An Approach for Design of Multi-element USBL Systems

Transcription

1 An Approach for Design of Multi-element USBL Systems MIKHAIL ARKHIPOV Department of Postgrauate Stuies Technological University of the Mixteca Carretera a Acatlima Km. 2.5 Huajuapan e Leon Oaxaca MEXICO mikhail@mixteco.utm.mx Abstract: - This paper presents an approach for esign of multi-element ultra-short baseline (USBL) systems with the goal to improve the coorinate etermination accuracy of the unerwater objects. The object location estimation is performe on the base of the measurement of the istance to the object an the angular object position. It is suppose that the object is equippe with a transponer that receives the interrogation acoustical impulse an sens an acoustical impulse in reply. The iea of the esign metho is to increase of the number of receiving elements of the USBL antenna with a corresponing increase of the number of the receiving bases with the ifferent spatial orientation. The arrangement of the array receiving elements is obtaine by the rotations of the basic (three-element) receiving arrays aroun the longituinal an lateral carrier axes (the case of the etermination of the Cartesian coorinates of the object in the carrier coorinate system is consiere). It is suppose that the control of the spatial orientation of the receiving USBL array is realize by means of measurement of its pitch an roll angles (in the carrier coorinate system). In the article the propose metho is applie for esign of the nine-element USBL system. The special coorinate etermination algorithm for the propose USBL system is esigne an teste. The simulation of the algorithm was realize with the assumption that the object can have an arbitrary location in the lower hemisphere an the receiving USBL antenna can have significant inclination. The coorinate etermination accuracy of the propose USBL system is evaluate. Key-Wors: - Ultra-short baseline (USBL) system unerwater object transponer carrier coorinate system local coorinate system pitch an roll angles. 1 Introuction The methos an the systems for etermining the position of unerwater objects are constantly being evelope with an intention to improve the reliability an accuracy of the object coorinate measurements. For many tasks in ocean engineering the accuracy of etermination of the position of the object may be critical. Very often the task of precise etermination of the position of an unerwater object it is necessary to provie for real severe marine conitions where sea surface roughness an strong currents take place. In long istances the acoustical methos for position etermination still remain the unique solution of the pose problem. In this paper we examine the esign metho for ultra-short baseline (USBL) systems where the location measurement of the unerwater objects is base on the etermination of the istance to the object an the object s angular position. The iea of the esign metho is to increase the number of receiving elements of the USBL array an use for the placement of receiving elements the natural rotations of the basic (three-element) receiving arrays aroun the longituinal an lateral carrier axes. In aition to the esign metho the coorinate etermination algorithm is propose. The esigne algorithm realizes the processing of multiple time elays obtaine with the multi-element USBL array. 2 Problem Formulation 2.1 Location problem In this article we will consier the position etermination of an unerwater object with USBL acoustic systems. The principle for measuring the object coorinates with USBL metho is well known an is escribe in etail in [1][2]. Object position etermination with this metho is realize by means of the measuring of the istance to the object an its angular position relative to the measuring system location. During the last ecaes the improvements in accuracy an reliability of object position etermination was the subject of investigation an evelopment of the USBL systems [3-7]. To improve the USBL systems various special signal processing techniques were utilize. In particular the chirp ISSN: Issue 8 Volume 8 August 2009

2 signals an greater inter-element array separation [3] were use. Also the acoustic igital sprea spectrum [4] an moulate Barker-coe signals [5] were applie. In [6] the USBL system with frequencyhoppe pulses was investigate. The problem of instability of the position of the receiving antenna was stuie in etail in [7]. In [8][9] the problem of low precision in coorinate etermination for the case of the object foun in the plane of receiving bases is stuie. In this paper the metho to esign a multi-element USBL is propose. The objective of this esign metho is the further improvement of accuracy for the USBL systems (as well increasing the reliability of USBL systems) by means of increasing the number of receiving elements an the number of elemental USBL systems with ifferent orientations in space. The esigne algorithm processes multiple time elays in the output of the propose multielement USBL receiving array. Furthermore in this paper we assume that the propagation meium is homogeneous an the multipath interference is absent. 