The Isotropic Decoupling of the Direct Kinematics
of Parallel Manipulators Under Sensor Redundancy
Luc Baron
Department of Mechanical Engineering
Ecole Polytechnique de Montreal
P.O. 6079, station Centre-Ville
Montreal, Quebec, Canada, H3C 3A7
[email protected]
Abstract
Using redundant sensors, the direct kinematics of
general parallel manipulators is here decoupled and reduced to an orientation problem only. The accuracy of
the decoupling depends on the condition number of a
decoupling matrix. The conditions under which this
matrix is isotropic are derived and illustrated with a
numerical example.
1 Introduction
1.1 Notation
A, B
ai ,bi
pi
fp; Rg
:
:
:
:
Frames xed to bodies A and B .
Position vectors of points Ai and Bi .
Vector directed from Ai to Bi .
Pose of B in A.
In general, parallel manipulators consist of two main
rigid bodies. Body A is xed and is called the base,
while body B is movable and is called the moving body.
The two main bodies are coupled via n legs attached
to points fAi gn1 and fBi gn1 , respectively. Each leg
is a simple kinematic chain, on which a set of sensors
measures information about the position vectors of the
end-points of the legs, pi . The pose, i.e., the position
and orientation, of body B in A, is described by p, the
position vector of the origin of B in A, and R, a 3 3
proper orthogonal matrix, expressing the orientation
of B in A. The direct kinematics (DK) associated
with the foregoing system is dened here as: For a
given set of sensor position measurements, determine
the pose of the moving body relative to the base. The
inverse kinematics (IK) is correspondingly dened as:
For a given pose of the moving body relative to the
base, determine the set of actuator positions.
Jorge Angeles
Department of Mechanical Engineering &
Centre for Intelligent Machines
McGill University, 817 Sherbrooke St. W.
Montreal, Quebec, Canada, H3A 2K6
[email protected]
1.2 Previous Work
The DK of parallel manipulators of general geometry using the six leg lengths as the set of sensed joints
has been the focus of intensive research. Raghavan [1]
reported numerical experiments showing up to forty
real and imaginary solutions. Although closed-form
solutions exist for some manipulators of special geometries [2, 3], in general, only numerical solutions
are possible [4, 5, 6]. However, the complexity of these
numerical procedures is not suitable for on-line implementation. Alternatively, the use of additional sensors like a passive instrumented leg [7], extra sensors
[8], nine sensors measuring the position of three leg
end-points [9], or redundant sensors [10], have shown
to greatly simplify the DK. In particular, the decoupling technique proposed in [10] has shown to provide a linear algebraic solution of the DK. However,
the numerical conditioning of the decoupling decreases
signicantly with the reduction of sensor redundancy.
This is so because the decoupling equations consider
only homogeneous sets of measurement subspaces and,
hence, force the use of the same number of sensors
for every leg of the manipulator. Here, using the linear kinematic model introduced in [11], and briey recalled below, we derive a general decoupling equation
for the non-homogeneous sets of measurement subspaces. On the other hand, the actuation accuracy has
been shown to be determined by the condition number of a Jacobian matrix that arises at the the velocity level [12, 13]. Similarly, the measurement accuracy
of the pose of the moving body is determined by the
condition number of a decoupling matrix that arises
at the displacement level. Here, using the decoupling
matrix of the general decoupling equation, we derive
the conditions under which an isotropic decoupling of
the DK of the parallel manipulator is obtained.
IEEE Int. Conf. on Robotics and Automation, Nagoya, Japan, May 21-27, 1995 Page 2
2 Linear Kinematic Model
with ei denoting a unit vector along the projection line
or, correspondingly, normal to the projection plane.
