Constraint Stabilization of Mechanical Systems in Ordinary Differential Equations Form

Abstract

This work discusses a simple means to add kinematic constraints to existing mechanical problems formulated in terms of ordinary differential equations. The constraints are expressed by algebraic relationships between the co-ordinates of the unconstrained problem. A solution projection approach ensures compliance of the solution with the derivatives of holonomic constraint equations up to second order within the desired accuracy. The structure of the unconstrained problem is not altered, resulting in a simple, little intrusive, yet effective means to enforce kinematic constraints into existing formulations and implementations originally intended to address differential problems, without the complexity of solving differential-algebraic problems or resorting to implicit numerical integration schemes and without altering the number and type of equations of the original unconstrained problem. The proposed formulation is compared with known approaches. Numerical applications of increasing complexity illustrate its distinguishing aspects

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

“…However, the absence of constraint equations of velocity and position will lead to the constraint violation caused by accumulated numerical integration errors. Some stabilization procedures 22–24 , mass-orthogonal projection method 11 , direct correction approach 25 or TLISMNI method 26 should be implemented to eliminate or minimum the constraint violation at position and velocity level. By adding the Baumgarte stabilization method (BSM) to equation (9), the dynamic equations of motion turns into the following form where α and β are two penalty factors which should be appropriately chosen as positive constants by experience.…”

Computed torque control of redundant manipulators using general-purpose software in real-time

“…needs to be solved. As shown in [31], weighting the norm of the velocity correction with the mass matrix minimizes (and guarantees to reduce, in case of scleronomic constraints) the correction of the kinetic energy.…”

A regularization method for solving dynamic problems with singular configuration

“…Based on Gauss principle, 23 Udwadia and Kalaba 24 and Fan et al 25 described the dynamic motion of mechanical systems in an explicit expression with Moore-Penrose pseudoinverse. In reference, 26,27 Baumgarte stabilization method is combined with Udwadia and Kalaba method to eliminate the constraint violation. However, these methods are presented mainly for the regular problems and may show some weaknesses to handle the singular problems.…”