arXiv:1012.2662v1  [cs.RO]  13 Dec 2010
Cusp points in the parameter space of
RPR-2PRR parallel manipulators
G. Moroz1,2, D. Chablat1, P. Wenger1, F. Rouiller2
1Institut de Rercheche en Communications et Cybern´etique de Nantes, France,
e-mail:
(Guillaume.Moroz,Damien.Chablat,Philippe.Wenger)@irccyn.ec-nantes.fr
2Laboratoire d’informatique de Paris, France, e-mail:
Fabrice.Rouiller@inria.fr
Abstract. This paper investigates the existence conditions of cusp points in the design parameter
space of the RPR-2PRR parallel manipulators. Cusp points make possible non-singular assembly-
mode changing motion, which can possibly increase the size of the aspect, i.e. the maximum sin-
gularity free workspace. The method used is based on the notion of discriminant varieties and
Cylindrical Algebraic Decomposition, and resorts to Gr¨obner bases for the solutions of systems of
equations.
Key words: Kinematics, Singularities, Cusp, Parallel manipulator, Symbolic computation
1 Introduction
It is well known that the workspace of a parallel manipulator is divided into
singularity-free connected regions [2]. These regions are separated by the so-called
parallel singular configurations, where the manipulator loses its stiffness and gets
out of control. The so-called cuspidal manipulators have the ability to change their
assembly-mode without running into a singularity, which thus may increase the size
of the singularity-free regions [9, 7]. The word “cuspidal” stems from the notion
of cuspidal configuration, defined as one configuration where three direct kinematic
solutions coalesce. A cuspidal configuration in the manipulator joint space allows
non-singular assembly-mode changing motions. Thus, determining cuspidal config-
urations is an important issue that has attracted the attention of several researchers
[9, 6, 13, 1]. In particular, [13] (resp. [1]) has analyzed the cuspidal configurations
of planar 3-RPR (resp. 3-PRR) manipulators1. More recently, [6] studied the RPR-
2PRR, a simpler planar 3-DOF manipulator that lends itself to algebraic calculus
[6]. In both papers, the cusp configurations were determined by looking for the
triple roots of a univariate polynomial. This approach may yield spurious solutions.
In this paper, the cuspidal configuration are determined directly from the Jacobian of
the whole set of geometric constraints of the robot, which guaranties that only true
solutions are obtained. Then, we classify the parameter space of a family of RPR-
2PRR manipulators according to the number of cuspidal configurations. It is shown
that these manipulators have either 0 or 16 cuspidal configurations. The proposed
1 The underlined letter means an actuated joint
1
2
G. Moroz1,2, D. Chablat1, P. Wenger1, F. Rouiller2
method is based on the notion of discriminant varieties and cylindrical algebraic de-
composition, and resorts to Gr¨obner bases for the solutions of systems of equations.
2 Mechanism under study
2.1 Kinematic equations
PSfrag replacements
A1
A2
A3
B1(x,y)
B2
B3
θ1
θ2
θ3
α
ρ1
ρ2
ρ3
a
b
L2
L3
Fig. 1 A RPR-2PRR parallel manipulators
with (a = 1, b = 2, L2 = 2, L3 = 2, x = 1/2,y =
1, θ = 0.2). The actuated joint symbols were
filled in gray.
A RPR-2PRR parallel manipulator is
shown in Fig. 1. This manipulator was an-
alyzed in [6]. It has 1 actuated prismatic
joint ρ1, and 2 passive prismatic joints
ρ2 and ρ3. The two revolute joints cen-
tered in A2 and A3 are actuated while the
ones centered in A1, B1, B2 and B3 are
passive. The pose of the moving platform
is described by the position coordinates
(x,y) of B1 and by the orientation α of
the moving platform B1B2. The input vari-
ables (actuated joints values) are defined
by ρ1, θ2 and θ3. The points B1, B2 and
B3 are aligned, a= (B1, B2), b= (B1, B3),
L2 = (A2,B2) and L3 = (A3,B3).
The geometric constraints can be expressed by the following 5 equations [6]:
f1 : ρ2
1 =x2 +y2
f2 : x =ρ2+L2 cos(θ2)−bcos(α)
f4 : x =L3 cos(θ3)−acos(α)
(1)
f3 : y =L2 sin(θ2)−bsin(α)
f5 : y =ρ3+L3 sin(θ3)−asin(α)
Without loss of generality, we fix a = 1 in the rest of the article.
2.2 An algebraic model
