Review
An unmanned aerial vehicle optimal selection methodology for object tracking
Advances in Mechanical Engineering 2018, Vol. 10(12) 1–12 Ó The Author(s) 2018 DOI: 10.1177/1687814018814085 journals.sagepub.com/home/ade
Antoni Kopyt1 , Janusz Narkiewicz1 and Paweł Radziszewski2
Abstract In this article, an optimal method of unmanned aerial vehicle selection for following a moving ground target is presented. The most suitable unmanned aerial vehicle from an aircraft operating fleet is selected for tracking task. The aircraft choice is done taking regarding the unmanned aerial vehicle distribution in the mission zone, individual performance of fleet platforms, and maneuverability of a ground platform. The unmanned aerial vehicle fleet members’ flying qualities are various, so an optimization process is used to assign an aircraft to follow the ground target. A target trajectory prediction is embedded in tracking algorithm. The simulation results demonstrated the effectiveness of algorithms developed, and in-flight demonstration proved the possibility of realization of computed trajectories by the unmanned aerial vehicle. Keywords Adaptive control, optimization, flight control system, simulation, control engineering
Date received: 19 April 2018; accepted: 29 October 2018 Handling Editor: Dumitru Baleanu
Introduction A tracking of moving objects by unmanned aerial vehicles (UAVs) is investigated by many authors.1–4 In this research, the tracking task was a part of wider mission that contains three phases: Locate, Target, and Track (LTT). The optimization of Locate and Target tasks was subject to other research optimized within the project OpUSS—‘‘Optimization of Unmanned System of Systems’’—performed at Warsaw University of Technology under a research grant agreement with Lockheed Martin Corporation. The main objective of OpUSS project was to create and implement a methodology that optimizes the performance of a UAV fleet for the LTT mission at each phase. It was realized using multilevel optimization methods for each task of the LTT mission. For the Track task, the optimization was used for the selection of the most proper and useful UAV for following the moving object and generation of tracking trajectory. The optimization of third— Track—task is presented in this article. The novelty of
the research shows multilevel optimization, taking into account as well the model of the UAVs and the estimation of the escaping object’s trajectory. The presented method considers the UAV’s performance and fuel/ energy resources. The main difficulty of the research was to apply such an algorithm that may be used onboard, so the calculation can be done in real time. The algorithm and method were developed and tested in virtual environment (computer simulations), but to 1
Division of Automation and Aeronautical Systems, Institute of Aeronautics Applied Mechanics, Faculty of Power and Aeronautical Engineering, Warsaw University of Technology, Warsaw, Poland 2 Faculty of Power and Aeronautical Engineering, Warsaw University of Technology, Warsaw, Poland Corresponding author: Antoni Kopyt, Division of Automation and Aeronautical Systems, Institute of Aeronautics Applied Mechanics, Faculty of Power and Aeronautical Engineering, Warsaw University of Technology, Nowowiejska 24, 00-665 Warsaw, Poland. Email:
[email protected]
Creative Commons CC BY: This article is distributed under the terms of the Creative Commons Attribution 4.0 License (http://www.creativecommons.org/licenses/by/4.0/) which permits any use, reproduction and distribution of the work without further permission provided the original work is attributed as specified on the SAGE and Open Access pages (https://us.sagepub.com/en-us/nam/ open-access-at-sage).
2 prove its applicability, the method was applied in real flight, with the use of set of various UAVs. One of the key elements of optimization was the selection of the best aircraft for tracking. The process takes into account the performance and maneuverability of each fleet member, actual positions of aircraft, and estimated available flight time durations based on remaining fuel/energy consumption. There are several references published, which concern target tracking. In some of them, stochastic methods1,3,5 were used for generating tracking trajectory; in others, a deterministic approach2,4 was investigated. To optimize the path of a tracking vehicle, various methods were used, such as Markov decision processes1 or dynamic programming.6–8 In this article, an approach used to solve the tracking problem is delivered. Next, the method of selection of an optimal aircraft for tracking is described, followed by a description of UAV/target models, track algorithm, and target motion prediction method. The simulation findings illustrate the applicability and effectiveness of the algorithms developed, and flight tests proved the applicability of the results in practice.
UAV optimal selection for tracking phase The non-moving object on the ground was searched in the mission zone and has been identified by a member of UAV fleet. Being found, the object starts moving on the ground. This is the initial situation for research described in this article. A decision must be undertaken which UAV from in-the-air operating fleet over
Figure 1. Search area definition.
