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(b) RAC without disturbance compensation

(b) RAC without disturbance compensation

Fig. 9. Experimental results of RAC method with and without disturbance compensation

5. Discrete-time RAC

In practical systems digital computers are utilized for controllers, but there is no discrete-time control method for UVMS except our proposed methods (Sagara, 2003; Sagara et al., 2004; Sagara et al., 2006; Yatoh & Sagara, 2008). In this section, we address discrete time RAC methods including the ways of disturbance compensation of the vehicle and avoidance of singular configuration of the manipulator.

5.1 Discrete-time RAC law

Discretizing Equation (16) by a sampling period T, and applying fi(k) and W(k) to the backward Euler approximation, the following equation can be obtained:

W (k)a(k -1) = T {(k) - v (k -1) + Ty{k) - [w (k) - W (k -1)])} (42)

where v = [vj vj ]T . Note that a computational time delay is introduced into Equation (42), and the discrete time kT is abbreviated to k . For Equation (42), the desired acceleration is defined as a (k) = T W #( k){ (k +1) - vd (k) +Aev( k) + Ty(k)} (43)

and Vd (k) is the desired value of v(k), A = diag^ } (i = 1, • • •, 12 ) is the velocity feedback gain matrix.

From Equations (42) and (43) we have

TW (k)ea(k -1) = ev(k) - ev(k -1) + Aev (k) - Tfr(k) - y(k -1)}+ {w (k) - W (k -1) jZ(t) (45)

Assuming W(k) « W(k -1) and y(k) « y(k -1) for one sampling period, Equation (45) can be rewritten as

where q is the forward shift operator. Since all elements of W(k) are bounded, if is selected to satisfy 0 < ^ < 1 and the convergence of ea(k) tends to zero as k tends to infinity, the convergence of ev(k) to zero as k tends to infinity can be ensured from Equation (46).

Moreover, the desired velocity of v(k) is defined as

Vd (k) = T S0e (k){ d (k) - x d (k -1)+rex (k -1)} (47)

where

r0 W0 Pe We and Xd(k) is the desired value of x(k)( = [xQ xq]T ), r = diagY} (i = 1, ••• ,12) is the position feedback gain matrix.

From Equations (44) - (47) the following equation can be obtained:

where v(k) is applied to the backward Euler approximation. From Equation (48), if yi is selected to satisfy 0 <y < 1 and the convergence of ev(k) tends to zero as k tends to infinity, the convergence of ex (k) to zero as k tends to infinity can be ensured. The configuration of the control system described in this subsection is shown in Figure 10.

Fig. 10. Control system of discrete-time RAC
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