A New Performance Metric for Search and Track ... - Semantic Scholar
12th International Conference on Information Fusion Seattle, WA, USA, July 6-9, 2009
A New Performance Metric for Search and Track Missions 2: Design and Application to UAV Search Donald DelBalzo Planning Systems Inc. Technical Solutions Group QinetiQ North America, Slidell, LA, USA [email protected]
Ryan R. Pitre and X. Rong Li∗ Electrical Engineering Department University of New Orleans New Orleans, LA, USA [email protected], [email protected] Abstract – This is a companion paper to the paper titled “A New Performance Metric for Search and Track Missions,” which is also published in these proceedings. This paper provides a realistic simulation using the performance index proposed in the aforementioned paper to show how the proposed performance measure scores missions consisting of a team of unmanned aerial vehicles searching for and tracking targets in a realistic search mission setting. Keywords: Detection, tracking, filtering, estimation, mission planning, path planning, resource management, UAV.
1 Introduction This is a companion paper to the paper titled “A New Performance Metric for Search and Track Missions” [1]. The purpose of this paper is to provide simulation results using our proposed performance metric in a realistic simulation. Our original intent was to write a single paper that provided a detailed overview of our new performance metric and its application to search and track missions but, due to the page limitations of this conference, the material needed to be separated into two papers. Our work in [1] proposed a novel performance metric that can be used to jointly optimize the objectives of detection and tracking. We created an objective function that has several desirable properties including: • It can be used to jointly optimize the objectives of detection and tracking. • It is a scalar, which makes it easy to compare and evaluate against candidate solutions. • It can be controlled to encourage early detection. • It has easily tunable parameters for different types of search and track missions. ∗ Research supported in part by ARO through Grant W911NF-08-10409, NAVO through Contract N62306-09-P-3S01, Navy through Planning Systems Contract N68335-05-C-0382, and Project 863 through Grant 2006AA01Z126.
978-0-9824438-0-4 ©2009 ISIF
The cornerstone of our approach is the introduction of a new performance index—information gain—which allows joint optimization of standard objectives for search and track missions. When a path maximizes our proposed metric, the vehicle following that path will be able to gain the most information, on the average, by detecting and tracking as many targets as possible and as accurately as possible during the course of the mission. Here, we use information in the Fisher sense, which roughly means that information is the inverse of uncertainty and thus, the reduction in uncertainty results in an increase in information [2]. In order to use our metric, we require that the distribution of the targets be available along with a well-defined detection function. Other work in mission planning has approached this problem from different perspectives, such as maximizing the total number of targets detected, minimizing the distance to complete objectives, or by maximizing the one-step information gain. See [3], [4], [5], [6] , [7], and [8]. None of these jointly optimize the objectives of detection and tracking over the entire mission, as was proposed in [1]. Furthermore, a more in-depth review of [3], [4], [5], [6] , [7], and [8] can be found in [1], which is the first part of this work. This paper is organized as follows. In Section 2, we review our proposed objective function and its parameters. Section 3 describes an illustrative scenario and three different search paths that will be used to search the region. In Section 4, we evaluate the three search paths using our metric for the search and track mission described in Section 3. Finally, Section 5 gives our conclusions.
2 Objective Function Defined Our objective function, namely, the expected information gain, was defined in [1] as "K N # XX G=E αn,k λn,k tr (In,k ) (1) k=1 n=1
where
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• E[·] is the expectation with respect to all randomness, e.g., target distributions and detections.
• K is the length of the mission measured by discrete time intervals.
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• N is the total number of targets detected during the mission, which is random.
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• αn,k is the importance factor for target n at time k. nmi
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• λn,k is the forgetting factor for tracking target n at time k once it has been detected.
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• tr(In,k ) is the trace of the information matrix In,k for target n at time k.
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Note that the objective function does not directly depend on the number of searchers involved in the mission; rather it only depends on how well they work together as a team. Based on the above objective function, the best solution is the one that has the largest value of G among all candidate search paths.
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Figure 1: Terrain map
3 Simulation
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In this section, we describe a scenario in which we generate a set of search paths by maximizing G. In Section 4, we show the results of this simulation. Mission planning begins with an objective, an environment, and a search team with a set of sensors. Our objective is to locate targets and then track them during the course of a two-hour mission in order to maximize G. We begin by describing the terrain where the mission takes place. Then, we describe the search vehicles and their sensor capabilities. Next, we explain how we model the sensor’s detection performance. Here, we assume complete prior knowledge of the distribution of target locations. Then, we describe the simplifications that we used to avoid having to use a target-tracking algorithm while our optimization algorithm generated search paths. After that, we describe the parameters used in our objective function. Finally, we conclude with a description of each set of search paths.
