AM  Vol.4 No.12 , December 2013
An Integrating Algorithm and Theoretical Analysis for Fully Rheonomous Affine Constraints: Completely Integrable Case
Author(s) Tatsuya Kai
ABSTRACT

This paper develops an integrating algorithm for fully rheonomous affine constraints and gives theoretical analysis of the algorithm for the completely integrable case. First, some preliminaries on the fully rheonomous affine constraints are shown. Next, an integrating algorithm that calculates independent first integrals is derived. In addition, the existence of an inverse function utilized in the algorithm is investigated. Then, an example is shown in order to evaluate the effectiveness of the proposed method. By using the proposed integrating algorithm, we can easily calculate independent first integrals for given constraints, and hence it can be utilized for various research fields.


Cite this paper
Kai, T. (2013) An Integrating Algorithm and Theoretical Analysis for Fully Rheonomous Affine Constraints: Completely Integrable Case. Applied Mathematics, 4, 1720-1725. doi: 10.4236/am.2013.412235.
References
[1]   J. Cortes, “Geometric, Control and Numerical Aspects of Nonholonomic Systems,” Springer-Verlag, Berlin, Heidelberg, 2002.

[2]   A. M. Bloch, “Nonholonomic Mechanics and Control,” Springer-Verlag, New York, 2003.
http://dx.doi.org/10.1007/b97376

[3]   F. Bullo and A. D. Rewis, “Geometric Control of Mechanical Systems,” Springer-Verlag, New York, 2004.

[4]   R. Montgomery, “A Tour of Subriemannian Geometries, Their Geodesics and Applications,” American Mathematical Society, Providence, 2002.

[5]   O. Calin and D. C. Change, “Sub-Riemannian Geometry: General Theory and Examples,” Cambridge University Press, Cambridge, 2009.
http://dx.doi.org/10.1017/CBO9781139195966

[6]   T. Kai and H. Kimura, “Theoretical Analysis of Affine Constraints on a Configuration Manifold—Part I: Integrability and Nonintegrability Conditions for Affine Constraints and Foliation Structures of a Configuration Manifold,” Transactions of the Society of Instrument and Control Engineers, Vol. 42, No. 3, 2006, pp. 212-221.

[7]   T. Kai, “Integrating Algorithms for Integrable Affine Constraints,” IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences, Vol. E94-A, No. 1, 2011, pp. 464-467.

[8]   T. Kai, “Mathematical Modelling and Theoretical Analysis of Nonholonomic Kinematic Systems with a Class of Rheonomous Affine Constraints,” Applied Mathematical Modelling, Vol. 36, 2012, pp. 3189-3200.
http://dx.doi.org/10.1016/j.apm.2011.10.015

[9]   T. Kai, “Theoretical Analysis for a Class of Rheonomous Affine Constraints on Configuration Manifolds—Part I: Fundamental Properties and Integrability/Nonintegrability Conditions,” Mathematical Problems in Engineering, Vol. 2012, 2012, Article ID: 543098.
http://dx.doi.org/10.1155/2012/543098

[10]   T. Kai, “Theoretical Analysis for a Class of Rheonomous Affine Constraints on Configuration Manifolds—Part II: Foliation Structures and Integrating Algorithms,” Mathematical Problems in Engineering, Vol. 2012, 2012, Article ID: 345942. http://dx.doi.org/10.1155/2012/345942

[11]   T. Kai, “On Integrability of Fully Rheonomous Affine Constraints,” International Journal of Modern Nonlinear Theory and Application, Vol. 2, No. 2, 2013, pp. 130-134.
http://dx.doi.org/10.4236/ijmnta.2013.22016

[12]   S. S. Sastry, “Nonlinear Systems,” Springer-Verlag, New York, 1999. http://dx.doi.org/10.1007/978-1-4757-3108-8

[13]   A. Isidori, “Nonlinear Control Systems,” 3rd Edition, Springer-Verlag, London, 1995.
http://dx.doi.org/10.1007/978-1-84628-615-5

[14]   S. Nomizu and K. Kobayashi, “Foundations of Differential Geometry (Volume II),” John Wiley & Sons Inc., New York, 1996.

[15]   S. Nomizu and K. Kobayashi, “Foundations of Differential Geometry (Volume I),” John Wiley & Sons Inc., New York, 1996.

 
 
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