Shih-Feng Chen
Imin Kao
Manufacturing and Automation Laboratory
Department of Mechanical Engineering
State University of New York at Stony Brook
Stony Brook, New York 11794-2300, USA
[email protected]
Abstract
Conservative
Congruence
Transformation for
Joint and Cartesian
Stiffness Matrices
of Robotic Hands
and Fingers
1. Introduction
In this paper, we develop the theoretical work on the properties and
mapping of stiffness matrices between joint and Cartesian spaces of
robotic hands and fingers, and propose the conservative congruence
transformation (CCT). In this paper, we show that the conventional
formulation between the joint and Cartesian spaces, Kθ = JθT Kp Jθ ,
first derived by Salisbury in 1980, is only valid at the unloaded equilibrium configuration. Once the grasping configuration is deviated
from its unloaded configuration (for example, by the application
of an external force), the conservative congruence transformation
should be used. Theoretical development and numerical simulation are presented. The conservative congruence transformation
accounts for the change in geometry via the differential Jacobian
(Hessian matrix) of the robot manipulators when an external force
is applied. The effect is captured in an effective stiffness matrix,
Kg , of the conservative congruence transformation. The results of
this paper also indicate that the omission of the changes in Jacobian in the presence of external force would result in discrepancy of
the work and lead to contradiction to the fundamental conservative
properties of stiffness matrices. Through conservative congruence
transformation, conservative and consistent physical properties of
stiffness matrices can be preserved during mapping regardless of the
usage of coordinate frames and the existence of external force.
KEY WORDS—stiffness control, conservative congruence
transformation, stiffness mapping, differential Jacobian
The International Journal of Robotics Research
Vol. 19, No. 9, September 2000, pp. 835-847,
©2000 Sage Publications, Inc.
The conventional formulation for the mapping of stiffness
matrices between the Cartesian and joint spaces,
Kθ = JθT Kp Jθ ,
(1)
was first derived by Salisbury (1980) and generally has been
accepted and applied. However, we will show in this paper
that eq. (1) is only valid when robotic manipulators are at
their unloaded equilibrium configuration, i.e., without external force. In this paper, we present the theoretical derivations
of the general relationship of congruence transformation to
replace eq. (1), as well as results of numerical simulation.
This paper builds on the previous results of properties of
grasp stiffness matrix via congruence transformation to discuss the fundamental properties of a grasp stiffness matrix
as applied in the analysis of grasping and dextrous manipulation in linear R3×3 spaces (Salisbury 1980; Cutkosky and
Kao 1989; Pigoski, Griffis, and Duffy 1992; Griffis and Duffy
1993; Ciblak and Lipkin 1994; Howard, Žefran, and Kumar
1998; Kao, Cutkosky, and Johansson 1997; Chen and Kao
1998; Kao and Ngo 1999). From the analysis of conservative properties (Kao and Ngo 1999) and the simulation results
of congruence transformation of stiffness matrices (Chen and
Kao 1998), the conventional formulation in eq. (1), defining
the relationship of stiffness matrices via the Jacobian matrix
between the joint and Cartesian spaces, is found to be a nonconservative mapping. This is proven by the work discrepancy of stiffness control due to the configuration-dependent
Jacobian matrix (Kao and Ngo 1999). That is, there will be
discrepancy between the work done in the joint and Cartesian spaces using the stiffness control in eq. (1), except in the
following special cases: (1) when the Jacobian and stiffness
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THE INTERNATIONAL JOURNAL OF ROBOTICS RESEARCH / September 2000
matrix are uncoupled such as that of a gantry (or Cartesian)
robot manipulator, (2) when the manipulator is maintained at
its unloaded position at all times (i.e., no external load), and
(3) special Kθ or Kp (see Kao and Ngo 1999).
In this paper, we will present the conservative congruence
transformation (CCT) as the correct and general relationship
for congruence mapping with stiffness control in robotics.
Theoretical derivation and numerical results are used to illustrate the CCT as the complete transformation. We will
also show that the conventional formulation in equation (1)
is not valid if external force is applied. The CCT accounts
for changes in grasping geometry as well as changes in forces
and torques due to stiffness matrices. The change in geometry under the presence of external load is captured by the
matrix, Kg , in the CCT that includes the differential Jacobian
(a Hessian–3D tensor) and external load. It will be shown
that the differential Jacobian plays an important role in determining the conservative property of stiffness control. In
addition, we will discuss the properties of stiffness matrices
when mapping between coordinate basis.
In 1980, Salisbury first derived a congruence transformation to describe the mapping between the Cartesian and the
joint stiffness matrices in robot manipulation. Žefran and Kumar (1997) presented an asymmetric connection and used the
congruence transformation to define an affine connection in
a symmetric Cartesian stiffness matrix. Kao and Ngo (1999)
defined two conservative criteria of a linear stiffness matrix,
which does not involve rotational components, to investigate
the conservative properties of the congruence transformation
and the corresponding control strategies for conservative control using the conventional formulation. Furthermore, twodimensional planar robot simulation results of nonconservative properties via congruence transformation were provided
by Chen and Kao (1998, 1999).
3. Theoretical Background
In this section, we will briefly describe the criteria for conservative stiffness matrix and the conventional formulation as
presented by Salisbury (1980). After that, the conservative
congruence transformation will be derived and discussed.
