@article {1314256,
	title = {Contact-Implicit Trajectory Optimization using Variational Integrators},
	journal = {International Journal of Robotics Research},
	year = {2019},
	note = {This paper is currently under review. Suggestions and comments welcome!\&nbsp;},
	abstract = {Contact constraints arise naturally in many robot planning problems. In recent years, a variety of contact-implicit trajectory optimization algorithms have been developed that avoid the pitfalls of mode pre-specification by simultaneously optimizing state, input, and contact force trajectories. However, their reliance on first-order integrators leads to a linear tradeoff between optimization problem size and plan accuracy. To address this limitation, we propose a new family of trajectory optimization algorithms that leverage ideas from discrete variational mechanics to derive higher-order generalizations of the classic time-stepping method of Stewart and Trinkle. By using these dynamics formulations as constraints in direct trajectory optimization algorithms, it is possible to perform contact-implicit trajectory optimization with significantly higher accuracy. For concreteness, we derive a second-order method and evaluate it using several simulated rigid body systems, including an underactuated biped and a quadruped. In addition, we use this second-order method to plan locomotion trajectories for a complex quadrupedal microrobot. The planned trajectories are evaluated on the physical platform and result in a number of performance improvements.},
	url = {https://journals.sagepub.com/doi/abs/10.1177/0278364919849235?journalCode=ijra},
	author = {Zachary Manchester and Neel Doshi and Wood, Robert J. and Scott Kuindersma}
}
