ASME Design Automation Conference, Minneapolis, Sept. 1994
THE MEASUREMENT SUBSPACES OF PARALLEL MANIPULATORS
UNDER SENSOR REDUNDANCY
Luc Baron and Jorge Angeles
Department of Mechanical Engineering and Centre for Intelligent Machines
McGill University
Montreal, Quebec
Canada
ABSTRACT
Using the concept of measurement subspaces of the
leg-end-point position of general parallel manipulators,
a linear kinematic model is derived. For a class of 6dof leg-kinematic architectures of general geometry, the
sets of sensed joints that lead to a measurement subspace are classied into 16 dierent types. For each of
these types, a geometric interpretation and the equations for the computation of the measurement subspace
from the readouts of a set of sensed joints, are provided.
The design rules on the sensing of parallel manipulators
in order to obtain the linear kinematic model are also
derived.
INTRODUCTION
Notation
A
B
Ai
Bi
ai
bi
pi
qi
p
R
: Reference frame xed on body A.
: Moving frame xed on body B .
: ith point of body A.
: ith point of body B .
: Position vector of Ai in A.
: Position vector of Bi in B.
: Vector directed from Ai to Bi in A.
: Noisy measurement of pi in A.
: Position vector of the origin of B in A.
: Orientation of frame B in A.
In general, parallel manipulators, as shown in Fig. 1,
consist of two main rigid bodies. Body A is xed and
is, hence, called the base, while body B irnTfy
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ASME Design Automation Conference, Minneapolis, Sept. 1994
has been the focus of intensive research. For example,
several numerical procedures have been proposed (Innocenti and Parenti-Castelli, 1991; Merlet, 1991; Zanganeh and Angeles, 1993). Raghavan (1993) reported
numerical experiments showing up to forty real and
imaginary solutions. However, the complexity of the
numerical procedures and the multiplicity of solutions
of the underlying system of quadratic kinematic equations to solve are not suitable for on-line implementation. On the other hand, Shi and Fenton (1991)
obtained a system of linear kinematic equations using a redundant set of nine sensors, i.e., three legs
along which three sensors measure the leg-end-point
position. Merlet (1993) showed that some other redundant sets of sensed joints lead to a unique solution
of the DK. Baron and Angeles (1994) proposed three
measurement conditions under which linear kinematic
equations are obtained. Although the DK is solved
directly|i.e., non-iteratively|using homogeneous sets
of sensed joints, the robustness of the estimation procedure decreases signicantly with the reduction of sensor
redundancy. No simulation results are reported for the
non-redondant case of homogenous 1-D measurements,
due to the lack of robustness of this case. Here, using the concept of measurement subspaces, we derive a
general linear kinematic model for the non-homogenous
set of measurement conditions. For a class of 6-dof legkinematic architectures of general geometry, the sets of
sensed joints that lead to a measurement subspace are
classied into 16 types, for each of which relations for
the computation of the measurement subspace from the
sensor readouts are provided.
Page 2
In this case, the solution of the IK|the computation of
fqi gn1 for a given pose fp; Rg|is straightforward, while
the solution of the DK|the computation of fp; Rg for
a given set of sensed joint positions fqi gn1 |is more
challenging, since this problem requires the solution
of a system of quadratic kinematic equations, namely,
eq.(3). However, the Linear Direct Kinematic Estimation (LDKE) Theorem (Baron and Angeles, 1994)
species the measurement conditions of the leg-endpoint positions under which a system of linear kinematic
equations is obtained, i.e.,
Linear Direct Kinematic Estimation Theorem:
The kinematic equations of parallel manipulators of leg
architectures Kp Kr , where Kp is a simple kinematic
chain composed of revolute and prismatic joints in series and Kr is a series of revolutes of intersecting axes,
are linear in the pose of the moving body under measurements of the position of: i) all joints of Kp ; or ii)
all joints of Kp other than a translation along a line;
or iii) all joints of Kp other than a motion in a plane.
