What is the damping ratio of the closed loop system and

Question

1. The open loop dynamics of a single direct-drive joint actuator for a mechanical manipulator is shown in Figure 1 below.

Fig 1: Open loop Dynamics

The transfer function is given by:

G(s) = θ^·(s)/u(s) = K_cK_eK_aK_t/(JLs² + J(R + K_i)s + K_tK_b)

If K_i is adjusted such that J(K_i + R)² >> 4K_TK_BL the transfer function may be approximated by:

G(s) = K_cK_eK_a/K_b/((τ_ms + 1)(τ_es + 1))

where τ_m = J(R + Ki)/K_TK_b and τ_e = L/(R + Ki)

The values of the parameters are given as:

K_a = 20 K_b = 0.5 (V/rad/s)
K_e = 159.15 (bit/rad) R = 1 Ω
K_c = 0.00448 V/bit L = 0.01 H
K_t = 0.5 Nm/A J = 0.02 Kgm²

a. Determine the value of Ki such that Τ_m = 10Τ_e. Hence we can ignore the electrical time constant (i.e. Τ_e ≈ 0). Determine the error proportional gain of Figure 2 to have closed loop natural frequency of 10 Hz.

b. What is the damping ratio of the closed loop system?

c. What is the steady state error for a constant velocity input of 10000 BLU's?

d. What is the peak magnitude error at steady-state for a sinusoidal input of the form 1000×sin(2Πt) BLU's?

e. In order to have the same natural frequency but increase the damping ratio of the closed loop system, and analog velocity feedback loop is included as shown in Figure 3. To have a damping ratio of 0.707 what should be the value of the velocity feedback gain? What are the new values for part c, and d?

Fig. 2: Error Proportional Controller

Fig. 3: Error Proportional Controller with Analog Velocity Feedback