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On Stabilization of an Unbalanced Lagrange Gyrostat

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Mathematical and Computational Approaches in Advancing Modern Science and Engineering

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Abstract

In this contribution we consider a chain of two coupled Lagrange gyrostats moving about a fixed point in a non-homogeneous gravitational field and show the existence of the following stabilization effect: for a gyrostat that is in its unstable equilibrium position, there is another gyrostat such that, when the two of them are coupled to form this chain, the rotation of the latter gyrostat can be used to stabilize the equilibrium of the former one. We establish and analyze stabilization conditions in the space of mechanical parameters characterizing the chain.

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Acknowledgements

Support for this project was provided by a PSC-CUNY Award, jointly funded by The Professional Staff Congress and The City University of New York (Award # 68091-00 46). The work of the second and third authors was supported by the NASA New York Space Grant CCPP Program.

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Authors and Affiliations

  1. City University of New York – LaGCC, Long Island City, NY, USA

    Dmitriy Chebanov, Natalia Mosina & Jose Salas

Authors
  1. Dmitriy Chebanov
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  2. Natalia Mosina
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  3. Jose Salas
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Corresponding author

Correspondence to Dmitriy Chebanov .

Editor information

Editors and Affiliations

  1. Department of Mathematics and Statistics, University of Montreal, Montreal, Québec, Canada

    Jacques Bélair

  2. Department of Mathematics, University of British Columbia, Vancouver, British Columbia, Canada

    Ian A. Frigaard

  3. Department of Mathematics and Statistics, University of Guelph, Guelph, Ontario, Canada

    Herb Kunze

  4. Department of Mathematics, Wilfrid Laurier University, Waterloo, Ontario, Canada

    Roman Makarov

  5. MS2Discovery Institute, Wilfrid Laurier University, Waterloo, Ontario, Canada

    Roderick Melnik

  6. Department of Computer Science, University of Saskatchewan, Saskatoon, Saskatchewan, Canada

    Raymond J. Spiteri

Appendix

Appendix

The coefficients μ 6, μ 5, …, μ 0 of \(\varLambda _{3}\) in (8) admit the following representation in terms of the dimensionless parameters κ 1, κ 2, κ 3, κ 4.