2.2 Basic USBL System The principle of operating of the USBL system is the following: the transmitter sens an interrogation acoustical impulse in the propagation meium where the object is locate in an accessible istance. It is suppose that the object is equippe with a transponer that receives the interrogation impulse an sens an acoustical impulse in reply. The istance to the object is etermine by the measurement of the values of propagation times of the interrogation impulse from the system an the transponer pulse response. The angular position of the object is etermine by the measurement of the phase ifference of the transponer pulse carrier frequency on the receiving array outputs. The minimum number of receiving elements for the USBL system for object coorinate etermination is three [1]. To improve the reliability an accuracy of coorinate etermination the number of elements of receiving antenna can be increase. This approach was initially propose in [8] an stuie in etail in [9] for the case of a five-element USBL system. The USBL system propose in [89] has being esigne by assembling of elemental (three-element) USBL arrays to single five-element USBL array. We consier briefly the principle of coorinate etermination for the three-element USBL system an then escribe the case for esigning of the multielement USBL array. Let =(0xyz) be the carrier coorinate system (lefthan) with the origin in the point O in such a way that the x-coorinate axis coincies with the carrier longituinal axis L-L' (the positive irection coincies with the irection of the straight arrowe line) the y-coorinate coincies with the carrier lateral axis B-B' an z-axis goes ownwars. Now we can efine the USBL array orientation in the introuce carrier coorinate system. Let the angle between the y-axis an base 1-2 (when it lies in horizontal plane) is 135 an the angle between the y-axis an the base 3-2 is 45. The geometry of the receiving antenna with the carrier longituinal an lateral axes is presente in Fig.1. We also efine a local coorinate system for this elemental (three-element) USBL system. Let 123 =(0x 123 y 123 z 123 ) be the local coorinate system for the consiere USBL system (with antenna elements 123). In Fig.1 we specify the angles ( an ) that efine the position of the unerwater object locate in point P (firstly the object position is efine in the 123 =(0x 123 y 123 z 123 ) coorinate system). We also assume that the USBL system is equippe with special unit to measure the pitch an roll angles of the receiving antenna. Let angles ξ an ζ be pitch an roll angles of the receiving USBL antenna (in the figure these angles show the rotations of the receiving three-element USBL array relative to the carrier lateral B-B' axis an the carrier longituinal L-L' axis). Let that the interrogation impulse has been sent an the reply impulse is being receive by antenna. The istance to the object is efine by measuring the propagation times of the interrogation an reply pulses. x x 123 y L' B' 3 x 123 x α z 123 USBL antenna O2 z 123 z 123 z γ β 1 L R Reply impulse y 123 y 123 y 123 B Interrogation impulse P(XYZ) Fig.1. Geometry of the three-element USBL receiving array an the carrier longituinal an lateral axes. ISSN: Issue 8 Volume 8 August 2009

3 Time elays on receiving elements efine the object s angular position. The time elays 12 an 32 at the outputs of the receiving elements of the base 1-2 an the base 3-2 (it is suppose that R>>) can be expresse in the following way: cos β c τ 12 = 32 cosα τ = c (1) where c is the spee of the soun in the water. With /c efine as we can write the irection cosines cos an cos : cos α τ 32 τ The thir irection cosine is: β τ τ = cos =. (2) 12 ( ) ( ) cos 2 2 γ 1 τ 32 τ τ 12 τ =. (3) If we know the incline istance R to the object (istance R is efine by measuring the propagation times of the interrogation an reply pulses) Cartesian coorinates of the point P in the 123 =(0x 123 y 123 z 123 ) coorinate system efine as: X = Rcosα 123 Y = Rcos β 123 Z = Rcosγ. (4) To obtain the coorinates of point P in the carrier coorinate system (coorinates of the point P in the =(0xyz) coorinate system see Fig.1) it is necessary realize the corresponing transformation of obtaine coorinates X 123 Y 123 Z 123. Let that the pitch an roll rotations of the receiving antenna take place. The pitch an roll rotations on the pitch angle ξ an on the roll angle ζ of the elemental three-element USBL antenna are shown in Fig.1. After the first rotation on pitch angle ξ relative to B- B' axis the receiving elements are isplacing to the points 1 an 3 respectively. After secon rotation on the angle ζ relatively L-L' axis the receiving elements are isplacing to the points 1 an 3 respectively (see Fig.1). With the rotations of receiving bases the corresponing transformations of coorinate systems from 123 =(0x 123 y 123 z 123 ) to 123=(0x 123y 123z 123) an to 123=(0x 123 y 123 z 123) are have taken place. Calculation expressions for the case of the pitch an roll of the three element receiving antenna with introuce orientation relative to the carrier have been obtaine in [8][9]. So we escribe the calculation proceure here very briefly. If we introuce vectors: p 123=[X 123 Y 123 Z 123] T an p 123=[X 123 Y 123 Z 123] T (vector p 123 represents the 123 coorinates of object in 123 coorinate system an vector p 123 represents the coorinates of object in 123 coorinate system) an the transformation matrix B=B[ζ 123( ) 123( ) 123( )] with irection cosines 123=cos(x 123L) 123=cos(y 123L) y 123.