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

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 uni...

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Published in:	Multibody system dynamics Vol. 38; no. 3; pp. 201 - 225
Main Authors:	Terze, Zdravko, Müller, Andreas, Zlatar, Dario
Format:	Journal Article
Language:	English
Published:	Dordrecht Springer Netherlands 01.11.2016 Springer Nature B.V
Subjects:	Algorithms Automotive Engineering Control Differential equations Dynamical Systems Electrical Engineering Engineering Kinematics Lie groups Mechanical Engineering Numerical integration Optimization Ordinary differential equations Quaternions Rotation Singularity (mathematics) Time integration Vibration Time integration schemes Special orthogonal group Lie groups Rotational quaternions Symplectic group Integration of quaternions Special unitary group Spatial rotations
ISSN:	1384-5640, 1573-272X
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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.
AbstractList	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. 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.
Author	Terze, Zdravko Müller, Andreas Zlatar, Dario
Author_xml	– sequence: 1 givenname: Zdravko surname: Terze fullname: Terze, Zdravko email: zdravko.terze@fsb.hr organization: Department of Aeronautical Engineering, Faculty of Mechanical Engineering and Naval Architecture, University of Zagreb – sequence: 2 givenname: Andreas surname: Müller fullname: Müller, Andreas organization: Institute of Robotics, Johannes Kepler University – sequence: 3 givenname: Dario surname: Zlatar fullname: Zlatar, Dario organization: Department of Aeronautical Engineering, Faculty of Mechanical Engineering and Naval Architecture, University of Zagreb
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Cites_doi	10.2514/3.55974 10.1142/p549 10.1002/zamm.200900383 10.1109/TSMC.1982.4308815 10.1007/BF02510919 10.1007/978-3-662-09156-2 10.2307/2689481 10.1016/S0045-7825(98)00031-0 10.1016/S0168-9274(98)00030-0 10.1364/JOSAA.4.000629 10.1098/rsta.1999.0360 10.1007/978-0-85729-760-0 10.1145/325165.325242 10.1007/978-3-642-52465-3_9 10.1007/978-94-007-0335-3 10.1147/rd.234.0424 10.1088/0004-6256/135/6/2298 10.1023/A:1009701220769 10.4173/mic.1997.1.4 10.1007/s11044-014-9439-2 10.1007/978-1-4612-5286-3 10.1002/nme.2586 10.1017/S0962492900002154 10.1016/S0045-7825(02)00520-0 10.1016/j.cam.2013.10.039 10.2514/6.1970-996 10.1007/978-3-319-07260-9_10
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Snippet	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...
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SubjectTerms	Algorithms Automotive Engineering Control Differential equations Dynamical Systems Electrical Engineering Engineering Kinematics Lie groups Mechanical Engineering Numerical integration Optimization Ordinary differential equations Quaternions Rotation Singularity (mathematics) Time integration Vibration
Title	Singularity-free time integration of rotational quaternions using non-redundant ordinary differential equations
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