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214 Autonomous Mobile Robots

(a) 2

1.5
1

0.5

0

–0.5
–1

–1.5
0 1 2 3 4 5 6
(b) 4
3
2
1
0
–1
–2
–3
–4
–5
–6
0 1 2 3 4 5 6

FIGURE 5.3 (a) Time histories of x(t) (solid), y(t) (dashed), φ(t) (dash-dotted), and
θ(t) (dotted). (b) Translational (solid) and rotational (dashed) velocity controls.

disturbances. In what follows we perform a very simple robustness analysis,
with reference to an interesting situation, namely in the presence of external
disturbances and model errors, and for a prototype system.
Consider a three dimensional chained system perturbed by a constant
nonzero disturbance entering the third equation, 17 that is,

˙ x 1 = u 1 , ˙ x 2 = u 2 , ˙ x 3 = x 2 u 1 + d (5.31)

17 The disturbance models a violation of the nonholonomic constraint, that is, x 2 ˙x 1 −˙x 3 = 0.

© 2006 by Taylor & Francis Group, LLC

FRANKL: “dk6033_c005” — 2006/3/31 — 16:42 — page 214 — #28