arXiv:0708.3920v1  [cs.RO]  29 Aug 2007
KINEMATIC ANALYSIS OF
THE 3-RPR PARALLEL MANIPULATOR
D. Chablat, Ph. Wenger
Institut de Recherche en Communications et Cybern´etique de Nantes, 1, rue de la No¨e,
44321 Nantes, France
Damien.Chablat@irccyn.ec-nantes.fr, Philippe.Wenger@irccyn.ec-nantes.fr
I. Bonev
D´epartement de g´enie de la production automatis´ee
´Ecole de Technologie Sup´erieure
1100 rue Notre-Dame Ouest, Montr´eal (Qu´ebec) Canada H3C 1K3
Ilian.Bonev@etsmtl.ca
Abstract
The aim of this paper is the kinematic study of a 3-RPR planar parallel
manipulator where the three fixed revolute joints are actuated. The di-
rect and inverse kinematic problem as well as the singular configuration
is characterized. On parallel singular configurations, the motion produce
by the mobile platform can be compared to the Reuleaux straight-line
mechanism.
Keywords: Kinematics, Planar parallel manipulators, Singularity
1.
Introduction
2.
Preliminaries
A planar three-dof manipulator with three parallel RRR chains, the
object of this paper, is shown in Fig. 1.
This manipulator has been
frequently studied, in particular in Merlet, 2000; Gosselin, 1992; Bonev,
2005. The actuated joint variables are the rotation of the three revolute
joints located on the base, the Cartesian variables being the position
vector p of the operation point P and the orientation φ of the platform.
The trajectories of the points Ai define an equilateral triangle whose
geometric center is the point O, while the points B1, B2 and B3, whose
geometric center is the point P, lie at the corners of an equilateral tri-
angle. We thus have ||a2 −a1|| = ||b2 −b1|| = 1, in units of length that
need not be specified in the paper. .
A1
B1(
)
x,y
B2
A2
A3
B3
θ3
θ1
θ2
φ
ρ3
ρ1
ρ2
Figure 1.
A three-DOF parallel manipulator
3.
Kinematics
The velocity ˙p of point P can be obtained in three different forms,
depending on which leg is traversed, namely,
˙p = ˙ρi
(bi −ai)
||bi −ai|| + ˙θiE(bi −ai) + ˙φE(p −bi)
(1)
with matrix E defined as
E =
 0
−1
1
0

We would like to eliminate the three idle joint rates ˙ρ1, ˙ρ2 and ˙ρ3 from
Eqs.(1), which we do upon dot-multiplying the former by (Evi)T , thus
obtaining
(Evi)T ˙P = ˙θiρi + ˙φ(Evi)T E(p −bi)
(2)
with
vi = (bi −ai)
||bi −ai|| =
 cos(θi)
sin(θi)

and
ρi = (Evi)T E(bi −ai)
Equations (2) can now be cast in vector form, namely,
At = B˙θ
with
t =
"
˙p
˙φ
#
and
˙θ =


˙θ1
˙θ2
˙θ3


(3)
with ˙θ 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
A =


(Ev1)T
−(Ev1)T E(p −b1)
(Ev2)T
−(Ev1)T E(p −b2)
(Ev3)T
−(Ev3)T E(p −b3)


B =


ρ1
0
0
0
ρ2
0
0
0
ρ3


(4)
When A and B are nonsingular, we obtain the relations
t = J ˙θ,
with J = A−1B
and
˙θ = Kt
with K denoting the inverse of J.
3.1
Parallel Singularities
Parallel singularities occur when the determinant of matrix A van-
ishes (Chablat, 1998 and Gosselin, 1990). At these configurations, it is
possible to move locally the operation point P with the actuators locked,
the structure thus resulting cannot resist arbitrary forces, and control is
lost. To avoid any performance deterioration, it is necessary to have a
Cartesian workspace free of parallel singularities.
For the planar manipulator studied, the direct-kinematic matrix can
be written as function of θi
A =


