“Optimal Statistical Operations for 3-Dimensional Rotational Data: Geometric Interpretations and Application to Prosthesis Kinematics”

Authors: Øyvind Stavdahl, Anne K. Bondhus, Kristin Y. Pettersen and Kjell E. Malvig,
Affiliation: NTNU, Department of Engineering Cybernetics and SINTEF
Reference: 2005, Vol 26, No 4, pp. 185-200.

Keywords: Orientation statistics, rotational data, rotation matrix, quaternion, Euler parameter, Euler angle, orientation vector, attitude vector kinematics, orthopaedics, prosthetics, biomechanics

Abstract: Rotational data in the form of measured three-dimensional rotations or orientations arise naturally in many fields of science, including biomechanics, orthopaedics and robotics. The cyclic topology of rotation spaces calls for special care and considerations when performing statistical analysis of rotational data. Relevant theory has been developed during the last three decades, and has become a standard tool in some areas. In relation to the study of human kinematics and motion however, these concepts have hardly been put to use. This paper gives an introduction to the intricacies of three-dimensional rotations, and provides a thorough geometric interpretation of several approaches to averaging rotational data A set of novel, simple operators is presented. Simulations and a prosthetics-related real-world example involving wrist kinematics illuminate important aspects of the results. Finally generalizations and related subjects for further research are suggested.

