Trajectory optimization with contact-rich behaviors has recently gained attention for generating diverse locomotion behaviors without pre-specified ground contact sequences. However, these approaches rely on precise models of robot dynamics and the terrain and are susceptible to uncertainty. Recent works have attempted to handle uncertainties in the system model, but few have investigated uncertainty in contact dynamics. In this study, we model uncertainty stemming from the terrain and design corresponding risk-sensitive objectives under the framework of contact-implicit trajectory optimization. In particular, we parameterize uncertainties from the terrain contact distance and friction coefficients using probability distributions and propose a corresponding expected residual minimization cost design approach. We evaluate our method in three simple robotic examples, including a legged hopping robot, and we benchmark one of our examples in simulation against a robust worst-case solution. We show that our risk-sensitive method produces contact-averse trajectories that are robust to terrain perturbations. Moreover, we demonstrate that the resulting trajectories converge to those generated by a traditional, non-robust method as the terrain model becomes more certain. Our study marks an important step towards a fully robust, contact-implicit approach suitable for deploying robots on real-world terrain.
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