Under the assumption of rigidity of the base and
Under these measurement conditions, the projection
the moving body, the position vectors of the leg endof pi onto Mi can be computed from the readouts of
points, pi , can be computed for a given pose fp; Rg
dierent sets of sensed joints, as described in [11], i.e.,
from the relations
[qi ]M = Mi qi ;
qi fi + gi;
(5)
pi = p + Rbi , ai ; for i = 1; :::; n:
(1)
where fi and gi are dened as vectors directed from
The set of sensed joints located along each leg of the
point Ai to a point of M?i and from another point of
manipulator provides complete or, more usually, parM?i to point Bi , respectively. In general, pi 6= qi , but
tial, information about pi . The Linear Direct Kinethey share the same projection onto Mi . Now, let
matic Estimation (LDKE) Theorem reported in [10]
xi = [pi]M , [qi ]M ;
(6)
species the measurement conditions under which a
linear kinematic model is obtained. For quick referdenote the error vector on the closure of the kinematic
ence, we recall this result below:
chain of leg i that arises from noisy measurements. FiTheorem 1 (Measurement Conditions):
nally, the linear kinematic model is obtained by subThe kinematic equations of parallel manipulators of leg
stituting eqs.(1), (2) and (5) into eq.(6), i.e.,
architectures KpKr , where Kp is a simple kinematic
xi = Mi(p + Rbi , ai) , Mi qi ; for i = 1; :::; n: (7)
chain composed of revolute and prismatic joints in series, while Kr is a series of revolutes of intersecting
axes, are linear in the pose of the moving body under
3 Direct Kinematics
measurements of the position of: i) all joints of Kp ;
or ii) all joints of Kp other than a translation along a
The unavoidable measurement noise can be ltered
line; or iii) all joints of Kp other than a motion in a
resorting to the redundancy of the measurements, the
plane.
DK solution then being obtained as:
For each condition, the position of the end-points
n
of the legs is completely measured into a dierent sub1X
RT R = 1 (8)
T xi ! min s:t:
3
z
=
x
space of R , called the measurement subspace Mi .
i
det(R) = +1
p;R
2 i=1
The measurement conditions are called 3-D, 2-D, or 1D, depending on the dimension of their corresponding
Since p is subjected to no constraint, it is possible
subspace Mi . Let us decompose pi into two orthogoto decouple the two unknowns of this problem, i.e., p
nal parts, [ ]M , the component lying on the measureand R, by deriving an equation expressing the leastment subspace Mi , and [ ]M ? , the component lying
square solution of z with respect to p for any given R,
on the subspace orthogonal to Mi , denoted with M?i .
namely, a decoupling equation.
By using the measurement projector Mi and an orthogonal complement of Mi , denoted with M?i , the
3.1 General Decoupling Equation
decomposition of pi takes the form
The decoupling equations for homogeneous sets of
pi = [pi ]M + [pi ]M ?
(2)
3-D,
2-D, and 1-D measurement subspaces have been
= Mi pi + M?i pi = (Mi + M?i )pi :
proposed in [10]. Here, we derive a general decoupling equation that considers not only homogeneous
It is apparent that Mi and M?i are related by Mi +
but also non-homogeneous sets of measurement
?
?
Mi = 1 and Mi Mi = O, where 1 and O are the sets,
subspaces.
Using the kinematic model of eq.(7), we
3 3 identity and zero matrices, respectively. These
have:
measurement projectors are given as
8
8
Corollary 1 (General Decoupling Equation):
i) 3 , D
< 1
< O
M be the mean value of the projectors Mi, while
Mi = : Pi ; M?i = : Li ii) 2 , D ; (3) Let
aM , B M and qM are the mean values of the projecLi
Pi iii) 1 , D
tions of ai , bi and qi onto the measurement subspace
Mi , for i = 1; :::; n, respectively. The decoupling equawhere the plane and line projectors, Pi and Li , respectively, are dened as
tion with non-homogeneous measurements is given by
T
+ B M r;
Pi 1 , Li ; Li eiei ;
(4)
aM + qM = Mp
(9)
IEEE Int. Conf. on Robotics and Automation, Nagoya, Japan, May 21-27, 1995 Page 3
where
and B0i and a0i are dened as
n
n
X
X
B0i Bi , M ,1B M ;
(17a)
aM n1 Mi ai ; qM n1 Miqi ; (10a)
0
,
1
ai ai + qi , M (aM + qM ): (17b)
i=1
i=1
n
n
X
X
solution of the constrained nonlinear least-square
M n1 Mi ; B M n1 MiBi ; (10b) The
problem
of eq.(15), R^ , is approximated by the orthogi=1
i=1
onal factor of the polar decomposition of the unconthe 3 9 mapping Bi and the 9{dimensional rotation
strained least-square solution. Equation (16) is thus
vector r being dened as
rewritten as a linear algebraic system, i.e.,
2
3
2
3
T
T
T
bi 0 0
r1
aM = BM r
(18)
Bi 4 0T bTi 0T 5 ; r 4 r2 5 ; (11)
where the 3n-dimensional vector aM and the 3n 9
0T 0T bTi
r3
matrix BM are dened as
2
2
where rTj denotes the j th row of R.
