Micro air vehicle path planning in fuzzy quadtree framework
Introduction
Micro air vehicles (MAVs) have gained considerable interest in recent times. Superior maneuverability, ability to make close surveillance, and less chance of being detected are some of the unique features which make MAVs suitable for dull, dirty, and dangerous missions. In most cases, such missions demand paths to be planned in congested urban or industrial environment. Further, in-flight detection of static pop-up obstacles makes MAV path planning a challenging task.
Unlike bigger unmanned air vehicles (UAVs), MAVs cannot afford much space for on-board processors. In fact, severe space restriction often necessitates the obstacle information to be transferred from MAV to the ground station. Then the planner runs at ground computer to plan and/or re-plan the path, thereafter, command is uploaded to the MAV. Hence, it is imperative that the MAV path planner must ensure time and space economy by utilizing small processor thereby saving onboard space. Classical approaches like visibility graph [1], Voronoi diagram [2], freeway net [3], potential field method [4] are computationally costly for MAV path planning problem, particularly in the context of urban area navigation. The model predictive control [5], vector calculus [6], receding horizon control [7], GPS feedback [8] and evolutionary algorithm [9], [10] are widely used for UAV path planning problem. However, the computational cost for these methods often outweighs their accuracy, and thereby, restricts their use to large platform where better computational facility can be embedded. On the other hand, MAVs have stringent budgetary constraint to the extent that they can be used as expendable vehicles. Therefore, application of such conventional methods is replaced by heuristic methods which can significantly reduce the computational complexity, hence relaxing the requirement of high processing speed and storage capacity of the system, the two major factors in MAV budget.
Quadtree and its variants are widely used for path planning [11], [12], [13], [14], [15]. For example, Chen et al. [12], [13] used framed-quadtree for path planning using cell decomposition and circular wave to find the shortest distance between two points. Even though it used a linear time algorithm for computing dynamic Voronoi diagram but the reported space and time complexity was still high. Jozef [14] used a weighted-region approach for coding the distance and image using matrix implementation of quadtree. Yahja et al. [15] suggested an incremental re-planning of optimal path using framed-quadtrees and an optimal algorithm (D*) without a priori or with partial information about a terrain. Mirolo and Pagello [16] suggested a new cell-subdivision approach for the translation of a convex polygon in a cluttered environment. Further, research [17] has shown that formulating one existing path planner in fuzzy logic framework yields better result than the original planner itself. To conclude, various implementations of quadtree for path planning has been used successfully in the past. Moreover, as observed above fuzzyfication of the original planner yields better result. Keeping in view these conclusions, this paper introduces a fuzzy quadtree based path planner for MAV navigation and shows its superiority over the quadtree planner theoretically, i.e. in the sense of amount of computation and storage involved. As the weight assignment rule is same for both and hence the path length will be same in both the cases. However, it is described later that because of some measures taken, even the conventional quadtree becomes efficient. This implies that if we remove the fuzzy part then quadtree is nothing but a conventional one. Therefore, length of the optimal path will be same but the amount of computation will differ and will yield identical result as in fuzzy quadtree.
The organization of this paper is as follows. In the next section, conventional quadtree planner is illustrated in detail. The concept of fuzzy quadtree planner is explained in Section 3, which is followed by a discussion of some critical aspects of the problem in Section 4. Simulation results for a real map are presented in Section 5. The last section concludes the paper.
Section snippets
Conventional quadtree planner
The path planner proposed in this paper is based on the assumption that the environment is structured, i.e. the map of the obstacles is known beforehand. Furthermore, it is assumed that the obstacles are static and the path planning amid such obstacles is assumed to be a two-dimensional problem. The rest of this paper will adhere to these assumptions. The only exception to these assumptions, which has been addressed in this paper, is the case of pop-up obstacles which were not anticipated a
Fuzzy quadtree planner
The fuzzy quadtree planner is achieved by truncating a full fledged quadtree at a desired level and then assigning suitable membership function values to the resultant nodes at the truncated level. For that, one needs to count the total number of black and white daughter nodes under that level of gray nodes in which the user wants to stop. The user imposed fuzzy leaf node (which actually was a gray node) will then have a fuzzy membership function value equal to the total number of black
Some critical aspects of the problem
A quadtree planner, in general, has less time complexity compared to other conventional planners like Voronoi diagram (quadtree planner requires O(N) time and Voronoi diagram needs O(N log N) time). Further, fuzzy quadtree planner allows the user to specify the level of quadtree that the planner will retain, but if any parent node is having gray daughter then it cannot be folded at the parent level. In the context of MAV navigation, the map of the static obstacles generally consists of some
Simulation result for a real map
To demonstrate the efficacy of the proposed algorithm, path planning has been performed for a real satellite image (of size 256 × 256 pixels) of a part of Vatican City. The original image (Fig. 8(a)) is processed digitally to remove noises from it. Various combinations of filters are applied for this purpose to yield an image (Fig. 8(b)) which has clear obstacles and obstacle free regions. The path planning algorithm is applied next to generate a quadtree (Fig. 8(c)) for this processed image.
Conclusion
A fuzzy quadtree path planner is formulated in this paper for MAV navigation. The formulation addresses how to specify the minimum quad length depending upon vehicle's turn rate kinematics and MAV dimension. This, in fact, avoids pixel level computation and takes much less execution time and storage space compared to a conventional quadtree planner. This makes it possible to plan a path for MAV, which otherwise would not be an easy task or may not be feasible at all keeping in view requirement
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