A path planning strategy for a search and coverage mission for a small UAV that maximizes the area covered based on stored energy and maneuverability constraints is presented. The proposed formulation has a high level of autonomy, without requiring an exact choice of optimization parameters, and is appropriate for realtime implementation. The computed trajectory maximizes spatial coverage while closely satisfying terminal constraints on the position of the vehicle and minimizing the time of flight. Comparisons of this formulation to a path planning algorithm based on those with time constraint show equivalent coverage performance but improvement in prediction of overall mission duration and accuracy of the terminal position of the vehicle.
The increased interest in UAVs has seen their implementation in military and civilian operations. Small inexpensive autonomous aerial vehicles are of great interest in search and coverage, surveillance, border patrol, and mapping missions [
The primary challenge in implementing small autonomous aerial vehicles for a search and coverage mission is planning the path of the vehicle that will effectively cover the specified region. This requires the development of an algorithm that will always generate trajectories to maximize the spatial coverage for any specified conditions. Different approaches exist to deal with the search and coverage problem. These fall broadly into two categories: standard search patterns and nonstandard search patterns. Standard patterns include those such as spiral and serpentine/grid (boustrophedon motion) [
The foundation for the primary contribution in this paper is the algorithm defined in [
This paper proposes a path planning formulation of a single UAV that can be implemented in realtime which was first presented in [
The rest of the paper is organized as follows: Section
Consider a UAV modeled as a nonholonomic point mass moving in a twodimensional plane at a constant velocity and no wind [
Problem of generating a trajectory that maximizes the area covered.
Area covered by sensor footprint for generated trajectory throughout the mission.
The procedure proposed in this paper to determine a path that maximizes coverage area depends on optimization across a range of turning rates based on the maximum load factor of the vehicle. In addition, the algorithm considers realistic power consumption, which is not constant and varies with the type of maneuver (parameterized by turn rate) executed by the vehicle. This enables the accurate determination of the mission duration and enforcing of the terminal position requirements of the vehicle for the energy available at the start of the mission. The high level of autonomy ensures that the algorithm also generates the most optimal trajectory for any conditions by avoiding previously covered area as much as possible. The formulation avoids the vehicle from going over previously covered area by redirecting it to portions of the bounded region that have not been previously covered. This feature of the proposed formulation is that it does not require choosing the density of the space discretization/decomposition. The density of the space discretization depends on the size of the discretized space and the distance between the center points of the discretized spaces. Without the need to evaluate the parameters for optimal space discretization, the computation process is faster while still providing an accurate calculation of the covered area.
The problem of interest is that of an unmanned aerial vehicle operating at a fixed altitude in a closed, bounded region. The goal is to find a feasible trajectory, defined by
The proposed formulation determines the maximum coverage trajectory for a range of turning rates. Equation (
The overall path planning formulation is discretized into multiple optimizations performed over successive time steps,
Modelbased optimization.
This formulation is expected to determine the turn rate from the allowable range of turning rates that minimizes the cost function. The performance index
The cost function primarily determines the percentage of area covered in each time interval by the possible paths. If the sensor covers area that was not previously covered then the cost function is equal to the inverse of the new area covered. The inverse of the new area covered is obtained since the optimization is minimizing the performance index, so the more area covered lowers the value of the index. However, if the sensor covers area that was previously covered without covering any new area then the cost function is equal to the distance from the current state to the centroid of uncovered area. The second condition in the cost function ensures that the algorithm always searches for area not previously covered, in order to maximize the percentage of area covered at the end of the mission. The cost function can thus be defined as
The priority function determines the immediate objective of the vehicle during the mission. It controls the priority of the vehicle to continue searching or redirect to the desired exit point. The function can be based on the time constraint as in [
The performance index of a receding horizon optimization problem estimates the costtogo from a selected terminal state to the final goal. Again, this terminal function can be based on time constraint [
The performance of the path planning algorithm is evaluated through its implementation on a highfidelity 6DOF nonlinear simulation of a UAV in the Matlab/Simulink environment. The UAV is assumed to be a 1/10th scale model of the Navion general aviation aircraft, allowing for the determination of its geometry and mass properties. The UAV has an elevator, ailerons, and rudder to control its motion. Additionally, it has a battery powered motor that produces a constant thrust throughout the mission. The aerodynamic coefficients that are required to determine the forces and moments acting on the UAV to complete the 6DOF simulation model were obtained using the USAF Datcom software [
The path planning procedure assumes that the aircraft is flying at a constant velocity and fixed altitude. Additionally, it assumes that the turn rate selected by the optimization can be achieved by the aircraft. This requires the UAV simulation to have a velocity hold, altitude hold, and a directional (heading hold) autopilot. The Simulink model of the Navion general aviation aircraft includes the 6DOF block and the three aforementioned autopilots. The input for the Simulink model of the Navion general aviation aircraft is the optimized turning rate, which is selected during the path planning phase based on a 3DOF model of the vehicle. From the turning rate and the turn duration the heading of the vehicle at the end of the turn duration is determined, which is then provided to the heading autopilot.
The block diagram of UAV simulation model is presented in Figure
Block diagram of 6DOF UAV simulation model.
The following subsections discuss the autopilots that have been developed to control the motion of the aircraft.
The altitude hold autopilot (Figure
Altitude hold autopilot.
The velocity hold autopilot is presented in Figure
Velocity hold autopilot.
Figure
Heading hold/direction autopilot.
From the desired turning rate (
The algorithm is implemented in four main steps as summarized in Figure
Summary of path planning implementation.
The simulation performed considers a single UAV navigating a specified region. The region is defined from the maximum area that the vehicle can observe, based on the vehicle specifications, assuming ideal conditions. The simulation plans the path using a threedegreeoffreedom model and applies the control input, the turning rate obtained in the threedegreeoffreedom model, to a sixdegreeoffreedom model of the small UAV.
The maximum area that the vehicle can cover assuming straight and steady level flight, is 598,100 m^{2} that will provide a square region of approximately 773.36 m by 773.36 m. The vehicle enters the region at
Properties of the vehicle for simulation.
Property  Value 