The singular and cuspidal equations were previously computed using the three fol-
lowing steps [6]:
1. Reduce the equation system to a polynomial equation depending on the articular
variables ρ1,θ1,θ2 and one pose variable α: g(ρ1,θ1,θ2,α) = 0 (this step is done
by eliminating variables, either with resultants or with Gr¨obner basis )
2. Add the constraint ∂g
∂α = 0 to define the parallel singularities
3. Add the constraint ∂g
∂α = 0 and ∂2g
∂α2 = 0 to define the cuspidal configurations
This approach has the advantage of reducing the problem of computing the cusp
configurations, to the problem of analysing the triple roots of a single polynomial.
However, this only gives a necessary condition for the manipulator to have cusp
configurations. In particular it is possible that 3 configurations of the robot coalesce
in one coordinate but not in the others.
Cusp points in the parameter space of RPR-2PRR parallel manipulators
3
Let us come back to the theoretical definitions, using Jacobian matrices to define
directly the triple roots of the original system of equations in all the input and output
variables. If P is a list of polynomials and X a list of variables, let Jk(P,X) be the
union of P = {p1,..., pm} and of all the k × k minors of the Jacobian matrix of the
pi with respect to the Xi.
For the analysis of the RPR–2PRR manipulator, we introduce:
Y := [x,y,α,ρ2,ρ3]
W := [b,L2,L3,ρ1,θ2,θ3]
S := { f1,..., f5}.
Using these notations, the parallel singularities of the manipulator are defined by
{v ∈R11, p(v) = 0,q(v) > 0,∀p ∈J5(S,Y),∀q ∈{b,L2,L3,ρ1} so that the cuspidal
configurations are fully characterized by :
S = {v ∈R11, p(v) = 0,q(v) > 0,∀p ∈J5(J5(S,Y),Y),∀q ∈{b,L2,L3,ρ1}}
3 Main tools from computational algebra
The algebraic problem to be solved is basically related to the resolution of polyno-
mial parametric systems.
More specifically, one needs to solve a system of the following form :
E = {v ∈Rn, p1(v) = 0,..., pm(v) = 0,q1(v) > 0,...ql(v) > 0}
where p1,..., pm,q1,...,ql are polynomials with rational coefficients depending
on the unknowns X = [X1,...,Xn] and on the parameters U = [U1,...,Ud].
There are numerous possible ways of solving parametric systems in general. Here
we focus on the use of Discriminant Varieties (DV, [8]) and Cylindrical Algebraic
Decomposition (CAD, [3]) for two reasons. It provides a formal decomposition of
the parameter space through an exactly known algebraic variety (no approximation).
It has been already successfully used for similar mathematical classes of problems
(see [4]).
To reduce the dimension of the parameter space to three so that it can be dis-
played, we set L2 = L3. Not that the proposed method can treat the general case
L2 ̸= L3 without any problems. When L2 = L3, the system to solve is S with the
unknowns [x,y,α,ρ2,ρ3,θ2,θ3] and the parameters [b,L2,ρ1].
3.1 Basic black-boxes
First experiments are often performed for specific values of the parameters, espe-
cially singular and/or degenerated cases. Here, we mainly use exact computations,
namely formal elimination of variables (resultants, Gr¨obner bases) and resolution of
systems with a finite number of solutions, including univariate polynomials.
Let us describe the global solver for zero-dimensional systems. It will be used as
a black box in the general algorithm we describe in the sequel.
Given any system of equations p1 = 0,..., pm = 0 of polynomials of Q[X1,...,Xn],
one first computes a Gr¨obner basis of the ideal < p1,..., pm > for any ordering.
At this stage, one can detect easily if the system has or has not finitely many
complex solutions.
4
G. Moroz1,2, D. Chablat1, P. Wenger1, F. Rouiller2
If yes, then compute a so called Rational Univariate Representation or RUR (see
[10]) of < p1,..., pm >, which is, in short, an equivalent system of the form :
{ f(T) = 0,X1 = g1(T)
g(T) ,...,Xn = gn(T)
g(T) }, T being a new variable that is indepen-
dent of X1,...,Xn, equipped with a so called separating element (injective on the
solutions of the system) u ∈Q[X1,...,Xn] and such that :
V(⟨p1,..., pm⟩) u−→
V(f)
u−1
−−→V(⟨p1,..., pm⟩)
(x1,...,xn)
7→β = u(x1,...,xn) 7→