=cos(z 123L) (matrix B transforms vector p 123 to vector p 123) we can write the equation: p = B p. (5) If we introuce vector p 123 =[X 123 Y 123 Z 123 ] T ( vector p 123 represents the coorinates of the object in 123 coorinate system) an the transformation matrix A= A[ ] with irection cosines 123 =cos(x 123 B) 123 =cos(y 123 B) 123 =cos(z 123 B) (matrix A transforms vector p 123 to vector p 123) we can write the equation: -1 ξ A. (6) p = p These two transformations can be combining in the equation: p = A B p. (7) In orer to obtain the coorinates of the object in a carrier coorinates system =(0xyz) it is necessary to make one more rotation of the coorinate system 123 aroun the axis z on the angle of 135 (see Fig.1). For the coorinate system we have the following irection cosines for the z-axis: cos(x z)=0 cos(yz)=0 cos(z z)=1. If we introuce vector p=[x Y Z] T (vector p represents the coorinates of the object in coorinate system) an the transformation matrix C (matrix C transforms vector p to vector p 123 ) the final equation to fin vector p be the next: p = C A B p. (8) Five-element USBL system Articles [8][9] note the problem of the low precision coorinate etermination for the cases when the controlle object is foun in planes of receiving bases of elemental USBL systems. To resolve this problem a five-element USBL system with ifferent spatial orientation of receiving bases was propose. First one more element (element number 4) was ae to the antenna array in the horizontal plane. This element is ae in such a way that a square is forme (the sies of the square form the receiving ISSN: Issue 8 Volume 8 August 2009

4 bases of the antenna). As result we have four basic three-element USBL arrays locate in a horizontal plane. The fifth element (element number 5) is place unerneath the four-element USBL antenna plane exactly unerneath its geometric center in such a way that the four obtaine incline bases woul be the same size as the size of the horizontal receiving bases. In this case the USBL system obtains two aitional three-element basic USBL arrays place in two orthogonal vertical planes. Finally we have the five-element USBL array forme with the six basic three-element USBL arrays with the six ifferent orientations: USBL 123 USBL 234 USBL 341 USBL 412 USBL 153 an USBL 254. (see Fig.2). For this five-element USBL system a special algorithm ha been esigne an investigate an significant coorinate etermination accuracy improvement was obtaine. At the same time the etaile analysis of the algorithm simulation results represente in [9] shows that some imperfections of the moernize USBL system still remain. Thus the algorithm simulation for some orientations of the receiving antenna relative to the object shows significant fluctuations of coorinate etermination errors [9]. The frequency of the time elay counter (160MHz) of the system looks unjustifiably high. The propose in [9] algorithm also has threshol level election sensitivity. In the present paper the aitional increase of the number of the receiving elements is propose (as a result the new receiving array has a larger number of elemental USBL systems with ifferent spatial orientations). In the paper the metho of esign of a nine-element USBL array is consiere. The metho is also allows us to realize the further increase of the number of the receiving elements for the USBL systems. Also the significant improvement of the coorinate etermination algorithm is realize. For example in new algorithm the Z-coorinate sign etermination is realize without utilizing any threshol value. In new multi-element USBL system the frequency of the time elay counter was significantly reuce. 3 Problem Solution 3.1 Nine-element USBL system The propose nine-element USBL array is shown in Fig.3. To emonstrate the evelopment of the USBL array we will use the five-element array as a reference (see Fig.2). Let that the elements of the three-element array an of the five-element array have the same location in the carrier coorinate system (see Figs.12). In this case the rotation aroun the lateral axis L-L' coincies with the rotation to pitch angle ξ an the rotation aroun the longituinal axis B-B' coincies with the rotation to roll angle ζ. The aitional new receiving elements (receiving elements with numbers ) can be forme by means of the rotation first of the three-element USBL 123 array (USBL 123 array is forme with the receiving elements 123) aroun the B-B' axis an then by means of the rotations of other three-element