Comparison of Selected Formulations for Multibody System Dynamics with Redundant Constraints

Abstract

This paper compares selected optimization-based methods for the analysis of multibody systems with redundant constraints. The following numerical schemes are examined: direct integration method, Udwadia-Kalaba formulation, two types of least-squares block solution method and Udwadia-Phohomsiri formulation. In order to compare efficiency of the algorithms, a series of simulations is performed on two exemplary McPherson struts. In the first variant, the mechanism has no redundant constraints whereas the other is overconstrained. Three constraint stabilization schemes are also compared in terms of integration errors.

Application of Gauss principle of least constraint in multibody systems with redundant constraints

“…Udwadia and Phohomsiri 12 presented a new, general, explicit dynamic equation, which is suitable for the systems with singular mass matrices, holonomic or non-holonomic redundant constraints. Based on D’Alembert’s principle, the work done by ideal constraint force under virtual displacement equals zero 27 . However, when it comes to non-ideal constraint force, the work done is hard to describe.…”

Udwadia-Kalaba Approach for Three Link Manipulator Dynamics With Motion Constraints

“…The constraints must be satisfied at each instant of time, including the initial time, while in practice, due to various factors, it is usually quite difficult for the initial conditions to satisfy the constraint equations, for this reason, many researches use famous Baumgarte's method [38] to correct numerical drift when the initial conditions are incompatible with the constrained equations, see refs [6], [39]- [41]. However, the introduced parameters, α and β must be carefully selected, since the selection can make the reformulated problem stiff.…”

The contact problem in Lagrangian systems with redundant frictional bilateral and unilateral constraints and singular mass matrix. The all-sticking contacts problem

“…KKT problems as (14) (a) have been studied using Moore-Penrose generalized inverses in [75] with full-rank mass matrix, where problems related to the dependence of numerical solutions to chosen units are analysed. Many other results for (26) may be found in [76,55,73,27,74], with both uniqueness of contact forces analysis and algorithms for solving the KKT problem.…”

“…This is the reason why we chose to work with (43) instead, with a matrix with a skew-symmetric part, enabling us to apply Proposition 16. This is a big difference with the bilaterally constrained case (26), which does not rely on any positivity of the matrixM b (q), but just on linear algebra arguments, so that the symmetric form in (26) is usually chosen [75,55,27,76]. But unilaterality yields complementarity problems which need some positivity.…”