1 
The Kinematic Analysis  
of a Symmetrical Three-Degree-of-Freedom  
Planar Parallel Manipulator 
Damien Chablat and Philippe Wenger 
Institut de Recherche en Communications et Cybernétique de Nantes1 
1, rue de la Noë, 44321 Nantes, France 
Damien.Chablat@irccyn.ec-nantes.fr 
 
 
1 IRCCyN: UMR CNRS 6596, Ecole Centrale de Nantes, Université de Nantes, Ecole des Mines de Nantes 
Abstract— Presented in this paper is the kinematic analysis 
of a symmetrical three-degree-of-freedom planar parallel 
manipulator. In opposite to serial manipulators, parallel 
manipulators can admit not only multiple inverse kinematic 
solutions, but also multiple direct kinematic solutions. This 
property produces more complicated kinematic models but 
allows more flexibility in trajectory planning. To take into 
account this property, the notion of aspects, i.e. the maximal 
singularity-free domains, was introduced, based on the notion 
of working modes, which makes it possible to separate the 
inverse kinematic solutions. The aim of this paper is to show 
that a non-singular assembly-mode changing trajectory exist 
for a symmetrical planar parallel manipulator, with equilateral 
base and platform triangle. 
Index Terms—Parallel manipulator, Singularity, Aspect, 
Assembly modes, Working modes. 
I. INTRODUCTION 
OR two decades, parallel manipulators have attracted the 
attention of more and more researchers who consider them 
as valuable alternative for robotic mechanisms [1-3]. As stated 
by a number of authors [4], conventional serial kinematic 
machines have already reached their dynamic performance 
limits, which are bounded by high stiffness of the machine 
components required to support sequential joints, links and 
actuators. Thus, while having good operating characteristics 
(large workspace, high flexibility and manoeuvrability), serial 
manipulators have disadvantages of low precision, low 
stiffness and low power. Also, they are generally operated at 
low speed to avoid excessive vibration and deflection. 
Conversely, parallel kinematic machines offer essential 
advantages over their serial counterparts (lower moving 
masses, higher rigidity and payload-to-weight ratio, higher 
natural 
frequencies, 
better 
accuracy, simpler modular 
mechanical construction, possibility to locate actuators on the 
fixed base) that should lead to higher dynamic capabilities. 
However, most existing parallel manipulators have limited and 
complicated workspace with singularities, and highly non-
isotropic input/output relations [5]. Hence, the performances 
may significantly vary over the workspace and depend on the 
direction of the motion.  
A well-known feature of parallel manipulators is the existence 
of multiple direct kinematic solutions (or assembly modes). 
That is, the mobile platform can admit several positions and 
orientations (or configurations) in the workspace for one given 
set of input joint values [6]. The dual problem arises in serial 
manipulators, where several input joint values correspond to 
one given configuration of the end-effector. To cope with the 
existence of multiple inverse kinematic solutions in serial 
manipulators, the notion of aspects was introduced [7]. The 
aspects were defined as the maximal singularity-free domains 
in the joint space. For usual industrial serial manipulators, the 
aspects were found to be the maximal sets in the joint space 
where there is only one inverse kinematic solution. Many other 
serial manipulators, referred to as cuspidal manipulators, were 
shown to be able to change solution without passing through a 
singularity, thus meaning that there is more than one inverse 
kinematic solution in one aspect. New uniqueness domains 
have been characterized for cuspidal manipulators [8], [9].  
A definition of the notion of aspect was given by [10] for 
parallel manipulators with only one inverse kinematic solution. 
These aspects were defined as the maximal singularity-free 
domains in the workspace. A second definition was given by 
[11] for parallel manipulators with several inverse kinematic 
solutions. These aspects were defined as the maximal 
singularity-free domains in the Cartesian product of the 
workspace with the joint space. 
However, it was shown in [12] that it is possible to link several 
direct kinematic solutions without meeting a singularity, thus 
meaning that there exists cuspidal parallel manipulators. This 
property was found for particular links lengths. However, [13] 
conjectured that such properties cannot exist for symmetrical 
parallel manipulator. The aim of this paper is to show that a 
symmetrical 3-DOF planar parallel manipulator can change 
assembly mode without meeting a singularity. We mean by 
symmetrical, a manipulator with equilateral base and platform 
triangles. 
This paper is organized as follows. Section II describes the 
planar 3-RRR parallel manipulator studied, which is used all 
along this paper. Section III recalls the notion of aspect for 
parallel 
manipulators. 
A 
non-singular 
assembly-mode 
changing trajectory is shown for the symmetrical planar 
parallel manipulator. The workspace and the generalized 
F 
 
2 
aspects are calculated using octree models.  
II. PRELIMINARIES 
A. Parallel manipulator studied 
The manipulator under study is a planar three-dof manipulator 
comprising three parallel RRR chains shown in Fig. 1. This 
manipulator is used to illustrate the example in this paper. This 
manipulator has frequently studied, in particular in [6-15].  
A1
A2 
C2
C3
C1
A3
B2
B3
B1
O
l1
l2
l3
m2
m3
P
m1
2
2
x
y
 
