Aircraft Localization Using a Passive Acoustic Method. Experimental Test 78 13 79 A,∗ B a a 14Q2 Sara R

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1Q1 Contents lists available at ScienceDirect 67 2 68 3 69 4 Aerospace Science and Technology 70 5 71 6 72 7 www.elsevier.com/locate/aescte 73 8 74 9 75 10 76 11 77 12 Aircraft localization using a passive acoustic method. Experimental test 78 13 79 a,∗ b a a 14Q2 Sara R. Martín , Meritxell Genescà , Jordi Romeu , Arnau Clot 80 15 81 a Laboratory of Acoustics and Mechanical Engineering (LEAM), Universitat Politècnica de Catalunya, C/Colom 11, 08222 Terrassa, Spain 16 82 b Acoustics Research Centre, Department of Electronics and Telecommunications, Norwegian University of Science and Technology, Trondheim, Norway 17 83 18 84 19 a r t i c l e i n f o a b s t r a c t 85 20 86 21 Article history: A passive acoustic method for aircraft localization is experimentally tested in this paper. The method 87 22 Received 23 January 2015 relies on the Doppler effect influencing the signals received by a mesh of microphones distributed over 88 23 Received in revised form 17 August 2015 the acoustic area of interest. The relative Doppler stretch factors between the microphone signals are 89 Accepted 13 November 2015 24 estimated using a one-dimensional version of the Ambiguity function. Then, a Genetic Algorithm is used 90 Available online xxxx 25 to solve the non-linear system of equations that relates the aircraft’s position and velocity to this relative 91 26 Keywords: stretch factors. This method is used in this study to locate a radio controlled airplane equipped with 92 27 Aircraft a Global Positioning System (GPS). Seven microphones are distributed in the airfield area. Although the 93 Localization localization errors are influenced by the uncertainty in the microphones position, the acoustic system 28 94 Wideband cross ambiguity function succeeds at locating the airplane. 29 95 Acoustical Doppler effect © 2015 Published by Elsevier Masson SAS. 30 96 31 97 32 98 33 99 34 100 35 101 36 1. Introduction array. Under these circumstances, a possible approach is to use a 102 37 distributed network of nodes – a node being in this context a sin- 103 38 The interest in passive acoustic aircraft localization systems gle acoustic sensor or a set of several acoustic sensors – where 104 39 arises when the performance of RADAR systems is reduced such every node provides an estimate of the aircraft bearing. The 3D 105 40 as non-line-of-sight tracking or when electromagnetic radiation is position of the aircraft can be calculated afterwards by triangula- 106 41 present. Acoustic aircraft localization systems can also be a cheaper tion of the bearing estimates from, at least, two nodes. Such an 107 42 alternative in small airports and, in addition, the data from the approach has the benefit that only the bearing data has to be 108 43 acoustic sensors can also be processed for source classification or transmitted to a central processing unit, but the disadvantage is 109 44 noise monitoring purposes. that the nodes are complex systems such as microphone arrays 110 45 Several acoustic methods have been developed to determine the [10–13] or, more recently, acoustic vector sensors [14]. 111 46 motion parameters such as height and speed of both jet and pro- An alternative approach to passive acoustic 3D localization that 112 47 peller driven aircraft flying in a straight line at constant flight level uses a network of single microphones was initially described in 113 48 and speed. The methods developed for propeller driven aircraft [15,16]. The methods were based on the estimation of time delay 114 49 take advantage of the Doppler effect and require one single micro- between the signals at the different microphones, and therefore 115 50 phone [1–3] or a distributed array of microphones [4]. The meth- they only obtain the bearing of the aircraft. 