12th IFToMM World Congress, Besançon (France), June18-21, 2007 
 
 
Singularity Surfaces and Maximal Singularity-Free Boxes in the Joint Space of 
Planar 3-RPR Parallel Manipulators 
 
Mazen ZEIN  
Philippe WENGER         Damien CHABLAT 
IRCCyN, Institut de recherche en communications et cybernéthique de Nantes 
1, rue de la Noe, 44322 Nantes 
 
 
 
Abstract— In this paper, a method to compute joint 
space 
singularity 
surfaces 
of 
3-RPR 
planar 
parallel 
manipulators is first presented. Then, a procedure to determine 
maximal joint space singularity-free boxes is introduced. 
Numerical examples are given in order to illustrate graphically 
the results. This study is of high interest for planning trajectories 
in the joint space of 3-RPR parallel manipulators and for 
manipulators design as it may constitute a tool for choosing 
appropriate joint limits and thus for sizing the link lengths of the 
manipulator. 
 
 
Keywords: 3-RPR parallel manipulators, singularity, 
singularity-free zones, joint space, joint limits. 
 
I. 
Introduction 
Most parallel manipulators have singularities that limit the 
motion of the moving platform. The most dangerous ones 
are the singularities associated with the direct kinematics, 
where two assembly-modes coalesce. Indeed, approaching 
such a singularity results in large actuator torques or 
forces, and in a loss of stiffness. Hence, these singularities 
are undesirable. There exists three main ways of coping 
with singularities, which have their own merits. A first 
approach consists in eliminating the singularities at the 
design stage by properly determining the kinematic 
architecture, the geometric parameters and the joint limits 
[4,4,7]. This approach is safe but difficult to apply in 
general and restricts the design possibilities. A second 
approach is the determination of the singularity-free 
regions in the workspace [5,15-17,20,24]. This solution 
does not involve a priori design restrictions but, because 
of the complexity of the singularity surfaces, it may be 
difficult to determine definitely safe regions. Finally, a 
third way consists in planning singularity-free trajectories 
in the manipulator workspace [2,6,19]. With this solution 
one is also faced with the complexity of the singularity 
equations but larger zones of the workspace may be 
exploited. 
In this paper, we choose to use the second approach by 
determining maximal joint space singularity-free boxes. 
This approach will help us determine appropriate joint 
limits and link dimensions. 
Planar parallel manipulators and particularly manipulators 
with three extensible leg rods, referred to as 3-RPR, have 
received a lot of attention because they have interesting 
potential applications in planar motion systems [9,21]. As 
shown in [18], moreover, the study of the 3-RPR planar 
manipulator may help better understand the kinematic 
behavior of its more complex spatial counterpart, the 6-
dof octahedral manipulator, which has also triangular base 
and platforms. 
The singularities of these manipulators have been most 
often represented in their workspace [13,14,18] but more 
rarely in their joint space [18,24,26]. 
Hunt and Primrose showed that 3-RPR planar manipulator 
could have up to 6 assembly-modes [12]. Mcaree and 
Daniel analyzed the joint space singularities through slices 
to explain non-singular changing trajectories [188], and 
Zein et al analyzed the topology of these slices in [26]. It 
was shown in [18,24,25] that, to change its assembly 
mode without meeting a singularity, a 3-RPR manipulator 
should encircle a cusp point in its joint space.  
In this paper, a method to compute and to represent joint 
space singularities of 3-RPR planar parallel manipulators 
is first proposed. A procedure is then provided to 
determine maximal joint space singularity-free boxes. 
This work is of a high interest for the determination of 
appropriate joint limits and for planning trajectories in the 
joint space.  
  
II. 
Manipulators under study 
The manipulators under study are 3-DOF planar parallel 
manipulators with three extensible leg rods (Fig.1). These 
manipulators have been frequently studied and have 
interesting potential applications in planar motion 
systems. The geometric parameters are the three sides of 
the moving platform d1, d2, d3 and the position of the base 
revolute joint centers defined by A1, A2 and A3. The 
reference frame is centered at A1 and the x-axis passes 
through A2. Thus, A1 = (0, 0), A2 = (A2x, 0) and A3 = (A3x, 
A3y). The parameter  is function of d1, d2 and d3.  
The joint space Q is defined by the vectors of the lengths 
of the three actuated extensible links 