arrays locate in a horizontal plane (the USBL 234 array with the receiving elements 234; the USBL 341 array with the receiving elements 341; an the USB 412 array with the receiving elements 412). We explain in etail only how the USBL 163 array was forme. The element 2 rotates aroun the axis B-B' on angle of 45 an moves to point 6 (6 is also the number of the new receiving element) whereas elements 1 an 3 are fixe. Rotation irection coincies with the curve arrow that inicates the positive irection of pitch angle. The plane of location of the new USBL 163 array is shown in Fig.3 with hatching. The other new three-element USBL arrays (the USBL 274 the USBL 381 an the USBL 492 ) are obtaine in the same way (see Fig.3). The obtaine locus of points (receiving elements 6789) forms the spherical surface of esigne nine-element USBL array. B' x L' B y 123 x 163 B' x L' y 163 B y 123 L L Fig.2. Five-element USBL array Fig.3. Nine-element USBL array ISSN: Issue 8 Volume 8 August 2009

5 3.2 Calculation expressions In the case of the nine-element array the USBL system consists of ten elemental tree-element USBL systems with ten ifferent orientations. These elemental USBL systems are: USBL 123 USBL 234 USBL 341 USBL 412 USBL 153 USBL 254 USBL 163 USBL 274 USBL 381 an USBL 492. We can ivie the three-element USBL systems in three groups: horizontal USBL systems (USBL 123 USBL 234 USBL 341 USBL 412 ) vertical USBL systems (USBL 153 USBL 254 ) an incline USBL systems (USBL 163 USBL 274 USBL 381 an USBL 492 ). The coorinates of the object are etermine iniviually in each elemental USBL system. The horizontal USBL 234 USBL 341 an USBL 412 systems iffer from the examine earlier USBL 123 system in their own values of the pitch an roll angles an in their own angles of rotation of each USBL antenna aroun the z-axis. The coorinate etermination for the vertical USBL 153 system is almost the same as for the USBL 123 system the ifference is that one aitional step is require to reuce the USBL 153 system coorinates to a horizontal plane (by rotation the USBL 153 system on a 90 angle). We have to o the same with coorinates obtaine with the USBL 254 system. The incline USBL 163 system iffers from the USBL 123 only in its aitional inclination on an angle of 45 (see Fig.3). We have the same for the USBL 274 an USBL 234 systems (for USBL 341 an systems USBL 381 ; an for the USBL 412 an USBL 492 systems). So for USBL 163 USBL 274 USBL 381 an USBL 492 systems we have to accomplish the aitional step (to realize the aitional rotation on an angle of 45 for each system). To carry out these aitional rotations for non horizontal USBL arrays we introuce for each system the transformation matrix D (for example for the USBL 163 system matrix D 163 transforms vector p 123 to vector p 163 ). In orer to istinguish the results of the measure coorinates by ifferent basic USBL systems we introuce the following esignations for the measure vectors: p p p p USBL 412 p USBL 153 p USBL 254 USBL 123 p USBL 163 USBL 234 p USBL 274 USBL 341 p USBL 381 p. USBL 492 We also introuce the corresponing inexes for the transformation matrixes for each particular threeelement USBL system. After introucing these esignations the calculation expressions for the horizontal USBL systems will be written as follows: USBL USBL USBL USBL p = C A B p ; p = C A B p ; p = C A B p ; p = C A B p ; (9) where the vector p is the vector that represents USBL 123 the object coorinates (in the carrier coorinate system) obtaine with the USBL 123 system the vector p is the vector obtaine with the with USBL 234 the USBL 234 system an so on. The calculation expressions for coorinate vectors obtaine with the vertical USBL systems are the next: USBL USBL p = C D A B p ; p = C D A B p. (10) The calculation expressions for coorinate vectors obtaine with the incline USBL systems are written in the following way: USBL USBL USBL USBL p = C D A B p ; p = C D A B p ; p = C D A B p ; p = C D A B p. (11) 3.3 Algorithm escription It is assume that the measure values are: an pitch an roll angles of the receiving nine-element antenna (see Fig.3); t interrogation pulse an response pulse separation; time elays for receiving bases of the corresponing USBL 123 USBL 234 USBL 341 USBL 412 USBL 153 USBL 452 USBL 163 USBL 274 USBL 381 an USBL 492 systems (twenty time elays are measure to provie the positive values of time elays the secon-inexe outputs are inverte). The time elays can be measure by the stanar igital metho (with the infilling of the interval corresponing to time elay with the high-frequency impulses). The time elays on each base can be expresse in number of impulses as follows: ISSN: Issue 8 Volume 8 August 2009

