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Bifurcations and symmetry in two optimal formation control problems for mobile robotic systems

Published online by Cambridge University Press:  14 July 2016

Baoyang Deng ,
Michael O'Connor  and
Bill Goodwine
Baoyang Deng
Affiliation:
Department of Aerospace & Mechanical Engineering, University of Notre Dame, Notre Dame, IN 46556USA E-mails: moconn10@alumni.nd.edu, deng_baoyang@cat.com
Michael O'Connor
Affiliation:
Department of Aerospace & Mechanical Engineering, University of Notre Dame, Notre Dame, IN 46556USA E-mails: moconn10@alumni.nd.edu, deng_baoyang@cat.com
Bill Goodwine*
Affiliation:
Department of Aerospace & Mechanical Engineering, University of Notre Dame, Notre Dame, IN 46556USA E-mails: moconn10@alumni.nd.edu, deng_baoyang@cat.com
*
*Corresponding author. E-mail: bill@controls.ame.nd.edu
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Summary

This paper studies bifurcations in the solution structure of an optimal control problem for mobile robotic formation control. In particular, this paper studies a group of mobile robots operating in a two-dimensional environment. Each robot has a predefined initial state and final state and we compute an optimal path between the two states for every robot. The path is optimized with respect to two factors, the control effort and the deviation from a desired “formation,” and a bifurcation parameter gives the relative weight given to each factor. Using an asymptotic analysis, we show that for small values of the bifurcation parameter (corresponding to heavily weighting the control effort) a single unique solution is expected, and that as the bifurcation parameter becomes large (corresponding to heavily weighting maintaining the formation) a large number of solutions is expected. Between the asymptotic extremes, a numerical investigation indicates a solution bifurcation structure with a cascade of increasing numbers of solutions, reminiscent, but not the same as, period-doubling bifurcations leading to chaos in dynamical systems. Furthermore, we show that if the system is symmetric, the bifurcation structure possesses symmetries, and also present a symmetry-breaking example of a non-holonomic system. Knowledge and understanding of the existence and structure of bifurcations in the solutions of this type of formation control problem are important for robotics engineers because common optimization approaches based on gradient-descent are only likely to converge to the single nearest solution, and a more global study provides a deeper and more comprehensive understanding of the nature of this important problem in robotics.

Keywords

Control of Robotic SystemsMobile RobotsMotion PlanningMulti-Robot SystemsPath PlanningRobot Dynamics
Type
Articles
Information
Robotica , Volume 35 , Issue 8 , August 2017 , pp. 1712 - 1731
Copyright
Copyright © Cambridge University Press 2016 

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