SPINAL INJURY CONTROL - Injury Biomechanics and Control: Optimal Protection from Impact - page 189

The performance criteria of Problem 5.3 in terms of these variables are
expressed as

J 1 = max

t ∈ [0 , ∞ ) |

x 2 (t) − x 1 (t) | , (5.65)

J 2 =

t ∈ [0 , ∞ ) | C y 1 (t) + Ky 1 (t) | ,

max (5.66)

=

t ∈ [0 , ∞ ) |

− |

J 3 max x 1 (t) x 0 (t) . (5.67)

The criteria J 1 and J 3 coincide with those of Eqs. (4.161) and (4.163) in
Problem 4.8, respectively. The criterion J 2 of Problem 5.3 can be identi-
fied among the criteria J s in Problem 4.8. The criterion J 2 of Problem 4.8
[Eq. (4.162)] is absent from Problem 5.3. Formally, this criterion can be
included among the constrained criteria in Problem 5.3, with an infinitely
large upper bound specified for it. Thus, Problem 5.3 is reducible to Prob-
lem 4.8 and, hence, can be solved by the technique that is described in
Section 4.2.3 and is based on Proposition 4.5. This reduction, however, is
unnecessary, and the solution can be described proceeding directly from
Problem 5.3.

Solution of Auxiliary Problem

Problem 5.4 Auxiliary Problem for Problem 5.3

List of Variables for Problem 5.4

State variables x Displacement of the upper torso

relative to the airframe

y Displacement of the lower torso

relative to the airframe

z Displacement of the airframe relative

to the inertial reference frame

Control variable F Force produced by the seat cushion

External disturbance v Shock acceleration pulse, the negative

of the acceleration of the airframe,
v =− z

Functions and
parameters of the model

C Damping coefficient of the pilot's

vertebral column

(continued)