RF Source Seeking using Frequency Measurements

Transcription

1 RF Soure Seeking using Frequeny Measurements Muhammed Faruk Genel, Upamanyu Madhow, João Pedro Hespanha Department of Eletrial and Computer Engineering University of California Santa Barbara Santa Barbara, California Abstrat In this paper, we onsider a problem motivated by searh-and-resue appliations, where an unmanned aerial vehile (UAV) seeks to approah the viinity of a distant quasistationary radio frequeny (RF) emitter surrounded by loal satterers. The UAV employs only measurements of the Doppler frequeny of the reeived RF signal, along with its own bearing, to ontinuously adapt its trajetory. We propose and evaluate a trajetory planning approah that addresses tehnial diffiulties suh as the unknown arrier frequeny offset between the emitter and the UAV s reeiver, the frequeny drifts of the loal osillators over time, the diretion ambiguity in Doppler, and the noise in the observations. For the initial trajetory, the UAV estimates the diretion of the emitter using a irular motion, whih resolves diretion ambiguity. The trajetory is then ontinuously adapted using feedbak from frequeny measurements obtained by perturbing the bearing around the urrent trajetory. We show that the proposed algorithm onverges to the viinity of the emitter, and illustrate its effiay using simulations. Index Terms Soure seeking, Maximum Doppler estimation, RF trail following, UAV navigation I. INTRODUCTION We onsider a senario motivated by searh-and-resue, or other emergeny appliations, where a UAV seeks to approah an RF soure, starting from an initially large distane.the UAV is equipped with a single omnidiretional antenna, and does not rely on GPS or on being able to deode messages from the emitter. The soure may be surrounded by loal satterers. In the approah proposed here, the UAV adapts its trajetory towards the emitter using frequeny measurements on the reeived beaon. In an ideal line of sight (LoS), a single omni-diretional antenna an extrat the angle of the arrival between the veloity vetor of the mobile node and LoS to the soure by measuring the Doppler frequeny f d = v os f, where v is the veloity and is the speed of light. Thus, a natural approah is for the UAV to follow the trajetory that maximizes the Doppler shift (whih orresponds to =). However, translating this intuition into a working approah requires that we address the following tehnial hallenges: 1) The sattering environment around the soure auses multipath fading, resulting in large spatial variations of the reeived signal power. This an often lead to errors in frequeny measurements, espeially at the low reeived signal-to-noise ratio (SNR) obtained at large distanes. ) The loal osillators at the emitter and UAV are not synhronized, and drift over time. Thus, the frequeny measurements made by the UAV are a sum of the Doppler shift and a slowly drifting arrier frequeny offset. 3) Even in ideal LoS settings, Doppler estimates have diretion ambiguity: if the trajetory makes an angle with the LoS, then the Doppler shift is proportional to os, and annot therefore enable us to distinguish between + and. 4) Any trajetory adaptation done by the UAV should be feasible, avoiding sharp diretion hanges. The main ontribution of this paper is to show that we an indeed overome the preeding diffiulties to obtain a sheme that reahes the viinity of the emitter, with net distane traversed being only a small fration (of the order of 1%) larger than the initial LoS distane between the UAV and the emitter. We onsider a small UAV that flies at around 1 m altitude, listening to a beaon in ommerial frequeny bands (the arrier frequeny is set to GHz in our numerial examples). The initial distane between the UAV and the soure is of the order of 5 km. In our proposed approah, the UAV obtains an initial trajetory estimate by finding the diretion of maximum Doppler when exeuting a irular motion. Subsequently, it employs feedbak ontrol to ontinuously adapt its trajetory, using the hange in measured frequeny offset as it exeutes designed pieewise linear deviations in bearing from the nominal trajetory. The proposed approah performs signifiantly better than RF soure following using reeived signal strength (RSS) measurements [1]. While RSS measurements are simpler to make and do not require oherent proessing at the reeiver, the sensitivity of RSS hange as a funtion of range to the emitter is small even in ideal settings, sine it is proportional to the inverse square of the range. In addition, loal satterers around the emitter lead to slow, and deep, spatial variations in RSS due to fading. The approah in [1] employs the observation that the rate of hange of RSS due to fading is minimum in the LoS diretion, along with a random walk inspired by baterial hemotaxis. For a setting similar to ours, the RSS-based sheme requires the UAV to traverse a distane that is about three times larger than the shortest path between the initial UAV loation and the emitter. Furthermore, the trajetories employed in [1], both for initialization and for the random walk, are non-smooth and diffiult to exeute. RSS measurements are more effetive if supplemented with diretional information. A rotating UAV was employed in [], with the angle of arrival to the emitter estimated as the diretion of maximum RSS. This approah is not appliable to fixed wing UAVs with omnidiretional antennas as onsidered here.

