Some Rational Vehicle Motions

  • J. M. SeligEmail author
Conference paper
Part of the Mechanisms and Machine Science book series (Mechan. Machine Science, volume 15)


It is observed that the kinematic equations of many vehicles take the same form. This form is that the body-fixed velocity twist of the vehicle lies in a fixed screw-system of a particular type. The Cayley map can be used to pull-back these equations to the Lie algebra of the group of rigid-body motions. Rational solutions to the equations can be found by the method of undetermined coefficients. Since the Cayley map is a rational map, mapping these rational solutions back to the group gives rational rigid-body motions. A 3-parameter family of rational Frenet-Serret motions is found in this way. Multiplying these motions by a rational roll-motion gives a 4-parameter family of aeroplane motions.


Rational rigid-body motions Cayley map Cars Aeroplanes. 



This work was has been much improved following the comment of the anonymous reviewers.


  1. 1.
    Eisenhart, L.P.: A Treatise on the Differential Geometry of Curves and Surfaces. Ginn and Co, Boston (1909)Google Scholar
  2. 2.
    Selig, J.M.: Cayley Maps for \(SE(3)\). In: The International Federation of Theory of Machines and Mechanisms 12th World Congress, Besançon (2007)Google Scholar
  3. 3.
    Selig, J.M.: Characterisation of Frenet-Serret and Bishop Motions with applications to Needle Steering, Robotica (to appear) 31(6), 981–992 (2013)Google Scholar
  4. 4.
    Wagner, M., Ravani, B.: Curves with rational Frenet-Serret motion. Comput. Aided Geom. Des. 15, 79–101 (1997)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Webster III, R.J., Kim, J.S., Cowan, N.J., Chirikjian, G.S., Okamura, A.M.: Nonholonomic modeling of needle steering. Int. J. Robot. Res. 25(5–6), 509–525 (2006)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Faculty of BusinessLondon South Bank UniversityLondonUK

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