The dependent coordinates in the linearization of constrained multibody systems: Handling and elimination

In this paper, the handling of the dependent coordinates in the linearization of constrained multibody systems is clarified. To this end, a brief but comprehensive overview of linearization approaches used in the field of Multibody System Dynamics is presented. These procedures are illustrated by using a common notation and classified into four groups, depending on the initial form of the nonlinear equations of motion and the selection of the generalized coordinates (a redundant or a minimal set). The handling of the dependent coordinates in each of the procedures is discussed. The linearized equations of motion of constrained multibody systems are of great importance in several applications, such as linear stability and modal analyses, the design of linear state feedback controllers or the building of state and input estimators, like Kalman filters. When modeling a constrained multibody system, a number of coordinates greater than the number of degrees of freedom of the system is usually used. While some linearization approaches make use of the complete set of coordinates, several procedures are based on a coordinate partition in terms of independent and dependent coordinates to reduce the number of linearized equations of motion. In this latter scenario, the role of the dependent coordinates in the linearization is important and, in a general case, these dependent coordinates cannot be ignored to obtain the correct linearized equations of motion. This aspect, which may seem obvious, is not straightforward and is addressed in this work. The importance of considering the set of dependent coordinates in the linearization is demonstrated with a well -acknowledged bicycle benchmark multibody model and an electric kickscooter multibody model with rear and front suspensions. To this end, the Jacobian matrix and the linear stability results of these case studies are computed by considering different choices of independent and dependent coordinates.