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Mathematical Problems in Engineering
Volume 2013, Article ID 613964, 9 pages
Research Article

Path Planning for Mobile Objects in Four-Dimension Based on Particle Swarm Optimization Method with Penalty Function

1Navigation College, Wuhan University of Technology, Wuhan, Hubei 430063, China
2Department of Mathematics, Islamic Azad University, Bojnourd Branch, Bojnourd 94186-54145, Iran
3Chutian College, Huazhong Agricultural University, Wuhan, Hubei 430205, China

Received 4 December 2012; Accepted 1 January 2013

Academic Editor: Yang Tang

Copyright © 2013 Yong Ma et al. 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.


We present one algorithm based on particle swarm optimization (PSO) with penalty function to determine the conflict-free path for mobile objects in four-dimension (three spatial and one-time dimensions) with obstacles. The shortest path of the mobile object is set as goal function, which is constrained by conflict-free criterion, path smoothness, and velocity and acceleration requirements. This problem is formulated as a calculus of variation problem (CVP). With parametrization method, the CVP is converted to a time-varying nonlinear programming problem (TNLPP). Constraints of TNLPP are transformed to general TNLPP without any constraints through penalty functions. Then, by using a little calculations and applying the algorithm PSO, the solution of the CVP is consequently obtained. Approach efficiency is confirmed by numerical examples.