3.1 Terrain The simulation takes place in a square region that is 60 nautical miles (nmi) wide. The resolution of the map is 162 cells by 162 cells. See Figure 1. This environment contains three different terrain types. They are mountain, desert, and forest. In the colorbar of the figure, the different colors labeled “Mount.” indicate different altitudes of the mountainous terrain. The region also contains a road network that is shown using grey lines. The altitude map, which can be seen in Figure 2, shows that the elevation of the terrain ranges from 856 m to 2833 m.
3.2 Search vehicles The search team consists of two unmanned aerial vehicles (UAVs). Searcher 1 begins the mission with initial coordinates of (1.25, 1.25) nmi, which is located near the bottom left of the map. Searcher 2 begins the mission with initial
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Figure 2: Altitude of the region in meters coordinates of (58.75, 1.25) nmi, which is located near the bottom right of the map. They will navigate the region at a constant altitude of 3,000 m above sea level and they fly at a constant velocity of 100 kt. They are limited to a maximum turning rate of two degrees per second. This leisurely turn rate was chosen to smooth the search paths and to help them appear to be less erratic. Because the search paths are composed of samples once every 30 seconds, this turn rate forces the searcher to take at least 3 minutes to make a 360 deg turn. The searchers are equipped with sensors as explained next.
3.3 Sensor model The searchers use a simple sensor that is aimed directly below their vehicle. The sensor takes a snapshot of the terrain once every 30 seconds. At each discrete time-step used in our objective function, an image is captured for the purpose of detecting targets. The sensor has a viewing angle, θ, which we set to be about 51 deg in every direction. This
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viewing angle is sufficient for the searcher to detect a target up to 2 nmi away while flying at 3000 m above the ground. Because we constrain our sensor with a maximum viewing angle rather than constraining the detection range, the detection range varies dramatically across this region. When searching the low lying areas, the detection range is 1.4 nmi. On the other hand, when searching the highest parts of the mountains, the detection range is only 0.1 nmi. The sensor cannot observe a target that is outside of the sensor’s instantaneous viewing angle. To determine if a target is within the detection range of a searcher, we need to determine the difference in their altitudes, ∆alt , which is the difference in altitude between the target and the searcher. See Figure 3 for a graph showing the detection range versus the difference in altitude between the target and the searcher. In order to have a better idea of the values for ∆alt pertaining to this mission, recall that the searcher flies at a constant altitude of 3000 m above sea level. Then, compare the searcher’s altitude to the altitudes found in Figure 2.
3.4 Detection function The detection function is expressed as ( PD,terrain + δr Br , θt < θ PD = 0, θt > θ
(3)
It is based on the simple sensor model described in the previous section. The expression in (3) is the sum of the probability of detection associated with the target’s terrain, PD,terrain , plus the probability bonus Br , which is added when the target is on a road. We assume that it is easier to detect a target when it is on a road. θ and θt are the sensor’s viewing angle and the angle to the target, respectively, which were defined in Section 3.3. The binary operator δr is ( 1, Target is on a road δr = (4) 0, Target is off the road See Table 1 for the probability of detection associated with each type of terrain. These values consider possible obstructions of view and the possibility of the target blending in with the surroundings.
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Table 1: Target detection probabilities
Detection range (nmi)
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PD,terrain 0.90 0.50 0.10
Road Bonus 0.05 0.25 0.40
PD,terrain +Road Bonus 0.95 0.75 0.50
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3.5 Initial target distribution
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Table 2: Percent of targets in each type of terrain
Figure 3: Detection range vs ∆alt
Terrain Type Mountain Desert Road Forest
Formally speaking, a target cannot be detected if the angle to the target, θt , is greater than the sensor’s maximum viewing angle θ. Formula (2) is used to calculate θt . θt = tan−1 (d/∆alt )
% of targets 90 07 02 01
(2)
where d is the distance along the ground between the searcher and the target. ∆alt is the difference in altitude between the target and the searcher. The detection function, explained next, is used to determine the probability of detecting a target when it is within the detection range. Clearly, the detection function should depend on whether or not the target is within the sensor’s field of view.
This means that for our simulations a target has a 90% chance to be located in the mountains. Figure 4 illustrates the probability density map for the distribution of a target’s location. In this figure, the percentages from Table 2 are uniformly distributed over their respective regions. The numbers appear to be very small because there are a total of 26,244 cells used to create the map, because the map is 162 cells by 162 cells. Summing the values from each cell does, in fact, result in unity.
