Nonholonomic systems are mechanical systems subjected to kinematic constraints that are not completely integrable. Typical examples are wheeled vehicles whose motions are constrained by the nonslip property of wheels rolling on the ground, according to which the velocity of a wheel’s center is parallel to the wheel’s plane and proportional to the wheel’s rotation velocity. While these constraints do not prevent the system from being controllable (a fortunate fact for car drivers) they reduce the possibilities of instantaneous displacements (i.e. the number of degrees of freedom) in the configuration space. As a consequence the kinematics equations of these systems are modelled by driftless systems of the form dx/dt=B(x)u, with the dimension of the control vector –u– smaller than the dimension of configuration state vector –x-.

A difficulty, well-known to control designers, is that the linearization of the kinematic equations of these systems at a fixed-point is not controllable. For instance, linearization of the above system at the point (x=0,u=0) yields the non-controllable linear system dx/dt=B(0)u. This already implies that classical linear and nonlinear control techniques do not provide solutions to the asymptotic stabilization (i.e. convergence and stability in the sense of Lyapunov) of fixed-points. As a matter of fact, as soon as the matrix B(0) is of full rank (the common case), a theorem due to Brockett (1983), and its extensions, forbid the existence of asymptotic stabilizers that depend on x only. A complementary result by Lizarraga (2004) indicates that all attempts to work out (causal) feedback controllers capable to stabilize asymptotically all feasible reference trajectories are doomed to failure.

The transverse function approach provides a way to circumvent Brockett’s obstruction by abandoning the stringent objective of asymptotic stabilization and replacing it with an objective of practical stabilization that ensures convergence to, and stability of, a neighborhood centered at the fixed-point of interest, which can be rendered arbitrarily small via the choice of the transverse function parameters. It also has the complementary advantage of allowing for the design of practical stabilizers that apply to all reference trajectories, feasible and non-feasible ones. Practical stabilization of non-feasible trajectories is of particular interest in the case of systems for which the determination of useful feasible trajectories is not simple.