P. Wenger, 
D. Chablat 
M. Zein 
Institut de Recherche en 
Communications et Cybernétique de 
Nantes 
1, rue la Noe – BP 92101 44321 Nantes 
Cedex 3, France 
Degeneracy study of the 
forward kinematics of planar 3-
RPR parallel manipulators  
This paper investigates two situations in which the forward kinematics of planar 3-RPR 
parallel manipulators degenerates. These situations have not been addressed before. The first 
degeneracy arises when the three input joint variables ρ1, ρ2 and ρ3 satisfy a certain 
relationship. This degeneracy yields a double root of the characteristic polynomial in 
 
tan( / 2)
t
ϕ
=
, which could be erroneously interpreted as two coalesce assembly modes. 
But, unlike what arises in non-degenerate cases, this double root yields two sets of solutions 
for the position coordinates (x, y) of the platform. In the second situation, we show that the 
forward kinematics degenerates over the whole joint space if the base and platform triangles 
are congruent and the platform triangle is rotated by 180 deg about one of its sides. For these 
“degenerate” manipulators, which are defined here for the first time, the forward kinematics 
is reduced to the solution of a 3rd-degree polynomial and a quadratics in sequence. Such 
manipulators constitute, in turn, a new family of analytic planar manipulators that would be 
more suitable for industrial applications. 
1 
Introduction 
Solving the forward kinematic problem of a parallel manipulator 
often leads to complex equations and non analytic solutions, even 
when considering planar 3-DOF parallel manipulators [1]. For 
these planar manipulators, Hunt showed that the forward 
kinematics admits at most 6 solutions [2] and several authors [3, 
4] have shown independently that their forward kinematics can be 
reduced as the solution of a characteristic polynomial of degree 6. 
In [3], a set of two linear equations in the position coordinates (x, 
y) of the moving platform is first established, which makes it 
possible to write x and y as function of the sine and cosine of the 
orientation angle ϕ of the moving platform. Substituting these 
expressions of x and y into one of the constraint equations of the 
manipulator and using the tan-half angle substitution leads to a 
6th-degree polynomial in  
tan( / 2)
t
ϕ
=
. Conditions under which 
the degree of this characteristic polynomial decreases were 
investigated in [5, 6]. Four distinct cases were found, namely, (i) 
manipulators for which two of the joints coincide (ii) manipulators 
with similar aligned platforms (iii) manipulators with nonsimilar 
aligned platforms and, (iv) manipulators with similar triangular 
platforms. For cases (i), (ii) and (iv) the forward kinematics was 
shown to reduce to the solution of two quadratics in cascade while 
in case (iii) it was shown to reduce to a 3rd-degree polynomial and 
a quadratic in sequence. To the best of the author’s knowledge, no 
other degenerate cases have been identified yet. In this paper, we 
show that the forward kinematics of planar 3-RPR1 manipulators 
degenerates under conditions that have not been identified before. 
More precisely, the system of linear equations in x and y that 
needs be established prior to the derivation of the characteristic 
polynomial becomes singular under certain conditions. Moreover, 
we show that the forward kinematics may degenerate over the 
whole joint space. 
                                                                 
1 The underlined letter refers to the actuated joint 
Next section recalls the kinematic equations and points out the 
singularity that may occur when solving the system of linear 
equations. Section 3 derives the first degeneracy condition, which 
arises when the input joint coordinates satisfy a certain 
relationship. The second degeneracy condition is set in section 4. 
This condition pertains to the geometry of the manipulator and is 
shown to define new analytic manipulators. These manipulators 
have a characteristic polynomial of degree 3 instead of 6, and 
feature more simple singularities. Last section concludes this 
paper. 
2 
Kinematic equations 
Figure 1 shows a general 3-RPR manipulator, constructed by 
connecting a triangular moving platform to a base with three RPR 
legs. The actuated joint variables are the three link lengths ρ1, ρ2 
and ρ3. The output variables are the position coordinates ( , )
x y  of 
the operation point P chosen as the attachment point of link 1 to 
the platform, and the orientation ϕ of the platform. A reference 
frame is centred at A1 with the x−axis passing through A2. 
Notation used to define the geometric parameters of the 
manipulator is shown in Fig. 1.  
x
θ
ρ1
l2
y
ϕ
β
ρ2
ρ3
l3
A3(c3, d3)
A2(c2, 0)
P(x, y)
A1(0, 0)
 
