A coordinate measuring machine (CMM) measures coordinates on three-dimensional objects. The accuracy of CMMs is affected by temperature changes as well as by mechanical resonances due to joint elasticity. These resonances are more pronounced when the machine has to go over abrupt changes of dimension, such as sharp corners at high speed. Each of the machine links can be controlled in a closed loop configuration, such as the one shown in Figure P9.12, designed for a specific machine with prismatic (sliding) links. In the figure, X_ref (s) is the commanded position and X(s) is the actual position. The minor loop uses a tachometer generator to obtain the joint speed, while the main loop controls the joint’s position.

a. Find the value ofKthat will result in a minor loop with ζ = 0.5.

b. Use a notch filter compensator, G_c(s), for the external loop so that it results in a closed-loop damping factor of ζ = 0.7 with T_s ≈4 seconds.

c. Use MATLAB to simulate the compensated system’s closed loop step response.

Plant Compensator Xef (s) + 574.98 X(s) GAs) K + 14.24 s+3447.91) Tachometer generator 0.0466 s(s2 + 1.15s + 0.33) FIGURE P9.12