D. Chablat, G. Moroz, P. Wenger
Abstract - Parallel robots admit generally several solutions to
the direct kinematics problem. The aspects are associated with
the maximal singularity free domains without any singular
configurations. Inside these regions, some trajectories are
possible between two solutions of the direct kinematic problem
without meeting any type of singularity: non-singular assembly
mode
trajectories.
An
established
condition
for
such
trajectories is to have cusp points inside the joint space that
must be encircled. This paper presents an approach based on
the notion of uniqueness domains to explain this behaviour. 1
I. INTRODUCTION
The direct and inverse kinematics problem of parallel
robots have been study in many papers to define first the
maximal numbers of solution for each problem and secondly
to characterize the joint space and workspace. The mobile
platform can admit several positions and orientations (or
configurations) in the workspace for one given set of input
joint values. Conversely, the robot can admit several input
joint values for a given mobile platform configurations.
The notion of assembly modes has been defined to
represent the different solutions to the direct kinematic
problem while the notion of working mode has been
introduced to separate the solutions to the inverse kinematic
problem [1].
To cope with the existence of multiple inverse kinematic
solutions in serial mechanisms, the notion of aspects was
introduced [2]. The aspects were defined as the maximal
singularity-free domains in the joint space. The same notion
was extended for parallel mechanism with several inverse
and direct kinematic solutions [1, 3].
For path planning, we need to define a one-to-one
mapping between the joint space and the workspace, which
makes it possible to associate one single solution to the
inverse and direct kinematic problem. One way to solve this
problem is to introduce the definition of the uniqueness
domains. Like for serial mechanisms, the aspects do not
define the uniqueness domains of the inverse and direct
kinematic problem because some parallel robots are able to
1 Manuscript received September 10, 2010. The research work reported
here was made possible by SiRoPa ANR Project.
D Chablat, G. Moroz and P. Wenger are with the IRCCyN, Nantes,
44321 France (corresponding author: +33240376958; fax: +33240376930;
e-mail: Damien.Chablat @irccyn.ec-nantes.fr).
change
assembly-mode
without
passing
through
a
singularity, thus meaning that there is more than one direct
kinematic solution in one aspect [4]. This feature was first
analyzed for the 3-RPR parallel robot and more recently for
other ones such as the RPR-2PRR [5].
The change of assembly-mode was first analyzed in the
joint space. However, it did not make it possible to explain
the non-singular assembly-mode phenomenon. To solve this
problem, a configuration-space was defined by the input
joint value plus one coordinate of the platform configuration
[6]. This approach makes it possible to show that a cusp
point is encircled during a non-singular assembly-mode
motion. A second problem is to find trajectories that induce
an assembly mode changing. This problem can be solved by
defining the uniqueness domains as defined for serial robots
in [7] and for parallel robots in [8].
To compute the aspects and the uniqueness domains, new
algebraic tools based on Groebner basis are introduced in
this paper. These tools make it possible to obtain the analytic
expression of the border of these domains and, by using a
cylindrical algebraic decomposition, they define completely
these regions. As a matter of fact, only numerical methods
have been used to generate these regions, such as Octree
models by the subdivision of the joint space and workspace).
The paper is organised as follows. In the next section, we
recall the notion of working mode, aspects and uniqueness
domains. Then, we will introduce the algebraic tools used for
the first time to describe these domains. Finally, assembly-
mode changing motions are analyzed with an example
trajectory.
II. DEFINITION OF THE UNIQUENESS DOMAINS
In this section, we recall briefly the definition of
uniqueness domains.
A. Definition of the kinematics
The vector of input variables q and the vector of output
variables X for a n-DOF parallel manipulator are linked by a
system of non linear algebraic equations:
,
F
q X
0
(1)
where 0 is the n-dimensional zero vector. Differentiating
Uniqueness domains and non singular assembly mode changing
trajectories
(1) with respect to time leads to the velocity model:
0
A X
B q
(2)
where A et B are n n Jacobian matrices. These matrices are
functions of q and X:
F
F
A
B
X
q
(3)
These matrices are useful for the determination of the
singular configurations [9].
