  In this paper we propose a method to improve the accuracy of trajectory
optimization for dynamic robots with intermittent contact by using orthogonal
collocation. Until recently, most trajectory optimization methods for systems
with contacts employ mode-scheduling, which requires an a priori knowledge of
the contact order and thus cannot produce complex or non-intuitive behaviors.
Contact-implicit trajectory optimization methods offer a solution to this by
allowing the optimization to make or break contacts as needed, but thus far
have suffered from poor accuracy. Here, we combine methods from direct
collocation using higher order orthogonal polynomials with contact-implicit
optimization to generate trajectories with significantly improved accuracy. The
key insight is to increase the order of the polynomial representation while
maintaining the assumption that impact occurs over the duration of one finite
element.
