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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 variationbased 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.

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This letter develops, theoretically justifies, and experimentally implements an optimization-based nonlinear control methodology for stabilizing quadrupedal locomotion. This framework utilizes virtual constraints and control Lyapunov functions (CLFs) in the context of quadratic programs (QPs) to robustly stabilize periodic orbits for hybrid models of quadrupedal robots. Properties of the proposed QP are studied wherein sufficient conditions for the continuous differentiability of the controller are presented. Additionally, this letter addresses the robust stabilization problem of the orbits based on the Poincar é sections analysis and input-to-state stability (ISS). The proposed controller is numerically and experimentally validated on the A1 quadrupedal robot with 18 degrees of freedom to demonstrate the robust stability of trotting gaits against external disturbances and unknown payloads.