B. Working modes
The notion of working modes was introduced in [1] for
parallel manipulators with several solutions to the inverse
kinematic problem and whose matrix B is diagonal.
A working mode, denoted by
i
Mf , is the set of robot
configurations for which the sign of
jj
B (
1,
,
j
n
for a
parallel manipulator with n degrees of freedom) does not
change and
jj
B
does not vanish. A robot configuration is
represented by the vector (X, q).
(
,
)
such that sign(
)=cst
for
1,
,
and det(
)
0
jj
i
W
Q
B
Mf
j
n
X q
B
Therefore, the set of working modes (
i
Mf , i
I
) is
obtained using all combinations of sign of each term
jj
B .
Changing working mode is equivalent to changing the
posture of one or several legs. A working mode is defined in
W
Q
because the terms of
jj
B
depend on both X and q.
For a working mode
i
Mf , we have only one inverse
kinematic solution. So, we can define an application that
maps X onto q:
ig
X
q
(4)
Then the images in W of a posture q in Q is denoted by:
1
\ (
,
)
i
i
g
M f
q
X
X q
(5)
C. Generalized aspect
The generalized aspects
ij
A were defined in [1] as the
maximal sets in W
Q
such that
is connected
,
\det
0
ij
ij
ij
i
A
W
Q
A
A
Mf
X q
A
(6)
The projection
W
of the generalized aspects
ij
A onto the
workspace are the regions
ij
W A
W
and are also connected.
These regions, called W-aspects, define the maximal
singularity-free regions of the workspace for a given
working mode Mfi.
The projection
Q
of the generalized aspects onto the
jointspace are the regions
ij
QA
Q
and are also connected.
These regions, called Q-aspects, define the maximal
singularity-free regions of the joint space for a given
working mode Mfi.
D. Characteristic surfaces
The characteristic surfaces were introduced in [10] to
define the uniqueness domains for serial cuspidal robots.
This definition was extended to parallel robots with one
inverse kinematic solution in [3] and to parallel robots with
several inverse kinematic solutions in [8].
Let
ij
W A be one W-aspect. The characteristic surfaces,
denoted by
(
)
C
ij
S
WA
, are defined as the preimage in
ij
W A of
the boundary
ij
WA
that delimits
ij
W A
1
C
ij
i
i
ij
ij
S
WA
g
g
WA
WA
(7)
where :
ig is defined in eq. (4)
1
ig
is a notation defined in eq. (5). Let C Q :
1 (
)
/
i
i
g
W
g
C
C
X
X
When the robot admits only two W-aspects for each
working mode, the characteristic surfaces coincide with the
pseudo-singularities defined by:
1
ij
i
i
ij
Sc WA
g
g
WA
(8)
E. Basic components and basic regions
Let
ij
W A be an W-aspect. The basic regions of
ijk
WA
,
denoted
,
ijk
WAb
k
K
, are defined as the connected
components of the set
ij
C
ij
W A
S
W A
( means the
difference between sets). The basic regions induce a
partition on
ij
W A :
ij
k
K
ijk
C
ij
WA
WAb
S
WA
Let
ijk
ijk
QAb
g WAb
,
ijk
QAb
is a domain in the
reachable joint space Q called basic components. Let
ij
W A
an W-aspect and
ij
Q A its image under g. The following
relation holds:
ij
k
K
ijk
C
ij
QA
QAb
g S
WA
F. Uniqueness domain
The uniqueness domains
il
Wu are the union of two sets,
(i) the set of adjacent basic regions
'
(
)
k
K
ijk
WAb
of the
same W-aspect
ij
W A whose respective preimages are disjoint
basic components, and (ii) the set of the characteristic
surfaces
C
ijk
S
WAb
for
'
k
K
which separate these basic
components:
'
il
k
K
ijk
C
ijk
Wu
WAb
S
WAb
(9)
with
'
K
K
such that
1
2
,
'
k
k
JK
,
1
2
ij
ij
g WAb
g WAb
.
