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Analytical Solutions in the Darboux Problem and the Bortz Equation and the Approach to Orientation Algorithms Based on Them

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Abstract

We consider the problem of determining the angular position of a rigid body in space from its known angular velocity and initial position (the Darboux problem) in the quaternion setting. Based on the exact solution of the Bortz approximate differential equation with respect to the orientation vector of the rigid body, we analytically solve the problem to determine the quaternion of the orientation of the rigid body for each arbitrary angular velocity and small rotation angle of the rigid body. Based on this solution, we propose an approach to design a new algorithm to compute the orientation of moving objects by means of strapdown inertial navigation systems.

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Funding

This work was supported by the Russian Foundation for Basic Research, project no. 19-01-00205.

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Authors and Affiliations

  1. Institute of Precision Mechanics and Control, Russian Academy of Sciences, 410028, Saratov, Russia

    A. V. Molodenkov & Ya. G. Sapunkov

  2. Gagarin Saratov State Technical University, 410054, Saratov, Russia

    T. V. Molodenkova

  3. Ishlinsky Institute for Problems in Mechanics, Russian Academy of Sciences, 119526, Moscow, Russia

    S. E. Perelyaev

Authors
  1. A. V. Molodenkov
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  2. Ya. G. Sapunkov
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  3. T. V. Molodenkova
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  4. S. E. Perelyaev
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Corresponding authors

Correspondence to A. V. Molodenkov, Ya. G. Sapunkov, T. V. Molodenkova or S. E. Perelyaev.

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Translated by A. Muravnik

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Molodenkov, A.V., Sapunkov, Y.G., Molodenkova, T.V. et al. Analytical Solutions in the Darboux Problem and the Bortz Equation and the Approach to Orientation Algorithms Based on Them. Mech. Solids 55, 1013–1020 (2020). https://doi.org/10.3103/S002565442007016X

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  • DOI: https://doi.org/10.3103/S002565442007016X

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