Advances in Mechanical Engineering the mission area will be used to track the running object. The followed object performance is known in general, but its trajectory is not known in advance by the tracking fleet and may vary during the Track phase. The main factors taken into account, which influence the selection of UAV for tracking, are as follows:
Fleet distribution in the mission area when the target is detected; Estimated remained flight duration of each UAV performing tracking, based on fuel/energy consumption and assumed the ‘‘worst’’ target escaping trajectory; UAV maneuverability depending on UAV type, which is attached to the UAV model; Target maneuverability, which is considered in an object model.
The main criterion for selection of an optimal aircraft is the estimated tracking time duration, depending on fuel/energy consumption. The fuel consumption is considered in both mission phases: reaching the target and tracking the target. An UAV and an object trajectory are considered as a movement on a two-dimensional (2D) space, which means that the UAV altitude is constant. An example of random fleet distribution over the mission area when an object was found is presented in Figure 1. The distances of UAVs to the object and its initial position are presented. The UAV are placed randomly, and their distances dAT, i to the target are
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dAT, i =
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðxai xT Þ2 + ðyai yT Þ2
ð1Þ
The fuel/electrical energy consumption by a UAV is estimated by the following formulas:
For piston engines DmFi = qei
dAT, i N ðvai Þ vai
ð2aÞ
For electrical engines DQ =
dAT, i N ðvai Þ vai hU ðQÞ
ð2bÞ
where qe is the specific fuel consumption. The N parameter represents the power required to maintain required speed, h is the efficiency of the electric drive, U is the voltage on energy source, and vopt is the UAV velocity selected to reach starting tracking position. The power that is required to maintain specified altitude and velocity is represented by an equation Nt, i =
rðva, i Þ3 SCD, i 2
Vehicle models In the literature, various models of vehicle motion may be found for trajectory optimization processes, like Dubins car,9,10 full dynamics aircraft model,11 or models based on transfer functions12–15 identified during tests in flight. In optimization processes, the model complexity influences the selection of an applied method and computation time. Usually, for simulation efficiency, the applied model is not very complex. This is the case in this research, but the software structure allows to make it more comprehensive in the future research. An UAV and the object to find are modeled as a maneuvering point. An UAV flies at a fixed altitude. Both—UAVs’ and object’s—trajectories are expressed by two parametric equations in horizontal plane of north-east-down (NED) coordinate system x a = x a ðt Þ y a = y a ðt Þ
ð3Þ
x_ a = va cos(ca ),
x_ T = vT cos(cT )
ð6Þ
y_ a = va sin(ca ),
y_ T = vT sin(cT )
ð7Þ
v_ T = aT
ð8Þ
vT c_ T = OT = RT
ð9Þ
v_ a = aa , va , c_ a = Oa = Ra
1 DmFi = rdAT, i ðva, i Þ2 qe, i SCD, i or 2 1 SCD, i DQ = rdAT, i ðva, i Þ2 2hU
ð5Þ
UAV and object motion are presented by kinematic equations
By inserting equation (3) into equation (4), the following expressions for fuel/energy consumption is delivered
xT = xT ð t Þ yT = yT ð t Þ
The UAV is subjected to constraints imposed on: ð4Þ
The crucial factors that have an impact on the fuel/ energy consumption are air density, distance flown, flight velocity va, i , and UAV characteristics like propulsion efficiency and aircraft aerodynamics. To begin tracking phase, the selected UAV has to get to the initial tracking position. In the research, it was assumed that the trajectory to the target will be a straight line and its velocity va, i will be constant. In this part of task, UAV would consume a specified amount of fuel/energy DmF, i or Qi . An estimated tracking time ttrack, i during which an UAV may follow the target depends on the target motion and the tracking strategy realized by aircraft, which result in a distance di flown by UAV with velocity va, i . To select the UAV which is best for tracking, the tracking time is estimated using the algorithm for target following (described in subsequent part of this article), which will be used to track the object, and its trajectory is chosen from a certain class of trajectories consistent with its maneuverability.
