Abstract
In the mid-nineteenth century, Donders had proposed that for every human head rotating away from the primary pointing direction, the rotational vectors in the direction of the corresponding axes of rotation, is restricted to lie on a surface. Donders’ intuition was that under such a restriction, the head orientation would be a function of its pointing direction. In this paper, we revisit Donders’ Law and show that indeed the proposed intuition is true for a restricted class of head-orientations satisfying a class of quadratic Donders’ surfaces, if the head points to a suitable neighborhood of the frontal pointing direction. Moreover, on a suitably chosen subspace of the 3D rotation group SO(3), we describe a head movement dynamical system with input control signals that are the three external torques on the head provided by muscles. Three output signals are also suitably chosen as follows. Two of the output signals are coordinates of the frontal pointing direction. The third signal measures deviation of the state vector from the Donders’ surface. We claim that the square system is locally feedback linearizable on the subspace chosen, and the linear dynamics is decomposed into parts, transverse and tangential to the Donders’ surface. We demonstrate our approach by synthesizing a tracking and path-following controller. Additionally, for different choices of the Donders’ surface parameters, head gaits are visualized by simulating different movement patterns of the head-top vector, as the head-pointing vector rotates around a circle.
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Acknowledgments
The second author would like to acknowledge the contribution of Dr. Stefan Glasauer and Dr. Indika Wijayasinghe for their collaboration on the head movement problem, especially understanding of the Donders’ constraint.
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This paper is dedicated to Professor T. J. Tarn on the occasion of his 80th birthday.
This work was supported by the Dick and Martha Brooks Professorship to Texas Tech University.
Takafumi OKI was born in Japan in 1980. He received his B.Sc. and M.Sc. degrees from Tokyo Denki University, Japan in 2004 and 2006 respectively, and his Ph.D. degree in Mathematics from Texas Tech University, Lubbock, Texas, U.S.A., in 2016. His primary research interests are in nonlinear feedback stabilization and tracking, linear and nonlinear optimization problems.
Bijoy K. GHOSH received his Ph.D. degree in Engineering Sciences from the Decision and Control Group of the Division of Applied Sciences, Harvard University, Cambridge, MA, in 1983. From 1983 to 2007 Bijoy was with the Department of Electrical and Systems Engineering, Washington University, St. Louis, MO, U.S.A., where he was a Professor and Director of the Center for BioCybernetics and Intelligent Systems. Currently, he is the Dick and Martha Brooks Regents Professor of Mathematics and Statistics at Texas Tech University, Lubbock, TX, U.S.A. He received the D. P. Eckmann award in 1988 from the American Automatic Control Council, the Japan Society for the Promotion of Sciences Invitation Fellowship in 1997, the Chinese Academy of Sciences Invitation Fellowship in 2016, the Indian Institute of Technology, Kharagpur, Distinguished Visiting Professorship in 2016. He became a Fellow of the IEEE in 2000, a Fellow of the International Federation on Automatic Control in 2014 and a Fellow of South Asia Institute of Science and Engineering in 2016. Bijoy had held visiting positions at Tokyo Institute of Technology, Osaka University and Tokyo Denki University, Japan, University of Padova in Italy, Royal Institute of Technology and Institut Mittag-Leffler, Stockholm, Sweden, Yale University, U.S.A., Technical University of Munich, Germany, Chinese Academy of Sciences, China and Indian Institute of Technology, Kharagpur, India. Bijoy’s current research interest is in BioMechanics, Cyberphysical Systems and Control Problems in Rehabilitation Engineering.
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Oki, T., Ghosh, B.K. A transverse local feedback linearization approach to human head movement control. Control Theory Technol. 15, 288–300 (2017). https://doi.org/10.1007/s11768-017-7034-9
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DOI: https://doi.org/10.1007/s11768-017-7034-9