  This work introduces an optimization-based locomotion control framework for
on-the-fly synthesis of complex dynamic maneuvers. At the core of the proposed
framework is a cascaded-fidelity model predictive controller (Cafe-Mpc).
Cafe-Mpc strategically relaxes the planning problem along the prediction
horizon (i.e., with descending model fidelity, increasingly coarse time steps,
and relaxed constraints) for computational and performance gains. This problem
is numerically solved with an efficient customized multiple-shooting iLQR
(MS-iLQR) solver that is tailored for hybrid systems. The action-value function
from Cafe-Mpc is then used as the basis for a new value-function-based
whole-body control (VWBC) technique that avoids additional tuning for the WBC.
In this respect, the proposed framework unifies whole-body MPC and more
conventional whole-body quadratic programming (QP), which have been treated as
separate components in previous works. We study the effects of the cascaded
relaxations in Cafe-Mpc on the tracking performance and required computation
time. We also show that the Cafe-Mpc, if configured appropriately, advances the
performance of whole-body MPC without necessarily increasing computational
cost. Further, we show the superior performance of the proposed VWBC over the
Riccati feedback controller in terms of constraint handling. The proposed
framework enables accomplishing for the first time gymnastic-style running
barrel rolls on the MIT Mini Cheetah. Video: https://youtu.be/YiNqrgj9mb8.