116 51Q5 ods developed for jet aircraft use the time differences between a In contrast, the method used in the present paper relies on both 117 52 microphone array [3,5], the interference between the direct and the time delay (retardation effect) and the time stretch (Doppler 118 53 ground reflected sounds [6,7] or both [8]. effect) [17,18] to obtain the source position and velocity. Seven mi- 119 54 Other techniques are used for the 3D localization of maneuver- crophones are distributed on the ground within the acoustic area 120 55 ing aircraft without limitations on the trajectory. For a low altitude of influence of the moving sound source. This acoustic method for 121 56 aircraft, the sound wavefronts can be considered spherical and the aircraft localization is experimentally tested in this paper. 122 57 bearing and distance of the aircraft can be estimated with a planar The rest of the paper is organized as follows. Section 2 de- 123 58 microphone array [9]. At larger distances, the front waves become scribes the acoustic localization method, Section 3 presents the 124 59 planar and only the bearing can be obtained with a microphone experimental setup, Section 4 describes the parameters used in the 125 60 implementation of the algorithm for the test, Section 5 shows the 126 61 experimental results, Section 6 discusses different sources of er- 127 62 * Corresponding author. rors, and finally Section ?? summarizes the main findings of this Q4 128 63 E-mail address: [email protected] (S.R. Martín). paper. 129 64 130 http://dx.doi.org/10.1016/j.ast.2015.11.023 65 131 1270-9638/© 2015 Published by Elsevier Masson SAS. 66 132 JID:AESCTE AID:3491 /FLA [m5G; v1.168; Prn:23/11/2015; 11:22] P.2(1-8) 2 S.R. Martín et al. / Aerospace Science and Technology ••• (••••) •••–•••

1 be easily shown that the signals received at two different micro- 67 2 phones are time shifted and time stretched to each other disre- 68 3 garding the amplitude. 69 4 The underlying idea of this localization method is that, since 70 5 the Doppler stretch term is related to the speed and position of 71 6 the aircraft, if the value of the Doppler stretch could be estimated 72 7 by comparing the signals of the different microphones, the aircraft 73 8 could be localized. 74 ∗ 9 The method is iterative, and for each iteration k ∈ N the initial 75 10 position p(to) of the aircraft at a time to needs to be known from 76 11 the previous iteration. To initialize the method it is necessary that 77 12 the real position of the aircraft is known at an arbitrary time. Let 78 13 pinitial at tinitial = 0sbe a known value such that 79 14 80 t 15 to = (k − 1) 81 16 2 82 17 initial = 83 o = p if k 1, 18 p(t ) − (3) 84 pk 1 if k = 1. 19 85 20 where t is the time interval between two successive position es- 86 − 21 timates, and pk 1 is the position estimate obtained in the previous 87 Fig. 1. Microphone set up and geometric variables relevant to the localization algo- 22 88 rithm. iteration. 23 The following steps are repeated in each iteration. 89 24 90 2. Localization algorithm 25 Step 1: Signals synchronization 91 26 92 This section describes the localization algorithm and its imple- This step consists in selecting the signal portion xn(t ) of every 27 o + 93 mentation. Further details of the theoretical background can be receiver signal yn(t t ) originated when the source was at posi- 28 o o 94 found in [17] and [19]. tion p(t ) and lasting for t. The signal originated at p(t ) reaches 29 | o | | − | 95 qn(t ) rn p(to) 30 The method requires at least seven microphones randomly dis- the receiver n after a time period = . Therefore 96 ∈{ } c c 31 tributed as showed in Fig. 1. Their positions rn for n 1, .., 7 97 =[ ] the set of synchronized signal portions xn(t ) are obtained as 32 with respect to an origin O 0, 0, 0 need to be known. 