinematiken f¨ur die Metallbearbeitung in der Automobil-
Massenproducktion” Proc. Working Accuracy of Parallel Kinematics, Chemnitz, pp. 273–
290, 2000.
10
arXiv:0707.3665v1  [cs.RO]  25 Jul 2007
A Comparative Study of Parallel Kinematic
Architectures for Machining Applications
Philippe Wenger1, Cl´ement Gosselin2 and Damien Chablat1
1Institut de Recherche en Communications et Cybern´etique de Nantes ∗
1, rue de la No¨e, 44321 Nantes, France
2D´epartement de g´enie m´ecanique, Universit´e Laval
Qu´ebec, Qu´ebec, Canada, G1K 7P4
Philippe.Wenger@irccyn.ec-nantes.fr
Abstract: Parallel kinematic mechanisms are interesting alternative designs for machining ap-
plications. Three 2-DOF parallel mechanism architectures dedicated to machining applications
are studied in this paper. The three mechanisms have two constant length struts gliding along
fixed linear actuated joints with different relative orientation. The comparative study is con-
ducted on the basis of a same prescribed Cartesian workspace for the three mechanisms. The
common desired workspace properties are a rectangular shape and given kinetostatic perfor-
mances. The machine size of each resulting design is used as a comparative criterion. The
2-DOF machine mechanisms analyzed in this paper can be extended to 3-axis machines by
adding a third joint.
1
Introduction
Most industrial 3-axis machine tools have a PPP kinematic architecture with orthogonal joint
axes along the x, y, z directions. Thus, the motion of the tool in any of these direction is linearly
related to the motion of one of the three actuated axes. Also, the performances (e.g. maximum
speeds, forces, accuracy and rigidity) are constant in the most part of the Cartesian workspace,
which is a parallelepiped. In contrast, the common features of most existing PKM (Parallel
Kinematic Machine) are a Cartesian workspace shape of complex geometry and highly non lin-
ear input/output relations. For most PKM, the Jacobian matrix which relates the joint rates
to the output velocities is not constant and not isotropic. Consequently, the performances may
vary considerably for different points in the Cartesian workspace and for different directions at
one given point, which is a serious drawback for machining applications [8]. To be of interest
for machining applications, a parallel kinematic architecture should preserve good workspace
properties (regular shape and acceptable kinetostatic performances throughout). It is clear that
some parallel architectures are more appropriate than others, as it has already been shown in
previous studies [6, 7].
The aim of this paper is to compare three parallel kinematic archi-
tectures. To limit the analysis, the study is conducted for 2-DOF mechanisms but the results
can be extrapolated to 3-DOF architectures. The three mechanisms studied have two constant
length struts gliding along fixed linear actuated joints with different relative orientation. Each
∗IRCCyN: UMR n◦6597 CNRS, ´Ecole Centrale de Nantes, Universit´e de Nantes, ´Ecole des Mines de Nantes
1
mechanism is defined by a set of three design variables. Given a prescribed Cartesian rectan-
gular region with given kinetostatic performances, we calculate the link dimensions and joint
ranges of each mechanism for which the prescribed region is included in a t-connected region of
the mechanism and the kinetostatic constraints are satisfied. Then, we compare the size of the
resulting mechanisms. The organisation of this paper is as follows. The next section presents
the mechanism studied. Section 3 is devoted to the comparison of three architectures. The last
section concludes this paper.
2
Kinematic study
2.1
Serial Topology with Three Degrees of Freedom
Most industrial machine tools use a simple PPP serial topology with three orthogonal prismatic
joint axes (Figure 1).
X
Z
Y
Figure 1: Typical industrial 3-axis machine-tool
For a PPP topology, the kinematic equations are :
J ˙ρ = ˙p
with
J = 13×3
where ˙p = [x y z]T is the velocity-vector of the tool center point P and ˙ρ = [ρ1 ρ2 ρ3]T is
the velocity-vector of the prismatic joints. The Jacobian kinematic matrix J being the identity
matrix, the ellipsoid of manipulability of velocity and of force [1] is a unit sphere for all the
configurations in the Cartesian workspace.
The problem of the PPP topology is that the
actuator controlling the Y axis supports at the same time the workpiece and the actuator
controlling the displacement of the X axis, which affects the dynamic performances. To solve
this problem, it is possible to use more suitable kinematic architectures like parallel or hybrid
topologies.
2.2
The Parallel Mechanisms Studied
We focus our study on the use of a 2-DOF parallel mechanism (Figure 2) for the motion of the
table of the machine tool depicted in (Figure 1).
The joint variables are the variables ρ1 and ρ2 associated with the two actuated prismatic
joints and the output variables are the position of the tool center point P = [x y]T .
The
mechanisms can be parameterized by the lengths L0, L1 and L2, the angles α1 and α2 and the
actuated joint ranges ∆ρ1 and ∆ρ2 (Figure 2). To reduce the number of design parameters,
2
P
A
C
B
D
r1
a2
q2
q1
a1
r2
L1
L2
Figure 2: Two degree-of-freedom paral-
lel mechanism
P
A
C
B
D
r1
r2
Figure 3: Two degree-of-freedom paral-
lel mechanism with control of the ori-
entation
we impose L1 = L2 and ∆ρ1 = ∆ρ2. This simplification also provides symmetry and, in turn,
reduces the manufacturing costs.
To control the orientation of the reference frame attached to the tool center point P, two
parallelograms can be used which also increase the rigidity of the structure (Figure 3).
2.3
Kinematics of the Parallel Mechanism Studied
The velocity ˙p of P can be written in two different ways. By traversing the closed-loop (ACP −
BDP) in the two possible directions, we obtain
˙p = ˙c + ˙θ1E(p −c)
(1a)
and
˙p = ˙d + ˙θ2E(p −d)
(1b)
where E is the rotation matrix,
E =
 0
−1
1
0

