  Planning multi-contact motions in a receding horizon fashion requires a value
function to guide the planning with respect to the future, e.g., building
momentum to traverse large obstacles. Traditionally, the value function is
approximated by computing trajectories in a prediction horizon (never executed)
that foresees the future beyond the execution horizon. However, given the
non-convex dynamics of multi-contact motions, this approach is computationally
expensive. To enable online Receding Horizon Planning (RHP) of multi-contact
motions, we find efficient approximations of the value function. Specifically,
we propose a trajectory-based and a learning-based approach. In the former,
namely RHP with Multiple Levels of Model Fidelity, we approximate the value
function by computing the prediction horizon with a convex relaxed model. In
the latter, namely Locally-Guided RHP, we learn an oracle to predict local
objectives for locomotion tasks, and we use these local objectives to construct
local value functions for guiding a short-horizon RHP. We evaluate both
approaches in simulation by planning centroidal trajectories of a humanoid
robot walking on moderate slopes, and on large slopes where the robot cannot
maintain static balance. Our results show that locally-guided RHP achieves the
best computation efficiency (95\%-98.6\% cycles converge online). This
computation advantage enables us to demonstrate online receding horizon
planning of our real-world humanoid robot Talos walking in dynamic environments
that change on-the-fly.
