488 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 52, NO. 3, MARCH 2007
Adaptive Regulation of Amplitude Limited Robot
Manipulators With Uncertain Kinematics and Dynamics
W. E. Dixon
Abstract—Common assumptions in most of the previous robot con-
trollers are that the robot kinematics and manipulator Jacobian are
perfectly known and that the robot actuators are able to generate the
necessary level of torque inputs. In this note, an amplitude-limited torque
input controller is developed for revolute robot manipulators with un-
certainty in the kinematic and dynamic models. The adaptive controller
yields semiglobal asymptotic regulation of the task-space setpoint error.
The advantages of the proposed controller include the ability to actively
compensate for unknown parametric effects in the dynamic and kinematic
model and the ability to ensure actuator constraints are not breached by
calculating the maximum required torque a priori.
I. I
NTRODUCTION
For a robotic system to interact with and execute tasks in the task-
space, a transformation between the position and orientation of the
robot with respect to objects located in the task-space is typically re-
quired. Since the robot is controlled through inputs to the link actua-
tors, the robot kinematics and manipulator Jacobian are used to relate
a task-space coordinate system to coordinate systems attached to each
actuator (i.e., the joint-space). A common assumption in most of the
previous robot control literature is that the kinematics and manipulator
Jacobian are perfectly known. This assumption limits robustness be-
cause uncertainty in the robot kinematics and Jacobian may lead to
degraded performance or unpredictable responses.
Uncertainty in the kinematic model of a robot is a practical problem.
The end-effector of a manipulator is often interchanged with other end
effector tools with different lengths; the manipulator picks up an ob-
ject with an unknown length with the intention of using it as a tool; or
generic robot data sheets may be used that are not specific to the partic-
ular robot. For example, the Puma 560 has well published kinematics
but many different versions of the robot are on the market that have
the same kinematic structure but different kinematic parameters (e.g.,
due to differences in the manufacturers). From these examples, it is
also clear that uncertainty in the kinematic model is a separate problem
from uncertainty in the dynamic model. In the invited paper [1], Ari-
moto describes the importance of the problem with uncertain kinematic
parameters and states that research which targets this problem is at the
beginning stages.
The observations by Arimoto are supported by a review of literature
which yields relatively few different approaches to address this topic. In
[28], Shahamiri and Jagersand used a nullspace-biased Newton-step vi-
sual servo strategy with a Broyden type Jacobian estimation for on-line
singularity detection and avoidance in an uncalibrated visual servo con-
trol problem. Based on a similar idea, Piepmeier
et al. published a se-
Manuscript received August 5, 2005; revised April 12, 2006 and August
31, 2006. Recommended by Associate Editor A. Astolfi. This work was
supported in part by the National Science Foundation under CAREER Award
CMS-0547448, by the Air Force Office of Scientific Research under Contracts
F49620-03-1-0381 and F49620-03-1-0170, by the AFRL under Contract
FA4819-05-D-0011, and by Research Grant US-3715-05 from BARD, the
United States–Israel Binational Agricultural Research and Development Fund
at the University of Florida.
The author is with the Mechanical and Aerospace Engineering Depart-
ment, the University of Florida, Gainesville, FL 32611-6250 USA (e-mail:
wdixon@ufl.edu).
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TAC.2006.890321
ries of papers (e.g., see [21]–[25]) that exploit dynamic quasi-Newton
recursive least-squares estimators to solve a variety of visual servo con-
trol problems despite an uncertain kinematic model and uncertainty in
the camera model.
Some results have been published by Cheah et al. in [4]–[6] that ad-
dress uncertainty in the kinematic model. Specifically, Cheah et al. de-
veloped several approximate Jacobian feedback controllers in [4]–[6]
that exploit a static, best-guess estimate of the manipulator Jacobian
to achieve task-space regulation objectives despite parametric uncer-
tainty in the manipulator Jacobian. As reported in [3], a drawback of
the controllers in [4]–[6] is that the task-space velocity of the robot
end-effector is required to be measurable, and the controller in [5] re-
quires the computation of an estimate for the Jacobian inverse. In [3],
Cheah et al. resolve these issues by developing a controller that exploits
a static, best-guess estimate of the manipulator Jacobian to achieve a
task-space regulation result.
