SPINAL INJURY CONTROL - Injury Biomechanics and Control: Optimal Protection from Impact - page 195

The optimal time history of the absolute acceleration of the seat pan to
minimize the peak magnitude of the spinal compressive force will be found,
provided that the maximum magnitude of the displacement of the seat pan
relative to the airframe, measured by

J 1 (u s ) = max

t

| y(t) | , (5.88)

does not exceed a prescribed quantity D . The corresponding optimal control
problem is stated as follows:

Problem 5.5 Optimal Control Problem for the MADYMO Model

List of Variables for Problem 5.5

State variables y Displacement of the seat pan relative

to the airframe

z Displacement of the airframe relative

to the inertial reference frame

Control variable u s Absolute acceleration of the seat pan

External disturbance v Shock acceleration pulse, the

negative of the acceleration of the
airframe, v =− z

Functions and
parameters of the model

V Initial velocity of the airframe

τ Duration of the shock acceleration

pulse

Performance index J 2 Maximum magnitude of the vertebral

column's compressive force

Performance criterion
subjected to a constraint

J 1 Maximum magnitude of the

displacement of the seat pan relative
to the airframe

Constraint D Maximum allowable value for the

criterion J 1

Optimal solution J 2 ( u s 0 ) Optimal value of the performance

index J 2

u s 0 Optimal control u s

For a system for which the dynamics are simulated by the MADYMO
model of a seated occupant supplemented with the equations

y ¨ +¨ z = u s , z ¨ =− v(t) (5.89)

(continued)