BASIC OPTIMAL SHOCK ISOLATION: SINGLE-DEGREE-OF-FREEDOM SYSTEMS - Injury Biomechanics and Control: Optimal Protection from Impact - page 86

an emergency landing. Suppose that the helicopter hits the ground with a
vertical velocity that is high enough to cause serious injuries to the pilot
and passengers but not high enough to destroy the vehicle. To reduce the
severity of injuries, it is reasonable to equip helicopter seats with actively
controlled impact isolators. In this case, it is possible to predict the velocity
and time of landing based on the measurement of the helicopter's height
and the rate of descent. The situation is similar for automobiles, when, for
instance, an obstacle that cannot be avoided is in the path of travel. In this
case, the only means to reduce the severity of injuries to the occupants
is effective isolation from the impact. The time and velocity of impact
can be calculated beforehand using the distance to the obstacle and the
velocity of the automobile relative to the obstacle. The automobile must
be equipped with appropriate sensors that can measure the distance to an
object that is in front of the automobile and the velocity of the automo-
bile relative to this object as well as with a microcomputer to process this
information and to form control signals that can activate the shock isolation
system.
To formulate optimal control problems for pre-acting shock isolators,
assume that the controller is able to start acting for a time t ∗ before the dis-
turbance occurs. Accordingly, the equation of motion of the object relative
to the base,

x ¨ = u + v(t), (3.125)

will be subjected to the initial conditions

− = ˙ − =

t ∗ ≥ x( t ∗ ) 0 , x( t ∗ ) 0 , 0 (3.126)

and the disturbance v(t) will be assumed to be identically zero on the time
interval − t ∗ ≤ t< 0, that is,

v(t) ≡ 0f r − t ∗ ≤ t< 0 . (3.127)

The last assumption implies that the time instant t = 0 is identified with

the time of occurrence of the disturbance.
If the disturbance is the instantaneous impact of Eq. (3.38), the relations
of Eqs. (3.125) - (3.127) become

x ¨ = u, (3.128)

x( − t ∗ ) = 0 , x( ˙ − t ∗ ) = 0 , t ∗ ≥ 0 , (3.129)

x( ˙ + 0 ) =˙ x( − 0 ) + V, (3.130)