Path Planning and State Estimation for Unmanned Aerial Vehicles in Hostile Environments

U NMANNED aerial vehicles (UAVs) are now widely used in antiterrorism activities and intelligence gathering to enhance mission performance and maximize safety. The susceptibility of these UAVs in hostile environments raises requirements for flight path planning. Path planning strategies in hostile environments are normally composed of two phases [1,2]. The first phase is a Voronoi graph search, which will generate polygonal graphs and will optimize a safety performance index. The second is to use the virtual forces emanated from the virtual field of each surveillance radar site to refine the generated Voronoi graphs. These virtual forces provide information that can be used to reduce the vertices of the Voronoi polygonal and greatly improve the UAV performance. But the curvature continuity of the refined graphs, which plays an important role in the stability of the UAVs’ turning maneuvers, often does not meet the requirements for a continuously flyable path. Many kinds of curves have been studied and designed for UAVs to accomplish their mission [3–5]. Dubins curves, first applied in robotics path planning, are curves along which the UAV can move forward. This kind of circle–line–circle curve has a jump discontinuity in curvature at the connection points between the circle and the line that will cause a robot to stop at these connection points when traveling through the whole path. Other curves, such as the Reeds– Shepp curves [6,7], also have curvature discontinuities at their joint points. An alternative choice, the composite clothoid–line–clothoid curves, can be well designed with curvature being zero at the joint points to eliminate the discontinuities. This allows one to generate a continuous-curvature path by using different kinds of simply shaped curves, although, under most circumstances, we are expecting more flexibility in the curve shape that will allow more space for change. Shanmugavel et al. [3–5]. proposed quintic Pythagorean hodograph (PH) curves for a flyable path, with ten parameters representing each curve. The PH curves are flexible in design and their curvatures are expressed in continuous polynomials. The parameter calculation of a PH curve is an iterative process in order to satisfy different constraints. Such kinds of curves can be further simplified with fewer parameters and a more efficient optimization algorithm. All the methods discussed leave room for improvement in the area of continuous-curvature path planning. The Cornu sprial (CS) [8,9], also known as a clothoid or Euler’s spiral, has wide application in highway and railroad construction, since it can be used to design gradual and smooth transitions in highway entrances or exits. Kelly and Nagy [10] used a parametric CS model to generate real-time nonholonomic trajectories for robotics to minimize the terminal posture error. Here, we consider this CSmodel for use in UAV path planning and investigate how this parametric CS curve works under different constraints. To generate a flyable and safe path with given starting and ending points for UAVs passing through areas covered, at least partially, by several radar sites, the path constraints considered here include 1) minimum accumulative exposure to all radar sites; 2) continuous curvature throughout its length, which will ensure a flyable path’ 3) maximum curvature corresponding to the maximum achievable lateral turn rate; and 4) initial and final boundary constraints. Unlike other path planning problems, including that of moving objects finding the final path, which normally result in motion planning or trajectory planning with system dynamics [11], the path considered here is in a static object environment without dynamic constraints. The work in this paper is based on the developed Voronoi graph; it refines the graph by proposing a generalized CS curve along with a simplified parameter-identification procedure. Most papers on the topic of path planning do not include the information about dynamic state variables of UAVs flying along the planned path. For control purposes, it is beneficial to estimate these state variables to construct complete information of the flight. Kalman filtering [12] has been widely used as an efficient tool in optimal filtering and prediction, especially in the field of state estimation of UAVs performing designated missions [13–17]. For example, Grillo and Vitrano [13] used an extended Kalman filter (EKF) to estimate the state variables and wind velocity for a nonlinear UAV model with Global Positioning System (GPS) measurements. Abdelkrim et al. [14] used an EKF and an H1filter to estimate the localization of UAVs for which the position, velocity, and attitude aremeasured by an inertial navigation system. Campbell andOusingsawat [15] used two different estimators to provide online state and parameter estimation for path planning in uncertain environments. The state estimations in these studies have a commonality, because the UAVs in both cases have sensors or other instruments to provide useful measurement information. If all of the Voronoi points on the initial path are fixed and expected to be followed as closely as possible, they can be assumed asmeasurement points. The generated CS curve is then treated as the reference solution so that the state variables can be estimated by the EKF. The following sections present the procedure for the pathinformation construction in three parts. The first part is the initial rough path of the Voronoi graph and the dynamic programming search algorithm. In the second part, CS curve expression and properties are introduced and different constraints and their mathematical expression are explained. This is followed by the systematicsolution nonlinear programming (NLP) solver. In the last part, the state variables are estimated based on the generated Voronoi points and the refined reference path generated in the first two parts. Simulation results are presented in each part separately.