OPTIMAL SHOCK ISOLATION FOR MULTI-DEGREE-OF-FREEDOM SYSTEMS - Injury Biomechanics and Control: Optimal Protection from Impact - page 139

4.1.4 Near-Optimal Control for Two-Body Model Based on
Optimal Control for the Rigid Model

If Assumptions 1 - 3 of Section 4.1.3 are not valid, the algorithm suggested
by Proposition 4.1 for the construction of the optimal control for Prob-
lem 4.2 does not apply. In this case, proceed either to solve Problem 4.2
numerically or to build an approximate, near-optimal control. In this section
it will be shown that the optimal control for Problem 4.4 (for the rigid-body
model) can be used as a near-optimal control for Problem 4.2 if the stiffness
coefficient k is relatively high. Use

˜

u 0 (t) and u 0 (t) to denote the optimal
controls for Problems 4.4 and 4.2, respectively.

Estimates for Approximation Errors

Proposition 4.2. The difference between the peak magnitudes of the dis-
placement of body 1 in the original two-degree-of-freedom system and in
its approximation by a rigid model satisfies the inequality

u 0 ) ≤

U
k .

J 1 (

J 1 (u 0 ) − ˜ (4.74)

Proof. From Eq. (4.32) it follows that

exp τ) dτ

1

exp

c t ,

t

U
c

k
c (t

U
k

k

| ξ(t) |≤ − − = − − (4.75)

0

where U is the maximum magnitude allowed for the control variable u .
Hence,

U
k .

| ξ(t) |≤ (4.76)

Thus, the magnitude of the relative displacement of bodies 1 and 2 does
not exceed the maximum static deformation of the viscoelastic element
connecting these bodies under the action of the control force of maximum
admissible magnitude.
In accordance with the definition of the quantity ξ in Eq. (4.12),

x = ξ + y, y = x − ξ. (4.77)