Proceedings of the 2001 IEEE
International Conference on Robotics & Automation
Seoul, Korea • May 21-26, 2001
On the Stiness Control and Congruence Transformation
Using the Conservative Congruence Transformation (CCT)
Yanmei Li
y
and Imin Kao
Department of Mechanical Engineering
SUNY at Stony Brook, Stony Brook, NY 11794-2300
Abstract
The conservative congruence transformation (CCT),
K Kg = J T Kp J , was proposed by Chen and Kao as
the correct congruence transformation to replace the conventional mapping, K = J T Kp J , proposed by Salisbury in 1980. The conventional mapping was shown to
lead to physically inconsistent results when external force
is present in stiness control. Theoretical proofs are also
provided to show the conservative nature of the CCT, and
the non-conservative property of the conventional mapping.
The CCT is established as the general and valid mapping
of the stiness matrices between the joint and Cartesian
spaces of robotic manipulators. In this paper, the work of
CCT is extended to a redundant planar manipulator. Numerical simulations are presented to illustrate issues related
to the application of generalized inverse in the analysis of
redundant manipulators.
1
Introduction
The new theory of the conservative congruence
transformation (CCT) between the joint space and
the R3x3 Cartesian space for stiness control was presented by Chen and Kao [1, 3, 4]. The CCT relates
the mapping between the stiness matrices of the joint
and Cartesian spaces, and can be expressed by the following equation
J T Kp J = K Kg
(1)
where J is the Jacobian matrix, Kp and K are
the Cartesian and joint stiness matrices, respectively,
and Kg is the eective stiness matrix due to the
change in geometry of grasping and manipulation under the presence of external force. In their paper [1],
the authors showed that the conventional formulation
for the mapping between the Cartesian and joint stiness matrices, given by the following equation [6]
J Kp J = K
Research Assistant; Email: [email protected]
y Associate Professor of Mechanical Engineering;
T
is in general an incorrect mapping that results in physical inconsistency in conservative properties of stiness
control, even though equation (2) has generally been
accepted and widely used. Chen and Kao [1] showed
that equation (2) is only valid when robotic manipulators are at their unloaded equilibrium conguration,
i.e., without external force.
In this paper, we will prove theoretically that the
CCT can preserve the conservative properties of stiness matrices between the joint and Cartesian spaces,
while the conventional formulation can only preserve
the conservative properties under special condition. In
addition, we extend the CCT work in [1, 3, 2] and the
analysis of conservative properties of stiness matrix
in [5] to further illustrate, with mathematical formulation, that the conventional formulation in equation (2)
is a non-conservative mapping. In addition, we provide a theoretical proof to show that the CCT is the
correct and general relationship for congruence mapping with stiness control in robotics. The simulation
of a three-link, redundant planar manipulator is used
to illustrate the CCT as the correct transformation.
2
The conventional mapping of stiness matrices between the joint space and the R33 Cartesian space
is given in equation (2). In this section, we will
show with mathematical formulation that the conventional mapping proposed by Salisbury is a nonconservative mapping. There are two typical stiness
control schemes by which we can show this conclusion:
(i) the Cartesian-based stiness control, and (ii) the
joint-based stiness control.
(i) the Cartesian-based stiness control: solv-
ing for the joint stiness matrix K with specied
Cartesian stiness matrix Kp , via equation (2)
For the Cartesian-based strategy, K is shown to
be generally non-conservative, even with a conservative Kp .
(2)
Email:
(ii) the joint-based stiness control: solving for
the Cartesian stiness matrix Kp with specied
[email protected]
0-7803-6475-9/01/$10.00 © 2001 IEEE
The Conventional Formulation
3937
joint stiness matrix K , via an inverted relationship given by equation (2)
For the scheme of joint-based stiness control, we
will show in the following that Kp is generally
non-conservative even when K is conservative.
In other words, the conventional formulation can
not preserve the conservative properties between the
joint stiness matrix K and the Cartesian stiness
matrix Kp , even though the symmetry of stiness matrix is preserved under equation (2). We will rst focus
on the joint-based control scheme in the following.