For each condition, the leg-end-point position is completely measured into a dierent subspace of R3 , called
the measurement subspace Mi . Under condition i), Mi
is R3 itself, and hence, qi is completely measured. Under condition ii), Mi is R2 , and hence, only the projection of qi onto the plane i perpendicular to the
unknown translation line Li is completely measured.
Under condition iii), Mi is R1 , and hence, only the
projection of qi along the line Li perpendicular to
the unknown motion plane i is completely measured.
Clearly, the components of the leg-end-point position qi
are completely measured in Mi and only partially measured in Mi . The measurement conditions are called:
i) 3-D measurements; ii) 2-D measurements; or iii) 1D measurements, depending on the dimension of their
corresponding measurement subspace Mi .
Let us decompose the measurement qi into two orthogonal parts, [ ]M , the component lying on the measurement subspace Mi , and [ ]M ? , the component lying on the subspace orthogonal to Mi , namely Mi ,
using the measurement projector Mi and an orthogonal
complement of Mi , namely Mi , as follows:
?
LINEAR KINEMATIC MODEL
Under the assumption of rigidity of the base and the
moving body, all the leg-end-point positions pi of the
manipulator can be computed from a given pose fp; Rg
as:
pi = p + Rbi , ai ; for i = 1; :::; n:
(1)
The set of sensed joints located along each leg of the manipulator provides complete or, more usually, partial,
information about the measured leg-end-point position,
namely qi . When noise is present on the measurements,
the actual and measured leg-end-point positions are related as
xi = pi , qi ; for i = 1; :::; n;
(2)
where xi is the error vector of the closure of the kinematic chain of leg i, that arises from noisy measurements. For example, when the sensed information is
the magnitude of qi , namely qi , the kinematic equations are quadratic in terms of the pose of the moving
body fp; Rg, i.e.,
qi2 = (p + Rbi , ai )T (p + Rbi , ai ):
(3)
?
?
qi = [qi ]M + [qi ]M ?
(4)
= Mi qi + Mi qi = (Mi + Mi )qi :
It is apparent that Mi and Mi are related by Mi +
Mi = 1 and MiMi = O, where 1 and O are the
3 3 identity and zero matrices, respectively. The orthogonal complement Mi thus dened is unique, and
hence, both Mi and Mi are projectors that verify the
?
?
?
?
?
?
?
properties below:
Symmetry:
[Mi ]T = Mi ; [Mi ]T = Mi
Idempotency: [Mi ]2 = Mi ; [Mi ]2 = Mi
Rank-complementarity: rank(Mi ) + rank(Mi ) = 3
?
?
?
?
?
ASME Design Automation Conference, Minneapolis, Sept. 1994
The measurement projector Mi , which projects vectors
onto the measurement subspace Mi , and the orthogonal measurement projector Mi , which projects vectors
onto the orthogonal measurement subspace Mi , are
given for the three measurement conditions as
8
8
i) 3 , D
< 1
< O
Mi = : Pi ; Mi = : Li ii) 2 , D ; (5)
Li
Pi iii) 1 , D
?
?
?
where the plane and line projectors, Pi and Li , respectively, are dened as
Pi 1 , Li ; Li eieTi ;
(6)
with ei denoting a unit vector along line Li . Since qi is
known only on the measurement subspace Mi , and not
on Mi , the unknown part of qi , [qi ]M ? , is eliminated
from eq.(2) by projecting each equation onto its own
measurement subspace Mi as follows:
?
projector pose model
}| { measurements
z }| {
zerror
z}|{ z }| { z }| {
[|xi ]M + {z [qi ]M } = Mi (| p + R{zbi , ai }) : (7)
subspace Mi
R3
It is apparent that only the part of the error xi lying
in the measurement subspace Mi , i.e., [xi ]M , is completely known, and hence, can be minimized. Using this
system of linear kinematic equations, the solution of the
DK can be estimated as the pose fp; Rg, that gives the
least-square errors f[xi ]M gn1 in the set of measurement
subspaces fMign1 . Althrough the projectors Mi are
rank-decient for 2-D and 1-D measurements, the pose
can be estimated if the set of subspaces spans R3 .
