1987 Harashima - Tracking Control of Robot Manipulators Using Sliding Mode.pdf

170 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. PE-2, NO. 2, APRIL 1987
T = [Tl, , Tn I" is the generalized external force. If the control inputs i1H and u; are selected such that si
The Lagrangian L is expressed as and si always have the opposite signs in the neighborhood
L = K(, O)-P(H) (2) of si = 0, that is,
>
where P(6) denotes the potential energy of the system , 0 lim i < , (9)
and K(6, 0) is the kinetic energy which is written as
then sliding mode will occur on the switching surface si
K = IOTR(6)b (3) = 0. Therefore, the state trajectories of the system will
slide and remain on the surface, which implies that the
where the inertia matrix R (6) is symmetric and positive motion of the system (6) is dominated by the sliding equadefinite
for all 0.
tion (7). Consequently, the original system, though it may
Substituting (2) and (3) into (1), the equation of motion be higher order or nonlinear, behaves like a lower order
for robotic manipulators can be written as and linear system defined by the sliding equation si = 0.
(fn M( \ To
I
guarantee the existence condition (7) of the sliding
R(H)H + L 0.
0 -
mode, one need only know the bounds of system param-
e _R(6)&
Ki=1 36j / 2 a6 eters for determining the control inputs (8). This clearly
a means that the system is insensitive with respect to pa-
+ - P(O) = T. (4) rameter uncertainties and other disturbances.
Note that the second and third terms of the left side of the TRACKING CONTROL USING SLIDING MODE
equation represent the Coriolis and centrifugal forces, and
the last term denotes the gravitational forces. Equation (4) Two kinds of motions, the point-to-point (PTP) motion
can take a more brief form and the tracking motion, should be considered. Usually
the PTP motion may be regarded as a special case of the
R(O)H + M(0) t + g(6) = T (5) tracking motion. Therefore, only the tracking problem is
where
considered here. The tracking capability of advanced ma-
=}[06 0 0 022
T nipulators is indispensable for the smooth motion required
g = [,3P/0l, * * *aP/aOn]
in various industrial tasks, such as in an assembly line
T process. In tracking motion the values of desired position
Od(t), desired velocity Ad(t), and the desired acceleration
M = {mi,(6)}nx
of the manipulator on the entire path are provided by a
supervisory computer.
1 = ln(n + 1). In this paper a new approach is developed to tracking
motion control of manipulators.- Each arm is directly con-
SLIDING MODE CONTROL trolled in the manipulator's coordinates (joint space).
Moreover, the proposed method can be easily imple-
Sliding mode control based on the theory of variable mented in a task-oriented space
structure systems has a number of attractive features [7]. When manipulator motion is in low speed the super-
Therefore, it is widely applied to solve various control visory computer is able to neglect those quadratic terms
problems [11]-[ 15]. Consider the following system: 6,6A of
i
. Then (5) would become
H =f(O, t) + B(H, t)u (6) R(60)0 + g(0) = T. (10)
where 6 e Rn x I, fe Rn x', B E Rn xm, and u E Rmxl. To The desired input torque is computed from
obtain the desired dynamic performance of the system, a
set of sliding mode equations is previously selected as T = R(6) {I -Kd -Kpe} + g(6) (11)
si(0, t) = 0, i = 1, * , m (7) where Kd and Kp are constant gain matrices. Equations
which defines the switching surfaces. Then the control in- (10) and ( 1) give
puts with discontinuities have been decided according to R(6) { e + Kde + Kp e } = 0 (12)
the switching logic as where e = 6 -
(u+(6, t), if s460, t) < 0 Since the inertia matrix R(6) is nonsingular except in
u1(6, t) = _.(8) some special cases, we may choose Kd and K,p such that
yu, (6, t), If si(6, t) > 0 the characteristic roots of (12) have negative real parts,
where u1 (6, t) is the ith component of u. The preceding and the error e approaches zero asymptotically.
system with discontinuity control is termed a variable Notice that the effectiveness of the preceding control
structure system since the feedback structure of the sys- algorithm is grounded on the premise that the manipulator
tem is switching alternatively according to the state of the is in low-speed motion, whereas in most cases the manipsystem.
ulator is required to track preplanned trajectories in high