Robot Trajectory Direct Collocation with Conditional Constraints for Optimal Phase Transitions

Incorporating phase transitions such as impulsive or non-impulsive contact interactions and subtasks into trajectory planning is a critical but challenging problem for robot control. This study introduces a unified, derivative-based, direct collocation framework with conditional constraints of general tasks and mathematical forms for optimal robot trajectories and phase transitions, without the need for the pre-specified sequence/timing, explicit contact forces, complementarity constraints, multi-level solvers, or additional variables/constraints present in most contact-implicit or logic-geometric schemes. Encoded conditions detect or plan the phase transitions and transform the relevant constraint bounds, which are equivalent to the equality/inequality form, tightening/relaxation, or addition/removal of the constraints. The formulations are embedded in a sequential quadratic programming algorithm, with the allowance of outer loops for hybrid dynamic systems, where the numerical convergence of the solution toward optimality is confirmed by iteration and sensitivity analyses. The framework is validated and demonstrated for three example tasks with distinct contact phases and boundary nonlinearities—bouncing ball, manipulation, and legged balancing with tipping—with different levels of system actuation (unactuated, under-actuated, and fully-actuated) in non-redundant problems for prediction and redundant problems for optimization.