Controllers that exploit combinations of proportional, integral, or
derivative feedback terms (i.e., PD and PID) such as the seminal work
in [19] can be used to achieve joint-space regulation in the presence
of uncertainty in the dynamic model; however, task-space regulation
has yet to be proven by such a controller when the kinematics and ma-
nipulator Jacobian are corrupted by uncertainty. A PID controller such
as [19] could be used to achieve joint-space regulation that is coupled
with an adaptive path planner that compensates for the kinematic uncer-
tainty. This is a valid approach that is also at the beginning stages of re-
search with some results presented in [7] and [14]; however, shifting the
ability to compensate for the kinematic uncertainty on the path planner
may not be reasonable for some applications. That is, efforts should
also be developed that compensate for the uncertainty through the con-
troller so that a generic path planner can be easily incorporated.
The controller developed in this note (and the preliminary work in
[12]) is the first adaptive result for the regulation problem under the
presence of kinematic and dynamic uncertainty. As in the recent re-
sult in [3], the controller does not require the task-space velocity of the
robot end-effector to be measurable, does not require the inverse of the
estimated Jacobian to be computed, and does not require exact model
knowledge of the robot dynamics. The current result actively compen-
sates for uncertainty in the gravity and static friction effects. In contrast
to the use of a static, best-guess estimate as in [3], the controller in this
note also exploits a feedforward control term that actively compensates
for the parametric uncertainty in the Jacobian. The adaptive controller
in this note provides the first result that eliminates the assumption (re-
quired by the previous robust controllers) that the mismatch between
the Jacobian and the Jacobian estimate be bounded as follows:
k
J
(
q
)
0
^
J
(
q
)
k
(1)
where
is some known positive constant. Moreover, the constraints
on the feedback gains that are imposed by (1) (i.e., high-gain robust
feedback) are eliminated by the proposed adaptive controller. The de-
velopment of the adaptive controller requires an innovative arrange-
ment of the uncertainty in the open-loop error system that is enabled
by using appropriate saturation methods to bound functions in a spe-
cific manner. Strategic use of the saturation functions coupled with an
innovative nonquadratic Lyapunov function also allows the control ef-
fort to be a priori bounded.
The fact that the controller can be a priori bounded is a significant
added advantage. The practical implications are that the actuators can
be appropriately sized (or the control gains set to specific thresholds)
without requiring an ad hoc saturation scheme (that would not be in-
cluded in the stability analysis) to protect the actuator. The common
assumption that the robot actuators are able to generate the necessary
0018-9286/$25.00 © 2007 IEEE
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 52, NO. 3, MARCH 2007 489
level of torque is limiting since robotic actuators have physical con-
straints. If the controller commands more torque than the actuators can
supply from typical control methods, degraded or unpredictable mo-
tion control and thermal or mechanical failure may result. For a re-
view of literature that addresses amplitude limited control designs, see
[8]–[10], [13], [16], [17], [26], [30], and the references within.
II. R
OBOT
MODEL
The dynamic model for a rigid
n
-link, serially connected, direct-
drive revolute robot is given as follows [15]:
M
(
q
)
q
+
V
m
(
q;
_
q
)_
q
+
G
(
q
)+
F
s
sgn( _
q
)=
:
(2)
In (2),
q
(
t
)
;
_
q
(
t
)
;
q
(
t
)
2
n
denote the link position, velocity, and
acceleration vectors, respectively,
M
(
q
)
2
n
2
n
represents the in-
ertia matrix,
V
m
(
q;
_
q
)
2
n
2
n
represents centripetal-Coriolis matrix,
G
(
q
)
2
n
represents gravity effects,
F
s
2
n
2
n
denotes the con-
stant diagonal static friction matrix,
sgn(
1
)
2
n
denotes the vector
signum function, and
(
t
)
2
n
represents the torque input vector. Let
x
(
t
)
2
m
(
m
n
)
represent a task-space vector that is related to the
robot joint-space as
x
=
h
(
q
)_
x
=
J
(
q
)_
q
(
t
)
(3)
where
h
(
q
)
2
m
denotes the differentiable forward kinematics of the
manipulator, and
J
(
q
)
(
@h=@q
)
2
m
2
n
denotes the differentiable
manipulator Jacobian.
