Quasi-coordinates based dynamics modeling and control design for nonholonomic systems

The paper presents dynamics modeling of nonholonomic systems in quasi-coordinates. These non-inertial coordinates are useful in motion description of constrained systems, since their selection is arbitrary and they result in a set of equations of motion in a reduced-state form. Modeling systems in quasi-coordinates may facilitate a subsequent controller design, e.g. for underactuated systems with passive wheels or when a control input is a composite quantity with respect to coordinates that describe motion. This is in contrast to most dynamics modeling which is based on generalized coordinates and Lagrange’s approach. Basic disadvantages of Lagrange’s approach are that it may include systems with constraints of first order and the number of unknowns that result from Lagrange’s equations increases. From the control oriented modeling perspective, Lagrange’s approach requires the elimination of the constraint reaction forces in order to obtain a dynamic control model. The paper proposes an approach to control oriented modeling based on the generalization of the Boltzmann–Hamel equations. These are the generalized programmed motion equations in quasi-coordinates. This dynamics framework yields equations of motion of a constrained system in a reduced-state form, from which the dynamic control model directly follows. The framework applies to fully actuated and underactuated systems, it is computationally efficient and may facilitate a subsequent controller design.