Safe Motion Planning with Tubes and Contraction Metrics

The recent proliferation of model predictive control (MPC) in safety-critical systems has placed additional emphasis on developing algorithms that have strict performance guarantees despite the presence of modeling error or external disturbances. This tutorial summarizes key results of combining contraction theory with MPC to enable provably-safe motion planning for robotic and aerospace systems. After a brief review of control contraction metrics, we summarize the fundamental result that any closed-loop contracting system has an associated invariant tube centered around a desired trajectory. It is then shown how these tubes can be systematically incorporated into the motion planning problem as an additional safety margin for systems with uncertain dynamics. Finally, several future research directions are discussed.