AUV Underwater Positioning Algorithm Based on Interactive ... - MDPI

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AUV Underwater Positioning Algorithm Based on Interactive Assistance of SINS and LBL Tao Zhang 1,2, *, Liping Chen 1,2 and Yao Li 1,2 Received: 27 October 2015; Accepted: 23 December 2015; Published: 30 December 2015 Academic Editor: Jaime Lloret Mauri 1 2

*

Key Laboratory of Micro-Inertial Instrument and Advanced Navigation Technology, Ministry of Education, Southeast University, Nanjing 210096, China; [email protected] (L.C.); [email protected] (Y.L.) School of Instrument Science and Engineering, Southeast University, Nanjing 210096, China Correspondence: [email protected]; Tel./Fax: +86-25-8379-3911

Abstract: This paper studies an underwater positioning algorithm based on the interactive assistance of a strapdown inertial navigation system (SINS) and LBL, and this algorithm mainly includes an optimal correlation algorithm with aided tracking of an SINS/Doppler velocity log (DVL)/magnetic compass pilot (MCP), a three-dimensional TDOA positioning algorithm of Taylor series expansion and a multi-sensor information fusion algorithm. The final simulation results show that compared to traditional underwater positioning algorithms, this scheme can not only directly correct accumulative errors caused by a dead reckoning algorithm, but also solves the problem of ambiguous correlation peaks caused by multipath transmission of underwater acoustic signals. The proposed method can calibrate the accumulative error of the AUV position more directly and effectively, which prolongs the underwater operating duration of the AUV. Keywords: LBL; TDOA; underwater positioning; information fusion

1. Introduction An AUV (autonomous underwater vehicle) is applied to execute all kinds of underwater tasks, including ocean exploration, underwater mine clearance and collecting bathymetry data of ocean and rivers [1–4]. In order to guarantee that underwater tasks will be completed smoothly and accurate underwater measurement data will be acquired, the AUV is required to have long-term autonomous high-precision positioning and navigation abilities and invisibility [5]. In an underwater environment, electromagnetic wave signals have the characteristic of serious attenuation. In deep sea or under an ice surface, adopting GPS and other radio positioning means cannot achieve ideal positioning effects. In order to meet the navigation requirements, DVL (Doppler velocity log) and SINS (strapdown inertial navigation system) are often used to integrate navigation [6], and the position will be estimated by dead reckoning. However, when this means is used for positioning, positioning errors will accumulate as time goes on [7]. When the AUV is performing tasks in shallow sea, it can adopt the navigation mode of “submerge, water surface calibration, submerge” to launch positioning and navigation; in other words, the AUV relies on SINS/DVL to launch positioning and navigation when navigating under water. After the AUV has been submerged under water for a certain time, in order to calibrate the accumulative errors, the AUV must emerge from the water, and the SINS/GPS integrated navigation system must be used to do the calibration [8]. Adopting this scheme can reach the goal of calibrating accumulative errors, but the AUV must be required to travel to and fro between the underwater operation position and the water surface, which will not only influence the working efficiency and increase the energy consumption, but also expose the position of the AUV. Especially when the AUV is operating in deep sea or under an ice surface, this scheme will be more impractical. Hence, it is very important to study a method in which reliable assistance positioning can be conducted for a long time Sensors 2016, 16, 42; doi:10.3390/s16010042

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underwater. This paper suggests an interactive assistance positioning method that integrates an LBL (long base line) underwater acoustic positioning system and an SINS/DVL/MCP (magnetic compass pilot) integrated navigation system, and this is a new idea for solving the above problems. An LBL underwater acoustic positioning system, usually consisting of a seabed transponder matrix and an interrogation responder with a base length of hundreds to thousands of meters [9,10], adopts distance information between the underwater objective and the seabed matrix element to solve the target position. It can provide accurate positioning of an underwater vehicle within a local area without accumulative errors. Hence, an LBL underwater acoustic positioning system is very applicable to an underwater AUV to launch assistance positioning. Among some of the research of predecessors, Liu, Y. puts forward an underwater AUV positioning and navigation algorithm, which adopts an LBL underwater acoustic positioning system, an ADCP (acoustic Doppler current profiler) and depthometer-assisting INS [11]. Miller, P.A. et al. puts forward a tight integrated system based on LBL/DVL/INS [3]. Cheng,W.H. proposes a modification method, which is based on the periodically-measured actual navigation distance and is associated with the TOA positioning algorithm [12]. Jakuba, M.V. et al. report results for LBL acoustic navigation during autonomous under-ice surveys near the seafloor and adaptation of the LBL concept for several typical operational situations, including navigation in proximity to the ship during vehicle recoveries [13]. Eustice, R.M. et al. report recent experimental results in the development and deployment of a synchronous-clock acoustic navigation system suitable for the simultaneous navigation of multiple underwater vehicles [14]. Chen, Y.M. et al. propose a near-real-time approach to underwater inertial navigation with LBL, which uses a ping-response protocol, resulting in asynchronous measurements [15]. Although these systems have reached a certain positioning effect, there are some deficiencies. Firstly, these systems do not explain how to solve the positioning difficulty brought by the multi-path transmission of the underwater acoustic signal. In addition, acoustic velocity is distributed unevenly with the change of underwater depth, and sound ray transmission is curved, which will result in big positioning errors; additionally, the above systems have not proposed any solution. When solving the target position, the LBL underwater acoustic positioning system can adopt the TOA (time of arrival) positioning algorithm and the TDOA (time difference of arrival) positioning algorithm. The equation set formulated by the TOA positioning algorithm can be directly transformed into a simple linear system of equations with a simple solution. However, strict time synchronization between the hydrophones and the sound source is required to measure a relatively accurate TOA value. It is very hard to do so in reality. The TDOA positioning algorithm acquires the TDOA value by conducting a generalized cross-correlation calculation of the signal received from one hydrophone and another, and then makes the positioning calculation. This method does not have to assure synchronization of the sound source and hydrophones, and the communication between them is quite simple. Hence, it is often adopted in wireless positioning. As acoustic signals will finally be in a coherence stack at one hydrophone through different paths, there will be multiple correlation peaks with approximate amplitudes in the generalized cross-correlation results, thus forming a phenomenon of ambiguous correlation peaks. Then, An, L. et al. came up with an ambiguity-solving algorithm based on underwater acoustic propagation characteristics [16]. This method, by studying the distribution rule of cross-correlation peaks forming the multipath transmission of underwater signal channels, tracks stable correlation peaks, which can effectively correct the miscalculation of the TDOA value caused by the ambiguity of the correlation peaks. However, under the condition that distances between fake peaks and the main peak do not differ that much, it is still very hard to accurately track, and the tracking error will be enlarged and finally diverge. In addition, underwater acoustic propagation channels will change as the underwater environment changes, so will the distribution rule of correlation peaks: if the former tracking strategy is still adopted, there will also be errors. Hence, the adaption of this tracking algorithm is not that strong. In this paper, we propose an underwater positioning method based on the interactive assistance of LBL/SINS/DVL for an AUV. This positioning system consists of SINS, DVL and a sound source