The dynamic model introduced in (2) has the following properties
that are used in the subsequent control development and analysis.
Property 1: The positive–definite and symmetric inertia matrix, sat-
isfies the following inequalities [11]:
m
1
k
k
2
T
M
(
q
)
m
2
k
k
2
8
2
n
(4)
where
m
1
;m
2
2
are known positive bounding constants, and
k1k
is the standard Euclidean norm.
Property 2: The time derivative of the inertia matrix and the cen-
tripetal-Coriolis matrix satisfy the following skew symmetric relation-
ship [11]:
T
1
2
_
M
(
q
)
0
V
m
(
q;
_
q
)
=0
8
2
n
:
(5)
Property 3: The unknown gravitational and static friction terms can
be linearly parameterized as follows [11]:
Y
(
q;
_
q
)
G
(
q
)+
F
s
sgn( _
q
)
(6)
where
2
p
contains unknown constant parameters, and the regres-
sion matrix
Y
(
q;
_
q
)
2
n
2
p
contains measurable functions of the
link position and link velocity. Lower and upper bounds denoted by
;
2
p
, respectively, are assumed to be known for each parameter
in
as follows:
i
i
i
8
i
=1
;
2
;
...
p
(7)
where
i
i
2
denote the
i
-th component of
and
, respectively,
and
i
2
denotes the
i
th component of
.
Property 4: The time derivative of the inertia matrix, the centripetal-
Coriolis matrix, the gravity vector, and the static friction matrix can be
upper bounded in the following manner:
k
_
M
(
q
)
k
i
1
m
k
_
q
kk
V
m
(
q;
_
q
)
k
i
1
c
k
_
q
k
k
M
(
q
)
k
i
1
m
2
k
G
(
q
)
k
g
k
F
s
k
f
(8)
where
g
;
f
;
c
;
and
m
;
m
2
2
are known positive constants, and
k1k
i
1
denotes the induced infinity norm of a matrix.
Property 5: Since the controller in this note is developed for revolute
robots, the terms
M
(
q
)
;V
m
(
q;
_
q
)
;G
(
q
)
, and
J
(
q
)
are bounded for all
possible
q
(
t
)
. That is, these terms only depend on
q
(
t
)
as arguments of
bounded trigonometric functions, and
k
J
(
q
)
k
i
1
<
1
(9)
where
1
2
is a known positive constant. The subsequent develop-
ment is also based on the assumption that all kinematic singularities
associated with
J
(
q
)
are assumed to always be avoided.
Property 6: The product of the manipulator Jacobian with any mea-
surable vector
(
t
)
2
m
can be linearly parameterized as
Y
J
(
q;
)
J
J
T
(
q
)
where
Y
J
(
q;
)
2
n
2
p
contains measurable functions of the link po-
sition and
, and
J
2
p
contains the unknown constant parameters
contained in the Jacobian matrix. Lower and upper bounds denoted by
J
;
J
2
p
, respectively, are assumed to be known for each param-
eter in
J
as
Jk
Jk
Jk
8
k
=1
;
2
;
...
p
2
where
Jk
;
Jk
2
denote the
k
-th component of
J
and
J
, respec-
tively, and
Jk
2
denotes the
k
th component of
J
.
III. C
ONTROL DEVELOPMENT
A. Control Objective
Many robotic tasks are naturally defined in terms of the task-space.
The objective for these tasks is to regulate the end-effector of a robot
manipulator to a desired task-space setpoint. To quantify this objective,
a task-space setpoint error denoted by
e
(
t
)
2
m
is defined as
e x
0
x
d
(10)
where
x
(
t
)
was introduced in (3), and
x
d
2
m
denotes the known,
constant desired setpoint.
Unlike typical task-space regulation results, a contribution of this
note is that the constant parameters in the manipulator Jacobian are
not assumed to be known (i.e., the task-space to joint-space relation-
ship may be corrupt). To develop a controller that allows for task-space
regulation in the presence of uncertain kinematic and dynamic parame-
ters, the subsequent development is based on the assumption that
x
(
t
)
;
q
(
t
)
, and
_
q
(
t
)
are measurable. Specifically,
q
(
t
)
and
_
q
(
t
)
can be ob-
tained from encoder/tachometer sensors, and
x
(
t
)
could be obtained
from a camera system [3].