III. ALGEBRAIC TOOLS
A. Projection or Groebner basis elimination
We use the Groebner basis theory to compute the
projections Q and W . Let P be a set of polynomials in the
variables X=(x1, .., xn) and q=(q1, .., qn). Moreover, let V be
the set of common roots of the polynomial in P, let W be the
projection of V on the workspace and Q the projection on the
joint space. It might not be possible to represent W (resp. Q)
by polynomial equations. Let W (resp. Q ) be the smallest
set defined by polynomial equations that contain W (resp.
Q). Our goal is to compute the polynomial equations
defining W (resp. Q ).
A Groebner basis P is a polynomial system equivalent to
P, satisfying some additional specific properties. The
Groebner basis of a system depends on the chosen ordering
on the monomials (cf [11], Chapter 3).
For the projection W , when we choose an ordering
eliminating q, the Groebner basis of P contains exactly the
polynomials defining W .
For the projection Q , when we choose an ordering
eliminating X, the Groebner basis of P contains exactly the
polynomials defining Q .
B. Discussing the number of solutions of the parametric
system
The joint space (resp. workspace) analysis requires the
discussion of the number of solutions of the parametric
system associated with the direct (resp. inverse) kinematics.
More precisely we want to decompose the joint space (resp.
workspace) in cells
1
k
,...,C
C
, such that:
i
C is an open connected subset of the joint space (resp.
workspace).
for all joint (resp. pose) values in
i
C , the direct (resp.
inverse) kinematics problem has a constant number of
solutions.
i
C is maximal in the sense that if
i
C is contained in a set
E, then E does not satisfy the first or the second condition.
This analysis is done in 3 steps:
computation of a subset of the jointspace (resp.
workspace) where the number of solutions changes: the
Discriminant Variety.
description of the complementary of the discriminant
variety in connected cells: the Generic Cylindrical Algebraic
Decomposition
connecting the cells that belong to the same connected
component of the complementary of the discriminant
variety: interval comparisons.
From a general point of view, the discriminant variety can
be defined for any system of polynomial equations and
inequalities. Let
1
1
, ...
,
, ...,
m
l
p
p
q
q
be polynomials with
rational coefficients depending on the unknowns
1, ...,
n
X
X
and on the parameters
1, ...,
d
U
U . Let us consider the
constructible set:
n+d
1
1
=
,
( )
0,...,
( )
0,
( )
0,...,
( )
0
m
l
p
p
q
q
v
v
v
v
v
C
C
If we assume that C is a finite number of points for almost
all the parameter values, a discriminant variety
D
V of C is
a variety in the parameter space
d
C such that, over each
connected open set U satisfying
D
V
U
, C defines
an analytic covering. In particular, the number of points of
C over any point of U is constant.
Let us now consider the following semi-algebraic set:
n+d
1
1
,
( )
0,...,
( )
0,
( )
0,...,
( )
0
m
l
p
p
q
q
v
v
v
v
v
S
C
If we assume that S has a finite number of solutions over
at least one real point that does not belong to
D
V , then
d
D
V
R
can be viewed as a real discriminant variety of S ,
with the same property: over each connected open set
d
U
R
such that
D
V
U
, C defines an analytic
covering. In particular, the number of points of R over any
point of U is constant.
Discriminant varieties can be computed using basic and
well known tools from computer algebra such as Groebner
bases (see [16]) and a full package computing such objects
in a general framework is available in Maple software
through the RootFinding[Parametric] package.
C. The complementary of a discriminant variety
At this stage, we know, by construction, that over any
simply connected open set that does not intersect the
discriminant variety (so-called regions), the system has a
constant number of (real) roots. The goal of this part is now
to provide a description of the regions for which the number
of solutions of the system at hand is constant. Accordingly,
we compute an open CAD [12, 13].