Oswald efficiency  1 
Motor efficiency  0.9 
Max load factor (n)  1.5 


Electric charge (mAh)  2200 
Voltage (V)  11.1 


Camera radius (m)  50 
The optimization routine used in the simulation is
The area covered by the vehicle as it navigates the planned path is determined as follows. At a given simulation time step, the contour of the sensor foot print (circle of radius 50 m centered at the position of the vehicle at that time) is approximated using a finite number of vertices. As the vehicle navigates the path, the sensor foot prints are continuously added together to determine the covered area while the union of vertices of the contours is recorded to update its overall boundary. It is ensured that any area outside the mission boundary as well as the area previously covered is excluded from this computation. The computation of these contours involve the union of sets of vertices (for combining covered regions) as well as its intersection (for excluding regions). Such operations are implemented by using available routines from the Mapping Toolbox in Matlab such as
Figure
Illustrative example of the area calculation. The contours are defined by a matrix of the vertices. Matlab routines are utilized to compute the area covered based on these vertices.
Figure
(a) Comparison of planned path and actual path. The actual path presented is the path from the 6DOF simulation generated from desired turning rate. The planned path is a path generated using the 3DOF Dubins model. (b) Area covered by the vehicle.
Percent tracking error for a mission with a 6second turn duration.
This section discusses the performance of proposed path planning algorithm based on the following factors:
Percentage of total area covered
Distance of the final position of the UAV from the desired exit point
The results provided explore the improvement in the performance parameters for each of the features of the formulation. These include the utilization of a range of turn rates based on maneuver capability of the vehicle, a novel formulation for computing the area covered, and an explicit constraint based on the total energy available for the mission.
Figure
Performance of discrete set and range of turn rates: (a) percent of area covered and (b) distance of the vehicle from the desired exit state at the end of the mission.
The primary cost function proposed in this paper allows the capability of the path planning algorithm to make an explicit decision to visit an uncovered area. Figure
Performance of algorithm for no decision capability and decision capability for range of turn rates: (a) coverage performance and (b) distance from desired exit state at the end of mission.
The final feature of the proposed path planning formulation is the explicit utilization of the available stored energy as a constraint and calculation of power consumed based on the maneuver. Figure
Algorithm performance with decision capability and energy optimization for a range of turn rates: (a) coverage of search region and (b) distance from the desired exit state.
Actual mission duration for time constraint based formulation with power consumption dependent on the maneuver.
The results presented in the previous section show the advantages of implementing the proposed features in the path planning of a UAV. All three features, including the maneuverability constraint based on design load factor of the vehicle, novel area formulation, and explicit modeling of energy constraint, allows improvement in the primary performance metrics of the path planning. This formulation can be extended to multiple UAVs with inclusion of obstacle avoidance. Additionally, without having to determine parameters such as a discrete set of possible turn rates or having a priori knowledge of the actual time, the stored energy will last; the proposed formulation (with optimization of the programming to improve computational efficiency) can be implemented as a realtime guidance algorithm for the UAV. Such an implementation will allow flexibility in the missions that the UAV can execute while still maintaining a high percentage of area covered and reliable recovery position.
This paper presents an efficient path planning algorithm for a UAV that maximizes the area covered with stored energy. An important feature of the path planning algorithm includes optimization over a range of turn rates defined by the maneuver capability of the vehicle in a sustained turn as compared to heuristically selecting discrete set of turn rates. Simulation results show that the novel formulation of the optimization problem does not degrade the area covered as compared to the typical optimization using a time constraint. Evaluation of the overall mission duration and final position of the UAV for both the energy constraint and the time constraint based optimization indicates that the formulation using time constraint calculates it incorrectly. Further, direct comparison of the final position of the vehicle when comparing the two optimization formulations shows that the energy constraint allows the vehicle to be recovered from a location closer to the desired exit point.
Area function
Aspect ratio
Cost function
Zerolift drag
Drag
Oswaldo efficiency factor
Energy
Acceleration due to gravity
Time step
Mass
Load factor
Power for the path
Reference area
Priority function
Thrust
Magnitude of Velocity
Weight of the vehicle
Air density
Turn rate
Dynamic pressure
Turn duration
Sensor footprint
Bounded search region.
The authors declare that there are no conflicts of interest regarding the publication of this paper.