)%0!,-%.2*"*''*)#*)#)%0!,-%.2*"*''*)#*)#
!-!,$)'%)!!-!,$)'%)!
/'.2*")#%)!!,%)#+!,-,$%0!
/'.2*")#%)!!,%)#) )"*,(.%*)
%!)!-
/#/-.
)$)! .%"")!--* !'%)# !).%4.%*)) $,.!,%3.%*)"*,**.)$)! .%"")!--* !'%)# !).%4.%*)) $,.!,%3.%*)"*,**.
)%+/'.*,-)%+/'.*,-
'%%
)%0!,-%.2*"*''*)#*)#
#/,-!'/*1! //
$%,%)3 !$
*)-$)%0!,-%.2
*''*1.$%-) %.%*)'1*,&-.$..+-,*/*1! //!)#++!,-
,.*".$!)#%)!!,%)#*((*)-
$..+-,*/*1! //!)#++!,-
!*((!) ! %..%*)!*((!) ! %..%*)
'%%) $%,%)3 !$)$)! .%"")!--* !'%)# !).%4.%*)) $,.!,%3.%*)"*,**.
)%+/'.*,-
$..+-,*/*1! //!)#++!,-
!-!,$)'%)!%-.$!*+!)!--%)-.%./.%*)',!+*-%.*,2"*,.$!)%0!,-%.2*"*''*)#*)#*,"/,.$!,%)"*,(.%*)
*)...$!%,,2,!-!,$+/-/*1! //
554 IEEE TRANSACTIONS ON ROBOTICS, VOL. 21, NO. 4, AUGUST 2005
Enhanced Stiffness Modeling, Identification and
Characterization for Robot Manipulators
Gürsel Alici and Bijan Shirinzadeh
Abstract—This paper presents the enhanced stiffness modeling
and analysis of robot manipulators, and a methodology for their
stiffness identification and characterization. Assuming that the
manipulator links are infinitely stiff, the enhanced stiffness model
contains: 1) the passive and active stiffness of the joints and 2) the
active stiffness created by the change in the manipulator configu-
ration, and by external force vector acting upon the manipulator
end point. The stiffness formulation not accounting for the latter
is known as conventional stiffness formulation, which is obviously
not complete and is valid only when: 1) the manipulator is in
an unloaded quasistatic configuration and 2) the manipulator
Jacobian matrix is constant throughout the workspace. The
experimental system considered in this study is a Motoman SK
120 robot manipulator with a closed-chain mechanism. While the
deflection of the manipulator end point under a range of external
forces is provided by a high precision laser measurement system,
a wrist force/torque sensor measures the external forces. Based on
the experimental data and the enhanced stiffness model, the joint
stiffness values are first identified. These stiffness values are then
used to prove that conventional stiffness modeling is incomplete.
Finally, they are employed to characterize stiffness properties of
the robot manipulator. It has been found that although the compo-
nent of the stiffness matrix differentiating the enhanced stiffness
model from the conventional one is not always positive definite, the
resulting stiffness matrix can still be positive definite. This follows
that stability of the stiffness matrix is not influenced by this stiff-
ness component. This study contributes to the previously reported
work from the point of view of using the enhanced stiffness model
for stiffness identification, verification and characterization, and
of new experimental results proving that the conventional stiffness
matrix is not complete and is valid under certain assumptions.
Index Terms—Compliance/force control, manipulator kine-
matic, stiffness identification, stiffness modeling.
I. INTRODUCTION
K
NOWING the stiffness or compliance of a robot manip-
ulator reflected at its end point is of prime importance
to successfully conduct contact and noncontact tasks. In fact,
the stiffness of robot manipulators generally represents the
accuracy required to satisfy the desired position and force
commands [1]–[3]. Further, it must be recalled that there is
Manuscript received May 4, 2004; revised August 4, 2004. This paper was
recommended for publication by Associate Editor R. Roberts and Editor I.
