Page 192 - Sensing, Intelligence, Motion : How Robots and Humans Move in an Unstructured World

MINIMUM TIME STRATEGY 167

of two one-dimensional problems, one for control p and the other for control q.
For details on obtaining such a solution and the proof of its sufficiency, refer to
Ref. 99.
The optimization problem is formulated based on the Pontryagin’s optimality
principle [100], with respect to the secondary frame (ξ, η). We seek to optimize
a criterion F, which signifies time. Assume that the trajectory being sought starts
at time t = 0 and ends at time t = t f (for “final”). Then, the problem at hand is
F(ξ(·), η(·), t f ) = t f → inf
ξ = p, p ≤ p max
¨
¨ η = q, q ≤ q max
ξ(0) = ξ 0 , η(0) = η 0 , ξ(0) = ξ 0 , ˙ η(0) =˙η 0
˙
˙
η(t f ) = η(t f ) = ξ(t f ) =˙η(t f ) = 0
˙
Analysis shows (see details in the Appendix in Ref. 99) that the optimal solu-
tion of each one-dimensional problem corresponds to the “bang-bang” control,
with at most one switching along each of the directions ξ and η, at times t s,ξ and
t s,η (“s” stands for “switch”), respectively.
The switch curves for control switchings are two connected parabolas in the
phase space (ξ, ξ),
˙
ξ ˙ 2
˙
ξ =− , ξ> 0
2p max
ξ ˙ 2
˙
ξ = , ξ< 0 (4.16)
2p max
and in the phase space (η, ˙η), respectively (see Figure 4.9),

˙ η 2
η =− , ˙ η> 0
2q max
˙ η 2
η = , ˙ η< 0 (4.17)
2q max
The time-optimal solution is then obtained using the bang-bang strategy for ξ
˙
and η, depending on whether the starting points, (ξ, ξ) and (η, ˙η), are above or
below their corresponding switch curves, as follows:

α 1 · p max , 0 ≤ t ≤ t s,ξ
ˆ p(t) =
α 2 · p max ,t s,ξ <t ≤ t f

α 1 · q max , 0 ≤ t ≤ t s,η
ˆ q(t) = (4.18)
α 2 · q max ,t s,η <t ≤ t f

where α 1 =−1,α 2 = 1 if the starting point, (ξ, ξ) s or (η, ˙η) s , respectively, is
˙
above its switch curves, and α 1 = 1,α 2 =−1 if the starting point is below its