  This paper presents a novel Representation-Free Model Predictive Control
(RF-MPC) framework for controlling various dynamic motions of a quadrupedal
robot in three dimensional (3D) space. Our formulation directly represents the
rotational dynamics using the rotation matrix, which liberates us from the
issues associated with the use of Euler angles and quaternion as the
orientation representations. With a variation-based linearization scheme and a
carefully constructed cost function, the MPC control law is transcribed to the
standard Quadratic Program (QP) form. The MPC controller can operate at
real-time rates of 250 Hz on a quadruped robot. Experimental results including
periodic quadrupedal gaits and a controlled backflip validate that our control
strategy could stabilize dynamic motions that involve singularity in 3D
maneuvers.