Horizontal velocity: minimal va, min because of the stall conditions and maximum va, max — power constrains va, min ł va ł va, max
Roll angle—stall constrains 0 ł jua j ł ua, max
ð10Þ
ð11Þ
The minimum turn radius calculated as v2a Ra, min = pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 g nmax 1
ð12Þ
Aircraft turn radius may also be taken from measures in experimental flights. Target velocity is constrained by maximum engine power vT ł vT , max
ð13Þ
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Figure 2. Bearing selection.
An algorithm for target following The object following algorithm is a deterministic one, based on the method developed in Rafi et al.2 and Ruangwiset.4 In Figure 2, parameters crucial for the tracking algorithm are illustrated. Point A(xa , ya ) is the aircraft actual position and point T (xT , yT ) is the target estimated position in the next time step. An object circle is drawn with the center in target position and radius rTARGET equals to a minimum aircraft turn radius Ra, min . The relative distance b of aircraft to the target is defined as b=
dAT^ rTARGET
ð14Þ
At each time step, calculations are done as follows:
Actual target position T (xT , t , yT , t ) is determined (using aircraft sensors) during the actual instant of time t; Predicted target position T^ (xT , est, t + 1 , yT , est, t + 1 ) in the next time step t + 1 is calculated using algorithm for prediction of target motion; Heading change Dca, t of the aircraft located in point A(xa, t , ya, t ) is calculated, which depends on predicted target position T^ (xT , est, t + 1 , yT , est, t + 1 ) and the distance to the predicted target position dAT^ , i , expressed in a non-dimensional form as parameter b; The UAV position in consequent step is calculated using aircraft model xat + Dt = xa, t + va cos ca, t + Dca, t Dt ya, t + Dt = ya, t + va sin ca, t + Dca, t Dt
value was assumed to improve the algorithm efficiency. For each instant of time one of the three cases may be encountered: 1.
2.
3.
The UAV is inside an object circle, that is, 0 \ b \ 1. If so, the heading rate is constant and does not change, Dca, t = 0. The UAV is outside the object circle, however closer than assumed relative distance b0, that is, if 1 \ b \ b0. In this case, the target tracking algorithm described below is applied. The UAV is further than b0, that is, b . b0. If so, the aircraft heading is selected straight to the target (a dotted line AT in Figure 3).
The object following algorithm developed here and applied in case 2 is described now. In Figure 4, point A(xa , ya ) is an actual UAV position, and point T (xTest , yTest ) is a predicted target position in a next instant of time. The target circle is drawn around the target predicted position T (xTest , yTest ). The lines AT1 and AT2 tangent to the target circle are drawn from actual aircraft position (point A(xa , ya )). The aircraft velocity vector is va . The UAV heading change is calculated as the lower value of one of the two bearings of the tangent lines t1 and t2 t = minðt 1 , t2 Þ
To calculate the angles t1 and t 2, the bearing d to predicted target position is calculated as sin d =
ð15Þ
ca, t + Dt = ca, t + Dca, t The heading changes along the UAV, and depends on the actual value of parameter b, that is, the relative distance between UAV and the object. The fixed b0
ð16Þ
v a 3 d
^ AT
va dAT^ = sin va , dAT^
va dAT^ sin va , dAT^ = va dAT^
Angles t1and t2 are calculated as t1 = d Dt,
t 2 = d + Dt
and the incremental change Dt is calculated as
ð17Þ
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Figure 3. Target tracking algorithm steps.
Figure 4. Mean and maximum distances to the target as a time step function.
rTARGET Dt = arcsin ð18Þ dAT 0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where dAT^ , t = (xT , est, t + 1 xa, t )2 + (yT , est, t + 1 ya, t )2 the distance dAT^ between the UAV actual position and object’s predicted position in the next time step. When the bearing is selected, an algorithm decides about the heading change in the current time step. And so, if the t(t)\Dca, max dependency is true, then Dca, t = 6 tt
ð19Þ
Dca, t = 6 Dca, max
ð20Þ
the heading change is t t . Otherwise, the UAV changes its heading Dca, max . The object position prediction in the following time step is an element of tracking algorithm, proposed in this study. There are many methods used to predict target motion, for instance.4,16 Usually, the past object’s motion knowledge is used. Thanks to the modular algorithm structure, the various prediction methods can be applied and tested. In presented research, a deterministic extrapolation is applied, where the last time step data are used to estimate the object’s trajectory using the object’s model
else
If the actual angle between aircraft velocity vector and the tangent direction is smaller than a maximum change of an aircraft turn angle in one time step, then
cT , est, t + 1 = cT , t
ð21Þ
xT , est, t + 1 = xT, t + vT , t cos cT , t Dt
ð22Þ
yT , est, t + 1 = yT , t + vT , t sin cT, t Dt
ð23Þ
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Table 1. Aircraft performance data.
