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REFERENCES 257

PROBLEMS

Section 4.2
4.2–1 Minimum-Time Control. Derive the minimum-time control switching time t s
[cf. (4.2.11)] when the initial and final velocities are not zero.
4.2–2 Polynomial Path Interpolation. It is desired to move a single joint from q(0)=0,
(0)=0 through the point q(l)=5, (l)=40 to a final position/velocity of (2)=10,
Determine the cubic interpolating polynomials required in this two-interval
path. Plot the path generated and verify that it meets the specified requirements
on q(t) and (t) Plot (t) versus q(t).
4.2–3 LFPB. Repeat Problem 4.2–2 using LFPB.

4.2–4 Polynomial Path for Acceleration Matching. Derive the interpolating
polynomial required to match positions, velocities, and accelerations at the
via points.
Section 4.3

4.3–1 Simulation of Flexible Coupling System. Use computer simulation to reproduce
the results for the motor with flexible coupling shaft in Example 3.6.1.

4.3–2 Simulation of Nonlinear System. The Van der Pol oscillator is a nonlinear
system with some interesting properties. The state equation is

Simulate the dynamics for initial conditions of x 1 (0)=0.1, x 2 (0)= 0.1. Use
values for the parameter of α=0.1 and then α=0.8. Plot x 1 (t) and x 2 (t), as well
as x 2 (t) vs. x 1 (t) in the phase plane. For each simulation you should clearly see
the limit cycle that is characteristic of the Van der Pol oscillator.

Copyright © 2004 by Marcel Dekker, Inc.