Lyapunov stability of a rigid body with two frictional contacts

Péter L. Várkonyi, Yizhar Or

Research output: Contribution to journalArticlepeer-review


Lyapunov stability of a mechanical system means that the dynamic response stays bounded in an arbitrarily small neighborhood of a static equilibrium configuration under small perturbations in positions and velocities. This type of stability is highly desired in robotic applications that involve multiple unilateral contacts. Nevertheless, Lyapunov stability analysis of such systems is extremely difficult, because even small perturbations may result in hybrid dynamics where the solution involves many non-smooth transitions between different contact states. This paper concerns Lyapunov stability analysis of a planar rigid body with two frictional unilateral contacts under inelastic impacts, for a general class of equilibrium configurations under a constant external load. The hybrid dynamics of the system under contact transitions and impacts is formulated, and a Poincaré map at two-contact states is introduced. Using invariance relations, this Poincaré map is reduced into two semi-analytic scalar functions that entirely encode the dynamic behavior of solutions under any small initial perturbation. These two functions enable determination of Lyapunov stability or instability for almost any equilibrium state. The results are demonstrated via simulation examples and by plotting stability and instability regions in two-dimensional parameter spaces that describe the contact geometry and external load.

Original languageEnglish
Pages (from-to)363-393
Number of pages31
JournalNonlinear Dynamics
Issue number1
StatePublished - 1 Apr 2017


  • Friction
  • Impacts
  • Lyapunov stability
  • Non-smooth mechanics
  • Poincaré map
  • Unilateral contacts

All Science Journal Classification (ASJC) codes

  • Control and Systems Engineering
  • Aerospace Engineering
  • Ocean Engineering
  • Mechanical Engineering
  • Applied Mathematics
  • Electrical and Electronic Engineering


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