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.