Figure 1: A 3-RPR parallel manipulator 
The three constraint equations of the manipulator can be 
expressed as [3] 
2
2
2
1
x
y
ρ =
+
 
(1) 
( )
(
)
( )
(
)
2
2
2
2
2
2
2
cos
sin
x
l
c
y
l
ρ
ϕ
ϕ
=
+
−
+
+
 
(2) 
(
)
(
)
(
)
(
)
2
2
2
3
3
3
3
3
cos
sin
x
l
c
y
l
d
ρ
φ
β
φ
β
=
+
+
−
+
+
+
−
 
(3) 
A system of two linear equations in x and y is first derived by 
subtracting Eq. (1) from Eqs. (2) and (3), thus obtaining 
0
Rx
Sy
Q
+
+
=
 
(4) 
0
Ux
Vy
W
+
+
=
 
(5) 
where, 
2
2
2
2 2
2
2
2
2
2
2
2
1
2 cos( )
2
2 sin( )
2
cos( )
R
l
c
S
l
Q
c l
l
c
ϕ
ϕ
ϕ
ρ
ρ
=
−
=
= −
+
+
−
+
        
3
3
3
3
3 3
3 3
2
2
2
2
2
3
3
3
3
1
2 cos(
)
2
2 sin(
)
2
2
sin(
)
2
cos(
)
U
l
c
V
l
d
W
d l
c l
l
c
d
ϕ
β
ϕ
β
ϕ
β
ϕ
β
ρ
ρ
=
+
−
=
+
−
= −
+
−
+
+
+
+
−
+
 
As pointed out in [7], x and y can be solved only if the 
determinant RV−SU is different from zero. If it is so, the 6th-
degree characteristic polynomial is obtained upon substituting the 
expressions of x and y into Eq. (1). The general expression of this 
characteristic polynomial is not reported here but can be found in 
[8]. Otherwise, the system degenerates and the forward kinematics 
cannot be solved this way. To the best of the authors’ knowledge, 
however, the degenerate case RV−SU =0 has never been 
examined. In the following sections, we will investigate the 
conditions under which the determinant of the linear system 
vanishes and will derive the forward kinematics equations 
associated with these conditions. 
3 
First degeneracy condition  
3.1 
Derivation of the condition 
Since RV−SU depends only on the geometric parameters of the 
manipulator and the orientation ϕ of the platform, RV−SU =0 
yields a condition on ϕ for the linear system to degenerate. This 
condition is 
2
3
2 3
2
3
2 3
2 3
2 3
sin( )
(
sin( ))cos( )
(
cos( ))sin( )
0
c d
l l
l d
c l
l c
c l
β
β
ϕ
β
ϕ
+
−
+
+
−
=
 
(6) 
Resorting to the tan-half angle substitution provides a quadratic 
in
tan( / 2)
t
ϕ
=
, which has the following form 
2
3
2
2
3
2
2
2 3
2 3
3
2
2
3
2
2
(
(
)
sin( )(
))
2(
cos( ))
(
)
sin( )(
)
0
d l
c
l
l
c
t
l c
c l
t
d l
c
l
l
c
β
β
β
+
+
+
+
−
−
−
+
−
=
 
(7) 
and may define two orientation angles of the platform. 
In order to be able to check the degeneracy condition while 
solving the forward kinematics, it is useful to set it in terms of ρ1, 
ρ2 and ρ3. When the determinant of the linear system of Eqs. (4, 5) 
vanishes, the Cramer’s rule tells us that an additional condition 
must be satisfied for a solution to exist, that is 
0
SW
VQ
−
=
, or, 
equivalently, 
0
RW
UQ
−
=
. When set in terms of  
tan( / 2)
t
ϕ
=
, 
this condition yields  
(
)
2
2
2
2
2
2
2
2
4
3
2 2
2
2
2
1
3
2 2
2
2
2
1
2
2
2
2
2
3 3
3
2
2
2
2
3
1
2
3
2
2
2
2
2
2
1
3
3
3
3
2
2
1
2 3
2
3
2
3 3
3
2
2
1
2
(2
)sin( )
(2
)
4
sin( )
2 (
2 (
)
)cos( )
2 (
)
4
(
2 )sin( )
8
cos( )
2
(
l
c l
c
l
d
c l
c
l
t
l d l
l l
c
l c
c
t
l
c
d
l
l l c
c
l d l
d l
c
ρ
ρ
β
ρ
ρ
β
ρ
ρ
β
ρ
ρ
β
β
ρ
ρ
+
−
+
+
+
+
−
+
+
⎛
⎞
−
+
+
−
+
−
+
+
⎜
⎟
⎜
⎟
+
−
+
+
+
⎝
⎠
−
−
−
+
+
+
−
(
)
2
2
2
2
2
2
2
3 3
3
2
2
2
2
3
1
2
2
2
2
2
2
2
1
3
3
3
3
2
2
2
2
2
2
2
2
3
2
2
2
1
2 2
3
2
2
2 2
1
2
)
4
sin( )
2 (
2 (
)
)cos( )
2 (
)
(
2
)sin( )
(
-2
)
0
t
l d l
l l
c
l c
c
t
l
c
d
l
l c
l
c l
d c
l
c l
β
ρ
ρ
β
ρ
ρ
ρ
ρ
β
ρ
ρ
+
⎛
⎞
−
−
+
+
−
+
−
+
+
⎜
⎟
⎜
⎟
−
+
+
+
⎝
⎠
−
−
+
+
−
+
+
+
−
=
 