Jij 1
@i
. Substituting these terms into equation
= @x
j
(8), we have
@K
@K
=
l=1 m=1
n
=
@Kik
@xj
1 i; j 3
(3)
1 i; j; k 3
(4)
where xi denotes the coordinates. Let Jij 1 denotes
the (i; j )th element of J 1 , K;ij denotes the (i; j )th
element of K , and Jij denotes the (i; j )th element of
J . Then the Cartesian stiness matrix Kp based on
Salisbury's congruence formulation can be written as
Kp = J
T
K J
1
Kp;ij
=
n X
n
X
l=1 m=1
Jmi 1 K;ml Jlj 1
Kp;ik =
l=1 m=1
Jmi
1
K;ml Jlk
1
(7)
p;ik
6= @K
; 1 i; j; k 3
@x
j
(8)
Following the denition of J , the (i; j )th element of
th element of J 1 is
i
J is Jij = @x
@j , and the (i; j )
@
m
K;ml l
@xi
@xj
=
@ 2 m
@l
K
@xk @xi ;ml @xj
@
@xj
=
{z
|
+
@l
m
K;ml
@xi
@xk
@
@
@xj
@m @K;ml @l
@xi
@xk @xj
} |
@
term1
@
{z
@l
m
K;ml
@xi
@xk
+
@l
m
K
@xj @xi ;ml @xk
|
{z
term1
+
{z
}
term3
(10)
@m @K;ml @l
@xi
@xj @xk
} |
0
o
(9)
m
l
K
@xi ;ml @xk
@2 @m
@ 2 l
K
@xi ;ml @xk @xj
} |
term2
{z
+
@2 @m
l
K
@xi ;ml @xj @xk
} |
0
term2
{z
}
0
term3
It is obvious that the third term (term3) in equation
(9) is equal to (term3') in equation (10) due to commutative property.
Next, we show that the second terms (term2) and
(term2') in equations (9) and (10) are equal to each
other. First, the term2 in equation (9) is
!
n
@m X @K;ml @p
@l
@m @K;ml @l
@xi
@xk
=
@xj
=
@xi
@p
p=1
@xl
@xj
n
X
@m @K;ml @p
@xi
p=1
@l
@xk @xj
@p
(11)
Similarly, term20 in equation (10) can be written as
!
n
@m @K;ml @l
@m X @K;ml @p
@l
@xi
@xj
=
@xk
=
@xi
@p
p=1
@xj
@xi
@p
@l
@xj @xk
From this we have
n X
n
n X
n X
n X
X
@m @K;ml
0
(term2 ) =
l=1 m=1
@xk
n
X
@m @K;ml @p
p=1
In order to prove that the conventional formulation is
non-conservative, we need to show that
@Kp;ij
@xk
l=1 m=1
@
@
@
@xj
m
l
K
@xi ;ml @xj
@xk
(6)
The (i; k )th element of Kp is
n X
n
X
@
@xk
n X
n
X
p;ik
@xj
Next, we calculate the two terms in the braces:
@
@
@
(5)
If Kp is non-conservative, it will violate either one of
the two conservative criteria in equations (3) and (4).
Since equation (2) preserves symmetry, Kp will also
be symmetric as long as K is symmetric.
What is left is to examine the exactness property
of Kp . The (i; j )th element of Kp is
@
m
K;ml l
@xi
@xj
l=1 m=1
According to the two criteria for a conservative matrix in [5], a stiness matrix K is conservative if it
satises the following two criteria:
= Kji
@
@
@xk
n n
XX
=
2.1 Joint-based stiness control using the
conventional formulation
Kij
@Kij
@xk
n X
n
X
p;ij
@xk
l=1 m=1 p=1
@xi
@p
@l @p
@xk @xj
(12)
(13)
When K is conservative, we require that
@K;mp
@l
So we have
n X
n
X
(term2)
l=1 m=1
3938
=
=
@K;ml
@p
n X
n X
n X
@m @K;ml @p
l=1 m=1 p=1
@xi
@p
@l
@xk @xj
=
n X
n X
n X
@m @K;mp @p
@xi
l=1 m=1 p=1
=
n X
n X
n X
@m @K;ml
@xi
p=1 m=1 l=1
=
n X
n
X
@l
@xk @xj
@l
@p
@l @p
@xk @xj
0
(term2 )
= K d
Kg d = (dJ T )f
d
(14)
l=1 m=1
In the derivation of equation (14), exchange of the
dummy indices is used. It is clear from equations (13)
and (14) that the sums of term2 and term20 are the
same. Next, let us examine the rst terms. From the
denition and properties of the Jacobian matrix, we
nd that in general
2
2 @ m
@l
@l
6= @ m
(15)
@xk @xi
@xj
@xj @xi
known relationships dened in the joint space as follows,
Equation (19) was derived in [1]. Our task is to show
that the formulation of CCT, J T Kp J = (K Kg ),
always yields the consistent Kp in the Cartesian space
as dened by Kp = ddxf . Since = J T f , we can
dierentiate this equation to render
d
6=
(16)
Hence, the rst term in equation (9) is not equal to
the rst term in equation (10), in general. Thus,
n X
n
n X
n
X
X
@l
@ 2 m
@l
@ 2 m
l=1 m=1
@xk @xi
K;ml
@xj
6=
l=1 m=1
@xj @xi
K;ml
@xk
2.2 Discussions
Based on the above derivation, we can identify special cases for which Kp is conservative. One such special case is when the Jacobian matrix J satises the
equality in equation (17), i:e:, the Jacobian matrix is
conguration-independent and constant, as described
in [1].
