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Trajectory optimization with contact-rich behaviors has recently gainedattention for generating diverse locomotion behaviors without pre-specifiedground contact sequences. However, these approaches rely on precise models ofrobot dynamics and the terrain and are susceptible to uncertainty. Recent workshave attempted to handle uncertainties in the system model, but few haveinvestigated uncertainty in contact dynamics. In this study, we modeluncertainty stemming from the terrain and design corresponding risk-sensitiveobjectives under the framework of contact-implicit trajectory optimization. Inparticular, we parameterize uncertainties from the terrain contact distance andfriction coefficients using probability distributions and propose acorresponding expected residual minimization cost design approach. We evaluateour method in three simple robotic examples, including a legged hopping robot,and we benchmark one of our examples in simulation against a robust worst-casesolution. We show that our risk-sensitive method produces contact-aversetrajectories that are robust to terrain perturbations. Moreover, we demonstratethat the resulting trajectories converge to those generated by a traditional,non-robust method as the terrain model becomes more certain. Our study marks animportant step towards a fully robust, contact-implicit approach suitable fordeploying robots on real-world terrain.

Robust Trajectory Optimization over Uncertain Terrain with Stochastic Complementarity