Nonholonomic virtual constraints for dynamic walking

Virtual constraints are functional relations (i.e., constraints) on the state variables of a robot's model that are achieved through the action of actuators and feedback control instead of physical contact forces. They are called virtual because they can be re-programmed on the fly without modifying any physical connections among the links of the robot or its environment. Previous analytical and experimental work has established that vector relative degree two virtual holonomic (i.e., only configuration dependent) constraints are a powerful means to synchronize the links of a bipedal robot so as to achieve walking and running motions over a variety of terrain profiles. This paper introduces a class of virtual nonholonomic constraints that depend on velocity through (generalized) angular momentum while maintaining the property of being relative degree two. This additional freedom is shown to yield control solutions that handle a wider range of gait perturbations arising from terrain variations and exogenous forces. Moreover, including angular momentum in the virtual constraints allows foot placement control to be rigorously designed on the basis of the full dynamic model of the biped, instead of on the basis of an inverted pendulum approximation of its center of mass, as is commonly done in the bipedal robotics literature. This new class of control laws is shown in simulation to be robust to a variety of common gait disturbances.