Aircraft type Propulsion type Best endurance velocity (m/s) Turn radius (m)
REKIN
CITABRIA
T-REX
Fixed-wing Electrical motor 22.8 40
Fixed-wing Internal combustion engine 25 70
Rotorcraft Electrical motor 5 10
The advantages of presented method are that it is not necessary for extensive onboard calculations and a fact that a deterministic approach assures the solution repeatability for constant parameters.
Simulation results In this section, the results of simulation of the tracking algorithm are summarized within two groups of calculations. First one concerns the assessment of dependence of the target tracking algorithm results on the algorithm internal parameters:
Time step Dt; Relative distance to the target b0 ; Target circle radius rTARGET .
The second group concerns sample computations of tracking paths for various UAVs and target motion parameters. Finally, comparison of simulated trajectories with experimental flight results is presented. All calculations were done using MATLAB R2014b software. To assess the quality of tracking process, two indicators were calculated for the whole duration of the tracking task—first one, a mean distance to the target T P
dmean =
dAT (t)
t=0
T
Dt
ð24Þ
where dAT (t) is a distance to the target at the moment t, T is the total tracking time and Dt is the time step size reflecting sampling time of the onboard sensors; the mean distance to the target reflects possibility and quality of monitoring the target behavior. And, the second indicator, a maximum distance to the target dmax = max ðdAT (t)Þ t2h0;T i
ð25Þ
which indicates whether the target is all the time within the sensor field of view. To test the algorithm properties, three different aircraft models were used, due to their availability for flight tests; the aircraft performance data are given in Table 1.
Table 2. Simulation parameters (time step). Parameter
Value
Selected aircraft Trajectory Target velocity Transition value of the relative distance
REKIN (fixed-wing) Straight line 5 m/s (constant) 2
Investigation of algorithm properties In this section, the target tracking algorithm sensitivity to its internal parameters is discussed. There are three algorithm internal parameters crucial for simulation efficiency: Dt is the time step size; b0 is the relative distance to the target, according to which an aircraft changes its tracking strategy; and radius of the target circle rTARGET . A size of a time step Dt reflects a sampling time of onboard sensor, which is also related to computing power needed for the realization of algorithm online. Minimum duration of the time step is selected depending on sensor sampling frequency, aircraft velocity, and time of computations to be performed in one time step. The last factor turned out to be non-crucial in our application (max computing time for one step was 0.07 (s)). Assuming a constant velocity of an aircraft, the time step was changed within range from 0.1 to 2 s. Other values set in simulation are given in Table 2. The time step duration significantly influences the maximal distance, but not so much the mean to the target (Figure 4). The longer is a time step, the greater is the maximum distance to the target; for time step of 2 s, it is about 56% larger than for time step of 0.1 s. Increasing the time step results in slightly larger mean distances to the target (23% more for 2 s than for 0.1 s). The tracking trajectories obtained for 0.1 and 2 s time steps are compared in Figure 5. The larger time step results in the deterioration of algorithm performance (longer trajectories), which is coherent with engineering understanding of the process. Such a situation may appear when a sampling time in an onboard sensor, which reflects the distance and relative position to the target, is high due to sensor performance or severe weather conditions.
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Figure 5. Aircraft trajectories for time step t = 0.1 s and t = 2 s.
Figure 6. Mean and maximum distances to the target in b0 function.