Walker upon evaluation of the reviewers’ comments. This work was supported
in part by the Australian Research Council (ARC) and Monash University
(SMURF2).
G. Alici is with the School of Mechanical, Materials and Mechatronics Engi-
neering, University of Wollongong, Wollongong, NSW 2522, Australia (e-mail:
gursel@uow.edu.au).
B. Shirinzadeh is with Robotics and Mechatronics Research Laboratory, De-
partment of Mechanical Engineering, Monash University, Melbourne, Victoria
3800, Australia (e-mail: bijan.shirinzadeh@eng.monash.edu.au).
Digital Object Identifier 10.1109/TRO.2004.842347
coupling among rotational and translational Cartesian move-
ments of a typical serially connected robot manipulator because
of the inconsistency between the task specification (Cartesian)
space and the actuation (joint) space. This coupling also shows
itself in the nondiagonal stiffness matrix seen at the end point
of the robot manipulator. As a result, the manipulator tool
frame rotates when an end point force is applied along one
of the degrees of freedom [1], [2]. If the stiffness seen at the
manipulator end point is modeled and identified accurately, it
would be possible to compensate for the coupling and posing
errors caused by the external forces. The model may also be
used to generate manipulator stiffness maps defining the ma-
nipulator end point stiffness as functions of the joint stiffness
and manipulator configurations. The minimum and maximum
stiffness values, which are the maximum and minimum eigen-
values of the stiffness matrix, and their directions, which are
the corresponding eigenvectors, can be known in advance.
Hence, the most appropriate configurations for certain tasks
can be selected. In this study, we, therefore, address enhanced
stiffness modeling, analysis, identification and characterization
for robot manipulators. The enhanced stiffness model differs
from the conventional stiffness model first derived by Mason
and Salisbury [4] such that it contains the stiffness component
due to the change in the manipulator configuration and the
external forces acting on the manipulator in addition to the
inherent stiffness of the system. The conventional stiffness
formulation is valid only when the manipulator is in a qua-
sistatic configuration with no loading, or when it has a constant
Jacobian matrix throughout its workspace such as a Cartesian
robot manipulator. Although the existence of the additional
stiffness component has been known for a long time [5], [6],
its importance for preserving the conservative and fundamental
properties of joint and Cartesian stiffness matrices has been
highlighted by Kao
et al. [9] and others [7], [8], and [10]
recently.
Stiffness modeling, analysis, synthesis and control have
attracted the attention of many researchers. Ang and Andeen
[11] reported on how to generate variable passive compliance
through the topology of robot manipulators with backdrivable
actuators. It has been concluded that a nondiagonal stiffness
matrix can be effective in preventing jamming and contact in-
duced vibrations. Ang et al. [12] also introduced a methodology
to model the end-effector stiffness due to link flexibilities of
serially connected robot manipulators. They demonstrated that
the methodology could be applied to all possible serial manip-
ulator topologies: 1) to describe their end-effector stiffness as
a function of manipulator configurations, and 2) to choose the
most appropriate manipulator configuration compatible with
the compliance requirements of the task at hand. Kao et al.