Let
d
1,...,
d
U
U
P
Q
be a set of polynomials. For
1...0
i
d
,
we
introduce
a
set
of
polynomials
i
1,...,
d
i
U
U
P
Q
defined by a backward recursion:
d
P : the polynomials defining the discriminant variety
d
P :
1
Discriminant
,
, LeadingCoefficient
,
,
Resultant
, ,
, \
,
i
i
i
i
p U
p U
p
q U
p q
P
We can associate to each
iP an algebraic variety of
dimension at most
1
i :
1
i
i
p
V
V
p
P p
The
i
V are used to define recursively a finite union of
simply connected open subsets of
i
R of dimension i:
1
,
in
k
i k
U
such that
,
i
i k
V
U
, and one point
,i k
u
with
rational coordinates in each
,i k
U
.
In order to define the
,i k
U
, we introduce the following
notations. If p is a univariate polynomial with n real roots:
if
0
Root(
, )
the
real roots of if 1
if
th
l
p l
l
p
l
n
l
n
\[
Moreover, if p is a n-variate polynomial, and v is a
1
n -
uplet, then p
V denotes the univariate polynomial where the
first
1
n variables have been replaced by v .
Roughly speaking, the recursive process defining the
,i k
U
is the following:
For
1
i
, let
1
i
p
p
P p . Taking
,
]Root( ,
);
i k
p k
U
Root( ,
1)[
p k
for k from 0 to n where n is the number of
real roots of
1
p , one gets a partition of R that fits the
above definition. Moreover, one can chose arbitrarily one
rational point
,i k
u
in each
,i k
U
.
Then, let
1
i
p
p
P p . The regions
,i k
U
and the points
,i k
u
are of the form:
1
1
1
1
,
,...,
,
|
:
,...,
Root(
, ) ; Root(
,
1)
i
i
i
i k
i
i
i
v
v
v
v
v
v
p
l
p
l
V
V
v
U
=|
,
1
1
,...,
,
i k
i
i
u
,
with
1
1
1,
1,
1,
,...,
Root(
, ) ; Root(
,
1)
i
i
i
i
j
u
j
u
j
i
i
i
u
p
l
p
l
where ,i
j are fixed integer.
D. Connecting the cells
Finally, we need to connect the cells that belong to the
same connected component in the complementary of the
discriminant variety. This property is represented by an
undirected unweighted graph G where each node represents
a cell: if an edge connects two nodes, then the corresponding
cells are adjacent (i.e. their closures have a non empty
intersection) and are in the same connected component in the
complementary of the discriminant variety.
We use the cylindrical shape of the cells output in the
Cylindrical Algebraic Decomposition to compute the edges
of G. Our method only works when the joint space (resp.
workspace) has dimension 2.
In this case, let
1
2,i
C U
and
2
2, j
C
U
be 2 cells of the
cylindrical algebraic decomposition computed in the
previous subsection. First, a necessary condition for these
cells to be adjacent is that their projection is adjacent. In the
case of dimension 2 cells, the projection of
1
C (resp.
2
C ) on
the horizontal axis is an interval
1I (resp.
2I ). Without loss
of generality, we can assume that the values in
2I are greater
that the values in
1I . In this case,
1I and
2I are adjacent if
and only if the right bound of
1I equals the left bound of
2I .
In the following, we denote this bound by b. Then, let e be a
real value small enough such that
2
b
e
I
. By the
cylindrical properties of
1
C and
2
C , the subset of
1
C (resp.
2
C ) that projects on b
e
(resp. b
e
) is an interval
1
J
(resp.
2
J ). If
1
J and
2
J overlap, then let
1
2
c
J
J
and let
1P (resp.
