Rolling with Planar Parametric Curves for Real-time Robot Locomotion Algorithms

Robots routinely encounter obstacles and rough terrain, but terrain curvature is seldom included in models for real-time algorithms. We present a closed-form dynamic model for rolling with two planar smooth curves, and apply it to sagittal-plane locomotion problems. We assumed that the body rolls without slip and maintains a single point of contact. Using an auxiliary coordinate system to define the rolling body and terrain as parametric curves, we derived rolling constraints and dynamic equations of motion for model-based control algorithms – specifically Operational Space Control. The formulation was used to simulate an arbitrarily curved rock rolling on undulating terrain and to generate control signals to stabilize it on parabolic terrain. The stabilization problem was solved as a quadratic program in < 3 ms which shows that our formulation is suitable for real-time control algorithms. We also applied this framework to dynamically balance an underactuated 2 degree-of-freedom leg on parabolic terrain and achieve prescribed locomotion tasks for a wheel-leg vehicle on sinusoidal terrain in simulation. A supplementary video is available at https://youtu.be/EtPQEzkqsK8.