Aspect Aware UAV Detection and Tracking

Transcription

1 Aspect Aware UAV Detection and Tracking Christian R. Berger, Shengli Zhou, and Peter Willett Department of Electrical and Computer Engineering University of Connecticut Storrs, Connecticut 6269, USA {crberger, shengli, ABSTRACT We consider target detection and tracking of stealthy targets. These targets can be characterized by a strong aspect dependance leading to difficult detectability without a multi-static setup. Even in a multi-static setup only sensors in a certain zone can detect the return signal, if the the aspect dependent return has a small bandwidth. We propose a solution based on a large number of simple sensor, as using many receivers increases the probability of detection. The sensors are simple in the sense that they only transmit binary detection results to a fusion center that has comparitively large capabilities, and they do not need to know their own position or communiate with other sensors. We characterize the target position estimation performance using the Cramér-Rao bound and simulation results, considering uncertainty in nuisance parameters as the sensor positions or the specifics of the aspect dependance. We suggest a data collection protocol which includes locating sensors that detect the target and has low communication complexity. As a novelty we also include information about non-localized sensors, as sensors which do not detect the target stay quiet to save bandwidth and energy, therefore are not known to the fusion center except knowledge of the deployed sensor density and deployment region.. INTRODUCTION Targets in the detection and tracking literature are known to feature strong aspect dependance in their return strength characteristic. Therefore these targets are often only detectable using a multi-static setup. Based on our previous work, which was targeted predominantly at shoreline protection via a rather static sonar sensor setup, we develop a new scenario of locating low visible ground or air targets using multiple UAVs (Unmanned Autonomous Vehicles) and a large number of rapidly deployed stationary sensors. Using a multi-static setup these targets are still only detectable by receivers in a certain region (see Fig. ), therefore the chance of detection increases considerably with the number of sensors and their positioning, as we can cover more ground. Also we want to enforce strict communication constraints, as with a growing number of sensors the resources have to be used itelligently. Our suggested protocol includes localizing the sensors or UAVs first, and accounts for uncertainty in their position estimates in the following target localization as we cannot assume a long setup phase to deliver highly precise sensor position information or frequent position updates of the UAVs. We require sophisticated sensors only on the main platform, while the distributed sensors can be rather simple. The sensors are only equipped with a detection mechanism activated by the direct blast and a clock or timer. The timer is necessary for their own localization, for which we use the round trip time when communicating the binary detection to the main platform, but can also be used to measure Time Difference of Arrival (TDoA) between the direct blast and possible detections. If the aspect dependancy of the target can be assumed known rather well, binary detection results deliver a good position estimate. On the other hand if we can t model this behavior accurately, we can achieve sufficient estimation precision by including the TDoA measurements, but incur increased communication overhead. For the case of direct acess to sensors signal energy measurements, the literature is rich, see, e.g., 2 6 and references therein. While for binary or more generally quantized energy measurements only and 7 have considered target localization. Our previous work in is the only work so far which considers aspect dependance and exploits it for localization. This work was supported by the Office of Naval Research.