2. Literature Survey
Robot compliance and stiffness have been studied by many
investigators. Dimentberg (1965) employed screw theory, introduced by Ball (1900), to describe the motion of a rigid
body in a general spatial potential field to derive a symmetric stiffness matrix at an unloaded equilibrium configuration.
Lončarić (1985) applied Lie algebra to investigate the properties of symmetric stiffness matrices.
Recently, Pigoski, Griffis, and Duffy (1992) studied a 6×6
asymmetric stiffness matrix with three elastic springs coupling between two planar rigid bodies. In addition, Griffis
and Duffy (1993) derived a 6 × 6 asymmetric Cartesian stiffness matrix in the study of the Stewart platform. Howard,
Žefran, and Kumar (1998) used the differential geometry and
properties of Lie groups to show that the 6 × 6 stiffness matrix is asymmetric in any conservative system subjected to a
nonzero external load.
In Mussa-Ivaldi, Hogan, and Bizzi (1985); Hogan (1990);
and Kao, Cutkosky, and Johansson (1997), a 2D grasp stiffness matrix was partitioned into two components pertaining to
the conservative gradient and nonconservative curl functions.
The stiffness matrices for human grasping tasks were obtained
via experiments and decomposed into symmetric (conservative) and skew symmetric (nonconservative) components in
R2×2 (Kao, Cutkosky, and Johansson 1997). Ciblak and
Lipkin (1994) developed a formulation to represent the skew
symmetric part of the 6 × 6 Cartesian stiffness matrix for conservative systems and showed that the skew-symmetric part
of a 6 × 6 stiffness matrix is related to the externally applied
force/moment. Huang and Schimmels (1998) studied the spatial compliance behavior that can be achieved through the use
of simple springs connected in parallel to a single rigid body.
3.1. Conservative Criteria of Stiffness Matrix
For a stiffness matrix to be conservative, it must satisfy the
following two conditions: (i) the force due to stiffness is integrable and conservative, and (ii) the work done by such force
is conservative. For a linear stiffness matrix, K ∈ R3×3 , these
two criteria can be used to deduce the following two conditions for conservative stiffness matrix (Kao and Ngo 1999).
THEOREM 1. A robotic system with stiffness matrix is considered conservative if and only if
1. symmetry: The stiffness matrix K is symmetric: kij =
kj i , and
2. exactness: The stiffness matrix K satisfies the exact dif∂k
ik
ferential condition: ∂xijk = ∂k
∂xj , where kij is the (i, j )-th
element of the stiffness matrix and 1 ≤ i, j, k ≤ 3.
An alternative proof of the above theorem using the tensor
analysis is presented in Section A of the appendix. Criterion
1 is a well-known fact about stiffness matrix. Criterion 2
is a result of (curl K) = K× ∇ = 0; that is, if the external
force f = K · dx is considered a point function, the exact
∂k
ik
differential condition ∂xijk = ∂k
∂xj should be satisfied, where
∂fi
f is a function of independent variables and kij = ∂x
. In
j
summary, Theorem 1 can also be stated as (1) (curl f)=0 and
(2) (curl K)=0.
This conservative property implies that the work done by a
conservative force f depends only on the end states, not on the
path. This is the basis of numerical integration in Section 4.2
to determine if a linear Cartesian stiffness is conservative.
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Chen and Kao / Conservative Congruence Transformation
Both conservative criteria can also be applied to any stiffness
matrix with generalized homogeneous coordinates.
3.2. Robot Stiffness and Conventional Congruence Mapping
In robot grasping and manipulation, we can think of the
grasped stiffness matrix, Kp , as a measure of interaction between the robot fingertip and object when external force is
applied to the end-effector. By using the Taylor’s
expan
∂f
1 ∂2f
sion, we can write df = ∂x dx + 2 ∂x2 dx dx + · · · =
∂K
Kp dx + 21 ∂xp dx dx + · · · , where df and dx are the force
and displacement vectors in the Cartesian space relative to
the robot’s base frame. If the higher-order small terms are
neglected, the above equation is reduced to the familiar definition of stiffness matrix
df = Kp dx
or
Kp =
df
.
dx
(2)
Note eq. (2) represents the stiffness as an infinitesimal relationship.1 In addition, the manipulation Jacobian matrix,
Jθ , relates the differential joint displacements to infinitesimal movement of the end-effector, i.e., dx = Jθ dθ, where
dθ = [dθ1 , . . . , dθn ]T is the displacement of the joints.
When we consider the forward kinematics of a serial robot
manipulating an object under stiffness control, the grasped
stiffness matrix associated with each manipulator can be derived from the structural compliance and the joint stiffness
matrix Kθ (Cutkosky and Kao 1989). If the stiffness control
is employed and the manipulator is always maintained at its
unloaded configuration, the relationship between the joint and
Cartesian stiffness matrices, defined in eq. (1), can be derived
by the principle of virtual work (Salisbury 1980; Mason and
Salisbury 1985).
3.3. Derivation of the Conservative Congruence
Transformation
From the principle of virtual work and the definition of the
Jacobian, we obtain the following relationship between the
joint torques and the force applied on the end-effector
τ=
JθT
f.
(3)
Eq. (4) includes two separate terms. The first term is related
to the changes in geometry; the second term is related to
differential external force. Using the definition of stiffness in
eq. (2), eq. (4) can also be written as
∂JθT
T
Jθ Kp dx = Kθ dθ −
dθ f,
(5)
∂θ
dτ = (dJθT ) f + JθT (df).