In general, [qi ]M can be computed from the readouts of a set of sensed joints, if there exit two points of
Mi that are at a known position relative to points Ai
and Bi , respectively. Let us dene fi and gi as vectors
directed from point Ai to a point of Mi and from another point of Mi to point Bi . The two points of Mi
may have dierent locations, but once projected onto
Mi , they collapse into a single point, and hence, the
projection of (fi + gi ) onto Mi gives [qi ]M , i.e.,
[qi ]M = Mi (fi + gi ):
(8)
When Mi is R3 , Mi is empty, and hence, point Bi is
at a known position relative to Ai . Below, we show how
to obtain [qi ]M for dierent leg-kinematic architectures
and sets of sensed joints.
?
?
?
?
?
MEASUREMENT SUBSPACES
Let us consider the class of 6-dof leg-kinematic architectures of simple open kinematic chains starting
with three joints, either revolute (R) or prismatic (P ),
and ending with a spherical (S ) joint. The eight kinematic chains of this class are: RRR, RRP , RPR, PRR,
Page 3
RPP , PRP , PPR and PPP , where the S joint is omitted for brevity. The sensed joints are denoted by R or
P , while the unsensed joints by R or P . Therefore, the
number of possible combinations of kinematic chains
and sets of sensed joints using a minimum of one sensor, is given for this class as: 82 , 8 = 56. In general, the
geometry of each leg i can be described by three sets of
four Hartenberg-Denavit parameters: fj ; j ; j ; j g31 ,
i.e., one for each joint j , where the time-dependent joint
variable is either j or j , for an R or a P joint, respectively. The three remaining parameters are known
constants for a given joint geometry. For example, j
is the distance along the common perpendicular of two
consecutive joint axes, while j is the angle of rotation
between these two axes around the common perpendicular. Moreover, sin j , cos j , sin j and cos j , are
denoted as, sj , cj , j and j , respectively. Furthermore, i, j and k are dened as three unit vectors along
the x-, y- and z -axis, respectively. Finally, [dj ]i and
[Rj ]i , i.e., the position of the origin and the orientation
of frame j relative to frame j , 1, are dened as follows:
2
3
2
3
j cj
cj ,j sj j sj
[dj ]i 4 j sj 5 ; [Rj ]i 4 sj j cj ,j cj 5 :
j i
0
j
j i
(9)
Since [dj ]i and [Rj ]i are completely known for a sensed
joint, while they are only partially known for an unsensed joint, we can decompose them into measured
and unmeasured parts as follows:
[dj ]i = [dj ]i + [dj ]i ; [Rj ]i = [Rj ]i [Rj ]i ;
0
where
2
[dj ]i 4
00
3
0
00
2
3
0
0 5;
j cj
j sj 5 ; [dj ]i 4
0 i
3
2
(10)
(11)
2
3
cj ,sj 0
1 0
0
[Rj ]i 4 sj cj 0 5 ; [Rj ]i 4 0 j ,j 5 :
0
0 1 i
0 j j i
0
00
0
j i
00
The joint information is classied, in Table 1, in known
and unknown entries relative to sensed and unsensed,
revolute and prismatic joints. Among the 56 possible
Joint j
known
Unknown
P
[dj ]i , [Rj ]i
{
R
[dj ]i , [Rj ]i
{
P
[dj ]i , [Rj ]i
[dj ]i
R
[dj ]i , [Rj ]i [dj ]i , [Rj ]i
TABLE 1: KNOWN AND UNKNOWN INFORMATION
0
00
00
00
0
0
combinations of leg-kinematic architectures and sets of
ASME Design Automation Conference, Minneapolis, Sept. 1994
sensed joints of the class under study, there are 38 that
lead to a measurement subspace, and hence, lead to the
aformentioned linear kinematic equation. Moreover, if
we consider legs of special geometries of two consecutive
joints, i.e., parallel axes and perpendicular axes, there
are 12 additional combinations. The other 18 combinations of this class, or 6 if we consider legs of special
geometries, have a quadratic kinematic equation, like
the most used leg-kinematic architecture of the RRP
type. Based on the location of the unsensed information along each leg, as described in Table 1, the 50
combinations that lead to a measurement subspace are
classied, in Table 2, within 16 dierent types. Using
only the known information of Table 1, the equations
for the computation of vectors ei , fi and gi , from the
readouts of the set of sensed joints, are also given in
Table 2.
