ABSTRACT
Sliding mode design [1-6] has several features that make it an attractive technique to solve tracking problems in motion control of robotic manipulators, the most important being its robustness to parametric uncertainties and unmodelled dynamics, and the computational simplicity of the algorithm. One way of looking at sliding mode controller design is to think of the design process as a two-step procedure. First, a region of the state space where the system behaves as desired (sliding surface) is defined. Then, a control action that takes the system into such surface and keeps it there is to be determined. Robustness is usually achieved based on a switching control law. The design of the control action can be based on different strategies, a straightforward one being to define a condition that makes the sliding surface an attractive region for the state vector trajectories. Consider a nonlinear affine system of the form:
x(ni )i = fi (x) + m∑
j=1 bi j (x)u j , i = 1, . . . , m, j = 1, . . . , m (18.1)
where u = [u1, ..., um]T is the vector of m control inputs, and the state x is composed of the xi coordinates to be tracked and their first (ni − 1) derivatives. Such systems are called square systems because they have as many control inputs u j as outputs to be controlled xi [3]. The motion control problem to be addressed is the one of making the state vector x track a desired trajectory r . Consider first the time-varying manifoldσ given by the intersection of the surfaces si (x , t) = 0, i = 1, . . . , m, specified by the components of
S(x , t) = [s1(x , t), . . . , sm(x , t)]T , where
s (x , t) = (
d
dt + λi
)ni −1 (xi − ri ) = 0 (18.2)
which can be computed from x andr , λι being some positive constant. Such manifold is usually called sliding surface; any state trajectory lying on it tracks the reference r since Equation (18.2) defines a differential equation on the error vector that stably converges to zero. An integral term can be incorporated to the sliding surface to further penalize tracking error. For instance, Equation (18.2) can be rewritten for ni = 3 as
s (x , t) = (
d
dt + λi
)2 t∫ 0
(xi − ri )dt = (x˙i − r˙i ) + 2λi (xi − ri ) + λ2i
(xi − ri )dt = 0 (18.3)
There are also several possible strategies to design the control action u j that takes the system to the sliding surface. One such strategy is to find u j in a way that each component of the sliding surface si is an attractive region of the state space by forcing the control action to satisfy a geometric condition such as
d
dt
( s 2i ) ≤ −ηi |si | ⇔ si dsi
dt ≤ −ηi |si | ≤ 0 (18.4)
whereηi is a strictly positive constant. This condition forces the squared distance to the surface (as measured by s 2i ) to decrease along all state trajectories [3]. In other words, all the state trajectories are constrained to point toward σ . A simple solution that satisfies the sliding condition (18.4) in spite of the uncertainty on the dynamics (18.1) is the switching control law:
usw = k(x , t)sgn(s ), sgn(s ) = {
1, s > 0
−1, s < 0 (18.5)
The gain k(x , t) of this infinitely fast switching control law increases with the extent of parametric uncertainty, introducing a discontinuity in the control signal called chatter. This is an obviously undesirable practical problem that will be discussed in Section 18.3.