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consecutive detections. For simplicity, we normalized tr(I) to unity and call it Φ. Without normalizing tr(Ik ), the values for G can get uncomfortably large to read when there are many targets. The second graph in Figure 5 provides a multiplier that is used to calculate the information obtained via prediction after consecutive misses. Recall that the Kalman filter is able to continue predicting a target’s state even if it does not receive new measurements. The prediction loss vs consecutive misses curve shows the ratio between the information gained via prediction to the information gained from the most recent detection. This can be interpreted as the quality of the predictions begins comparable to the quality of the most recent detection but, as the amount of time increases since the last observation, the quality of the predictions begins to deteriorate. 1
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Figure 4: Probability density map for a target’s location 0.5
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Figure 5: Gain/Loss functions used to replace the Kalman filter
3.8 Objective function design parameters In this simulation, we set the importance factor to unity for all of the targets and define the forgetting factor as an exponential function that decays at a rate such that the value of old information is reduced by 50% after half of the two-hour mission. The forgetting factor is given by # " log(1 − p) (5) (K − k)Ts , λk = exp τ
Table 3: Target speed (knots) in various types of terrain. Terrain Type Road Desert Mountain Forest
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The targets move at a constant speed of 35 kt on a road and travel more slowly when they are off of the road. See Table 3 for the target’s velocity when traveling in the various terrain types. Road targets are allowed to leave the map when they encounter the edge of the map. Upon leaving the map, they are replaced with a new target headed in the opposite direction, which is back into the search region. If an offroad target encounters the edge of the map, it turns around at some random angle. These steps guarantee that there is a constant number of targets on the map at all times. Furthermore, the targets must also remain in their initial terrain type. Otherwise, the targets would leave the target rich terrain types and enter into other types of terrain, which would cause the values in Table 2 to vary over time.
Vt 35.00 33.25 8.75 3.50
3.7 Target tracker In order to reduce the computation time of our optimization algorithm, we did not use a Kalman filter to actively track the targets and to gain information. Instead, we created a look-up table that returns the trace of an information matrix proportional to the number of samples that the target has been tracked. Without this shortcut, it would have added considerable computation time when evaluating solutions. Our look-up tables are plotted in Figure 5. The first graph shows the normalized trace of the information matrix vs
where k is the sample number at which the information was obtained. The input parameter p is a value between 0 and 1, indicating how much to decay the information after τ time has passed. In this simulation, we chose p to be 0.5. K is the total number of samples in the mission and Ts is the sampling interval. This two-hour scenario has a sampling rate of twice per minute resulting in a total of 240 samples. Figure 6 shows the relation between λk and k for our chosen parameters. Note that λk is small for early detections and gradually increases toward unity at the end of the mission because the purpose of the forgetting factor in this scenario is to reduce the value of old information.
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Figure 6: Forgetting factor vs time (120 samples = 1 hour)
Figure 7: Search paths generated using PSO
3.9 Search paths In this scenario, we evaluate three paths using our proposed metric. The first path was generated using a modified particle swarm optimization (PSO) algorithm and the other two paths are simple ladder patterns. The paths are explained next.
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We used a modified PSO algorithm to generate the search paths shown in Figure 7. PSO is a form of evolutionary computation because it iteratively improves its solution. The modified PSO uses the prior distribution of the target’s spatial density not only to evaluate a solution but also as a guide when creating the next generation of solutions. Details of the modified PSO algorithm can be found in [9]. For further reading about PSO, see [10], [11], and [12]. The two-searcher solution generated by our PSO algorithm is clearly focused on searching the low lying, mountainous region because of a combination of two factors. The first factor is that there is an abundance of targets in the mountains and the second is that the detection range in low lying areas is better than the detection range in high altitude regions. Both of these reasons are favorable conditions for detecting targets. The PSO solutions appear to have the searchers revisit previously searched areas for track maintenance. 3.9.2 Ladder pattern search paths Ladder patterns are commonly used search paths and are simple to generate. Figures 8 and 9 show two examples of a latter pattern. Figure 8 shows a pair of ladder patterns that use the prior knowledge of the possible target locations. This solution was generated by hand so that the searchers could hunt for targets in the mountainous region where 90% of the targets are known to be located. Figure 9 shows a pair of ladder patterns that were generated by hand assuming that the searchers have no prior knowledge of the spatial distribution of the targets.
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Figure 8: A ladder solution with prior
4 Results and analysis In this section we present the results for the three search paths in the simulated two-hour mission that was described in the previous section. All of the values in Table 4 were obtained by averaging over 100 runs using 1000 targets per run. The G column is the average value for our proposed metric. The “Targets” column indicates the average number of unique targets detected per run. The “Detections” column contains the average total number of detections, including repeated detections of the same target. Dividing the number in the detections column by 1000 is equivalent to the cumulative detection probability (CDP). The final column is the average number of detections per target, which is calculated by dividing the total number of detections by the number of unique targets that were detected. The solution generated by using the PSO algorithm has the highest value for G among the three solutions. Com-
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where the UAV height above the ground is reduced. Also, the searcher’s speed can affect the searcher’s ability to make multiple consecutive detections by moving the searcher out of detection range before the next measurement is taken.
