  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.
