Synthesis of virtual holonomic constraints for 3-DOF mechanical systems

The paper presents a new method for determining virtual holonomic constraints for a mechanical system. We focus on systems with 3 DOF and 2 controls. The main result of the paper shows that, if the distribution generated by the vector fields representing the accelerations due to the inputs is not involutive near a reference configuration, then it is possible to find a closed virtual holonomic constraint defined in an arbitrarily small neighborhood of the reference configuration, which is controlled invariant and stabilizable. We present a constructive proof that provides a method for the numerical computation of the constraint. As an application example, we present some families of virtual holonomic constraints for an underactuated double pendulum on a cart.