Convex optimization of nonlinear feedback controllers via occupation measures

Anirudha Majumdar, Ram Vasudevan, Mark M. Tobenkin, Russ Tedrake

Research output: Contribution to journalArticlepeer-review

81 Scopus citations


The construction of feedback control laws for underactuated nonlinear robotic systems with input saturation limits is crucial for dynamic robotic tasks such as walking, running, or flying. Existing techniques for feedback control design are either restricted to linear systems, rely on discretizations of the state space, or require solving a nonconvex optimization problem that requires feasible initialization. This paper presents a method for designing feedback controllers for polynomial systems that maximize the size of the time-limited backwards reachable set (BRS). In contrast to traditional approaches based on Lyapunov's criteria for stability, we rely on the notion of occupation measures to pose this problem as an infinite-dimensional linear program which can then be approximated in finite dimension via semidefinite programs (SDPs). The solution to each SDP yields a polynomial control policy and an outer approximation of the largest achievable BRS which is well suited for use in a trajectory library or feedback motion planning algorithm. We demonstrate the efficacy and scalability of our approach on six nonlinear systems. Comparisons to an infinite-horizon linear quadratic regulator approach and an approach relying on Lyapunov's criteria for stability are also included in order to illustrate the improved performance of the presented technique.

Original languageEnglish (US)
Pages (from-to)1209-1230
Number of pages22
JournalInternational Journal of Robotics Research
Issue number9
StatePublished - Aug 11 2014
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Software
  • Modeling and Simulation
  • Mechanical Engineering
  • Electrical and Electronic Engineering
  • Artificial Intelligence
  • Applied Mathematics


  • Nonlinear control
  • optimization
  • semidefinite programming
  • underactuated robots


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