Singularity-free time integration of rotational quaternions using non-redundant ordinary differential equations

Terze, Zdravko and Mueller, Andreas and Zlatar, Dario (2016) Singularity-free time integration of rotational quaternions using non-redundant ordinary differential equations. = Singularity-free time integration of rotational quaternions using non-redundant ordinary differential equations. Multibody system dynamics, 37 (3). pp. 1-25. ISSN 1384-5640. Vrsta rada: ["eprint_fieldopt_article_type_article" not defined]. Kvartili JCR: Q1 (2016). Točan broj autora: 3.

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Official URL: http://link.springer.com/article/10.1007%2Fs11044-...

Abstract

A novel ODE time stepping scheme for solving rotational kinematics in terms of unit quaternions is presented in the paper. This scheme inherently respects the unit-length condition without including it explicitly as a constraint equation, as it is common practice. In the standard algorithms, the unit-length condition is included as an additional equation leading to kinematical equations in the form of a system of differential-algebraic equations (DAEs). On the contrary, the proposed method is based on numerical integration of the kinematic relations in terms of the instantaneous rotation vector that form a system of ordinary differential equations (ODEs) on the Lie algebra so(3) of the rotation group SO(3). This rotation vector defines an incremental rotation (and thus the associated incremental unit quaternion), and the rotation update is determined by the exponential mapping on the quaternion group. Since the kinematic ODE on so(3) can be solved by using any standard (possibly higher-order) ODE integration scheme, the proposed method yields a non-redundant integration algorithm for the rotational kinematics in terms of unit quaternions, avoiding integration of DAE equations. Besides being ‘more elegant’—in the opinion of the authors—this integration procedure also exhibits numerical advantages in terms of better accuracy when longer integration steps are applied during simulation. As presented in the paper, the numerical integration of three non-linear ODEs in terms of the rotation vector as canonical coordinates achieves a higher accuracy compared to integrating the four (linear in ODE part) standard-quaternion DAE system. In summary, this paper solves the long-standing problem of the necessity of imposing the unit-length constraint equation during integration of quaternions, i.e. the need to deal with DAE’s in the context of such kinematical model, which has been a major drawback of using quaternions, and a numerical scheme is presented that also allows for longer integration steps during kinematic reconstruction of large three-dimensional rotations.

Item Type: Article (["eprint_fieldopt_article_type_article" not defined])
Keywords (Croatian): time integration schemes, spatial rotations, rotational quaternions, integration of quaternions, lie groups, special orthogonal group SO(3), symplectic group Sp(1), special unitary group SU(2)
Subjects: NATURAL SCIENCES > Mathematics
TECHNICAL SCIENCE > Mechanical Engineering
TECHNICAL SCIENCE > Aviation, rocket and space technology
Divisions: 1300 Department of Aeronautical Engineering > 1320 Chair of Aircraft Dynamics
Indexed in Web of Science: Yes
Indexed in Current Contents: Yes
Quartiles: Q1 (2016)
Date Deposited: 22 Sep 2016 13:26
Last Modified: 15 Mar 2018 12:37
URI: http://repozitorij.fsb.hr/id/eprint/6963

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