g1(β)
g(β) ,..., gn(β)
g(β)

defines a bijection between the (real) roots of the system (denoted by V(p1,..., pm))
and the (real) roots of the univariate polynomial (denoted by V(f)).
We then solve the univariate polynomial f, computing so called isolating in-
tervals for its real roots, say non-overlapping intervals with rational bounds that
contain a unique real root of f (see [11]). Finally, interval arithmetic is used for
getting isolating boxes of the real roots of the system (say non overlapping products
of intervals with rational bounds containing a unique real root of the system), by
studying the RUR over the isolating intervals of f. In practice, we use the function
RootFinding[Isolate] from Maple software, which performs exactly the computa-
tions described above.
For example, with (b = 2, L2 = 2, L3 = 3, ρ1 = 2), the polynomial system S
defining the cuspidal configurations has 16 real solutions. One of these solutions is
(ρ2 = 2.30,ρ3 = 1.86,φ = −2.36,x = 1.79,y = 0.885,θ2 = −2.88,θ3 = −0.999).
We observe the three coalescing configurations around this root by calculating the
direct kinematic solutions with θ2 = −2.892 and θ3 = −1.007. These solutions are
shown in Figure 2.
PSfrag replacements
A1
A3
A2
B1
B2
B3
PSfrag replacements
A1
A1
A1
A3
A3
A3
A2
A2
A2
B1B1
B1
B2
B2
B2
B3B3
B3
Fig. 2 A cuspidal configuration (left), and the three converging configurations (right).
3.2 Discriminant varieties
The above method allows one to study instances of the problem and may be used
together with a discretization of the parameter space to get a first idea of the com-
plexity of the general problem to be solved. But the true issue addressed in this paper
is to find criteria on the parameters that allow classifying the configurations to be
studied (for example to distinguish the manipulators having cuspidal configurations
from the others). This leads to a more general problem since one then has to study
non zero-dimensional, semi-algebraic sets.
Let p1,..., pm,q1,...,ql be polynomials with rational coefficients depending on
the unknowns X1,...,Xn and on the parameters U1,...,Ud. Let us consider the con-
structible set :
Cusp points in the parameter space of RPR-2PRR parallel manipulators
5
C = {v ∈Cn ,
p1(v) = 0,..., pm(v) = 0,q1(v) ̸= 0,...ql(v) ̸= 0}
If we assume that C is a finite number of points for almost all the parameter
values, a discriminant variety VD of C is a variety in the parameter space Cd such
that, over each connected open set U satisfying U ∩VD = /0, 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 :
S = {v ∈R11, p(v) = 0,q(v) > 0,∀p(v) ∈J5(J5(S,Y),Y),∀q(v) ∈{b,L2,L3,ρ1}}
If we assume that S has a finite number of solutions over at least one real point
that does not belong to VD, then VD ∩Rd can be viewed as a real discriminant variety
of S , with the same property : over each connected open set U ⊂Rd such that
U ∩VD = /0, 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 Gr¨obner bases (see [8]) and a full package computing
such objects in a general framework is available in Maple software through the
RootFinding[Parametric] package. Figure 3 represents the discriminant variety of
the cuspidal configurations of the RPR-2PRR manipulator.
3.3 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. For that, we compute an open
CAD ([3, 5]).
Let Pd ⊂Q[U1,...,Ud] be a set of polynomials. For i = d −1...0, we introduce
a set of polynomials Pi ⊂Q[U1,...,Ud−i] defined by a backward recursion:
• Pd : the polynomials defining the discriminant variety
• Pi : {Discriminant(p,Ui),LeadingCoefficient(p,Ui), Resultant(p,q,Ui),
p,q ∈Pi+1}
We can associate to each Pi an algebraic variety of dimension at most i−1 : Vi =
V(∏p∈Pi p). Figure 3 and 4 represent respectively V3 and V2 for the manipulator at
hand.
The Vi are used to define recursively a finite union of simply connected open
subsets of Ri of dimension i: ∪ni
k=1Ui,k such that Vi ∩Ui,k = /0, and one point ui,k
with rational coordinates in each Ui,k.
In order to define the Ui,k, we introduce the following notations. If p is a univari-
ate polynomial with n real roots:
Root(p,l) =