Figure 1: The 3-RRR parallel manipulator studied 
The actuated joint variables are the rotation of the three 
revolute joints located on the base (
1
2
3
,
,
). The Cartesian 
variables are the coordinate ( , )
x y  of the operation point P and 
the orientation  of the platform. The passive and actuated 
joints will always be assumed unlimited in this study. Points 
A1, A2 and A3, (respectively C1, C2 and C3) lie at the corners of 
an equilateral triangle, whose geometric center is 0 
(respectively P). Moreover, 
1
2
3
6
l
l
l
l




, with li denoting 
the length of AiBi, 
1
2
3
6
m
m
m
m




, with mi denoting the 
length of BiCi, 
1
2
3
10
r
r
r
r




, with ri denoting the length 
of AiOi and 
1
2
3
5
s
s
s
s




, with si denoting the length of 
CiP, in units of length that need not be specified in the paper.  
B. Kinematic Relations 
The velocity p of point P can be obtained in three different 
forms, depending on which leg is traversed, namely, 
 = 
(
) + 
(
)
(
),
[1,3]
i
i
i
i
i
i
i
i








p
E b
a
E c
b
E p
c




 
(1) 
with matrix E,  
0
1
1
0







E
 
We would like to eliminate the three idle joint rates 
1
, 
2
 
and 
3
 from eq. (1), which we do by dot-multiplying eq. (1) 
by (
)T
i
i

c
b
, thus obtaining 
(
)
=(
)
(
)+(
)
(
),
[1,3]
T
T
T
i
i
i
i
i
i
i
i
i
i
i








c
b
p
c
b
E b
a
c
b
E p
c



 
(2) 
Equation (2) can now be cast in vector form, namely, 


1
2
3
 with 
 and
T
T











At
Bρ
t
p
ρ







 
(3) 
with ρ thus being the vector of actuated joint rates. Moreover, 
A and B are, respectively, the direct-kinematics and the 
inverse-kinematics matrices of the manipulator, defined as 
1
1
1
1
1
2
2
2
2
2
3
3
3
3
3
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
T
T
T
T
T
T























c
b
c
b
E p
c
A
c
b
c
b
E p
c
c
b
c
b
E p
c
 
(4a) 
1
1
1
1
2
2
2
2
3
3
3
3
(
)
(
)
0
0
0
(
)
(
)
0
0
0
(
)
(
)
T
T
T

















c
b
E b
a
B
c
b
E b
a
c
b
E b
a
 
(4b) 
C. Singularities 
For the planar manipulator studied, such configurations are 
reached whenever the axes B1C1, B2 C2 and B3C3 intersect 
(possibly at infinity), as depicted in Fig. 2. In the presence of 
such configurations, the manipulator cannot resist any torque 
applied at the intersection point I. 
A1
A2 
C2
C3
C1
A3
B2
B3
B1
P
I
 
Figure 2: Example of parallel singularity 
For the manipulator under study, the serial singularities occur 
whenever (
) (
)
T
i
i
i
i
i
i
l m



b
a
c
b
 for at least one value of i, 
as depicted in Fig. 3 for i = 1, i.e. whenever the points Ai, Bi, 
and Ci are aligned. 
D. Working modes 
The notion of working modes was introduced in [11] for 
parallel manipulators with several solutions to the inverse 
kinematic problem and whose matrix B is diagonal. 
A working mode, denoted 
i
Mf , is the set of mechanism 
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 mechanism configuration 
is represented by the vector (X, q), which permits us to locate 
the mobile platform as well as all the links.. 
(
, )
 such that sign(
)=cst for 
1,
,  
and det( )
0
jj
i
X q
W
Q
B
j
n
Mf
B











 
Therefore, the set of working modes (
i
Mf , j
I

) is obtained 
while using all permutations of sign of each term 
jj
B . 
The manipulator under study has eight working modes, as 
depicted in Fig. 4, that we call now (a), (b), ..., (h). Each 
working mode is defined according to the sign of 
jj
B  as is 
given is in table 1. 
 
