Nonlinear Geometric Control of Mechanical Systems: Virtual Constraints

This chapter reviews basic results on virtual holonomic constraints (VHCS) for mechanical control systems. The control framework of VHCS is that of output stabilization, and this chapter reviews this problem in the context of relative degree and zero dynamics. The concept of VHCS is introduced first in this context, and then understood geometrically. For mechanical control systems with degree of underactuation one, a detailed characterization is given of the constrained dynamics arising from a VHC, and conditions under which such dynamics are Lagrangian are provided. It is shown that constrained dynamics exhibits an “abundance” of two different types of closed orbits, oscillations and rotations. In the non-Lagrangian case, conditions are presented under which the constrained dynamics have a stable limit cycle. The ideas of the chapter are finally generalized in the coordinate-free context of mechanical control systems on Riemannian manifolds using the concept of affine connection. Various examples illustrate the theoretical ideas.