Google Research

Safe Motion Planning with Tubes and Contraction Metrics

  • Sumeet Singh
  • Brett Lopez
  • Hiroyasu Tsukamoto
  • Jean-Jacques Slotine
  • Soon-Jo Chung


The recent widespread use 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 model error or external disturbances. This tutorial summarizes the key theoretical results of combining contraction theory with MPC to enable provably-safe motion planning for robotic and aerospace systems. The first approach presented establishes the fundamental result that any closed-loop contracting system has an associated state and control input invariant tube which can serve as a safety margin within the motion planning problem. This result is leveraged in an alternative approach that utilizes neural networks and imitation learning to offloading the computational complexity of online motion planning while maintaining strong safety guarantees. Finally, current challenges and future research directions, e.g., online model learning, are discussed.

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