2 While we onsider the problem of approahing the emitter, there is a signifiant literature on loalizing the emitter using a mobile platform [3], [4] or multiple mobile platforms [5], [6]. Partile filter based algorithms for traking the posterior distribution of the emitter are investigated in [7], [8]. In addition to having a different design goal from ours, it is worth noting that these approahes require that the mobile platform always knows its own absolute loation, say using GPS. Our problem formulation requires the UAV to trak hanges in its own bearing, but does not require that it know its absolute loation, and hene is appliable even in GPS-denied environments. Fig. 1. D system model, with satterers in a disk around the soure II. SYSTEM MODEL The problem of drawing the UAV to the emitter loation is three-dimensional, but we restrit the problem to two dimensions for simpliity. As shown in Figure 1, the soure is loated at the the origin of the D plane, and is surrounded by L loal satterers. The UAV oordinates at any given time are denoted by p = (x, y) with veloity omponents v x = v os( ) and v y = v sin( ). The distane between the mobile reeiver and the soure is d. The satterers are inside an annulus with outer radius R and inner radius R in, and the angles { i } are uniformly distributed between and for eah satterer. We onsider a narrowband flat fading hannel at arrier frequeny f, and assume that the reeiver will ompute its estimates based on known pilot signals transmitted by the soure. The omplex baseband hannel seen by the mobile node an be expressed as the sum of LoS and sattered omponents and an be written as s K h h(t) = K +1 ej( f d,max os( )t+ f o(t)t+ ) + s LX h e j( f d,max os( s i )t+ f o(t)t+ i) + n(t) K +1 i=1 (1) where K is the ratio of power between the diret path and the sattered paths, f d,max = vf is the maximum Doppler frequeny, f o (t) is the arrier frequeny offset drifting over time, and i are the phase of LoS and sattered signal omponents respetively, h is the reeived signal power and n(t) represents the additive noise at the reeiver with the variane n. The reeived signal strength h is governed by the distane between soure and mobile reeiver d, along with spatial variations due to multipath fading. The effet of Doppler frequeny on the reeived signal profile is negligible at these low speeds sine f f d,max. We model the reeived signal strength by modeling the eletri field at the mobile node at a point P in polar oordinates (d, ) as [1]: EF(d, ) = e j d d + LX i=1 ie j (di+ri) () d i + r i where = and is the wavelength, d i + r i is the total distane of the path that goes from the soure to reeiver through i th satterer and i is the refletion oeffiient for the i th satterer [9]. Frequeny estimation auray in a flat fading hannel, assuming all paths have the same frequeny offset, is proportional to the reeived SNR [1]. This assumption is a good approximation for our model when d R. We use lowomplexity single tone frequeny estimation [11], seleting the maximum frequeny over the DFT grid, and then interpolating using a quadrati fit. As the UAV gets loser to the soure, eah path sees a different frequeny offset due to the differene in the refletion angles, and the frequeny estimate degrades. Estimation auray in this region ould potentially be further improved with frequeny estimates derived from seond order statistis [1], or by employing super-resolution tehniques [13]. However, at shorter ranges, more sophistiated frequeny estimation should be oupled with more detailed anisotropi refletion models, hene we leave this as an interesting topi for future work (e.g., on how to trak a moving emitter in urban anyons). III. ALGORITHM FOR SOURCE SEEKING In this setion, we desribe and justify the strategy for planning the UAV trajetory by using frequeny measurements and show that the proposed algorithm will onverge to the true soure diretion with no prior information on the soure loation. The purpose is to draw UAV to the viinity of the emitter as quikly as possible. We set d v d as the required distane between UAV and the soure at whih we delare

