A New Performance Metric for Search and Track Missions

Transcription

1 12th International Conference on Information Fusion Seattle, WA, USA, July 6-9, 9 A New Performance Metric for Search and Track Missions X. Rong Li Ryan R. Pitre Vesselin P. Jilkov Huimin Chen Department of Electrical Engineering University of New Orleans New Orleans, LA 7148, USA {xli,rrpitr,vjilkov,hchen2}@uno.edu Abstract The joint optimization of conflicting objectives is usually challenging because there is no clear-cut way to do so. When planning for search and track missions, one must choose between the conflicting objectives of detecting new targets and tracking previously detected targets. This paper proposes a novel metric that integrates the objectives of target detection, target tracking, and vehicle survivability nicely into an integrated single scalar index that can be used to optimize paths for joint detection and tracking. The cornerstone is the introduction of a new performance index, that is, information gain, which permits joint optimization of the objectives for search and track missions. Several examples are provided to illustrate how to use our proposed metric. Keywords: Detection, tracking, filtering, estimation, mission planning, path planning, resource management, UAV. I. INTRODUCTION In order to mitigate the loss of human life, many agencies have removed man from the driver s seat and have begun employing unmanned aerial vehicles (UAVs). As the demand for unmanned vehicles increases, the research community has been called upon to solve many issues that arise from using autonomous systems. This paper proposes a metric that can be used to jointly optimize the objectives for search and track missions. Previous work in mission planning for unmanned vehicles has approached this problem from different angles such as maximizing the total number of targets detected [4], [5], shortest distance to complete objectives [2], [3], [7], [11], [12], or by maximizing the one-step information gain [13]. A commonly used measure of performance is the cumulative detection probability (CDP) over the total mission; that is, the probability that the searcher will have detected the target(s) by the end of the mission. CDP is equivalent to the expected number of targets detected during the mission and is the basis for search theory and most search techniques. This measure emphasizes the detection of targets but does not reward early detection it does not distinguish between early detections and late detections. Furthermore, it completely ignores frequent observations of the same targets, which is necessary for tracking targets. Research supported in part by ARO through Grant W911NF , NAVO through Contract N626-9-P-3S1, and Project 863 through Grant 6AA1Z126. Search path planning has received considerable attention. Flint et al. proposed in [4], [5] a method for cooperative search using multiple UAVs with limited communication. They try to maximize a search-to-go gain, which sums the one-step gains over the planning horizon while considering the UAV s survival in addition to the effects of interference by a team member. Pongpunwattana et al. proposed a method [11], [12] that assigns and schedules tasks in order to maximize the team s overall score. This method focuses on completing tasks, such as visiting a target or a site, which are known prior to the mission. It places a special emphasis on the communication structure, which they refer to as a market-based approach. It allows each team member to bid on a task and then the task is assigned to the member that will best improve the team s score. The method of [2] maximizes an objective function that takes into consideration survival of the team, the value associated with completing certain tasks, and the time needed to complete the tasks. An interesting point from this work is the idea of phasing, which allows team members to begin and end the mission at different times rather than at the same time. The approach of [13] maximizes the information gained from an environment while searching for new targets and tracking known targets. This idea of information gain, which inspired our work, is based on the information filter. However, this approach is myopic in that it only plans one time-step into the future whereas our approach can maximize information gain over the entire mission. [3] introduces a probabilistic approach to path planning assuming the locations of the targets are known and fixed. It jointly minimizes the threat to the UAV and the total time to arrive at the target by using a probabilistic threat map containing the locations of the threats and their probabilities of disabling the UAV. The probability that a threat can kill a UAV is a two-dimensional Gaussian function centered at the threat s location. [7] proposes a path planner for UAVs based on a modified particle swarm optimization algorithm. To generate the best path between a fixed starting location and a fixed terminal location, they jointly minimize the distance traveled, the average altitude above sea level, and the danger from threats while constraining the UAV s turning radius. [14] proposes a look-ahead policy for autonomous cooperative surveillance. This work ISIF 11

