• bipedal locomotion
• contact modeling
• humanoid robots
• model predictive control
• optimal control
• trajectory planning
• uneven terrain

This thesis presents a predictive control approach to generate trajectories for humanoid robots. To plan whole-body dynamic movements, one is often faced with a choice between full models that take into account the entire robot dynamics, including the information relative to the inertial properties of each link and the torques developed by each actuated joint, or reduced models that substantially simplify the robot structure, reducing it, for example, to a linear inverted pendulum or a reaction mass pendulum. The former are extremely complex to handle, while the latter do not provide sufficiently expressive descriptions. A middle ground is the model that incorporates the full kinematics and only considers the centroidal dynamics, which is the time evolution of the total angular momentum and the system center of mass position. This approach has been shown to be very effective for planning.

In this work, an MPC-based planner is proposed, which predicts future states of the robot and calculates the optimal action to take based on this kind of intermediate model. The optimization problem underlying the MPC is equipped with contact conditions for the interaction of the robot with rigid ground. In order to obtain movements on ramps, stairs or more generic terrain, a continuous and differentiable model is proposed to approximate their geometry. The interaction between the robot and the ground is modeled explicitly via new conditions, that extend the validity domain from a flat and horizontal surface to any smooth two-dimensional surface. The planner was first tested in simulation on a simplified system consisting only of the robot foot, modeled as a single rigid body. It successfully generates trajectories that allow the foot to land on a generic terrain, minimizing the distance from a desired pose set by the user. Once the planner was validated on the simplified model, it began to be tested on the complete kinematic model of the iCub humanoid robot. In this context, it was necessary to develop an algorithm for setting the desired initial and target configurations for the robot. The current implementation generates movements that shift the center of mass such that its projection falls within the support polygon of the stance foot, in order to maintain balance while lifting the swinging foot.