We can show similar results using the Cartesianbased control scheme. That is, when Kp is conservative, we can prove that K is in general nonconservative unless a special condition similar to that
in equation (17) is met.
3
=
The Conservative Property of CCT
The theory of the conservative congruence transformation (CCT) between joint space and the linear R33
Cartesian space is given by equation (1). Our objective is to show that equation (1) always preserves the
conservative property of stiness control, and is consistent with the denition of stiness, namely, K = ddxf ,
where f represents the bias force (external force) acting on the end-eector and dx denotes the displacement of the end-eector. Let us proceed with the
df
df dx
d = J T
d
d
dx d
df
J T J d
dx
(21)
Substituting equation (18), (19) and (21) into equation
(20), we have
K d = Kg d + J T df
df
= Kg d + J T J d
dx
(17)
Therefore, Kp is non-conservative, even with conservative K , using the conventional mapping.
(20)
J T df = J T
@xk
@ 2 m
@l
K;ml
@xj @xi
@xk
= (dJ T )f + J T (df )
The second term in equation (20) can be rewritten in
alternate mathematical expression as
This results in
@ 2 m
@l
K;ml
@xk @xi
@xj
(18)
(19)
(22)
Thus, we can write
K Kg = J T df J
dx
(23)
Comparing equation (23) with the denition of CCT
in equation (1), it is clear that
Kp =
df
dx
(24)
Therefore, with the combination of both K and Kg ,
the Cartesian stiness matrix is shown to be the standard denition. Thus, the Kp obtained from the CCT
is consistent with the fundamental property of a conservative system, and that the CCT establishes equivalent stiness matrices in both the joint space and
Cartesian space.
4
Simulation of a Three-link Planar
Manipulator
In this section, we will consider a three-link redundant planar manipulator to illustrate the conservative
property of the stiness control, using the Cartesianbased stiness control strategy via the CCT in equation (1). The incorrect results of the conventional formulation in equation (2) will also be computed and
compared with the results of the CCT equation.
3939
Parameters:
L1= 0.29 m
L2 = 0.23 m
L3 = 0.05 m
L3
trajectory
2.5
fx
fy
2
P(x,y)
L2
Initial states:
o
1= 21.8
Y
2
L1
3=
60.9o
f 0 = [20
x0 = [0.45
X
O
= 39.1o
25]T N
0.33]T m
Work done over circular path (Joules)
f=
Figure 1: A three-link redundant planar manipulator
for simulation.
For a two-link planar manipulator, the orientation
of the end-eector is dened by the orientation of the
second link, and can not be controlled; whereas, we
can control the orientation of the end-eector of a
three-link planar manipulator through the extra degree of freedom of the redundant joint. Thus, this
three-link manipulator is called a \redundant manipulator" if we are concerned with only the linear fx; y g
coordinates with three joint parameters, f1 ; 2 ; 3 g.
The geometry of the redundant manipulator is
shown in Figure 1, where three serial links are congured with the reference point at the end-eector at
P (x; y ), and the robot base frame is at O-XY . We denote the position of the end-eector as x = [x; y ]T in
the Cartesian space and = [1 ; 2 ; ; 3 ]T are the joint
parameters. From Figure 1, the Cartesian position of
the end-eector can be written as
= L1 c1 + L2 c12 + L3 c123
y = L1 s1 + L2 s12 + L3 s123
x
J22
1
0.5
0
work in Cartesian space
-0.5
J23
where J11 = (L1 s1 + L2 s12 + L3 s123 ), J12 =
(L2 s12 + L3 s123 ), J13 = L3 s123 , J21 = L1 c1 +
L2 c12 + L3 c123 , J22 = L2 c12 + L3 c123 , and J23 =
L3 c123 .
4.1 Cartesian-based stiness control
The initial orientation of the third link is maintained in horizontal orientation, as shown in Figure 1.
After that, the orientation of the third link is specied
to align with the radial direction of the trajectory, as
shown in the shaded third link in Figure 1, so as to
align the end-eector with the orientation of the grasp
object for grasping tasks and stable prehension. This
work in joint space
-1
-1.5
-2
0
100
200
300
360
degrees
Figure 2: Results of simulation, using the CCT, for
the Cartesian-based stiness control.
gives rise to the advantage of the three-link redundant
manipulator over the two-link manipulator.