−sin θ1
cos θ1
0
−sin θ2
cos θ2
cos(θ2)
−sin θ3
cos θ3
cos(θ3)/3 + sin(θ3)
√
3/2


(5)
and its determinant is the follows,
det(A)
=
−2 sin(θ3 −θ1 −θ2) −sin(θ3 −θ1 + θ2) −sin(θ3 + θ1 −θ2)
+
√
3 cos(θ3 + θ1 −θ2) −
√
3 cos(θ3 −θ1 + θ2)
(6)
The parallel singularities occur whenever the three axes normal to the
prismatic joint intersect. In such configurations, the manipulator cannot
resist a wrench applies at the intersecting point P, as depicted in Fig. 2
A special case exists when θ1 = θ2 = θ3 because the intersection point
is in infinity. Thus, the mobile platform can translate along the axes of
the prismatic joints, as depicted in Fig. 3.
3.2
Serial Singularities
Serial singularities occur when det(B) = 0. In the presence of theses
singularities, there is a direction along which no Cartesian velocity can
be produced. Serial singularities define the boundary of the Cartesian
A1
B1
B2
A2
A3
B3
v1
v3
v2
P
Figure 2.
Parallel singularity when the three axes normal to the prismatic joint
intersect
A1
B1
B2
A2
A3
B3
Figure 3.
Parallel singularity when θ1 = θ2 = θ3
workspace. For the topology under study, the serial singularities occur
whenever at least one ρi = 0 (Figure 4). When ρ1 = ρ2 = ρ3 = 0 not
any motion can be produce by the actuated joints.
3.3
Direct kinematics
The position of the base joints are defined in the base reference frame,
a1 =
 0
0

a2 =
 1
0

a3 =
"
1/2
√
3/2
#
(7)
A1
B1
B2
A2
A3
B3
v1
Figure 4.
Serial singularity when ρ1 = 0
In the same way, the position of the mobile platform joints are defined
in the mobile reference frame,
b′
1 =
 0
0

b′
2 =
 1
0

r′
3 =
"
1/2
√
3/2
#
(8)
The rotation matrix R describes the orientation of the mobile frame
with respect to the base frame.
R =
 cos(φ)
−sin(φ)
sin(φ)
cos(φ)

The position of p can be obtain in three different ways:
 x = cos(θ1)ρ1
y = sin(θ1)ρ1
(9a)
or
 x = −cos(φ) + cos(θ2)ρ2 + 1
y = −sin(φ) + sin(θ2)ρ2
(9b)
or
(
x = −cos(φ + π/3) + cos(θ3)ρ3 + 1/2
y = −sin(φ + π/3) + sin(θ3)ρ3 +
√
3/2
(9c)
To remove ρi from the previous equations, we multiply sin(θi) (respec-
tively cos(θi)) the equations in x (respectively in y) and we subtract the
first one to the second one, to obtain three equations,
sin(θ1)x −cos(θ1)y
=
0
(10a)
sin(θ2)x −cos(θ2)y + sin(θ2 −φ) −sin(θ2)
=
0
(10b)
sin(θ3)x −cos(θ3)y −cos(θ3 −φ + π/6)
−sin(θ3)/2 + cos(θ3)
√
3/2
=
0
(10c)
We obtain x and y as function of φ by using Eqs. 10a-b
x = sin(θ2 −φ + θ1) −sin(θ1 −θ2 + φ) −sin(θ1 + θ2) + sin(θ1 −θ2)
2(sin(θ1 −θ2))
y = cos(θ1 −θ2 + φ) −cos(θ2 −φ + θ1) −cos(θ1 −θ2) + cos(θ1 + θ2)
2(sin(θ1 −θ2))
Thus, we substitute x and y in Eq. 10c to obtain
m cos(φ) + n sin(φ) −m = 0
(12a)
with
m
=
sin(θ3) cos(θ2) sin(θ1) +
√
3 cos(θ3) sin(θ2) cos(θ1)
−
2 cos(θ3) sin(θ2) sin(θ1) −
√
3 cos(θ3) cos(θ2) sin(θ1)
+
sin(θ3) sin(θ2) cos(θ1)
(12b)
n
=
cos(θ3) sin(θ2) cos(θ1) +
√
3 sin(θ3) sin(θ2) cos(θ1)
−
2 sin(θ3) cos(θ2) cos(θ1) −
√
3 sin(θ3) cos(θ2) sin(θ1)
+
cos(θ3) cos(θ2) sin(θ1)
(12c)
Equation 12a admits two roots
φ = 0
and
φ = tan−1
 
2 nm
n2 + m2 , −−m2 + n2
n2 + m2
!
(13)
A trivial solution is φ = 0 which exists for any values of the actuated
joint values where x = y = 0. This means that when configuration of the
mobile platform associated to the trivial solution to the direct kinematic
problem is also a serial singularity because ρ1 = ρ2 = ρ3 = 0.
Such a behavior is equivalent to that of the agile eye....
3.4
Inverse kinematics
From Eqs. 10, we can easily solve the inverse kinematic problem and
find two real solutions in ] −π π] for each leg,
θ1
=
tan−1
y
x