3 the traking proess suessful. We assume onstant speed through the trajetory of the UAV and use a feasible trajetory for the motion of UAV. The goal is to minimize flight time. We assume prior knowledge of the emitted signal arrier frequeny f, but not of the arrier frequeny offset f o, whih also drifts over time. The pilot beaon for a given frequeny measurement ontains N symbols, with symbol period T s, so that the measurement interval for frequeny estimation is T = NT s. The pilot beaons are repeated with the period of T slot. For the nth reeived beaon, frequeny measurements are obtained by applying FFT to the N omplex baseband samples of (1), and the peak frequeny ˆ! i i 1,,N FFT is refined by using a quadrati interpolation with adjaent samples: ˆ! i 1 ˆ! i+1! n =ˆ! i + (3) (ˆ! i 1 +ˆ! i+1 ˆ! i ) T s N FFT Thus, we obtain a noisy estimate of the sum of arrier frequeny offset, Doppler frequeny and the frequeny drift. We model this, together with the bearing angle measured by the UAV sensors, as setting. The smallest frequeny measurements orresponds to the f d,max at the diretion of. The SNR is low when UAV is very distant, and multipath fading may oasionally result in large outliers in the frequeny measurements! n. We apply outlier rejetion to the frequeny measurements as (! n,! n! n 1 < 4 vf ˇ! n = (5)! n 1, otherwise and apply a moving average filter with length 15 T slot. Then, the initial diretion for UAV is determined by finding the diretion of maximum frequeny estimate as follows i = argmax ˇ! n n = i. (6)! n =! n + n!,n n = n + n,n. (4) where n!,n and n,n are frequeny measurement and bearing measurement noises, modeled as zero mean independent Gaussian random variables with varianes!,n and,n, respetively. The bearing measurement error variane,n is assumed to be onstant throughout the flight. However, the frequeny measurement error variane!,n an vary: it inreases as RSS drops during fades, and as Doppler spread inreases as the UAV approahes the emitter. While we do not model the UAV dynamis, we will restrit the algorithm desribed in the next setion to use trimming trajetories, whih have the desirable property that the traking error dynamis and kinematis is time invariant and for whih there are well-developed trajetory traking ontrollers [14] [16]. The orientation traking error about the desired trimming trajetory is inluded in error parameter n,n. A. Trajetory adaptation The soure traking algorithm an be divided into two stages, disussed in more detail below. In the first stage, the UAV gets a rough estimate of the diretion of the soure by doing a irular motion. This stage is optional, and an be removed at the expense of some ineffiieny in flight time. The seond stage involves pieewise linear trajetories with perturbations of bearing, with hange in Doppler providing feedbak signal for ontinuous trajetory orretions. Stage 1 - Cirular motion for initial trajetory estimate: The UAV piks a random point at a distane R and follows a irular trajetory, as shown in Figure, saving the frequeny measurements! n with orresponding bearing measurements n. The largest frequeny measurement orresponds to the maximum Doppler f d,max and bearing angle that orresponds to the desired diretion is approximately + in an ideal Fig.. Initial irular motion of the UAV Stage - Continuous updates: In this stage, the UAV derives information for feedbak ontrol of its trajetory in disrete time steps spanning MT slot for eah step. If the estimated diretion towards the emitter is k from the previous diretion, the UAV moves in the diretion k + k for a time interval with length MT slot, yielding