2 uses a layered decision framework rather than integrating multiple objectives into a single objective function. In other words, the objectives are prioritized and then evaluated in order of priority, a well-known strategy in multi-objective optimization. If the minimum criterion of an objective is not satisfied, the next objective will not be evaluated. In no particular order, their mission objectives are detecting new targets, classification, tracking, and UAV safety. [6] approaches the path planning problem for search and track missions by maximizing the peak value of the predicted pdf for the possible target locations. In other words, a search path is chosen that minimized the predicted estimation error covariance over their look-ahead time. [9] presents a path planning strategy for a UAV to track a previously detected ground vehicle. The system was tested on a real UAV and was shown to successfully track a ground target. The purpose of our work is to propose a metric that can be used to jointly optimize the objectives for search and track missions. We boil down mission planning into a path-planning problem that integrates the objectives of target detection, target tracking, and team survivability into an integrated single scalar metric. The beauty of this scalar metric is that it greatly simplifies determining which path, among many candidates, is the best solution for the problem at hand. When a path maximizes our proposed metric, it can be said that the vehicle following that path will be able to gain the most information, on the average, by 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 is roughly the inverse of uncertainty, and thus the reduction in uncertainty results in an increase in information [1]. In order to use our metric, we require that the distribution of the targets be known (to within a small set), along with a well-defined detection function. In the search and track mission, detection is obviously an essential objective. Clearly, to begin acquiring information about a target, the searcher must be able to detect the target. To do this, the searcher must first get within detection range and then make a detection. This requires a path that can put the searcher in the right place at the right time, which is vital when finding targets with unknown locations. Naturally, we desire a path that can satisfy the detection objective by placing our searcher in the best position to make detections. The second primary objective included in our metric is our intent to track targets. Once a target has been detected, tracking begins. By making additional observations of the target, we can improve our estimate of the target s state and therefore increase the amount of information that we have about the target. In search and track missions, it is desirable for a searcher to be able to make new detections in addition to being able to keep tracking the targets detected. We can think that we have more information about a target when we know better its state (i.e., a good estimate). Overall, we want paths that will allow our searchers to detect new targets as well as to continue observing previously detected targets. These are the key ingredients of search and track missions. Another objective for the search and track mission is team survivability. The likelihood that the team members will survive is indirectly included in our proposed objective function. Obviously, a searcher needs to be alive in order to gain information about targets, and a searcher that lives longer will have more opportunities to detect and track targets. Clearly, an early loss of a team member is worse for the mission than a late loss because the team will have one less member participating in the search. In summary, we created a metric for search and track missions that can be used to jointly optimize the objectives of detection and tracking by maximizing the information gain. The paper is organized as follows. In Section II, we define our proposed objective function and describe its parameters. Section III highlights the desirable properties of our metric. Section IV modifies our metric to include a penalty for a loss of a team member. In Section V, we analyze several examples and evaluate our metric for simplified search and track missions. Finally, Section VI contains our conclusions. Also refer to the companion paper in [1] for an application to UAV search in a realistic scenario. II. OBJECTIVE FUNCTION Our objective function is the expected information gain [ K ] N G = E α n,k λ n,k tr (I n,k ) (1) where k=1 n=1 E[ ] is the expectation with respect to all randomness, including target distributions and detections. K is the length of the mission measured by discrete time intervals. N is the total number of targets detected during the mission, which is random. α n,k is the importance factor for target n at time k. λ n,k is the forgetting factor for tracking target n at time k once it has been detected. tr(i n,k ) is the trace of the information matrix I n,k for target n at time k. 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 this objective function, the best solution is the one that has the largest value of G among all candidate search paths. Next, we explain each of these components in detail. A. Expectation operator In our objective function, we take the expectation of the total information that is gained using a particular solution. This means that G is the average amount of information that will be gained by choosing the path being evaluated. By comparing the G values from two different paths, one can tell which of the paths is better on the average. 111