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5 Conclusions This is a companion paper to [1], which can be found in these proceedings. It provides simulation results using our proposed performance metric in a realistic simulation of a search and track mission. Using G, we have provided simulation results to compare three pairs of paths for a team of cooperative unmanned vehicles. Furthermore, we have shown that our metric prefers a solution that jointly optimizes the objectives of detection and tracking. Table 4 shows that the solution that gained the most information was able to detect almost as many targets as the solution that detected the most targets and yet it was able to track them better by averaging more detections per target.
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Figure 9: A ladder solution without prior Table 4: Tabulated results. Figure 7 Figure 8 Figure 9
G 170 160 106
Targets 188 199 145
Detections 346 308 224
References detections target
[1] X. R. Li, R. R. Pitre, V.P. Jilkov, H. Chen, “A New Performance Metric for Search and Track Missions,” The 12th International Conference on Information Fusion, submitted for review, March 2009.
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paring the value of the index G from the solution in Figure 9 with the other solutions shows how a poor solution will score. This solution detected far fewer targets and then poorly tracked them because it did not use the prior knowledge that was available to the other two solutions. The PSO solution in Figure 7 found 11 fewer targets on the average than the ladder solution in Figure 8 but it still managed to have a higher value of G. This is due to the repeated detections of previously detected targets, which is necessary to improve tracking accuracy. Table 4 shows that the solution with the best G value had 38 more detections than the ladder solution that used prior knowledge. Multiple detections of the same target are necessary for tracking and the “Detections” column shows that the PSO solution maximized G by reacquiring targets. This brings to light the trade-off between conflicting objectives. One solution detected more targets but took fewer measurements of them. The other solution detected fewer targets but tracked each target better. The solution with the most detections (and higher G) concentrated the search in the low lying, mountainous regions of the map because of the combination of two factors mentioned above. Both factors are favorable conditions for tracking targets, especially because they make it easier for the searchers to reacquire previously detected targets. In each of the solutions, the average number of detections per target is low because the sampling rate is low for that speed and the detection range is low in the mountains,
[2] Y. Bar-Shalom, X. R. Li, T. Kirubarajan, Estimation with Applications to Tracking and Navigation, New York: Wiley, 2001. [3] M. Flint, M. Polycarpou, and E. FernandezGaucherand, “Cooperative Control for Multiple Autonomous UAV’s Searching for Targets,” Proc. of the 41st IEEE Conference on Decision and Control, pp. 2823–2828 December 2002. [4] M. Flint, E. Fernandez-Gaucherand, and M. Polycarpou, “Cooperative Control for UAV’s Searching Risky Environments for Targets,” Proc. of the 42nd IEEE Conference on Decision and Control, pp. 3567–3572 December 2003. [5] A. Pongpunwattana, “Real-Time Planning for Teams of Autonomous Vehicles in Dynamic Uncertain Environments,” Ph.D. Dissertation , University of Washington, 2004. [6] A. Pongpunwattana and R. Rysdyk, “Real-Time Planning for Multiple Autonomous Vehicles in Dynamic Uncertain Environments,” Journal of Aerospace Computing, Information, and Communication, Vol. 1, 2005. [7] J. Bellingham, M. Tillerson, M. Alighanbari, and J. How, Cooperative Path Planning for Multiple UAVs in Dynamic and Uncertain Environments, Proc. of the 41st IEEE Conference on Decision and Control, pp. 2618– 2622 December 2002.
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[8] A. Sinha, T. Kirubarajan and Y. Bar-Shalom, Autonomous Surveillance by Multiple Cooperative UAVs, Proc. of SPIE Signal and Data Processing of Small Targets, December 2005. [9] R. R. Pitre, “Modified Particle Swarm Optimization for Search Missions,” 40th Southeastern Symposium on System Theory, pp. 362–365, 2008. [10] J. Kennedy, R. Eberhart, “Particle swarm optimization,” Proc. IEEE International Conference on Neural Networks, pp. 1942-1948 vol. 4, 1995. [11] R. Eberhart, Y. Shi, “Particle swarm optimization: developments, applications and resources,” Proc. of the 2001 Congress on Evolutionary Computation, pp. 81– 86 vol. 1, 2001. [12] X. Hu, Y. Shi, and R. Eberhart, “Recent advances in particle swarm,” 2004 Congress on Evolutionary Computation, pp. 90–97 vol. 1, 2004.
