Page 320 - Autonomous Mobile Robots

308 Autonomous Mobile Robots

8.3.1 Kinematics-Based Tracking Controller
The target performance of the vehicle tracking maneuvers can be specified by
a first-order system for the closed-loop output tracking error

˙
˜ z + λ˜z = 0 (8.29)
where the convergence rate λ> 0 can be specified for a desired target
performance.
Equation (8.29) can then be rewritten equivalently as

˙ z =˙z d − λ˜z (8.30)

Time differentiation of (8.4) leads to

v + f d cos φ − fd( ˙ θ + ˙ φ) sin φ
 ˙ 
T
˙ z d = R (θ)  1 + f  (8.31)
˙
v tan γ + f d sin φ + fd( ˙ θ + ˙ φ) cos φ
2
By substituting (8.6), (8.13), and (8.31) into (8.30), we have

E(θ, γ)µ = F kin (θ, v, γ , d, d, φ, ˙ φ) (8.32)
˙
where

T
¯
˙
F kin =˙z d − λ˜z = R (θ)F kin (v, γ , d, d, φ, ˙ φ) (8.33)
with
v − λl cos pγ ˙
 
T
¯
F kin =  1 + f  + fR (φ) d + λd (8.34)
v tan γ − λl sin pγ d( ˙ θ + ˙ φ)
2
Multiplying the orthogonal matrix R(θ) to both sides of (8.32) produces
˙
¯
¯
E(γ )µ = F kin (v, γ , d, d, φ, ˙ φ) (8.35)
With the parameters l and p chosen satisfying Lemma 8.2, the resultant
nonlinear kinematics-based controller can be obtained from (8.35) by using
(8.15) and (8.34)

˙
¯
¯
µ input = E −1 (γ )F kin v, γ , d, d, φ, ˙ φ (8.36)

© 2006 by Taylor & Francis Group, LLC

FRANKL: “dk6033_c008” — 2006/3/31 — 16:43 — page 308 — #14