3 B. Total number of targets detected, N The total number of detected targets, N, is random before the mission due to the uncertainty in the locations and detection of the targets. This is the most unusual ingredient of our objective function. The expectation is used to average out this randomness when calculating G for a particular path. Not surprisingly, different paths may have different average values for N. This can be attributed to a heterogeneous target distribution and/or that the probability of detecting a target is dependent on the environment itself. For example, some types of targets may be more difficult to detect in a forest than in an open field. That is why N would vary for different search paths. It should be clear from Eq. (1) that as more targets are detected, N will increase, which results in more terms being included in the summation. Since all of the terms are positive valued, more terms will result in a higher score. We emphasize that the objective function is only interested in maximizing the total information gained and is not directly concerned with the total number of targets detected. It is possible for a solution that poorly tracks many targets to have a lower score than a solution that perfectly tracks a few targets. With this said, maximizing the number of detected targets is not critical when maximizing our metric because maximizing G in (1) is not the same as maximizing N. In summary, our proposed objective function does not care directly about the total number of targets detected. It only guarantees that the most information will be gained on the average. C. Importance factor α n,k The importance factor is a design parameter that is necessary when the targets are not equally important, such as when high-profile targets and/or major threats are involved. When using the importance factor for real search missions, sensors capable of recognizing target class or type information are very helpful but not necessary. The importance factor is mission dependent and directly depends on what is known about the targets. For example, a bicycle may not be as important as a tank. Therefore, the importance factor should be used to encourage the team to gain information about a particular target or type of target. The importance factor may also be a function of a target s kinematic state, such as location or velocity. For example, a target may be more valuable in one location than in another; a moving target could have a higher importance than a stationary target. Lastly, the importance factor can be time dependent a particular target may be more valuable at a particular time. As previously stated, it is possible for our objective function to award a higher score to a path that accurately tracks a few targets than to a second path that poorly tracks many targets. This is a conflict between two competing objectives, specifically, detection and tracking. If the user is more interested in detecting targets than tracking them, the importance factor for a target can be reduced once that target has been detected. This immediately decreases the attractiveness of previously detected targets and results in solutions that seek out additional targets. If targets become completely unimportant after the first time that they are detected, then maximizing G is equivalent to maximizing the total number of detected targets, N, which is the same as maximizing the CDP discussed above. D. Forgetting factor λ n,k The forgetting factor is used to specify how valuable information is as a function of time. It is a design parameter that can reduce the value of the information based on the age of the information. Reducing the value of old information also agrees with the common sense that newer information almost always has a higher value than dated information. For example, it may be desired for information obtained during the middle of the mission to be only half as valuable as information obtained at the end of the mission. If that were the case, the forgetting actor can be set accordingly. There are two extreme cases for the forgetting factor. One case forgets nothing and the other forgets everything but the very last sample. For surveillance missions, which are missions interested in recording a complete history of target behavior, the forgetting factor might be set such that all information is equally important, which would result in nothing being forgotten. The forgetting factor can also be set such that it is only important to maximize the information gained in the very last sample, which can be interpreted as maximizing the most up-to-date information. This results in solutions that can provide the best estimates of the target s state at the very end of the mission. The forgetting factor does not necessarily have to have the heaviest weights towards the end of the mission. It can be used to emphasize information gained at any time during the mission rather than just at the end. For example, a particularly interesting event may occur at the midway point of the mission and it is important to maximize the information gained during this period. Note that the importance factor and forgetting factor appear in G only as a product. As such, their effect on G can be lumped into a single factor, but the use of such two intuitively appealing factors greatly simplifies the design of this single total factor. E. Information matrix I n,k The term information quantifies the accuracy of our estimates. In the Fisher sense, information is the inverse of uncertainty and a reduction in uncertainty results in an increase in information [1]. By improving the accuracy of the state estimates, we are increasing the amount of information that we have about the target. In other words, we have the most information when our estimates of the target s position and velocity are most accurate. It should be emphasized that the information matrix I n,k is the combined information about target n at time k from all of the UAVs that are estimating the state of target n. When multiple UAVs are 112

