Optimal control of systems with unilateral constraints

Problems in robotics and biomechanics such as trajectory planning or resolution of redundancy can be effectively solved using optimal control. Such systems are often subject to unilateral constraints. Examples include tasks involving contacts (e.g., walking, running, multifingered or multiarm manipulation), and other tasks that may not involve contacts but in which the system state or the inputs must satisfy inequality conditions (e.g., limits on actuator forces). This paper shows how problems of optimal control in robotics that involve unilateral constraints can be efficiently solved by first formulating the constrained optimal control problem as an unconstrained problem of the calculus of variations and then solving it using an integral formulation. This method has several advantages over the Pontryagin minimum principle which is traditionally employed to solve such problems. An example of two-arm manipulation with inequality constraints due to Coulomb friction is used to demonstrate the formulation of the problem and the algorithms.