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A New Performance Metric for Search and Track Missions 2: Design and Application to UAV Search Donald DelBalzo Planning Systems Inc. Technical Solutions Group QinetiQ North America, Slidell, LA, USA [email protected]
Ryan R. Pitre and X. Rong Li∗ Electrical Engineering Department University of New Orleans New Orleans, LA, USA [email protected], [email protected] Abstract – This is a companion paper to the paper titled “A New Performance Metric for Search and Track Missions,” which is also published in these proceedings. This paper provides a realistic simulation using the performance index proposed in the aforementioned paper to show how the proposed performance measure scores missions consisting of a team of unmanned aerial vehicles searching for and tracking targets in a realistic search mission setting. Keywords: Detection, tracking, filtering, estimation, mission planning, path planning, resource management, UAV.
1 Introduction This is a companion paper to the paper titled “A New Performance Metric for Search and Track Missions” [1]. The purpose of this paper is to provide simulation results using our proposed performance metric in a realistic simulation. Our original intent was to write a single paper that provided a detailed overview of our new performance metric and its application to search and track missions but, due to the page limitations of this conference, the material needed to be separated into two papers. Our work in [1] proposed a novel performance metric that can be used to jointly optimize the objectives of detection and tracking. We created an objective function that has several desirable properties including: • It can be used to jointly optimize the objectives of detection and tracking. • It is a scalar, which makes it easy to compare and evaluate against candidate solutions. • It can be controlled to encourage early detection. • It has easily tunable parameters for different types of search and track missions. ∗ Research supported in part by ARO through Grant W911NF-08-10409, NAVO through Contract N62306-09-P-3S01, Navy through Planning Systems Contract N68335-05-C-0382, and Project 863 through Grant 2006AA01Z126.
978-0-9824438-0-4 ©2009 ISIF
The cornerstone of our approach is the introduction of a new performance index—information gain—which allows joint optimization of standard objectives for search and track missions. When a path maximizes our proposed metric, the vehicle following that path will be able to gain the most information, on the average, by detecting and tracking as many targets as possible and as accurately as possible during the course of the mission. Here, we use information in the Fisher sense, which roughly means that information is the inverse of uncertainty and thus, the reduction in uncertainty results in an increase in information [2]. In order to use our metric, we require that the distribution of the targets be available along with a well-defined detection function. Other work in mission planning has approached this problem from different perspectives, such as maximizing the total number of targets detected, minimizing the distance to complete objectives, or by maximizing the one-step information gain. See [3], [4], [5], [6] , [7], and [8]. None of these jointly optimize the objectives of detection and tracking over the entire mission, as was proposed in [1]. Furthermore, a more in-depth review of [3], [4], [5], [6] , [7], and [8] can be found in [1], which is the first part of this work. This paper is organized as follows. In Section 2, we review our proposed objective function and its parameters. Section 3 describes an illustrative scenario and three different search paths that will be used to search the region. In Section 4, we evaluate the three search paths using our metric for the search and track mission described in Section 3. Finally, Section 5 gives our conclusions.
2 Objective Function Defined Our objective function, namely, the expected information gain, was defined in [1] as "K N # XX G=E αn,k λn,k tr (In,k ) (1) k=1 n=1
where
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• E[·] is the expectation with respect to all randomness, e.g., target distributions and detections.
• K is the length of the mission measured by discrete time intervals.
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• N is the total number of targets detected during the mission, which is random.
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• αn,k is the importance factor for target n at time k. nmi
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• λn,k is the forgetting factor for tracking target n at time k once it has been detected.
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• tr(In,k ) is the trace of the information matrix In,k for target n at time k.
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Note that the objective function does not directly depend on the number of searchers involved in the mission; rather it only depends on how well they work together as a team. Based on the above objective function, the best solution is the one that has the largest value of G among all candidate search paths.
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Figure 1: Terrain map
3 Simulation
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In this section, we describe a scenario in which we generate a set of search paths by maximizing G. In Section 4, we show the results of this simulation. Mission planning begins with an objective, an environment, and a search team with a set of sensors. Our objective is to locate targets and then track them during the course of a two-hour mission in order to maximize G. We begin by describing the terrain where the mission takes place. Then, we describe the search vehicles and their sensor capabilities. Next, we explain how we model the sensor’s detection performance. Here, we assume complete prior knowledge of the distribution of target locations. Then, we describe the simplifications that we used to avoid having to use a target-tracking algorithm while our optimization algorithm generated search paths. After that, we describe the parameters used in our objective function. Finally, we conclude with a description of each set of search paths.