The parameter b0 is a relative distance to the target with respect to the aircraft turn radius, influencing aircraft tracking strategy. As the value of this parameter is chosen arbitrary, only detailed simulations for the specific cases may provide conclusive results of the impact of this parameter on tracking quality. It is expected that for each UAV—target case, it is possible by simulations to find an optimal value of b0 , which ensures the best tracking performance. An influence of b0 was checked for fixed-wing UAV (REKIN) and rotorcraft (T-REX). The simulation parameters for investigating the influence of b0 are given in Table 3, and simulation results are presented in Figure 6. For both cases, the influence of this parameter on tracking efficiency is not substantial. For REKIN aircraft, larger b0 gives smaller maximum (1.5% less for maximum values) and smaller mean distance to the target (6.7% less). However, there is a transition range of b0 where the values of these quality indicators are significantly higher (up to 12.5% more) in the region of 1.5 relative distance to the target. It seems that for a particular case, such a region should be identified and
Table 3. Simulation parameters (relative distance). Parameter
Value
Selected aircraft
REKIN (fixed-wing) and T-REX (rotorcraft) Straight line 5 m/s 0.1 s
Trajectory Target velocity Time step
avoided. For rotorcraft, increasing the relative transition distance results in a slightly greater tracking maximum distance (2.3% more) and in a smaller mean distance to the target (3.4%). The parameter rTARGET is the radius of the ‘‘target circle,’’ which was assumed to be equal to the minimal turn radius of the tracking aircraft. It was interesting to assess the algorithm efficiency dependence on the ratio of turn radius to target circle radius ratio Rmin =rTARGET . It may be expected that the best value of these ratios may be found in terms of fuel/energy consumption and total tracking time. To
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Figure 7. Mean and maximum distances to the target in P function.
Figure 8. Trajectories for straight-line target trajectory: T-REX—green, REKIN—blue, and CITABRIA—yellow.
Table 4. Simulation parameters (target circle radius). Parameter
Value
Selected aircraft Trajectory Target velocity Time step
REKIN (fixed-wing) Straight line 5 m/s 0.1 s
check the impact of the target circle radius, it was combined with aircraft turn radius as a non-dimensional parameter P=
rTARGET Ra, min
Figure 9. Mean and maximum distances to the target for all UAVs for straight-line target trajectory.
ð26Þ
The value of P used in the previous simulations was equal to 1; therefore, current set of simulations tested the algorithm performance for other values of P. The simulation parameters for P simulation are given in Table 4, and the simulation results are presented in Figure 7.
When increasing the value of P parameter, the maximum distance to the target is slightly changed with a tendency to increase for greater P values. The value of the mean distance to the target is increasing, until P = 1:8, then it drops about 12% and stays constant. Comparing to the case of the value P = 1 assumed previously, the values of mean distances are smaller for
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Figure 10. Trajectories for zig-zag target trajectory: T-REX—green, REKIN—blue, and CITABRIA—yellow.
Table 5. UAV performance used in simulation. Aircraft type
Performance
T-REX (rotorcraft) REKIN (fixed-wing) CITABRIA (fixed-wing)
va = 6 m=s, Ra = 10 m va = 6 m=s, Ra = 40 m va = 6 m=s, Ra = 70 m
UAV: unmanned aerial vehicle.
P 2 h0:3; 0:9i. This indicates that by the selection of ‘‘tracking radius,’’ the tracking performance may be influenced and improved. However, it still requires simulations to get better insight into the impact of P for other target trajectories and other UAVs.
UAV/target motion parameters In this section, the influence of target trajectories and target/aircraft velocity ratio vT =va is investigated. A target trajectory influenced significantly the aircraft trajectory, as there is close relation between these two motions. The simulations were performed for various aircraft models and for two trajectories: straight line and zig-zag composed of straight segments. The simulation was performed for three different UAVs (see Table 5). The tracking trajectories are presented in Figure 8. T-REX (helicopter) has available velocity only slightly larger than the target; therefore, it is compensated by making sinusoidal movements, sufficient to directly follow the target. The airplanes (fixed-wing): REKIN and CITABRIA due to much higher flight velocities compared to the target velocity perform loiter around moving point. In Figure 9, the tracking quality indicators are presented for each case shown in Figure 9. The higher mean and maximum distances to the target for quicker flying aircraft are intuitively explainable, which prove that the algorithm operations are proper. A similar analysis was done for zig-zag trajectory. In Figure 10, trajectories for all three UAVs are shown. The zig-zag ‘‘non-smooth’’ target trajectory has larger influence on rotorcraft than on fixed-wing UAVs, which results from velocity ratios. In the
Figure 11. Mean and maximum distances to the target for all UAVs for zig-zag target trajectory.
zig-zag corners, helicopter needs more time to reach the target than the fixed-wing loitering around the target. The comparison of maximal and mean distances to the target for zig-zag trajectories, presented in Figure 11, reveals the same trends as in the straight-line case. But, there is largest relative change for T-REX, which suggests that the quality of tracking by rotorcraft is influenced by the shape of trajectories more than fixed-wing.
