Controllability and motion algorithms for underactuated Lagrangian systems on Lie groups

Francesco Bullo, Naomi Ehrich Leonard, Andrew D. Lewis

Research output: Contribution to journalArticlepeer-review

154 Scopus citations


In this paper, we provide controllability tests and motion control algorithms for underactuated mechanical control systems on Lie groups with Lagrangian equal to kinetic energy. Examples include satellite and underwater vehicle control systems with the number of control inputs less than the dimension of the configuration space. Local controllability properties of these systems are characterized, and two algebraic tests are derived in terms of the symmetric product and the Lie bracket of the input vector fields. Perturbation theory is applied to compute approximate solutions for the system under small-amplitude forcing; in-phase signals play a crucial role in achieving motion along symmetric product directions. Motion control algorithms are then designed to solve problems of point-to-point reconfiguration, static interpolation and exponential stabilization. We illustrate the theoretical results and the algorithms with applications to models of planar rigid bodies, satellites and underwater vehicles.

Original languageEnglish (US)
Pages (from-to)1437-1454
Number of pages18
JournalIEEE Transactions on Automatic Control
Issue number8
StatePublished - Aug 2000

All Science Journal Classification (ASJC) codes

  • Control and Systems Engineering
  • Computer Science Applications
  • Electrical and Electronic Engineering


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