−∞if l ≤0
the lth real roots of p if 1 ≤l ≤n
+∞if l > n
6
G. Moroz1,2, D. Chablat1, P. Wenger1, F. Rouiller2
PSfrag replacements
ρ1
L2
b
0
1
2
3
4
5
6
7
8
10
Fig. 3 Discriminant Variety of the cuspidal
configurations
PSfrag replacements
0
1
2
3
4
5
6
7
8
10
ρ1
L2
b
Fig. 4 V2 for the cuspidal configurations of
the manipulator.
Moreover, if p is a n-variate polynomial, and v is a n −1-uplet, then pv denotes
the univariate polynomial where the first n −1 variables have been replaced by v.
Roughly speaking, the recursive process defining the Ui,k is the following:
• For i = 1, let p1 = ∏p∈P1 p. Taking U1,k =]Root(p,k);Root(p,k+1)[ for k from
0 to n where n is the number of real roots of p1, one gets a partition of R that fits
the above definition. Moreover, one can chose arbitrarily one rational point u1,k
in each U1,k.
• Then, let pi = ∏p∈P1 p. The regions Ui,k and the points ui,k are of the form:
Ui,k = {(v1,...,vi−1,vi) |v := (v1,...,vi−1) ∈Ui−1,j,
vi ∈]Root(pv
i ,l),Root(pv
i ,l + 1)[}
ui,k = (β1,...,βi−1,βi), with
(β1,...,βi−1) = ui−1,j
βi ∈]Root(p
ui−1,j
i
,l),Root(p
ui−1,j
i
,l + 1)[
where j,l are fixed integer.
For our example, we get for p3 a trivariate polynomial of degree 33, for p2 a
bivariate polynomial of degree 113, and for p1 a univariate polynomial of degree
59. The zero-dimensional solver then provides the positive real roots of p1 (Table
1), from which we easily deduce the open intervals u1,k. We then use the zero-
dimensional solver to solve every p2(u1,k,U2) and deduce all the tests points of
the U2,k′ in each cells of Figure 4. Finally we use the zero-dimensional solver to
solve every p3(u2,k,U3) and deduce all the tests points of the U3,k′, describing so
the complementary of the discriminant variety.
Table 1 Numerical values of the positive roots of p1
b b1
b2
b3
b4
b5
b6
b7
b8
b9
b10
b11
b12
0.0 0.533 0.564 0.617 0.656 0.707 1.0 1.41 1.52 1.62 1.77 1.88
3.4 Discussing the number of solutions of the parametric system.
At this stage, we have a full description of the complementary of the discriminant
variety of the system to be solved : a recursive process for the construction of each
cell Ud,k and a test point (with rational coordinates) in each of these cells. By def-
inition of the discriminant variety, we know that the system has a constant finite
Cusp points in the parameter space of RPR-2PRR parallel manipulators
7
number of solutions over each of these cells and computing this number for each
cell is the only remaining step. This can be done simply by solving all the systems
S|U=ud,k, k = 1...nd using the zero-dimensional solver.
For our example, the process described in 3.3 returns 344 cells of dimension 3
( U3,1,...,U3,344 ). We solve the system S for each of the 344 associated sample
points, and we get always either 0 or 16 solutions. By selecting only the cells where
the manipulator has 16 cuspidal configurations, we obtain the 58 cells shown in Fig-
ure 5. Table 2 provides the different formula bounding the three dimensional cells
U3,1,...,U3,344 and Table 3 represents the 58 cells of Figure 5, where the manipula-
tor has 16 cuspidal configurations.
Table 2 Formula describing the boundaries of the cells in Table 3.
b1 = 0,b2 = Root(8b6 −11b4 +6b2 −1,2)
L21 (b) =
Root