3 
A1
A2 
C2
C3
C1
A3
B2
B3
B1
P
 
Figure 3: Example of serial singularity when 
1A , 
1B  and 
1
C  are aligned 
(g)
(h)
(e)
(f)
(c)
(d)
(a)
(b)
 
Figure 4: The eight working modes 
 
Figure 4 
 (a)   (b)   (c)  
 (d)   (e)   (f)   (g)   (h)  
11
B  
P 
P 
P 
P 
N 
N 
N 
N 
22
B  
P 
N 
P 
N 
N 
P 
N 
P 
33
B  
P 
P 
N 
N 
P 
P 
N 
N 
Table 1: The eight working modes of the manipulators studied with N 
(resp. P) denoted negative values of 
ii
B  (resp. positive values) 
According to each working mode, the parallel singularity locus 
changes in the workspace, as shown in Fig. 5. In generally, the 
ability of a parallel manipulator to change its inverse kinematic 
solution depends on the bounds in the passive and actuated 
joints. This problem is not taken into account in our study 
since unlimited joints are assumed. 
A1
A2 
C2
C3
C1
A3
B2
B3
B1
P
A1
A2 
C2
C3
C1
A3
B2
B3
B1
P
 
Figure 5: The same platform configuration with two joint 
configurations (singular on the left and none singular on the 
right) 
E. Octree Models 
Octree models are hierarchical data structures based on 
recursive subdivision of the space, respectively [16]. They are 
useful for representing complex 3 dimensional shapes like 
workspaces [10]. A close method is used in [17] that divides 
the workspace into boxes. This method does not use recursive 
subdivision but interval analysis methods [18] to build the 
dextrous workspace. However, it does not make it possible to 
perform Boolean operations or to make path-connectivity 
analysis easily. The first method permits us to calculate easily 
all kind of space and the computing time is limited as a 
function of the accuracy. The second one is more exact but 
requires more ability to be implemented. In both cases, we can 
characterize spaces whose dimensions are either lengths or 
angles. 
Since the structure of the octree model has an implicit 
adjacency graph, path-connectivity analyses and trajectory 
 
4 
planning can be carried out naturally. The optimal construction 
method of a 2k -tree is derived from the shape, which recalls 
the tree. The most interesting approach consists in testing 
successively all the nodes present in the maximal depth, 
following an order of numbering which quickly allows nodes 
to be grouped and thus, simplifies the 2k -tree. The order of 
numbering for this algorithm is based on Morton's sweeping 
[19]. The inverse or direct kinematic model is used to calculate 
2k -tree.  
The figure 6 represents the octree model of its Cartesian 
workspace where the first and the second axis represent the 
position and the third axis the orientation of the mobile 
platform. 
 
x
y

 
Figure 6: The Cartesian workspace 
III. WORKSPACE ANALYSIS 
A. Aspect definitions 
The notion of aspect was introduced by [7] to cope with the 
existence of multiple inverse kinematic solutions in serial 
manipulators. Recently, the notion of aspect was defined for 
parallel manipulators with only one inverse kinematic solution 
to cope with the existence of multiple direct kinematic 
solutions [10] and for parallel manipulators with multiple 
inverse and direct kinematic solutions (the generalized aspects 
[11]). For the manipulator studied, we use the second 
definition. 
The generalized aspects 
ij
A  are defined as the maximal sets in 
W
Q

 so that 
 
ij
A
W
Q


; 
 
ij
A  is connected; 
 


(
, )
  such that det( )
0
ij
i
A
X q
Mf



A
  
In other words, the generalized aspects 
ij
A  are the maximal 
singularity-free domains of the Cartesian product of the 
reachable workspace (called W) with the reachable joint space 
(called Q). 
The projection of the generalized aspects onto the workspace 
yields the parallel aspects 
ij
WA  so that, 
 
ij
WA
W

; 
 
ij
WA  is connected. 
The parallel aspects are the maximal singularity-free domains 
in the workspace for one given working mode. 
The projection of the generalized aspects onto the joint space 
yields the serial aspects 
ij
QA  so that, 
 
ij
QA
Q

; 
 
ij
QA  is connected. 
The serial aspects are the maximal singularity-free domains in 
the joint space for one given working mode.  
For each working mode, there exists, at least, one aspect where 
det( )
A  is positive and another one where det( )
A  is negative. 
However, such regions can be disjoint. In table 2, we 
associated the aspects with a working mode for which det( )
A  
is positive. For the working mode (a), there exist four aspects 
and for the other ones, there is only one aspect. Due to the 
symmetrical properties of the mechanism, there exist also 11 
aspects where det( )
A  is negative.  
Working modes 
(a) 
(b) (c) (d) 
(e) (f) 
(g) 
(h) 
N° figure 
7 
8 
9 
10 
11 
12 
13 
14 
Table 2: The projection of the generalized aspects on the workspace when 
det( )
0

A
 for each working mode 
 
 
Figure 7: The four parallel aspects for the working mode (a) and 
det( )
0

A
 
 
 
 
Figure 8: The parallel aspect for 
the working mode (b) and det( )
0

A
 
Figure 9: The parallel aspect for 
the working mode (c) and det( )
0

A
 
 
 
 
Figure 10: The parallel aspect for 
the working mode (d) and det( )
0

A
 
Figure 11: The parallel aspect for 
the working mode (e) and det( )
0

A
 
 
 
5 
 
 
Figure 12: The parallel aspect for 
the working mode (f) and det( )
0

A
 
Figure 13: The parallel aspect for 
the working mode (g) and det( )
0

A
 
 
 
