A contribution on the modular modelling of multibody systems

Abstract

Over the past half-century, the increasing use of computational tools for mathematical modelling and simulation was responsible for significant advances in the area of Multibody System Dynamics. However, there is still a high dependence on the use of proprietary software in this area. Noticing that most of the complex multibody systems share many components and subsystems, this paper aims to propose a modular modelling methodology in which the starting points are some already known mathematical models of subsystems and the corresponding descriptions of the constraints existing among them. The proposed algorithm is based on the computation of some orthogonal complements of Jacobian matrices, derived from the constraint equations among the subsystems, leading to a minimal system of equations without requiring the use of undetermined multipliers or generalized constraint forces. Such an algorithm can be implemented using general-purpose (eventually open source) software packages or programming languages. Another remarkable advantage of this methodology stems from the fact that even when different (Classical or Analytical Mechanics) formalisms have been used in the modelling of subsystems, it is still possible to use the proposed algorithm. Well-known examples and a rederivation of the Whipple bicycle model are used to illustrate applications of this novel methodology.

Recursive modular modelling methodology for lumped-parameter dynamic systems

“…A remarkable result that came along with the development of this new methodology is the equivalence between the formulations by Orsino [1] and Orsino & Hess-Coelho [2] and the formulation by Udwadia & Phohomsiri [4]. The investigation concerning the derivation of a recursive algorithm based on the Udwadia-Kalaba equation, under the same conditions adopted for the general methodology, revealed an even more relevant result: the algorithm is equivalent to the one of a Kalman filter for the estimation of the state of a static process in which the constraint equations play the role of the observation ones, and the associated orthogonal projectors play the role of the covariance matrices.…”

“…If S r is chosen to be an orthogonal complement of A r , equation (3.4) becomes the one addressed in previous developments of a modular modelling metodology for multibody systems by Orsino [1] and Orsino & Hess-Coelho [2]. 12 On the other hand, if S r = I − A + r A r , with A + r denoting the Moore-Penrose pseudoinverse 13 of A r , equation (3.4) becomes the one addressed by Udwadia & Phohomsiri [4] in their derivation of explicit equations of motion for constrained mechanical systems with singular mass matrices (which led to applications also to the modular modelling of multibody systems).…”

“…Therefore, a modular methodology might not be understood simply as another alternative for the already extensive framework of existing ones, but rather as a methodology that aims to conciliate the use of others from which the models of the modules may have been obtained. Some examples of modular modelling methodologies originally developed for multibody systems can be found on works by Orsino [1], Orsino & Hess-Coelho [2] and Udwadia et al [3][4][5][6][7][8]. The latter works are typically referred to in the literature as the Udwadia-Kalaba equation.…”

“…(i) the formalisms used in the derivation of the models at the inferior level of a hierarchy (the models may either follow from the application of physical principles using some analytical formalism or can be phenomenological); (ii) the order of the differential equations involved, which can be higher than two; 2 (iii) the criteria for selecting variables, which allows exploration of the advantages of formulations based on the use of redundant variables and on reparameterizations of time rates [1,2]; (iv) the nature of the constraints and the techniques adopted for their description; and (v) how the user chooses to conceive the system modularly and how many levels there are in a given hierarchical representation.…”

“…There is a unique generalized inverse matrix that satisfies the four properties simultaneously: it is called the Moore-Penrose pseudoinverse of A and is specially denoted by A + [3,18]. Finally, it is worth commenting that the notation adopted in this text is not only compatible with the desired generality for the methodology proposed, but also tries to resemble as much as possible the notations from previous works from the author [1,2] and from texts on the Udwadia-Kalaba equation and on the Kalman filter. Once all the variables in the presented equations are matrices, no distinctive boldface notation is adopted.…”