We consider the problem of bridging the gap between geometric tracking control theory and implementation of model predictive control (MPC) for robotic systems operating on manifolds. We propose a generic on-manifold MPC formulation based on a canonical representation of the system evolving on manifolds. Then, we present a method that solves the on-manifold MPC formulation by linearizing the system along the trajectory under tracking. There are two main advantages of the proposed scheme. The first is that the linearized system leads to an equivalent error system represented by a set of minimal parameters without any singularity. Secondly, the process of system modeling, error-system derivation, linearization and control has the manifold constraints completely decoupled from the system descriptions, enabling the development of a symbolic MPC framework that naturally encapsulates the manifold constraints. In this framework, users need only to supply system-specific descriptions without dealing with the manifold constraints. We implement this framework and test it on a quadrotor unmanned aerial vehicle (UAV) operating on $SO(3) times mathbb{R}^n$ and an unmanned ground vehicle (UGV) moving on a curved surface. Real-world experiments show that the proposed framework and implementation achieve high tracking performance and computational efficiency even in highly aggressive aerobatic quadrotor maneuvers.
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