Figure 14: The parallel aspect for the working mode (h) and det( )
0

A
 
The calculation of the generalized aspects can be performed by 
using a 
62  octree model or two octree models. We use the 
second method. The first one is the projection of the 
generalized aspect onto the reachable workspace and the 
second one the projection onto the reachable joint space for a 
given working mode and a given sign of det( )
A constant. To 
obtain these results with an accuracy of 0.093 for the position 
and 1,4 degrees for the orientation, the computing times is 90 
seconds with an AMD Athlon XP processor 2500+ and the 
maximum memory used is 180 Mb. The connectivity analysis 
of each domain requires 20 seconds. 
B. Non-singular posture changing trajectories 
In [12], a non-singular posture changing trajectory was found 
for a 3-RPR planar manipulator. However, it appears that this 
trajectory passes close to a singular configuration. This 
property was confirmed in [10] for the same manipulator and 
for a 3-RRR planar manipulator with non-symmetrical 
geometry [20].  
According to the assumption in [13], we can think that such 
properties may not exist for the mechanism studied. However, 
we shown that a non-singular configuration changing 
trajectories exists. 
For the following input joint values: 
1
2
3
=5.862610, 
=1.277470, 
= 5.213885



 
Four direct kinematic solutions are found (Figure 15 and Table 
3). We notice that solutions 1 and 4 are in the same 
generalized aspect (The parallel aspect associated is depicted 
in the figure 8).  
 
 
Posture (1) 
Posture (2) 
 
 
Posture (3) 
Posture (4) 
Figure 15: The four direct kinematic solutions for 
1=5.862610

, 
2
3
=1.277470, 
= 5.213885


 
Posture N° 
x 
y 
 in degrees 
(1) 
1.102  
1.956 
57.50 
(2) 
0.705  
2.751 
46.85 
(3) 
4.638  
-5.413 
32.35 
(4) 
-0.357  
2.720 
26.51 
Table 3: Four direct kinematic solutions for the same joint values 
 
A first method to confirm this property is to evaluate the 
determinant of A and 
ii
B  (Table 4) and to find out a trajectory 
between these two postures. 
 
det(A) 
11
B  
22
B  
33
B  
Posture (1) 
307.990 
-34.132 
-34.008 
31.827 
Posture (4) 
522.868 
-33.023 
-35.997 
21.7203 
Table 4: Evaluation of det(
)
A  and 
ii
B  for the two pose in the same 
generalized aspect  
We find a non-singular continuous trajectory between postures 
(1) and (4) by passing through an intermediate posture (5) 
(Figure 16) whose position is (-0.987; 1.930)  and orientation 
is 12.35 degrees. Between these three postures, a linear 
interpolation is defined to stay in the same generalized aspect. 
The values of det(A), 
11
B , 
22
B  and 
33
B  are evaluated and 
each value of these indices is normalized by its maximum 
value as it is shown in figure 17.  
 
6 
 
Figure 16: The intermediate posture (5) for the non-singular changing 
trajectory 
With this result, we have proofed that a non-singular assembly 
mode trajectory is possible for a symmetrical planar 3-RRR 
parallel manipulator. Inside such trajectory, not any kinematic 
index, derived from the Jacobian matrices, permits us to 
recognize such property. 
0
0.2
0.4
0.6
0.8
1
1.2
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
B11
det( )
A
B12
B33
Posture (1)
Posture (5)
Posture (4)
t
 
Figure 17: Variations of the normalized values of det(A), 
11
B , 
22
B
 and 
33
B  along the trajectory (t) between postures (1) and (4) 
 
C. Characteristic surfaces 
To separate the direct and inverse kinematic solutions, the 
uniqueness 
domains 
are 
determined 
for 
the 
parallel 
manipulator with one inverse kinematic solution in [10] and 
for parallel manipulator with several inverse kinematic 
solutions in [20]. The boundaries of the uniqueness domains 
are defined by the characteristic surfaces [20]. For the 
generalized aspect (b), we can compute the characteristic 
surface that permits us to isolate the assembly modes where it 
is possible to realize non-singular assembly mode changing 
trajectories (Figure 18). 
 
Figure 18: The parallel singularities and the characteristic 
surfaces associated with the generalized aspect (b) 
IV. SUMMARY AND CONCLUSIONS 
A kinematic analysis of a planar 3-RRR parallel manipulator 
with symmetrical properties was presented in this paper. The 
eight working modes have been characterized and 22 
generalized aspects have been found out. Inside such domains, 
any continuous trajectories are possible. In such domains, 
there are non-singular changing trajectories but not any 
kinematic index can recognize such property. An example of 
non-singular changing trajectory is given and the characteristic 
surface are computed which permit, in a future works, to 
define closely the uniqueness domains of the manipulator 
studied. 
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Canada (1966).  
[20] D. Chablat and Ph. Wenger, “Les Domaines d’Unicité 
des 
Manipulateurs 
Pleinement 
Parallèles,” 
Journal 
of 
Mechanism and Machine Theory, Vol 36/6, pp. 763-783, 
(2001). 
 