(4)
1. In some literature, the definition of df = −Kp dx is used, indicating that
a restoring force is generated with the application of displacement. In this
paper, we use the definition of df = Kp dx, where the increment of force
df is seen as an addition to the external force f. Either definition will yield
consistent overall results although certain terms in respective equations will
differ by a sign.
dτ
df
where the
∂JθT
∂θ
and
∂JθT
∂θ
dθ are third-order and second-order
tensors (or matrices), respectively,
and dx =
Jθ dθ . Thus, we
can simplify the first-order tensor,
∂JθT
∂θ
dθ f, by the indicial
notation (see Section B of the appendix for details) as follows:
n ∂JT
∂JθT
θ
f
dθi
dθ f =
∂θ
∂θi
i=1
n
∂JθT
f dθi
=
∂θi
i=1
∂JθT
∂JθT
∂JθT
= (
f) (
f) · · · (
f) dθ. (6)
∂θ1
∂θ2
∂θn
Eq. (5) can be rearranged to become
∂JθT
∂JθT
∂JθT
T
Jθ Kp Jθ = Kθ − (
f) (
f) · · · (
f)
∂θ1
∂θ2
∂θn
(7)
or
JθT Kp Jθ = Kθ − Kg ,
(8)
∂JT
where ( ∂θθi f) is an n × 1 column vector, with i = 1 . . . n, and
n is the number of joints. The n × n matrix Kg , defining the
changes in geometry via the differential Jacobian, is
∂JθT
∂JθT
∂JθT
∂JθT
Kg = (
f) (
f) · · · (
f) =
f , (9)
∂θ1
∂θ2
∂θn
∂θn
n×n
∂JθT
∂θn
f denotes an n × n matrix with the ith column
∂JθT
element being ∂θi f and i = 1 · · · n (see Section B of the
where
It is an important force/torque relation that enables us to determine the resulting torques, τ , reflected at the joints as a
result of the force applied at end-effector f, and vice versa.
By the chain rule, we can differentiate eq. (3),
837
appendix). We call equation (8) the conservative congruence
transformation (CCT). Therefore, the inverse of the CCT is
Kp = Jθ−T (Kθ − Kg )Jθ−1 ,
(10)
where the inverse is taken as the Penrose-Moore generalized
inverse. Note that the conventional formulation in equation
(1) did not consider the change of geometry resulting from
the differential Jacobian and the applied load. Hence, it is
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THE INTERNATIONAL JOURNAL OF ROBOTICS RESEARCH / September 2000
true only at unloaded equilibrium configurations when f =
0. Since robot manipulators normally carry external load,
equation (1) is not valid in general.
From the CCT defined in eq. (8), we conclude the following
corollaries. Proofs of Corollaries 3 and 4 are in Section C of
the appendix.
COROLLARY 1. When there is no externally applied force,
i.e., f = 0, the conservative congruence transformation of
eq. (8) becomes identical to the conventional formulation in
eq. (1). Namely, eq. (1) is a special case of the conservative
congruence transformation, and it is only valid at unloaded
configuration of a robotic manipulator.
COROLLARY 2. When the robot manipulator has a
configuration-independent Jacobian, e.g., constant Jacobian
matrix of a Cartesian gantry robot, the changes in Jacobian
∂JθT
∂θi
are equal to zero, i.e.,
= 0. The conservative congruence
transformation can be reduced to the conventional formulation. This is also a special case.
COROLLARY 3. The conservative congruence transformation results in a symmetric mapping between the joint and
Cartesian spaces. That is, the symmetry is preserved in the
conservative congruence transformation and its inverse.
COROLLARY 4. The positive
definiteness of the stiffness matrices, Kp and Kθ − Kg , is preserved in both the conservative congruence transformation and the inverse congruence
transformation.
3.4. Properties of Kg Matrix
Next, we need to prove that the Kg matrix of the conservative
congruence transformation defined in eq. (9) is also conservative based on the two criteria in Theorem 1. First of all,
from eq. (9), we have
Kg − KgT =
T
∂JθT
∂JθT
f −
f .
∂θn
∂θn
(11)
Let us denote the (i, k)th element of Jθ as Jik . Thus, the
(i, j )th element of (Kg − KgT ) is
m ∂Jki
k=1
∂Jkj
fk − f k
∂θj
∂θi
m 2
∂ xk
∂ 2 xk
=
fk = 0,
−
∂θj ∂θi
∂θi ∂θj
k=1
(12)
where
m is the number of degrees of freedom of f, m
k=1
∂Jki
∂θj fk represents the ith element of the j th column of Kg ,
∂Jkj
and m
is the (i, j )th element of KgT . Therefore,
k=1 fk ∂θi
we conclude from eq. (12) that Kg is symmetric.
Second, we can examine the exactness of the Kg matrix by
∂
∂θl
=
m
∂Jki
k=1
∂θj
m k=1
fk
∂
−
∂θj
m
∂Jki
k=1
∂θl
∂ 3 xk
∂ 3 xk
−
∂θl ∂θj ∂θi
∂θj ∂θl ∂θi
fk
(13)
fk = 0.