frequeny measurements {ˇ! m,m = 1,...,M}, and then in the diretion k k for the same duration, yielding measurements {ˇ! m,m = M +1,...,M}. The differene between these two sets of frequeny measurements is used to update k, as follows: k+1 = k + 1 M MX X M ˇ! m m=1 m=m+1 ˇ! m k f d,max. (7) Taking the differene in this fashion allows signifiant redution of the effet of arrier frequeny offset and drift, whih vary slowly relative to the iteration step duration MT slot. Additional robustness against measurement noise an be obtained by inreasing the perturbation k, at the ost of inreased travel distane. B. Analysis We now analyze the onvergene of the algorithm in stage. Straightforward trigonometry shows that the update step in

4 equation (7) with a onstant k = an be written as k+1 = k + (os( k k + ) os( k k )) = k sin( k k)sin( ). Let the error term k = k k. For d p k+1 p k (range muh larger than the distane between onseutive iterations), we have k k+1, whih yields (8) k+1 = k sin( k )sin( ) (9) Intuitive insight is obtained for small k and by using the approximation sin x x: k+1 k (1 ), orresponding to exponential derease in estimation error. For a rigorous proof of onvergene, we pik k as a Lyapunov funtion. From (9), we obtain that the hange in one time step is given by k+1 k = 4 sin k ( k sin k ) where = sin. Note that sin x(x sin x) > for < x < and < <1. Thus, k is a stritly dereasing funtion for <1, whih provides a wide range of hoies for. Parameters Parameter Symbol Value d 5 m R m R in 1 m f GHz v 1 m/s n -7 db T slot 5 ms T 1 ms T s 1 us N FFT o d v m M TABLE I SIMULATION PARAMETERS 3 times of the shortest path. The proposed algorithm works even if we disard stage 1 and use a random initial diretion. Figure 7 shows the histogram of the total distane traveled with Monte Carlo simulations of 1 runs with only Stage of the algorithm. The average traking distane is now 6 km, whih is 1.5 times the shortest path approah Soure Mobile node initial Satterers Mobile node path System model 5 y (m) Fig. 3. Trajetory updates using diretion perturbations ± k IV. SIMULATION RESULTS The simulation parameters are given in Table I. We apply N FFT = 496 point FFT to N = 1 data hunks in every T slot = 5ms for frequeny estimation. The average reeived SNR at the initial distane of 5 km is set to db. Figure 4 shows an example UAV trajetory. Figure 5 shows the estimated frequeny in the presene of multipath, CFO and frequeny drift for that partiular trajetory. Figure 5 also shows the reeived signal power profile through the trajetory and the spatial variations at the reeived power. We observe that the frequeny estimation error inreases as the UAV gets loser to the soure due to inreased Doppler spread. Figure 6 shows the histogram of the total distane traveled with Monte Carlo simulations of 1 runs for the same senario. The average distane traveled is 5.5 km, whih is 1.15 times the shortest path to get within the desired distane of the target. This signifiantly outperforms the RSS based algorithm [1], for whih the average distane traveled is about x (m) Fig. 4. Example trajetory ( db average initial SNR, initial distane 5 km). V. CONCLUSIONS We have shown that a UAV with a single omnidiretional antenna an approah an RF soure using only frequeny and bearing measurements, in a manner that is robust to multipath fading (via rejetion of outliers in frequeny measurements) and arrier frequeny offset and drift (via averaging and differening frequeny measurements over relatively short intervals). Our analysis shows exponential onvergene to the orret approah angle towards the soure. While the reeiver operations required are more sophistiated than required for extrating RSS, the performane is far superior to that of a previously proposed RSS-based sheme. There are several interesting diretions for future work, inluding improved algorithms for determining the maximum Doppler, espeially in the presene of Doppler spread; detailed modeling of