98 o o Let se(t + t) be the signal emitted by the aircraft t seconds |q (t )| 33 = o + + n ∈[ ; ] 99 after a certain time to, this signal is received at the microphone n xn(t ) yn(t (t )) with t 0 t . (4) 34 c 100 at the time to + t , which accordingly to Fig. 1 is 35 Combining Eq. (2) into Eq. (4) it comes out that the relation 101 36 102 | o + | between the synchronized signal portions and the emitted signal qn(t t) 37 t = t + , (1) is 103 38 c 104 c 39 | o + x (t ) = ρ · s (to +[ ]t ). (5) 105 where c is the sound speed in an isospeed medium and qn(t n n e −| | o 40 o c v cos(αn(t )) 106 t)| =|rn − p(t + t)| is the position vector from the source to the 41 receiver as shown in Fig. 1. Eq. (5) shows that the set of synchronized signal portions are 107 42 108 If t is so small that, during the interval [to; to + t], the distance time stretched and attenuated versions of each other. 43 109 traveled by the aircraft is much smaller than the distance between 44 Step 2: Computation of the relative Doppler Effect 110 the aircraft and the receiver and also the aircraft velocity v can be 45 111 considered a constant, then the relationship between the received This step consists of the calculation of the relative Doppler 46 112 and the emitted signal is [18] stretch δ fmn between each pair of synchronized signal portions re- 47 113 ceived at microphones m and n. Since 48 | o | 114 o o qn(t ) c o 49 yn(t +t ) = ρn ·se(t +[t − ]·[ ]), (2) c −|v|·cos(αn(t )) 115 c c −|v| cos(α (to)) δ f = , (6) 50 n mn o 116 c −|v|·cos(αm(t )) 51 117 where yn corresponds to the acoustic signal received by the n-th 52 the search domain can be limited to 118 microphone, ρn represents the amplitude attenuation factor due to 53 c −|v | c +|v | 119 the sound propagation and αn is the angle between the aircraft max ≤ ≤ max 54 δ f , (7) 120 velocity v and the pathlength vector qn pointing from the aircraft c +|vmax| c −|vmax| 55 to the receiver n. 121 | o | where |vmax| is the maximum possible speed reached by the air- 56 qn(t ) 122 The term in Eq. (2) is the time that the signal takes craft. 57 c 123 58 to propagate from the initial position of the source to the re- To obtain the value of δ fmn, ﬁrst the discrete Fourier trans- 124 c form X ( f ) of x (t ) is calculated for all the receivers. Note that 59 ceiver, and the term is the Doppler effect due n n 125 o 60 c −|v| cos(αn(t )) both t and f are discrete variables since xn(t ) is a digital sig- 126 61 to the movement of the source called further on absolute Doppler nal. Second, the following discrete 1-dimensional version of the 127 62 stretch δ fn. Therefore, the received signal yn at a microphone n ambiguity function is calculated over a set of attempted relative 128 63 is the emitted signal se shifted in time by the propagation time Doppler stretches δ f 129 | o | qn(t ) 64 and also stretched – i.e. expanded or contracted – in time L 130 c δ f 65 c χ(δ f ) = |X (l · f )|·|X (δ f · l · f ))| (8) 131 by the Doppler effect term . By recursion, it can + n m 66 o L 1 132 c −|v| cos(αn(t )) l=0 JID:AESCTE AID:3491 /FLA [m5G; v1.168; Prn:23/11/2015; 11:22] P.3(1-8) S.R. Martín et al. / Aerospace Science and Technology ••• (••••) •••–••• 3

1 where L is the available number of frequency bins. Since the 67 2 Nyqvist theorem predicts that spectral information is limited up 68 3 to half the sampling frequency fm, then 69 4 ⎧ 70 ⎪ fm/2 · δ f 5 ⎨⎪ if δ f < 1, 71 6 f 72 L = 7 ⎪ fm/2 73 ⎩ if δ f ≥ 1. 8 f 74 9 75 10 where f is the frequency resolution of the frequency spectra 76 11 Xn( f ). 77 12 The ambiguity function in Eq. (6) reaches its absolute maximum 78 13 value at δ f = δ fmn since the frequency spectra satisfy |Xn( f )| = 79 14 γmn|Xm( f · δ fmn))| for an amplitude factor γmn . Therefore, the rel- 80 15 ative Doppler stretch δ fmn between the signals at two different 81 16 microphones can be deduced as the argument of the maximum of 82 17 Eq. (8). 83 18 Step 3: Calculation of the position estimate 84 19 Fig. 2. The radio controlled airplane frequency spectrum. 85 20 The third step is to obtain an estimate of the aircraft position 86 21 from the relative Doppler stretch estimates. Eq. (6) can alterna- this outdoor test is to compare the airplane trajectory estimated 87 22 tively be expressed as by the acoustic method and the trajectory given by a Global Posi- 88 23 tioning System (GPS) mounted on the airplane. 