$$ \displaystyle\begin{array}{rcl} \mu _{6}& =& \kappa _{4}^{2} - 4, {}\\ \mu _{5}& =& -2\kappa _{4}\left [\left (\kappa _{1} - 2\kappa _{2}\left (1 +\kappa _{ 3}^{2}\right ) + 1\right )\kappa _{ 4}^{2} + 9\kappa _{ 2}\left (1 +\kappa _{ 3}^{2}\right ) - 2\left (2\kappa _{ 1} -\kappa _{2}\kappa _{3} + 2\right )\right ], {}\\ \mu _{4}& =& \left [8\kappa _{2}\kappa _{3}^{2}\left (2\kappa _{ 2} - 1\right ) +\kappa _{1}\left (\kappa _{1} - 8\kappa _{2} + 4\right ) + 1\right ]\kappa _{4}^{4} - 2\bigl [12\kappa _{ 2}^{2}\kappa _{ 3}^{3} + 3\kappa _{ 1}\kappa _{2}\kappa _{3}^{2} {}\\ & & +40\kappa _{2}^{2}\kappa _{ 3}^{2} - 8\kappa _{ 1}\kappa _{2}\kappa _{3} + 12\kappa _{2}^{2}\kappa _{ 3} - 19\kappa _{2}\kappa _{3}^{2} +\kappa _{ 1}^{2} - 19\kappa _{ 1}\kappa _{2} - 8\kappa _{2}\kappa _{3} {}\\ & & +12\kappa _{1} + 3\kappa _{2} + 1\bigr ]\kappa _{4}^{2} - 27\kappa _{ 2}^{2}\kappa _{ 3}^{4} + 36\kappa _{ 2}^{2}\kappa _{ 3}^{3} + 36\kappa _{ 1}\kappa _{2}\kappa _{3}^{2} - 2\kappa _{ 2}^{2}\kappa _{ 3}^{2} {}\\ & & -40\kappa _{1}\kappa _{2}\kappa _{3} + 36\kappa _{2}^{2}\kappa _{ 3} - 12\kappa _{2}\kappa _{3}^{2} - 8\kappa _{ 1}^{2} - 12\kappa _{ 1}\kappa _{2} - 27\kappa _{2}^{2} - 40\kappa _{ 2}\kappa _{3} {}\\ & & +32\kappa _{1} + 36\kappa _{2} - 8, {}\\ \mu _{3}& =& -2\kappa _{4}\bigl [\left (\kappa _{1}^{2} +\kappa _{ 1} - 2\kappa _{1}^{2}\kappa _{ 2} - 2\kappa _{2}\kappa _{3}^{2}\right )\kappa _{ 4}^{4} +\bigl ( 64\kappa _{ 2}^{3}\kappa _{ 3}^{3} + 8\kappa _{ 1}\kappa _{2}^{2}\kappa _{ 3}^{2} - 40\kappa _{ 2}^{2}\kappa _{ 3}^{3} {}\\ & & +2\kappa _{1}^{2}\kappa _{ 2}\kappa _{3} - 40\kappa _{1}\kappa _{2}^{2}\kappa _{ 3} + 7\kappa _{1}\kappa _{2}\kappa _{3}^{2} + 8\kappa _{ 2}^{2}\kappa _{ 3}^{2} +\kappa _{ 1}^{3} + 7\kappa _{ 1}^{2}\kappa _{ 2} + 24\kappa _{1}\kappa _{2}\kappa _{3} {}\\ & & +7\kappa _{2}\kappa _{3}^{2} - 9\kappa _{ 1}^{2} + 7\kappa _{ 1}\kappa _{2} + 2\kappa _{2}\kappa _{3} - 9\kappa _{1} + 1\bigr )\kappa _{4}^{2} + 48\kappa _{ 2}^{3}\kappa _{ 3}^{4} + 6\kappa _{ 1}\kappa _{2}^{2}\kappa _{ 3}^{3} {}\\ & & -160\kappa _{2}^{3}\kappa _{ 3}^{3} - 18\kappa _{ 2}^{2}\kappa _{ 3}^{4} + 3\kappa _{ 1}^{2}\kappa _{ 2}\kappa _{3}^{2} - 38\kappa _{ 1}\kappa _{2}^{2}\kappa _{ 3}^{2} + 48\kappa _{ 2}^{3}\kappa _{ 3}^{2} + 130\kappa _{ 2}^{2}\kappa _{ 3}^{3} {}\\ & & -2\kappa _{1}^{2}\kappa _{ 2}\kappa _{3} + 130\kappa _{1}\kappa _{2}^{2}\kappa _{ 3} - 