Semialgebraic Regularization of Kinematic Loop Constraints in Multibody System Models

Redundant constraints in multibody system (MBS) models, reflected by a singular constraint Jacobian, impair the efficient dynamics simulation. In particular, kinematic loop constraints are often found to be permanently redundant. This problem is commonly attacked numerically by decomposing the constraint Jacobian either at every simulation time step or beforehand in an admissible assembly (assuming that the redundancy is permanent). This paper presents a method for the elimination of permanently redundant loop cloture constraints, which, instead of numerically decomposing the constraints, relies on the geometric characterization of kinematic loops comprising lower kinematic pairs. In particular, the invariant vector space of velocities of a kinematic loop is taken into account, which can be determined as the sum of Lie (screw) algebras of two sub-chains of a kinematic loop. The actual reduction is achieved by restricting the constraints to this space. The presented method does not interfere with the actual generation of constraints but can be considered as a preprocessing step of MBS models. It is numerically robust and only uses a geometrically exact model. The method is able to completely eliminate redundant loop constraints for "nonparadoxical" single-loop mechanisms and applies conservatively to multiloop MBS. The presented method only requires information (vectors, matrices) that is readily available in any MBS simulation package. The only numerical operations involved are cross products and a singular value decomposition of a low dimensional matrix.