3.1 Terrain The simulation takes place in a square region that is 60 nautical miles (nmi) wide. The resolution of the map is 162 cells by 162 cells. See Figure 1. This environment contains three different terrain types. They are mountain, desert, and forest. In the colorbar of the figure, the different colors labeled “Mount.” indicate different altitudes of the mountainous terrain. The region also contains a road network that is shown using grey lines. The altitude map, which can be seen in Figure 2, shows that the elevation of the terrain ranges from 856 m to 2833 m.
3.2 Search vehicles The search team consists of two unmanned aerial vehicles (UAVs). Searcher 1 begins the mission with initial coordinates of (1.25, 1.25) nmi, which is located near the bottom left of the map. Searcher 2 begins the mission with initial
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Figure 2: Altitude of the region in meters coordinates of (58.75, 1.25) nmi, which is located near the bottom right of the map. They will navigate the region at a constant altitude of 3,000 m above sea level and they fly at a constant velocity of 100 kt. They are limited to a maximum turning rate of two degrees per second. This leisurely turn rate was chosen to smooth the search paths and to help them appear to be less erratic. Because the search paths are composed of samples once every 30 seconds, this turn rate forces the searcher to take at least 3 minutes to make a 360 deg turn. The searchers are equipped with sensors as explained next.
3.3 Sensor model The searchers use a simple sensor that is aimed directly below their vehicle. The sensor takes a snapshot of the terrain once every 30 seconds. At each discrete time-step used in our objective function, an image is captured for the purpose of detecting targets. The sensor has a viewing angle, θ, which we set to be about 51 deg in every direction. This
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viewing angle is sufficient for the searcher to detect a target up to 2 nmi away while flying at 3000 m above the ground. Because we constrain our sensor with a maximum viewing angle rather than constraining the detection range, the detection range varies dramatically across this region. When searching the low lying areas, the detection range is 1.4 nmi. On the other hand, when searching the highest parts of the mountains, the detection range is only 0.1 nmi. The sensor cannot observe a target that is outside of the sensor’s instantaneous viewing angle. To determine if a target is within the detection range of a searcher, we need to determine the difference in their altitudes, ∆alt , which is the difference in altitude between the target and the searcher. See Figure 3 for a graph showing the detection range versus the difference in altitude between the target and the searcher. In order to have a better idea of the values for ∆alt pertaining to this mission, recall that the searcher flies at a constant altitude of 3000 m above sea level. Then, compare the searcher’s altitude to the altitudes found in Figure 2.
3.4 Detection function The detection function is expressed as ( PD,terrain + δr Br , θt < θ PD = 0, θt > θ
(3)
It is based on the simple sensor model described in the previous section. The expression in (3) is the sum of the probability of detection associated with the target’s terrain, PD,terrain , plus the probability bonus Br , which is added when the target is on a road. We assume that it is easier to detect a target when it is on a road. θ and θt are the sensor’s viewing angle and the angle to the target, respectively, which were defined in Section 3.3. The binary operator δr is ( 1, Target is on a road δr = (4) 0, Target is off the road See Table 1 for the probability of detection associated with each type of terrain. These values consider possible obstructions of view and the possibility of the target blending in with the surroundings.
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Table 1: Target detection probabilities
Detection range (nmi)
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Desert Mountain Forest
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PD,terrain 0.90 0.50 0.10
Road Bonus 0.05 0.25 0.40
PD,terrain +Road Bonus 0.95 0.75 0.50
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3.5 Initial target distribution
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Table 2: Percent of targets in each type of terrain
Figure 3: Detection range vs ∆alt
Terrain Type Mountain Desert Road Forest
Formally speaking, a target cannot be detected if the angle to the target, θt , is greater than the sensor’s maximum viewing angle θ. Formula (2) is used to calculate θt . θt = tan−1 (d/∆alt )
% of targets 90 07 02 01
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where d is the distance along the ground between the searcher and the target. ∆alt is the difference in altitude between the target and the searcher. The detection function, explained next, is used to determine the probability of detecting a target when it is within the detection range. Clearly, the detection function should depend on whether or not the target is within the sensor’s field of view.
This means that for our simulations a target has a 90% chance to be located in the mountains. Figure 4 illustrates the probability density map for the distribution of a target’s location. In this figure, the percentages from Table 2 are uniformly distributed over their respective regions. The numbers appear to be very small because there are a total of 26,244 cells used to create the map, because the map is 162 cells by 162 cells. Summing the values from each cell does, in fact, result in unity.