26\kappa _{1}\kappa _{2}\kappa _{3}^{2} - 38\kappa _{ 2}^{2}\kappa _{ 3}^{2} - 4\kappa _{ 1}^{3} + 7\kappa _{ 1}^{2}\kappa _{ 2} {}\\ & & -18\kappa _{1}\kappa _{2}^{2} - 92\kappa _{ 1}\kappa _{2}\kappa _{3} + 6\kappa _{2}^{2}\kappa _{ 3} + 7\kappa _{2}\kappa _{3}^{2} + 20\kappa _{ 1}^{2} - 26\kappa _{ 1}\kappa _{2} - 2\kappa _{2}\kappa _{3} + 20\kappa _{1} {}\\ & & +3\kappa _{2} - 4\bigr ], {}\\ \mu _{2}& =& \kappa _{1}^{2}\kappa _{ 4}^{6} - 2\bigl (12\kappa _{ 1}^{2}\kappa _{ 2}^{2}\kappa _{ 3} + 40\kappa _{1}\kappa _{2}^{2}\kappa _{ 3}^{2} + 12\kappa _{ 2}^{2}\kappa _{ 3}^{3} + 3\kappa _{ 1}^{3}\kappa _{ 2} - 8\kappa _{1}^{2}\kappa _{ 2}\kappa _{3} {}\\ & & -19\kappa _{1}\kappa _{2}\kappa _{3}^{2} +\kappa _{ 1}^{3} + 38\kappa _{ 1}^{2}\kappa _{ 2} - 8\kappa _{1}\kappa _{2}\kappa _{3} + 3\kappa _{2}\kappa _{3}^{2} + 12\kappa _{ 1}^{2} +\kappa _{ 1}\bigr )\kappa _{4}^{4} {}\\ & & +\bigl (256\kappa _{2}^{4}\kappa _{ 3}^{4} - 64\kappa _{ 1}\kappa _{2}^{3}\kappa _{ 3}^{3} - 320\kappa _{ 2}^{3}\kappa _{ 3}^{4} - 66\kappa _{ 1}^{2}\kappa _{ 2}^{2}\kappa _{ 3}^{2} - 320\kappa _{ 1}\kappa _{2}^{3}\kappa _{ 3}^{2} {}\\ & & +76\kappa _{1}\kappa _{2}^{2}\kappa _{ 3}^{3} - 64\kappa _{ 2}^{3}\kappa _{ 3}^{3} - 2\kappa _{ 2}^{2}\kappa _{ 3}^{4} - 4\kappa _{ 1}^{3}\kappa _{ 2}\kappa _{3} + 76\kappa _{1}^{2}\kappa _{ 2}^{2}\kappa _{ 3} + 40\kappa _{1}^{2}\kappa _{ 2}\kappa _{3}^{2} {}\\ & & +696\kappa _{1}\kappa _{2}^{2}\kappa _{ 3}^{2} + 76\kappa _{ 2}^{2}\kappa _{ 3}^{3} +\kappa _{ 1}^{4} + 40\kappa _{ 1}^{3}\kappa _{ 2} - 2\kappa _{1}^{2}\kappa _{ 2}^{2} - 108\kappa _{ 1}^{2}\kappa _{ 2}\kappa _{3} + 76\kappa _{1}\kappa _{2}^{2}\kappa _{ 3} {}\\ & & -192\kappa _{1}\kappa _{2}\kappa _{3}^{2} - 66\kappa _{ 2}^{2}\kappa _{ 3}^{2} - 4\kappa _{ 1}^{3} - 192\kappa _{ 1}^{2}\kappa _{ 2} - 108\kappa _{1}\kappa _{2}\kappa _{3} + 40\kappa _{2}\kappa _{3}^{2} + 102\kappa _{ 1}^{2} {}\\ & & +40\kappa _{1}\kappa _{2} - 4\kappa _{2}\kappa _{3} - 4\kappa _{1} + 1\bigr )\kappa _{4}^{2} + 4\bigl (64\kappa _{ 2}^{4}\kappa _{ 3}^{5} - 24\kappa _{ 1}\kappa _{2}^{3}\kappa _{ 3}^{4} - 128\kappa _{ 2}^{4}\kappa _{ 3}^{4} {}\\ & & -72\kappa _{2}^{3}\kappa _{ 3}^{5} - 6\kappa _{ 1}^{2}\kappa _{ 2}^{2}\kappa _{ 3}^{3} - 56\kappa _{ 1}\kappa _{2}^{3}\kappa _{ 3}^{3} + 36\kappa _{ 1}\kappa _{2}^{2}\kappa _{ 3}^{4} + 64\kappa _{ 2}^{4}\kappa _{ 3}^{3} + 152\kappa _{ 2}^{3}\kappa _{ 3}^{4} {}\\ & & +\kappa _{1}^{3}\kappa _{ 2}\kappa _{3}^{2} + 40\kappa _{ 1}^{2}\kappa _{ 2}^{2}\kappa _{ 3}^{2} + 152\kappa _{ 1}\kappa _{2}^{3}\kappa _{ 3}^{2} + 52\kappa _{ 1}\kappa _{2}^{2}\kappa _{ 3}^{3} - 56\kappa _{ 2}^{3}\kappa _{ 3}^{3} - 12\kappa _{ 2}^{2}\kappa _{ 3}^{4} {}\\ & & +6\kappa _{1}^{3}\kappa _{ 2}\kappa _{3} - 14\kappa _{1}^{2}\kappa _{ 2}^{2}\kappa _{ 3} - 49\kappa _{1}^{2}\kappa _{ 2}\kappa _{3}^{2} - 72\kappa _{ 1}\kappa _{2}^{3}\kappa _{ 3} - 192\kappa _{1}\kappa _{2}^{2}\kappa _{ 3}^{2} - 24\kappa _{ 2}^{3}\kappa _{ 3}^{2} {}\\ & & -14\kappa _{2}^{2}\kappa _{ 3}^{3} -\kappa _{ 1}^{4} - 15\kappa _{ 1}^{3}\kappa _{ 2} - 12\kappa _{1}^{2}\kappa _{ 2}^{2} + 26\kappa _{ 1}^{2}\kappa _{ 2}\kappa _{3} + 52\kappa _{1}\kappa _{2}^{2}\kappa _{ 3} + 31\kappa _{1}\kappa _{2}\kappa _{3}^{2} {}\\ & & +40\kappa _{2}^{2}\kappa _{ 3}^{2} + 12\kappa _{ 1}^{3} + 31\kappa _{ 1}^{2}\kappa _{ 2} + 36\kappa _{1}\kappa _{2}^{2} + 26\kappa _{ 1}\kappa _{2}\kappa _{3} - 6\kappa _{2}^{2}\kappa _{ 3} - 15\kappa _{2}\kappa _{3}^{2} {}\\ & & -22\kappa _{1}^{2} - 49\kappa _{ 1}\kappa _{2} + 6\kappa _{2}\kappa _{3} + 12\kappa _{1} +\kappa _{2} - 1\bigr ), {}\\ \mu _{1}& =& -2\kappa _{4}\bigl [\left (9\kappa _{1}^{3}\kappa _{ 2} + 2\kappa _{1}^{2}\kappa _{ 2}\kappa _{3} + 9\kappa _{1}\kappa _{2}\kappa _{3}^{2} - 4\kappa _{ 1}^{3} - 4\kappa _{ 1}^{2}\right )\kappa _{ 4}^{4} +\bigl ( 48\kappa _{ 1}^{2}\kappa _{ 2}^{3}\kappa _{ 3}^{2} {}\\ & & -160\kappa _{1}\kappa _{2}^{3}\kappa _{ 3}^{3} + 48\kappa _{ 2}^{3}\kappa _{ 3}^{4} + 6\kappa _{ 1}^{3}\kappa _{ 2}^{2}\kappa _{ 3} - 38\kappa _{1}^{2}\kappa _{ 2}^{2}\kappa _{ 3}^{2} + 130\kappa _{ 1}\kappa _{2}^{2}\kappa _{ 3}^{3} - 18\kappa _{ 2}^{2}\kappa _{ 3}^{4} {}\\ & & +3\kappa _{1}^{4}\kappa _{ 2} - 18\kappa _{1}^{3}\kappa _{ 2}^{2} - 2\kappa _{ 1}^{3}\kappa _{ 2}\kappa _{3} + 130\kappa _{1}^{2}\kappa _{ 2}^{2}\kappa _{ 3} + 7\kappa _{1}^{2}\kappa _{ 2}\kappa _{3}^{2} - 38\kappa _{ 1}\kappa _{2}^{2}\kappa _{ 3}^{2} {}\\ & & +6\kappa _{2}^{2}\kappa _{ 3}^{3} - 4\kappa _{ 1}^{4} - 26\kappa _{ 1}^{3}\kappa _{ 2} - 92\kappa _{1}^{2}\kappa _{ 2}\kappa _{3} - 26\kappa _{1}\kappa _{2}\kappa _{3}^{2} + 20\kappa _{ 1}^{3} + 7\kappa _{ 1}^{2}\kappa _{ 2} + 3\kappa _{2}\kappa _{3}^{2} {}\\ & & -2\kappa _{1}\kappa _{2}\kappa _{3} + 20\kappa _{1}^{2} - 4\kappa _{ 1}\bigr )\kappa _{4}^{2} + 4\bigl (8\kappa _{ 1}^{2}\kappa _{ 2}^{3}\kappa _{ 3}^{3} - 32\kappa _{ 1}\kappa _{2}^{3}\kappa _{ 3}^{4} + 8\kappa _{ 2}^{3}\kappa _{ 3}^{5} + 2\kappa _{ 1}^{3}\kappa _{ 2}^{2}\kappa _{ 3}^{2} {}\\ & & -32\kappa _{1}^{2}\kappa _{ 2}^{3}\kappa _{ 3}^{2} - 2\kappa _{ 1}^{2}\kappa _{ 2}^{2}\kappa _{ 3}^{3} + 96\kappa _{ 1}\kappa _{2}^{3}\kappa _{ 3}^{3} + 20\kappa _{ 1}\kappa _{2}^{2}\kappa _{ 3}^{4} - 32\kappa _{ 2}^{3}\kappa _{ 3}^{4} - 13\kappa _{ 1}\kappa _{2}\kappa _{3}^{2} {}\\ & & -2\kappa _{1}^{3}\kappa _{ 2}^{2}\kappa _{ 3} - 3\kappa _{1}^{3}\kappa _{ 2}\kappa _{3}^{2} + 8\kappa _{ 1}^{2}\kappa _{ 2}^{3}\kappa _{ 3} + 54\kappa _{1}^{2}\kappa _{ 2}^{2}\kappa _{ 3}^{2} - 32\kappa _{ 1}\kappa _{2}^{3}\kappa _{ 3}^{2} - 92\kappa _{ 1}\kappa _{2}^{2}\kappa _{ 3}^{3} {}\\ & & +8\kappa _{2}^{3}\kappa _{ 3}^{3} + 20\kappa _{ 2}^{2}\kappa _{ 3}^{4} - 3\kappa _{ 1}^{4}\kappa _{ 2} + 20\kappa _{1}^{3}\kappa _{ 2}^{2} - 4\kappa _{ 1}^{3}\kappa _{ 2}\kappa _{3} - 92\kappa _{1}^{2}\kappa _{ 2}^{2}\kappa _{ 3} - 13\kappa _{1}^{2}\kappa _{ 2}\kappa _{3}^{2} {}\\ & & +54\kappa _{1}\kappa _{2}^{2}\kappa _{ 3}^{2} - 2\kappa _{ 2}^{2}\kappa _{ 3}^{3} + 4\kappa _{ 1}^{4} - 13\kappa _{ 1}^{3}\kappa _{ 2} + 20\kappa _{1}^{2}\kappa _{ 2}^{2} + 72\kappa _{ 1}^{2}\kappa _{ 2}\kappa _{3} - 2\kappa _{1}\kappa _{2}^{2}\kappa _{ 3} {}\\ & & +2\kappa _{2}^{2}\kappa _{ 3}^{2} - 4\kappa _{ 1}^{3} - 13\kappa _{ 1}^{2}\kappa _{ 2} - 4\kappa _{1}\kappa _{2}\kappa _{3} - 3\kappa _{2}\kappa _{3}^{2} - 4\kappa _{ 1}^{2} - 3\kappa _{ 1}\kappa _{2} + 4\kappa _{1}\bigr )\bigr ], {}\\ \mu _{0}& =& -4\kappa _{1}^{3}\kappa _{ 4}^{6} +\bigl ( -27\kappa _{ 1}^{4}\kappa _{ 2}^{2} + 36\kappa _{ 1}^{3}\kappa _{ 2}^{2}\kappa _{ 3} - 2\kappa _{1}^{2}\kappa _{ 2}^{2}\kappa _{ 3}^{2} + 36\kappa _{ 1}\kappa _{2}^{2}\kappa _{ 3}^{3} - 27\kappa _{ 2}^{2}\kappa _{ 3}^{4} {}\\ & & +36\kappa _{1}^{4}\kappa _{ 2} - 40\kappa _{1}^{3}\kappa _{ 2}\kappa _{3} - 12\kappa _{1}^{2}\kappa _{ 2}\kappa _{3}^{2} - 8\kappa _{ 1}^{4} - 12\kappa _{ 1}^{3}\kappa _{ 2} - 40\kappa _{1}^{2}\kappa _{ 2}\kappa _{3} {}\\ & & +36\kappa _{1}\kappa _{2}\kappa _{3}^{2} + 32\kappa _{ 1}^{3} - 8\kappa _{ 1}^{2}\bigr )\kappa _{ 4}^{4} +\bigl ( 256\kappa _{ 1}^{2}\kappa _{ 2}^{4}\kappa _{ 3}^{3} - 512\kappa _{ 1}\kappa _{2}^{4}\kappa _{ 3}^{4} + 256\kappa _{ 2}^{4}\kappa _{ 3}^{5} {}\\ & & -96\kappa _{1}^{3}\kappa _{ 2}^{3}\kappa _{ 3}^{2} - 224\kappa _{ 1}^{2}\kappa _{ 