  A new method is developed for accurately approximating the solution to
state-variable inequality path constrained optimal control problems using a
multiple-domain adaptive Legendre-Gauss-Radau collocation method. The method
consists of the following parts. First, a structure detection method is
developed to estimate switch times in the activation and deactivation of
state-variable inequality path constraints. Second, using the detected
structure, the domain is partitioned into multiple-domains where each domain
corresponds to either a constrained or an unconstrained segment. Furthermore,
additional decision variables are introduced in the multiple-domain
formulation, where these additional decision variables represent the switch
times of the detected active state-variable inequality path constraints. Within
a constrained domain, the path constraint is differentiated with respect to the
independent variable until the control appears explicitly, and this derivative
is set to zero along the constrained arc while all preceding derivatives are
set to zero at the start of the constrained arc. The time derivatives of the
active state-variable inequality path constraints are computed using automatic
differentiation and the properties of the chain rule. The method is
demonstrated on two problems, the first being a benchmark optimal control
problem which has a known analytical solution and the second being a
challenging problem from the field of aerospace engineering in which there is
no known analytical solution. When compared against previously developed
adaptive Legendre-Gauss-Radau methods, the results show that the method
developed in this paper is capable of computing accurate solutions to problems
whose solution contain active state-variable inequality path constraints.
