Surprising smoothness in common multi-contact collisions

ABSTRACT: In many practical situations in robotics and biomechanics, objects collide along multiple contact points which form contacts in a state-dependent order.  Examples include a human catching a basketball — 10 contacts forming in an unknown order — or a quadrupedal robot trotting — forming two contacts in unknown order.  Modeled as hybrid systems, these examples represent a challenge because the ensemble of trajectories spans an area where the guard functions representing the contact events intersect, meaning that over an infinitesimal change in initial conditions the system could potentially be subject to vastly different dynamics.  The hybrid systems that appear in these collision problems are not arbitrary; they have the property that the equations of motion depend only on the set of contacts created but not on the order of creation of those contacts.  These “Event Selected Systems (ESS)” generally lead to continuous, piecewise differentiable dynamics, with many desirable properties that are not shared by more general classes of hybrid systems, such as perturbation expansions.

Surprisingly, in recent work we have shown that an ESS where forces change because of position dependent guards is even more regular — it will always have a continuously differentiable state-space flow.  We demonstrated this experimentally by building a three legged hopper whose legs generate measurably distinct dynamics individually, yet the arbitrarily ordered triple contact leads to the same outcome up to affine approximation.  In other words, even though each leg moves the robot in a different way as the contacts build up, correctly suggesting a classically non-differentiable flow after the first impact, the dynamics from aerial phase to after all three legs have landed are smooth — continuous and differentiable.

The implication of our results is that many multi-contact problems that may seem non-smooth and difficult to control are in fact smooth and may be amenable to conventional non-linear control approaches.  The talk will introduce the theory of event selected systems, and be suitable for an audience familiar with multivariate calculus and differential equations. The work was funded by ARO MURI W911NF-17-1-0306, NSF CPS 2038432, and the D. Dan and Betty Kahn Michigan-Israel Partnership for Research and Education Autonomous Systems Mega-Project.

BIO: Shai Revzen is an Associate Professor of Electrical Engineering and Computer Science department of the University of Michigan, and holds a courtesy appointment in the Department of Ecology and Evolutionary Biology, and in the Applied Physics Program.  Shai is the founder and principal investigator of the Biologically Inspired Robotics and Dynamical Systems (BIRDS) lab in ECE, focusing on fundamental science in robotics and its transformative potential for future technology.   Shai holds PhD in Integrative Biology (Berkeley, 2009), an M.Sc. in computer science (Hebrew U., Jerusalem, 2002) and did postdoctoral work in robotics at the GRASP Lab, U. Penn before joining the faculty at Michigan in 2012, where he was a core founding member of the Robotics Institute.  Shai has extensive industry experience as Chief Architect R&D (Harmonic Lightwaves; NASDAQ HLIT), co-founder and Chief Science Officer of Acculine Medical, and General Manager of a consulting company, Izun, Inc.  Shai has over 30 publications in top scientific journals in biology, robotics, and mathematics.  In his copious free time he is pursuing a law degree (JD), and serving five cats.

***Event will take place in hybrid format. The location for in-person attendance will be room 1500 EECS.   Attendance will also be possible via Zoom. The Zoom link and password will be distributed to the Controls Group e-mail list-serv.

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