M1a01 3
M1B01 3
The normality condition of z with respect to p is
aM 64 ... 75 ; BM 64 ... 75 : (19)
derived as
n
X
n
X
dz
dp = ( i=1 Mi )p + i=1 Mi Rbi
,
n
X
i=1
Mi ai ,
n
X
i=1
Miqi = 0:
(12)
However, the rotation matrix R cannot be directly
factored out of the summation, as can the position
vector p. Nevertheless, if we introduce a 3 9 linear
mapping Bi and a 9{dimensional rotation vector r as
follows:
Rbi = Bir;
(13)
where Bi and r are dened as in eq.(11), then we
can derive eq.(9) by substituting eq.(13) into eq.(12).
Moreover, the Hessian matrix of z with respect to p,
namely,
n
d2 z = X
Mi = nM
(14)
dp2
i=1
is positive denite for the direct sum of measurement
subspaces, M1 ::: Mn yields R3 , thereby completing the proof of Corollary 1. Notice that the three
decoupling equations proposed in [10] are particular
cases of eq.(9).
3.2 Orientation Problem
xi = Mi (B0i r , a0i );
decomposition theorem [14]. The constrained leastsquare solution of the DK is approximated by
R^ = Q; p^ = M ,1(aM + qM , B M ^r): (21)
3.3 Numerical Conditioning
Apparently, from eqs.(17) and (21), the decoupling
. Thereof the DK needs the inversion of matrix M
fore, the accuracy of the decoupling, and consequently,
of the solution of the decoupled DK, depends on
, (M
). Since M
is the
the condition number of M
sum of symmetric, positive-semidenite matrices, i.e.,
fMign1 , it is also symmetric, and positive-semidenite;
hence, its eigenvalues i are real, while its eigenvectors are mutually orthogonal. The condition number
is then calculated as:
of M
) = 1 ; where 1 2 3 0: (22)
(M
3
Using this general decoupling equation, the DK of
eq.(8) can now be rewritten as a simple problem of
orientation estimation, i.e.,
n
X
RT R = 1 (15)
s
:
t
:
z = 12 xTi xi ! min
det(R) = +1
R
i=1
where
Mna0n
MnB0n
The vector form ~r of the unconstrained least-square
solution R~ is symbolically obtained via the generalized
inverse of BM , ByM , as follows:
~r = ByM aM ; ByM (BTM BM ),1 BTM :
(20)
Then, R~ is decomposed as R~ = QW, using the polar-
(16)
This number expresses \how well" the set of measurement subspaces fMi gn1 allows to span R3 or, in
other words, how much relative emphasis is placed on
the errors in all directions. Matrices with the minimum condition number of unity are called isotropic,
an isotropic decoupling being thus one that has equal
accuracy in all directions. The measurement conditions under which an isotropic decoupling is obtained
are derived below.
4
IEEE Int. Conf. on Robotics and Automation, Nagoya, Japan, May 21-27, 1995 Page 4
Isotropic Decoupling
where, obviously, the rst term in the right hand side
is isotropic, the second being isotropic as well, by
is isotropic i it veries
The decoupling matrix M
virtue of Theorem 2. The sum is, therefore, isotropic.