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installed on the AUV and an LBL underwater matrix located at the seabed. The hydrophones of LBL receive signals source and conduct calculation and acquire the installed on thefrom AUVthe andsound an LBL underwater matrixcross-correlation located at the seabed. The hydrophones of TDOA value.signals We adopt model solve the position of thecalculation sound source the LBL receive fromthe thehyperbolic sound source andto conduct cross-correlation and(namely, acquire the positionvalue. of theWe AUV), correct accumulative errors of SINS/DVL, use of resolving results of SINS/DVL to TDOA adopt the hyperbolic model to solve the position the sound source (namely, the assist in solving ambiguous correlation peaks when LBL is launching underwater acoustic position of the AUV), correct accumulative errors of SINS/DVL, use resolving results of SINS/DVL to positioning, estimate the TDOA valuepeaks and improve theissolution accuracy of LBL positioning. This assist in solving ambiguous correlation when LBL launching underwater acoustic positioning, scheme can not only directly correct accumulative errors caused by the dead reckoning algorithm, estimate the TDOA value and improve the solution accuracy of LBL positioning. This scheme can not but also solves the problem of ambiguous correlation peaks caused by multipath of only directly correct accumulative errors caused by the dead reckoning algorithm, buttransmission also solves the underwater acoustic signals. Therefore, it is quite applicable to underwater positioning and problem of ambiguous correlation peaks caused by multipath transmission of underwater acoustic navigation of an AUV. signals. Therefore, it is quite applicable to underwater positioning and navigation of an AUV. The structure structure of of this this paper paper is is as as follows: follows: Firstly, we introduce introduce the the principle principle and and structure structure of of the the The Firstly, we underwater assistance assistance positioning positioning system introduce key key technologies technologies of of the the system, system, such such underwater system and and then then introduce as underwater acoustic propagation channel modeling, the calculation of time delay differences, the as underwater acoustic propagation channel modeling, the calculation of time delay differences, the Taylor series series expansion expansion algorithm, algorithm, the the TDOA TDOA position position solution solution method method and and the the interactive interactive assistance assistance Taylor algorithm. Finally, we verify the effectiveness of the algorithm through a simulation experiment. algorithm. Finally, we verify the effectiveness of the algorithm through a simulation experiment. 2. Principle Principleand andStructure Structureof ofthe theSystem System 2.1. Placement and Positioning of the Hydrophone The underwater LBL system needs to use a seabed hydrophone to confirm the position of the vehicle. The calculated position coordinates are the ones corresponding to the seabed seabed hydrophone hydrophone matrix. Hence, the hydrophone fixed on the seabed should be positioned firstly, firstly, and then, its absolute geographic position should be calculated. calculated. As shown in Figure 1, we install the hydrophone reception matrix at the bottom of mother ship (at hydrophones, usually moremore than three), receive from the hydrophone (at least leastthree three hydrophones, usually than which three), will which willsignals receive signals from the (with the sound source) underwater then calculate the three-dimensional position coordinates of hydrophone (with the sound source)and underwater and then calculate the three-dimensional position each underwater hydrophone to the hydrophone at the bottom coordinates of each underwatercorresponding hydrophone corresponding to thematrix hydrophone matrix of at the mother bottom ship according to short base linetopositioning The GPS, IMU andThe compass are installed on the of the mother ship according short base principles. line positioning principles. GPS, IMU and compass mother ship to the accurate geographic position (longitude, depth) oflatitude the mother are installed onprovide the mother ship to provide the accurate geographiclatitude positionand (longitude, and ship, asofwell the attitude such as this factors, information ship depth) the as mother ship, asangle. well asWe thecombine attitudefactors, angle. We combine such of as the this mother information and themother installation errors, calculateerrors, the absolute geographic positiongeographic of each hydrophone of the ship and the and installation and calculate the absolute position ofunder each geodetic coordinates. The underwater AUV canunderwater adopt theseAUV hydrophones (their accurate positions are hydrophone under geodetic coordinates. The can adopt these hydrophones (their already launch theacquired) local areatopositioning itself. accurateacquired) positionsto are already launch theof local area positioning of itself.

Figure 1. Positioning of the seabed hydrophone.

2.2. LBL Positioning Model Model Based Based on on TDOA TDOA 2.2. LBL Underwater Underwater Positioning TDOA positioning positioning is is aa method method that that adopts adopts delay delay inequality inequality to to perform perform the the positioning. positioning. By By TDOA measuring the the time time difference difference of of aa signal signal reaching reaching different different hydrophones, hydrophones, the the distance distance difference difference measuring between the signal source and different hydrophones can be acquired. As shown in Figure 2, suppose:

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hydrophones Ti (i source 0,1, 2) and located at three different positions; a sound the AUV sends a between the signal different hydrophones can be acquired. Assource shownon in Figure 2, suppose: signal, and theTtransmission time of at thethree signal reaching three hydrophones is ti (on i  the 0,1, 2) ; thesends sounda hydrophones different positions; a sound source AUV i pi “ 0, 1, 2q located signal, and transmission time of the signal velocity is athe steady-state value (suppose it is c);reaching then: three hydrophones is ti pi “ 0, 1, 2q; the sound velocity is a steady-state value (suppose it is c); then: R2  R0  c(t2  t0 )  ct20 (1) R2 R ´1  R0R0“cpt c (t21 ´  tt00)q“cc∆t t10 20

(1) (2)

Ri (i  0,1, 2) represents the distance the source and the hydrophone Ti . The R1 ´ Rbetween t0 qsound “ c∆t10 (2) 0 “ cpt1 ´ above two equations respectively represent a hyperbolic curve n, which takes T0 and T2 as focal Ri pi “ 0, 1, 2q represents the distance between the sound source and the hydrophone Ti . The points, andequations hyperbolic curve m, which takes T0 and T1curve as focal points; their intersection is above two respectively represent a hyperbolic n, which takes T0point and Tof 2 as focal points, the of the sound source. As Ta0certain exists in the measured distance difference, there andposition hyperbolic curve m, which takes and T1 error as focal points; their point of intersection is the position may be a condition with no solution. Hence, in view of this condition of multiple hydrophones placed of the sound source. As a certain error exists in the measured distance difference, there may be a on the seabed, redundant is usually to acquire the position closestplaced to the on actual condition with no solution.information Hence, in view of this used condition of multiple hydrophones the position. seabed, redundant information is usually used to acquire the position closest to the actual position.

Figure 2. Schematic diagram of the TDOA positioning model.

2.3. of the the System System 2.3. Working Working Process Process of Figure is aa schematic schematic diagram diagram of of AUV AUV positioning positioning based based on on LBL, LBL, with with multiple multiple fixed fixed Figure 33 is hydrophones on the in the the diagram); diagram); aa sound at the hydrophones on the seabed seabed (there (there are are four four hydrophones hydrophones in sound source source fixed fixed at the bottom of the AUV will send sound signals; first, accurate positioning of the hydrophones through bottom of the AUV will send sound signals; first, accurate positioning of the hydrophones through sensors, GPS, IMU IMU and and compass, compass, is is performed, performed, and and the the absolute absolute geographic geographic coordinates coordinates of of sensors, like like the the GPS, each each hydrophone hydrophone are are acquired, acquired, which which is is in in preparation preparation for for solving solving the the position position of of the the sound sound source; source; then, a generalized correlation calculation of the sound source signals received by each hydrophone then, a generalized correlation calculation of the sound source signals received by each hydrophone is is done. signalswill willbeberefracted refractedand andreflected reflectedduring duringthe thetransmission, transmission,multiple multiple correlation correlation peaks peaks done. AsAs signals will be the ambiguity of correlation peaks. Directed at thisatproblem, this paper will be generated, generated,resulting resultinginin the ambiguity of correlation peaks. Directed this problem, this adopts SINS position assistance to estimate the time difference of sound source signals reaching each paper adopts SINS position assistance to estimate the time difference of sound source signals reaching hydrophone, solves the distance difference according to to thethe time equivalent each hydrophone, solves the distance difference according timedifference differenceand and the the equivalent transmission velocity of the signal and, finally, calculates the position of the sound source according transmission velocity of the signal and, finally, calculates the position of the sound source according to to the hyperbolic positioning model. Hence, the interactive assistance positioning technology of SINS SINS the hyperbolic positioning model. Hence, the interactive assistance positioning technology of and is an an innovation innovation point point of of this this paper. paper. and LBL LBL is Figure 4 is the operating block diagram of the Figure 4 is the operating block diagram of the system. system. The The positioning positioning system system mainly mainly consists consists of of LBL, an SINS/DVL/MCP SINS/DVL/MCP integrated made LBL, an integratedsystem systemand andaadata dataprocessing processing unit. unit. The The solution solution will will be be made according the hydrophones hydrophones in in the the LBL LBL system system receive receive according to to the the sequence sequence number number in in the the box; box; firstly, firstly, the signals (Box 1) from the sound source on the AUV and conduct a generalized correlation calculation signals (Box 1) from the sound source on the AUV and conduct a generalized correlation calculation of of received signal hydrophone i andhydrophone hydrophonej,j,and andthe thecalculation calculationresult result is is aa group xi (t ) j,ptq) x j (t of ) ) of received signal (x(i ptq,x hydrophone i and group of ambiguous correlation peaks (Box 2). 2). Then, Then, the the current current AUV AUV position position information information PSI PSINS NS and the absolute position of the hydrophones (Box 3) according to the SINS/DVL/MCP integrated system absolute position of the hydrophones (Box 3) according to the SINS/DVL/MCP integrated system are '

acquired, and the delay inequality tij (Box 4) of the sound source signal reaching hydrophone i and

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1 (Box are acquired, jjand the delay inequality 4) of the peak soundscreening source signal reaching hydrophone is We module again to ij hydrophone is calculated. calculated. We adopt adopttthe the correlation correlation peak screening module again hydrophone to screen screen the thei and hydrophone j is calculated. We adopt the correlation peak screening module again to screen the (Box former acquired ambiguous correlation peaks and acquire the actual delay inequality 5), former acquired ambiguous correlation peaks and acquire the actual delay inequality ttijij (Box 5), former acquired ambiguous correlation peaks and acquire the actual delay inequality t (Box 5), then ij then then we we acquire acquire the the distance distance difference difference by by combining combining the the sound sound velocity velocity correction correction algorithm, algorithm, we acquireathe distance solution difference by combining the sound velocity correction algorithm, formulate formulate positioning equation, take as the initial iterative position, adopt the P formulate a positioning solution equation, take PSINS as the initial iterative position, adopt the Taylor Taylor SINSinitial iterative position, adopt the Taylor series a positioning solution equation, take PSI NS as the series expansion algorithm to solve AUV position (Box 6) in LBL positioning and, finally, P series expansion algorithm to solve position (Box 6) in LBL positioning and, finally, input input PLBL expansion algorithm to solve AUV AUV position PLBL (Box LBL 6) in LBL positioning and, finally, input the the difference value of and as the external observation information into a Kalman filter; P P difference valuevalue of PLBL Pand as external observation information into a Kalman filter; velocity the difference of and the external observation information into a Kalman filter; PSINStheas PLBL LBL SI NS SINS velocity by the DVL heading provided the MCP are information providedprovided by the DVL provided by the MCPby are also taken an velocity information information provided byand the heading DVL and andinformation heading information information provided by the MCP areasalso also taken as an observed quantity. Filtering results will correct the errors of SINS, and navigation results, observed quantity. Filtering results will correct the errors of SINS, and navigation results, such as the taken as an observed quantity. Filtering results will correct the errors of SINS, and navigation results, such accurate position ,, velocity and of the finally accurate position PAUV , velocity attitude ofattitude the AUV, finallywill be acquired. P such as as the the accurate position velocity and attitude of will the AUV, AUV, will finally be be acquired. acquired. PAUVand AUV

Figure Figure 3. 3. Schematic Schematic diagram diagram of of AUV AUV underwater underwater LBL LBL positioning. positioning.

