The varied and complex dynamics of deformable terrain are significant impediments toward real-world viability of locomotive robotics, particularly for legged machines. We explore vertical jumping on granular media (GM) as a model task for legged locomotion on uncharacterized deformable terrain. By integrating (Gaussian process) GP-based regression and evaluation to estimate ground forcing as a function of state, a one-dimensional jumper acquires the ability to learn forcing profiles exerted by its environment in tandem to achieving its control objective. The GP-based dynamical model initially assumes a baseline rigid, non-compliant surface. As part of an iterative procedure, the optimizer employing this model generates an optimal control to achieve a target jump height while respecting known hardware limitations of the robot model. Trajectory and forcing data recovered from evaluation on the true GM surface model simulation is applied to train the GP, and in turn, provide the optimizer a more richly informed dynamical model of the environment. After three iterations, predicted optimal control trajectories coincide with execution results, within 1.2% jumping height error, as the GP-based approximation converges to the true GM model.

This work demonstrates fast motion planning for robot locomotion that is optimized for terrain with complex dynamics, specifically, rapid penetration of granular media. Gait planning is critical for many legged locomotion control approaches, but they typically assume rigid ground contact. We aim to extend these planning methods to include terrain dynamics we see in the natural world, like sand and dirt, which can both deform and fluidize. Using an added-mass description of collective grain motion, we formulated a model of hydrostatic and hydrodynamic terrain effects that is both principled and representable with closed-form dynamics. As a result, we present a model and fast optimization formulation which solves accurate motion plans on granular media with tractable solving times (6.4-3.8 seconds). For validation, we optimized open-loop motor trajectories for a testbed jumping robot to jump to a target apex height from a bed a loosely packed poppy seeds, a model granular medium. While jumps optimized for rigid ground were anemic on granular media, terrain-aware trajectories hit within 6% of their target. This demonstrates the potential for robot locomotion which meets practical task demands, all while being aware of the terrain beneath it.