M = 1; > 0:
(23)
The decoupling is said isotropic if we nd at least one
is isotopic. In
measurement condition under which M
order to nd this condition, let us group the measurement projectors fMi gn1 into m sums of homogeneous
measurement projectors fHi gm
1 such that
n
m
X
X
M = 1 M = 1 H with m n; (24)
n i=1
i
i
n i=1
where Hi is either the sum of 3-D, 2-D or 1-D measurement projectors, namely, [Hi ]3-D , [Hi ]2-D or [Hi ]1-D .
Moreover, let us rst look at the isotropy conditions
of these three homogeneous sums. Matrix [Hi ]3-D is,
obviously, always isotropic, since it is a sum of identity matrices, while matrices [Hi ]1-D and [Hi ]2-D are
isotropic under the conditions below:
Theorem 2 (Homogeneous 1-D Measurements):
Let fei gn1 be a set of vectors directed from the centroid
to the vertices of a regular polyhedron. Then, matrix
H1-D dened as
H1-D =
n
X
i=1
eieTi
(25)
is isotropic.
Proof:
Theorem 2 can be proven most simply for each of
the Platonic solids, namely, the tetrahedron T , the
cube C , the octahedron O, the dodecahedron D and
the icosahedron I . To this end, we use the values
of the position vectors of the corners of the Platonic
solids inscribed inP
a sphere of unit radius. With these
values, the sum n1 ei eTi is computed in the form
1-D 1. Table 1 records the number of vertices n of
each Platonic solid and the values of 1-D and 2-D .
As a consequence of Theorem 2, we have:
Corollary 2 (Homogeneous 2-D Measurements):
For fei gn1 dened as in Theorem 2, matrix H2-D dened as
n
X
H2-D = (1 , ei eTi )
(26)
i=1
is isotropic.
Proof:
Note that the following sum can be written as
n
X
n
X
1
1
(1 , ei eTi ) = (n)1 ,
eieTi
(27)
n
T
C
O
D
I
4
8
6 20 12
1,D 8/3 8/3 2 20/3 4
2,D 16/3 16/3 4 40/3 8
Table 1: Characteristics of Platonic solids
is the sum of one
Since the decoupling matrix M
or several, homogeneous sums Hi , it is isotropic under
the conditions stated below:
Theorem 3 (Non-Homogeneous Sets):
Let fHi gm
1 be a set of homogeneous sums of 3-D, 2D and 1-D measurement projectors. The decoupling
dened as
matrix M
M = n1
m
X
i=1
Hi with m n;
(28)
is isotropic if, for all i = 1; :::; m, i) Hi is isotropic, or
ii) Hi is an orthogonal complement of Hj , for i 6= j .
The rst condition is obtained under the measurement conditions specied by Theorem 2, while the second condition is obtained as follows: Since Hj is an
orthogonal complement of Hi , we have:
Hi Hj = O33
(29)
and hence, Hj can be expressed as
Hj = H?i = 1 , Hi:
(30)
Therefore, the sum of Hi and Hj is isotropic, since
Hi + Hj = Hi + (1 , Hi) = 1:
(31)
As an example, let us assume that Hi and Hj are,
respectively, a single plane and line projector, i.e.,
Hi = 1 , ei eTi and Hj = ej eTj :
Since Hi is an orthogonal complement of Hj ,
Hi Hj = (1 , eieTi ) ej eTj
= ej eTj , ei eTi ej eTj = O33 :
(32)
(33)
The only condition under which eq. (33) holds is ei =
ej , and hence, we have Hi + Hj = 1.
IEEE Int. Conf. on Robotics and Automation, Nagoya, Japan, May 21-27, 1995 Page 5
5 Numerical Example
Using the geometry of an industrial parallel manipulator and a non-homogeneous combination of nine
sensors, with a minimum of one sensor per leg, the
isotropic decoupling is shown to be reached at a particular pose of the moving body and for a particular
set of orientations of the universal joints located at
points fAi gn1 . The geometry of the parallel manipulator is chosen as that of the ight simulator series 500
from CAE Electronics Ltd., as shown in Fig. 1 and
described in Table 2.