1552-3098/$20.00 © 2005 IEEE
ALICI AND SHIRINZADEH: ENHANCED STIFFNESS MODELLING ROBOT MANIPULATORS 555
[13] have pointed out that the linear 3 3 stiffness matrices for
grasping can be decomposed into symmetric and nonsymmetric
components such that they provide better physical insights into
stiffness control and grasping. The stiffness components are
calibrated using experimental data from some grasping tasks. It
has been concluded that the stiffness matrix has a significant ef-
fect on grasp stability and robustness against slipping. Hang and
Schimmels [14] demonstrated that spatial compliance behavior
could be accomplished using a number of springs connected in
parallel to a rigid body. In a study on global stiffness modeling
of compliant couplings, Griffis and Duffy [15] pointed out
that a 6
6 general stiffness matrix was not symmetric for a
conservative system. Ciblak and Lipkin [16] later demonstrated
that the skew-symmetric part of such a matrix was due to the
externally applied force/moment vectors. Ciblak and Lipkin
[17] also employed the spatial vector (screw) algebra to demon-
strate that the Cartesian stiffness of robotic systems could be
realized by a number of springs. Chen and Kao [7] have recently
reported that the conventional stiffness formulation derived by
Mason and Salisbury [4] is not valid and a new conservative
congruence transformation must be used as the generalized
relationship between the joint and Cartesian stiffness matrices
in order to preserve the fundamental properties of the stiffness
matrices. Based on this finding, a number of publications on
the use of this new transformation have been made available
by the same group of researchers [7]–[10], simulation results
for a planar revolute-jointed manipulator have been provided
to demonstrate its validity. With this in mind, it is believed that
the work presented in this paper is the first to verify enhanced
stiffness modeling through experimental results provided by a
high-resolution laser-based position sensing and measurement
system.
The sources of the stiffness of a typical robot manipulator
are the compliance of its joints, actuators and other transmis-
sion elements, geometric and material properties of the links,
base, and the active stiffness provided by its position control
system. For the purpose of this study, we assume that: 1) the
compliance in actuators and transmission elements is the dom-
inant source of the stiffness, and it can be represented by a
linear torsional spring for each joint; 2) the active compliance
in actuators due to a robot position control system provided by
the original equipment manufacturer does not vary with time
though an integral controller can increase the active compli-
ance, depending on the positioning error; and 3) the links are
infinitely stiff. The experimental system employed here in this
work is a Motoman SK 120 robot manipulator, which is not a
simple open chain robot manipulator, but rather contains a par-
allel five-bar mechanism to increase the structural stiffness of
the system [18]. In order to identify the joint stiffness values, a
number of manipulator configurations are heuristically selected
to acquire experimental position and force data, which are pro-
vided by a laser-based sensing and measurement system, and
a wrist force/torque (F/T) sensor, respectively. The laser-based
sensing system has an accuracy of
ppm m/m , a coordi-
nate repeatability of
ppm m/m and a distance resolution
of 1.26
m, and can measure the position of any target along the
three orthogonal axes. A classical nonlinear least square estima-
tion algorithm is used to estimate the joint stiffness values. The
estimated joint stiffness matrix is verified experimentally. Also,
the work done in Cartesian and joint spaces are computed for
both enhanced and conventional stiffness formulations. It has
been found that the conventional stiffness formulation does not
satisfy the principle of the conservation of energy. It has also
been demonstrated that this arises from the existence of the ad-
ditional stiffness component. Further, based on the estimated
joint stiffness constants, the limits and conditions of the posi-
tive definiteness (stability) of the Cartesian stiffness matrix are
evaluated. It has been found that although the additional stiff-
ness component differentiating the enhanced formulation from
the conventional formulation is not always positive definite, the
resulting Cartesian stiffness matrix can still be positive definite.
The main contribution of this study is the new theoretical and ex-
perimental results supporting the enhanced stiffness modeling
(or conservative congruence transformation) through stiffness
identification, verification, and characterization.
II. K
INEMATIC
ANALYSES AND
JACOBIAN
MATRIX
The schematic of the robot manipulator and the coordinate
frames needed to generate a kinematic model based on De-
navit–Hartenberg parameters is depicted in Fig. 1. The manip-
ulator possesses a parallel five-bar mechanism. Therefore, the
kinematic model of the parallel five-bar mechanism is also de-
rived. The Denavit Hartenberg parameters for the manipulator
and the parallel mechanism are given in Table I.
