P6.8 A very interesting and useful velocity control system has been designed for a wheelchair control system. A proposed system utilizing velocity sensors mounted in a headgear is shown in Figure P6.8. The headgear sensor provides an output proportional to the magnitude of the head movement. There is a sensor mounted at 90° intervals so that forward, left, right, or reverse can be commanded. Typical values for the time constants are τ τ 1 2 = = 0.5 s, 1 s, and

τ3 = 1 4 s.

(a) Determine the limiting gain K K = 1 2 K K3 for a stable system.

(b) When the gain K is set equal to one-third of the limiting value, determine whether the settling time (to within 2% of the final value of the sys

tem) is Ts ≤ 4 s.

(c) Determine the value of gain that results in a sys

tem with a settling time of Ts ≤ 4 s. Also, obtain the value of the roots of the characteristic equa

tion when the settling time is Ts ≤ 4 s.

Sensor Wheelchair Amplifier in hat dynamics Head Desired nod K K3 K velocity 75+1 (T25+ 1)(738+1) Velocity FIGURE P6.8 Wheelchair control system.