89 v · (rn − p) 24 c − 90 |r − p| 25 δ f = n , (9) 3.1. Airfield 91 mn v · (r − p) 26 − m 92 c | − | 27 rm p The experimental test has been carried out in the airfield of the 93 28 showing that the aircraft velocity v and position p can be deter- Aeronautic Club Egara in Terrassa (Spain). It is placed at the out- 94 29 mined by solving the system of equations defined by Eq. (9) for at skirts of the city in an area where the background noise is low. It 95 30 least six independent pairs of microphones. This means that seven is not surrounded by any building that could introduce reflections 96 31 microphones are enough to obtain the aircraft position and veloc- of the sound emitted by the radio controlled airplane. The airfield 97 32 ity, p and v. Here, all the possible pair combinations out of the is placed in a more or less rectangular flat area of 120 m long by 98 33 seven microphones are considered to form a system of 42 non- 45 m wide of sandy soil. The runway is made of artificial grass and 99 34 linear equations defined by Eq. (9), which is solved by using a it is 72 m long by 15 m wide. 100 35 genetic algorithm. The following least squares problem is stated 101 36 ⎛ ⎞ 3.2. Radio controlled airplane 102 2 37 v · (rn − p) 103 c − 38 N ⎜ | − | ⎟ A radio controlled airplane with a 4-stroke engine has been 104 ⎜ − rn p ⎟ 39 min ⎝δ fmn · − ⎠ (10) used for the test. Fig. 2 shows the frequency spectrum of the air- 105 (p, v) ∈ I v (rm p) 40 m = 1n = m c − plane. The airplane was held while the engine was running (with 106 |r − p| 41 m the wheels not touching the ground) to record its spectrum with a 107 42 where I ⊆ R6 is the search space for the three dimensional posi- microphone located 2 m far from the airplane. It can be seen from 108 43 tion and velocity of the aircraft. The kth iteration of the location Fig. 2 that it is a tonal spectrum including multiples of the en- 109 44 k k k k k k ∈ gine speed. Therefore, the frequency spectrum is highly dependent 110 algorithm gives (px, p y, pz, vx, v y, v z) I and the search space 45 6 on the engine speed. The maximum speed of such an airplane is 111 I =[Ix, I y, Iz, I v , I v , I v ] ⊆ R is defined as follows: 46 x y z about 33 m/s. 112 47 t t 113 = k−1 −| |· ; k−1 +| |· Iu pu vmax,u pu vmax,u (11a) 48 2 2 3.3. Microphone distribution 114 49 115 I =[v ; v ] (11b) 50 vu min,u max,u To distribute the different microphones along the airfield, two 116 51 where u = x, y, z and k > 0. different observations from the simulations published on [17] and 117 52 The position and speed estimates obtained from solving the [19] have been considered: 118 53 system of equations are then associated to the aircraft states at 119 54 · t k = · t k = · t 1. when the distance between the microphones is much smaller 120 k 2 , i.e., p p(k 2 ) and v v(k 2 ). 55 121 Note however, that the Doppler stretch values δ fmn used to es- than the distance between the microphones and the source, 56 timate each aircraft position are calculated using signal portions of the value of all the relative Doppler stretches δ fmn tends to 122 57 length t. These stretch values correspond indeed to any time t the same value of 1 and the localization fails, 123 58 within [0; t] since Doppler stretches are not constant during t 2. the higher the difference between all relative Doppler stretches, 124 59 due to the continuous aircraft motion. Therefore, the estimated po- the higher the accuracy on the localization results. 125 60 sition obtained from that set of stretches can correspond as well to 126 61 any position of the aircraft movement during t. Both remarks are directly dependent on the airplane trajectory. 127 62 Therefore, to distribute the microphones accordingly, the airplane 128 63 3. Experimental setup has been assumed to ideally fly-over along the runway at a con- 129 64 stant altitude of 33 m, and a constant speed of 26 m/s (typical 130 65 The acoustic localization method described in section 2 is ex- values for this particular kind of airplane). Note that this assump- 131 66 perimentally tested using a radio controlled airplane. The goal of tion is only used to select the positions of the sensors and should 132 JID:AESCTE AID:3491 /FLA [m5G; v1.168; Prn:23/11/2015; 11:22] P.4(1-8) 4 S.R. Martín et al. / Aerospace Science and Technology ••• (••••) •••–•••

