3D Path Planning in a Threat Environment Boris Miller, Karen Stepanyan, Alexander Miller, and Michael Andreev Abstract—We address optimal path planning in three di- mensional space for an unmanned aerial vehicle (UAV) in the stationary risk environment. We separate the task into two stage, in the first one we determine the risk optimal 2D path for fixed time problem. Then we solve the series of BVPs (Boundary Value Problems) with different UAV speeds and determine the admissible 2D path, which satisfies the time and risk constraints. In the last step one takes into account the relief along the chosen path and determine the approximated 3D path, which minimizes 2D threat along the path and satisfies other constraints. I. I NTRODUCTION Problem of the path planning in a threat environment is well known but stills to be in the focus of research related particularly to the mission planning of autonomous UAV. The main difficulty of the problem is the absence of exact information about the probabilities distributions of risks, such as possibility of detection of the UAV by the enemy sensor or/and radars, hitting the UAV by means of air defense, collision with obstacles, which are rather difficult to identify in exact mathematical terms. Meanwhile all these problems have to be solved by a mission planer usually in short time of the operation planning. One can mention the article [7], where authors search for the optimal 2D path of the minimal risk of the UAV detection in the presence of multiple radars. Their approach is based on the search for on-line solution with the aid of the spline trajectoty approximation. So, the idea of our work is to join together exact mathematical tools and interactive approaches. In the first stage one have to take into account the spatial distribution of the risks centers and to find the path which comes from initial to terminal point at given speed and minimizes the total specific risk of the mission failure. Then the choice of the speed will change the flight time, real value of the probability of the This work was supported in part by Australian Research Council Grant DP0988685 and by Russian Basic Research Foundation Grant 10-01-00710. Boris Miller: School of Mathematical Sciences, Monash University, Clayton, 3800, Victoria, Australia and Institute for Information Transmis- sion Problems, Russian Academy of Sciences, Moscow, Russia, E-mail: boris.miller@monash.edu Karen Stepanyan: Institute for Information Transmission Problems, Russian Academy of Sciences, Moscow, Russia. E-mail: KVStepanyan@iitp.ru. This work was partially supported by Monash University during his stay in the School of Mathematical Sciences, Monash University, Clayton, 3800, Victoria, Australia Alexander Miller: Institute for Information Transmission Problems, Russian Academy of Sciences, Moscow, Russia. E-mail: amiller@iitp.ru. This work was partially supported by Monash University during his stay in the School of Mathematical Sciences, Monash University, Clayton, 3800, Victoria, Australia Michael Andreev: Institute for Information Transmission Problems, Russian Academy of Sciences, Moscow, Russia. E-mail: andreevm@iitp.ru mission failure, and dynamic parameters such as the angular and linear accelerations along the paths. This is enough for the mission planer to make a decision and get the optimal 2D path of UAV. Then he has to take into account the relief along the path. Of course, it would be nice to design the path in 3D imme- diately, taking into account the relief as a threat, distributed along the altitude, but this 3D optimal control problem be- comes rather unstable and gives rather sophisticated solutions which don’t have the real physical meaning [15]. So the easy approach is to get the altitude curve along the desired path and manually put some values of the altitudes desired from the planner point of view, afterwards the trajectory may be represented as some polynomial, approximating the 3D path and used by autopilot for UAV control. Our approach is focused on numerical procedures, so in the next Section 2 we consider the different threats’ description and the problem statement. In Section 3 we consider the decomposition of the general problem into series of more simple, but realistic from applied point of view problems. At first stage we discuss the solution of the optimal control problem which leads to 2D BVP (Boundary Value Problem). In Section 4 we use this solution to get the admissible path and calculate 3D path and to approximate it by polynomial or spline. Discussion of the results are given in Section 5. II. PROBLEM STATEMENT.DESCRIPTION OF THREATS’ RELIEF. Problem of the trajectory planning is known long ago [8], [16] and is still in the focus of researchers’ efforts [1], [5], [6], [12], [17], [18] particularly with respect to the autonomous UAV mission planning. From the very beginning [8], the problem is stated as a problem of determining the path satisfies given initial and terminal conditions, which minimizes some integral functional (the integral of the hazard rate along the path), which corresponds to the probability of the mission failure in the case of Markov hazard model [4]. A. Models for threats’ relief Application of the optimal control methods requires the hazard relief and its derivatives. Typical risk distribution is described by following parameters: • coordinates of threats’ centers (x i ,y i ),i =1,...,M • spatial distribution of the hazard rate f i (x,y), where (x,y) are the coordinates in the plane. In the literature one can find different functions f i used for the description of the risk distribution: 2011 50th IEEE Conference on Decision and Control and European Control Conference (CDC-ECC) Orlando, FL, USA, December 12-15, 2011 978-1-61284-799-3/11/$26.00 ©2011 IEEE 6864