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312 Autonomous Mobile Robots

˙
Having substituted ˜z in (8.6), ¨z in (8.44), ¨z d in (8.47), and ˜z in (8.50) into
(8.43), we obtain

˙ ¨
E(θ, γ)u = F dyn (θ, v, ˙v, γ , ω, d, d, d, φ, ˙ φ, ¨ φ) (8.51)

where

2
˙
F dyn =¨z d − 2ξλ˜z − λ ˜z − H(θ, γ)µ
T ˙ ¨
¯
= R (θ)F dyn (v, ˙v, γ , ω, d, d, d, φ, ˙ φ, ¨ φ) (8.52)
with

   2 2 
˙ v ( ˙ θ + pω) − λ
T
¯
F dyn =  1 + f  + lR (pγ)  vω 
˙ v tan γ − 2 − 2ξλ( ˙ θ + pω)
2 a cos γ
2 2
¨
d + 2ξλd + λ d − d( ˙ θ + ˙ φ)
˙
T
+ fR (φ) (8.53)
˙
d( ¨ θ + ¨ φ) + 2(d + ξλd)( ˙ θ + ˙ φ)
Multiplying the orthogonal matrix R(θ) to both sides of (8.51) produces
˙ ¨
¯
E(γ )u = F dyn (v, ˙v, γ , ω, d, d, d, φ, ˙ φ, ¨ φ) (8.54)
¯
Conditions that satisfy (8.39) for the look-ahead tracking mode or (8.40)
for the look-behind tracking mode guarantee that the decoupling matrices E(γ )
¯
and E(θ, γ) are invertible. Under those conditions, the dynamics-based vehicle-
following controller can be achieved

−1 ˙ ¨
¯
u input = E ¯ (γ )F dyn (v, ˙v, γ , ω, d, d, d, φ, ˙ φ, ¨ φ) (8.55)
T
where u input =[u m u s ] , with
cos(((1 + f )/2)γ ) 2 2
u m =˙v + {l[( ˙ θ + pω) − λ ]
cos[(p − ((1 + f )/2))γ ]
2
2
˙
¨
+ f [d + 2ξλd + λ d − d( ˙ θ + ˙ φ) ] cos(pγ − φ)
˙
+ f [d( ¨ θ + ¨ φ) + 2(d + ξλd)( ˙ θ + ˙ φ)] sin(pγ − φ)} (8.56)

© 2006 by Taylor & Francis Group, LLC

FRANKL: “dk6033_c008” — 2006/3/31 — 16:43 — page 312 — #18