Collocation Solution of Index-1 BVP-DAEs Arising from Constrained Trajectory Optimization

Brian C. Fabien


Trajectory optimization, optimal control, collocation, indirect method


This paper presents a collocation method for the solution of two-point boundary value problems associated with the solution of constrained trajectory optimization of robotic systems. Here, the trajectory optimization is formulated as an optimal control problem with state and control variable inequality constraints. The indirect approach to solving this problem leads to a two-point boundary value problem involving index-1 differential-algebraic equations and inequality constraints due to the complementarity conditions. In the numerical algorithm presented here, the differential and algebraic variables of the BVP-DAEs are approximated using piecewise polynomials on a mesh that may be nonuniform. The collocation method is realized by approximating the BVP-DAE at Lobatto points within each interval of the mesh. An interior point Newton's method is used to solve the collocation equations, and maintain feasibility of the inequality constraints. Two examples are presented to show the effectiveness of the numerical algorithm. In the first example the algorithm is used to determine the time optimal point to point trajectory for a 3 degree of freedom floating robot. The second example determines the optimal trajectory of a two-link R-R robot that is subject to end effector constraints.

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