B. Control Development
Based on the control objective and the subsequent stability analysis,
the following adaptive controller is developed:
=
Y
(
q;
_
q
)
^
0
k
p
^
J
T
(
q
)Tanh(
e
)
0
k
v
Tanh( _
q
)
(11)
where
k
v
;k
p
2
denote constant control gains,
^
(
t
)
2
p
denotes a
subsequently designed parameter estimate,
^
J
(
q
)
2
m
2
n
represents
a subsequently designed Jacobian estimate, and the vector functions
Tanh(
e
)
2
m
and
Tanh( _
q
)
2
n
are defined as
Tanh(
e
) [tanh(
e
1
)
;
tanh(
e
2
)
;
...
;
tanh(
e
m
)]
T
Tanh( _
q
) [tanh(_
q
1
)
;
tanh(_
q
2
)
;
...
;
tanh(_
q
n
)]
T
:
490 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 52, NO. 3, MARCH 2007
To facilitate the design of the Jacobian estimate, the linear parameteri-
zation
Y
J
(
q;e
)
J
introduced in Property 6 is specifically defined as
Y
J
(
q;e
)
J
J
T
(
q
)Tanh(
e
)
(12)
where
Y
J
(
q;e
)
2
n
2
p
contains measurable functions of the link
position and task-space setpoint error, and
J
2
p
contains the un-
known constant parameters contained in the Jacobian matrix. An esti-
mate for (12) is developed as
Y
J
(
q;e
)
^
J
^
J
T
(
q
)Tanh(
e
)
(13)
where
^
J
(
t
)
2
p
denotes a parameter estimate. Based on the sub-
sequent stability analysis, the parameter estimates
^
(
t
)
and
^
J
(
t
)
are
generated from the following adaptation laws:
_
^
i
=proj
f
oi
g
_
^
Jk
(
t
) = proj
f
1
k
g
(14)
where
oi
(
q;
_
q;e
)
and
1
k
(
q;
_
q;e
)
denote the
i
th and
k
th compo-
nent of
o
(
q;
_
q;e
)
and
1
(
q;
_
q;e
)
, respectively,
8
i
=1
;
2
;
...
p
and
8
k
=1
;
2
;
...
p
2
, where the auxiliary terms
o
(
q;
_
q;e
)
2
p
and
1
(
q;
_
q;e
)
2
p
are defined as
o
(
q;
_
q;e
)
0
0
0
Y
T
(
q;
_
q
)( _
q
+
"Y
J
(
q;e
)
^
J
)
1
(
q;
_
q;e
)
k
p
0
1
Y
T
J
(
q;e
)_
q:
(15)
For the adaptation laws given in (14) and (15),
0
0
2
p
2
p
and
0
1
2
p
2
p
denote constant, diagonal positive definite adaptation gain ma-
trices,
"
2
is a positive, constant adaptation weighting gain, and the
function
proj
f1g
is defined as follows:
proj
f
oi
g
oi
;
if
^
i
>
i
oi
;
if
^
i
=
i
and
oi
0
0
if
^
i
=
i
and
oi
<
0
0
;
if
^
i
=
i
and
oi
>
0
oi
if
^
i
=
i
and
oi
0
oi
;
if
^
i
<
i
i
^
i
(0)
i
(16)
where
^
i
(
t
)
denotes the
i
th component of
^
(
t
)
. The
proj
f
1
k
(
q;
_
q;e
)
g
is defined in the same manner as in (16)
with regard to
^
Jk
(
t
)
. Based on the aforementioned assumption that
all kinematic singularities associated with
J
(
q
)
are always be avoided,
the development for
^
J
(
q
)
can be used to determine that
^
J
(
q
)
is also
nonsingular; hence, [20, Th. 5.9] can be used to conclude that
0
<
min
f
^
J
(
q
)
^
J
T
(
q
)
g
:
The closed-loop error system can be determined after substituting
(11) into (2) as follows:
M
(
q
)
q
+
V
m
(
q;
_
q
)_
q
=
0
Y
(
q;
_
q
)
~
0
k
p
Y
J
(
q;e
)
^
J
0
k
v
Tanh( _
q
)
(17)
where (6) and (13) have been utilized. In (17), the parameter estimation
error signal
~
(
t
)
2
p
is defined as
~
0
^
(18)
and a parameter estimation error signal
~
J
(
t
)
2
p
is also defined for
the mismatch between the actual and estimated Jacobian parameters as
follows:
~
J
J
0
^
J
:
(19)
Based on the control development introduced in (11)–(16), the fol-
lowing properties can be developed to facilitate the stability analysis.
