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.

Tracks, wheels, and legs are all useful locomotion modes for Unmanned Ground Vehicles (UGVs), and ground robots that combine these mechanisms have the potential to climb over large obstacles. As robot morphologies include more degrees of freedom and obstacles become increasingly large and complex, UGVs will need to rely on automatic motion planning to compute the joint trajectories for traversal. This article presents a trajectory optimization formulation for multibody UGVs with combined wheel-leg and track-leg designs. We derive the dynamics and constraints for rolling wheels and circulating elliptical tracks. Using direct collocation, we formulate a model-based trajectory optimization where all constraints and objectives are written in closed-form with smooth and exact derivatives for tractable computation times with existing large-scale nonlinear optimization solvers (<1 minute). We demonstrate the trajectory optimization on numerous simulated planar wheel-leg and track-leg morphologies completing locomotion tasks, demonstrating full body dynamic coupling for the multibody system. Future work will extend this formulation to 3D and include contact planning.