+ kπ
θ2
=
tan−1

y + sin(φ)
x + cos(φ) −1

+ kπ
θ3
=
tan−1
 
y + sin(φ + π/3) −
√
3/2
x + cos(φ + π/3) −1/2
!
+ kπ
for k = 0, 1 and from Eqs. 9, we can easily find ρi,
ρ1
=
q
x2 + y2
ρ2
=
q
(x + cos(φ) −1)2 + (y + sin(φ))2
ρ3
=
q
(x + cos(φ + π/3) −1/2)2 + (y + sin(φ + π/3) −
√
3/2)2
3.5
Full cycle motion: Cardanic curve
A planar Cardanic curve is obtained by the displacement of one point
of one body whose position is constrained by making two of its points lie
on two coplanar lines. This curve is obtained when we set for example θ1
and θ2 and we observe the location of B3. A geometrical method to solve
the direct kinematic problem is find the intersection points between this
curve and the line passing through the third prismatic joint whose ori-
entation is given by v3. For the manipulator under study, the Cardanic
curve always intersects A3 and, apart from the singular configurations,
I.
The loop (A1, B1, B2, A2) can be written,
ρ1 cos(θ1) + cos(φ) −1 −ρ2 cos(θ2)
=
0
(16a)
ρ1 sin(θ1) + sin(φ) −ρ2 sin(θ2)
=
0
(16b)
We solve the former system to have ρ1 and ρ2 as a function of φ and θi
ρ1
=
−cos(φ) sin(θ2) + sin(θ2) + cos(θ2) sin(φ)
cos(θ1) sin(θ2) −cos(θ2) sin(θ1)
(17a)
ρ2
=
−sin(θ1) cos(φ) + sin(θ1) + sin(φ) cos(θ1)
cos(θ1) sin(θ2) −cos(θ2) sin(θ1)
(17b)
The position of B3 in the base frame can be written as follows,
xB3
=
ρ1 cos(θ1) + cos(φ + π/3)
(18a)
yB3
=
ρ1 sin(θ1) −sin(φ + π/3)
(18b)
Thus, by substituting the values of ρ1 and ρ2 obtained in Eqs. 17a-b in
Eqs. 18, we obtain the equations of the Caradanic curve,
xB3
=
 
−cos(θ1) cos(θ2)
cos(θ2) sin(θ1) −cos(θ1) sin(θ2) −
√
3
2
!
sin(φ)
+

cos(θ1) sin(θ2)
cos(θ2) sin(θ1) −cos(θ1) sin(θ2) + 1
2

cos(φ)
−

cos(θ1) sin(θ2)
cos(θ2) sin(θ1) −cos(θ1) sin(θ2)

(19a)
yB3
=

−cos(θ2) sin(θ1)
cos(θ2) sin(θ1) −cos(θ1) sin(θ2) + 1
2

sin(φ)
+
 
sin(θ1) sin(θ2)
cos(θ2) sin(θ1) −cos(θ1) sin(θ2) +
√
3
2
!
cos(φ)
−
sin(θ1) sin(θ2)
cos(θ2) sin(θ1) −cos(θ1) sin(θ2)
(19b)
Cardanic curve
I
A1
B1
B2
A2
B3
A3
v3
Figure 5.
Cardanic curve of B3 when B1 and B2 sliding along v1 and v2, respectively
The Cardanic curve degenerates for specific actuated joint values (
Hunt, 1982). In this case, the manipulator under study is equivalent
to a Reuleaux straight-line mechanism (Nolle, 1974). This mechanism
is composed by two prismatic joint and a mobile platform assembled
with the prismatic joint via two revolute joints as depicted in Fig. 6.
The angle between the axes passing through the prismatic joints is π/3.
The mobile platform is an unit equilateral triangle. The displacement
made by P is a straight line whose length is two. The magnitude of this
displacement is the same for A1 and A2.
Whenever θ2 −θ1 = π/3 or θ1 −θ2 = π/3 and θ3 −θ1 = π/3 or
θ1 −θ3 = π/3 with θ1 ̸= θ2 ̸= θ3, there exists a infinity of solutions
to the direct kinematic problem. This feature exists whenever we have
the same condition to have a parallel singularity and the three vector vi
intersect in one point. The displacement of Ai are around this point P
and is magnitude is 4
√
3/3.
4.
Conclusions
A kinematic analysis of a planar 3-RPR parallel manipulator was pre-
sented in this paper. The parallel and serial singularities have been char-
π/3
P
A1
A2
Figure 6.
Reuleaux straight-line mechanism
A1
B1
B2
A2
A3
B3
P
Figure 7.
Degenerated Cardanic curve when θ2 −θ1 = π/3
acterized as well as the direct and inverse kinematics. This mechanism
features two direct kinematic solutions whose one is a trivial singular
configuration. For some actuated joint values associated to a parallel
singularity, the motion made by the mobile platform is equivalent to a
Reuleaux straight-line mechanism where the amplitude of the motion is
well known.
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A1
B1
B2
A2
A3
B3
P
–3
–2
–1
0
1
2
3
–2
–1
1
2
–2
–1
0
1
2
–2
–1
1
2