Philip Holmes
Department of Mechanical and Aerospace Engineering and
Program in Applied and Computational Mathematics,
Princeton University



Piecewise-holonomic mechanics, hybrid dynamical systems, and escaping cockroaches

I will discuss joint work with John Schmitt, Raffaele Ghigliazza and Justin Seipel, in which nonlinear mechanics and hybrid dynamical systems meet biology. Motivated by Robert Full's experimental studies of insects at UC Berkeley, we propose a mechanical model for the dynamics of legged locomotion in the horizontal plane. Our three-degree-of freedom rigid body model with massless, compliant legs in intermittent contact with the ground allows for passive and prescribed (active muscle) force and torque generation. Starting with energetically conservative bipedal models (each leg corresponding to the front/rear/opposite-middle tripod used in rapid running by many insect species), we move on to include active muscles and a central pattern generator of bursting neurons, and begin to develop hexapedal models with more realistic leg geometries. We show that piecewise holonomic mechanics due to intermittent foot contacts can confer strong asymptotic stability, and compare our models' behaviors with experiments on running insects. We stress the relevance of simple models, and show how phase reductions and averaging allow significant simplifications of complex neuromechanical models.