From eqs. (11) to (13), we conclude that the Kg matrix satisfies both symmetric and exact differential criteria; hence,
Kg is conservative. Note that in the above proof, the generalized coordinates of the joint space, θ, satisfy the commutative
property since they are independent parameters representing
the displacements of each joint in the joint space.
3.5. Positive Definite Properties between Kθ
and Kp in CCT
We have proven that the positive definite properties are preserved between Kp and (Kθ − Kg ) in Section C of the appendix. While the effective stiffness matrix Kg is a function
of the geometry and external force, a positive definite joint
stiffness matrix, Kθ > 0, can be chosen or prescribed in stiffness control. The issue of concern here is the positive definite
properties between the Kθ and Kp matrices.
Since the Kg matrix depends on changes in geometry and
external force, it can be positive definite or nonpositive definite. Based on eq. (10), we can rewrite eq. (38) as
yT Kp y = zT Kθ z − zT Kg z,
(14)
where z = Jθ−1 y. Note that zT Kg z can assume positive or
negative values, although we can prescribe positive definite
joint control such that zT Kθ z > 0. Thus, it is obvious that
the positive definiteness between Kθ and Kp is not necessarily
preserved from eq. (14). We will discuss the following cases.
(i) If (zT Kθ z − zT Kg z) > 0, then yT Kp y > 0. Thus,
Kp is positive definite. Note that the magnitude and
sign of zT Kθ z − zT Kg z may depend on the choice of
z. For Kp to be positive definite, the above term has to
be positive for any arbitrary nontrivial vector z.
(ii) If (zT Kθ z − zT Kg z) ≥ 0, then Kp is positive
semidefinite.
(iii) If the external force and the geometry of the manipulator are arranged (for example, large external force
applied in unfavorable direction) such that they render
a Kg matrix that results in (zT Kθ z − zT Kg z) < 0,
then Kp is nonpositive definite. That is, a positive definite joint stiffness control can result in a nonpositive
definite Cartesian stiffness matrix due to external force
and geometry that is manifested in the effective stiffness matrix Kg . This result suggests profound implication the external force and geometry could have on
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Chen and Kao / Conservative Congruence Transformation
839
4.1. Analysis of Conservative Property of the CCT
In this section, we will verify the conservative properties of the
conservative congruence transformation using the joint-based
control scheme, i.e., obtaining Kp of the Cartesian space with
known Kg and Kθ in the joint space. We need to show, via
the inverse congruence transformation of eq. (10), that the
resulting Kp matrix from the known Kθ and Kg satisfies the
two conservative criteria in Theorem 1.
First of all, it is obvious that Kg is symmetric from eq.
(15). Consequently, the resulting matrix Kp from eq. (10) is
symmetric as long as Kθ is also symmetric. By substituting
eq. (15) and a conservative jointstiffness3 into
eq. (10), we
k11 k12
obtain the components of Kp =
, as follows:
k12 k22
Fig. 1. A two-link planar robot manipulator.
the stiffness control of a system. Similar effects are
documented in the literature such as the “snap instability” due to large normal force in grasping with stiffness
control (Cutkosky and Kao 1989).
k11
=
4. Analysis and Simulation of a Two-link Planar
Manipulator via the Conservative Congruence
Transformation
In this section, we will use a two-link planar manipulator to
illustrate the conservative property of the Cartesian stiffness
matrix, transformed from the joint stiffness matrix, via the
CCT in eqs. (8) and (10) versus the conventional formulation
in eq. (1). Numerical simulation will be provided to demonstrate the differences between them.
In Figure 1, a two-link planar manipulator is illustrated
with the end-effector at P (x, y) and the robot base frame
at O(XY ). We define x = [x y]T and θ = [θ1 θ2 ]T for
the planar robot. For the two-link manipulator, the stiffness
matrix Kg defined in eq. (9) is
∂JθT
−(L1 c1 + L2 c12 )fx − (L1 s1 + L2 s12 )fy
Kg =
f =
(−L2 c12 fx − L2 s12 fy )
∂θn
(−L2 c12 fx − L2 s12 fy )
,
(−L2 c12 fx − L2 s12 fy )
2. The joint-based stiffness control is specified with a prescribed Kθ in the
joint space. The Cartesian stiffness, Kp , is then obtained based on Kθ and
Kg using the inverse CCT equation in (10).
+
2
kθ11 − 2kθ12 + kθ22 c12
L21
s22
−
kθ12 θ1 + kθ22 θ2 c12 c2
L22
s23
+
2 c
kθ11 θ1 − kθ12 (θ1 − θ2 ) − kθ22 θ2 c12
2
L21
s23
+
kθ11 θ1 − kθ12 (θ1 − θ2 ) − kθ22 θ2 c12
L1 L2
s23
2
kθ12 θ1 + kθ22 θ2 c12
L1 L2
s23
−kθ12 + kθ22 c12 s1
kθ22 c1 s1
+ 2 2
2
L1 L2
s2
L2 s 2
−
k12
=
(15)
where c1 = cos θ1 , s1 = sin θ1 , c12 = cos(θ1 + θ2 ), and
s12 = sin(θ1 + θ2 ). Next, we will employ the conservative
criteria to examine the stiffness matrices of the planar robot
via the conservative congruence transformation. In addition,
the numerical simulation of forces and work done under jointbased stiffness control2 scheme will be obtained and compared, based on both the CCT and conventional formulation.
kθ22 c12
2(kθ12 − kθ22 ) c1 c12
−
2
2
L 1 L2
L2 s 2
s22
k21
=
(16)
−
kθ12 θ1 + kθ22 θ2 c1 c2 s1
L22
s23
+
kθ11 θ1 − kθ12 (θ1 − θ2 ) − kθ22 θ2 c1 s1
L1 L2
s23
−
kθ12 − kθ22 c1 s12
kθ11 − kθ12 + kθ22 c12 s12
+
2
L1 L 2
s2
L21
s22
+
kθ11 θ1 − kθ12 (θ1 − θ2 ) + kθ22 θ2 c2 c12 s12
L21
s23
−
kθ12 θ1 + kθ22 θ2 c12 s12
L1 L2
s23
k12
3. We use a symmetric and constant Kθ =
convenience of illustration.