2
P ) be the point (
, )
b
e c
(resp. (
, )
b
e c
). Then
the two cells
1
C and
2
C are adjacent and belong to the same
connected component of the complementary of the
discriminant variety if the line segment
1
2
[
]
P P
does not cross
the discriminant variety. This property can be checked by
using Descartes' rule of sign [14] or Sturm's theorem [15].
IV. MECHANISM UNDER STUDY
A. Kinematic equations
The aim of this section is to recall briefly the kinematic
equations of the RPR-2PRR mechanism [5].
B 1
B
x y
2( , )
B 3
A 3
A 1
A 2
y
x
C 2
C 3
l
l
a
b
Figure 1 : RPR-2PRR mechanism with
2
3
3
l
l
and
1
a
b
The kinematic equations are defined in [5]
2
2
2
2
2
2
2
2
1
3
3
3
3
3
cos(
)
0
sin(
)
0
(
cos( ))
(
sin( )) -
0
cos(
)
cos( )
0
sin(
)
sin( )
0
l
x
l
y
x
a
y
a
l
b
x
l
b
y
(10)
In the following, we have fixed
2
3
3
l
l
and
1
a
b
in certain units of length that we need not specify. The
position of the end-effector in B2 is such that for a given
value of
y,
we have
either
2
2
arcsin(
/
)
y l
or
2
2
arcsin(
/
)
y l
. This allows us to study the
mechanism in a 2D slice of the workspace or in the joint
space.
B. Singularity analysis
Matrices A and B can be derived from eq. (10). The roots
of the determinant of these matrices define the parallel and
serial singularities. The serial singularities, denoted by
S
S ,
are defined by
1 2 3
2
3
:
cos(
) sin(
)
0
S
l l
S
This
singularity
occurs
when
2
/ 2
k
or
3
0
k
. The parallel singularities, denoted by
P
S , are
defined by
:
cos( )
sin( )
sin( )
sin( ) cos( )
0
P
ya
xa
b
x
ab
S
This singularity occurs whenever the axes (A1B1), (A2B2)
and (A3C3) intersect (possibly at infinity). The parallel
singularities do not depend on the choice of the inverse
kinematic solution.
C. Projection of the singularities into the workspace and
joint space
To determine the polynomial equations that characterize
the serial and parallel singularities in the joint space and
workspace, we use the operators W and Q ,
8
2
2
6
P
1
2
3
1
2
2
2
3
2
3
2
4
4
4
2
3
2
1
4
2
6
3
3
2
4
2
2
3
2
2
(
) :
(42 cos(
)
52
12 cos(
) )
(468 cos(
)
960
1584 cos(
)
558 cos(
)
cos(
)
18 cos(
)
657 cos(
) )
( 2988 cos(
)
5760 cos(
)
4536 cos(
)
2430 cos(
) cos(
)
7168
18432 cos(
)
1
Q S
4
6
2
2
3
3
2
2
4
2
4
2
3
2
1
2
2
2
2
4
3
2
2
3
2
2
2
2
3
2
3
5840 cos(
)
324 cos(
)
13320 cos(
)
cos(
)
7290 cos(
) cos(
) )
(9 cos(
)
18 cos(
) cos(
)
24 cos(
)
9 cos(
)
12 cos(
)
16)(36 cos(
)
32
9 cos(
) )
0
(11)
3
0,
3
0,
(
) :
(2(cos( )
3
)) /(cos( )
1)
0,
(2(cos( )
3
)) /(cos( )
1)
0
S
y
y
x
x
W S
Figure 2 and 3 depict a slice of the workspace for y=1/2
and the joint space, respectively, with in red the serial
singularities and in blue the parallel singularities.
x
Figure
2:
The
workspace
analysis for y=1/2 with in blue the
parallel singularities and in red the
serial singularities
Figure 3: The joint space analysis
for
2
arcsin(1 / 6)
with in blue
the parallel singularities and in red
the serial singularities
For the joint space analysis, the sample plot was obtain for
2
arcsin(1/ 6)
, i.e. another working mode. It can be
noticed as is written in [5] that there exist four cusp points in
this cross-section.