Property 7: The projection algorithm in (16) ensures that the fol-
lowing inequalities are satisfied (for further details, see [2] and [18]):
i
^
i
(
t
)
i
Jk
^
Jk
(
t
)
Jk
:
(20)
Based on (14)–(16), the following inequality can also be shown to hold:
k
_
^
J
kk
1
(
q;
_
q;e
)
k
k
p
max
f
0
1
gk
Y
J
(
q;e
)
k
i
1
k
_
q
k
(21)
where
max
f1g
denotes the maximum eigenvalue of a matrix.
Property 8: From (9), (12), (13), and (20), the estimated Jacobian
matrix can be upper bounded as
k
^
J
(
q
)
k
i
1
<
2
(22)
where
2
2
denotes a known positive constant (i.e.,
2
>
0
).
Property 9: The time derivative of (12) can be upper bounded as
d
dt
(
Y
J
(
q;e
)
^
J
(
t
))
J
k
_
q
k
(23)
where
J
2
denotes a known positive constant.
Property 10: The following inequalities can be shown to hold for
all
e
(
t
)
2
m
and
_
q
(
t
)
2
n
[10]
2
m
i
=1
ln(cosh(
e
i
))
k
Tanh(
e
)
k
2
tanh
2
(
k
e
k
)
(24)
k
_
q
k
+1
k
_
q
k
tanh(
k
_
q
k
)
(25)
k
Tanh(
e
)
kk
Tanh(_
q
)
kk
Tanh(
e
)
k
2
+
k
Tanh( _
q
)
k
2
(26)
k
Tanh(
e
)
kk
_
q
kk
Tanh(
e
)
k
2
+
k
_
q
k
2
(27)
k
_
q
kk
Tanh( _
q
)
k
(28)
where
ln(
1
)
denotes the natural logarithm.
Property 11: The control effort can be upper bounded in terms of a
priori known terms as
k
kk
Y
k
i
1
k
k
+
k
p
k
Y
J
k
i
1
k
J
k
+
k
v
p
n
(29)
where the control gains
k
v
and
k
p
can be made arbitrarily small pro-
vided some relative magnitudes are maintained as subsequently de-
scribed.
Property 12: Based on (13) and (14)–(16), the following inequality
can be developed [3]:
min
f
^
J
(
q
)
^
J
T
(
q
)
gk
Tanh(
e
)
k
2
Tanh
T
(
e
)
^
J
(
q
)
^
J
T
(
q
)Tanh(
e
)
8
e
(
t
)
2
m
(30)
where
min
f1g
denotes the minimum eigenvalue of the argument.
IV. STABILITY ANALYSIS
Theorem 1: Given the robotic system defined by (2) and (3), the con-
trol torque input given in (11), along with the adaptation law given in
(14)–(16) ensures semi-global asymptotic regulation of the task-space
error in the sense that
k
e
(
t
)
k!
0
;
as
t
!1
:
(31)
The result in (31) is valid, provided the control gains
k
p
and
k
v
given
in (11)-(15), and the adaptation weighting gain
"
defined in (15) are
chosen to satisfy the following sufficient conditions:
k
p
>k
v
2
min
f
^
J
(
q
)
^
J
T
(
q
)
g
>
0
(32)
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 52, NO. 3, MARCH 2007 491
Fig. 1. Task-space position error.
"<
min
m
1
2
2
m
2
;
k
p
2
2
m
2
;
1
2
2
(33)
and
k
v
(1
0
2
"
2
)
2
"
2
(0)
1
2
m
1
0
"
2
m
2
+1
2
(34)
where
m
1
and
m
2
are defined in (4),
2
was defined in (22),
2
denotes a positive bounding constant, and
2
(
t
)
2
denotes a subse-
quently defined state-dependent positive bounding function.
