Nonlinear Optimization Using Discrete Variational Mechanics for Dynamic Maneuvers of a 3D One-Leg Hopper

We present an optimization-based motion planning framework for producing dynamically rich and feasible motions for a 3D one-leg hopper in challenging terrains. We formulate dynamic motion planning as a nonlinear optimization problem that computes position and orientation of the centroidal model, position of the limb, contact forces, contact locations, and timings of the gait in one unified framework. The dynamics are represented as a single rigid body, while the equations of motion are derived using discrete mechanics with a variational quaternion-based integrator for the orientation. We validate the capabilities by planning complex motions in three challenging tasks: jumping over an obstacle, leaping over a gap, and performing a somersault. All contact forces generated by the proposed optimization are verified with accurate numerical simulation to prove the feasibility of the generated agile motions with respect to the kinematic, dynamic, and environmental constraints.