In this paper, we propose a novel controller that achieves trajectory following vithout requiring exact knowledge of the nonlinear dynamics described by
Lyapunov function candidate
(2 .6)
X + *K t *I1
£TP£
I.
Our control has the form
fc K(q)('%d + KvS + Kpe)
-
+ V(q,q)q +
Fdq
(2.1)
where
p- [.x aI 5
(i)
where K}
and
Kp
l
qd- q.
(2.2)
After
.i2
<TP < A2
0l12
Note that our controller has compensated for all of the nonlinear dynamics except for the gravity, static friction, and disturbance After substituting the terms since all these terms are bounded. control (2.1) into the system equation (1.1), we obtain the error system j- Ag +B
are nxn positive definite matrices defined as
By Lema D.1 (Appendix D) if condition (2.5) is satisfied then:
V is positive definite since the symetric matrix P is positive definite, and
A
(ii) the bound A3 defined by
ATQ
V
with
|
<
-
A3 Iiit
in (2-4)
is as gi
From (2.7), we can state that
-3
t2 citTPit c
FP2 A
2:15
SDK
FIIfD LTAPDNOV-MSD
CJUOLLU
FM
A ROBOT
M
PUIATOR
D. M. Dawson and F. L. Lewis School of Electrical Engineering Georgia Institute of Technology Atlanta, GA 30332 404-894-2994
Theorea 2.1
Given the system (1.1) with control (2.1) and with e(t0) - 0, the velocity and position errors can be bounded withinius R. This bound is given by
Appendix A
In this appendix, we obtain condition on k , k and a to ensure that the mtrix P is positive definite and to establish a values for A1 and 12 which are defined by
on
the controller
where
(0
I kp > 1,
a c 1,
and
kv >
1
(2.5)
and a is an arbitrary scalar chosen to meet the above conditions.
II. au.j.a n
cDTNID Cs
Select the
V
-
In [4], Craig notes that the gravity and static friction terms in (1.1) are bounded independently of q and 4. In this section, we design a controller that exploits this known bound.
A3
lL. IT I.
+ F5(q)
+
TO | | [ I]M-(q)[G(q) Al - G1 - *)/2, A2 (kP + kv + a)/2,
-
Td14|11
and
13
gains
-
min(akp kv
- a)
Furthermore, we require the follow conditions for the above bound to be valid:
f
-
If the control gains in (2.3) are selected such that the error system is asymtotically stable, the tracking error will be bounded since the input is bounded (i.e. bounded input-bounded output). In certain applications, it may advantageous to obtain a quantitative A measure of the measure of the maxima error during tracking. tracking error can be obtained by applying the following theorem.
I. XITIc5
A
u-
. p
]
.-[:I,
B
-
and
M-l(G(q) + F (q) +Td).
If all the dynmics are exactly known for the control of a robot mipulator, the computed-torque controller is known to be asymptotically stable in following a desired trajectory [1]. On the other hand, all of the dynamics are not needed if the control objective is only to drive the robot to a fixed final position without specifying a trajectory. Indeed, in 12J a controller of siplified structure was designed for a desired setpoint control that did not require any of the nonlinear terms except for the gravity terms.
(i) V is negative definite since the syeetric matrix Q it positive definite, and
[7] S. Barnett, Matrices in Control Theory, Malabar Fea.: Robert E. Krieger Publishing Company, 1984.
Kv -kI v
and
K
p
-
p.
I,
f is a nxl input control vector, q is a nil vector representing the desired trajectory, and the joint error e is defined as
e
-
(ii) the bounds Al and 12 defined by
u
are as given in (2.4)
taking the derivative of (2.6) along the solution, we obtain
(2 .7)
2K-CB w V~- -aTO +T2pBu l her
(2.3)
where
where
Q
-
0
Kv
- al
2337
By LesD.1 (Appendix D) if condition (2.5) is satisfied then:
g(t)
where
s R
for every
t e
NI(q)q
+
V(q,q)q
+
G(q)
+
Fdq
+
F(q)
+ Td
(1. 1)
[tot t1j
(2.4)
R
-
where M(q) is anm inertia matrix (bounided, syetric, and positive matrix containing the centrifugal and definite), V(q,q. is a mxn Coriolis terms, 0(q) is a nxl vector containing the gravity terms, Fd is a constant diagonal nxn matrix containing the dynamic coefficients of friction, F (q) is a nil vector containing the static friction terms, Td is an nxl vector representing an unknown bounded disturbance, q(t) is a il joint variable vector, and f is a nxl input vector. Equation (1.1) describes robot mnipulators in the Lagrange- Euler formulation [3].