1 
The Kinematic Analysis  
of a Symmetrical Three-Degree-of-Freedom  
Planar Parallel Manipulator 
Damien Chablat and Philippe Wenger 
Institut de Recherche en Communications et Cybernétique de Nantes1 
1, rue de la Noë, 44321 Nantes, France 
Damien.Chablat@irccyn.ec-nantes.fr 
 
 
1 IRCCyN: UMR CNRS 6596, Ecole Centrale de Nantes, Université de Nantes, Ecole des Mines de Nantes 
Abstract— Presented in this paper is the kinematic analysis 
of a symmetrical three-degree-of-freedom planar parallel 
manipulator. In opposite to serial manipulators, parallel 
manipulators can admit not only multiple inverse kinematic 
solutions, but also multiple direct kinematic solutions. This 
property produces more complicated kinematic models but 
allows more flexibility in trajectory planning. To take into 
account this property, the notion of aspects, i.e. the maximal 
singularity-free domains, was introduced, based on the notion 
of working modes, which makes it possible to separate the 
inverse kinematic solutions. The aim of this paper is to show 
that a non-singular assembly-mode changing trajectory exist 
for a symmetrical planar parallel manipulator, with equilateral 
base and platform triangle. 
Index Terms—Parallel manipulator, Singularity, Aspect, 
Assembly modes, Working modes. 
I. INTRODUCTION 
OR two decades, parallel manipulators have attracted the 
attention of more and more researchers who consider them 
as valuable alternative for robotic mechanisms [1-3]. As stated 
by a number of authors [4], conventional serial kinematic 
machines have already reached their dynamic performance 
limits, which are bounded by high stiffness of the machine 
components required to support sequential joints, links and 
actuators. Thus, while having good operating characteristics 
(large workspace, high flexibility and manoeuvrability), serial 
manipulators have disadvantages of low precision, low 
stiffness and low power. Also, they are generally operated at 
low speed to avoid excessive vibration and deflection. 
Conversely, parallel kinematic machines offer essential 
advantages over their serial counterparts (lower moving 
masses, higher rigidity and payload-to-weight ratio, higher 
natural 
frequencies, 
better 
accuracy, simpler modular 
mechanical construction, possibility to locate actuators on the 
fixed base) that should lead to higher dynamic capabilities. 
However, most existing parallel manipulators have limited and 
complicated workspace with singularities, and highly non-
isotropic input/output relations [5]. Hence, the performances 
may significantly vary over the workspace and depend on the 
direction of the motion.  
A well-known feature of parallel manipulators is the existence 
of multiple direct kinematic solutions (or assembly modes). 
That is, the mobile platform can admit several positions and 
orientations (or configurations) in the workspace for one given 
set of input joint values [6]. The dual problem arises in serial 
manipulators, where several input joint values correspond to 
one given configuration of the end-effector. To cope with the 
existence of multiple inverse kinematic solutions in serial 
manipulators, the notion of aspects was introduced [7]. The 
aspects were defined as the maximal singularity-free domains 
in the joint space. For usual industrial serial manipulators, the 
aspects were found to be the maximal sets in the joint space 
where there is only one inverse kinematic solution. Many other 
serial manipulators, referred to as cuspidal manipulators, were 
shown to be able to change solution without passing through a 
singularity, thus meaning that there is more than one inverse 
kinematic solution in one aspect. New uniqueness domains 
have been characterized for cuspidal manipulators [8], [9].  
A definition of the notion of aspect was given by [10] for 
parallel manipulators with only one inverse kinematic solution. 
These aspects were defined as the maximal singularity-free 
domains in the workspace. A second definition was given by 
[11] for parallel manipulators with several inverse kinematic 
solutions. These aspects were defined as the maximal 
singularity-free domains in the Cartesian product of the 
workspace with the joint space. 
However, it was shown in [12] that it is possible to link several 
direct kinematic solutions without meeting a singularity, thus 
meaning that there exists cuspidal parallel manipulators. This 
property was found for particular links lengths. However, [13] 
conjectured that such properties cannot exist for symmetrical 
parallel manipulator. The aim of this paper is to show that a 
symmetrical 3-DOF planar parallel manipulator can change 
assembly mode without meeting a singularity. We mean by 
symmetrical, a manipulator with equilateral base and platform 
triangles. 
This paper is organized as follows. Section II describes the 
planar 3-RRR parallel manipulator studied, which is used all 
along this paper. Section III recalls the notion of aspect for 
parallel 
manipulators. 
A 
non-singular 
assembly-mode 
changing trajectory is shown for the symmetrical planar 
parallel manipulator. The workspace and the generalized 
F 
 
