This letter presents a trajectory planner for humanoid robots to perform dynamic motions on complex, parameterizable terrains using a contact-implicit whole-body optimization framework. The planner generates feasible movements without predefined contact sequences, considering the robot's full dynamics and kinematics. To plan bipedal locomotion on steps, ramps, and other terrains, we propose a mathematical model to approximate ground geometry with a smooth surface. The rigid interaction between the robot and the ground is modeled via novel contact constraints, extending the domain of applicability from flat and horizontal surfaces to any arbitrary smooth surface. Effectiveness is demonstrated using nonlinear programming to plan motions on various terrain topologies. We also explore dynamic trajectory generation by embedding high-level requirements in the cost function and the initial guess. Preliminary feasibility tests were conducted over non-coplanar terrains, both in simulation and on the real robot ErgoCub.

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