b2 +1
3 L6
2 −3

b2 +3b +1

b2 −3b +1

(b −1)2 (b +1)2 L4
2+
b3 = Root(4b2 +b6 −3b4 −1,2), b4 = Root(b8 +3b6 +3b4 +b2 −1,2)
3

b2 +1

(b −1)4 (b +1)4 L2
2 −(b −1)6 (b +1)6 ,2

b5 = Root(−2b4 +b6 +3b2 −1,2), b6 = 1/
√
2, b7 = 1, b8 =
p
(2)
L22 (b) =
1 −b2, L23 (b) = 1/
p
1 −b2, L24 (b) = (1 −b2)/b, L25 (b) = ∞
b9 = Root(2b2 +b6 −3b4 −1,2), b10 = Root(b8 −b6 −3b4 −3b2 −1,2) L26 (b) =
b2/
p
1 −b2, L27 (b) = 1/
p
b2 −1, L28 (b) = b2/
p
b2 −1
b11 = Root(−4b4 +b6 +3b2 −1,2)
L29 (b) =
(b2 −1)/b, L210 (b) = b2 −1
b12 = Root(b6 −6b4 +11b2 −8,2), b13 = ∞
ρ11 (b,L2) = Root(−ρ6
1 b6 +3b4 
L2
2b2 +1 −L2
2