Under 3-D measurements, i.e., condition i) of the
LDKE Theorem, all joints of leg i are sensed, and
hence, the eight combinations of type a) can be used, for
which an example is shown in Fig. 2. Under 2-D measurements, i.e., condition ii) of the LDKE Theorem,
all joints of leg i are sensed other than a translation
along a line. This unsensed translation along a line can
only be obtained with an unsensed P joint, and hence,
the twelve combinations are classied into types b), c)
and d), based on the location of the unsensed P joint
along the leg, for each of which an example is shown in
Fig. 3. Under 1-D measurements, i.e., condition iii) of
the LDKE Theorem, all joints of leg i are sensed other
than a motion in a plane. The unit vector ei is normal
to the unsensed motion plane, and hence, is orthogonal
to Mi and lies in Mi . Therefore, ei spans the measurement subspace Mi . For legs of general geometries,
the unsensed motion in a plane can be obtained in two
dierent ways. First, using two unsensed P joints, the
six combinations are classied into types e), f ) and g),
based on the location of the two unsensed P joints along
the leg, for each of which an example is shown in Fig. 4.
Second, using a single unsensed R joint, the twelve combinations are classied into types h), i) and j ), based on
the location of the unsensed R joint along the leg, for
each of which an example is shown in Fig. 4. Although
an R joint does not generate the full motion in a plane,
its motion lies completely in a plane, and hence, the 1-D
measurement applies to it as well. For legs of special geometries, the unsensed motion in a plane is obtained in
two dierent ways. First, using two parallel unsensed R
joints, i.e., fRRg , the four combinations are classied
into types k) and l), based on the location of fRRg
along the leg, for each of which an example is shown in
Fig. 5. Second, using two perpendicular PR and RP
joints, i.e., fPRg and fRP g , the eight combinations
are classied into types m), n), o) and p), based on the
location of fPRg and fRP g along the leg, for each
?
k
k
?
?
?
?
Page 4
of which an example is shown in Fig. 5. Since vector
ei spans Mi, vectors Mifi and Migi can be reduced to
simpler expressions, using this vector as a basis.
Table 2 can be used to answer questions like: Where
and how many sensors should I place on an RRP leg in
order to obtain a linear kinematic equation? There are
,
seven possible sets of sensed joints, i.e., R R P , RRP
P , RRP , RRP
, RRP and RRP
. The rst four
RR
combinations are listed in Table 2, as legs of general
geometry, and hence, always lead to a linear kinematic
equation. The following two combinations are listed in
Table 2, as leg of special geometry, i.e., RfRP g and
fRRg P , and hence, lead to linear kinematic equations
only if the special geometry is met. Finally, there is only
one combination that never leads to a linear kinematic
equation, i.e., the last one in the list. It is noteworthy
that the most used set of sensed joints, for this leg architecture, is RRP . It is listed in Table 2, as fRRg P ,
for which the special geometry is not usually met, and
hence, leads to a quadratic kinematic equation.