2}^{3}\kappa _{ 3}^{3} + 608\kappa _{ 1}\kappa _{2}^{3}\kappa _{ 3}^{4} - 288\kappa _{ 2}^{3}\kappa _{ 3}^{5} - 24\kappa _{ 1}^{4}\kappa _{ 2}^{2}\kappa _{ 3} {}\\ & & -288\kappa _{1}^{3}\kappa _{ 2}^{3}\kappa _{ 3} + 160\kappa _{1}^{3}\kappa _{ 2}^{2}\kappa _{ 3}^{2} + 608\kappa _{ 1}^{2}\kappa _{ 2}^{3}\kappa _{ 3}^{2} - 56\kappa _{ 1}^{2}\kappa _{ 2}^{2}\kappa _{ 3}^{3} - 224\kappa _{ 1}\kappa _{2}^{3}\kappa _{ 3}^{3} {}\\ & & -48\kappa _{1}\kappa _{2}^{2}\kappa _{ 3}^{4} - 96\kappa _{ 2}^{3}\kappa _{ 3}^{4} + 4\kappa _{ 1}^{5}\kappa _{ 2} + 144\kappa _{1}^{4}\kappa _{ 2}^{2} + 24\kappa _{ 1}^{4}\kappa _{ 2}\kappa _{3} + 208\kappa _{1}^{3}\kappa _{ 2}^{2}\kappa _{ 3} {}\\ & & -60\kappa _{1}^{3}\kappa _{ 2}\kappa _{3}^{2} - 768\kappa _{ 1}^{2}\kappa _{ 2}^{2}\kappa _{ 3}^{2} + 208\kappa _{ 1}\kappa _{2}^{2}\kappa _{ 3}^{3} + 144\kappa _{ 2}^{2}\kappa _{ 3}^{4} - 4\kappa _{ 1}^{5} - 196\kappa _{ 1}^{4}\kappa _{ 2} {}\\ & & -48\kappa _{1}^{3}\kappa _{ 2}^{2} + 104\kappa _{ 1}^{3}\kappa _{ 2}\kappa _{3} - 56\kappa _{1}^{2}\kappa _{ 2}^{2}\kappa _{ 3} + 124\kappa _{1}^{2}\kappa _{ 2}\kappa _{3}^{2} + 160\kappa _{ 1}\kappa _{2}^{2}\kappa _{ 3}^{2} - 24\kappa _{ 2}^{2}\kappa _{ 3}^{3} {}\\ & & +48\kappa _{1}^{4} + 124\kappa _{ 1}^{3}\kappa _{ 2} + 104\kappa _{1}^{2}\kappa _{ 2}\kappa _{3} - 196\kappa _{1}\kappa _{2}\kappa _{3}^{2} - 88\kappa _{ 1}^{3} - 60\kappa _{ 1}^{2}\kappa _{ 2} {}\\ & & +24\kappa _{1}\kappa _{2}\kappa _{3} + 4\kappa _{2}\kappa _{3}^{2} + 48\kappa _{ 1}^{2} - 4\kappa _{ 1}\bigr )\kappa _{4}^{2} + 256\kappa _{ 1}^{2}\kappa _{ 2}^{4}\kappa _{ 3}^{4} - 512\kappa _{ 1}\kappa _{2}^{4}\kappa _{ 3}^{5} {}\\ & & +256\kappa _{2}^{4}\kappa _{ 3}^{6} - 128\kappa _{ 1}^{3}\kappa _{ 2}^{3}\kappa _{ 3}^{3} - 512\kappa _{ 1}^{2}\kappa _{ 2}^{4}\kappa _{ 3}^{3} - 128\kappa _{ 1}^{2}\kappa _{ 2}^{3}\kappa _{ 3}^{4} + 1024\kappa _{ 1}\kappa _{2}^{4}\kappa _{ 3}^{4} {}\\ & & +512\kappa _{1}\kappa _{2}^{3}\kappa _{ 3}^{5} - 512\kappa _{ 2}^{4}\kappa _{ 3}^{5} - 256\kappa _{ 2}^{3}\kappa _{ 3}^{6} + 16\kappa _{ 1}^{4}\kappa _{ 2}^{2}\kappa _{ 3}^{2} - 128\kappa _{ 1}^{3}\kappa _{ 2}^{3}\kappa _{ 3}^{2} {}\\ & & +128\kappa _{1}^{3}\kappa _{ 2}^{2}\kappa _{ 3}^{3} + 256\kappa _{ 1}^{2}\kappa _{ 2}^{4}\kappa _{ 3}^{2} + 1152\kappa _{ 1}^{2}\kappa _{ 2}^{3}\kappa _{ 3}^{3} - 128\kappa _{ 1}^{2}\kappa _{ 2}^{2}\kappa _{ 3}^{4} - 512\kappa _{ 1}\kappa _{2}^{4}\kappa _{ 3}^{3} {}\\ & & -1536\kappa _{1}\kappa _{2}^{3}\kappa _{ 3}^{4} + 256\kappa _{ 2}^{4}\kappa _{ 3}^{4} + 512\kappa _{ 2}^{3}\kappa _{ 3}^{5} + 128\kappa _{ 1}^{4}\kappa _{ 2}^{2}\kappa _{ 3} - 16\kappa _{1}^{4}\kappa _{ 2}\kappa _{3}^{2} {}\\ & & +512\kappa _{1}^{3}\kappa _{ 2}^{3}\kappa _{ 3} - 64\kappa _{1}^{3}\kappa _{ 2}^{2}\kappa _{ 3}^{2} - 1536\kappa _{ 1}^{2}\kappa _{ 2}^{3}\kappa _{ 3}^{2} - 640\kappa _{ 1}^{2}\kappa _{ 2}^{2}\kappa _{ 3}^{3} + 1152\kappa _{ 1}\kappa _{2}^{3}\kappa _{ 3}^{3} {}\\ & & +512\kappa _{1}\kappa _{2}^{2}\kappa _{ 3}^{4} - 128\kappa _{ 2}^{3}\kappa _{ 3}^{4} - 16\kappa _{ 1}^{5}\kappa _{ 2} - 128\kappa _{1}^{4}\kappa _{ 2}^{2} - 128\kappa _{ 1}^{4}\kappa _{ 2}\kappa _{3} - 256\kappa _{1}^{3}\kappa _{ 2}^{3} {}\\ & & -640\kappa _{1}^{3}\kappa _{ 2}^{2}\kappa _{ 3} + 192\kappa _{1}^{3}\kappa _{ 2}\kappa _{3}^{2} + 512\kappa _{ 1}^{2}\kappa _{ 2}^{3}\kappa _{ 3} + 1632\kappa _{1}^{2}\kappa _{ 2}^{2}\kappa _{ 3}^{2} - 128\kappa _{ 1}\kappa _{2}^{3}\kappa _{ 3}^{2} {}\\ & & -640\kappa _{1}\kappa _{2}^{2}\kappa _{ 3}^{3} - 128\kappa _{ 2}^{3}\kappa _{ 3}^{3} - 128\kappa _{ 2}^{2}\kappa _{ 3}^{4} + 16\kappa _{ 1}^{5} + 192\kappa _{ 1}^{4}\kappa _{ 2} + 512\kappa _{1}^{3}\kappa _{ 2}^{2} {}\\ & & +128\kappa _{1}^{3}\kappa _{ 2}\kappa _{3} - 640\kappa _{1}^{2}\kappa _{ 2}^{2}\kappa _{ 3} - 352\kappa _{1}^{2}\kappa _{ 2}\kappa _{3}^{2} - 64\kappa _{ 1}\kappa _{2}^{2}\kappa _{ 3}^{2} + 128\kappa _{ 2}^{2}\kappa _{ 3}^{3} - 64\kappa _{ 1}^{4} {}\\ & & -352\kappa _{1}^{3}\kappa _{ 2} - 128\kappa _{1}^{2}\kappa _{ 2}^{2} + 128\kappa _{ 1}^{2}\kappa _{ 2}\kappa _{3} + 128\kappa _{1}\kappa _{2}^{2}\kappa _{ 3} + 192\kappa _{1}\kappa _{2}\kappa _{3}^{2} + 16\kappa _{ 2}^{2}\kappa _{ 3}^{2} {}\\ & & +96\kappa _{1}^{3} + 192\kappa _{ 1}^{2}\kappa _{ 2} - 128\kappa _{1}\kappa _{2}\kappa _{3} - 16\kappa _{2}\kappa _{3}^{2} - 64\kappa _{ 1}^{2} - 16\kappa _{ 1}\kappa _{2} + 16\kappa _{1}. {}\\ \end{array} $$

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Chebanov, D., Mosina, N., Salas, J. (2016). On Stabilization of an Unbalanced Lagrange Gyrostat. In: Bélair, J., Frigaard, I., Kunze, H., Makarov, R., Melnik, R., Spiteri, R. (eds) Mathematical and Computational Approaches in Advancing Modern Science and Engineering. Springer, Cham. https://doi.org/10.1007/978-3-319-30379-6_6

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