Remark 1: The adaptive weighting gain
"
can be selected arbitrarily
small to satisfy the conditions given in (32)–(34). Hence, the magnitude
of
k
v
(and thus
k
p
) can be made arbitrarily small, provided the combi-
nation of the gains is selected according to the initial conditions of the
states of the system due to the term
2
(0)
in (34). Since the selection
of the control gains depends on the initial conditions of the states, the
result in (31) is semiglobal.
Proof: Let
V
(
t
)
2
denote the following nonnegative function
(i.e., a Lyapunov function candidate):
V
(
t
)
1
2
_
q
T
M
(
q
)_
q
+
"
Tanh
T
(
e
)
^
J
(
q
)
M
(
q
)_
q
+
m
i
=1
k
p
ln(cosh(
e
i
)) +
1
2
~
T
0
0
1
0
~
+
1
2
~
T
J
0
0
1
1
~
J
:
(35)
The Raleigh–Ritz Theorem [11] can be used along with (4), (24), and
(27) to bound (35) by the following inequalities:
1
(
t
)
V
(
t
)
2
(
t
)
:
(36)
In (36), the positive functions
1
(
t
)
;
2
(
t
)
2
are defined as
1
(
t
)
1
2
m
1
0
"
2
m
2
k
_
q
k
2
+
m
i
=1
(
k
p
0
2
"
2
m
2
)ln(cosh(
e
i
))
+
1
2
min
0
0
1
0
k
~
k
2
+
1
2
min
0
0
1
1
k
~
J
k
2
(37)
2
(
t
)
1
2
+
"
2
m
2
k
_
q
k
2
+
m
i
=1
(
k
p
+2
"
2
m
2
)ln(cosh(
e
i
))
+
1
2
max
0
0
1
0
k
~
k
2
+
1
2
max
0
0
1
1
k
~
J
k
2
:
(38)
Based on (37), it is straightforward that if
"
is selected according to
(33), then
1
(
t
)
0
; hence, from (36)
V
(
t
)
0
.
After taking the time derivative of (35), the following simplified ex-
pression can be obtained:
_
V
(
t
)=
0
_
q
T
Y
~
0
"
Tanh
T
(
e
)
^
J
(
q
)
Y
~
+
k
p
_
q
T
Y
J
~
J
0
k
v
_
q
T
Tanh( _
q
)+
"
0
"k
p
^
T
J
Y
T
J
Y
J
^
J
0
"k
v
^
T
J
Y
T
J
Tanh( _
q
)
0
~
T
0
0
1
0
_
^
0
~
T
J
0
0
1
1
_
^
J
(39)
where (5), (12), (13), (17), and (18) were utilized, and the auxiliary
term
(
t
)
2
is defined as
d
dt
(
Y
J
^
J
)
M
(
q
)_
q
+
^
T
J
Y
T
J
(
_
M
(
q
)
0
V
m
(
q;
_
q
)) _
q:
(40)
Based on the form of (40), Properties 4, 5, and 8–10 can be utilized to
show that
k
k
k
_
q
k
2
(41)
where the positive constant
was introduced in (34). After utilizing
(13), (15), (22), (26)–(30), and (41), the following expression can be
obtained:
_
V
(
t
)
0
"k
p
min
f
^
J
^
J
T
gk
Tanh(
e
)
k
2
0
k
v
k
Tanh( _
q
)
k
2
+
"k
v
2
(
k
Tanh(
e
)
k
2
+
k
Tanh( _
q
)
k
2
)+
"
k
_
q
k
2
+
~
T
0
0
1
0
(
o
0
_
^
)+
~
T
J
0
0
1
1
(
1
0
_
^
J
)
(42)
492 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 52, NO. 3, MARCH 2007
Fig. 2. Control torque inputs.