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consecutive detections. For simplicity, we normalized tr(I) to unity and call it Φ. Without normalizing tr(Ik ), the values for G can get uncomfortably large to read when there are many targets. The second graph in Figure 5 provides a multiplier that is used to calculate the information obtained via prediction after consecutive misses. Recall that the Kalman filter is able to continue predicting a target’s state even if it does not receive new measurements. The prediction loss vs consecutive misses curve shows the ratio between the information gained via prediction to the information gained from the most recent detection. This can be interpreted as the quality of the predictions begins comparable to the quality of the most recent detection but, as the amount of time increases since the last observation, the quality of the predictions begins to deteriorate. 1
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Figure 5: Gain/Loss functions used to replace the Kalman filter
3.8 Objective function design parameters In this simulation, we set the importance factor to unity for all of the targets and define the forgetting factor as an exponential function that decays at a rate such that the value of old information is reduced by 50% after half of the two-hour mission. The forgetting factor is given by # " log(1 − p) (5) (K − k)Ts , λk = exp τ
Table 3: Target speed (knots) in various types of terrain. Terrain Type Road Desert Mountain Forest
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The targets move at a constant speed of 35 kt on a road and travel more slowly when they are off of the road. See Table 3 for the target’s velocity when traveling in the various terrain types. Road targets are allowed to leave the map when they encounter the edge of the map. Upon leaving the map, they are replaced with a new target headed in the opposite direction, which is back into the search region. If an offroad target encounters the edge of the map, it turns around at some random angle. These steps guarantee that there is a constant number of targets on the map at all times. Furthermore, the targets must also remain in their initial terrain type. Otherwise, the targets would leave the target rich terrain types and enter into other types of terrain, which would cause the values in Table 2 to vary over time.
Vt 35.00 33.25 8.75 3.50
3.7 Target tracker In order to reduce the computation time of our optimization algorithm, we did not use a Kalman filter to actively track the targets and to gain information. Instead, we created a look-up table that returns the trace of an information matrix proportional to the number of samples that the target has been tracked. Without this shortcut, it would have added considerable computation time when evaluating solutions. Our look-up tables are plotted in Figure 5. The first graph shows the normalized trace of the information matrix vs
where k is the sample number at which the information was obtained. The input parameter p is a value between 0 and 1, indicating how much to decay the information after τ time has passed. In this simulation, we chose p to be 0.5. K is the total number of samples in the mission and Ts is the sampling interval. This two-hour scenario has a sampling rate of twice per minute resulting in a total of 240 samples. Figure 6 shows the relation between λk and k for our chosen parameters. Note that λk is small for early detections and gradually increases toward unity at the end of the mission because the purpose of the forgetting factor in this scenario is to reduce the value of old information.
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Figure 6: Forgetting factor vs time (120 samples = 1 hour)
Figure 7: Search paths generated using PSO
3.9 Search paths In this scenario, we evaluate three paths using our proposed metric. The first path was generated using a modified particle swarm optimization (PSO) algorithm and the other two paths are simple ladder patterns. The paths are explained next.
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We used a modified PSO algorithm to generate the search paths shown in Figure 7. PSO is a form of evolutionary computation because it iteratively improves its solution. The modified PSO uses the prior distribution of the target’s spatial density not only to evaluate a solution but also as a guide when creating the next generation of solutions. Details of the modified PSO algorithm can be found in [9]. For further reading about PSO, see [10], [11], and [12]. The two-searcher solution generated by our PSO algorithm is clearly focused on searching the low lying, mountainous region because of a combination of two factors. The first factor is that there is an abundance of targets in the mountains and the second is that the detection range in low lying areas is better than the detection range in high altitude regions. Both of these reasons are favorable conditions for detecting targets. The PSO solutions appear to have the searchers revisit previously searched areas for track maintenance. 3.9.2 Ladder pattern search paths Ladder patterns are commonly used search paths and are simple to generate. Figures 8 and 9 show two examples of a latter pattern. Figure 8 shows a pair of ladder patterns that use the prior knowledge of the possible target locations. This solution was generated by hand so that the searchers could hunt for targets in the mountainous region where 90% of the targets are known to be located. Figure 9 shows a pair of ladder patterns that were generated by hand assuming that the searchers have no prior knowledge of the spatial distribution of the targets.
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Figure 8: A ladder solution with prior
4 Results and analysis In this section we present the results for the three search paths in the simulated two-hour mission that was described in the previous section. All of the values in Table 4 were obtained by averaging over 100 runs using 1000 targets per run. The G column is the average value for our proposed metric. The “Targets” column indicates the average number of unique targets detected per run. The “Detections” column contains the average total number of detections, including repeated detections of the same target. Dividing the number in the detections column by 1000 is equivalent to the cumulative detection probability (CDP). The final column is the average number of detections per target, which is calculated by dividing the total number of detections by the number of unique targets that were detected. The solution generated by using the PSO algorithm has the highest value for G among the three solutions. Com-
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where the UAV height above the ground is reduced. Also, the searcher’s speed can affect the searcher’s ability to make multiple consecutive detections by moving the searcher out of detection range before the next measurement is taken.
