Contact Implicit Control for the Vertical Hopper

Legged robots bear significant potential for robotic applications that deal with diverse and challenging environments where wheeled robots may struggle to move effectively. Nonetheless, the promise of extended mobility comes at the cost of increased complexity due to the inherently
nonlinear dynamics and hybrid nature with continuous dynamics and discrete switching between contact modes. 
Model Predictive Control (MPC), a control scheme based on iterative,finite-horizon optimization of a model, has become a heavily utilized method. However MPC for contact mode planning is computational very expensive due to the high non linearity of the contact Models, hence it is often convexified using predefined contact modes. Here the disadvantage is that the MPC can not plan the contact itself losing its advantages on optimality. One way of encoding the contact modes is by using linear complementary constraints (LCC) resulting from the Signorini condition. These come with increased computational burdens due to exponential contact mode combinations. This problem can be circumvented using contact-implicit control, a method that includes the complementary constraints implicitly into the systems dynamics and thereby reduces the associated numerical complexity. However, the strict linear complementarity constrains introduce further problems, often leading to the system being stuck at the initial contact condition and preventing contact breakage. Relaxation and penalty methods help to facilitate dynamic discovery of new contact modes, though also may reduce physical realism by allowing ground penetration or contact forces at a non zero distance. 
The goal of the thesis is the employment of a contact implicit controller for the vertical hopper that does not depend on predefined contact mode sequences or foothold positions but rather facilitating real-time discovery of contact modes.
The chosen approach uses a contact- implicit multiple shooting iterative linear quadratic regulator (iLQR) that combines a hard contact model with an explicit Signorini condition in the roll out and smooth gradient with a relaxation variable in the backwardspass, thereby balances out the weaknesses of the different methods.