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.