2 stealth target UAV emitted signal detectable zone receiver Figure. Strongly aspect dependent targets can only be detected by a receiver in a certain zone. The rest of this paper is structured as follows, in Section 2 we explain the scenario and our proposed algorithm. In Section 3 we go over the general system model, in Section 4 we develop the Cramér-Rao bounds that are used to gain insight into the performance. In Section 5 we look into a numerical example and Section 6 concludes this paper. 2. Basic setup 2. SCENARIO We consider a scenario, in which a sensor field is used to detect and localize a strongly aspect dependent (stealth) target. The term sensor field indicates that differing from a sensor network, the sensors cannot communicate with each other, and are also generally very limited in their capacities. The scenario is described by the following: The sensors are equipped with a simple detection device (e.g., an energy detector), and can send a simple signal to the main plattform (binary, ON/OFF). The main plattform (a UAV) is assumed to have sophisticated processing capabilities, i.e., receive array to determine angle of arrival (AoA) and coputational capacities to process and fuse the sensor reports. The main plattform tracks the sensor locations, based on their emitted signals. In this way the plattform can utilize the sensor position information, even though the sensors have no way to transmit their position information or even to be aware of it. 2.2 Rapid deployment In earlier work, the positions of the sensors were assumed to be perfectly known, a reasonable assumption in case the sensors are stationary and an intialization period has passed (assuming during the initialization all sensors emit randomly, a mobile plattform could locate the stationary sensors based on AoA only, see e.g. 8,9 ). We want to relax these assumptions to include a more dynamic scneario. The rapid deployment scenario is characterized by the following (see Fig. 2), Sensor positions are only known with limited accuracy described by a covariance matrix. Some sensors which have not recently emitted any signal are non-localized, i.e., there position is unkown except as limited by the drop-zone. The number of deployed (or functional) sensor can be unknown, in this case the number of non-localized sensors is treated as a random variable.

3 narrow reflection rapid deployment platform broad beam drop zone zone of detection Figure 2. In the rapid deployment scenario the sensor positions are no longer assumed to be known, furthermore sensors with a missed detection result are non-localized, i.e., we only know they are somewhere within the deployment corridor. 2.3 Algorithm We suggest the following target detection/localization scheme based on a multi-static setup using the sensor field.. The main plattform emits a probing signal, this signal can be isotropic or focussed to some degree 2. The sensors are activated by the direct blast of the probing signal and start a timer (the direct blast can be assumed significantly stronger than reflection off targets). 3. During a detection interval, the sensors monitor the received energy to detect possible targets. The length of the detection interval is linked to the observation region via the bi-static round-trip-time (RTT). 4. After a sensor s timer indicates the end of the detection interval, it emits a signal if a target was detected (we don t consider multiple targets). 5. The main plattform can locate sensors based on their emitted signal; the AoA is obtained via the receiver array, while the range can be determined usint the RTT of the direct blast and the detection result (the plattform knows the detection interval length). After these steps, the main plattform localizes has collected all the detection results and knows the location of reporting sensors with a known accuracy, determined by the receive array and the delay estimate of the RTT. We next specify how the target can be localized and develop the Cramér-Rao bound (CRB) on the estimation accuracy. 3. Measurements 3. SYSTEM MODEL We assume there are N s sensors in the sensor field, with positions (x n, y n ), n =,...,N s. The target position is denoted as (x t, y t ). After the direct blast, each sensor monitors the received energy. Denote z n as the detection In case the probing signal is focussed, the possible target locations can be restriced in the localization step.

4 (x t,y t ) ψ s,t φ t ψ n α (x n,y n ) (x s,y s ) Figure 3. The plattform at (x s, y s) emits a probing signal, which is reflected off the target along an angle α measured relative to some vertical line. result for the nth sensor, if the measured energy is larget than a certain threshold, the nth sensor declares detection by setting z n =. Otherwise it declares no-detection by setting z n =. Denoting the received signal-to-noise ratio (SNR) at the nth sensor as γ n, the probability of detection assuming a Swerling I target can be expressed as ( P D,n = p (z n = ) = exp Γ ) th () + γ n where Γ th is the detection threshold at the sensor. If there is no target present, the probability of false alarm is The dependance of γ n on the target position can be specified as follows, P fa = exp ( Γ th ). (2) γ n = c f (r n )f 2 (ψ n α), (3) where r n is the distance between the nth sensor and the target, while ψ n and α are specified with respect to some reference, see Fig. 3. To be specific, r n = (x t x n ) 2 + (y t y n ) 2 (4) ( ψ n = arctan x ) n x t. (5) y n y t Next we define the attenuation with distance f (r n ) and aspect dependance f 2 (ψ n α). Assuming free-space attenuation, we have f (r n ) = rn P (6) where P = correponds to cylindrical spreading and P = 2 is spherical spreading (sometimes larger P are used to model loss due to shadowing, e.g., P = 3.5). As a simple example we use a Butterworth filter to model the aspect dependance, f 2 (ψ n α) = ) 2K, (7) + where 2W s the 3 db bandwidth and K is the filter order. ( ψn α W Given the previous definitions the detection result z n is a binary random variable with probability mass function p(z n = ) = P D,n, p(z n = ) = P D,n. (8)