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(17)
(18)
kθ11
kθ21
kθ12
kθ22
matrix for
840
k22
THE INTERNATIONAL JOURNAL OF ROBOTICS RESEARCH / September 2000
=
kθ22 s12
kθ12 θ1 + kθ22 θ2 c2 s12
−
L22 s22
L22
s23
In this simulation, the initial joint torque is given as τ0 =
[10 20]T N · m, resulting in the initial bias force of f0 =
[73.1 96.7]T N at the initial configuration of θ1 = 1.21rad
and θ2 = −1.09rad. The force and work done in the Cartesian space are computed by employing the two-dimensional
trapezoidal rule
+
kθ11 θ1 − kθ12 (θ1 − θ2 ) − kθ22 θ2 s12
L1 L2
s23
−
2
2(kθ12 − kθ22 ) s1 s12
kθ11 − 2kθ12 + kθ22 s12
+
L1 L2
s22
L21
s22
+
2
kθ11 θ1 − kθ12 (θ1 − θ2 ) + kθ22 θ2 c2 s12
L21
s23
−
2
kθ12 θ1 + kθ22 θ2 s12
.
L1 L1
s23
fi = fi−1 + Kp dx
(22)
1
Wi = Wi−1 + dW = Wi−1 + fiT dx + dxT Kp dx.
2
(23)
(19)
Thus, the exactness (i.e., criterion 2 in Theorem 1) of the
Cartesian stiffness matrix can be examined by substituting
eqs. (16) through (19) into the following equations:
∂k11
∂k12
∂k11 ∂θ2
∂k11 ∂θ1
−
=
+
∂y
∂x
∂θ1 ∂y
∂θ2 ∂y
∂k12 ∂θ1
∂k12 ∂θ2
−
+
=0
(20)
∂θ1 ∂x
∂θ2 ∂x
∂k21
∂k22
∂k21 ∂θ2
∂k21 ∂θ1
−
=
+
∂y
∂x
∂θ1 ∂y
∂θ2 ∂y
∂k22 ∂θ1
∂k22 ∂θ2
−
+
= 0,
(21)
∂θ1 ∂x
∂θ2 ∂x
∂θi
−1
i
where the terms ∂θ
∂x and ∂y are the entries of the Jθ matrix.
Eqs. (20) and (21) can be verified by symbolic software such
as Mathematica or Maple V.
From the above analysis, we conclude that the Cartesian
stiffness matrix transformed from a constant and symmetric
joint stiffness matrix via the CCT is conservative because it
satisfies both criteria of symmetry and exactness.
4.2. Numerical Simulation of Congruence Transformation
In the following sections, we will present and compare the
results of numerical simulation of both congruence transformations. Since Kg is shown to be conservative in Section 3.4,
we consider in the following sections three cases: (i) conservative Kθ , (ii) nonconservative Kθ , and (iii) special case.
4.2.1. Case 1: Conservative Kθ
We will use the example of the two-link manipulator in Figure 1 with link lengths, L1 = 0.29m and L2 = 0.23m, under
joint-based stiffness control with a prescribed conservative
joint stiffness matrix
10 0
Kθ =
N · m/rad.
0 20
An alternative derivation of dW in eq. (23) is in Section D
of the appendix. The last quadratic term in eq. (23) usually
has a smaller contribution than the second term. The path
of integration is a circle with a radius of 0.08m centered at
(x, y) = (0.25, 0.30)m, starting from the initial end-effector
position at x0 = 0.33m and y0 = 0.3m, tracing a counterclockwise sense. If the system is conservative, we expect that
the net work done over the closed circular path is zero and the
net change in force is also zero. Similarly, the work done in
the joint space is calculated by
1
Wi = Wi−1 + dW = Wi−1 + τiT dθ + dθ T Kθ dθ.
2
(24)
Note that the work done in the Cartesian and joint spaces
is only integrated from the force/torque and Kp /Kθ , respectively. The difference between the two congruence transformations is that Kp in the CCT is obtained from both Kθ and
Kg . The results of numerical simulation are shown in Figure 2(a) and 2(b) for the conventional formulation and CCT,
respectively.
A few observations are in order:
• In Figure 2(a), the conventional formulation yields net
changes in forces after integrating over the closed path,
i.e., %fx = 0 and %fy = 0. This shows that the conventional formulation is not a conservative mapping.
• In addition, the net work in the Cartesian space for
the conventional formulation in Figure 2(a) is not zero,
whereas the net work done in the joint space is zero.
