SPINAL INJURY CONTROL - Injury Biomechanics and Control: Optimal Protection from Impact - page 186

to the seat pan, provided that the peak magnitude of the spinal compressive
force and the peak magnitude of the displacement of the seat pan relative
to the airframe do not exceed prescribed values. There are no constraints
imposed on the control forces F and F 1 . The corresponding optimal control
problem is formulated as follows.

Problem 5.3 Optimal Control Problem for a System
with Two Shock Isolators

List of Variables for Problem 5.3

State variables x Displacement of the upper torso

relative to the airframe

x s Displacement of the seat pan 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 variables F Force produced by the seat cushion

F 1 Force produced by the shock isolator

between the seat pan and the airframe

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

K Stiffness coefficient of the pilot's

vertebral column

m 1 Mass of the pilot's upper torso

m 2 Mass of the pilot's lower torso

m s Mass of the seat pan

V Initial velocity of the airframe

τ Duration of the shock acceleration

pulse

Performance index J 1 Maximum magnitude of the

displacement of the lower torso
relative to the seat pan

Performance criteria
subjected to a constraint

J 2 Maximum magnitude of the vertebral

column's compressive force