4 tracking a target, their individual estimates need to be fused to have a single, more accurate estimate. The information matrix is obtained by inverting the estimation error covariance matrix P. When a target is initially detected, the estimate of its state has the accuracy at the level of the measurement noise inherent to the sensor. With this relatively low level of accuracy, there is not much information in the estimate compared with that in an accurate measurement. Therefore, using our metric, it is in the best interest of the team to track many targets as accurately as possible. Lastly, even after observations of a target have not been made for some time, information will continue to be gained from that target because of the Kalman filter s ability to predict the target s state without needing new observations of the target. We chose to use the information rather than using the error covariance directly because this approach has some simplifying properties. The first of these properties is that information obtained from independent observations is additive, which is why we are able to use summations in Eq. (1). Another property is that it is easier to think of having zero information about a target s state than it is to think of having an infinite estimation error covariance before detecting a target, which is useful when implementing the simulation software. A seeming reasonable objective function is ] [ K [ K G = P D,k ]E k=1 k=1 λ k tr(p 1 k k ) which is the product of those for target detection and target tracking, respectively. Our function (1) reduces to this performance index if all targets found in the mission are first detected at the same time, all target tracks have identical accuracy, and all targets are equally important. Evidently, these conditions are very restrictive and thus this index is clearly inferior to our objective function. III. DESIRABLE PROPERTIES OF G This section highlights some of the desirable properties of our metric. A. Jointly optimizes detection and tracking Jointly optimizing conflicting objectives is difficult [8]. Our objective function contains elements of both detection and tracking and can be used to optimize these objectives jointly. B. Easily compares different solutions When using our metric to evaluate different solutions, all that needs to be done is to compare the scalar value associated with each solution. This greatly simplifies determining which path, among many candidate paths, is the best solution for the problem at hand. C. Promotes early detection Our metric usually encourages early detection of targets in search and track missions, although it can be used to encourage late detection as well. When tracking targets, it is usually better to make an earlier detection than a later detection because there is more time to improve the state estimates and more time for action. When targets are detected early in a mission, they can be tracked for a longer time, which will increase the total amount of information gained from the targets. This is supported by the Kalman filter s ability to continue estimating the state of the target long after the last time that the target was observed. D. Encourages repeated observations of the same targets Frequent observations of the same targets is beneficial because the targets can be tracked more accurately, which increases the total amount of information gained. To maximize G, we need to select a solution that enables us to detect more targets (i.e., a larger N) and in the meantime track the targets more accurately so that the estimation error associated with those targets is small. As such, other things equal, it is certainly better to keep tracking a target for a longer time than for a shorter time. Our index also encourages this since it will lead to a larger information gain as a result of more terms over the time in the summation. For example, our index differentiates three cases: never found (i.e., detected), found and lost, found and tracked (up to now). E. Useful for resource management When using a team of UAVs, the quantities used in our index refer to those from the entire team. For instance, N is the total number of targets detected by all UAVs; I n,k is the information matrix of target n using data from all sensors onboard all UAVs; that is, it is a result of data fusion. This new index also reflects the effect of many other factors of a UAV mission on joint detection and tracking. For example, given a UAV path, different resource allocation strategies under energy or time constraints will yield different index values. The one with a higher value is better and so the index can be used as a basis for resource allocation. IV. ACCOUNTING FOR TEAM SURVIVAL A UAV may be lost if it is too close to a threat or if it hits an obstacle or the ground. It would not be able to contribute to the mission if it is lost, not to mention the financial cost of the UAV system. To penalize the selection of paths that are too risky in terms of losing UAVs, the survivability of the UAVs is better accounted for in the objective function. Our objective function indirectly promotes team survival. Unfortunately, our metric views a path that does not find a target as being equivalent to a solution that loses that searcher. In other words, it is unable to differentiate a path that loses a UAV from a path that does not detect a target because there is no penalty term in our measure for the loss of a UAV. Our proposed objective function does not care directly about how many UAVs survive. Rather, it only 113