Since we are considering the Cartesian-based stiness control scheme, the stiness matrix, Kp , and the
bias force, f , in Cartesian space should be specied
rst. In this simulation, the prescribed conservative
Cartesian stiness matrix and the bias force are
300 0
Kp = 0 250 N=m
f = [20 25]T N
The matrix Kp satises the two conservative criteria in
equations (3) and (4). The initial joint torque based on
f is calculated to be = [ 17:25 8:66 1:25]T N m. The stiness matrix Kg dened in equation (1) is
2
3
Kg = 4
(25)
(26)
where c1 = cos 1 , s1 = sin 1 , c12 = cos (1 + 2 ), etc.
Thus, the Jacobian matrix becomes
J11 J12 J13
J =
(27)
J21
1.5
Kg;11
Kg;21
Kg;31
Kg;12
Kg;22
Kg;32
Kg;13
Kg;23
Kg;33
5
(28)
where
Kg;11 = (L1 c1 + L2 c12 + L3 c123 )fx
(L1 s1 +
L2 s12 + L3 s123 )fy , Kg;12 =
(L2 c12 + L3 c123 )fx
(L2 s12 + L3 s123 )fy , Kg;13 = L3 c123 fx
L3 s123 fy ,
Kg;21 =
(L2 c12 + L3 c123 )fx
(L2 s12 + L3 s123 )fy ,
Kg;22 =
(L2 c12 + L3 c123 )fx
(L2 s12 + L3 s123 )fy ,
Kg;23 = (L2 c12 + L3 c123 )fx (L2 s12 + L3 s123 )fy , Kg;31 =
L3 c123 fx L3 s123 fy , Kg;32 = L3 c123 fx L3 s123 fy , and
Kg;33 = L3 c123 fx L3 s123 fy .
The trajectory of the end-eector is a circle with
a radius of R = 0:08m, centered at (x; y ) =
(0:37; 0:33)m. The point P of the end-eector moves
along the circular trajectory in counterclockwise sense,
starting from the initial position at (x0 ; y0 ) =
(0:45; 0:33)m and nally returning to the same initial
position. The force and net work done in the Cartesian space are calculated using the following equations
for numerical integration
fi = fi 1 + Kp dx
Wc;i = Wc;i 1 + f T dx
3940
(29)
(30)
Work done over circular path (Joules)
14
120
12
100
Work done over circular path (Joules)
10
work in joint space
8
6
4
80
60
40
20
2
0
0
-2
work in Cartesian space
−20
0
100
200
300
360
degrees
0
50
100
150
200
degree
250
300
350
400
Figure 3: Results of simulation, using the conventional
mapping, for the Cartesian-based stiness control.
Figure 4: Results of simulation, using the CCT, for
the joint-based stiness control.
After Kg matrix from equation (28) and K matrix
from equation (1) are computed, the torque and work
done in joint space can be calculated using the following equations of numerical integration
in Cartesian space and in joint space are not identical. The discrepancy reveals the inconsistency
of the conventional mapping.
i
Wc;i
= i 1 + K d
= Wc;i 1 + T d
(31)
(32)
In equations (29) to (32), the second-order and higher
terms are neglected since they have little contribution
in numerical integration. Furthermore, the integrations of both the net work and force over the circular
path should be zero if the system is conservative. The
results of numerical simulation are shown in Figures 2
and 3.
4.2 Discussions
From Figures 2 and 3, the following observations
are in order:
When the CCT is employed, the net works done
in both the Cartesian and joint spaces of this redundant manipulator are identical. Since we are
considering a conservative system (Kp is conservative), so we expect that the work calculated in
both Cartesian space and joint space should be
the same. Form Figure 2, we can see that the
net work done in both Cartesian space and joint
space are equal to zero when returned to the starting point after a full circle. This indicates that
energy is conservative when CCT is used.
When the incorrect conventional formulation is
used, the net work done in the joint space in Figure 3 is not zero, although the net work done in
the Cartesian space remains zero. That indicates
that energy is not conservative using the conventional formulation. Furthermore, the work done
The two work proles of the conservative stiness
control should be identical to each other, regardless of which coordinate system the work is integrated with respect to. Therefore, only the CCT
should be used.
The work prole depends on the initial bias force
and the Kp matrix. However, the conservative
characteristics of the system does not depend on
the initial bias force, nor the trajectory as long as
it is a continuous closed path.
It is important to note that equation (1) used in
this simulation does not involve the generalized
inverse of J { an important subject to be discussed in Section 4.3.