Figure 4. Operation principle block diagram of the SINS, strapdown inertial navigation Operationprinciple principle block diagram of system. the system. system. strapdown navigation Figure 4.4.Operation block diagram of the SINS,SINS, strapdown inertialinertial navigation system; system; DVL, Doppler velocity log; MCP, magnetic compass pilot. system; DVL, Doppler log; MCP, magnetic pilot. DVL, Doppler velocity velocity log; MCP, magnetic compass compass pilot.

3. of the Interactive Assistance Positioning Algorithm of SINS/DVL/MCP/LBL 3. Principle Principle 3. Principleof ofthe theInteractive InteractiveAssistance Assistance Positioning Positioning Algorithm Algorithm of of SINS/DVL/MCP/LBL SINS/DVL/MCP/LBL This section introduces the realization principles of the algorithms in Figure 4, including This section section introduces introduces the the realization realization principles principles of of the the algorithms algorithms in in Figure Figure 4, 4, including including This generalized cross-correlation calculation of hydrophone receiving signals; SINS assists in seeking the generalized cross-correlation cross-correlation calculation calculation of of hydrophone hydrophone receiving receiving signals; signals; SINS SINS assists assists in in seeking seeking the the generalized ideal time differences and AUV position calculation based on TDOA. ideal time differences and AUV position calculation based on TDOA. ideal time differences and AUV position calculation based on TDOA. 3.1. 3.1. Generalized Generalized Cross-Correlation Cross-Correlation Calculation Calculation of of Hydrophone Hydrophone Receiving Receiving Signals Signals x(t) x(t) represents represents the the sound sound source source signal; signal; suppose suppose that that the the signal signal received received by by No. No. ii hydrophone hydrophone is: is:

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x(t) represents the sound source signal; suppose that the signal received by No. i hydrophone is:

(t ) “  x(t´τiiq)`nniiptq (t ) xxi ptq αiixpt

(3) (3)

The signal signal received received by by No. No. jj hydrophone hydrophone is: is: The

x j (t )   j x(t   j )  n j (t )

(4) (4)

x j ptq “ α j xpt ´ τj q ` n j ptq

 and  j are attenuation coefficients of sound signals propagating underwater; ni (t ) and attenuation coefficients of sound signals propagating underwater; ni ptq and n j ptq are αi i and α j are noise signals; and  j aretime. propagation time. n j (t ) are non-correlative non-correlative noise signals; τi and τj are ipropagation andx j ptq The ross-correlation function of xxi ptq x j (is: t ) is: i (t )and

) “EErx [ x11(ptqx (t ´ )] “ R xRi xxji x(j pτq t ) x2˚ pt  τqs 

11 T´  τ

ż TT

( t ) jxpt (´t   )dt τqdt τ xxi ptqx i

j

(5) (5)

 i τirepresents representsTDOA; TDOA; TT isis observation observation time. time. According According to to the the characteristics characteristics of the τ “ jτj ´ correlation function,ififthe thepeak peakvalue value is found, the corresponding the right x(ix j)pτq  is τtheis right correlation function, of ofRxR is found, thenthen the corresponding time ixj time difference. difference. 3.2. Multi-Path Effect of Sound Signals Underwater 3.2. Multi-Path Effect of Sound Signals Underwater Figure 5 is a simplified multi-path underwater sound propagation model. Place a sound source Figure 5 is a simplified multi-path underwater propagation Place a sound source and two hydrophones R1 and R2; simply consider sound nonstop path (Pid, model. i=1,2), sea surface reflection and two R1 and R2; simply nonstop path (Pid, i=1,2),signal sea surface path (Pis,hydrophones i=1,2) and seabed reflection pathconsider (Pib, i=1,2); set the sound source as x(t); reflection then, the path (Pis, i=1,2) and seabed reflection path (Pib, i=1,2); set the sound source signal as x(t); then, the reception model of the hydrophone is as shown in Equation (6): reception model of the hydrophone is as shown in Equation (6): # x1 ptq ´ τ1D q ` α1S xpt ´ τ1S q ` α1B xpt ´ τ1B q ` n1 ptq (t )α1Dxpt  x1“ 1D x(t   1D )  1S x(t   1S )  1B x(t   1B )  n1 (t ) (6)  “ α2D xpt ´ τ2D q ` α2S xpt ´ τ2S q ` α2B xpt ´ τ2B q ` n2 ptq x2 ptq (6) x ( t )  x ( t  )  x ( t  )  x ( t  ) n ( t )        2D 2D 2S 2S 2B 2B 2  2

Figure multi-path effect. effect. Figure 5. 5. Simplified Simplified model model of of the the underwater underwater multi-path

 iD ,  iS and  iB are respectively attenuation coefficients of Pid path, Pis path and Pib path αiD , αiS and αiB are respectively attenuation coefficients of Pid path, Pis path and Pib path (i = 1,2).  ,  iS and  iB are respectively the propagation time of Pid path, Pis path and Pib path. (i = 1,2). τiDiD , τiS and τiB are respectively the propagation time of Pid path, Pis path and Pib path. t ) and n2 (ptq t ) and t) ( t ) is is irrelevant irrelevant to and that Suppose that Suppose that sound soundsource sourcesignal signal xxptq to noise noise nn11(ptq and noise noise n that nn11(ptq 2 t ) , then the t ) and (t ) is is as as shown in is irrelevant to to noise noise nn22(ptq, the cross-correlation cross-correlation function function of of xx11(ptq and xx22 ptq R (  ) x ( t ) isaaself-correlation self-correlation function function of xptq. It . Itcan canbe beseen seenfrom fromEquation Equation(7) (7) that: that: peak Equation (7). R xx xx pτq is values of the the cross-correlation cross-correlationfunctions functionsofof x11(ptq t ) and xx22ptq (t ) occur occurrespectively respectivelyon onnine nine time time delay of arrival ´ τ D ,, τ1D ´ τ and τ1D ´ τ (nine peak values will occur when the nine arrival points, points, such such as as τ1D 1D   22D 1 D   2 S2S and 1D 2 B2B (nine peak values will occur when the nine points equality situation situation among points exists, exists, then then there there will will be be an an overlapping overlapping points are are unequal; unequal; if if an an equality among nine nine points phenomenon, and the number of peak values will reduce); the peak value will be decided by the corresponding attenuation coefficient. The specific effect is as shown in Figure 6. Under a practical situation, we only need to calculate the time difference of arrival of a nonstop path (main peak), so other peak values will interfere with confirming the main peak, which makes it impossible to accurately estimate the TDOA of signals.

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Rx 1x2

phenomenon, and the number of peak values will reduce); the peak value will be decided by the corresponding attenuation coefficient. The specific effect is as shown in Figure 6. Under a practical situation, we only need to calculate the time difference of arrival of a nonstop path (main peak), so Sensors 2016, 16, 42 7 of 22 other peak values will interfere with confirming the main peak, which makes it impossible to accurately estimate the TDOA of signals. Rx1 x2 ( )  E[ x1 (t ) x2 (t   )] ˚ R x1 x2 pτq “ Erx 1 ptqx 2 pt ´ τqs 1 D  2 D Rxx ( 1D   2 D   ) “ α1D α2D R xx pτ1D ´ τ2D ´ τq 1D 2 S Rxx ( 1D   2 S   ) `α1D α2S R xx pτ1D ´ τ2S ´ τq 1D 2 B Rxx ( 1D   2 B   ) `α1D α2B R xx pτ1D ´ τ2B ´ τq 1Sα 2 DRRxx (pτ  S ´  2τD  ´) τq `α 2D 1S 2D xx 11S (7) (7) 1S1Sα2S2 SRRxx  11S  2τS2S ´) τq `α pτ xx ( S ´ `α pτ 1S1Sα2B2BRRxx  11S  2τB2B ´) τq xx ( S ´ `α α R pτ ´ τ  ´) τq 1B1B2D xx ( 11B B   2 D2D 2 D Rxx `α pτ1B ´ τ2S ´ τq 1B1Bα2S2 SRRxx xx ( 1B   2 S   ) `α1B α2B R xx pτ1B ´ τ2B ´ τq 1B 2B Rxx ( 1B   2 B   )

Figure Figure 6. 6. Ambiguity Ambiguity phenomenon phenomenon of of correlation correlation peaks. peaks.