T
1
2
3




q



. 
12th IFToMM World Congress, Besançon (France), June18-21, 2007 
 
 
x
A1
A2
B2
B3
B1
q
q2
d1
d3
d2
q1
y
A3
q3
a

2
3
1
 
Fig. 1. 
A 3-RPR parallel manipulator 
III. 
Kinematics of 3-RPR parallel manipulators 
The relation between the joint space Q and the workspace 
W can be expressed as a system of non-linear algebraic 
equations, which can be written as: 
(
)
0
F

x , q
 
(1) 
where x and q are respectively the vectors of the 
workspace and joint space variables. 
 Differentiating equation (1) with respect to time leads to 
the velocity model: 

At +Bq
0

 
(2) 
where 


T
w

t
 c
(w is the scalar angular velocity and c is 
the two-dimensional velocity vector of the operational 
point B1 of the platform if we used the first workspace 
parameters), A and B are 33 Jacobian matrices which are 
configuration dependent, and 


T
1
2
3




q



 is the 
joint velocity vector. 
 
IV. 
Joint space singularities of 3-RPR parallel 
manipulators 
The singularities of 3-DOF planar parallel manipulators 
have been extensively studied (see for example 
[3,8,14,18,22]). They were defined in the workspace (x, y, 
a and to the author‟s knowledge, there exist a small 
number of works dealing with the singular configurations 
in the manipulators joint space (1, 2, 3). 
 In a parallel singularity, matrix A is singular. To derive 
the singularity equations, it is usual to expand the 
determinant of A. We use rather a geometric approach that 
does not involve complicated algebraic calculus. The 3-
RPR parallel manipulator is in a singular configuration 
whenever the axes of its three legs are concurrent or 
parallel [11] (Fig. 2). 
x
A1
A2
B1
q2
q1
y
A3
q3
B2
B3
d1
d3
d2
a

Axis 1
Axis 3
Axis 2
2
1
3
 
Fig. 2. 
A 3-RPR parallel manipulator on a singular configuration. 
In order to derive this geometric condition, we derive the 
condition for the three leg axes to intersect at a common 
point (possibly at infinity). We first write the equations of 
the three leg axes: 
1
1
2
2
2
3
3
3
3
3
(Axis 1) : cos( )
sin( )
(Axis 2) : cos(
)
(
)sin(
)
(Axis 3) : cos(
)
(
)sin(
)
cos(
)
x
x
y
y
x
y
x
A
y
x
A
A
q
q
q
q
q
q
q











 
(3) 
Eliminating x and y yields the following singularity 
equation in the task parameters (q1, q2, q3): 


2
2 31
3
3
3
3
12
0
x
x
y
A s s
A s
A c
s



 
(4) 
where 

sin
i
i
s
q

, 

cos
i
i
c
q

 and 


sin
ij
i
j
s
q
q


. 
It is possible to express Eq. (4) as a function of the joint 
space parameters 1, 2 and 3 by using the constraint 
equations of the 3-RPR manipulator. However, the 
resulting equation would be too complicated to yield real 
insights, and difficult to handle. 
Our approach to compute the singular configurations in 
the joint space consists in reducing the dimension of the 
problem by first considering two-dimensional slices of the 
configuration space by fixing the first leg rod length 1. 
The singular surfaces in the full joint space are then 
calculated by “stacking” the slices. 
 Step 1: We rewrite Eq. (4) as a function of 1, a and q1 
using the constraint equations of the manipulator. 






2
2
2
1
1
1
2
2
1
1
1
3
3
3
1
1
3
3
3
3
1
1
3
c
c
cos
0
s
s
sin
0
c
c
cos
0
s
s
sin
0
x
x
y
A
d
d
A
d
A
d


a


a


a



a
























 
(5) 
The first (respectively last) two equations make it possible 
to express 2 (respectively 3) as function of 1, a and q1. 
Then, c2 and s2 (respectively c3 and s3) are calculated as 
function of 1, a and q1 from the first (respectively last) 
12th IFToMM World Congress, Besançon (France), June18-21, 2007 
 
 
two equations of (5) and their expressions are input in Eq. 
(4), which, now, depend only on L1, a and q1. 
Step 2: We fix a value for 1, so Eq. (4) depends now only 
on a and q1. By varying a or q1, we compute the roots of 
the equation, to obtain the singular configurations (as, q1s) 
for a fixed 1s. 
Step 3: For every singular configuration computed in the 
space (a, q1) in the second step of the approach, we 
calculate the corresponding values 2s and 3s using the 
equation system (6). We have thus the singular 
configurations curves in a slice of the joint space (2, 3) 
with 1 fixed. 








