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168 ACCOUNTING FOR BODY DYNAMICS: THE JOGGER’S PROBLEM

x
Switch h
curve
Start
position

Point of
switching
dx dh
dt dt
Final
Final position
position Point of
switching
Start
position
Switch
curve
Figure 4.9 (a) The start position in the phase space (ξ, ˙ ξ) is above the switch curves.
(b) The start position in the phase space (η, ˙η) is under the switch curves.

switch curves. For example, if the initial conditions for ξ and η are as shown in
Figure 4.9, then

−p max , 0 ≤ t ≤ t s,ξ
ˆ p(t) =
+p max ,t s,ξ <t ≤ t f

+q max , 0 ≤ t ≤ t s,η
ˆ q(t) = (4.19)
−q max ,t s,η <t ≤ t f
where the caret sign (ˆ) refers to the parameters under optimal control. The
time, position, and velocity of the control switching for the ξ components are
described by

( ˙ ξ 0 ) 2 ˙
2 + ξ 0 p max + ξ 0
t s,ξ =
p max
ξ ˙ 2 0 ξ 0
ξ s = +
4p max 2

ξ ˙ 2
˙
ξ s =− 0 + ξ 0 p max
2p max
and those for the η components are described by

(˙η 0 ) 2
− η 0 q max −˙η 0
2
t s,η =
q max