Euler parameters kinetic singularity

Abstract

Euler parameters do not suffer from the kinematic singularity associated with the three-parameter representation of the rotation. For this reason, the four Euler parameters are widely used in describing the motion of rigid bodies in space. These parameters are related by an algebraic equation which must be satisfied at the position, velocity, and acceleration levels. However, the constraint forces associated with Euler parameter constraints are always zero and lead to a kinetic singularity when a certain solution procedure is used to solve for the accelerations. It is shown in this paper that when the equations are formulated in terms of Euler parameters, the use of a solution procedure for the accelerations that requires the elimination of Lagrange multipliers leads to a singular coefficient matrix. Since Lagrange multipliers associated with Euler parameters constraints always define zero constraint forces, a nonsingular coefficient matrix can be identified and used to solve for Lagrange multipliers associated with other joint constraints. The proposed solution procedure requires the inverse of a 3 × 3 constant matrix for each rigid body in the system and the LU solution of a system of algebraic equations that has dimension equal to the number of constraint equations excluding Euler parameters constraints.