2
2
1
1
1
2
2
1
1
1
2
3
1
1
3
3
2
3
1
1
3
cos(
)
cos( )
sin(
)
sin( )
cos(
)
cos(
)
sin(
)
sin(
)
x
x
y
A
d
d
A
d
A
d

q
a


q
a

q
a



q
a























 
(6) 
Figure 3 shows a slice of the joint space singular 
configurations for 1=17 obtained for the same 3-RPR 
manipulator used in [14,18,21]. We refer only to this 
manipulator in this paper in order to illustrate our work. 
The geometric parameters of this manipulator are recalled 
below in an arbitrary length unit: 
A1=(0, 0) 
d1=17.04 
A2=(15.91, 0) 
d2=16.54 
A3=(0, 10) 
d3=20.84 
 
3
2
 
Fig. 3. 
Singular configurations in (2, 3) for 1=17. 
Step 4: We compute the joint space singularity slices for a 
number of 1 values, to do this we have to repeat steps 2 
and 3 while varying 1. 
Finally, we collect all the computed slices in one file to 
obtain the singularities in the joint space (1,2, 3). 
Figure 4 represents the singularities in the joint space of 
the manipulator studied when 1 varies from 0 to 50. To 
obtain this surface, we have imported the solutions 
obtained in step 4 into a CAD software, and we have 
meshed them together. 
Obviously, there is continuity between the singularities 
slices, one can claim, without any doubt, that there is 
singularity between the different slices. 
The surface depicted in Fig. 4 is of interest: 
i. for planning trajectories in the joint space because it 
shows clearly the joint space regions that are free of 
singularities.  
ii. for manipulator design, because it constitutes a tool for 
defining the values of the joint limits such that the joint 
space is a singularity-free box. 
2
1
3
0
10
20
30
70
60
50
40
60
70
 
Fig. 4. 
Joint space singularity surfaces of the 3-RPR manipulator 
studied when 1 varies from 0 to 50. 
 
V. 
Maximal joint space singularity-free boxes 
In a context of design and/or trajectory planning, an 
important problem is to find singularity-free zones in the 
joint space. 
In this section, we introduce a new procedure to determine 
maximal singularity-free boxes in the joint space of 3-
RPR manipulators. These singularity-free boxes will help 
us fix the manipulator joints limits. 
 Two numerical examples are provided to illustrate the 
effectiveness of the procedure. 
A. Procedure 
Step 1: We choose an initial joint space configuration 
Q0(10,20,30). This configuration can be chosen 
according to several considerations, for example choosing 
Q0 as the image through the inverse kinematics of a 
prescribed workspace center, or choosing it directly in the 
joint space as the center of a large singularity-free zone. 
 Step 2: We calculate the largest singularity-free cube 
centered at Q0(10,20,30). 
12th IFToMM World Congress, Besançon (France), June18-21, 2007 
 
 
To do this, we calculate the infinity norm distance d, also 
known as Chebyshev distance, between the center point 
Q0(10,20,30) and each of the joint space singular points 
Qs(1s,2s,3s) computed in Section IV: 


10
1
20
2
30
3
max
,
,
s
s
s
d










 
(7) 
and we keep the minimal distance dmin found over all, 
because we are searching for the distance between the 
closest joint space singularity configuration Qs from the 
center point Q0. 
The length of the singularity-free cube edge a will be: 
min
2
a
d


 
(8)  
Step 3: The 
choice 
of 
the 
initial 
center 
point 
Q0(10,20,30) does not lead to an optimized solution, in 
other words varying lightly the center point position may 
lead to a largest singularity-free cube. Thus, the position 
of the initial point must be optimized, which we have 
done using a Hooke and Jeeves optimization scheme [10]. 
Note that the solution found is a local optimum. 
Step 4: The cube found in step 3 touches the closest 
singular 
configuration 
to 
the 
center 
point.  
In order that the cube does not touch the singularities 
surface and for more security we subtract a small security 
value s from the distance dmin. Such a value can be related 
to laws of command to stop the motion when the joint 
velocity is maximum. 
 