5 3.2 ML estimator The unknown target position and the propagation angle of the reflected signal are collected into a vector θ = (x t, y t, α). With the detection results from N s sensors the optimal likelihood estimator is ˆθ = argmax θ N s n= p (z n θ) (9) where p (z n θ) is as defined in Section 3.. This had already been derived equivalently in our previous work, now we would like to extend this to account for additional unknowns, e.g., the target positions (x n, y n ) or the aspect dependency of the target as specified via c, W and K. In this case the maximization in (9) has to be performed over a larger dimension which could be impractical. In any case we only analyze the achievable performance as characterized by the CRB; as a practical suboptimal estimator of lower complexity, if the parameter uncertainty is small, we suggest maximizing only over θ = (x t, y t, α) and using the mean for all other values. 4. CRAMÉR-RAO BOUND The FIM of the binary detection z n has been calculated in as N s ( J = + P D,n n= P D,n ) f n f n () where we define f n := θ P D,n as the gradient vector of the detection probability with respect to the target parameters. To come to a matrix notation, we define a diagonal matrix [( ) ( )] D = diag +,, + () P D, P D, P D,Ns P D,Ns and a Jacobian matrix F := θ P D of the vector P D = [P D,,...,P D,Ns ]. With this we can write the FIM as 4. Uncertainty in nuisance parameters J = FD F. (2) If there is uncertainty in other parameters which are necessary to evaluate the measurements, e.g., the sensor positions, we need to estimate them as well. To this end we enlarge the FIM by the dimension of unknown parameters and after inversion evaluate the CRB via the appropriate block. We call the vector of all nuisance parameters, e.g., the sensor positions or the parameters of the aspect dependance in (7), ξ R ν, where ν is the number of uncertain parameters. Usually we can assume some prior about the nuisance parameters, e.g., the sensor locations could be known within some accuracy or the aspect dependency of typical targets could be characterized. We denote the information using a second Fisher information matrix P, which is an inverted covariance matrix in case of a Gaussian prior. Using the fact that information is additive, the Fisher information matrix for the extendend vector of unknowns (θ, ξ) can be written as [ F J = G ] D [ F G ] + [ ] P [ ], (3) where G := ξ P D is the Jacobian of the measurements with respect to ξ. We rewrite (3) as a block matrix [ ] FD J = F FD G GD F GD G + P (4)