?
k
k
CONCLUSIONS
Using the concept of measurement subspaces, we derived a linear kinematic model for the DK of general
parallel manipulators. For a class of 6-dof leg architectures of general geometry, the sets of sensed joints that
lead to a measurement subspace are classied into 16
types, for each of which a geometric interpretation and
equations for the computation of the measurement subspace from the sensor readouts are provided. Table 2
can be used in deciding on the sensing layout for parallel manipulators in order to obtain linear kinematic
models.
ACKNOWLEDGEMENTS
The research work reported here was supported
by NSERC (Natural Sciences and Engineering Research Council, of Canada) under Grants OGP0004532,
STR0100971 and EQP00-92729. The rst author
was supported by Ecole Polytechnique de Montreal
and FCAR (Formation de Chercheurs et l'Aide a la
Recherche, of Quebec) under a Graduate Scholarship.
REFERENCES
Zanganeh, K. E. and Angeles, J., 1993, \The Semigraphical Solution of the Direct Kinematics of General
Platform-Type Parallel Manipulators", Computational
Kinematics, Kluwer Academic Publishers, pp. 165-173.
Baron, L. and Angeles, J., 1994, \The Decoupling of
the Direct Kinematics of Parallel Manipulators Using
Redundant Sensors", IEEE Int. Conf. on Robotics and
Automation.
Innocenti, C. and Parenti-Castelli, V., 1991, \A novel
ASME Design Automation Conference, Minneapolis, Sept. 1994
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ASME Design Automation Conference, Minneapolis, Sept. 1994
ei
Mi
gi
R
e2
R
e2
Mi
?
fi
R
Page 7
gi
Mi
ei
Mi
?
R
R
Type k): RfRRg
fi
R
Type l): fRRg R
k
k
e2
R
e2
Mi
Mi
ei
Mi
?
P
ei
R
e1
gi
R
gi
fi
e1
Mi
?
R
Type m): RfPRg
?
P
Type n): fPRg R
?
ASME Design Automation Conference, Minneapolis, Sept. 1994
Page 8
Subspace Type Kinematic chains
ei
fi
gi
3-D
a) R RR , R RP , RP R , P RR ,
|
0
d1 + R1(d2 + R2d3)
RP P , P RP , P P R, P P P
, R PP
, P RP
, P PP
b) R RP
R1R2k d1 + R1d2
R1R2d3
R, RP
P , PP
R, PP
P
2-D
c) RP
R1k
d1
R1(d2 + R2d3)
d) P RR, P RP , P P R, P P P
k
0
d1 + R1(d2 + R2d3)
, PPP
e) RPP
R1R2i
d1
R1(d2 + R2d3)
, P PP
f ) P RP
e1 e3
d1
R1(d2 + R2d3)
1-D
g) PP R, PP P
R1i
d1
R1(d2 + R2d3)
, R PR
, P RR
, P PR
h) R RR
R1R2k d1 + R1d2
R1R2d3
i) RRR, RRP , PRR, PRP
R1k
d1
R1(d2 + R2 d3)
j ) RRR, RRP , RP R, RP P
k
0
d1 + R1 (d2 + R2d3)
k) R fRRg , P fRRg
R1k d1 + R1d2
R1d3
l) fRRg R, fRRg P
k
d1
d2 + R2 d3
1-D
m) R fPRg , P fPRg
R1R2k
d1
R1R2d3
special
n) fPRg R, fPRg P
R1k
0
R1(d2 + R2 d3)
o) RfRP g , P fRP g
R1k d1 + R1d2
R1R2 d3
p) fRP g R, fRP g P
k
d1
R1 (d2 + R2d3 )
TABLE 2: CLASSIFICATION OF THE MEASUREMENT SUBSPACES
0
0
0
0
0
0
0
0
0
00
00
00
k
k
?
00
00
00
?
00
?
?
?
00
00
k
?
00
00
k
00
00
?
?
00
00
00
00
0
0