An L-shaped apparatus with a longitudinal extension of
mm from the manipulator tool plate and a vertical offset
of
mm from the longitudinal axis of the tool plate
is connected to the manipulator tool plate in order to secure
the retroreflector of the measurement system to the robot via
a 3-point-contact magnetic fixture. With reference to Fig. 1, it
must be noted that, for the five-bar mechanism,
,
, , , and . The homogeneous
transformation matrix between the frames 1 and 2 of the five bar
is described in terms of the coordinate frames fixed to the links
of the five-bar mechanism as
(1)
The overall transformation matrix between the base coordi-
nate frame and the frame fixed to the manipulator end point is
written as
(2)
where is the homogeneous transformation matrix be-
tween two consecutive coordinate frames j and
based
on Denavit–Hartenberg convention [18]. From the first three
elements of the last column of
, the generalized relationship
between the input velocity vector
and the
output velocity vector
is obtained as
(3)
where
is the manipulator Jacobian matrix, see Appendix I for
analytical expressions of its elements.
556 IEEE TRANSACTIONS ON ROBOTICS, VOL. 21, NO. 4, AUGUST 2005
Fig. 1. Schematic representation of the robot manipulator with D-H convention and parameters. Note that the coordinate frames
O
and
O
, and
O
and
O
are
located at the same point.
TABLE I
R
ESULTS OF D-H P
ARAMETERS FOR MOTOMAN 120 SK M
ANIPULATOR AND
PARALLEL MECHANISM.SHADED ROWS ARE FOR PARALLEL MECHANISM
Using the duality between the generalized relationships for
motion and force transfer between the actuator and operation
spaces, the following force relationship is obtained:
(4)
where
represents the 3 1 vector of external
forces acting at the manipulator end point, and
denotes the 3 1 vector of the actuator forces/torques needed
to balance the external forces, and
denotes transposition.
III. S
TIFFNESS FORMULATION
The sources of stiffness for a typical serially connected robot
manipulator include the base stiffness, joints’ stiffness, link
stiffness, and active stiffness due to position feedback. It is
assumed that the primary source of the stiffness is the active
and passive joint stiffness in the axial direction of the actuation
torque, and it is lumped into a single constant stiffness value
for each joint. However, it must be noted that, depending
on positioning error and the integral gain that is usually much
smaller than the proportional gain in a typical proportional
integral derivative (PID) controller, the active stiffness due
to a robot position control system provided by the original
equipment manufacturer can vary with time. In this study, for
the sake of simplicity without loosing generality, it is assumed
that this variation is negligibly small. The actuator force/torque
needed to cause a change
in the angular position of the
joints from an unloaded position is
(5)
where
is the joint stiffness matrix,
and
is the three-dimensional vector of
the change in the joints position. It must be noted that such a
matrix is positive definite and symmetric. Similarly, the force
needed to cause the manipulator end point to experience a small
change
in its position is given by
(6)
where
is a symmetric 3 3 matrix representing the Carte-
sian stiffness of the manipulator, and
is the
vector of the change in the manipulator end point position. The
partial differentiation of (4) with respect to
leads to the fol-
lowing relationship:
(7)
For the static case of unloaded manipulator configuration, the
first term on the right-hand side of (7) is zero, and the stiffness
ALICI AND SHIRINZADEH: ENHANCED STIFFNESS MODELLING ROBOT MANIPULATORS 557
seen at the end point of the manipulator can be expressed
as
(8)
Equation (8) is the formulation of active stiffness control
[4], which regulates the apparent stiffness of a manipulator
end point in order to control the nominal position of the end
point. Stiffness values are changed in the software to satisfy
the desired position commands. It must be noted that
calculated from (8) is the Cartesian stiffness matrix based
on the conventional stiffness formulation, which neglects the
stiffness component
due to external loading and change in
the manipulator configuration.