3.3. SINS/DVL Assistance in Seeking the Ideal Delay Inequality 3.3. SINS/DVL Assistance in Seeking the Ideal Delay Inequality Because of the multi-path effect, multiple correlation peaks will appear in the cross-correlation Because of the multi-path effect, multiple correlation peaks will appear in the cross-correlation function, and these peaks differ a little in value; it is very difficult to judge which one corresponds to function, and these peaks differ a little in value; it is very difficult to judge which one corresponds to the most ideal time difference. In multi-path time delay estimation, the methods that can be adopted the most ideal time difference. In multi-path time delay estimation, the methods that can be adopted are usually the self-adaptation method, the generalized cross-correlation method, the are usually the self-adaptation method, the generalized cross-correlation method, the auto-correlation auto-correlation method, the cepstrum method, etc. Although the self-adaptation method is of high method, the cepstrum method, etc. Although the self-adaptation method is of high estimation accuracy estimation accuracy and a strong resolution ratio, its search scope is quite broad, which makes it hard and a strong resolution ratio, its search scope is quite broad, which makes it hard to guarantee to guarantee convergence and results in a large calculated quantity and poor instantaneity; besides, convergence and results in a large calculated quantity and poor instantaneity; besides, it has a certain it has a certain requirement for the signal-to-noise ratio. The generalized cross-correlation method is requirement for the signal-to-noise ratio. The generalized cross-correlation method is of simple of simple calculation, but the main correlation peak is not obvious; and there are certain deficiencies calculation, but the main correlation peak is not obvious; and there are certain deficiencies in its in its performance. Resolution ratios of the auto-correlation method and the cepstrum method are performance. Resolution ratios of the auto-correlation method and the cepstrum method are used used for calculating the time delay, as they need a relatively larger signal bandwidth [17–20]. This for calculating the time delay, as they need a relatively larger signal bandwidth [17–20]. This paper paper opens up a new path by putting forward a method that selects the main correlation peak based opens up a new path by putting forward a method that selects the main correlation peak based on on SINS/DVL positioning assistance after combining the advantages and disadvantages of the generalized cross-correlation method. This algorithm is simple and easy to realize; besides, it can estimate the time differences of multiple paths. Even with a low signal-to-noise ratio, it can effectively do the estimation; in the meantime, it can solve the problem of selecting the main correlation peak with high accuracy when correlation peaks are ambiguous. Now, the principle of the algorithm will be introduced in detail:

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SINS/DVL positioning assistance after combining the advantages and disadvantages of the generalized cross-correlation method. This algorithm is simple and easy to realize; besides, it can estimate the time differences of multiple paths. Even with a low signal-to-noise ratio, it can effectively do the estimation; in the meantime, it can solve the problem of selecting the main correlation peak with high accuracy when correlation peaks are ambiguous. Now, the principle of the algorithm will be introduced in detail: In the LBL underwater acoustics positioning system, set the position of No. i hydrophone as Pi pxi , yi , zi q and the AUV position output by the SINS integrated system at time k as PSI NS pxSI NS pkq, ySI NS pkq, zSI NS pkqq; adopt PSI NS pxSI NS pkq, ySI NS pkq, zSI NS pkqq to calculate the distance between the hydrophone and AUV at time k as: b Di pkq “ pxi ´ xSI NS pkqq2 ` pyi ´ ySI NS pkqq2 ` pzi ´ zSI NS pkqq2

(8)

The distance difference between two random hydrophones i and j and the AUV at time k is: Dij pkq “ Di pkq ´ D j pkqpi ‰ jq

(9)

Then, the calculation of the time difference of the two hydrophones receiving signals is: ∆t1ij pkq “

Dij pkq cij pkq

(10)

cij pkq is the equivalent sound velocity of signals corresponding to time difference ∆t1ij pkq at time k. As at time k ´ 1, the surrounding environment of the AUV is not changed that much at time k, the change of the sound ray structure is little. Hence, the equivalent sound velocity of the last time cycle can be used as the current equivalent acoustic velocity; in other words, the velocity can be acquired by using the distance difference between hydrophones i and j and the sound source at time k ´ 1 to divide the time difference. The specific calculation is as follows: At time k ´ 1, the corrected position of AUV by LBL is PLBL{SI NS px LBL{SI NS pk ´ 1q, y LBL{SI NS pk ´ 1q, z LBL{SI NS pk ´ 1qq, and the distance of hydrophone i and the sound source is: Ri pk ´ 1q “

b pxi ´ x LBL{SI NS pk ´ 1qq2 ` pyi ´ y LBL{SI NS pk ´ 1qq2 ` pzi ´ z LBL{SI NS pk ´ 1qq2

(11)

The distance difference between hydrophones i and j and the sound source is: ∆Rij pk ´ 1q “ Ri pk ´ 1q ´ R j pk ´ 1q

(12)

If the time difference of hydrophones i and j (which have been screened out) receiving the signal at time tk´ 1 is ∆tijpk´1q , then the current equivalent acoustic velocity is: cij pkq “

∆Rij pk ´ 1q ∆tij pk ´ 1q

(13)

Substitute the computed results of Equations (9) and (13) into Equation (10), and time difference can be calculated. Seek the peak value that is the most proximate to ∆t1ij pkq among a group of ambiguous correlation peaks in Equation (7), and take the time difference corresponding to this peak value as the more accurate time difference ∆tij pkq. ∆t1ij pkq

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3.4. TDOA Three-Dimensional Positioning Algorithm Based on SINS/DVL Assistance Three-Dimensional Positioning Algorithm Base on Taylor Series Expansion For the moment, there are many algorithms that use TDOA measured to perform positioning, such as Chan’s algorithm, the Taylor algorithm, the Friedlander algorithm, etc. Chan’s algorithm has strict requirements for the measurement accuracy of the time difference and is more applicable to a line-of-sight transmission channel environment. Under a non-line-of-sight transmission condition, as for the measurement errors of the time difference of the signal arrival, the positioning errors of the algorithm will be large and will be easily influenced by the effects of reflection, scattering and refraction. What the Friedlander algorithm acquires is only the second-best solution. The Taylor algorithm has no special requirements for the statistic property of measuring errors or a priori information, and it can provide a higher positioning degree on a certain Gaussian noise level. However, this algorithm is an iterative one without a final expression solution, and algorithmic convergence needs to be guaranteed by an initial position that is not far from the actual position [21,22]. Hence, this paper suggests the TDOA positioning algorithm based on SINS/DVL assistance, and the algorithm takes the positioning results of SINS/DVL as the iterative initial value of the Taylor algorithm, which not only guarantees algorithmic convergence, but also reduces the iterations. It is very applicable to underwater positioning. This algorithm will be introduced in detail as follows: Suppose that there are n hydrophones in the matrix, then formulate (n – 1) equations according to the hyperbolic positioning model: Ri1 “ Ri ´ R1 “ ci1 ∆ti1 pi “ 2, 3, 4, . . . , nq

(14)

Ri1 is the function of x, y, z, xi , yi , zi ; px, y, zq represents the position of the AUV; pxi , yi , zi q represents the position of No. i hydrophone; then, it can be expressed as f i px, y, z, xi , yi , zi q “ Ri ´ R1 ; set it as the objective function. Suppose that the measured value of objective function f i p˚q (namely f i px, y, z, xi , yi , zi q) is mi “ ci1 ∆ti1 , and the actual value is ui , ui “ mi ´ ei , ei is the measuring error. ˆ y, ˆ zq, ˆ and they meet x “ xˆ ` ∆ x, y “ yˆ ` ∆y, z “ zˆ ` ∆z; then, expand objective Adopt initial value px, ˆ y, ˆ zq ˆ according to the Taylor series as the following Equation (15): function f i p˚q in px, ˆ y, ˆ z, ˆ xi , yi , zi q ` p∆x f i px, y, z, xi , yi , zi q “ f i px,

B B B ˆ y, ˆ z, ˆ xi , yi , zi q ` ∆y ` ∆z q f i px, Bx By Bz

B B B 2 1 ˆ y, ˆ z, ˆ xi , yi , zi q ` ¨ ¨ ¨ p∆x ` ∆y ` ∆z q f i px, 2! Bx By Bz n 1 B B B ˆ y, ˆ z, ˆ xi , yi , zi q ` p∆x ` ∆y ` ∆z q f i px, n! Bx By Bz 1 B B B n`1 ` p∆x ` ∆y ` ∆z q f i pxˆ ` ξ∆x, yˆ ` ξ∆y, zˆ ` ξ∆z, xi , yi , zi q, p0 ă ξ ă 1q pn ` 1q! Bx By Bz `

(15)