4.3 Joint-based stiness control
The geometry of this simulation is similar to that in
Section 4.1. The initial orientation of the third link is
also maintained in horizontal orientation. After that,
the orientation of the third link is specied to align
with the radial direction of the trajectory for the upper
half circle (0 ! ). When the end-eector moves to
the lower half of the circle, the orientation of the third
link changes from ! 0. With the joint-based control
scheme, the joint stiness matrix K and the joint
torque are specied rst:
" 300 0 0 #
K =
0 250 0 N m = [10 5 5]T N m
0
0
5
The matrix K is conservative. The joint stiness
of link three is kept low (K;33 = 5) due to its larger
3941
matrix is currently being explored.
200
5
Work done over circular path (Joules)
150
100
50
0
−50
0
50
100
150
200
degree
250
300
350
400
Figure 5: Results of simulation, using the conventional
mapping, for the joint-based stiness control.
movement than the other two links. The initial bias
force is calculated to be f = [45:1 53:1]T N . The Kg
matrix is calculated using equation (28). The center
of the trajectory circle is at (x; y ) = (0:30; 0:30)m,
with a radius of 0:15m. The initial position of the
end-eector is at (x0 ; y0 ) = (0:45; 0:30)m.
The algorithm to calculate the work is the same as
that in the previous sections. The simulation results
are plotted in Figure 4 (using the CCT) and Figure 5
(using conventional formulation).
Unlike the Cartesian-based control scheme, the
joint-based control strategy in such redundant manipulator involves the Penrose-Moore generalized inverse
of the non-square Jacobian matrix in order to obtain
the Cartesian stiness matrix, i.e.,
Kp = (JT ) (K Kg )(J )
where `*' denotes the generalized inverse. Consequently, the inverse CCT equation will utilize the generalized inverse that masks part of the Kp components
that belongs to the null space of the generalized inverse
of the Jacobian matrix.
Due to the generalized inverse of the non-square
Jacobian matrix, part of the information for stiness
control is lost; thus, from Figure 4 (CCT is used) we
can see that there is a discrepancy between the work
done in the Cartesian space and joint space. But the
tendency of these two curves are similar, and the net
work done in the joint space is zero and is almost zero
in the Cartesian space. In Figure 5 (conventional mapping), it is obvious that even though the work done in
joint space returns to zero, as expected, there is a large
dierence between these two curves and the net work
done in Cartesian space is far from zero.
Simulation and derivation of a methodology to recover the lost information for stiness control due to
the application of generalized inverse of the Jacobian
Conclusions
In this paper, we prove theoretically that the conventional formulation of stiness mapping between the
Cartesian and joint spaces is generally incorrect, only
valid under special conditions. The conservative nature of the CCT is proven theoretically, and illustrated
with simulation results of a redundant planar manipulator. The results suggest that the stiness matrix
obtained through the CCT is consistent with the fundamental property of a conservative system. Results
of numerical simulation for the work done over a circular path using both equations (1) and (2) are presented
and compared.
The detrimental eect in the application of the generalized inverse of the Jacobian matrix for redundant
manipulator is reported. Future work is needed to resolve the issue of using the generalized inverse, in order
to deal with the inevitable loss of part of stiness control information due to the masking of the null space
of generalized inverse of the Jacobian.
6
Acknowledgment
This research has been supported by the National
Science Foundation under grant number IIS9906890.
References
[1] S.-F. Chen and I. Kao. Conservative congruence transformation for joint and cartesian stiness matrices of
robotic hands and ngers. the Int'l J. of Robotics Research, 19(9):835{847, September 2000.
[2] S.-F. Chen and I. Kao. Geometrical methods for modeling the asymmetric 66 cartesian stiness matrix. In
Proc. of IROS'00, pp. 1217-1222, Japan, Nov. 2000.
[3] S.-F. Chen and I. Kao. Simulation of conservative congruence transformation: Conservative properties in the
joint and cartesian spaces. In IEEE Int'l Conf. on
Robotics and Automation, pp 1283{1288, April 2000.
[4] Shih-Feng Chen. Fundamental Properties Of Stiness
Control In Grasping And Dextrous Manipulation In
Robotics. PhD thesis, State University of New York
at Stony Brook, May 2000.
[5] I. Kao and C. Ngo. Properties of grasp stiness matrix and conservative control strategy. the Int'l J. of
Robotics Research, 18(2):159{167, Febrary 1999.
[6] J. K. Salisbury. Active stiness control of a manipulator in cartesian coordinates. In Proc. 19th IEEE Conf.
on Decision and Control, pp 87{97, NM, Dec. 1980.
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