6 which we can invert to find the bound on our estimation accuracy. Since we are only interested in the top left corner of the inverted matrix, we use block-by-block inversion which leads to the following bound on any unbiassed estimator ˆθ [ E (ˆθ θ) 2] [FD F FD G ( GD G + P ) GD F ] (5) = [J J r ] (6) where J r is the reduction of information and J is defined in (2). Since J r is a positive definite matrix, this is always a loss of information or equivalently an increase in estimation uncertainty. Remark It is intuitive to understand that estimating a larger number of unknowns with the same information will lead to some loss of estimation accuracy. In one extreme P is zero, i.e., there is no prior information available, and the available information is spread out thinner over a larger number of unknowns; in the other extreme P is infinite and J r equals zero therefore the FIM reduces to the original form without uncertainty. 4.2 Non-localized sensors In the previous section sensor position uncertainty was adressed and as a special case P was considered as having no prior information. In our case, even if a sensor has not send any signal so far, we still know it is confined within the drop-zone. In this case the CRB has to be formulated with some modifications. Assume N number of sensors have send a signal either in this detection cycle or recently enough, and accordingly there are N = N s N non-localized sensors. The likelihood function can be written as, Λ (θ) = N n= p (z n θ, x n, y n ) N s n=n + A p (z n θ, x n, y n ) p(x n, y n )dx n dy n, (7) where p(x n, y n ) is the prior distribution we assume on the sensors. In case of a Gaussian prior the formula in the previous section applies where P is the covariance matrix. If the prior is diffuse, i.e., uniform within a limited interval, the CRB is not well defined. First we simplify the likelihood function to, Λ (θ) = N n= ( ) Ns N p (z n θ, x n, y n ) p (z o θ, x o, y o ) dx o dy o. (8) A A where A is the drop-zone and all non-localized sensors are combined into the diffuse information. The tricky part is that without any prior knownledge of the sensor position, its observation is meaningless, as I can always place the sensor into the center of any assumed propagation angle α in case of a detection result or arbitrarily far from it for a non-detection result, and a uniform prior gives zero information about the target position in the Fisher sense, since the distribution has no curviture. Re-analyzing the previous statement, since the drop-zone is not infinitely large, this is not completely correct. We might not be able to estimate the sensor location by its prior, but it still holds information about the target. The approach we choose differs from the previous approach, as we do not estimate the sensor positions of the non-localized sensors, but average over them instead correponding to calculating the integral in (8). The log-lieklihood function is N [ ( )] log Λ (θ) = log [p (z n θ, x n, y n )] + (N s N ) log p (z o θ, x o, y o ) dx o dy o A A n= where the influence of all non-localized sensors is combined. We define, P D := p (z o = θ, x o, y o ) dx o dy o (2) A A (9)

7 y coordinate [m] 5 5 CRLB VAR(e) plattform target estimate ""s ""s x coordinate [m] Figure 4. Simulation results for the general setup; the full lines indicate the reflected signal off the target at the 3 db bandwidth. which are the average probabilities. With this the non-localized sensors have all the same relationship to the unkown θ and result in one extra term in the CRB with weight N s N, N ( J = + P D,n n= P D,n ) ( f n f n + (N s N ) + P D,n P D,n and correspondingly we defined f o := θ PD as the gradient of the average probabilities. ) fo f o (2) As in practice we can not calculate (2) in a closed form, we have to use numerical integration. In terms finding the ML estimate via (8), the numerical integration has to be performed once for each function evaluation. Writing the numerical integral as a finite sum, we have [ ] N N A log Λ (θ) = log [p (z n θ, x n, y n )] + (N s N ) log p (z n θ, x n, y n ), (22) n= N A n= which has an interesting structure, as for the localized sensors the averaging is performed outside the logarithm, while for the non-localized sensors it is performed inside. By the Jensen inequality, this means the non-localized sensors allways have a reduced impact, which seems intuititve. 5. Implementation 5. SIMULATION RESULTS To give some numerical examples, we look at a scenario where the sensors are deployed within a strip of 2 km by.5 km, see Fig. 4. The target is located 5 m outside the sensor field and is illuminated by a probing signal, the reflection hitting the sensor field. Without sensor position uncertainty, there are three unknowns, which can be limited within some, interval since we assume the taret is within a maximum observable field (the plotted area). We implement an exhaustive search, which can be improved using, e.g., the genetic algorithm as the likelihood surface is smooth (but multimodal). Fig. 4 shows some simulation results, comparing the position accuracy with the CRB as in (). The target is parameterized with W = 7.5 o, K = 4 and c such that the SNR is about 2 db. We observe that for this SNR and sensor field density plenty of sensors detect the target leading to a good localization result.