Ignore high-order terms above the quadratic term in the expanded section, then the above equation can be expressed as: ˆ y, ˆ z, ˆ xi , yi , zi q ` p∆x f i px, y, z, xi , yi , zi q “ ui « f i px,

B B B ˆ y, ˆ z, ˆ xi , yi , zi q ` ∆y ` ∆z q f i px, Bx By Bz

(16)

It can be acquired according to the hyperbolic positioning model that: f i px, y, z, xi , yi , zi q “

b b px ´ xi q2 ` py ´ yi q2 ` pz ´ zi q2 ´ px ´ x1 q2 ` py ´ y1 q2 ` pz ´ z1 q2

(17)

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Expand Equation (17) according to Equation (16) and acquire: xˆ ´ x1 yˆ ´ y1 zˆ ´ z1 xˆ ´ xi yˆ ´ yi zˆ ´ zi ´ ´ ´ f i px, y, z, xi , yi , zi q « Rˆ i ´ Rˆ 1 ` ∆xp q ` ∆yp q ` ∆zp q Rˆ i Rˆ 1 Rˆ i Rˆ 1 Rˆ i Rˆ 1

where:

b pxˆ ´ x1 q2 ` pyˆ ´ y1 q2 ` pzˆ ´ z1 q2 b Rˆ i “ pxˆ ´ xi q2 ` pyˆ ´ yi q2 ` pzˆ ´ zi q2

Rˆ 1 “

(18)

(19) (20)

ˆ y, ˆ z, ˆ xi , yi , zi q, then: Set fˆi “ f i px, ai1 “

ai2 “

ai3 “

xˆ ´ xi xˆ ´ x1 B f i px, y, z, xi , yi , zi q | “ ´ ˆ ˆ x “ x Bx Ri Rˆ 1 y “ yˆ z “ zˆ

(21)

B f i px, y, z, xi , yi , zi q yˆ ´ yi yˆ ´ y1 | “ ´ ˆ x “ x By Rˆ i Rˆ 1 y “ yˆ z “ zˆ

(22)

B f i px, y, z, xi , yi , zi q zˆ ´ zi zˆ ´ z1 | “ ´ x “ xˆ Bz Rˆ i Rˆ 1 ˆ y“y z “ zˆ

(23)

and then: fˆi ` ai1 ∆x ` ai2 ∆y ` ai3 ∆z « mi ´ ei

(24)

For the n matrix elements, there will be: ε « h ´ Gδ where:

» — — ε“— — – »

» — — G“— — –

a21 a31 .. . an1

a22 a32 .. . an2

e2 e3 .. . en

(25)

fi ffi ffi ffi ffi fl

fi » m2 ´ fˆ2 c21 ∆t21 ´ p Rˆ 2 ´ Rˆ 1 q — ffi — — m3 ´ fˆ3 ffi — c31 ∆t31 ´ p Rˆ 3 ´ Rˆ 1 q ffi “ — h“— .. .. — ffi — – fl – . . ˆ mn ´ f n cn1 ∆tn1 ´ p Rˆ n ´ Rˆ 1 q » xˆ ´ x2 xˆ ´ x1 yˆ ´ y2 yˆ ´ y1 ´ ´ fi — Rˆ Rˆ 1 Rˆ 2 Rˆ 1 2 — a33 — ˆ ˆ ˆ ˆ x ´ x x ´ x y ´ y y ´ y1 3 3 1 ffi — ´ ´ a33 ffi — ˆ ˆ ˆ ˆ R3 R1 R3 R1 .. ffi ffi ““ — — .. .. . fl — . . — an3 – xˆ ´ xn xˆ ´ x1 yˆ ´ yn yˆ ´ y1 ´ ´ Rˆ n Rˆ 1 Rˆ n Rˆ 1

(26)

fi ffi ffi ffi ffi fl

(27)

zˆ ´ z2 zˆ ´ z1 ´ Rˆ 2 Rˆ 1 zˆ ´ z3 zˆ ´ z1 ´ Rˆ 3 Rˆ 1 .. . zˆ ´ z3 zˆ ´ z1 ´ Rˆ n Rˆ 1

fi ffi ffi ffi ffi ffi ffi ffi ffi ffi fl

(28)

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»

fi ∆x — ffi (29) δ “ – ∆y fl Sensors 2016, 16, 42 11 of 22 ∆z Take  as an unknown variable; suppose that Q is the covariance matrix of  ; then adopt Take δ as an unknown variable; suppose that Q is the covariance matrix of ε; then adopt the the method of weighing least squares estimation; it can be acquired that: method of weighing least squares estimation; it can be acquired that: x »  fi ∆x y     [G T Q 1G ]1 G T Q 1h (30) ´1 —  ffi (30) – ∆y fl “ δ “ rG T Q´1 Gs G T Q´1 h  z  ∆z The calculation process of the Taylor algorithm can be concluded as follows: The calculation process of the Taylor algorithm can be concluded as follows: (1) Select an initial value ( xˆ , yˆ , zˆ ) ; zˆ ) into ( xˆ, yˆ ,value ˆ Equations ˆ zq; ˆ (1) Select an initial px, y, (20) and (28) to calculate G and Rˆ ; (2) Substitute i

ˆ ˆi1 ,y, ˆ zq ˆ t21into (2) Substitute pcx, Equations and (28) Rˆi into (20) (3) Substitute and Equation (27)totocalculate calculateG hand ; Ri ; ˆ (3) Substitute ci1 , ∆t21 and RQ into Equation (27) to calculate h; b  ; if ( x ) 2  (y ) 2  (z ) 2   , (4) Substitute G , h and i into Equation (30) and update 2 (4) Substitute G, h and Q into Equation (30) and update δ; if p∆xq ` p∆yq2 ` p∆zq2 ă η, η is a very is a very small threshold value, then the iteration ends; ( xˆ , yˆ , zˆ ) is the final positioning result. ˆ y, ˆ zq ˆ is the final positioning result. Otherwise, small threshold value, then the iteration ends; px, Otherwise, update ( xˆ , yˆ , zˆ ) according to Equation (31), and repeat ( xˆ , yˆ , zˆ ) until the above ˆ y, ˆ zq ˆ according to Equation (31), and repeat px, ˆ y, ˆ zq ˆ until the above conditions are met. update px, conditions are met. $ xˆ Ð &xˆ  xˆ xˆ`x ∆x ’  yˆ Ð yˆ ` ∆y (31) yˆ  y (31) ’ %yˆ  ˆ ˆ z Ð z ` ∆z 

 zˆ  zˆ  z

error (m)

The algorithm needs an estimated position value as as an an initial initial value value to to make make iterations. iterations. The accuracy of the initial position value has great great influence influence on on the the convergence convergence of of the the algorithm. algorithm. As ´−7 7 shown in Figure 7, m. When the 7, the the convergence convergence error errorthreshold thresholdfor forthe theiterative iterativealgorithm algorithmisis1010 initial position error is 80 m, the iterative iterative result cannot cannot converge. When the initial position error is 35 m, the iterative result converges, but the number of iterations is 468 steps. While the initial position error is set as 10 m, the number of iterations requires only 364 steps. Thus, the higher the accuracy of the initial position position value, value, the the faster fasterthe theconvergence convergencerate. rate.This Thispaper paperselects selectsthe theoutput outputposition positionPP SISINS NS of SINS/DVL asthe theinitial initial position position value value of of the the iteration, iteration, which which can can not not only satisfy the convergence SINS/DVL as of the algorithm, but also greatly reduce reduce the the iteration iteration steps. steps.

Figure initial position position values. values. Figure 7. 7. The The iteration iteration results results with with different different initial

3.5. SINS/DVL/MCP/LBL Integrated System Modeling When the AUV does not enter the effective signal scope of LBL, adopt the SINS/DVL/MCP integrated navigation system as shown in Figure 8 to perform the navigation and positioning. When the AUV enters the effective signal scope of LBL, adopt the SINS/DVL/MCP/LBL integrated navigation system as shown in Figure 9 to perform the navigation and positioning.

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3.5. SINS/DVL/MCP/LBL Integrated System Modeling When the AUV does not enter the effective signal scope of LBL, adopt the SINS/DVL/MCP integrated navigation system as shown in Figure 8 to perform the navigation and positioning. When the AUV enters the effective signal scope of LBL, adopt the SINS/DVL/MCP/LBL integrated navigation Sensors 2016, 2016, 16, 42 42 in Figure 9 to perform the navigation and positioning. 12 of of 22 22 system as shown Sensors 16, 12

Figure8.8. 8.SINS/DVL/MCP SINS/DVL/MCP integrated integrated navigation system. Figure SINS/DVL/MCP Figure integratednavigation navigationsystem. system.

Figure9.9. 9.SINS/DVL/MCP/LBL SINS/DVL/MCP/LBL integrated integrated navigation system. Figure SINS/DVL/MCP/LBL Figure integratednavigation navigationsystem. system.