D. The generalized aspects
Form the definition of the parallel and serial singularities
in the workspace, and thanks to the property that the location
of the parallel singularities does not depend on the working
mode, we can define 2x4 generalized aspects.
x
Figure 4: A slice of the
workspace for y=1/2 with 2 W-
aspects
Figure 5: A slice of the joint
space for
2
arcsin(1 / 6)
with 2
Q-aspects
Each W-aspect of Fig. 4 is described by 41 cells. Same for
each Q-aspect of Fig. 5. We do a connectivity analysis to
extract the W-aspects from the set of cells obtained by the
cell decomposition. Then, we add the constraint on the sign
of
W
P
S
to isolate the red and blue regions.
3
2
1
0
-1
-2
-3 -2 -1
0
3
2
1
4
3
2 1
-2
-1
-3
x
y
0
Figure 6: The two W-aspects associated with a same working mode
Figure 6 represents the W-aspects obtained with the CAD
decomposition. The borders are the projection onto the
workspace of the serial and parallel singularities. Each W-
aspect is described by 411 cells. For instance, the cell 116 is
defined by the set of points x0, y0, 0:
2
3
4
0
2
2
2
2
2
0
3
4
2
2
0
2
2
in
Root 4
,1 ;Root 28
22
8
17
,1
Root
3, 1 ;
in
9
6
72
198
189
Root
72
9
, 1
Root 4
2
,1 ;
in
Root 4
2
, 2
x
x
x
x
x
x
y
y
y
y x
x y
x
x
x
x
TanHalfphi
x
xTanHalfphi
TanHalfphi
x
xTanHalfphi
The main benefit of the formulation is that we have the
complete definition of the space and we can find easily in
which cell a given point belongs.
E. Characteristic surface
The characteristic surface is defined by:
4
3
2
2
2
2
2
2
: 4
36 sin( )
(32
35 cos( )
108
184
cos( ))
6 sin( )(cos( )
18
14
cos( )
40
)
(cos( )
4
3)(cos( )
4
3)
(cos( )
2 )
0
C
S
y
y
x
x
y
x
x
y
x
x
x
with
. Figure 7 represents the singularities and the
characteristic surfaces. The projections of the cusps points
lie on the intersections of the parallel singularities and the
characteristic surfaces.
Figure 7: The joint space analysis for
2
arcsin(1 / 6)
with in red the
serial singularities, in blue the parallel singularities and in green the
characteristic surface
The two expressions defining the parallel singularities and
the characteristic surfaces can be used to study the kinematic
equations of the robot defined in Eq. (10). As the sign of the
two expressions can be positive or negative, we obtain four
regions in the workspace
0 0
0 0
0 0
0 0
4 2
4 2
4
4
4
4
4
4
4
4
Figure 8 : The workspace analysis for
2
arcsin(1 / 6)
with in red the
number of inverse kinematic solutions and in blue the number of direct
kinematic solution associated with each image
The cell decomposition and the connectivity analysis yield
10 regions, as shown in Fig. 8. The analysis of the inverse
kinematic solution allows us to compute the basic regions.
Figure 9 represents the 4 uniqueness domains for each
working mode.
x
(a)
x
(b)
x
(c)
x
(d)
Figure 9 : The four uniqueness domains with in red and in blue the
common regions.
F. Application to trajectory planning
For any trajectory inside a uniqueness domain do no
change of assembly mode occurs. Figure 10 (a) shows a
trajectory defined between two regions in green. As the end
points of the trajectory are in two separate uniqueness
domains, a non-singular assembly mode changing trajectory
occurs. We can also notice that in Fig. 10 (b), the two
images of the trajectory in the joint space encircle a cusp
point.