5 guarantees that the most information will be gained on the average. An expression that explicitly accounts for the loss of a team member is provided next. Including a survivability factor While the actual cost of a UV varies so much that a reasonable and general way for its direct use in an objective function would not seem possible, in our framework of information gain, it is reasonable to treat the value of a lost UV system for a mission as the expected information gain it would provide were it not lost. With this idea, (1) is modified as K N M G s = E α n,k λ n,k tr(i n,k ) (2) k=1 n=1 j=1 G j,kj Here, G j,kj is the expected information gain (from time k j on) of UAV j lost at k j 1, and M is the total number of UAVs lost during the mission, which is random before the mission. When there is no knowledge about G j,kj, we may choose it to be equal to the difference in the contributions of the jth UAV to the expected information gain if the UAV is not lost and if it is lost at time k j. Clearly, an early loss of a UAV is worse for the mission than a late loss. This is reflected in G because an early loss of a UAV results in less opportunities to make detections and observations. It is even more reflected in G s since G j,kj would be greater as k j decreases. G s also appraises the loss of different UVs differently. V. EXAMPLES AND ANALYSIS In this section, we present five simple scenarios to illustrate how different solutions score when using our metric. These examples highlight the desirable properties of our metric. In each figure, the solution that tracks multiple targets is plotted using a blue (solid) line and the solution that tracks a single target is plotted using a red (dot-dash) line. The first example compares two solutions that track a single target in order to show that early detection is better when the forgetting factor is set to unity. The second example demonstrates how the forgetting factor can encourage late detections. The next set of examples shows that our metric prefers tracking multiple targets to a single target given that both cases have an identical number of total observations on the targets. The final example demonstrates how the importance factor can encourage tracking multiple targets rather than tracking a single target. In all the examples, the targets move at a constant velocity (CV) in two dimensional space, and a Kalman filter with a CV model is used to track the targets. Each simulation takes place over a 1 second time period and is sampled at a rate of 1 samples per second for a total of 1 samples per scenario. The actual paths in the examples are not important and are therefore not provided. For the diligent reader, these examples are easily reproducible because the estimation error covariance does not depend on the actual measurements. In fact, no measurements were simulated in these examples since they do not affect error covariance. The motion and measurement models are x k = Fx k 1 + Gw k 1 z k = Hx k + v k (3) with w and v being i.i.d. zero-mean Gaussian variables with variances of Q and R, respectively. The matrices F, G, Q, R, and the sampling interval T are given by 1 1/T T 2 /2 F = 1 1 1/T, G = T T 2 /2 1 T [ ] [ ] 1 1 m 2 H =, Q = 1 1 s [ ] 4 4 R = m 2, T =.1s 4 When viewing the figures in this section, the label on the y-axis is the trace of the information matrix at sample k: = tr(i k ), which is the information gain at k. However, if multiple targets are included in an example, then = N n=1 tr(i n,k), unless otherwise stated. Furthermore, we always set tr(i n,k ) = before target n has been detected. Lastly, the legend in each figure shows each solution s score using our metric G, with G = 1 k=1, that is, the area under the curve. A. G prefers early detections rather than late detections This example shows that our metric encourages detection of a target early, rather than later, in the mission when the forgetting factor and the importance factor are set to unity. In Figure 1, two sets of scores are shown. Note that the second case (red, dot-dash line) is a delayed (by samples) version of the first case (blue, solid line). The first solution, in blue, detects a target at sample and then tracks it until sample 7. The second solution detects a target at sample number 4 and then tracks it until sample 9. In both cases, the target is observed and tracked for 5 samples and then tracked using prediction until sample 1. Note that in the blue case, the value of begins to decrease after sample 7 but it is still positive valued. Even though the blue case last observed the target at sample 7, it is still able to estimate the target s state and is therefore able to continue gaining information about the target by prediction. The solution in blue has a higher score (G = 16, 848) than that in red (G = 16, 57) because the former was able to take advantage of the Kalman filter s ability to estimate the target s state using prediction. By detecting the target earlier, the blue solution was able to predict the target s state for samples after the last time the target was observed whereas the second case was only able to do so for 1 samples. B. Use forgetting factor to encourage late information gain This example uses the same scenario as in Figure 1 except that here the forgetting factor λ k in (1) is not unity (see 114