(8) 
Let t1 and t2 define the two solutions of the quadratic defined by 
Eq. (7). Substituting t1 and t2 into Eq. (8) yields two degeneracy 
conditions D1 and D2 that depend only on ρ1, ρ2 and ρ3 and on the 
geometric parameters. Since Eq. (8) is a polynomial of degree 2 in 
ρ1, ρ2 and ρ3 and since these variables do not appear in Eq. (7), D1 
and D2 are also polynomials of degree 2 in ρ1, ρ2 and ρ3. 
Expressions of the two degeneracy conditions D1 and D2 are quite 
large when the geometric parameters are left as variables. When 
rational values are assigned to the geometric parameters of the 
manipulator, however, D1 and D2 are simple quadratic equations, 
which can be solved accurately without any difficulties. A 
particular case with a simple geometric interpretation will be 
investigated in § 3.3. 
3.2 
Degenerate forward kinematic solutions  
If the input variables satisfy one of the two degeneracy conditions 
D1 and D2, the inverse kinematic problem needs special attention. 
First of all, the 6th-degree characteristic polynomial P(t) can still 
be used to calculate all orientation solutions, provided that no 
division by RV−SU be performed while deriving P(t). Eqs (4, 5) 
can always be rewritten as x(RV−SU) = SW−VQ and y(RV−SU) = 
RW−UQ. Multiplying both sides of Eq. (1) by (RV−SU)2 and 
substituting x2(RV−SU)2 and y2(RV−SU)2 by (SW−VQ)2 and 
(RW−UQ)2, respectively, yields ρ1
2(RV−SU)2 = (SW−VQ)2 + 
(RW−UQ)2, which, after applying the tan-half substitution 
tan( / 2)
t
ϕ
=
, yields the 6th-degree characteristic polynomial 
P(t). If one of the two degeneracy conditions D1 and D2 is 
satisfied, then RV−SU = 0, SW−VQ = 0 and RW−UQ = 0 and, thus, 
P(t) vanishes. This means that the degenerate roots defined by 
Eq. (7) are also roots of P(t). Moreover, these roots are double 
roots of P(t) because RV−SU, SW−VQ and RW−UQ are squared in 
P(t). However, unlike what arises in non-degenerate situations, the 
existence of this double root does not mean that the manipulator 
admits two coalesce assembly modes. In effect, Eqs. (4,5) cannot 
be used to calculate x and y since RV−SU = 0, and we show below 
that the double root is associated with two sets of position 
coordinates (x, y).  
To know which equations should be used for the calculation of x 
and y, several cases need be considered. If R≠0, x can be 
expressed as function of y from Eq. (4). This expression is then 
reported into Eq. (1), which yields a quadratics in y and two 
solutions for y. Then each solution is reported in Eq. (4), which is 
solved for x. In this case we have two sets of position coordinates 
(x, y) with two distinct values of x and two distinct values of y. If 
R=0 and U≠0, the same procedure can be used by using Eq. (5) 
instead of Eq. (4). If R=0, U=0 and S≠0 (resp. V≠0),  y is directly 
calculated from Eq. (4) (resp. from Eq. (5)). Its solution is then 
reported in Eq. (1), which gives two values for x. In this case we 
have two sets of position coordinates (x, y) with two distinct 
values of x but only one for y. Finally, if R=U=S=V=0, then Eqs. 
(4,5) show that Q and W must also equal 0, and thus Eq. (2) yields 
x2+y2=0, that is, x=y=0. Then Eq. (1) implies that ρ1=0. Equating 
R, U, S, V, Q and W to zero implies that ρ2=0 and ρ3=0. This 
situation is possible only for a manipulator with congruent 
platform and base triangles. 
3.3 
A degeneracy condition with geometric interpretation  
We investigate now a particular case where D1 simplifies and 
yields a simple geometric interpretation. Assume that l2 = c2 (the 
base and platform triangles of the manipulator have the same base 
length). Then, t = 0 is a root of Eq. (7) because the constant term 
vanishes. Substituting t = 0 into Eq. (8) simplifies considerably 
this equations as only the constant term remains. This constant 
term can be factored as (d3 − l3sin(β))((c2 − l2)2 + ρ1
2 − ρ2
2) and 
when l2 = c2, this term simplifies and becomes (d3 − l3sin(β))(ρ1
2- 
ρ2
2). Assuming that l2 = c2 is the only geometric condition, that is, 
if d3 − l3sin(β) ≠ 0, then the first degeneracy condition, D1, is 
simply ρ1
2 − ρ2
2 = 0, or ρ1 = ρ2 if only positive joint values are 
assumed. The geometric interpretation of t = 0, l2 = c2 and ρ1 = ρ2 
is that the four-bar mechanism defined by disconnecting leg-rod 3 
while keeping ρ1, ρ2 constant, is a parallelogram. The coupler 
curve of the revolute joint centre of the moving platform 
associated with leg-rod 3 is a circle. For a constant value of ρ3, the 
tip of leg-rod 3 traces a circle too and its intersection with the one 
generated by the parallelogram defines the two assembly modes of 
the 3-RPR manipulator. This situation is illustrated in the 
numerical example below. 
3.4 
Numerical example 
Let us study the manipulator defined by c2 = l2 = 2, c3 = 1/2, d3 = 1, 
l3 = 3/2 and β = π/3. In this case the first degeneracy condition is 
ρ1 = ρ2 (we are in the situation described in § 3.3). For joint values 
that satisfy ρ1 = ρ2, the polynomial characteristics factors with t2 as 
the first factor and the following quartic as the second factor: 
(
)
(
)
(
)
4
2
2
4
2
3
2
3
4
2
3
2
2
3
2
3
4
2
2
2
4
2
2
3
2
3
3
2
2
2
3
4
2
2
3
2
16
(196
32
48 3)
16
(24
48 3)
348 3
805
(64
24 3)
(48 3
32)
12 3
368
32
(64
96 3
112)
64
142
120 3
32
(64
24 3)
(48 3
32)
180 3
304
16
(84
48 3
32
)
t
t
t
t
ρ
ρ
ρ
ρ
ρ
ρ
ρ
ρ
ρ
ρ
ρ
ρ
ρ
ρ
ρ
ρ
ρ
⎛
⎞
−
+
+
+
+
⎜
⎟
+
⎜
⎟
−
+
+
⎝
⎠
−
+
−
−
−
+
−
+
+
−
−
+
+
+
−
+
−
−
+
+
+
−
−
2
4
2
3
3
+16
(48 3
88)
132 3
229
0
ρ
ρ
+
−
−
+
=
 