The net work calculated in the joint space is zero because this is a joint-based control scheme (Kao and Ngo
1999), and the work is conservative as long as both
the Kθ and Kg matrices are conservative. However,
the Kp matrix is obtained through the nonconservative
mapping of eq. (1), thus resulting in discrepancy in the
work done. It is obvious that the two trajectories for
work done in Figure 2(a) should have been the same.
This further shows that the conventional formulation
does not yield correct results when external force is
applied.
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Chen and Kao / Conservative Congruence Transformation
841
Fig. 2. Simulation of joint-based control system with a conservative Kθ : The parameters are given in Section 4.2.1. The plots
show the Cartesian force f = [fx fy ]T and the work done over the closed circular path in counterclockwise sense for (a) the
conventional formulation and (b) the conservative congruence transformation (CCT).
• The profiles of forces in Figure 2(a) are erroneous due
to the employment of the conventional formulation in
calculating the Cartesian forces.
nonconservative joint stiffness matrix via the conventional
formulation and CCT, respectively.
We observe the following:
• When applying the conventional formulation, the discrepancy in work will become more pronounced when
the external force is larger. The magnitude of this discrepancy also depends on the stiffness matrix and the
differential Jacobian.
• Since this is a nonconservative system with a nonconservative Kθ , we expect that the net work done is
nonzero. Figure 3 clearly shows the net works are not
zero.
• In Figure 2(b), the CCT yields identical work profiles
with zero net work in both joint and Cartesian spaces
and results in zero changes
in forces over the closed
path, i.e., dW = 0 and df = 0. There is no discrepancy in work done, showing that the CCT yields the
exact results and is a conservative mapping between the
joint and Cartesian spaces.
4.2.2. Case 2: Nonconservative Kθ
Contrary to the previous section, we will use an asymmetric
(nonconservative) joint stiffness matrix to perform the simulation. First of all, we know both congruence mappings will result in an asymmetric Cartesian stiffness matrix, even though
the Kg matrix is still a conservative matrix (Section 3.4).
Thus, we obtain a nonconservative Cartesian stiffness matrix transformed from a nonconservative joint stiffness via
both congruence transformations. The nonconservative joint
stiffness matrix used in the simulation is
10 5
Kθ =
N ·m/rad.
0 20
The setup of simulation is the same as those in Section 4.2.1.
Figures 3(a) and (b) present the simulation results of the above
• Similar to Figure 2(a), Figure 3(a) gives incorrect results for the force components, fx and fy .
• Again, the work profiles of Figure 3(b) are identical,
indicating that the CCT is a correct mapping between
the joint and Cartesian spaces. On the other hand, Figure 3(a) shows the discrepancy in works done.
• The discrepancy of work profiles for Figure 3(a) becomes more pronounced as the external force becomes
larger.
• It is interesting to note that the net change in force in
Figure 3(b) is zero even though the system is nonconservative. This is because the exactness criterion in
Theorem 1 is met; hence, the CCT under this circumstance will result in %fx = %fy = 0. On the contrary,
Figure 3(a) yields %fx = 0 and %fy = 0, which violates the expectation because both Kθ and Kg satisfy
the exactness criterion.
4.2.3. Case 3: Special Configuration-Dependent Kθ
In addition to the two cases in Sections 4.2.1 and 4.2.2, we
consider the special Kθ in Kao and Ngo (1999). In the analysis
of Kao and Ngo and the simulation of Chen and Kao (1998),
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THE INTERNATIONAL JOURNAL OF ROBOTICS RESEARCH / September 2000
Fig. 3. Simulation of joint-based control system with an asymmetric and nonconservative Kθ : The parameters are given
in Section 4.2.2. The plots show the Cartesian force f = [fx fy ]T and the work done over the closed circular path in
counterclockwise sense for (a) the conventional formulation and (b) the conservative congruence transformation (CCT).
they presented a special form of configuration-dependent Kθ
matrix for a two-link manipulator that renders conservative
Kp matrix via the conventional formulation. When L1 = L2 ,
the special joint stiffness matrix is chosen as
2k k Kθ =
,
cos θ2
k
1+cos θ2 k
where k is a constant and is taken as 7.5 in the following simulation. Figures 4(a) and 4(b) show the results of simulation,
with L1 = L2 = 0.29m via the conventional formulation and
CCT, respectively. Other parameters and conditions are held
the same as those in Sections 4.2.1 and 4.2.2.
The results show that while both Figures 4(a) and 4(b) yield
the same end states (i.e., %fx = %fy = %W = 0), there is a
difference between the profiles. In particular, the discrepancy
in work profiles exists in Figure 4(a). Nevertheless, the CCT
still yields identical work profiles and correct results, as shown
in Figure 3.
4.3. Summary
We can summarize the results from the proceeding analysis
and numerical simulation.
1. The conventional formulation always produces discrepancies between the profiles of work done in the Cartesian and joint spaces. This is because the conventional
mapping is not a conservative mapping (Kao and Ngo
1999) and does not yield correct results unless the grasp
is always maintained at its unloaded configuration.
2. The discrepancies of force and work between the joint
and Cartesian spaces of the conventional formulation
become more pronounced as the influence of the external force becomes larger.
3. In the CCT, the work done in both joint and Cartesian
spaces is always identical and consistent. Thus, the
CCT is a correct and complete transformation of stiffness matrices between the joint and Cartesian spaces.
