Page 322 - Autonomous Mobile Robots

310 Autonomous Mobile Robots

• Backward tracking: f =−1


v d < 0


λ> 0

(8.40)
−d max < l < 0


 π
− < p < 0
2γ max
Proof The conditions in Lemmas 8.1 and 8.2 guarantee the existence of the
control laws (8.37) and (8.38).
Combining the target performance specification, the conditions for tracking
convergence equivalence in Lemma 8.1, the conditions for the existence of
control input in Lemma 8.2, and the necessary conditions for the tracking
stability in Lemma 8.3, we obtain


Target dynamics ⇒ {λ> 0 
π
  λ> 0

 
  |p| <  π
 
Lemma 8.1 ⇒ 2γ max  |p| <



2γ max
 

  
0 < fl < d max
 
 
 
 0 < fl < d max
lp = 0
⇒ 1 + f π
Lemma 8.2 ⇒ 1 + f π  p− <


 
  p − <  2 2γ max
 

 2 2γ max 
 
 lp > 0

lp > 0

 
 
 Lemma 8.3 ⇒  fv d > 0

fv d > 0

(8.41)
For f = 1, conditions (8.41) will lead to (8.39)
For f =−1, conditions (8.41) will lead to (8.40)
8.3.2 Dynamics-Based Tracking Controller
If the access to the torques/forces, or their corresponding convertible accelera-
tions, of the vehicle control is available, the controller can be developed based
on the dynamic model (8.3).
In this case, the target performance of the vehicle tracking maneuvers can
be specified by a second-order system for the closed-loop output tracking error
2
˙
¨ ˜ z + 2ξλ˜z + λ ˜z = 0 (8.42)
where the natural frequency λ> 0 and the damping ratio ξ> 0.
© 2006 by Taylor & Francis Group, LLC
FRANKL: “dk6033_c008” — 2006/3/31 — 16:43 — page 310 — #16