c and d represent the position vector of the points C and D, respectively.
Moreover, the velocity ˙c and ˙d of the points C and D are given by,
˙c =
c −a
||c −a|| ˙ρ1 =
 cos(α1)
sin(α1)

˙ρ1
,
˙d =
d −b
||d −b|| ˙ρ2 =
 cos(α2)
sin(α2)

˙ρ2
The two unactuated joint rates ˙θ1 and ˙θ2 can be eliminated from equations (1a) and (1b) by
dot-multiplying the former by p −c and the latter by p −d, thus obtaining
(p −c)T ˙p = (p −c)T
c −a
||c −a|| ˙ρ1
(2a)
(p −d)T ˙p = (p −d)T
d −b
||d −b|| ˙ρ2
(2b)
Equations (2a) and (2b) can now be cast in vector form, namely,
A ˙p = B ˙ρ
3
where A and B are, respectively, the parallel and serial Jacobian matrices, defined as
A =
 (p −c)T
(p −d)T

,
B =
 (p −c)T ((c −a)/||c −a||)
0
0
(p −d)T ((d −b)/||d −b||)

and with ˙ρ defined as the vector of actuated joint rates and ˙p defined as the vector of velocity
of point P:
˙ρ =

˙ρ1
˙ρ2

,
˙p =

˙x
˙y

When A and B are not singular, we can study the Jacobian kinematic matrix J [2],
˙p = J ˙ρ
with
J = A−1B
(3a)
or the inverse Jacobian kinematic matrix J−1, such that
˙ρ = J−1 ˙p
with
J−1 = B−1A
(3b)
2.4
Parallel Singularities
The parallel singularities occur when the determinant of the matrix A vanishes [3, 4], i.e. when
det(A) = 0. In this configuration, it is possible to move locally the tool center point whereas
the actuated joints are locked.
These singularities are particularly undesirable, because the
structure cannot resist any force and control is lost. To avoid any deterioration, it is necessary
to eliminate the parallel singularities from the workspace.
For the mechanism studied, the parallel singularities occur whenever the points C, D, and P
are aligned (Figure 4), i.e. when θ1 −θ2 = kπ, for k = 1, 2, ....
P
A
C
B
D
r1
r2
Figure 4: Parallel singularity
P
A
C
B
D
r1
r2
Figure 5: Structural singularity
They are located inside the Cartesian workspace and form the boundaries of the joint workspace.
Moreover, structural singularities can occur when L1 is equal to L2 (Figure 5). In these config-
urations, the control of the point P is lost.
2.5
Serial Singularities
The serial singularities occur when the determinant of the matrice B vanishes, i.e.
when
det(B) = 0. When the manipulator is in such configurations, there is a direction along which no
Cartesian velocity can be produced. The serial singularities define the boundary of the Cartesian
workspace [Merlet 97].
For the topology studied, the serial singularities occur whenever θ1 −α1 = π/2 + kπ, or
θ2 −α2 = π/2 + kπ, for k = 1, 2, ... (Figure 6), i.e whenever AC is orthogonal to CP or BD is
orthogonal to DP.
4
P
A
C
B
D
r1
r2
Figure 6: Serial singularity
2.6
Application to Machining
For a machine tool with three axes as in (Figure 1), the motion of the table is performed
along two perpendicular axes.
The joint limits of each actuator give the dimension of the
Cartesian workspace. For the parallel mechanisms studied, this transformation is not direct. The
resulting Cartesian workspace is more complex and its size smaller. We want to have a Cartesian
workspace which will be close to the Cartesian workspace of an industrial serial machine tool.
For our 2-DOF mechanisms, we will prescribe a rectangular shape Cartesian workspace. In
addition, the workspace must be reduced to a t-connected region, i.e. a region free of serial and
parallel singularities [9]. Finally, we want to prescribe relatively stable kinetostatic properties
in the workspace.
2.7
Velocity Amplification Study
In order to keep reasonable and homogeneous kinetostatic properties in the Cartesian workspace,
we study the manipulability ellipsoids of velocity defined by the inverse Jacobian matrix J−1
[1]. For the mechanisms at hand, the inverse Kinematic Jacobian matrix J−1 given in equation
(3b) is simple. In this case, the matrices B and J−1 are written simply,
B = 1
L1

