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260 REFERENCES

s instead of 2 s. Plot as well the Cartesian position (x 2(t), y 2(t)) of the end
effector in base coordinates.
(b) Redo the simulation deleting the lines in Figure 4.5.7 that zero the initial
velocity estimates.
(c) Try to simulate the digital CT controller using the alternative technique
to compute k from and v k, as given in equation (1) in the example.
4.5–2 Digital Control Simulation. Convert the PD-gravity CT controller in Example
4.4.3 to a digital controller. Try several sample periods.
4.5–3 Error Dynamics for Digital Control. Find the error system in Table 4.4.1
using digital control of the form (4.5.3).
4.5–4 Antiwindup Protection. In Example 4.4.4 we saw the deleterious effects in
robot control of integrator windup due to actuator saturation. In Example
4.5.2 we showed how to implement antiwindup protection on a simple PI
controller. Implement antiwindup protection on the robot controller in
Example 4.4.4. The issue is determining the limits on the integrator outputs
given the motor torque limits. Successful and thorough completion of this
problem might lead to a nice conference paper.
Section 4.6
4.6–1 Optimal LQ Outer-Loop PD Gains. Verify (4.6.15). To do this, select the
Riccati solution matrix as

with P i R Substitute P, A, B, Q, R into the Riccati equation (4.6.6). You will
n
obtain three n×n equations that can be solved for P i . Now use (4.6.5).
4.6–2 Robust Control Using LQ Outer-Loop Design. Redo the PDgravity simulation
in Example 4.4.3 using PD gains found from LQ design. Does the LQ
robustness property improve the responses found in Example 4.4.3 using
nonoptimal gains? Try various choices for Q p , Q v , and R, both diagonal and
nondiagonal.
Section 4.7
4.7–1 Direct Cartesian Computed-Torque Design. Begin with the Cartesian
dynamics in Section 3.5 and design a computed-torque controller. Compare
it to (4.7.10).
4.7–2 Approximate Cartesian Computed-Torque. Derive the error system dynamics
associated with the approximate control law (4.7.11).
4.7–3 Cartesian PD-Plus-Gravity Control. Repeat Example 4.4.3 using Cartesian
computed-torque control, where the trajectory is given in workspace
coordinates.

Copyright © 2004 by Marcel Dekker, Inc.