Page 106 - Handbook of Biomechatronics

102 Naser Mehrabi and John McPhee

environment is reasonably predictable. For example, in functional electrical
stimulation (FES) of foot drop, a predefined sequence of electrical impulses
stimulates the appropriate muscles to raise the forefoot at the appropriate
time during a gait cycle (when a trigger is activated) (Stein et al., 2006). Foot
drop is a pathological gait disorder in which the forefoot drags on the ground
during walking. Foot drop usually occurs because of muscle weakness or
neuromuscular disorders.
The control sequence can be achieved through trial and error experi-
ments or by using a DO method. For example, for FES of gait, an optimal
sequence of muscle activations can be achieved through DO of a biome-
chanical model of gait. The major advantage of this method over the trial
and error approach is that in DO, a criterion such as applied electrical stim-
ulation can be minimized so that the onset of muscle fatigue occurs later in
therapy.
A DO can be solved through direct and indirect optimal control
methods. An indirect method finds an optimal solution by reformulating
the original control problem such that the necessary conditions of the opti-
mality are satisfied. In the indirect methods (optimize and then discretize),
the optimal control problem is converted to a two-point boundary value
problem (2PBVP) by applying Pontryagin’s minimum principle. The solu-
tion of the 2PBVP provides an optimal solution for the original problem. In
a typical direct solution (discretize and then optimize), the dynamic equa-
tions are discretized using a numerical integrator; combined with the cost
function, the result is a relatively large nonlinear programming (optimiza-
tion) problem, or NLP. These NLPs can be solved using specially designed
optimizers (e.g., IPOPT (Wachter and Biegler, 2006) and SNOPT (Gill
et al., 2005)) that exploit the sparsity pattern that exists in such problems.
Although this is one of the most common techniques for formulating a direct
optimal control problem, there are many other methods (e.g., multiple-
shooting and direct collocation) that exist in the literature. Overall, indirect
methods may be very sensitive to the initial values and to the changes of the
unspecified boundary conditions in the 2PBVP. In contrast, direct methods
usually have better convergence properties, and the user doesn’t need to
worry about the costate variables that appear in indirect methods. However,
in the presence of many local extrema, direct methods may converge to a
local extremum (Betts, 1998). Although these approaches employ different
philosophical approaches, the techniques may ultimately merge. Interested
readers are referred to Rao (2009) for more information about indirect and
direct optimal control techniques.