A4
A5
3
5
A
A3
2
A2
1
6
A6
A1
Figure 1: Flight simulator geometry
The nine-sensor layout is chosen as:
R fRP g?; R RP;
R fRP g?; R RP;
R fRP g?g;
fR RP;
where J denotes a joint with a position sensor and
fJJ g? two joints with perpendicular axes. The spherical joint S located at the end of each leg is omitted
for brevity. Now, let us denote by p and q the oddand even-numbered legs, respectively. The sensing of
. The plane
leg p is under 2-D measurement, i.e. R RP
normal to the leg is the measurement subspace Mp ,
while the line along the leg is the orthogonal measurement subspace M?p , as shown in Fig. 2a. The
orientation of these measurement subspaces depends
only on the pose of the moving body, and hence, the
isotropic decoupling is obtained by nding a suitable
pose. The sensing of leg q is under 1-D measurement,
i.e. R fRP g?. The line perpendicular to the leg axis
and the axis of the unique sensed joint, namely, zq , is
the measurement subspace Mq , while the plane normal to that line is the orthogonal measurement subspace M?q , as shown in Fig. 2b. Here, the orientation of these measurement subspaces does not depend
only on the pose of the moving body, but also on the
M?q
eq
M?p
Mp
R
Mq
zq
R
R
R
(a) Leg p: R RP
(b) Leg q: RfRP g?
Figure 2: The Universal joints located to points Ai .
i
B
4
ep
x
ai
y
z
x
bi
y
z
1 0.0953 -1.9682 0 1.7780 -1.3638 0
2 1.7521 0.9016 0 2.0701 -0.8579 0
3 1.6568 1.0666 0 0.2921 2.2217 0
4 -1.6568 1.0666 0 -0.2921 2.2217 0
5 -1.7521 0.9016 0 -2.0701 -0.8579 0
6 -0.0953 -1.9682 0 -1.7780 -1.3638 0
m
m
m
m
m
m
Table 2: Flight simulator dimensions
orientation of the axis of the sensed joint zq . Therefore, the isotropic decoupling is obtained by nding a
suitable set of orientation axes zq together with the
pose fp; Rg. The selection of the set of orientation
axes is particularly interesting, from a design point
of view, because it does not change the geometry of
bodies A and B , or even the legs of the manipulator,
but rather, it changes the orientation of the measurement subspaces Mq by orienting the yokes through
which the legs are xed on the base A. Therefore, the
designer has complete freedom to orient these measurement subspaces so as to obtain isotropy, as we do
below.
For this manipulator, it is not possible to reach
isotropy with a homogeneous combination of 2-D measurement Mp , i.e., condition i) of Theorem 3. Instead, condition ii) of Theorem 3 is used. The sensed
joint axes are oriented so that: Mq = M?p , for
q = p +1, i.e., eq = ep . This condition holds whenever
zq lies in Mp. One possible solution is as follows:
p = 0; 0; 0:7267 T ; R = 1
(34)
with the sensed joint axes oriented as:
z2 = 0:3380; ,0:9412; 0 T
z4 = 0:6461; 0:7632; 0 T
z6 = ,0:9840; 0:1780; 0 T
(35a)
(35b)
(35c)
IEEE Int. Conf. on Robotics and Automation, Nagoya, Japan, May 21-27, 1995 Page 6
the measurement subspaces being described by
e1 = e2 = 0:8719; 0:3132; 0:3765 T (36a)
e3 = e4 = ,0:7071; 0:5985; 0:3765 T (36b)
e5 = e6 = 0:7071; 0:5985; 0:3765 T (36c)
fo9
hit
ws that M1 + M2 = 1, M3 + M4 = 1
and M5 + M6 = 1, and hence,