ISSN 1064-2307, Journal of Computer and Systems Sciences International, 2012, Vol. 51, No. 2, pp. 328–338. © Pleiades Publishing, Ltd., 2012. Original Russian Text © M.A. Andreev, A.B. Miller, B.M. Miller, K.V. Stepanyan, 2012, published in Izvestiya Akademii Nauk. Teoriya i Sistemy Upravleniya, 2012, No. 2, pp. 166–176. 328 INTRODUCTION The problem of path planning for an unmanned aerial vehicle (UAV) under risks has been known since long ago [1, 2] and has remained the subject of research in recent years, especially as applied to path plan- ning for autonomous UAV [3–7]. Starting from original works on path planning, this problem is stated as the problem of determining the path of a dynamic system with the given initial and terminal conditions that minimizes some functional characterizing integral risk and terminal miss. Applying methods of opti- mal control theory requires knowing hazard rate and its derivatives. At this stage, the problem of deter- mining the optimal admissible path for the given stationary risk distribution and given local relief is solved. The UAV is assumed to move at a constant linear velocity, with the path chosen so that to minimize the integral risk. To solve the problem, we apply numerical methods based on solving the boundary problem resulted from the necessary optimality condition in the form of the maximum principle. 1. TYPICAL RISK DISTRIBUTION The typical risk distribution is characterized by the following set of parameters: coordinates of hazard centers, spatial risk distribution. For the given coordinates of the center of the i-th source of hazards ( ), the spatial risk distribution can be of different types such as Gaussian [3], i.e., the spatial risk distribution is described by the probability density of a Gaussian ran- dom variable with the mathematical expectation at the point and some root-mean-square devia- tions, rational [4], which is closer to natural physical models and is given by the relation where is the distance from the current point to the center of the i-th source of hazards and V i is the relative velocity of UAV and the i-th source of hazards to the power of m, modified rational [7] where depends on the object orientation and is equivalent to the efficient reflection coefficient. , ( ) i i x y =, , 1 i … M , ( ) i i x y , = , ρ , 2 ( ) ( ) m i i i V f xy xy ρ , ( ) i xy , = θ , ρ , 2 ( ) () ( ) m i i i V f xy C xy θ () C Path Planning for Unmanned Aerial Vehicle under Complicated Conditions and Hazards M. A. Andreev, A. B. Miller, B. M. Miller, and K. V. Stepanyan Institute for Information Transmission Problems, Bolshoy Karetny per. 19, Moscow, 127994 Russia Monash University, Melbourne, Australia Received July 5, 2011 Abstract—The flight of an unmanned aerial vehicle under complicated conditions and hazards is con- sidered. Hazards are given in terms of 2D relief. To find the optimal 2D path minimizing the risk given constraints on flight time and velocity, the problem with the non-given boundary condition is trans- formed to the problem with the fixed flight time and the boundary problem solved numerically. The found 2D path is used to construct the polynomial approximation of the 3D path taking into account the local relief. DOI: 10.1134/S1064230712010030 CONTROL SYSTEMS OF MOVING OBJECTS