Abstract
This paper addresses computational aspects of the Jacobian motion planning algorithms for non-holonomic robotic systems. The motion planning problem is formulated in terms of a control problem in the control affine system representing the system’s kinematics. Jacobian motion planning algorithms are derived by means of the continuation (homotopy) method applied to the inverse kinematics problem in the space of control functions. The solution of the motion planning problem is obtained as the limit solution of a functional differential equation involving the control function. Two methods of representing the control functions are studied: parametric and non-parametric. The parametric method parametrizes the control functions by truncated orthogonal series. The non-parametric method can manage without the parametrization. The functional differential equation can be solved using either the Euler method of integration or higher order methods. The paper focuses on the non-parametric Jacobian pseudo inverse motion planning algorithms incorporating a higher order integration method. Performance of this algorithm is illustrated by the numeric solution of a motion planning problem for the rolling ball kinematics.
Article PDF
Similar content being viewed by others
Explore related subjects
Discover the latest articles, news and stories from top researchers in related subjects.Avoid common mistakes on your manuscript.
References
Tchoń, K.: Continuation method in robotics. In: Proc. 7th Conf. Computer Methods and Systems, pp. 17–24. Cracow, Poland (2007)
Sussmann, H.J.: A continuation method for nonholonomic path finding problems. In: Proc. 32nd IEEE CDC, pp. 2718–2723. San Antonio (1993)
Divelbiss, A.W., Wen, J.T.: A path space approach to nonholonomic motion planning in the presence of obstacles. IEEE Trans. Robot. Autom. 13, 443–451 (1997)
Divelbiss, A.W., Seereeram, S., Wen, J.T.: Kinematic path planning for robots with holonomic and nonholonomic constraints. Essays on Mathematical Robotics, pp. 127–150. Springer-Verlag, New York (1998)
Chitour, Y.: A continuation method for motion-planning problems. ESAIM 12, 139–168 (2006)
Alouges, F., Chitour, Y., Long, R.: A motion planning algorithm for the rolling-body problem. IEEE Trans. Robotics 26, 827–836 (2010)
Tchoń, K., Muszyński, R.: Instantaneous kinematics and dexterity of mobile manipulators. In: Proc. 2000 IEEE Int. Conf. Robot. Automat., pp. 2493–2498. San Francisco (2000)
Tchoń, K., Jakubiak, J.: Endogenous configuration space approach to mobile manipulators: a derivation and performance assessment of Jacobian inverse kinematics algorithms. Int. J. Control 76, 1387–1419 (2003)
Deuflhard, P.: Newton Methods for Nonlinear Problems. Springer-Verlag, Berlin (2004)
Arimoto, S., Kawamura, S., Miyazaki, F.: Bettering operation of robots by learning. J. Robot. Syst. 1, 123–140 (1984)
Tchoń, K.: Iterative learning control and the singularity robust Jacobian inverse for mobile manipulators. Int. J. Control 83, 2253–2260 (2010)
Jakubiak, J., Tchoń, K.: Fourier vs. non-Fourier band-limited Jacobian inverse kinematics algorithms for mobile manipulators. In: Proc. 10th IEEE Int. Conf. on Methods and Models in Automation and Robotics, pp. 1005–1010. Szczecin, Poland (2004)
Jung, A., Wen, J.T.: Nonlinear model predictive control for the Swing-Up of a rotary inverted pendulum. ASME J. Dyn. Syst. Meas. Control. 126, 666–673 (2004)
Tchoń, K., Jakubiak, J.: Jacobian inverse kinematics algorithms with variable steplength for mobile manipulators. Advances in Robot Kinematics, pp. 465–472. Springer, Dordrecht (2006)
Ratajczak, A.: Motion planning of underactuated robotic systems. Ph.D. Dissertation, Wrocław University of Technology (2011)
Chelouch, A., Chitour, Y.: Motion planning of rolling surfaces. Forum Math. 15, 727–258 (2003)
Sontag, E.D.: Mathematical Control Theory. Springer-Verlag, New York (1990)
Zadarnowska, K., Tchoń, K.: Kinematic dexterity of mobile manipulators: an endogenous configuration space approach. Robotica 21, 521–530 (2003)
Ważewski, T.: Sur l’évaluation du domaine d’existence des fonctions implicites réelles ou complexes. Ann. Soc. Pol. Math. 20, 81–120 (1947)
Davidenko, D.F.: On a new method of numerically integrating a system of nonlinear equations. Dokl. Akad. Nauk SSSR 88, 601–603 (1953)
Dormand, J.R., Prince, P.J.: A family of embedded Runge-Kutta formulae. J. Comput. Appl. Math. 6, 19–26 (1980)
Murray, R.M., Li, Z., Sastry, S.S.: A Mathematical Introduction to Robotic Manipulation. CRC Press, Boca Raton (1994)
Janiak, M., Tchoń, K.: Constrained motion planning of nonholonomic systems. Systems Control Lett. 60, 625–631 (2011)
Author information
Authors and Affiliations
Corresponding author
Additional information
This work was supported by the Wrocław University of Technology under a statutory grant.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.
About this article
Cite this article
Ratajczak, A., Tchoń, K. Parametric and Non-parametric Jacobian Motion Planning for Non-holonomic Robotic Systems. J Intell Robot Syst 77, 445–456 (2015). https://doi.org/10.1007/s10846-013-9880-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10846-013-9880-0