5 Frequeny measurements over time 35 Mean path length = meters 1 frequeny (Hz) -1 - frequeny estimate frequeny drift time (s) - -4 Reeived signal power over time 15 reeived power (db) time (s) distane traveled (m) Fig. 5. Frequeny measurements and RSS for the route in Figure 4 Fig. 7. Histogram of total distane traveled to get to within m of the soure without stage 1 (mean 6km) 8 Mean path length = meters distane traveled (m) Fig. 6. Histogram of total distane traveled to get to within m of the soure (mean 5.5km) the propagation environments at shorter range, in order to understand the impat on both the frequeny measurements and the trajetory updates (e.g., if the LoS is bloked, the UAV may follow a strong refleted path until it sees a LoS path again); and more detailed aounting of UAV dynamis. ACKNOWLEDGMENTS This researh was supported by the National Siene Foundation under grant CCF REFERENCES [1] A. Wadhwa, U. Madhow, J. Hespanha, and B. M. Sadler, Following an RF trail to its soure, in Communiation, Control, and Computing (Allerton), 11 49th Annual Allerton Conferene on. IEEE, 11, pp [] S. Venkateswaran, J. T. Isaas, K. Fregene, R. Ratmansky, B. M. Sadler, J. P. Hespanha, and U. Madhow, RF soure-seeking by a miro aerial vehile using rotation-based angle of arrival estimates, in Amerian Control Conferene (ACC), 13. IEEE, 13, pp [3] K. Beker, Passive loalization of frequeny-agile radars from angle and frequeny measurements, IEEE Transations on Aerospae and Eletroni Systems, vol. 35, no. 4, pp , [4] N. H. Nguyen and K. Doğançay, Single-platform passive emitter loalization with bearing and doppler-shift measurements using pseudolinear estimation tehniques, Signal Proessing, vol. 15, pp , 16. [5] A. Amar and A. J. Weiss, Loalization of narrowband radio emitters based on doppler frequeny shifts, IEEE Transations on Signal Proessing, vol. 56, no. 11, pp , 8. [6] A. Tahat, G. Kaddoum, S. Yousefi, S. Valaee, and F. Gagnon, A look at the reent wireless positioning tehniques with a fous on algorithms for moving reeivers, IEEE Aess, vol. 4, pp , 16. [7] H. Witzgall, B. Pinney, and M. Tinston, Doppler geoloation with drifting arrier, in MILITARY COMMUNICATIONS CONFERENCE, 11-MILCOM 11. IEEE, 11, pp [8] H. Witzgall, J. Covington, and A. Piere, Single airraft passive doppler loation of radios, in Aerospae Conferene, 15 IEEE. IEEE, 15, pp [9] A. Aragon-Zavala, Antennas and propagation for wireless ommuniation systems. John Wiley & Sons, 8. [1] V. M. Baronkin, Y. V. Zakharov, and T. C. Tozer, Cramer-rao lower bound for frequeny estimation in multipath rayleigh fading hannels, in Aoustis, Speeh, and Signal Proessing, 1. Proeedings.(ICASSP 1). 1 IEEE International Conferene on, vol. 4. IEEE, 1, pp [11] D. R. Brown III, Y. Liao, and N. Fox, Low-omplexity real-time singletone phase and frequeny estimation, IEEE Military Communiation, 1. [1] M. Souden, S. Affes, J. Benesty, and R. Bahroun, Robust doppler spread estimation in the presene of a residual arrier frequeny offset, IEEE Transations on Signal proessing, vol. 57, no. 1, pp , 9. [13] B. Mamandipoor, D. Ramasamy, and U. Madhow, Newtonized orthogonal mathing pursuit: Frequeny estimation over the ontinuum. IEEE Trans. Signal Proessing, vol. 64, no. 19, pp , 16. [14] I. Kaminer, A. Pasoal, E. Hallberg, and C. Silvestre, Trajetory traking for autonomous vehiles: An integrated approah to guidane and ontrol, Journal of Guidane, Control, and Dynamis, vol. 1, no. 1, pp. 9 38, [15] D. R. Nelson, D. B. Barber, T. W. MLain, and R. W. Beard, Vetor field path following for miniature air vehiles, IEEE Transations on Robotis, vol. 3, no. 3, pp , 7. [16] P. Sujit, S. Saripalli, and J. B. Sousa, An evaluation of UAV path following algorithms, in Control Conferene (ECC), 13 European. IEEE, 13, pp