1/c1
0
0
1/c2

and J−1 = 1
L1

(1/c1)(p −c)T
(1/c2)(p −d)T

with ci = cos(θi −αi) i = 1, 2
The square roots γ1 and γ2 of the eigenvalues of (JJT )−1 are the values of the semi-axes
of the ellipse which define the two factors of velocity amplification (from the joint rates to the
output velocities), λ1 = 1/γ1 and λ2 = 1/γ2, according to these principal axes. To limit the
variations of this factor in the Cartesian workspace, we pose the following constraints,
1/3 < λi < 3
(4)
This means that for a given joint velocity, the output velocity is either at most three times
larger or, at least, three times smaller. This constraint also permits to limit the loss of rigidity
(velocity amplification lowers rigidity) and of accuracy (velocity amplification also amplifies the
encoder resolution). The values in equation (4) were chosen as an example and should be defined
precisely as a function of the type of machining tasks.
5
3
Comparative Study
3.1
The Three Parallel Mechanism Architectures Studied
The three parallel mechanism architectures studied are the following:
1. The biglide1 mechanism with α1 = 0 and α2 = π (Fig. 7)
2. The biglide2 mechanism with α1 = π/2 and α2 = π/2 (Fig. 8)
3. The orthoglide mechanism with α1 = π/4 and α2 = 3π/4 (Fig. 9)
The biglide1 mechanism studied (Fig. 7) has been used for example in the hexaglide and in
the triglide [10].
P
A
C
B
D
r1
r2
L
L0
L
Figure 7: The biglide1 mechanism
The biglide2 mechanism (Fig. 8) has been used in the Linapod and in [10, 12].
P
A
C
B
D
r1
r2
L
L0
L
Figure 8: The biglide2 mechanism
L0
r1
r2
C
A
B
D
P
L
L
Figure 9: The orthoglide mechanism
The third mechanism (Fig. 9) was introduced in [11] and extended to 3-DOF in [10]. The
main constraint of this design is AC ⊥BD, which makes it isotropic, i.e. the Jacobian matrix
of this mechanism is isotropic in some configurations.
3.2
Determination of the Mechanism Dimensions
To determine the mechanism dimensions, we proceed in several steps as follows. Let L = L1 = L2
be the common link lengths, let L0 be the distance between the attachment points A and B of
the prismatic joints and let ∆ρ be the range of the actuated joints. The length L0 and the joint
6
range ∆ρ are determined consecutively as function of L by using the condition on the velocity
amplification factor (see section 2.7). For all mechanisms, we can show that the maximal (resp.
minimal) velocity amplification factor is reached at the configuration for which the distance
between C and D is maximal (resp. minimal). For the biglide1 and for the orthoglide, the
maximal (resp. minimal) velocity amplification factor is reached at the configuration where C
is on A and D is on B (resp. where C is on C’ and D is on D’) (figures 10 and 12).
A
r1
L
L0
P
B
r2
L
C’
D’
Figure 10: The biglide1 mechanism
P
A
C’
B
r1
r2
L
L
L0
D’
Figure 11: The biglide2 mechanism
L0
r1
r2
A
B
D’
P
L
L
C’
Figure 12: The orthoglide mechanism
By first writing that the maximal factor must be smaller than 3 in the first configuration, we
can calculate L0. Then ∆ρ is calculated by writing that in the opposite configuration the velocity
amplification factor must be larger than 1/3. For the biglide2, the maximal (resp. minimal)
amplification factor is reached at the configuration where C is on A and D is on D’(resp. where