Velocity ratio In this section, the algorithm behavior is investigated for various target/UAV velocity ratios vT =va . From this part of the simulations, additional guidance for selecting the aircraft best suited to track the specified target may be expected. It was assumed that the constant UAV velocity is 22.8 m/s, and the target velocity is changed from 1 to 21 m/s every 1 m/s. In Figure 12, mean and maximum distances to the target are presented as a function of relative target/aircraft velocity ratio. For the velocity ratio vT =va = 0:65, the tracking quality parameters encounter substantial change, which results from changing the tracking strategies. From this plot, some practical conclusions may be drawn. It is important to select the aircraft velocity to be comparable with the target velocity. In the case where it is not possible, aircraft velocity should be as high as possible. It may be recommended to avoid velocity ratio in range of 0.2–0.75. The conclusions above depend on aircraft
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Figure 12. Mean and maximum distances to the target as a velocity ratio function.
Figure 13. Simulated and experimental trajectories comparison.
were calculated for rotorcraft UAV (T-REX) flying with horizontal velocity of 4.5 m/s, tracking the target moving along the straight line with constant velocity of 3 m/s. On the simulated trajectories, the waypoints were selected, with mean distance between two simulated waypoints were about 10–15 m and downloaded to rotorcraft autopilot. The helicopter flight trajectories with respect to required waypoints are presented in Figure 13. The rotorcraft followed the waypoints with assumed accuracy of 15 m. This result proves the applicability of computed for real tracking flight. In Figure 14, the distance is shown between points of helicopter trajectory obtained from simulation and nearest points from experiment. The mean distance to the reference (simulation) trajectory was 2.33 m and the maximum inaccuracy was 6.64 m.
and target performances, so such calculations should be done to define the best aircraft velocity from the mean and the maximum distances to the target.
Conclusion
Comparison between simulation and experimental flights
The tracking of a ground-moving platform by an UAV is considered in the article. The tracking algorithm starts with the first step—a method for selection of the best UAV from the fleet distributed throughout the mission zone, based on one-step optimization. The kinematic models of aircraft and target are used, including their maneuverability. The objective of optimization
In this section, the simulated trajectories are compared with demonstrations in flight. The tracking trajectories
Figure 14. Distance between simulated and experimental trajectory points.
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is to achieve the longest tracking time taking into account the actual fuel/energy consumption. The tracking algorithm was developed considering the aircraft maneuverability, which significantly have an impact on the tracking trajectory. The algorithm properties were investigated showing efficiency in performing tracking for various types of UAVs. Having such strong results, the simulation could be taken into the real flight tests. The presented method’s complexity is not high, so its applicability in real flight was possible. Due to the algorithm simplicity, the calculation could be performed onboard. The test on an airfield has been performed and the applicability of simulation results was proved by demonstration in flight with the use of various UAVs fleet. Further research would consider applying the method and integrating the estimating the trajectory along with the sensors feedback (cameras and lasers). Furthermore, the method’s applicability should be tested for a wider spectrum of various UAVs.
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Declaration of conflicting interests The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.
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Funding The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This paper was prepared within project OpUSS— Optimization of Unmanned System of Systems—performed at Warsaw University of Technology under research grant agreement with Lockheed Martin Corporation supported by Drs Derreck Holian and Karen Duneman.
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ORCID iD Antoni Kopyt
https://orcid.org/0000-0003-1503-963X
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Appendix 1 Notation a(:) CD CL dAT dmean g m mF mF1 mF2 N n QQ qe R rFOV rTARGET S t, Dt U Vx , Vy v(:)
acceleration airplane drag coefficient airplane lift coefficient target–airplane distance mean target–airplane distance gravity airplane mass fuel mass available mass of fuel (phase 1) mass of fuel at the (phase 2) no. of UAV fleet aircraft load factor charge level in the energy source airplane fuel consumption turn radius sensor field of view radius radius target circle surface area of airplane wing time, time step voltage on the energy source wind velocities components velocity
best endurance velocity coordinates in ONED coordination system
Subscripts
a i min , max T , TARGET t
aircraft object minimum/maximum value target time step
b b0 d
distance to the target (relative) relative distance transition value angle between airplane velocity vector and aircraft-to-target bearing efficiency of the electric drive relative target circle radius air density angles between airplane heading and bearing lines roll angle heading (yaw angle) yaw angular velocity
h P r t1 , t2 f c(:) O(:)