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References:
[1] ANDREWS, J. G. (1984). On the specification of joint configurations and motions, I Biomech., 17(2), pp. 155-158 doi:10.1016/0021-9290(84)90133-7
[2] ARIEL, G. B., BUIJS, R. J. C., PENNY, A. CHUNG, S. G. (2000). Visualizing orientation using quaternions, Proc of the Sixth International Symposium on the 3D Analysis of Human Movement, pages 21-24.Cape Town, South Africa, May 2000, International Society of Biomechanics pp. 21-24.
[3] BRUMFIELD, R., NICKEL, V. NICKEL E. (1966). Joint motion in wrist flexion and extension, South Med. J., 59, pp. 909-910.
[4] CHAO, E. Y. S. (1980). Justification of triaxial goniometer for the measurement of joint rotation, J. Biomech., 13, pp. 989-1006 doi:10.1016/0021-9290(80)90044-5
[5] DIACONIS, P. (1988). Group Representations in Probability and Statistics, volume 11 of Lecture Notes Monograph Series.Institute of Mathematical Statistics, Hayward, CA, 1988.
[6] DOWNS, T. D. (1972). Orientation statistics, Biometrika, 59, pp. 665-676 doi:10.1093/biomet/59.3.665
[7] GROOD, E. S. SUNTAY, W. J. (1983). A joint coordinate system for the clinical description of three-dimensional motions: Application to the knee, J. Biomech. Eng., 10.2, pp. 136-144.
[8] HUMBERT, M., GEY, N., MULLER, L ESLING, C. (1996). Determination of a mean orientation from a cloud of orientations, Application to electron back-scattering pattern measurements. J. Appt Crystallogr., 29, pp. 662-666 doi:10.1107/S0021889896006693
[9] HUMBERT, M., GEY, N., MULLER, J. ESLING, C. (1998). Response to Morawiec´s, 1998 Comment on determination of a mean orientation from a cloud of orientations. Application to electron back-scattering pattern measurements, J. Appl. Crystallogr., 31, p. 485 doi:10.1107/S0021889898004014
[10] HUMPHREYS, F. J., BATE, P. S. HURLEY, P.J. (2001). Orientation averaging of electron backscattered diffraction data, I. Microsc., 20.Pt 1, pp. 50-58 doi:10.1046/j.1365-2818.2001.00777.x
[11] KHATRI, C.G. MARDIA, K.V. (1977). The von Mises-Fisher matrix distribution in orientation statistics, J. Roy. Stat. Soc., Ser B, 3.1, pp. 95-106.
[12] LANDRY, J.S. BIDEN, E.N. (2002). Optimal fixed wrist alignment for below-elbow, powered, prosthetic hands, MEC´2002 Conference Proceedings.Inst. of Biomedical Engineering, Univ. of New Brunswick, Fredericton, NB, Canada, August 2002 pp. 12-14.
[13] KRIEGER LASSEN, N. C., JUUL JENSEN, D. CONRADSEN, K. (1994). On the statistical analysis of orientation data, Acta-Crystallogr. A50, pp. 741-748 doi:10.1107/S010876739400437X
[14] PALMER, A. K., WERNER, F. W., MURPHY, D. GLISSON, R. (1985). Functional wrist motion: A biomechanical study, J. Hand. Surg. ´Aug 10.1. pp. 39-46.
[15] PRENTICE, M. J. (1986). Orientation statistics without parametric assumptions, J. Roy. Stat. Soc. Ser B. 4.2, pp. 214-222.
[16] RANCOURT, D., RIVEST, L.-P. ASSELIN, J. (2000). Using orientation statistics to investigate variations in human kinematics, Appl. Statist., 49, pp. 81-94 doi:10.1111/1467-9876.00180
[17] RUY, J., COONEY, W. P., ASKEW, L. J., AN, K.-N. CHAO, E. Y. S. (1991). Functional ranges of motion of the wrist joint, J. Hand. Surge [Am.] 16.3, pp. 409-419.
[18] SARRAFIAN, S. K., MELAMED, J. L. GOSHGARIAN, G. M. (1977). Study of wrist motion in flexion and extension, Clin. Orthop, 126, pp. 153-159.
[19] SHEEHAN, F. T. MITIGUY, P. (1999). In regards to the ISB recommendations for standardization in the reporting of kinematic data, J. Biomech.. 32(10). pp. 1135-1136 (Letter to the Editor).
[20] SPONG, M. W. VIDYASAGAR., M. (1989). Robot Dynamics and Control, John Wiley and Sons. 1989.
[21] STAVDAHL, O. (2002). Optimal wrist prosthesis kinematics: three-dimensional rotation statistics and parameter estimation, Ph.D. thesis.Norwegian University of Science and Technology, Trondheim, Norway, 2002. ISBN 82-471-5526-5.
[22] STEPHENS, M. A. (1979). Vector correlation, Biometrika, 6.1, pp. 41-48 doi:10.1093/biomet/66.1.41
[23] VELDPAUS, F.E., WOLTRING, H. J. DORTMANS, L.J.M.G. (1988). A least-squares algorithm for the equiform transformation from spinal marker co-ordinates, J. Biomech., 21(1), pp. 45-54 doi:10.1016/0021-9290(88)90190-X
[24] WOLTRING, H. J., HUISKES. R. DE LANGE, A. (1985). Finite centroid and helical axis estimation from noisy landmark measurements in the study of human joint kinematics, J. Biomech., 18(5), pp. 379-389 doi:10.1016/0021-9290(85)90293-3
[25] WOLTRING, H. J. (1994). 3-D attitude representation of human joints: a standardization proposal, J. Biotnech., 2.12, pp. 1399-1414.
[26] WU, G. CAVANAGH, P.R. (1995). ISB recommendations for standardization in the reporting of kinematic data, I Biomech., 28, pp. 1258-1261.
[27] YOUM, Y., McMURTRY, R.Y., FLATT, A.E. GILLESPIE, T.E. (1978). Kinematics of the wrist, I: An experimental study of radioulnar deviation and flexion-extension, J. Bone Joint Surg., 60-.4, pp. 423-431.


BibTeX:
@article{MIC-2005-4-1,
  title={{Optimal Statistical Operations for 3-Dimensional Rotational Data: Geometric Interpretations and Application to Prosthesis Kinematics}},
  author={Stavdahl, Øyvind and Bondhus, Anne K. and Pettersen, Kristin Y. and Malvig, Kjell E.},
  journal={Modeling, Identification and Control},
  volume={26},
  number={4},
  pages={185--200},
  year={2005},
  doi={10.4173/mic.2005.4.1},
  publisher={Norwegian Society of Automatic Control}
};