The state equation of the integrated system is: The state equation of the integrated system is:   FX  GW (32) X .X  FX  GW X “ FX ` GW (32) where X is the state variable, F is the state-transition matrix, G is the transition matrix of the process noise and W is systematic noise. where X is the state variable, F is the state-transition matrix, G is the transition matrix of the process the error, attitude error, accelerometer zero offset and gyroscopic drift as state noiseSelect and W is velocity systematic noise. vector X: Select the velocity error, attitude error, accelerometer zero offset and gyroscopic drift as state TT vector X:  byby  bzbz  X   VEE  VNN  VUU EE NN UU  L   h bx by bzbz  bx (33) bx by bx ” ıT X“ (33) δλ δhof the ∇bx directions ∇by ∇bz ε bx north ε by εand VEE ,,δV VNNE,, V VδV U φE φ Ntheφvelocity U δL errors bz the local V V are δV respectively of east, U N U vertical(up). EE ,,NN ,,UU are respectively the misalignment angles of the directions of east, north and δVE , δVN , δVU are respectively the velocity errors of the directions of east, north and the local the local vertical(up).  L,  ,  h are respectively the errors of latitude, longitude and altitude. vertical(up). φE , φN , φU are respectively the misalignment angles of the directions of east, north bx by bzbz are respectively biased errors of the three axial directions of the accelerator.  bx  ,,  by bx ,,  by ,,  by ,,  bz bz and the local vertical(up). δL, δλ, δ h are respectively the errors of latitude, longitude and bxaltitude. are respectively the drifts of the three axial directions of the gyroscopes. F can be confirmed by the ∇bx , ∇by , ∇bz are respectively biased errors of the three axial directions of the accelerator. ε bx , ε by , ε bz SINS error equation. are respectively the drifts of the three axial directions of the gyroscopes. F can be confirmed by the The measuring SINS error equation.equation of the integrated system is:

VSINS SINS

VSINSE SINSE     VSINSN SINSN  VSINSU SINSU  

(34)

SINS   LSINS    is the position information output of the SINS  Z is the observation vector; PSINS SINS SINS  SINS   hSINS SINS  

LBL   LLBL

SINSE  VSINSE

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The measuring equation of the integrated system is: »

VSINS

fi VSI NSE — ffi “ – VSI NSN fl VSI NSU

(34)

»

Z is the observation vector; PSINS Sensors 2016, 16, 42

fi LSI NS — ffi “ – λSI NS fl is the position information output of the 13 of 22 hSI NS

»

fi L LBL — ffi VDVLEoutput  SINS system; P LBL “ – λ LBL fl is the position information by the LBL system; VSINS “   velocity information output by the SINS system; VDVL  VDVLN  is the velocity information output h LBL » fi » fi VDVLU  VDVLE VSI NSE — ffi — ffi by λthe DVLflsystem; U is theinformation heading information by the SINS system; is the fl heading is the velocity output byoutput the SINS system; VDVL “ –MCPVDVLN is the – SI NSN VDVLU hSI NSU output by the MCP system; V is the observation noise vector; and H is the information measurement velocity information output by the DVL system; ϕU is the heading information output by the SINS matrix, which satisfies: system; ϕ MCP is the heading information output by the MCP system; V is the observation noise vector; 0 0 satisfies: 0 0 0 1 0 0 0 0 0 0 0 1 0 which and H is the measurement matrix,  H» 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0  fi (35) 10 00 01 00 00 01 0 10 0 10 00 00 00 00 00 — ffi H “ – 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 fl (35) 0 0 1 0 0 1 0 0 0 1 0 0 0 0 0 4. Simulation and Experiment 4. Simulation and Experiment 4.1. Underwater Sound Signal Propagation Channel Modeling 4.1. Underwater Sound Signal Propagation Channel Modeling The underwater acoustic channel is a time varying and space varying random channel of high environmental noise, acoustic a narrowchannel channelisbandwidth, largeand transmission loss and a serious multi-path The underwater a time varying space varying random channel of high effect. Because of the slow underwater the AUV, the acoustic channel can be seen as a environmental noise, a narrow channel movement bandwidth,oflarge transmission loss and a serious multi-path slow time-varying system canmovement be approximated as a LTI linear time-invariant effect. Because of system. the slowThe underwater of the AUV, the( acoustic channel can )system. be seen Suppose there are system. n paths for acoustic signal to transmitasfrom the( linear soundtime-invariant source to the as a slowthat time-varying Thethe system can be approximated a LTI hydrophones, andthat then, theare unit impulse of the multi-path channel from the sound )system. Suppose there n paths for response the acoustic signal to transmit from the sound sourcesource to the to the hydrophones will be: hydrophones, and then, the unit impulse response of the multi-path channel from the sound source to the hydrophones will be: N N a  (t   ) h (t )  ÿ (36) i i hptq “ i 1 ai δpt ´ τi q (36)



i “1

theattenuation attenuationcoefficient coefficientofofthe theNo. No.i itransmission transmissionpath; path;τ is ithe is relative the relative delay of aai i isisthe timetime delay of the i the transmission along i transmission path. underwater acoustic signal transmission transmission along the the No.No. i transmission path. TheThe underwater acoustic signal transmission cancan be be simplified as the model is shown in Figure simplified as the model thatthat is shown in Figure 10: 10: Sound source signal x(t)

Underwater sound channel h(t)

Hydrophones Receive Signal y(t)

Figure 10. A underwater sound sound transmission. transmission. Figure 10. A simplified simplified model model of of underwater

Then, the hydrophones receive signal y(t) as a convolution of sound source signal x(t) and the Then, the hydrophones receive signal y(t) as a convolution of sound source signal x(t) and the unit impulse response h(t), namely: unit impulse response h(t), namely: yptq (37) y (t ) “ xptq x (t )˚*hptq h (t ) (37) Porter, have developed developed BELLHOP BELLHOP software, software, which which simulates simulates the the marine marine environment environment Porter, M.B. M.B. et et al. al. have according to this model. This software can acquire amount N, the angle of incidence, the range andand the according to this model. This software can acquire amount N, the angle of incidence, the range the time delay of intrinsic sound rays and provide the unit impulsive response of the system by inputting marine environment parameters [23]. This paper, according to the sound velocity distribution curve under a lake, which is as shown in Figure 11, adopts the BELLHOP software to establish an underwater sound signal transmission channel model, which will describe the sound field of the underwater environment and calculate the sound ray transmission path by setting the

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time delay of intrinsic sound rays and provide the unit impulsive response of the system by inputting marine environment parameters [23]. This paper, according to the sound velocity distribution curve under a lake, which is as shown in Figure 11, adopts the BELLHOP software to establish an underwater sound signal transmission channel model, which will describe the sound field of the underwater environment and calculate the sound ray transmission path by setting the positions of the sound source and hydrophones, as shown in Figure 12; then, it will solve the unit impulse response function h(t) of the system and, finally, acquire the received signals of the hydrophones by the convolution Sensors 2016, and 16, 42simulation. 14 of 22 operation 0 5 10 15 20 25 30 35 40 1440

1450

1460

1470

1480

1490

1500

1510

1520

Sound Speed (m/s)

Figure Figure 11. 11. Underwater Underwater sound sound velocity velocity distribution distribution curve curve of of aa lake. lake.

Figure Figure 12. 12. Sound Sound ray ray transmission transmission path. path.

The simulated sound source signal adopts the amplitude-modulated signal expressed in The simulated sound source signal adopts the amplitude-modulated signal expressed in Equation Equation (38). Its bandwidth is 50 kHz, and its center frequency is 25 kHz. Suppose that the noise (38). Its bandwidth is 50 kHz, and its center frequency is 25 kHz. Suppose that the noise model of model of the facility is a white noise model; then, the waveform of the sound source signal will be as shown in Figure 13. N

sigIN  ( a sin(2 i t )) cos(2c t ) i

(38)

Calculate the unit impulse response as shown in Figure 14 according to the propagating sound rays, and the received signal can be calculated as shown in Figure 15 according to Equation (37). Adopt the sound field model established by BELLHOP and the underwater sound velocity distribution to simulate the situation of reflection and refraction, which may occur in the acoustic

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the facility is a white noise model; then, the waveform of the sound source signal will be as shown in Figure 13. N ÿ sigI N “ p asinp2πiωtqqcosp2πωc tq (38) i

amplitude (mV)

Calculate the unit impulse response as shown in Figure 14 according to the propagating sound rays, and the received signal can be calculated as shown in Figure 15 according to Equation (37). Adopt the sound field model established by BELLHOP and the underwater sound velocity distribution to simulate the situation of reflection and refraction, which may occur in the acoustic signal transmission process, as well as the signals received by the hydrophones, all of which will be used for the simulation Sensors 2016, 2016, 16, 16, 42 42 15 of of 22 22 Sensors 15 and positioning calculation.