1 67 2 68 3 69 4 70 5 71 6 72 7 73 8 74 9 75 10 76 11 77 12 78 13 79 14 80 15 81 16 82 17 83 18 84 19 85 20 Fig. 4. Time evolution of the absolute Doppler stretches along the expected trajec- 86 21 tory of the radio controlled airplane. 87 22 Fig. 3. Microphone distribution together with the expected aircraft trajectory which 88 23 consists on ﬂy-over at constant speed in one direction. 89 24 90 25 Table 1 91 26 Coordinates of each microphone n ∈{1, ..., 7} with the origin of coordinates set to 92 the position of microphone 1. 27 93 28 n 1234567 94

29 rnx (Easting) [m] 0 −27 −23 −58 −73 −94 −104 95 30 rny (Northing) [m] 0 6 −27 1 −51 −60 −54 96 31 97 32 98 not be interpreted as a movement restriction for the implementa- 33 99 tion of the acoustic method. 34 100 Fig. 3 shows the final microphone distribution together with 35 101 this expected ideal airplane trajectory. The fly-over is consid- 36 102 ered to be in one direction at constant speed from the xy-point 37 103 (41.08; 43.57) mto (−148.7; −138.9) m. Table 1 lists the micro- 38 104 phones coordinates. The position of microphone 1 has been chosen 39 105 to be the origin of coordinates. Fig. 5. Microphone distribution together with the radio controlled airplane trajectory 40 106 According to the trajectory in Fig. 3, the evolutions of the ab- provided by the GPS mounted on it. 41 107 solute Doppler stretches δ f for all microphones n ∈{1, ..., 7} are 42 n ∈{ } 108 shown in Fig. 4. Fig. 4 shows that the absolute Doppler stretches frequency rate fm for all microphones n 1, ..., 7 has been of 43 109 δ f for all microphones n are different enough up to t = 8s. 5kHz. 44 n 110 Note that due to the small distances between the microphones Regarding the time interval of the signal portions t, on the 45 111 themselves as well as the microphones and the radio controlled one hand, as said in Section 2, it has to be small enough so that 46 112 airplane, the experimental test carried out in this paper is a de- the distance traveled by the airplane is much smaller than the 47 113 manding test. The values of the absolute Doppler stretches are distance between the airplane and the receiver and also so that 48 114 between 0.95 and 1.08 or equivalently, the values of the relative the airplane velocity v can be considered a constant. On the other 49 115 Doppler stretches are between 0 95 1 08 = 0 88 and 1 08 0 95 = hand, it should be large enough to obtain an acceptable frequency 50 . / . . . / . 116 1.13 and this range is much smaller than the typical range ob- resolution f when the Fourier Transform of the synchronized 51 signals is computed. In the present test, the value t = 0.15 s sat- 117 52 tained in a full scale application of the method which goes from 118 0.5 to 1.5 (see Ref. [19]). isfies all these conditions simultaneously. 53 Then, the frequency resolution of the Fourier Transform of all 119 54 receivers signal is f = 6.6 Hz which is an acceptable resolution 120 3.4. Parameters 55 taking into account the frequency spectrum emitted by the air- 121 56 plane (Fig. 2). As a consequence of this choice, the acoustic method 122 57 To initialize the acoustic location method, an initial position of t 123 initial initial locates the radio controlled airplane every = 0.075 s. 58 the airplane p at some time t needs to be known. The 2 124 59 coordinates of this initial position are given by the GPS mounted 125 60 on the airplane. In the same system of coordinates than the mi- 4. Results 126 61 crophone positions listed in Table 1, these airplane coordinates are 127 62 pinitial = (41.4, 43.9, 35.1) m and the time corresponding to this This section presents the results for the acoustic localization 128 63 position is set to tinitial = 0s. of the radio controlled airplane. Fig. 5 shows the trajectory fol- 129 64 The sampling rate fm needs to be chosen taking into account lowed by the airplane during the experimental test provided by 130 65 that the significant frequency content of the radio controlled air- the GPS mounted on it. This trajectory will be named from now 131 66 plane spectrum in Fig. 2 goes up to 2.5 kHz. Therefore, the sample on real trajectory. It is worth remarking that the accuracy of the 132 JID:AESCTE AID:3491 /FLA [m5G; v1.168; Prn:23/11/2015; 11:22] P.5(1-8) S.R. Martín et al. / Aerospace Science and Technology ••• (••••) •••–••• 5

