A novel Quaternion integration approach for describing the behaviour of non-spherical particles

Abstract

There are three main frameworks to describe the orientation and rotation of non-spherical particles: Euler angles, rotation matrices and unit Quaternions. Of these methods, the latter seems the most attractive for describing the behaviour of non-spherical particles. However, there are a number of drawbacks when using unit Quaternions: the necessity of applying rotation matrices in conjunction to facilitate the transformation from body space to world space, and the algorithm integrating the Quaternion should inherently conserve the length of the Quaternion. Both drawbacks are addressed in this paper. The present paper derives a new framework to transform vectors and tensors by unit Quaternions, and the requirement of explicitly using rotation matrices is removed altogether. This means that the algorithm derived in this paper can describe the rotation of a non-spherical particle with four parameters only. Moreover, this paper introduces a novel corrector-predictor method to integrate unit Quaternions, which inherently conserves the length of the Quaternion. The novel framework and method are compared to a number of other methods put forward in the literature. All the integration methods are discussed, scrutinised and compared to each other by comparing the results of four test cases, involving a single falling particle, nine falling and interacting particles, a 2D prescribed torque on a sphere and a 3D prescribed torque on a non-spherical particle. Moreover, a convergence study is presented, comparing the rate of convergence of the various methods. All the test cases show a significant improvement of the new framework put forward in this paper over existing algorithms. Moreover, the new method requires less computational memory and fewer operations, due to the complete omission of the rotation matrix in the algorithm.

Improved quaternion-based integration scheme for rigid body motion

“…However, for objects which experience a non-constant torque, we will show that the accuracy of the original scheme of Ref. [17] is not second order, as claimed by the authors, but closer to first order.…”

“…In principle an evolution equation could be derived for the quaternions, but instead we will make use of the following property for a quaternion q (see Ref. [17] for more details):…”

“…(4) is equal to 1 × 10 5 N m. The time step for the simulation is equal to 1 × 10 −4 s. This test case is equal to the third test case presented by the authors of Ref. [17]. The torque will result in an accelerating rotation (see Fig.…”

“…It is better to only use the rotation quaternions in the update of the rigid body orientation. Zhao and Van Wachem [17] introduced a quaternion-based integration scheme that uses no rotation matrices. The method is based on a predictor-corrector scheme to update the angular velocity and rotation quaternion.…”

“…In this paper, we propose an improvement of the scheme of Zhao and Van Wachem [17], such that the method retains the intrinsic quaternion norm conservation, can be used in cases where the forces and torques are a function of position, orientation, linear and angular velocity. Most importantly, we will show that the improved method is second order accurate in time, allowing for much larger time steps to be used than in the original scheme.…”