2 
aspects are calculated using octree models.  
II. PRELIMINARIES 
A. Parallel manipulator studied 
The manipulator under study is a planar three-dof manipulator 
comprising three parallel RRR chains shown in Fig. 1. This 
manipulator is used to illustrate the example in this paper. This 
manipulator has frequently studied, in particular in [6-15].  
A1
A2 
C2
C3
C1
A3
B2
B3
B1
O
l1
l2
l3
m2
m3
P
m1
2
2
x
y
 
Figure 1: The 3-RRR parallel manipulator studied 
The actuated joint variables are the rotation of the three 
revolute joints located on the base (
1
2
3
,
,
). The Cartesian 
variables are the coordinate ( , )
x y  of the operation point P and 
the orientation  of the platform. The passive and actuated 
joints will always be assumed unlimited in this study. Points 
A1, A2 and A3, (respectively C1, C2 and C3) lie at the corners of 
an equilateral triangle, whose geometric center is 0 
(respectively P). Moreover, 
1
2
3
6
l
l
l
l




, with li denoting 
the length of AiBi, 
1
2
3
6
m
m
m
m




, with mi denoting the 
length of BiCi, 
1
2
3
10
r
r
r
r




, with ri denoting the length 
of AiOi and 
1
2
3
5
s
s
s
s




, with si denoting the length of 
CiP, in units of length that need not be specified in the paper.  
B. Kinematic Relations 
The velocity p of point P can be obtained in three different 
forms, depending on which leg is traversed, namely, 
 = 
(
) + 
(
)
(
),
[1,3]
i
i
i
i
i
i
i
i








p
E b
a
E c
b
E p
c




 
(1) 
with matrix E,  
0
1
1
0







E
 
We would like to eliminate the three idle joint rates 
1
, 
2
 
and 
3
 from eq. (1), which we do by dot-multiplying eq. (1) 
by (
)T
i
i

c
b
, thus obtaining 
(
)
=(
)
(
)+(
)
(
),
[1,3]
T
T
T
i
i
i
i
i
i
i
i
i
i
i








c
b
p
c
b
E b
a
c
b
E p
c



 
(2) 
Equation (2) can now be cast in vector form, namely, 


1
2
3
 with 
 and
T
T











At
Bρ
t
p
ρ







 
(3) 
with ρ thus being the vector of actuated joint rates. Moreover, 
A and B are, respectively, the direct-kinematics and the 
inverse-kinematics matrices of the manipulator, defined as 
1
1
1
1
1
2
2
2
2
2
3
3
3
3
3
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
T
T
T
T
T
T























c
b
c
b
E p
c
A
c
b
c
b
E p
c
c
b
c
b
E p
c
 
(4a) 
1
1
1
1
2
2
2
2
3
3
3
3
(
)
(
)
0
0
0
(
)
(
)
0
0
0
(
)
(
)
T
T
T

















c
b
E b
a
B
c
b
E b
a
c
b
E b
a
 
(4b) 
C. Singularities 
For the planar manipulator studied, such configurations are 
reached whenever the axes B1C1, B2 C2 and B3C3 intersect 
(possibly at infinity), as depicted in Fig. 2. In the presence of 
such configurations, the manipulator cannot resist any torque 
applied at the intersection point I. 
A1
A2 
C2
C3
C1
A3
B2
B3
B1
P
I
 
Figure 2: Example of parallel singularity 
For the manipulator under study, the serial singularities occur 
whenever (
) (
)
T
i
i
i
i
i
i
l m



b
a
c
b
 for at least one value of i, 
as depicted in Fig. 3 for i = 1, i.e. whenever the points Ai, Bi, 
and Ci are aligned. 
D. Working modes 
The notion of working modes was introduced in [11] for 
parallel manipulators with several solutions to the inverse 
kinematic problem and whose matrix B is diagonal. 
A working mode, denoted 
i
Mf , is the set of mechanism 
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 mechanism configuration 
is represented by the vector (X, q), which permits us to locate 
the mobile platform as well as all the links.. 
(
, )
 such that sign(
)=cst for 
1,
,  
and det( )
0
jj
i
X q
W
Q
B
j
n
Mf
B











 
Therefore, the set of working modes (
i
Mf , j
I

) is obtained 
while using all permutations of sign of each term 
jj
B . 
The manipulator under study has eight working modes, as 
depicted in Fig. 4, that we call now (a), (b), ..., (h). Each 
working mode is defined according to the sign of 
jj
B  as is 
given is in table 1. 
 