6 5 4 G = G = G = 8694 G = Fig. 1. Encourage early detection: scores for tracking a target first detected at samples and 4, respectively. Fig. 3. Encourage later detection: scores for tracking a target first detected at samples and 4, respectively. λ k Fig. 2. Forgetting factor vs. time. Figure 2). This λ k means that information gained towards the end of the mission is more valuable. Figure 3 shows the forgetting factor applied to the scenario from Figure 1. Here, = λ k tr(i k ). The latter solution has a greater value for G showing that the forgetting factor can encourage later information gain. C. G prefers multiple targets to a single target In each of the following examples, we compare two solutions. The first example compares tracking two targets with tracking a single target. The two-target case tracks the two targets contiguously in time and the single-target case tracks its target over the same interval. The second example compares tracking the targets over different intervals. In these examples, each solution is allowed a maximum number of observations on the targets. In other words, the tracker is only allowed to observe the targets for a fixed total number of measurements, whether observing a single target or multiple targets. This will demonstrate that it is better to distribute the fixed number of observations among multiple targets than it is to focus on a single target. This is only valid with equally important targets. Therefore, we set the forgetting factor and the importance factor to unity. Our metric prefers solutions that track multiple targets because the covariance of the estimation error is lower bounded and there comes a point when additional observations do not reduce the estimation error (noticeably). However, if multiple targets have been tracked, even if I n,k reaches its upper bound, information can still be gained from the previously detected targets. Figures 4 and 5 graph the information gained vs. time for both of the multi-target examples. These examples show why our metric prefers detecting multiple targets more than it values detecting a single target. Here, we plot the multiple target case in blue and we let = N n=1 tr(i n,k). Figure 4 compares a solution that tracks two targets with one that tracks a single target. The blue line shows the twotarget case, where = 2 n=1 tr(i n,k); the red line shows the single-target case. Both cases are given a total of 4 observations. The two-target case tracks target A for samples and then tracks target B for samples, whereas the single-target case only tracks target A for 4 samples. In the two-target case, target A is observed and tracked from sample 1 to sample. From samples to 1, target A s state is estimated using only prediction without observations. Then, target B is observed and tracked from sample to sample 5. From samples 5 to 1, target B s state is estimated using only prediction. For the single-target case, target A is tracked over the interval from sample 1 to sample 5. However, as reflected in G, the information gained in the single target case is less than that in the two-target case because it does not get the full benefit of predicting the state of multiple targets. Again, in both cases, the trackers are given the same number of observations but the one that is tracking two targets is able to gain more information. In fact, 115

7 5 4 G = G = G = G = Fig. 4. Prefer multiple targets to single target: comparison of two cases with same total number of observations. Fig. 5. Prefer multiple targets to single target: re-acquiring an old target (dot-dash) vs. starting tracking a new target (solid). is equal for both cases until the second target is detected, which is when the benefit of prediction becomes apparent. From sample 5 to sample 1, is always greater for the two-target case. Having a larger G, the two-target case is the preferred solution. The next multi-target example is shown in Figure 5. In this example, we again show that our metric prefers a solution that detects multiple targets to one that detects a single target when both cases have the same total number of observations. This time, however, there is a gap in time between detections. At sample 1, a searcher detects and actively tracks a target for a duration of 1 samples. Then, the searcher does not make another detection until sample number 7. In the meantime, the searcher continues to track the first target using prediction. At sample 7, the searcher can either begin tracking a new target, as shown in blue, or reacquire the original target, as shown in red. Regardless of the case, the second sequence of observations continues until sample 9, after which there are no additional observations. Note from Figure 5 that the two- and single-target cases are equal before the second