4. In addition to the consistency of the conservative property in net work, the relation between torque and force
is also verified in the CCT. The simulation results are
shown in Figure 5, where the parameters of simulation
for case (1), with a conservative Kθ in Section 4.2.1, are
used. In the figure, we compute the torque, τ1 and τ2 ,
based on the numerical simulation in the joint space, as
well as the torque calculated from the Cartesian force
using τ = JθT f. We expect that the two sets of torques
should be identical if the transformation equation between the Kθ and Kp is correct. It is clear from the
figure that the torques computed from the CCT are
consistent and identical. On the other hand, the conventional formulation always yields erroneous results
in both Cartesian forces and torques calculated from
forces in the joint-based stiffness control system.
5. It is essential that the conservative congruence transformation is employed whenever external force is
expected.
6. The Kg matrix captures the effect due to the changes
in geometry under the presence of external force. It
also accounts for the discrepancy of work profiles in
the simulation of conventional formulation.
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Chen and Kao / Conservative Congruence Transformation
843
Fig. 4. Simulation of joint-based control system with the special Kθ (Kao and Ngo 1999): The parameters are given in Section
4.2.3. The plots show the Cartesian force f = [fx fy ]T and the work done over the closed circular path in counterclockwise
sense for (a) the conventional formulation and (b) the conservative congruence transformation (CCT).
Fig. 5. Simulation results of torque integrated directly in the joint-based stiffness control system, and the torques calculated
from the Cartesian force using τ = JθT f, where Jθ is updated for each configuration in the simulation. The parameters used
are those of Case 1 in Section 4.2.1.
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THE INTERNATIONAL JOURNAL OF ROBOTICS RESEARCH / September 2000
7. The graphical illustration of the CCT is offered in Figure 6, where Kθ in the joint space is combined with
the effective stiffness matrix, Kg , for the conservative
mapping.
8. The elbow-out configuration of the manipulator was
used in the simulation, as shown in Figure 1. The CCT
can be equally applied to the elbow-in configuration,
with different numerical results. And yet, the conclusions of the conservative properties of force and work
remain the same.
5. Stiffness Control of Robot Grasping
Manipulation via the Conservative
Congruence Transformation
From the preceding analysis, we find that the conventional
formulation in eq. (1) is a special case of the conservative
congruence transformation when the system is always maintained at its unloaded equilibrium configuration. Equations
can be formulated for displacements and force/torque by tracing the matrix terms in square boxes. If the direction of an
arrow is followed, the matrix term in the box will be used,
whereas if the arrow is traced in the reversed direction, the
inverse of the matrix term should be employed.
Figure 7 illustrates the relationship of the stiffness matrices
between the joint and Cartesian spaces based on the conservative congruence transformation. From Figure 7, we find
the relation between displacements of the joints, dθ, and the
changes of externally applied forces, df, as
df = Jθ−T Kθ − Kg dθ.
(25)
In addition, we can write the relationship between the changes
in the joint torques, dτ , and the changes in the Cartesian
forces, df, as
df = Jθ−T Kθ − Kg Kθ−1 dτ
(26)
= Jθ−T dτ − Jθ−T Kg Kθ−1 dτ.
The first term of eq. (26), df = Jθ−T dτ , relates the infinitesimal changes of df and dτ when there is no initial
bias force/torque. Nevertheless, when relating the changes
in torques/forces in the joint/Cartesian spaces, the relationship of eq. (26) should be used to include the effect of load
and changes in geometry. Other relationships of stiffness control and mapping between the joint and Cartesian spaces can
be traced using Figure 7, along with eqs. (25) and (26).
between the joint and Cartesian spaces and should be employed to replace the conventional formulation that is only
valid at the unloaded equilibrium configuration. The conservative congruence transformation accounts for the effect of
change of robot grasping geometry under the presence of external force. It is shown that the Cartesian stiffness must be
obtained from two parts: one from the active joint stiffness
control Kθ and the other from the effective stiffness matrix
resulting from the changes in geometry as represented by the
Kg matrix. The conservative congruence transformation also
preserves the symmetric, positive-definite, and conservative
properties of stiffness matrices when mapping between the
Cartesian and joint spaces.
Appendix
Alternative Derivation for the Conservative Criteria of the
Stiffness Matrix
Here, we employ the indicial notation (Malvern 1969) to derive the conservative criteria for the stiffness matrix (Ngo
1998). We consider the external force f as a vector field, and
the curl of the vector field f is denoted as (curl f). If the vector field f is the gradient of a scalar function, then (curl f)=0.
Conversely, it can be proved that if (curl f)=0 throughout a
simply connected region, then f is the gradient of a potential
function φ defined by
T
φ = f dr = f1 dx1 + f2 dx2 + f3 dx3 ,
(27)
C
C
which is a line integral along any curve C from an arbitrary
initial point in the region to a different destination point of the
region. The necessary and sufficient conditions for φ to be a
point function is (curl f)=0, which makes the line integral of
eq. (27) independent of path on a simply connected region in
which f and (curl f) are continuous.
