This section presents a nonlinear control method, apparently first proposed in (Paul, 1972) and named the computed torque method in (Markiewicz, 1973) and (Bejczy, 1974). This method is based on using the dynamic model of the system in the control law formulation. Such a control formulation yields a controller that suppresses disturbances and tracks desired trajectories uniformly in all configurations of the system (Craig, 1988).

Suppose that the system's dynamics is governed by Eq. (1). The control objective is to track a desired trajectory qd . Such a trajectory may be preplanned by several well-known schemes (Craig, 1989). We define a tracking error q q = qd - q> (2)

and make the following proposition. Proposition 2.1 The control law

can track any desired trajectory qd, as long as the matrices H, C, and G are known to the designer. The servo law, u , is given by u = qd+Kvq + Kpq, (4)

where Kv and Kp are called servo gain matrices. Proof.

Substituting the proposed control law into the equation of motion, Eq. (1), we obtain

H(q)q + C(q, q)q + G(q) = H(q)(qd + Kvq + Kpq) + C(q, q)q + G(q), which yields the following error dynamics q +Kvq + Kpq = 0.

A proper choice of the servo gain matrices will lead to a stable error dynamics. One such example is given by the following matrices

where Aj are adjustable design parameters. ■■

It can be seen that this control formulation exhibits perfect tracking for any desired trajectory. But this desired performance is based on the underlying assumption that the values of parameters appearing in the dynamic model in the control law match the parameters of the actual system, which makes the implementations of the computed torque control less than ideal due to the inevitable uncertainties of the system, e.g. resulting from unknown hydrodynamic coefficients. In the existence of uncertainties, the control law (3) must be modified to

where [•] denotes the estimation of matrix [•] . One can show that substitution of the above control law into the equation of motion will lead to the following error dynamics q + Kvq + Kpq = H-1T, (8)

where T = Hq + Cq + G , and the tilde matrices are defined by [•] = [•] - [•]. Since the right hand side of the error dynamics is not zero anymore, this method becomes inefficient in the presence of uncertainties. This problem is conquered by the adaptive counterpart of the computed torque control method.

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