The manipulator joint limits corresponding to the cube 
found can be easily computed as follows: 




min
0
min
max
0
min
i
i
i
i
d
s
d
s












       with i=1,2,3 
(9) 
B. Application of the procedure 
In this section, two examples are provided in order 
illustrate the application of the procedure. 
 
Example 1: 
For the same manipulator studied, we consider the center 
point Q0(35,25,45) in the joint space. This point was 
chosen in the center of a large singularity-free zone in the 
joint space. By computing the Chebyshev distances 
between each joint space singular point Qs computed in 
section IV, and Q0, the minimal distance obtained is 
dmin=5.3, so the edge length of the singularity-free cube is 
a = 10.6. 
By running the optimization algorithm, we find a maximal 
value dmin=7.175 for a center point Q(41.625, 24.875, 
44.125). 
We subtract a security value of 0.1 from dmin which 
becomes dmin=7.075. 
Figure 5 shows the joint space singularity surface and the 
maximal joint space singularity-free cube centered at 
Q(41.625, 24.875, 44.125) for the manipulator studied. 
 
1
2
3
 
 
Fig. 5. 
Joint space singularity surfaces and maximal joint space 
singularity-free cube centered at Q(41.625, 24.875, 44.125). 
Figures 6 shows the images through the direct kinematics 
of the maximal joint space singularity-free cube, which 
are two separate singularity-free components, each of 
them being located in an aspect of the workspace [24]. 
The projections of these two components onto the (x,y) 
plane are plotted in gray in Figure 6. 
a
y
x
 
Fig. 6. 
Images by direct kinematics of the joint space singularity-free 
cube (in black), and their projection on the (x,y) plane (in 
gray). 
Figures 7 shows the two workspace components and the 
workspace singularities of the 3-RPR manipulator studied, 
the singularities are plotted in color. 
 
12th IFToMM World Congress, Besançon (France), June18-21, 2007 
 
 
y
x
a
 
Fig. 7. 
Singularity-free components with workspace singularities. 
Example 2: 
For the same manipulator, we consider in this example the 
center point Q0(30,50,35) in the joint space. By computing 
the Chebyshev distances between each joint space 
singular point Qs computed in Section IV and Q0, the 
minimal distance obtained is dmin=4, so the edge length of 
the singularity-free cube is a = 8. 
By running the optimization algorithm, we find a maximal 
value dmin=5.794 for a center point Q(38.125, 50, 33). 
We subtract a security value of 0.1 from dmin, which 
becomes dmin=5.694. 
Figure 8 shows the joint space singularity surface and the 
maximal joint space singularity-free cube centered at 
Q(38.125, 50, 33) for the manipulator studied. 
 
1
2
3
 
Fig. 8. 
Joint space singularity curves and maximal joint space 
singularity-free cube centered at Q(38.125, 50, 33). 
Figures 9 displays the images by the direct kinematics of 
the maximal joint space singularity-free cube, which are 
two separate singularity-free components, one in each 
aspect of the workspace. The projections of these two 
components onto the (x,y) plane are plotted in gray in 
Figure 9. This figure is displayed with the same viewing 
angle as in Figure 6 to show the difference between the 
two examples. 
x
y
a
 
Fig. 9. 
Images by direct kinematics of the joint space singularity-free 
cube (in black), and their projection on the (x,y) plane (in 
gray). 
C. Future works 
We can see in figures 6 and 9 that the components in the 
workspace do not have regular forms. Because this study 
is carried out in the joint space only, it cannot take into 
account any of the properties, in the workspace, of the 
image by direct kinematics of the singularity-free cube 
found. 
This work will be extended by taking into account the 
largest regular volume (cube, cylinder…) inside the 
workspace components images of the singularity-free 
cube. The idea will then be to optimize the location of the 
initial point Q0(10,20,30) such that the image of the 
maximal singularity-free cube in the workspace generates 
a regular volume of maximal size. 
 
VI. 
Conclusion 
A procedure for computing joint space singularities of 3-
RPR parallel manipulators has been presented firstly in 
this paper. Secondly, a procedure for the determination of 
maximal joint space singularity-free boxes has been 
provided. 
These two procedures are of interest for planning 
trajectories in the joint space, and for manipulators design 
because they provide a tool for choosing the values of the 
joint limits. 
Future work will optimize the choice of the cube center 
point Q0 in the joint space in order to maximize the 
volumes of the workspace components images of the 
singularity-free cube. 
12th IFToMM World Congress, Besançon (France), June18-21, 2007 
 
 
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