On-Manifold Model Predictive Control for Trajectory Tracking on Robotic Systems

Abstract

Model predictive control (MPC) has the ability to incorporate state and input constraints and extendability to systems with multiple inputs and outputs, making it a popular technique for robot control and other real-world applications. However, robotic systems usually evolve on manifolds, which are often over-parameterized or minimally parameterized (but with singularity) in a MPC. How to naturally incorporate the system manifold constraints without over-parameterization or singularity is a fundamental problem when deploying MPC on robotic systems. In this paper, we propose a unified on-manifold MPC framework to address the mentioned issue. This framework first formulates the MPC based on a canonical representation of on-manifold systems. Then, the on-manifold MPC formulation is solved by linearizing the system at each point 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 singularities. Secondly, the process of system modeling, error-system derivation, linearization and control has the manifold constraints completely decoupled from the system descriptions, enabling to develop a symbolic MPC framework that naturally encapsulates the manifold constraints. In this symbolic framework, one need only to supply system-specific descriptions without dealing with the manifold constraints. To validate the generality of the proposed framework, we implement it on two different robotic platforms, a quadrotor unmanned aerial vehicle (UAV) evolving on a Lie group, and an unmanned ground vehicle (UGV) moving on a curved surface with non-Lie group structure. To validate the non-singularity and tracking performance of the framework, we test it in tracking aggressive UAV trajectories. Experimental results show that with a single, global on-manifold MPC controller, the quadrotor tracks highly aggressive trajectories with large actuator efforts and attitude variation (360 degrees) in both roll and pitch directions, without controller switching.