Figure 13. Oscillogram of the sound source signal. Figure Figure13. 13.Oscillogram Oscillogramof ofthe thesound soundsource sourcesignal. signal. 2.5 2.5

-3 10-3 10

22 1.5 1.5 11

h

0.5 0.5 00 -0.5 -0.5 -1 -1 -1.5 -1.5 -2 -2 00

0.005 0.005

0.01 0.01

0.015 0.015

0.02 0.02

0.025 0.025

(s) tt (s)

amplitude (mV)

Figure 14. 14. Unit Unit impulse impulse response response h(t) h(t) Figure h(t).

0.03 0.03

0.035 0.035

-1.5 -2

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

t (s)

Figure 14. Unit impulse response h(t)

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amplitude (mV)

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Figure Figure15. 15.Oscillogram Oscillogramof ofreceived receivedsignals. signals.

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latitude (°)

4.2. Simulation of SINS Assistance in the LBL Positioning Algorithm 4.2. Simulation of SINS Assistance in the LBL Positioning Algorithm As shown in Figure 16, set the positions of the hydrophones and the AUV; place five As shown in Figuretheir 16, positions set the positions of the with hydrophones the AUV; five hydrophones underwater; will be expressed longitudeand and latitude, andplace they are hydrophones underwater; their positions be expressed with longitude latitude, are respectively(118.01°, 32°), (118.01°, 32.01°),will (118.01°, 32.02°), (118.02°, 32.01°),and (118°, 32.01°);and all they of their ˝ , 32˝ ), (118.01˝ , 32.01˝ ), (118.01˝ , 32.02˝ ), (118.02˝ , 32.01˝ ), (118˝ , 32.01˝ ); all of respectively(118.01 depths are 30 m. Suppose that the current actual position of the AUV is (118°, 32°) with a depth of their are 30 m. acoustic Supposevelocity that the distribution current actual position of the AUV (118˝ , 32˝ ) on with a depth 10 m. depths The underwater data are acquired by anisexperiment this lake. of 10 m. The underwater acoustic velocity distribution data are acquired by an experiment on Simulate the acoustic signals received by the hydrophones according to the method introducedthis in lake. Simulate acoustic received by hydrophones according to thethe method Section 4.1, andthe calculate thesignals time difference of the different hydrophones receiving signal.introduced in Section 4.1, and calculate the time difference of different hydrophones receiving the signal.

Figure Figure16. 16.Layout Layoutofofthe thehydrophones hydrophonesand andthe theAUV. AUV.

4.2.1. 4.2.1.Simulation Simulationof ofSINS SINSAssisting AssistingLBL LBLin inTracking Tracking the the Optimal Optimal Time Time Difference Difference As sound source signal will go go through multiple paths, there willwill be Asshown shownininFigure Figure17, 17,asasthe the sound source signal will through multiple paths, there also multiple correlation peaks in in thethe generalized be also multiple correlation peaks generalizedcross-correlation cross-correlationresult. result.The Thecorrelation correlationpeaks peaks whose amplitude is in the top twenty can be selected as alternative correlation peaks, whose amplitude is in the top twenty can be selected as alternative correlation peaks,while whileothers others are areneglected neglecteddue dueto toexcessive excessivesignal signalattenuation. attenuation.The Thedistance distancebetween betweenthe thetime timedifference differencewith withthe the highest amplitude (Point A in Figure 17) and the true time difference (the red circle in Figure 17) highest amplitude (Point A in Figure 17) and the true time difference (the red circle in Figure 17)isis relatively soPoint PointAA cannot selected as true the time true difference. time difference. Using the proposed relatively larger, larger, so cannot be be selected as the Using the proposed method, method, the time difference (symbol × in Figure 16), which is calculated through Equation (10) by the aiding of the SINS/DVL integrated navigation, better approaches the truth value. Therefore, the alternative correlation peak (Point B in Figure 17), which is closest to the symbol “×”, is selected as the ideal time difference, because it has the smallest distance to Point “A”. Traditional algorithms directly take the correlation peak that is corresponding to the maximum peak value as the main

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Rxy

the time difference (symbol ˆ in Figure 16), which is calculated through Equation (10) by the aiding of the SINS/DVL integrated navigation, better approaches the truth value. Therefore, the alternative correlation peak (Point B in Figure 17), which is closest to the symbol “ˆ”, is selected as the ideal time difference, because it has the smallest distance to Point “A”. Traditional algorithms directly take the correlation peak that is corresponding to the maximum peak value as the main correlation peak to calculate the time difference; there will be larger errors under the multi-path effect. The improved algorithm adopted in this paper selects the main correlation peak through the positions obtained by the SINS/DVL integrated navigation system. A comparison of the calculation results and errors of the two algorithms is as shown in Tables 1 and 2 which show that: the improved algorithm reduces the interference of multiple correlation peaks with time difference estimation under the multi-path effect and makes the calculation value accuracy of the final time difference superior to that 17 ofofthe Sensors 2016, 16, 42 22 traditional algorithm.

Figure Figure 17. 17. Screening Screening out out of of the the main main correlation correlation peak. Table Table 1. Comparison Comparison of of the the calculation calculation results of the time differences.

Time Difference of Two

Time Difference of Two Traditional Improved Truth Traditional Improved Hydrophones Hydrophones Receiving Truth Value/s Algorithm/s Algorithm/s Value/s Algorithm/s Algorithm/s theSignal Signal Receiving the

Ts1−Ts0 Ts1´Ts0 Ts2−Ts0 Ts2´Ts0 Ts3´Ts0 Ts3−Ts0 Ts4´Ts0 Ts4−Ts0

´0.3532 0.5080 0.6623 ´0.2388

−0.3532 ´0.3436−0.3436 0.5080 0.5043 0.5043 0.6623 0.6499 0.6499 −0.2388 ´0.2297−0.2297

−0.3470 ´0.3470 0.5044 0.5044 0.6492 0.6492 ´0.2297 −0.2297

Note: The expression (Ts1´Ts0) represents the time difference from the sound source from the sound source Note: The expression (Ts1-Ts0) represents the time difference from the sound source from the sound propagating to Hydrophone 1 and Hydrophone 0, and so on. source propagating to Hydrophone 1 and Hydrophone 0, and so on.

Table 2. Error comparison of the calculation values of the time difference. Table 2. Error comparison of the calculation values of the time difference. Time Difference of Two Time Difference of Two Hydrophones Receiving the Signal

Hydrophones Receiving Ts1´Ts0 the Signal Ts2´Ts0 Ts1−Ts0 Ts3´Ts0 Ts2−Ts0 Ts4´Ts0 Ts3−Ts0 Ts4−Ts0

Error ofError the Traditional of the Algorithm/s

Traditional 0.0062 Algorithm/s 0.0036 0.0062 0.0132 0.0036 0.0091 0.0132 0.0091

Error of the Error of Improved the Algorithm/s

Improved 0.0034 Algorithm/s 0.0001 0.0034 0.0007 0.0001 0.0001 0.0007 0.0001

4.2.2. Simulation of the Acoustic Velocity Correction Algorithm The traditional algorithm directly adopts acoustic velocity in a traditional sense to calculate the distance difference. The improved algorithm adopted in this paper calculates equivalent acoustic velocity, which will be used to calculate the distance difference. A comparison of the calculation results and errors of the two algorithms is as shown in Tables 3 and 4, which show that: as the

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4.2.2. Simulation of the Acoustic Velocity Correction Algorithm The traditional algorithm directly adopts acoustic velocity in a traditional sense to calculate the distance difference. The improved algorithm adopted in this paper calculates equivalent acoustic velocity, which will be used to calculate the distance difference. A comparison of the calculation results and errors of the two algorithms is as shown in Tables 3 and 4 which show that: as the improved algorithm adopts the equivalent acoustic velocity to calculate the distance difference, it corrects the sound velocity, under situations of the multi-path effect and sound ray curve. It greatly enhances the accuracy of the calculation values of the distance difference. Therefore, adopting the time difference algorithm and distance difference algorithm in this paper can greatly reduce the errors of the time difference and distance difference, which play a significant role in improving the AUV underwater positioning accuracy. Table 3. Comparison of the calculation results of the distance difference. Distance Difference of Two Hydrophones Receiving the Signal

Traditional Algorithm/m

Improved Algorithm/m

Ideal Distance Difference/m

Ds1´Ds0 Ds2´Ds0 Ds3´Ds0 Ds4´Ds0

´523.8619 739.6344 973.0620 ´360.3024

´509.5792 734.1894 954.7788 ´346.6323

´510.0840 734.3538 954.1334 ´346.1796

Note: The expression (Ds1´Ds0) represents the distance difference from the sound source propagating to Hydrophone 1 and Hydrophone 0, and so on.

