(b) RAC without disturbance compensation

(b) RAC without disturbance compensation

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

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.

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.

Was this article helpful?

This first volume will guide you through the basics of Photoshop. Well start at the beginning and slowly be working our way through to the more advanced stuff but dont worry its all aimed at the total newbie.

## Post a comment