In indicial notation, the curl operator from the right on a
vector given in rectangular Cartesian space yields
f × ∇ = (fr ir ) × ( ∂ p ip ) = fr ∂ p (ir × ip )
where erpq is the alternating (or permutation, or Levi-Civita)
∂f
∂fi
symbol. Thus, (curl f) = 0 implies kij = ∂x
= ∂xji = kj i .
j
Next, we need to prove that the force, f = K · dx, is
also a point function. The curl operator from the right for a
second-order tensor yields
K × ∇ = (krs ir is ) × ( ∂ p ip ) = krs ∂ p ir (is × ip )
6. Conclusions
The conservative congruence transformation (CCT) presented
in this paper is the correct and complete stiffness mapping
(28)
= erpq fr,p iq ,
= espq krs,p ir iq ,
with typical components:
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(29)
Chen and Kao / Conservative Congruence Transformation
Fig. 6. Graphical illustration of the conservative congruence transformation and its inverse.
Fig. 7. Relationships of stiffness control in robotic grasping and manipulation.
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845
846
THE INTERNATIONAL JOURNAL OF ROBOTICS RESEARCH / September 2000
(K× ∇ )11
(K× ∇ )12
(K× ∇ )13
(K× ∇ )21
(K× ∇ )22
(K× ∇ )23
(K× ∇ )31
(K× ∇ )32
(K× ∇ )33
=
∂k12
∂k13
−
;
∂x3
∂x2
=
∂k13
∂k11
−
;
∂x1
∂x3
=
∂k11
∂k12
−
∂x2
∂x1
=
∂k22
∂k23
−
;
∂x3
∂x2
=
∂k23
∂k21
−
;
∂x1
∂x3
=
∂k21
∂k22
−
∂x2
∂x1
=
∂k32
∂k33
−
;
∂x3
∂x2
=
∂k33
∂k31
−
;
∂x1
∂x3
=
∂k31
∂k32
−
.
∂x2
∂x1
where dθk and fj are first-order tensors. On the right-hand
side of the arrows, we draw the equivalence of the tensor
analysis to the terms used in Section 3.3. Thus, eq. (35) can
be rewritten as
n
m
ci =
aij k dθk fj
j =1
(30)
k=1
=
(31)
(32)
(33)
We can define a second-order tensor as
∂JθT
dθ,
∂θ
where aij k represents the components of
∂JθT
∂θ
(34)
and bij rep
∂JT
resents the components of second-order tensor in ∂θθ dθ .
The symbol “⇐⇒” is used to denote equivalence. Similarly,
for a first-order tensor, we can write
m
∂JθT
ci =
bij fj ⇐⇒
dθ f,
(35)
∂θ
j =1

aij 1 fj  dθ1 + · · · + 
m

aij n fj  dθn
j =1
⇐⇒
∂JθT
f dθn ,
∂θn
(36)
Proofs of Corollaries 3 and 4
Derivations for the CCT
k=1

where n = number of joints, m = number of degrees of
freedom in the Cartesian space, and aij 1 . . . aij n are the entries
of the third-order tensor, aij k .
for 1 ≤ i, j, k ≤ 3.
aij k dθk ⇐⇒
m
The curl operator from the left will give rise to the same results
as above (see Ngo 1998).
Summary: (curl f)=0 and (curl K)=0 are the criteria for
conservative stiffness.
n

j =1
j =1
bij =

n
m

=
aij k fj  dθk

It is obvious that for all of the components of (K× ∇ ) to be
equal to zero, we must have
∂kij
∂kik
=
∂xk
∂xj
k=1

Proof of Corollary 3. For joint-based stiffness control, Kg is
symmetric based on eqs. (11) and (12), and Kθ is symmetric in
a conservative stiffness control system. The resulting stiffness
matrix in the Cartesian space can be found using the inverse
CCT in eq. (10). Hence,
T T
KpT = Jθ−1
KθT − KgT Jθ−T
=
Jθ−T (Kθ
− Kg )Jθ−1
(37)
= Kp .
That is, Kp is also symmetric. Similarly, we can prove that
the symmetry is preserved for the Cartesian-based stiffness
control by using the CCT in eq. (8).
Proof of Corollary 4. First, let us regard (Kθ − Kg ) as one
entity to relate with Kp . Next, we multiply both sides of eq.
(10) by an arbitrary vector y to render
yT Kp y = yT Jθ−T (Kθ − Kg )Jθ−1 y = zT (Kθ − Kg )z,
(38)
where z = Jθ−1 y. The vector y is arbitrary and so is z.
From eq. (38), we can find that if Kp is positive definite (i.e.,
yT Kp y > 0), then zT (Kθ − Kg )z > 0, which means that
(Kθ − Kg ) is also positive definite. On the other hand, if
(Kθ − Kg ) is positive definite, then Kp will also be positive
definite. This proof can be extended to positive semidefinite
matrices.
Therefore, it is clear that the positive definite properties
are preserved between Kp and (Kθ − Kg ).
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Chen and Kao / Conservative Congruence Transformation
Equation of Work Integration
Here, we present a different derivation for eq. (23) using the
Taylor’s expansion. For stiffness controlled system, the potential energy is the work. Therefore, the work of this stiffness
system is W = U . We can write
dU =
∂U
1
∂ 2U
dx + dxT
dx + · · · .
∂x
2
∂x2
We note that f T = ∂U
∂x and K =
order small terms, we have
∂2U
.
∂x2
(39)
Neglecting the higher-
1
dW = f T dx + dxT K dx,
2
(40)
which renders the same equation as (23), which was presented
using the more well-known numerical algorithm (trapezoidal
rule).
Acknowledgments
The research was supported by the NSF/ARPA Grant IRI9309823 and IIS9906890.
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