sequence of detections begins. Then, from sample 7 to the end of the scenario, more information is gained in the two-targets case, which is due to the Kalman filter s ability to predict the state of the first target. In this figure, = 2 n=1 tr(i n,k). Again, the twotarget case is the preferred solution. D. Use importance factor to encourage finding more targets A solution that continuously tracks a target for the entire mission may have a higher score than a solution that tracks a few targets for a relatively shorter time. This is because continuous tracking a single target will result in many more positively-valued terms in the summation of Eq. (1). In the previous section, we compared cases under the constraint that the total number of observations were equal, whether they originated from the same target or not. If it is undesirable to continuously track a single target and the user would rather seek out new targets, the importance factor α n,k can be used to encourage the searchers to seek out new targets rather than continuously track a single target. In this case, the importance factor can be used to reduce the importance of previously detected targets. Figure 6 shows the information gained from continuous tracking of a single target, in red, compared with a solution that briefly tracks two different targets, drawn in blue. Note that the solution in red has a much larger value for G G = 857 G = Fig. 6. Encourage following a single target: comparison of one- (dot-dash) and two-target (solid) cases. Figure 7 shows the effect that an importance factor has when applied to the scenario depicted in Figure 6. Here, = 2 n=1 α n,ktr(i n,k ). The importance factor in Figure 7 was arbitrarily designed to linearly reduce the importance of a target over the samples after that target s first detection, which would encourage tracking multiple targets by decreasing the importance of previously detected targets. After the importance factor was applied, the score for the two-target case is larger than the single-target case. This is 116

8 an example of how the importance factor can influence the decision to track multiple targets. VI. CONCLUSIONS This paper proposes a novel metric for mission planning that integrates the objectives of target detection, target tracking, and vehicle survival nicely into a single scalar index, that is, information gain for the search and track mission. Because it is a scalar, it is convenient to use when jointly optimizing the conflicting objectives of detection and tracking. We have provided several simple examples to highlight the attributes and versatility of our performance index. They also illustrated how the proposed metric evaluates different situations and they showed how the parameters of the index affect the value of the performance index. Although, as a weakness, the index is probably too complex for analytic solutions, it serves well optimization by evolutionary (e.g., particle swarm, ante colony, and genetic) algorithms as a fitness function. This is illustrated in the companion paper [1] which presents a simulated search and track mission in a more realistic setting. REFERENCES [1] Y. Bar-Shalom, X. R. Li, T. Kirubarajan, Estimation with Applications to Tracking and Navigation, New York: Wiley, 1. [2] 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 December 2. [3] A. Dogan, Probabilistic Approach in Path Planning for UAVs, Proc. of the 3 IEEE International Symposium on Intelligent Control, October 3. [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 December 3. [5] M. Flint, M. Polycarpou, and E. Fernandez-Gaucherand, Cooperative Control for Multiple Autonomous UAV s Searching for Targets, Proc. of the 41st IEEE Conference on Decision and Control, pp December 2. [6] T. Furukawa, F. Bourgault, B. Lavis, and H. Durrant-Whyte, Recursive Bayesian Search-and-Tracking Using Coordinated UAVs for Lost Targets, of the 6 IEEE International Conference on Robotics and Automation, May 6. [7] Z. Hongguo, C. Changwenl, H. Xiaohui, and L. Xiang, Path Planner for Unmanned Aerial Vehicles Based on Modified PSO Algorithm, Proc. of the 8 IEEE International Conference on Information and Automation, June 8. [8] V. P. Jilkov, X. R. Li, and D. DelBalzo, Best Combination of Multiple Objectives for UAV Search & Track Path Optimization, Proc. 1th International Conference on Information Fusion, July 7, Québec City, Canada. [9] J. Lee, R. Huang, A. Vaughn, X. Xiao, K. Hedrick, M. Zennaro, R. Sengupta, Strategies of path-planning for a UAV to track a ground vehicle, AINS Conference, 3. [1] R. R. Pitre, X. R. Li, and D. DelBalzo, A New Performance Metric for Search and Track Missions 2: Design and Application to UAV Search, The 12th International Conference on Information Fusion, submitted for review, March 9. [11] A. Pongpunwattana, Real-Time Planning for Teams of Autonomous Vehicles in Dynamic Uncertain Environments, Ph.D. Dissertation, University of Washington, 4. [12] 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, 5. [13] 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 5. [14] X. Tian, Y. Bar-Shalom, and K. R. Pattipati, Multi-step Look-Ahead Policy for Autonomous Cooperative Surveillance by UAVs in Hostile Environments, Proc. of the 47th IEEE Conference on Decision and Control, Dec G = 4982 G = Fig. 7. Use importance factor to discourage following a single target: comparison of one- (dot-dash) and two-target (solid) cases. 117