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258 REFERENCES

Section 4.4
4.4–1 Prove (4.4.32).
4.4–2 PD Computed-Torque Simulation. Repeat Example 4.4.1 using various values
for the PD gains. Try both critical damping and underdamping to examine
the effects of overshoot on the joint trajectories.
4.4–3 Classical Joint Control. Prove (4.4.55), (4.4.57), (4.4.60), and (4.4.62). See
[Franklin et al. 1986].
4.4–4 PD Computed Torque with Payload Uncertainty. The CT controller is
inherently robust. In Example 4.4.1, suppose that m 2 changes from 1 kg to 2
kg at t=5 s, corresponding to a payload mass being picked up. The CT
controller, however, still uses a value of m 2=1. Use simulation to plot the
error time history. Does the performance improve with larger PD gains?
4.4–5 PID Computed Torque with Payload Uncertainty. Repeat Problem 4.4–4
using a PID outer loop. Does the integral term help in rejecting the mass
uncertainty?
4.4–6 PD Computed Torque with Friction Uncertainty. Repeat Problem 4.4–4
assuming now that m 2=1 kg stays constant and is known to the controller.
However, add friction of the form F(q, ) = Fv +K dsgn( )(see Table 3.3.1) to
the arm dynamics, but not to the CT controller. Use v i=0.1, k i=0.1. Simulate
the performance for different PD gains.
4.4–7 PID Computed Torque with Friction Uncertainty. Repeat Problem 4.4–6
using a PID outer loop.
4.4–8 PD Computed Torque with Actuator Dynamics
(a) Design a CT control law for the two-link planar elbow arm with actuator
dynamics (Section 3.6) of the form

Take the link masses and lengths as 1 kg, 1 m. Take motor parameters
2
of J m =0.1 kg-m , b m =0.2 N-m/rad/s, and R=5Ω, Set the gear ratio

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