3 
A1
A2 
C2
C3
C1
A3
B2
B3
B1
P
 
Figure 3: Example of serial singularity when 
1A , 
1B  and 
1
C  are aligned 
(g)
(h)
(e)
(f)
(c)
(d)
(a)
(b)
 
Figure 4: The eight working modes 
 
Figure 4 
 (a)   (b)   (c)  
 (d)   (e)   (f)   (g)   (h)  
11
B  
P 
P 
P 
P 
N 
N 
N 
N 
22
B  
P 
N 
P 
N 
N 
P 
N 
P 
33
B  
P 
P 
N 
N 
P 
P 
N 
N 
Table 1: The eight working modes of the manipulators studied with N 
(resp. P) denoted negative values of 
ii
B  (resp. positive values) 
According to each working mode, the parallel singularity locus 
changes in the workspace, as shown in Fig. 5. In generally, the 
ability of a parallel manipulator to change its inverse kinematic 
solution depends on the bounds in the passive and actuated 
joints. This problem is not taken into account in our study 
since unlimited joints are assumed. 
A1
A2 
C2
C3
C1
A3
B2
B3
B1
P
A1
A2 
C2
C3
C1
A3
B2
B3
B1
P
 
Figure 5: The same platform configuration with two joint 
configurations (singular on the left and none singular on the 
right) 
E. Octree Models 
Octree models are hierarchical data structures based on 
recursive subdivision of the space, respectively [16]. They are 
useful for representing complex 3 dimensional shapes like 
workspaces [10]. A close method is used in [17] that divides 
the workspace into boxes. This method does not use recursive 
subdivision but interval analysis methods [18] to build the 
dextrous workspace. However, it does not make it possible to 
perform Boolean operations or to make path-connectivity 
analysis easily. The first method permits us to calculate easily 
all kind of space and the computing time is limited as a 
function of the accuracy. The second one is more exact but 
requires more ability to be implemented. In both cases, we can 
characterize spaces whose dimensions are either lengths or 
angles. 
Since the structure of the octree model has an implicit 
adjacency graph, path-connectivity analyses and trajectory 
 
4 
planning can be carried out naturally. The optimal construction 
method of a 2k -tree is derived from the shape, which recalls 
the tree. The most interesting approach consists in testing 
successively all the nodes present in the maximal depth, 
following an order of numbering which quickly allows nodes 
to be grouped and thus, simplifies the 2k -tree. The order of 
numbering for this algorithm is based on Morton's sweeping 
[19]. The inverse or direct kinematic model is used to calculate 
2k -tree.  
The figure 6 represents the octree model of its Cartesian 
workspace where the first and the second axis represent the 
position and the third axis the orientation of the mobile 
platform. 
 
x
y

 
Figure 6: The Cartesian workspace 
III. WORKSPACE ANALYSIS 
A. Aspect definitions 
The notion of aspect was introduced by [7] to cope with the 
existence of multiple inverse kinematic solutions in serial 
manipulators. Recently, the notion of aspect was defined for 
parallel manipulators with only one inverse kinematic solution 
to cope with the existence of multiple direct kinematic 
solutions [10] and for parallel manipulators with multiple 
inverse and direct kinematic solutions (the generalized aspects 
[11]). For the manipulator studied, we use the second 
definition. 
The generalized aspects 
ij
A  are defined as the maximal sets in 
W
Q

 so that 
 
ij
A
W
Q


; 
 
ij
A  is connected; 
 


(
, )
  such that det( )
0
ij
i
A
X q
Mf



A
  
In other words, the generalized aspects 
ij
A  are the maximal 
singularity-free domains of the Cartesian product of the 
reachable workspace (called W) with the reachable joint space 
(called Q). 
The projection of the generalized aspects onto the workspace 
yields the parallel aspects 
ij
WA  so that, 
 
ij
WA
W

; 
 
ij
WA  is connected. 
The parallel aspects are the maximal singularity-free domains 
in the workspace for one given working mode. 
The projection of the generalized aspects onto the joint space 
yields the serial aspects 
ij
QA  so that, 
 
ij
QA
Q

; 
 
ij
QA  is connected. 
The serial aspects are the maximal singularity-free domains in 
the joint space for one given working mode.  
For each working mode, there exists, at least, one aspect where 
det( )
A  is positive and another one where det( )
A  is negative. 
However, such regions can be disjoint. In table 2, we 
associated the aspects with a working mode for which det( )
A  
is positive. For the working mode (a), there exist four aspects 
and for the other ones, there is only one aspect. Due to the 
symmetrical properties of the mechanism, there exist also 11 
aspects where det( )
A  is negative.  
Working modes 
(a) 
(b) (c) (d) 
(e) (f) 
(g) 
(h) 
N° figure 
7 
8 
9 
10 
11 
12 
13 
14 
Table 2: The projection of the generalized aspects on the workspace when 
det( )
0

A
 for each working mode 
 
 
Figure 7: The four parallel aspects for the working mode (a) and 
det( )
0

A
 
 
 
 
Figure 8: The parallel aspect for 
the working mode (b) and det( )
0

A
 
Figure 9: The parallel aspect for 
the working mode (c) and det( )
0

A
 
 
 
 
Figure 10: The parallel aspect for 
the working mode (d) and det( )
0

A
 
Figure 11: The parallel aspect for 
the working mode (e) and det( )
0

A
 
 
 