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5 Conclusions This is a companion paper to [1], which can be found in these proceedings. It provides simulation results using our proposed performance metric in a realistic simulation of a search and track mission. Using G, we have provided simulation results to compare three pairs of paths for a team of cooperative unmanned vehicles. Furthermore, we have shown that our metric prefers a solution that jointly optimizes the objectives of detection and tracking. Table 4 shows that the solution that gained the most information was able to detect almost as many targets as the solution that detected the most targets and yet it was able to track them better by averaging more detections per target.
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Figure 9: A ladder solution without prior Table 4: Tabulated results. Figure 7 Figure 8 Figure 9
G 170 160 106
Targets 188 199 145
Detections 346 308 224
References detections target
[1] X. R. Li, R. R. Pitre, V.P. Jilkov, H. Chen, “A New Performance Metric for Search and Track Missions,” The 12th International Conference on Information Fusion, submitted for review, March 2009.
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paring the value of the index G from the solution in Figure 9 with the other solutions shows how a poor solution will score. This solution detected far fewer targets and then poorly tracked them because it did not use the prior knowledge that was available to the other two solutions. The PSO solution in Figure 7 found 11 fewer targets on the average than the ladder solution in Figure 8 but it still managed to have a higher value of G. This is due to the repeated detections of previously detected targets, which is necessary to improve tracking accuracy. Table 4 shows that the solution with the best G value had 38 more detections than the ladder solution that used prior knowledge. Multiple detections of the same target are necessary for tracking and the “Detections” column shows that the PSO solution maximized G by reacquiring targets. This brings to light the trade-off between conflicting objectives. One solution detected more targets but took fewer measurements of them. The other solution detected fewer targets but tracked each target better. The solution with the most detections (and higher G) concentrated the search in the low lying, mountainous regions of the map because of the combination of two factors mentioned above. Both factors are favorable conditions for tracking targets, especially because they make it easier for the searchers to reacquire previously detected targets. In each of the solutions, the average number of detections per target is low because the sampling rate is low for that speed and the detection range is low in the mountains,
[2] Y. Bar-Shalom, X. R. Li, T. Kirubarajan, Estimation with Applications to Tracking and Navigation, New York: Wiley, 2001. [3] M. Flint, M. Polycarpou, and E. FernandezGaucherand, “Cooperative Control for Multiple Autonomous UAV’s Searching for Targets,” Proc. of the 41st IEEE Conference on Decision and Control, pp. 2823–2828 December 2002. [4] M. Flint, E. Fernandez-Gaucherand, and M. Polycarpou, “Cooperative Control for UAV’s Searching Risky Environments for Targets,” Proc. of the 42nd IEEE Conference on Decision and Control, pp. 3567–3572 December 2003. [5] A. Pongpunwattana, “Real-Time Planning for Teams of Autonomous Vehicles in Dynamic Uncertain Environments,” Ph.D. Dissertation , University of Washington, 2004. [6] A. Pongpunwattana and R. Rysdyk, “Real-Time Planning for Multiple Autonomous Vehicles in Dynamic Uncertain Environments,” Journal of Aerospace Computing, Information, and Communication, Vol. 1, 2005. [7] J. Bellingham, M. Tillerson, M. Alighanbari, and J. How, Cooperative Path Planning for Multiple UAVs in Dynamic and Uncertain Environments, Proc. of the 41st IEEE Conference on Decision and Control, pp. 2618– 2622 December 2002.
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[8] A. Sinha, T. Kirubarajan and Y. Bar-Shalom, Autonomous Surveillance by Multiple Cooperative UAVs, Proc. of SPIE Signal and Data Processing of Small Targets, December 2005. [9] R. R. Pitre, “Modified Particle Swarm Optimization for Search Missions,” 40th Southeastern Symposium on System Theory, pp. 362–365, 2008. [10] J. Kennedy, R. Eberhart, “Particle swarm optimization,” Proc. IEEE International Conference on Neural Networks, pp. 1942-1948 vol. 4, 1995. [11] R. Eberhart, Y. Shi, “Particle swarm optimization: developments, applications and resources,” Proc. of the 2001 Congress on Evolutionary Computation, pp. 81– 86 vol. 1, 2001. [12] X. Hu, Y. Shi, and R. Eberhart, “Recent advances in particle swarm,” 2004 Congress on Evolutionary Computation, pp. 90–97 vol. 1, 2004.
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