Applied Mathematics Seminar | Manfredi Maggiore, An Introduction to Virtual Holonomic Constraints

In Lagrangian mechanics, constraints that can be expressed in the form of equations involving only configuration variables, and not their derivatives, are called ``holonomic.'' For example, a particle constrained to move on the surface of a sphere is subjected to a holonomic constraint. In the case of Lagrangian control systems, one may use feedback to emulate the presence of holonomic constraints. For example, one may make a platoon of vehicles move in rigid formation by emulating the presence of distance constraints among the vehicles. Such emulated constraints are called “virtual holonomic constraints” (VHCs). In robotics, VHCs have become a popular tool to induce stable walking gaits in biped robots, and there is a growing body of work suggesting that VHCs might represent a universal paradigm for locomotion. 

From a theoretical viewpoint, there are a number of interesting questions arising in the context of VHCs. One of them is whether or not the motion of a Lagrangian control system subjected to a VHC is still Lagrangian. In this lecture I will show that, in contrast with classical mechanics, the answer to this question is ``typically no.'' For underactuated Lagrangian control systems with underactuation degree one, I will give necessary and sufficient conditions guaranteeing that the constrained dynamics arising from a VHC are Lagrangian. I will show experimental results illustrating VHCs in action and giving an intuitive feel of the significance of Lagrangian constrained dynamics.

Biography

Manfredi Maggiore was born in Genoa, Italy. He received the Laurea degree in Electrical Engineering in 1996 from the University of Genoa and the PhD degree in Electrical Engineering from the Ohio State University, USA, in 2000. Since 2000 he has been with the Edward S. Rogers Sr. Department of Electrical and Computer Engineering, University of Toronto, Canada, where he is currently Professor. He has been a Visiting Professor at the University of Roma Tor Vergata (2001) and the University of Bologna (2007-2008). His research focuses on mathematical nonlinear control, and relies on methods from dynamical systems theory and differential geometry.