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Mathematical Problems in Engineering
Volume 2014 (2014), Article ID 602424, 13 pages
Research Article

Stability Analysis and Variational Integrator for Real-Time Formation Based on Potential Field

School of Aerospace Engineering, Beijing Institute of Technology, Beijing 100081, China

Received 9 September 2013; Revised 5 December 2013; Accepted 9 December 2013; Published 2 March 2014

Academic Editor: Tadeusz Kaczorek

Copyright © 2014 Shengqing Yang and Jianqiao Yu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


This paper investigates a framework of real-time formation of autonomous vehicles by using potential field and variational integrator. Real-time formation requires vehicles to have coordinated motion and efficient computation. Interactions described by potential field can meet the former requirement which results in a nonlinear system. Stability analysis of such nonlinear system is difficult. Our methodology of stability analysis is discussed in error dynamic system. Transformation of coordinates from inertial frame to body frame can help the stability analysis focus on the structure instead of particular coordinates. Then, the Jacobian of reduced system can be calculated. It can be proved that the formation is stable at the equilibrium point of error dynamic system with the effect of damping force. For consideration of calculation, variational integrator is introduced. It is equivalent to solving algebraic equations. Forced Euler-Lagrange equation in discrete expression is used to construct a forced variational integrator for vehicles in potential field and obstacle environment. By applying forced variational integrator on computation of vehicles' motion, real-time formation of vehicles in obstacle environment can be implemented. Algorithm based on forced variational integrator is designed for a leader-follower formation.