5 
 
 
Figure 12: The parallel aspect for 
the working mode (f) and det( )
0

A
 
Figure 13: The parallel aspect for 
the working mode (g) and det( )
0

A
 
 
 
Figure 14: The parallel aspect for the working mode (h) and det( )
0

A
 
The calculation of the generalized aspects can be performed by 
using a 
62  octree model or two octree models. We use the 
second method. The first one is the projection of the 
generalized aspect onto the reachable workspace and the 
second one the projection onto the reachable joint space for a 
given working mode and a given sign of det( )
A constant. To 
obtain these results with an accuracy of 0.093 for the position 
and 1,4 degrees for the orientation, the computing times is 90 
seconds with an AMD Athlon XP processor 2500+ and the 
maximum memory used is 180 Mb. The connectivity analysis 
of each domain requires 20 seconds. 
B. Non-singular posture changing trajectories 
In [12], a non-singular posture changing trajectory was found 
for a 3-RPR planar manipulator. However, it appears that this 
trajectory passes close to a singular configuration. This 
property was confirmed in [10] for the same manipulator and 
for a 3-RRR planar manipulator with non-symmetrical 
geometry [20].  
According to the assumption in [13], we can think that such 
properties may not exist for the mechanism studied. However, 
we shown that a non-singular configuration changing 
trajectories exists. 
For the following input joint values: 
1
2
3
=5.862610, 
=1.277470, 
= 5.213885



 
Four direct kinematic solutions are found (Figure 15 and Table 
3). We notice that solutions 1 and 4 are in the same 
generalized aspect (The parallel aspect associated is depicted 
in the figure 8).  
 
 
Posture (1) 
Posture (2) 
 
 
Posture (3) 
Posture (4) 
Figure 15: The four direct kinematic solutions for 
1=5.862610

, 
2
3
=1.277470, 
= 5.213885


 
Posture N° 
x 
y 
 in degrees 
(1) 
1.102  
1.956 
57.50 
(2) 
0.705  
2.751 
46.85 
(3) 
4.638  
-5.413 
32.35 
(4) 
-0.357  
2.720 
26.51 
Table 3: Four direct kinematic solutions for the same joint values 
 
A first method to confirm this property is to evaluate the 
determinant of A and 
ii
B  (Table 4) and to find out a trajectory 
between these two postures. 
 
det(A) 
11
B  
22
B  
33
B  
Posture (1) 
307.990 
-34.132 
-34.008 
31.827 
Posture (4) 
522.868 
-33.023 
-35.997 
21.7203 
Table 4: Evaluation of det(
)
A  and 
ii
B  for the two pose in the same 
generalized aspect  
We find a non-singular continuous trajectory between postures 
(1) and (4) by passing through an intermediate posture (5) 
(Figure 16) whose position is (-0.987; 1.930)  and orientation 
is 12.35 degrees. Between these three postures, a linear 
interpolation is defined to stay in the same generalized aspect. 
The values of det(A), 
11
B , 
22
B  and 
33
B  are evaluated and 
each value of these indices is normalized by its maximum 
value as it is shown in figure 17.  
 
6 
 
Figure 16: The intermediate posture (5) for the non-singular changing 
trajectory 
With this result, we have proofed that a non-singular assembly 
mode trajectory is possible for a symmetrical planar 3-RRR 
parallel manipulator. Inside such trajectory, not any kinematic 
index, derived from the Jacobian matrices, permits us to 
recognize such property. 
0
0.2
0.4
0.6
0.8
1
1.2
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
B11
det( )
A
B12
B33
Posture (1)
Posture (5)
Posture (4)
t
 
Figure 17: Variations of the normalized values of det(A), 
11
B , 
22
B
 and 
33
B  along the trajectory (t) between postures (1) and (4) 
 
C. Characteristic surfaces 
To separate the direct and inverse kinematic solutions, the 
uniqueness 
domains 
are 
determined 
for 
the 
parallel 
manipulator with one inverse kinematic solution in [10] and 
for parallel manipulator with several inverse kinematic 
solutions in [20]. The boundaries of the uniqueness domains 
are defined by the characteristic surfaces [20]. For the 
generalized aspect (b), we can compute the characteristic 
surface that permits us to isolate the assembly modes where it 
is possible to realize non-singular assembly mode changing 
trajectories (Figure 18). 
 
Figure 18: The parallel singularities and the characteristic 
surfaces associated with the generalized aspect (b) 
IV. SUMMARY AND CONCLUSIONS 
A kinematic analysis of a planar 3-RRR parallel manipulator 
with symmetrical properties was presented in this paper. The 
eight working modes have been characterized and 22 
generalized aspects have been found out. Inside such domains, 
any continuous trajectories are possible. In such domains, 
there are non-singular changing trajectories but not any 
kinematic index can recognize such property. An example of 
non-singular changing trajectory is given and the characteristic 
surface are computed which permit, in a future works, to 
define closely the uniqueness domains of the manipulator 
studied. 
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