If any positive definite and symmetric matrix
is subjected
to a transformation operation in the form of
, the re-
sulting matrix will still be positive definite and symmetric, as
long as the transformation matrix
is nonsingular [19]. This
means that the stiffness matrix
evaluated from (8) is posi-
tive definite and symmetric. As provided in [7], [20], the matrix
resulting from the first part of (7) is symmetric, but depending
on the external forces/payloads and the configuration of the ma-
nipulator, it can be positive definite or not. With regard to the
second part of (7), the fundamental properties of
such as
its definiteness and symmetry are preserved provided that the
Jacobian matrix J is not singular. With this in mind, (7) always
gives a symmetric stiffness matrix
, but not necessarily a pos-
itive definite
. However, as explained above, the joint stiff-
ness matrix is the natural entity of a robot manipulator and does
not change with the manipulator configuration. Further, it is a di-
agonal and positive definite matrix. Therefore, the fundamental
properties of the Cartesian stiffness matrix can be evaluated for
any existing robot manipulator.
IV. E
STIMATION OF
JOINT STIFFNESS
The enhanced stiffness formulation given by (7) differs from
the conventional stiffness formulation in the sense that its first
part stands for the case when the manipulator is externally
loaded and/or the manipulator Jacobian changes with its con-
figuration. This part of (7) completes the conventional joint
stiffness matrix and therefore, we call it the
complementary
stiffness matrix
, and for a manipulator actuated through
three joints, it can be written as
(9)
where
is a 3 1 column vector. From (7), the
complete stiffness matrix of the manipulator in Cartesian space
is obtained as
(10)
The force vector
can be a dynamic force or a static force
such as a payload carried by the manipulator. In this study, it is
assumed that it is in the form of
. Such a force
vector will generate a deflection of
, which
is a positioning error. If the joint stiffness
values are iden-
tified accurately using some experimental deflection and force
data, the stiffness
can subsequently be calculated from (7)
for any external force vector without needing any other exper-
imental data. By substituting (10) into (6) and solving for
gives
(11)
For a given manipulator configuration where a force vector
causes the deflection vector , (11) is nonlinear with un-
known joint stiffness
values, which can be estimated using
a least squares estimation algorithm. Since the stiffness of a
joint is a local property, and the topology of a revolute joint
does not change with the movement of the manipulator, it is
assumed that the stiffness matrix
is constant for revolute
jointed manipulators.
A. Nonlinear Least Square
Based on the nonlinear deflection model
of the ma-
nipulator expressed with (11), the joint stiffness values are
estimated by minimizing the summed square of the error vector
associated with
number of measurements
(12)
where
is the error vector given by
(13)
is the measured (true) end point deflection vector under
a range of payloads, and
is the deflection vector cal-
culated from (11) for the same payload. This is basically
a nonlinear least square optimization problem that can be
solved using either the interior-reflective Newton method or
Levenburg–Marquardt algorithm. These are two efficient op-
timization algorithms for large-scale nonlinear problems. The
former, which is based on the method of preconditioned con-
jugate gradients, can solve difficult nonlinear problems more
efficiently than the latter [21]. In this study, both algorithms
are implemented for the nonlinear least square estimation of
the parameters. The solutions converged to the same numerical
values. The procedure was realized iteratively until the deflec-
tion error was small enough to meet a termination condition of
.
B. Experimental Setup and Results
The key elements of the experimental setup depicted in
Fig. 2 are the laser tracker, retroreflector, the robot (Motoman
SK120), and a fixture connected to the manipulator end point in
order to exert forces (loads) along the three orthogonal axes of
the manipulator end point. As shown in Fig. 3, a cable-pulley
system was used to generate the needed force vector. The
deadweights in the range of 0–50 kg with 10-kg increments
were suspended to the free end of the cable. The three resulting
force components were measured via a wrist force/torque
sensor. The laser-based dynamic measurement system was
calibrated to measure the manipulator end point with respect to
the manipulator base frame. The manipulator was commanded
to 20 different well-spaced configurations within the manip-
ulator workvolume, which had been determined heuristically
to cover the range of motion of all the active joints of the