ρ4
1 −3b2 
−7L2
2b2 +7L2
2 +L4
2 +L4
2b4 −2L4
2b2 +1

ρ2
1 +

L2
2b2 +1 −L2
2
3 ,2)
ρ12 (b,L2) = Root(ρ6
1 +

−3b4 +3L2
2b2 −3L2
2

ρ4
1 +

21L2
2b6 +3L4
2b4 −6L4
2b2 +3b8 −21L2
2b4 +3L4
2

ρ2
1 +

L2
2b2 −b4 −L2
2
3 ,2)
ρ12 (b,L2) = b2, ρ13 (b,L2) = 1/b
Table 3 Cells of R3 where the manipulator has cuspidal configurations.
]b1 b2[
(]L21 L22 [,]ρ11 ρ12 [),(]L22 L23 [,]ρ13 ρ12 [),(]L23 L24 [,]ρ13 ρ12 [),(]L24 L25 [,]ρ13 ρ14 [)
]b2 b3[
(]L21 L26 [,]ρ11 ρ12 [),(]L26 L22 [,]ρ11 ρ12 [),(]L22 L23 [,]ρ13 ρ12 [),(]L23 L24 [,]ρ13 ρ12 [),(]L24 L25 [,]ρ13 ρ14 [)
]b3 b4[
(]L21 L26 [,]ρ11 ρ12 [),(]L26 L22 [,]ρ11 ρ12 [),(]L22 L24 [,]ρ13 ρ12 [),(]L24 L23 [,]ρ13 ρ14 [),(]L23 L25 [,]ρ13 ρ14 [)
]b4 b5[
(]L21 L26 [,]ρ11 ρ12 [),(]L26 L22 [,]ρ11 ρ12 [),(]L22 L24 [,]ρ13 ρ12 [),(]L24 L23 [,]ρ13 ρ14 [),(]L23 L25 [,]ρ13 ρ14 [)
]b5 b6[
(]L21 L22 [,]ρ11 ρ12 [),(]L22 L26 [,]ρ13 ρ12 [),(]L26 L24 [,]ρ13 ρ12 [),(]L24 L23 [,]ρ13 ρ14 [),(]L23 L25 [,]ρ13 ρ14 [)
]b6 b7[
(]L21 L22 [,]ρ11 ρ12 [),(]L22 L24 [,]ρ13 ρ12 [),(]L24 L26 [,]ρ13 ρ14 [),(]L26 L23 [,]ρ13 ρ14 [),(]L23 L25 [,]ρ13 ρ14 [)
]b7 b8[
(]L21 L29 [,]ρ12 ρ11 [),(]L29 L210 [,]ρ14 ρ11 [),(]L210 L27 [,]ρ14 ρ13 [),(]L27 L28 [,]ρ14 ρ13 [),(]L28 L25 [,]ρ14 ρ13 [)
]b8 b9[
(]L21 L29 [,]ρ12 ρ11 [),(]L29 L27 [,]ρ14 ρ11 [),(]L27 L210 [,]ρ14 ρ11 [),(]L210 L28 [,]ρ14 ρ13 [),(]L28 L25 [,]ρ14 ρ13 [)
]b9 b10[
(]L21 L27 [,]ρ12 ρ11 [),(]L27 L29 [,]ρ12 ρ11 [),(]L29 L210 [,]ρ14 ρ11 [),(]L210 L28 [,]ρ14 ρ13 [),(]L28 L25 [,]ρ14 ρ13 [)
]b10 b11[
(]L21 L27 [,]ρ12 ρ11 [),(]L27 L29 [,]ρ12 ρ11 [),(]L29 L210 [,]ρ14 ρ11 [),(]L210 L28 [,]ρ14 ρ13 [),(]L28 L25 [,]ρ14 ρ13 [)
]b11 b12[
(]L21 L27 [,]ρ12 ρ11 [),(]L27 L29 [,]ρ12 ρ11 [),(]L29 L28 [,]ρ14 ρ11 [),(]L28 L210 [,]ρ14 ρ11 [,(]L210 L25 [,]ρ14 ρ13 [)
]b12 b13[
(]L21 L29 [,]ρ12 ρ11 [),(]L29 L28 [,]ρ14 ρ11 [),(]L28 L210 [,]ρ14 ρ11 [),(]L210 L25 [,]ρ14 ρ13 [)
4 Conclusion
We have proposed a general method to describe rigorously the design parameters
for which a manipulator has cuspidal configurations. This method can be applied
directly to other mechanisms, such as the ones studied in [4, 12] for example. The
tools used to perform the computations were implemented in a Maple library called
Siropa2. For 3D illustration purposes, we have detailed the main computations to
be performed with manipulators satisfying L2 = L3. However, the proposed method
allows one directly to solve the general case (L2 ̸= L3) by computing a discrimi-
nant variety of the system with 4 parameters b,L2,L3,ρ1, and by decomposing R4
with a CAD adapted to the discriminant variety. This description generalizes and
2 http://www.irccyn.ec-nantes.fr/ moroz/siropa/doc
8
G. Moroz1,2, D. Chablat1, P. Wenger1, F. Rouiller2
completes the analyse done in [6]. There is still some limitations though. In particu-
lar, when the system that defines the cuspidal configurations has no solution, it may
mean that there exists a manipulator with no cuspidal configurations, but it may also
mean that no manipulator can be assembled with these design parameters. Thus it
is essential to be able to describe precisely the set of design parameter values for
which a manipulator can be assembled.
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Fig. 5 The cells of R3 where the manipulator admits cuspidal configurations, front view (left) and
back view (right).
Acknowledgements The research work reported here was made possible by SiRoPa ANR Project.
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