By utilizing (14)–(16), the following advantageous expression can be
developed for the upper bound of (42):
_
V
(
t
)
0
k
v
2
k
Tanh( _
q
)
k
2
0
"
(
k
p
min
f
^
J
^
J
T
g0
k
v
2
)
k
Tanh(
e
)
k
2
0
k
v
2
0
"k
v
2
k
Tanh( _
q
)
k
2
+
"
k
_
q
k
2
:
(43)
Provided that the condition given in (32) and the following inequality
are both satisfied:
0
k
v
2
0
"k
v
2
k
Tanh( _
q
)
k
2
+
"
k
_
q
k
2
0
(44)
the expression in (43) can be used to prove that
_
V
(
t
)
0
. To facili-
tate further analysis, (25), (36), and (37) are used to obtain a sufficient
condition for (44) as
(
k
v
0
2
"k
v
2
)
2
"
V
(
t
)
1
2
m
1
0
"
2
m
2
+1
2
:
(45)
If the conditions in (32), (33), and (45) are satisfied, the inequality
in (43) can be used to obtain the following inequality:
_
V
(
t
)
0
k
k
2
(46)
where
2
is a positive constant, and
(
t
)
2
m
+
n
is
[Tanh
T
(
e
) Tanh
T
(_
q
)]
T
:
(47)
From (46), it is clear that
_
V
(
t
)
0
; therefore
V
(
z
(
t
)
;t
)
V
(
z
(0)
;
0)
2
(
z
(0)
;
0)
8
t
0
(48)
where
2
(
t
)
was defined in (38), and
z
(
t
)
2
4
is given by
z
k
_
q
k
2
m
i
=1
ln(cosh(
e
i
))
k
~
k
2
k
~
J
k
2
T
:
(49)
Based on (48), the final sufficient condition for (45) can be expressed
by the inequality in (34).
From (48), it is clear that
V
(
t
)
2L
1
; hence,
_
q
(
t
)
;e
(
t
)
;
~
(
t
)
;
~
J
(
t
)
;
(
t
)
2L
1
. Since
e
(
t
)
2L
1
, and the desired setpoint is
constant, it is clear that
x
(
t
)
2L
1
. Since the development is di-
rected at revolute robots,
q
(
t
)
only appears in
h
(
q
)
in (3) as an argu-
ment of bounded trigonometric functions; hence, it is typically unclear
how the boundedness of
q
(
t
)
can be proven. However, the bounded-
ness of
q
(
t
)
is typically not a concern since
q
(
t
)
only appears as an
argument of bounded trigonometric functions in the controller. From
(18), (14)–(16), (7), and the preceding arguments, it is clear that
_
^
(
t
)
;
^
(
t
)
;
_
^
J
(
t
)
;
^
J
(
t
)
;
(
t
)
2L
1
. Moreover, (3) and the facts that
_
q
(
t
)
;
J
(
q
)
2L
1
can be used to prove that
_
x
(
t
)
;
_
e
(
t
)
2L
1
; hence,
e
(
t
)
is uniformly continuous. From (46), (47), and the properties of the
hyperbolic tangent, it is clear that
_
q
(
t
)
;e
(
t
)
2L
2
[11]. Since
e
(
t
)
;
_
e
(
t
)
2L
1
and
e
(
t
)
2L
2
, Barbalat’s Lemma [29] can be invoked to
conclude the result in (31).
V. S
IMULATION
The controller developed in (11) and the adaptation law given in
(14)–(16) were simulated for a two-link robot planar manipulator. The
manipulator Jacobian is given by
J
=
0
L
1
sin(
q
1
)
0
L
2
sin(
q
1
+
q
2
)
0
L
2
sin(
q
1
+
q
2
)
L
1
cos(
q
1
)+
L
2
cos(
q
1
+
q
2
)
L
2
cos(
q
1
+
q
2
)
(50)
where
L
1
and
L
2
denote unknown link lengths. The dynamics of the
planar manipulator are
=
M
(
q
)
q
+
V
m
(
q;
_
q
)_
q
+
F
s
sgn(_
q
)
:
(51)
The vectors of uncertain constant kinematic and dynamic parameters,
J
2
2
and
2
2
, respectively, was found to be
J
=[
L
1
L
2
]
=[
F
s
1
F
s
2
]
where
F
s
1
;F
s
2
denote diagonal elements of
F
s
, and the initial param-
eter estimates were selected to be 20% of the actual values. The desired