V. CONCLUSIONS
In this paper, the notion of uniqueness domains and non-
singular assembly-mode changing motion was revisited and
exemplified using a RPR-2PRR parallel robot. The implicit
definition of the parallel and serial singularities as well as
the characteristic surface were obtained with a new approach
based on algebraic tools such as Discriminant Varieties and
Cylindrical Algebraic Decompositions. Moreover, this
allowed us to get algebraic formula describing the basic
regions, the basic components and the uniqueness domains.
x
(a)
(b)
Figure 10 : Trajectory for y=1/2 and [x,]= [[-1,1], [0,1/2], [1,-1],[1/2,-
2]] (a) in the workspace and (b) in the joint space
VI. REFERENCES
[1] Chablat D., Wenger Ph., “Working Modes and Aspects in Fully-Parallel
Manipulator”, Proceeding IEEE International Conference on Robotics and
Automation, pp. 1964-1969, May 1998
[2] Borrel Paul, “A study of manipulator inverse kinematic solutions with
application to trajectory planning and workspace determination”, Proc.
IEEE Int. Conf on Rob. And Aut., pp 1180-1185, 1986.
[3] Wenger Ph., Chablat D., “Definition Sets for the Direct Kinematics of
Parallel Manipulators”, 8th International Conference in Advanced Robotics,
pp. 859-864, 1997
[4] Innocenti C., Parenti-Castelli V., “Singularity-free evolution from one
configuration to another in serial and fully-parallel manipulators”, Robotics,
Spatial Mechanisms and Mechanical Systems, ASME 1992.
[5] Hernandez, A., Altuzarra, O, Petuya, V., and Macho, E., “Defining
conditions for non singular transitions between assembly modes”, IEEE
Transactions on Robotics, 25:1438–1447, Dec. 2009.
[6] Zein M., Wenger P. and Chablat D., “Non-Singular Assembly-mode
Changing Motions for 3-RPR Parallel Manipulators”, Mechanism and
Machine Theory, Vol 43(4), pp 480-490, 2008.
[7] Wenger, P., “Uniqueness domains and regions of feasible paths for
cuspidal manipulators”, IEEE Transaction on Robotics, Vol. 20(4), pp. 745-
750, Aug. 2004.
[8] Chablat D., Wenger P., “Séparation des solutions aux modèles
géométriques direct et inverse pour les manipulateurs pleinement
parallèles”, Mechanism and Machine Theory, Vol 36/6, pp. 763-783, 2001
[9] Sefrioui J., Gosselin C., “Singularity analysis and representation of
planar parallel manipulators”, Robots and autonomous Systems 10, pp. 209-
224, 1992
[10] Wenger P., “A new general formalism for the kinematic analysis of all
nonredundant manipulators”, IEEE Robotics and Automation, pp. 442-447,
1992.
[11] Cox D., Littleand J., O’Shea, D., “Ideals, Varieties, and Algorithms”,
Undergraduate Texts in Mathematics, Springer Verlag, 1992.
[12] Collins G.E., “Quantifier Elimination for Real Closed Fields by
Cylindrical Algebraic Decomposition”, Springer Verlag, 1975.
[13] Dolzmann A., Seidl A., and Sturm T, “Eficient projection orders for
CAD”, In Jaime Gutierrez, editor, ISSAC, pp. 111-118, ACM, 2004.
[14] Collins G. E. and, Akritas A. G., “Polynomial real root isolation using
descarte's rule of signs”, In SYMSAC'76: Proceedings of the third ACM
symposiumon Symbolic and algebraic computation, pp. 272-275, NewYork,
NY, USA, 1976.
[15] Sturm C., “Mémoire sur la résolution des équations numériques”, In
Jean-Claude Pont, editor, Collected Works of Charles François Sturm, pp.
345-390, Birkhäuser Basel, 2009.
[16] Lazard D. and Rouillier F., “Solving parametric polynomial systems”,
J. Symb. Comput., Vol. 42(6), pp. 636-667, 2007.