1 67 2 68 3 69 4 70 5 71 6 72 7 73 8 74 9 75 10 76 11 77 12 78 13 79 14 80 15 81 16 82 17 83 18 84 19 85 20 86 21 87 22 88 23 89 24 90 Fig. 8. Time evolution of the absolute localization error along the eight seconds 25 91 fly-over for the three spatial coordinates: (a) x (Easting), (b) y (Northing) and (c) z Fig. 6. Radio controlled airplane localization provided by the GPS (p = (px, p y , pz)) 26 (Altitude). 92 and estimated by the acoustic method (p = (px, p y , pz)). 27 93 28 94 Table 2 29 Maximum and mean localization errors for each spatial coordinate x, y, z while 95 30 t ∈[0; 8]. 96 31 97 u max(Erru ) [m] Erru [m] 32 98 x 18.2 7.17 33 y 13.63 5.21 99 34 z 14.82 5.48 100 35 101 36 102 the acoustic method) and real positions of the airplane and Fig. 7 37 103 shows the estimated and real velocities. The gray curve in both 38 104 figures corresponds to p = (p , p , p ) and v = (v, v , v) which 39 x y z x y z 105 are the estimated airplane states provided by the acoustic method. 40 106 The black curve corresponds respectively to p = (p , p , p ) and 41 x y z 107 v = (v , v , v ) which are the real position and velocity provided 42 x y z 108 by the GPS. 43 109 During the first fly-over for t ∈[0; 8] s, the radio controlled air- 44 110 plane flew along 210 m. Fig. 8 depicts the time evolution of the 45 111 absolute error Err =|p − p | where p and p are the estimated 46 u u u u u 112 and real airplane position respectively for each spatial coordinate 47 113 u = x y z. Table 2 lists the maximum and mean errors for the 48 , , 114 time interval t ∈[0; 8] s, i.e., max Err and Err where 49 ( u) u 115 50 K |k − k | 116 k=1 pu pu 51 Erru = , (12) 117 52 K 118 53 Fig. 7. Velocity of the radio controlled airplane provided by the GPS (v = and K is the maximum number of executed iterations of the track- 119 = 54 (vx, v y , v z)) and obtained from the acoustic method (v (vx, v y , v z)). ing algorithm. 120 55 Note that Fig. 8 shows that the maximum errors in Table 2 for 121 56 GPS mounted above the radio controlled airplane is 3 m. There- each spatial coordinate are reached at different time instants and 122 57 fore what is called real position in the text is indeed an estimation that the errors obtained are not cumulative. The acoustic method 123 58 of the real position with a 3 m accuracy. This uncertainty is dis- overcomes by itself these error peaks and tends to reach the real 124 59 regarded in this paper and the real position is assumed to be an trajectory of the aircraft afterwards. Moreover, the mean error dur- 125 60 error-free result. ing 210 m of flight does not exceed 7.2 m in the x-coordinate and 126 61 During the 30 s of signal analyzed here, the airplane flew over 5.5 m for both the y-coordinate and the altitude of the airplane. 127 62 the runway in one direction for t ∈[0; 8] s, turned around sharply Section 5 focuses on this first fly-over and discusses different pos- 128 63 for t ∈]8; 18] s, and carried out a second fly-over in the opposite sible sources of this error. 129 64 direction for t ∈]18; 30]. The first fly-over is the main concern of For t ∈]8; 18] s when the airplane was turning sharply, the 130 65 discussion since the microphones have been distributed based on acoustic localization is not expected to be accurate since the setup 131 66 it (see section 3.3). Fig. 6 represents the estimated (obtained from has been optimized to locate the airplane during the first fly-over 132 JID:AESCTE AID:3491 /FLA [m5G; v1.168; Prn:23/11/2015; 11:22] P.6(1-8) 6 S.R. Martín et al. / Aerospace Science and Technology ••• (••••) •••–•••

