Contact Implicit Control for the Vertical Hopper

Legged robots offer significant potential for tasks in challenging environ- ments, but their control remains complex due to highly nonlinear dy- namics and hybrid contact behaviors. Model Predictive Control (MPC) is a promising approach, yet traditional methods often rely on predefined contact modes, sacrificing optimality and limiting adaptability. This the- sis explores a contact-implicit MPC framework for a vertical hopper to enable autonomous contact discovery and achieve desired jump heights without pre-planned trajectories. The developed controller is iLQR based and uses a relaxed contact model. It was successfully validated on an ex- ternal MuJoCo simulation, demonstrating its ability to generate smooth jumping motions. However, it could be observed that the relaxation used in the contact model introduced inaccuracies, particularly in impulse cal- culation, which led to performance degradation with larger timesteps and violated strict Signorini conditions. Recommendations for future work include implementing more sophisticated optimization methods like the bisection method, employing higher-order numerical integrators such as Runge-Kutta, and optimizing computational efficiency through matrix caching. This research underscores the functionality of contact-implicit control while identifying critical areas for improvement to achieve robust, real-time performance for dynamic legged systems.