The first factor t2 gives the degenerate root t=0. The position 
coordinates are calculated as described in §3.2 (note that here 
R=0). For ρ1 = ρ2 =1 and ρ3 = 7/10, the quartic admits four real 
roots. The associated platform orientation angles and positions are 
given in Tab. (1), along with the two platform positions associated 
with the degenerate root t = 0. Figure 2 shows the associated 6 
assembly-modes. One can easily verify that the two assembly 
modes associated with the degenerate root t = 0 are distinct. 
 
Quartic root #1  Quartic root #2 Quartic root #3 Quartic root #4 
Degenerate root t=0 
ϕ  
-43.8049 deg 
-6.6271 deg 
23.6384 deg 
58.4876 deg 
0 
x 
-0.3395 
-0.9849 
0.9768 
0.6632 
-0.1394 
-0.9499 
y 
0.9406 
0.1728 
-0.2141 
-0.7485 
-0.9902 
-0.3126 
Table 1: The six sets of solutions for a manipulator defined by c2=l2=2, c3=1/2, d3=1, l3=3/2, β=π/3 with inputs ρ1=ρ2=1 and 
ρ3=3/5. 
#2
-1
0
1
-1
2 x
y
-1
0
1
-1
2
#3
x
y
#4
-1
0
1
-1
2 x
y
-1
0
1
-1
2 x
y
-1
0
1
-1
2
x
y
1
#1
x
y
-1
0
1
-1
2
 
Figure 2: The six assembly-modes corresponding to Tab. 1. The last two ones correspond to the degenerate root.   
 