1 67 2 68 3 69 4 70 5 71 6 72 7 73 8 74 9 75 10 76 11 77 12 78 13 79 14 80 15 81 16 82 17 83 18 84 19 85 Fig. 10. Expected (in black) and real (in gray) trajectories of the radio controlled 20 = 86 Fig. 9. Time evolution of the 42 values of relative Doppler stretches δ fmn for m n, aircraft together with the distributed microphones. 21 m, n ∈{1, ..7}. 87 22 88 5.2. Influence of the sudden variations in the real trajectory of the 23 ∈[ ; ] 89 for t 0 8 (see section 3). Fig. 6 shows that both real and esti- aircraft 24 mated locations differ the most. Actually, the Doppler effect does 90 25 no longer affect in a different manner the signals received by the 91 Fig. 10 shows in gray color the airplane trajectory provided by 26 microphones. This means that δ f = 1for all m, n ∈{1, .., 7} as 92 mn the GPS and in black the expected trajectory. The real trajectory 27 can be seen in Fig. 9 that shows the time evolution of the 42 the- 93 presents sudden oscillations that influence notably the value of the 28 oretical values of relative Doppler stretches during the flight. 94 relative Doppler stretches. 29 Despite of the localization failure between t = 8 s and t = 18 s, 95 Fig. 11 represents the time evolution of the 42 relative Doppler 30 the estimated position of the airplane tends to the real trajec- 96 stretches used for localizing the airplane in flight considering the 31 tory from t = 18 s onwards as shown in Fig. 6. The method seems 97 airplane travels along the real trajectory (Fig. 11a), and the ex- 32 to continue locating the airplane even if previously the localiza- 98 pected trajectory (Fig. 11b) described in Section 3.3. 33 tion has failed. Nevertheless, the error for t ∈[8; 18] s influences 99 The acoustic localization method used here assumes that the 34 mainly the signals synchronization for t > 18 s (Step 1 detailed in 100 Doppler stretches can be considered constant during t, but these 35 section 2) and thus, the accuracy of this second flyover is poorer. 101 sudden oscillations of the value of the relative Doppler stretches 36 Notice that this trajectory of the radio controlled airplane is not 102 weaken this assumption introducing error in the estimates of the 37 a common trajectory for a standard commercial aircraft when it 103 δ f . A wrong statement of the independent terms of the system 38 approaches or leaves the airport. A full scale test of the acous- mn 104 of non-linear equations, i.e., δ f in Eq. (9), will lead to an inaccu- 39 tic localization method would not have two different directions of mn 105 rate estimation of the airplane position. 40 flight neither a sharp turn. This is why the discussion in section 5 106 41 focuses only on the first fly-over. 107 42 5.3. Influence of the inaccuracy of the microphones positioning 108 43 5. Discussion 109 44 The inaccuracy of the GPS used to obtain the microphone loca- 110 45 The following section focuses on the results for t ∈[0; 8] intro- tions (an Etrex Venture by Garmin) is up to 3 m. To evaluate the 111 46 duced in section 4. This section discusses the influence of different influence of this uncertainty on the final position estimates, an er- 112 47 sources of error such as the inherent error of the method due to ror between 0 and 3 m has been randomly added to all receiver 113 48 the non-stop airplane motion during the time interval t (sec- coordinates in Table 1. Then, the least squares problem stated in 114 49 tion 5.1), the sudden variation and uncertainty of the real position Eq. (10) has been solved using the real relative Doppler stretches 115 50 116 p = (px, p y, pz) (section 5.2), and the uncertainty of the exact showed in Fig. 11a, the real airplane trajectory in Fig. 10, and the 51 positions of the microphones due to the inaccuracy of the GPS microphone positions in Table 1. 