Abstract
This paper addresses the problem of normal forms and singularities of non-holonomic robotic systems represented by control-affine systems. By means of the concept of the end-point map of the system, and of the system’s Jacobian, the configuration singularities have been defined as the control functions for which the Jacobian is not surjective. The presence of these singularities impairs performance of Jacobian motion planning algorithms. Being the singular optimal controls, the configuration singularities can be examined using the tools from the optimal control theory. The main idea of this paper is to rely the analysis of configuration singularities on normal forms of robotic systems. This idea has been applied to the dynamics of a space manipulator. Normal forms of this manipulator under the feedback equivalence have been obtained, and exploited in the analysis of its configuration singularities.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Tchoń, K., Muszyński, R.: Singularities of nonredundant robot kinematics. Int. J. Robot. Res. 16, 60–76 (1997)
Tchoń, K., Muszyński, R.: Singular inverse kinematic problem for robotic manipulators: a normal form approach. IEEE Trans. Robot. Autom. 14, 93–104 (1998)
Tchoń, K.: Endogenous configuration space approach: an intersection of robotics and control theory. In: Nonlinear Systems, pp. 209–234. Springer (2017)
Tchoń, K., Ratajczak, J.: Kinematic and dynamic singularities of non-holonomic robotic systems. In: Proc. 11th Int. Workshop RoMoCo, pp 148–153. Własowo Palace, Poland (2017)
Bonnard, B.: Feedback equivalence for nonlinear systems and the time optimal control problem. SIAM J. Control Optim. 29, 1300–1321 (1991)
Jakubczyk, B.: Critical hamiltonians and feedback invariants. In: Jakubczyk, B., Respondek, W. (eds.) Geometry of Feedback and Optimal Control, pp 219–256. Marcel Dekker, New York-Basel (1998)
Respondek, W., Zhitomirski, M.: Feedback clasification of nonlinear systems on 3-manifolds. Math. Contr. Signals Syst. 8, 299–333 (1995)
Yoshida, K., Wilcox, B.: Space robots and systems. In: Springer Handbook of Robotics, pp. 1031–1063. Springer (2008)
Johnson, C.D., Gibson, J.E.: Singular solutions in problems of optimal control. IEEE Trans. Autom. Contr. 8(1), 4–15 (1963)
Bonnard, B., Chyba, M.: Singular Trajectories and Their Role in Control Theory. Springer, Paris (2003)
Chitour, Y., Jean, F., Trélat, E.: Singular trajectories of control-affine systems. SIAM J. Contr. Opt. 47, 1078–1095 (2008)
L’Affito, A., Haddad, W.M.: Abnormal optimal trajectory planning of Multi-Body systems in the presence of holonomic and nonholonomic constraints. J. Intell. Robot. Syst. 89, 51–67 (2018)
Sontag, E.D.: Mathematical Control Theory. Springer, Berlin (1990)
Tchoń, K., Ratajczak, J.: General Lagrange-type jacobian inverse for nonholonomic robotic systems. IEEE Trans. Robot. 34, 256–263 (2018)
Rybus, T., Seweryn, K.: Planar air-bearing microgravity simulators: review of applications, existing solutions and design parameters. Acta Astronaut. 120, 239–259 (2016)
Rybus, T.: Control of a Satellite Manipulator During the Interception Maneuver and the Motion Stabilization. Ph.D. dissertation, Wrocław University of Science and Technology (2016)
Zhitomirski, M., Respondek, W.: Simple germs of corank one distributions. In: Singularities Symposium, Banach Center Publications, vol. 44, pp. 259–266. Warsaw
Zhitomirskii, M.: Typical singularities of differential 1-Forms and pfaffian equations, Transl. Math Monographs 113. Amer. Math. Soc., Providence (1992)
Acknowledgments
The authors are very much indebted to anonymous reviewers for their criticism that has allowed to improve both the contents as well as the presentation of this paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Research of authors no 1 and 3 was supported by the Wrocław University of Science and Technology under a statutory research project.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Tchoń, K., Respondek, W. & Ratajczak, J. Normal Forms and Configuration Singularities of a Space Manipulator. J Intell Robot Syst 93, 621–634 (2019). https://doi.org/10.1007/s10846-018-0883-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10846-018-0883-8