4 
Degeneracy over the whole joint space 
4.1 
Condition on the manipulator geometry 
We would like to know if it is possible to find manipulators for 
which the system of linear equations (4,5) degenerates for any 
input joint values. For such manipulators, Eq. (7) must be satisfied 
for any value of t. Thus, the following three conditions must be 
simultaneously satisfied  
3
2
2
3
2
2
(
)
sin( )(
)
0
d l
c
l
l
c
β
+
+
+
=
 
(9) 
2 3
2 3 cos( )
0
l c
c l
β
−
=
 
(10) 
3
2
2
3
2
2
(
)
sin( )(
)
0
d l
c
l
l
c
β
−
−
+
−
=
 
(11) 
Eqs. (9) and (11) yield 
2
2
l
c
=
 and 
3
3
sin( )
/
d
l
β = −
. Substituting 
2
2
l
c
=
 into Eq. (10) yields 
3
3
cos( ) 
 
/
c
l
β
=
. Thus, one must have 
3
2
2
3
3
d
c
l
+
=
. It can be easily verified that the geometric 
interpretation of these conditions is that the base and the platform 
triangles are congruent and the platform triangle is rotated by 180 
deg about the side l2. Note that if we had had 
3
3
sin( )
/
d
l
β =
 
instead of 
3
3
sin( )
/
d
l
β = −
, we would have got manipulators with 
congruent base and platform triangles that exhibit a passive 
translational motion when ϕ=0 [5, 6].  
4.2 
Forward kinematics 
Since the system of linear equations is always singular, this means 
that RV−SU=0 independently of ρ1, ρ2 and ρ3. Thus, SW−VQ must 
be equal to zero for any value of ρ1, ρ2 and ρ3. This means that t 
must satisfy Eq. (8). When substituting 
2
2
l
c
=
, 
3
3
sin( )
/
d
l
β = −
, 
3
3
cos( ) 
 
/
c
l
β
=
 and 
3
2
2
3
3
d
c
l
+
=
 into Eq. (8), the coefficient of 
highest degree vanishes and we get the following 3rd-degree 
polynomial in  
tan( / 2)
t
ϕ
=
: 
2
2
2
2
2
3
3
1
2
2
3 2
2
3
1
2
2
2
2
3
3 2
2
2
1
2
2
2
2
2
2
2
3
1
2
3
2
3
2
1
2
3
2
1
(
(
4
4
)
(
))
(8
4
)
(
(
)
4
)
(
)
0
c
c
c c
c
t
d
c c
c
t
c
c
d c
c t
d
ρ
ρ
ρ
ρ
ρ
ρ
ρ
ρ
ρ
ρ
ρ
ρ
−
+
−
+
−
+
−
+
−
+
−
+
−
−
+
−
=
 
(12) 
which is the characteristic polynomial to be solved for this family 
of manipulators. Thus, there exists a new family of analytic 
manipulators, namely, those that have congruent base and 
platform triangles with the platform triangle rotated by 180 deg 
about the side l2. The position coordinates of the platform are then 
calculated by solving in cascade a quadratic and a linear equation 
as explained in §3.2. These analytic manipulators may have up to 
6 assembly-modes but, unlike general 3-RPR manipulators, they 
have only three distinct platform orientations and each platform 
orientation is associated with two distinct positions. Because their 
forward kinematics degenerates over the whole joint space, we 
call these manipulators degenerate manipulators.  
4.3 
Numerical example 
A numerical example is now presented to show the different 
assembly modes of a given degenerate manipulator. The 
geometric parameters are c2 = l2 = 1, c3 = 0, d3 = 1, l3 = 1 and β = 
−π/2. These parameters satisfy the geometric conditions for the 
manipulator to be degenerate.  
The direct kinematics is now calculated for ρ3 = 4/5, ρ2 = ρ3 =3/2. 
The 3rd-degree polynomial is: 
161t3 - 239t2 - 239t + 161= 0 
(13) 
The six sets of solutions are reported in Tab. 2. 
 
