This course presents recent developments on control of underactuated mechanical systems, focusing on the notion of virtual constraint. Traditionally, motion control problems in robotics are partitioned in two parts: motion planning and trajectory tracking. The motion planning algorithm converts the motion specification into reference signals for the robot joints. The trajectory tracker uses feedback control to make the robot joints track the reference signals. There is an emerging consensus in the academic community that this approach is inadequate for sophisticated motion control problems, in that reference signals impose a timing on the control loop which is unnatural and inherently non robust.

The virtual constraint technique does not rely on any reference signal, and does not impose any timing in the feedback loop. Motions are characterized implicitly through constraints that are enforced via feedback. Through judicious choice of the constraints, one may induce motions that are surprisingly natural and biologically plausible. For this reason, the virtual constraints technique has become a dominant paradigm in bipedal robot locomotion, and has the potential of becoming even more widespread in other area of robot locomotion.

The virtual constraint approach is geometric in nature. This course presents the required mathematical tools from differential geometry and surveys the basic results in this emergent research area. Topics covered will include: Differentiable manifolds and basic operations. Controlled invariant manifolds and zero dynamics of nonlinear control systems. Euler-Lagrange robot models and models of impulsive impacts. Virtual holonomic constraints (VHCs). Constrained dynamics resulting from VHCs, and conditions for existence of a Lagrangian structure. Virtual constraint generators. Stabilization of periodic orbits on the constraint manifold. Virtual constraints for walking robots.