C and D lie on an horizontal line) (figure 11). In this case, we first calculate L0 at the minimal
factor configuration and ∆ρ is then calculated at the maximal factor configuration. The values of
L0 and ∆ρ obtained for all mechanisms are given in the first two rows of table 1. All derivations
and computations have been obtained with MAPLE.
Then, for each mechanism, we determine the maximum rectangular surface S which can be
included in the Cartesian workspace (figures 13 to 15). We have used the parametric sketcher
of a CAD system to perform this task. The area of the surfaces S obtained are given in the last
row of table 1. The last step is the scaling of the mechanism link dimensions and joint ranges in
7
Mechanism
L0
∆ρ
S
Biglide1
1.946L
0.547L
0.107L
Biglide2
0.458L
0.529L
0.249L
Orthoglide
1.961L
1.109L
0.885L
Table 1: Dimensions and rectangular Cartesian workspace surface as function of L
order to have a same Cartesian workspace rectangular for all mechanisms. The first three rows
of table 2 give the resulting link dimensions and joint ranges.
y
x
Figure 13:
The Cartesian
workspace of the Biglide1
x
y
Figure 14:
The Cartesian
workspace of the Biglide2
x
y
Figure 15:
The Cartesian
workspace of the Orthoglide
3.3
Comparison of the Mechanism Size Envelopes
Table 2 provides the mechanism dimensions and envelope sizes for the three parallel mechanisms
studied, for a prescribed rectangular Cartesian workspace surface of 1 m2.
Mechanism
L0
L
∆ρ
Mechanism envelope surface
Biglide1
5.95
3.05
1.67
16.45
Biglide2
0.92
2.00
1.06
8.50
Orthoglide
2.08
1.06
1.18
3.91
Table 2: Mechanism dimensions and envelope sizes for a same rectangular Cartesian workspace
Figs. 16, 17 and 18 show the three mechanisms along with their Cartesian workspace and the
same rectangular surface in it. We can notice that the orthoglide mechanism has smaller lengths
struts i.e. smaller mass in motion and thus higher dynamic performances than the other two
mechanisms. The biglide2 and the orthoglide mechanisms have similar values of ∆ρ. It should
be noticed, also, that the Cartesian workspace of the biglide2 includes a rectangulle which is far
from a square, whereas it is an exact square for the other two mechanisms. We have calculated
the dimensions of the biglide2 for a square of 1 m2 in its workspace and we have obtained
L0 = 1.075, L = 2.348 and ∆ρ = 1.242.
4
Conclusions
Three 2-DOF parallel mechanisms dedicated to machining applications have been compared in
this paper. The link dimensions and the actuated joint ranges have been calculated for a same
8
y
x
Figure 16: Workspace of the biglide1 mechanism
x
y
Figure 17: Workspace of the biglide2 mech-
anism
x
y
Figure 18:
Workspace of the orthoglide
mechanism
prescribed rectangular Cartesian workspace with identical kinetostatic constraints. The machine
size of each resulting design was used as a comparative criterion. One of the mechanisms, the
orthoglide, was shown to have lower dimensions than the other two mechanisms. This result
shows that the isotropic property of the orthoglide induces interesting additional features like
better compactness and lower inertia. In the future, the comparative study will be continued
using dynamic performance indices.
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