Table 4. Error comparison of the calculation values of the distance difference. Distance Difference of Two Hydrophones Receiving the Signal

Error of the Traditional Algorithm/m

Error of the Improved Algorithm/m

Ds1´Ds0 Ds2´Ds0 Ds3´Ds0 Ds4´Ds0

13.7779 5.2806 18.9286 14.1228

0.5048 0.1644 0.6454 0.4527

4.2.3. Simulation of the TDOA Positioning Algorithm In order to verify the positioning effect of the algorithm, we conduct a simulation under the MATLAB environment. Adopt the BELLHOP model to simulate the acoustic signal receiving from the hydrophones, and calculate the time difference and distance difference; finally, adopt the position obtained from SINS/DVL integrated navigation system as the iterative initial value of the Taylor series expansion method to solve the positioning results; then obtain the longitude, latitude and depth of AUV, and make a comparison with the positioning results of the traditional algorithm. The result is as shown in Figure 17. It can be seen from Figure 18 that: the positioning result of the traditional algorithm has far deviated from the actual position, while that of the improved algorithm is very proximate to the actual value. This is because the error of the distance difference of the traditional algorithm is larger, which results in the non-convergence of the positioning result or big error; then, an acceptable positioning result cannot be acquired. However, the error of the distance difference of the improved algorithm is less than 1 m, and a more accurate positioning result can be acquired finally.

latitude (°)

It can be seen from Figure 18 that: the positioning result of the traditional algorithm has far deviated from the actual position, while that of the improved algorithm is very proximate to the actual value. This is because the error of the distance difference of the traditional algorithm is larger, which results in the non-convergence of the positioning result or big error; then, an acceptable positioning result cannot be acquired. However, the error of the distance difference of the improved Sensors 2016, 16, 42 19 of 22 algorithm is less than 1 m, and a more accurate positioning result can be acquired finally.

Figure Figure 18. 18. Comparison Comparison of of the the positioning positioning results. results.

4.3. Dynamic Simulation of AUV Integrated Navigation System Based on SINS/LBL/DVL/MCP In order to further verify the effectiveness of this algorithm when the AUV is in dynamic operation, a dynamic simulation has been performed on this algorithm. Place five hydrophones underwater; their positions are the same as the above simulation. Suppose that the AUV starts from (118˝ , 32˝ ) and moves along north by east 45˝ , which is as shown in Figure 19. The random shift and constant value shift of the gyroscope is 50 µg; the constant value bias is 50 µg. The initial misalignment angles are respectively: pitching angle 1.5˝ , roll angle 1.5˝ and course angle 1.5˝ . The velocity error of DVL is 0.2 m/s. The heading error of MCP is 0.3˝ . The velocity of the AUV is 1 m/s. Adopt the algorithm in this paper to perform the positioning calculation, and integrate the positioning results with the SINS/DVL/MCP integrated system to calibrate accumulative errors; simulation time is three hours. In order to enhance the fault-tolerant ability of the system, if the positioning results are not converged, then the positioning results of LBL will not be used for the calibration. Figure 19 indicates that: when the AUV enters the scope of the hydrophones, the trajectory of the improved algorithm is fundamentally overlapping with the ideal trajectory, and that of the traditional algorithm deviates from the ideal trajectory. When the AUV leaves the scope of the hydrophones, the trajectory of both the traditional and improved algorithms deviates from the ideal trajectory. Figure 19 is a comparison graph of the positioning errors of the traditional and improved algorithms. The positioning errors will be expressed by the distance between the positioning results and the actual position. Figure 20 indicates that: within 0~1000 s, the positioning errors of both the traditional and improved algorithms will gradually enlarge, because within this period, the AUV is somewhat far away from the scope of the hydrophones; it takes a long time for the hydrophones to receive the sound source signal, and there will be great delay for the positioning results. Hence, LBL will not be used to calibrate within this period; instead, the SINS/DVL/MCP integrated navigation system will be used to perform the positioning. When the AUV approaches the scope of hydrophones, namely after 1000 s, we adopt LBL to perform the calibration. As the errors of adopting the traditional algorithm to calculate are larger, the LBL positioning results will not converge, which has nearly no contribution to calibrating for the SINS/DVL/MCP integrated navigation system. Within 1000 s to 3000 s, the positioning errors enlarge gradually, with the maximum reaching 30 m or so. However, the improved algorithm in this paper greatly reduces the errors of the calculation values of the distance difference. The LBL positioning results can calibrate the accumulative errors of the integrated navigation system, which will control the positioning errors within 2 m. When the AUV leaves the scope of the hydrophones, the LBL positioning system will lose its effect; however, in general, the precision is superior to the system that has not been calibrated by LBL. The above analysis shows that: under actual dynamic operation, when

has nearly no contribution to calibrating for the SINS/DVL/MCP integrated navigation system. Within 1000 s to 3000 s, the positioning errors enlarge gradually, with the maximum reaching 30 m or so. However, the improved algorithm in this paper greatly reduces the errors of the calculation values of the distance difference. The LBL positioning results can calibrate the accumulative errors of the integrated navigation system, which will control the positioning errors within 2 m. When the Sensors 2016, 16, 42 20 of 22 AUV leaves the scope of the hydrophones, the LBL positioning system will lose its effect; however, in general, the precision is superior to the system that has not been calibrated by LBL. The above analysis that: under actual dynamic operation, the AUV approaches of the the AUVshows approaches the scope of the hydrophones, thewhen improved algorithm can be the usedscope to perform hydrophones, improvedand algorithm can be used to can perform thetopositioning calculation, and the the positioningthe calculation, the calculation results be used calibrate the SINS/DVL/MCP calculation results cansystem, be usedwhich to calibrate the SINS/DVL/MCP integrated navigation which integrated navigation will greatly improve the positioning accuracy of thesystem, AUV when it will greatly improve the positioning accuracy of the AUV when it navigates underwater. navigates underwater. 32.07 32.0488 32.06

32.0487 32.0486

32.05

latitude (°)

118.0571 118.0571 118.0572 118.0572 32.04

32.0092 32.0091

32.03

32.0090 118.0106 118.0107 118.0107 118.0108

32.02 32.01 32 118

118.01

118.02

118.03

118.04

Matrix Form Hydrophone Ideal Trajectory Starting Point Trajectory of Improved Algorithm Trajectory of Traditional Algorithm 118.05

118.06

118.07

118.08

longitude (°)

Figure Figure 19. 19. Dynamic Dynamic simulation simulation of of the the AUV AUV integrated integrated navigation navigation system. system.

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error (m)

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Figure 20. Error comparison of the positioning results. Figure 20. Error comparison of the positioning results.

5. Conclusions 5. Conclusions This paper, directed at deficiencies in existing underwater positioning technology, proposes an This paper, directed at deficiencies in existing underwater positioning technology, proposes an underwater positioning system based on the mutual assistance of SINS/DVL/MCP and LBL. The underwater positioning system based on the mutual assistance of SINS/DVL/MCP and LBL. The whole system consists of an SINS/DVL/MCP integrated navigation system and an LBL underwater whole system consists of an SINS/DVL/MCP integrated navigation system and an LBL underwater acoustic positioning system. The latter adopts the TDOA positioning algorithm based on Taylor acoustic positioning system. The latter adopts the TDOA positioning algorithm based on Taylor series series expansion, as well as the positioning results provided by the SINS/DVL/MCP integrated expansion, as well as the positioning results provided by the SINS/DVL/MCP integrated navigation navigation system, which will assist in calculating the delay inequality and distance difference. In the system, which will assist in calculating the delay inequality and distance difference. In the meantime, meantime, the positioning results will also be used as the iterative initial value of the positioning the positioning results will also be used as the iterative initial value of the positioning algorithm. algorithm. The final positioning results will be used to calibrate the accumulative errors of the The final positioning results will be used to calibrate the accumulative errors of the position of the position of the SINS/DVL/MCP integrated navigation system. SINS/DVL/MCP integrated navigation system. The final simulation results show that: compared to the traditional algorithm, this scheme can not only directly correct accumulative errors caused by the dead reckoning algorithm, but also solves the problem of ambiguous correlation peaks caused by the multipath transmission of underwater acoustic signals. This algorithm can greatly improve the AUV underwater positioning accuracy. Only when the AUV installed with the SINS/DVL integrated navigation system approaches the scope of the hydrophones can accumulative errors be effectively calibrated. Hence, it has strong practicability.

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The final simulation results show that: compared to the traditional algorithm, this scheme can not only directly correct accumulative errors caused by the dead reckoning algorithm, but also solves the problem of ambiguous correlation peaks caused by the multipath transmission of underwater acoustic signals. This algorithm can greatly improve the AUV underwater positioning accuracy. Only when the AUV installed with the SINS/DVL integrated navigation system approaches the scope of the hydrophones can accumulative errors be effectively calibrated. Hence, it has strong practicability. Acknowledgments: This study is supported in part by the National Natural Science Foundation of China (Grant No. 51375088), the Foundation of Key Laboratory of Micro-Inertial Instrument and Advanced Navigation Technology of the Ministry of Education of China (201403), the Fundamental Research Funds for the Central Universities (2242015R30031) and the Key Laboratory fund of the Ministry of Public Security based on large data structure (2015DSJSYS002). Author Contributions: Tao Zhang: Design of system structure and interactive assistance and writing of the paper. Liping Chen: Implementation of algorithm and simulation analysis and writing of the paper. Yao Li: Scientific advising and writing. Conflicts of Interest: The authors declare no conflict of interest.

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