#1 
#2 
#3 
#4 
#5 
#6 
ϕ 
(deg)
-90 
-90 
53.6102 53.610 126.389 126.389 
x 
0.6547 
-0.459 
0.3963 
-0.794 0.6950 
0.0933 
y 
-0.4597 0.6547 
0.6950 
0.0933 0.3963 
-0.7945 
Table 2: The six sets of solutions for a degenerate manipulator 
defined by c2=l2=1, c3=0, d3=1, l3=1 and β=−π/2 with inputs 
ρ1=4/5 and ρ2=ρ3=3/2. 
5 
Conclusions 
In this paper, the general 6th-degree characteristic polynomial of  
planar 3-RPR parallel manipulators was shown to degenerate in 
situations that had not been investigated in the past.  
The first degeneracy may occur for any manipulator geometry. 
There may exist two platform angles for which the system of 
linear equations in x and y that needs be established prior to the 
derivation of the characteristic polynomial, becomes singular. 
Each of these two angles defines a relationship between the input 
joint values. When solving the forward kinematics, if the input 
joint values satisfy one of these two relationships, the general 6th-
degree characteristic polynomial gives a double root, which is a 
degenerate root for which the aforementioned system of linear 
equations is singular, and which could be erroneously interpreted 
as two coalesce assembly modes and, thus, as a kinematic 
singularity. But one gets an unusual situation in which there is one 
platform orientation associated with two distinct sets of positions 
(x, y).  The second degenerate situation has an interesting practical 
consequence. It occurs for any input joint value when the 
manipulator has a special geometry, namely, congruent base and 
platform triangles with a rotation of 180 deg of the platform 
triangle about one of its sides. The forward kinematics of such 
manipulators can be solved with a 3rd-degree characteristic 
polynomial and a quadratic in sequence. Thus, these manipulators, 
which we have called degenerate manipulators, define a new 
family of analytic manipulators that would be more suitable for 
industrial applications. 
These degenerate manipulators are now being examined in more 
details by the authors of this paper to evaluate their potential 
interests for industrial applications. First results show that their 
singularity locus is much simpler than the one of general 
manipulators. In addition, contrary to the analytic manipulators 
with similar platform and base triangles studied in [5, 9] that have 
also a simple singularity locus, the degenerate manipulators 
defined here are not singular at ϕ = 0 or ϕ = π throughout the 
plane and we have observed that they can execute non-singular 
assembly-mode changing motions [8, 10]. 
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#1
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#2
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#3
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#5
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#6
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#4
 
Figure 3: The six assembly-modes corresponding to Tab. 2. 
 
6 
References 
[1] Hunt K.H., Kinematic Geometry of Mechanisms, Oxford 
University Press, Cambridge, 1978. 
[2] Hunt K.H., “Structural kinematics of in-parallel actuated robot 
arms,” J. of Mechanisms, Transmissions and Automation in 
Design 105(4):705-712, March 1983. 
[3] Gosselin C., Sefrioui J. and Richard M. J., “Solutions 
polynomiales au problème de la cinématique des manipulateurs 
parallèles plans à trois degrés de liberté,” Mechanism and 
Machine Theory, 27:107-119, 1992. 
[4] Pennock G.R. and Kassner D.J., “Kinematic analysis of a 
planar eight-bar linkage: application to a platform-type robot,” 
ASME Proc. of the 21th Biennial Mechanisms Conf., pp 37-43, 
Chicago, September 1990. 
[5] Gosselin C.M. and Merlet J-P,  “On the direct kinematics of 
planar parallel manipulators: special architectures and number of 
solutions,” Mechanism and Machine Theory, 29(8):1083-1097, 
November 1994. 
[6] Kong X. and Gosselin C.M., “Forward displacement analysis 
of third-class analytic 3-RPR planar parallel manipulators,” 
Mechanism and Machine Theory, 36:1009-1018, 2001. 
[7] Merlet J-P, Parallel robots, Kluwer Academic Publ., 
Dordrecht, The Netherland, 2000. 
[8] P.R. Mcaree, and R.W. Daniel, “An explanation of never-
special assembly changing motions for 3-3 parallel manipulators,” 
The International Journal of Robotics Research, vol. 18( 6): 556-
574. 1999. 
[9] Kong X. and Gosselin C.M., “Determination of the uniqueness 
domains of 3-RPR planar parallel manipulators with similar 
platforms,” Proc.  ASME 26th Biennial Mechanisms and Robotics 
Conference, Baltimore, USA, No. MECH-14094, September 
2000. 
[10] C. Innocenti, and V. Parenti-Castelli, “Singularity-free 
evolution from one configuration to another in serial and fully-
parallel manipulators. ASME Journal of Mechanical Design. 
120:73-99. 1998.