8 2 8 6 x error [m] y error [m] α error [deg] 4 RMS error sensor position std [m] Figure 5. The localization accuracy of the sensor position has a direct impact on the target position estimation error. 5.2 Sensor position uncertainty Since we assume that the sensor positions are determined by the main platform, the accuracy is naturally limited by the ability of the receiver array, the precision of the time of arrival (ToA) estimation and the accuracy of the timer used in the sensors. Using the formulation developed in Section 4., we use the CRB to analyze the target location estimation degradation caused by sensor position uncertainty. Accordingly ξ = (x, y,...,x Ns, y Ns ). We assume the sensor locations are known within some accuracy, for tractability we assume they are measured with Gaussian measurement noise of a known covariance matrix P = σ 2 ξ I N s (usually determined by the capabilities of the receive array). In Fig. 5, we plot the resulting bounds on the target position estimation error. Since the vector of nuisance parameters ξ is of higher dimension than the number of measurements, the problem becomes ill-conditioned when σ 2 ξ. For σ ξ < 2 m, the effect is negligible, while for larger σ ξ the target position root-mean-square (RMS) error increases proportionally. 5.3 Uncertainty in aspect dependance Since different targets are characterized by different aspect dependance, some uncertainty seems reasonable. Also since the target orientation is not known, there are many possible observed patterns, which are best characterized by a histogram or probability density function. As a simple example we consider Gaussian distribution with varying standard deviation. We analyze the performance degradation due to this added uncertainty using the CRB formulation in Section 4.. The nuisance parameter is ξ {c, W, K} and P = σξ 2. Fig. 6(a) shows the increase in target position estimation error, where normalized standard deviation is σ ξ /ξ. We observe that the increase in target position error is bounded, as we have enough sensor measurements. Fig. 6(b) shows the error elipses for P = (one parameter at a time). 5.4 Non-localized sensors Typically we assume the sensors are distributed on some regular pattern, e.g., a grid with certain inter-sensor spacing. Of course, this does not reflect well reality, as sensors are usually deployed via some random process, e.g., deployment by dropping from the air or attaching them to some easily accessable position. Also if in fact sensors where located on a grid, there would hardly be any uncertainty in their positions. We are interested to see how large the performance degradation is, if sensors are deployed randomly. Again for mathematical tractability we adopt a model, here a Poisson field, to characterize the randomness of the sensor

9 RMS error y coordinate [m] 5 5 c W K none c 5 W K uncertainty in normalized std (a) x coordinate [m] Figure 6. As the aspect dependance of a target depends on the unknown orientation, it is best modeled including some degree of uncertainty; even if some parameters are completely unknown, the target location does not become infeasible. (b) positions. The average density of sensor deployment is defined as λ per unit of area (we assume a two-dimensional setup as in ) and therefore the probability to find k sensors in an area of size A is p (N s = k) = λa k! e λa. (23) For simplicity, we assume that sensors with detection results are perfectly localized, while sensors with nondetection results are non-localized. The average sensor density is equal to the grid pattern. In Fig. 7(a) the sensor positions are random, but we assume to know the total number of sensors N s, while in Fig 7(b) we assume the number of sensors is only known in the average sense. Again we implement an exhaustive search using the genetic algorithm, where for each possible target position evaluation the numerical integral in (8) has to be approximated using a sum. We see that in both cases, the performance is not significantly degraded. This is encouraging as it allows us to employ large numbers of sensors without having to localize each of them and at the same time limiting the amount of necessary communication. 6. CONCLUSION We have investigated the feasability of target localization using binary detection results of a sensor field under added uncertainty in the aspect dependance and/or the sensor positions. Our results indicate that the target aspect dependance can feature a significant degree of uncertainty with limited degradation in target localization. Furthermore the sensors that do not detect the target do not need to be located precisely, in fact, as long as we can limit them to a certain zone these non-localized sensors still contribute to target location estimation. REFERENCES. S. Zhou and P. Willett, Submarine location estimation via a network of detection-only sensors, IEEE Trans. Signal Processing, vol. 55, no. 6, pp , Jun J. Chen, R. Hudson, and K. Yao, Maximum-likelihood source localization and unknown sensor location estimation for wideband signals in the near-field, IEEE Trans. Signal Processing, vol. 5, no. 8, pp , Aug B. Sadler, R. Kozick, and L. Tong, Multi-modal sensor localization using a mobile access point, in Proc. of Intl. Conf. on ASSP, vol. 4, Philadelphia, PA, Mar. 25, pp

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