Recursive modular modelling methodology for lumped-parameter dynamic systems

Abstract

This paper proposes a novel approach to the modelling of lumped-parameter dynamic systems, based on representing them by hierarchies of mathematical models of increasing complexity instead of a single (complex) model. Exploring the multilevel modularity that these systems typically exhibit, a general recursive modelling methodology is proposed, in order to conciliate the use of the already existing modelling techniques. The general algorithm is based on a fundamental theorem that states the conditions for computing projection operators recursively. Three procedures for these computations are discussed: orthonormalization, use of orthogonal complements and use of generalized inverses. The novel methodology is also applied for the development of a recursive algorithm based on the Udwadia-Kalaba equation, which proves to be identical to the one of a Kalman filter for estimating the state of a static process, given a sequence of noiseless measurements representing the constraints that must be satisfied by the system.

Modular methodology applied to the nonlinear modeling of a pipe conveying fluid

“…For each time step, the use of Eq. (35) to obtain the values of the generalized accelerations associated to the unconstrained system (level 0 of the hierarchy), is followed by a constraint enforcement algorithm proposed by Orsino [12], based on the modular modeling methodology and on Udwadia-Kalaba equations [19,20] that estimates the corresponding values of € q satisfying the boundary and inextensibility conditions. The applied algorithm includes a constraint stabilization strategy based on Baumgarte's technique [1], which improves the estimation.…”

“…1) is obtained by C rþ1 ¼ ðI À B þ rþ1 B rþ1 Þ, with I representing an identity operator and ðÁÞ þ standing for the Moore-Penrose pseudo-inverse. [12] proved that an algorithm based on such a procedure can be put in the explicit form detailed below, which is analogous to the algorithm of a Kalman filter for estimating the state of a static process from a set of noiseless measurements.…”

“…1 are given by € q ¼ Nz 0 . As shown in Orsino [12], the variables z r , from which the generalized accelerations at the level r of the hierarchy could be determined as € q ¼ Nz r , are computed according to the following algorithm (r ¼ 0; 1; 2):…”

“…The only restriction lies on the fact that the constraints of the model must be described by known invariants. For lumped parameter systems, the modular modeling methodology is particularly useful in the development of subsystem-based modeling strategies [11][12][13]16]. In these cases, the constraints are relaxed to ''disassemble'' a complex system into several simpler subsystems whose models either are already known or can be derived directly from some conventional formalism.…”

A hierarchical approach for rigid-body dynamics model simplification of a high-speed parallel robot by considering kinematics performance

“…Considering this, many scholars have done a lot of research on the simplification method of the dynamics model. To propose a general and concise dynamics modeling method, Orsino et al [13][14][15] proposed a modular modeling methodology for parallel robots, where any restriction concerning either the space index or the number of degrees of freedom does not need to be considered. Considering the contribution of different torque components, Carbonari 16 proposed a polynomial simplified dynamics model for the 3-central processing unit class of parallel robots by fitting the actual behavior of the robot prototype, which shows substantial reliability of the model in non-static conditions and advantages in terms of computation time.…”