Geometric Modeling of Dubins Airplane Movement and its Metric

Document Type : Research Article

Authors

1 Corresponding Author, B. Bidabad, Department of Mathematics and Computer Sciences, Amirkabir University of Technology, Tehran, Iran. (e-mail:bidabad@aut.ac.ir

2 M. Sedaghat, Department of Mathematics and Computer Sciences, Amirkabir University of Technology, Tehran, Iran. (e-mail: m.sedaghat@aut.ac.ir)

Abstract

The time-optimal trajectory for an airplane from some starting point to some final point is studied by many authors. Here, we consider the extension of robot planer motion of Dubins model in three dimensional spaces. In this model, the system has independent bounded control over both the altitude velocity and the turning rate of airplane movement in a non-obstacle space. Here, in this paper a geometrization of time-optimal trajectory of Dubins airplane is also obtained. More intuitively, the metric related to this phenomenon is described as a geometry in space. It is shown that the distance traveled in movement of an airplane obeys certain conditions of a well-known geometry called Finsler geometry. Moreover, it is proved that the geometry of movement of an airplane is a special Finsler metric known as Randers metric, and therefore, time-optimal paths are geodesics of Randers metric.

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References

[1]     B. Bidabad, M. Rafie-rad. Pure pursuit Navigation on Riemannian Manifolds. Nonlinear Analysis: Real World Applications 10 (2009) pp. 1265-1269.
[2]     D. Bao, S. S. Chern, Z. Shen. An Intoduction to Riemann-Finsler Geometry, Springer-Verlag, 2000.
[3]     S. S. Chern and Z. Shen, Riemann-Finsler Geometry, World Scientific Publishers, 2005.
[4]     Y. Zheng and P. Moore, Time-Optimal trajectories for two-wheel driven cars tracking a moving target. In proc. of IEEE international conf. on automation and robotics, 1996.
[5]     D. J. Balkcom and M. T. Mason. Time-Optimal trajectories for bounded velocity differential drive vehicles. Int. J. Robot. Res,21(3), 2002.
[6]     H. Chitsaz, S. M. Lavalle, D. J. Balkcom and M. T. Mason. An explicit characterization of minimum wheel-rotation paths for differential drives. In 12th IEEE international conference on Methods and Models in automation and robotics, 2006.
[7]     H. Chitsaz, S. M. Lavalle, D. J. Balkcom and M. T. Mason. Minimum wheel-wheel-rotation paths for differential drive mobile robots. In 12th IEEE international conference on automation and robotics, 2006.
[8]     D. Bao, C. Robles, Z. Shen. Zermelo Navigation on Riemannian Manifols. Journal of Differential Geometry,vol. 66,pp.391-449, 2004.
[9]     Chachuat, B.C.,   Nonlinear and dynamic optimization: From theory to practice. IC-32: Ecole Polytechnique Federale de Lausanne, (2010).
[10]  L. E. Dubins, On curve of minimal length with a constraint on average curvature, and with prescribed initial and terminal positions and tangents. American Journal of mathematics, 1957.
[11]  H. Chitsaz and S. M. Lavalle, Time-Optimal paths for a Dubins airplane, Proceedings of the 46th IEEE Conference on Decision and Control New Orleans, LA, USA, Dec. 12-14, 2007.
[12]  Antonelli, P.L., Iingarden, R.S., Matsumoto, M. The theory of Sprays and Finsler spaces with applications in Physics and Biology, Kluwer Academic Publishers. (1993).
[13]  S. Hota and D. Ghose,  A modified Dubins Method for optimal path planning of a miniature air vehicle converging to a straight line path, American control conference, ACC 09, 2397-2402, (2009).

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