117 52 (section 5.3). Table 3 shows the averaged maximum localization error 118 53 max(Erru) and the mean localization error (Erru ) obtained for each 119 54 5.1. Influence of the continuous airplane motion coordinate u = x, y, z averaged over 100 realizations. Notice that 120 55 the absolute error for each coordinate is calculated from the abso- 121 56 As detailed in section 2, each airplane position provided by lute value of the difference between the location estimation given 122 57 the acoustic method is estimated by using signal portions of du- by the acoustic method and the real trajectory given by the GPS 123 58 ration t. The estimated position is assigned to p(t0 + t/2), mounted on the airplane which already contains an uncertainty 124 59 but indeed this estimated position belongs to the spatial inter- of 3 m. The location errors from Table 3 are of the same order, 125 60 val [p(t0); p(t0 + t)] due to the continuous motion of the air- even slightly higher, than those from Table 2. This result shows 126 61 0 127 plane during t. The maximum possible difference between pu (t ) that the GPS inaccuracy that affects both the real position of the 62 0 128 and pu(t + t) for the spatial coordinates u = x, y is at most aircraft and the position of the microphones accounts for a sub- 63 129 |vmax| · t. Thus, in this particular experimental test, the maxi- stantial part of the total error. Notice that for the implementation 64 mum inherent error of the acoustic method due to the airplane of the method in an airport, the distances between microphones 130 65 motion during t is |vmax| · t = 4.95 m. This error can contribute and aircraft will be of several kilometers and the GPS uncertain- 131 66 significantly to the maximum errors in Table 2. ties will remain of 3 m. Therefore, the influence of the inaccuracy 132 JID:AESCTE AID:3491 /FLA [m5G; v1.168; Prn:23/11/2015; 11:22] P.7(1-8) S.R. Martín et al. / Aerospace Science and Technology ••• (••••) •••–••• 7

1 An experimental test has been carried out using a radio con- 67 2 trolled airplane and seven microphones distributed over the air- 68 3 field area to validate the acoustic method. The main purpose of 69 4 the test is to show the applicability of the acoustic localization 70 5 method used. To do so the acoustically estimated airplane trajec- 71 6 tory is compared to the trajectory provided by a Global Positioning 72 7 System mounted on the airplane. 73 8 The present experiment is a highly demanding test and has 74 9 several constraints. For instance, the 3 m uncertainty of the GPS 75 10 devices used to locate both the radio controlled airplane and the 76 11 set of seven microphones have a significant impact on the results 77 12 taking into account the small distances between the sound source 78 13 and receivers. Despite of these conditions, the localization errors 79 14 provided by the acoustic method do not exceed the expected er- 80 15 rors arising from the inaccuracy of the GPS devices used, but at 81 16 the same time, their magnitude does not permit to quantify the 82 17 exact error of the acoustic method. Therefore, it can be concluded 83 18 that the method is capable of doing the measurements it has been 84 19 developed for. 85 20 The experimental verification presented in this paper is a pre- 86 21 liminary test of a full scale experiment. The influence of the dif- 87 22 ferent sources of errors discussed in this paper is expected to be 88 23 notably smaller taking into account that the distances for a real 89 24 scenario are at least 10 times longer and the velocity of the aircraft 90 25 will be higher. Therefore, the results obtained in this experimental 91 26 test are promising to move one step forward and test the acoustic 92 27 method in an airport with commercial aircrafts. 93 28 94 29 95 Conflict of interest statement 30 96 31 Q3 97 32 None declared. 98 33 99 34 References 100 35 101 36 [1] B. Ferguson, A ground-based narrow-band passive acoustic technique for esti- 102 37 mating the altitude and speed of a propeller-driven aircraft, J. Acoust. Soc. Am. 103 38 92 (3) (1992) 1403–1407, http://dx.doi.org/10.1121/1.403934. 104 [2] B. Ferguson, B. Quinn, Application of the short-time Fourier transform and the 39 105 Wigner–Ville distribution to the acoustic localization of aircraft, J. Acoust. Soc. 40 Am. 96 (2) (1994) 821–827. 106 41 107 Fig. 11. Time evolution of the 42 relative Doppler stretches δ fmn for m = n, m, n ∈ [3] B. Fergusson, K. 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