6 n ij =f c ( ij +T/2) where f c is the frequency of the time elay counter ij is the time elay between the receiving elements of the USBL array (j is the inex of the common receiving element) T is the perio of the transponer pulse carrier frequency. First the vector p 153=[X 153Y 153Z 153] T ) is calculate an the sign of the Z 153 coorinate is etermine. The coorinates X 153Y 153Z 153 are calculate in accorance with the calculation formulas for Cartesian coorinates in the 153=(0x 153 y 153 z 153) coorinate system. The sign of the Z 153 coorinate is efine by utilizing the time elay values obtaine for the USBL 452 system (values ). If 45 > 25 the sign of the Z 153 coorinate is assume to be negative. If is assume the Z 153 coorinate is positive. With the obtaine values of the vector p 153=[X 153Y 153 Z 153] T the values of the time elays ' 25 an ' 45 are calculate for the USBL 452 system. The moules of the ifferences 1= ' an 2= ' are then calculate (the values 25 an 45 are obtaine through measurement; an the values of ' 25 an ' 45 are calculate). Then the value 1=( ) 0.5 is calculate. The same proceure is repeate with the opposite sign of Z 153 coorinate (with calculation of the corresponing values of ' 25 an ' 45 the ifferences 1= ' an 2= ' an the corresponing value 2). If 1> 2 the latter sign is assume as correct. Otherwise the initial sign value of Z 153 coorinate is assume as correct. Then the vector p 254=[X 254 Y 254Z 254] T is calculate an the sign of the Z 254 coorinate is etermine. To efine the sign of the Z 254 coorinate the proceure is escribe above is applie to the vector p 254=[X 254 Y 254Z 254] T. The ifference is that in this time the measure time elays 15 an 35 are utilize to compare with the calculate values ' 15 an ' 35 for the ifferent signs of the Z 254 coorinate. The important part of the algorithm is solving the problem of the low precision of coorinate etermination in cases when an object is foun in the plane (or near to the plane) of the receiving bases. This problem was investigate in etail in [9] an it was foun that the ata obtaine with some elemental USBL system can be utilize in calculation if the latitue angle to object is more than 10. Otherwise (if the latitue angle to the object relative to plane of the measuring antenna is equal or less than 10 ) the calculate values are iscare an o not participate in the calculation of the final means of the coorinates of the object. Through the next stage of the algorithm is the calculation of the values of the angles between the planes of the measuring three-element antennas an the irections to the object (latitue angles to object). For that the coorinate vectors in the spherical coorinate can be expresse as follows: ς ξ ς ξ ς T ( R ψ ϕ ) ς ξ ς ξ ς T q = ; q = ( R ψ ϕ ) (12) where are the polar an azimuth angles in the USBL 153 spherical coorinate system an are the polar an azimuth angles in the USBL 254 spherical coorinate system. The values of polar angles ( 153 an 254) for each USBL system efine the ecision to utilize or no utilize this system in the calculation of coorinate means. So if the values of the corresponing polar angles of both systems are foun outsie [ ] iapason the values of both systems are utilize. If the value of the polar angle of one of the USBL systems lies outsie the iapason [ ] an the value of the polar angle of the other system belongs to the [ ] then the value of the first system is utilize an the value obtaine from another system is not taken into account. If the values of the polar angles of both systems are foun within [ ] iapason the coorinates obtaine with the both vertical USBL system are iscare. The case when the values of the polar angles for the both vertical USBL systems lie within [ ] iapason correspons to the location of the object in the narrow cone uner the USBL array an are measure with goo accuracy with the horizontal an incline elemental USBL arrays. The final step in the coorinate etermination of this part of the algorithm is the calculation (if it is necessary) of the Cartesian coorinates of the object in the coorinate system of the carrier (calculation of the vectors p an p ). These calculations USBL 153 USBL 254 for the USBL 153 an USBL 254 systems are carrie out accoring to the formulas (10). In the secon stage of the algorithm the object coorinates are calculate using the time elays obtaine by the horizontal measuring systems (USBL 123 USBL 234 USBL 341 USBL 412 ). It is also suppose that the receiving USBL array can have pitch an roll inclinations. We consier the USBL 123 system in orer to escribe this proceure (for the other horizontal USBL systems this part of the algorithm functions in the same way). First we calculate the coorinates of the object base on the elays measure with the USBL 123 system (p 123=[X 123 Y 123 Z 123] T ) an then the sign of the Z 123 coorinate is etermine. The sign of the Z 123 coorinate is obtaine using the values of time ISSN: Issue 8 Volume 8 August 2009

7 elays of the USBL 153 system an the values of the time elays of the USBL 254 system. For the horizontal USBL systems we initially assume that the value of the Z coorinate (for the USBL 123 system it is the Z 123 coorinate) is positive. Just as in the first stage of the algorithm the values of the components of the vector p 123=[X 123 Y 123 Z 123] T allow us to calculate the values of the elays ' 15 an ' 35 for the USBL 153 system an the values of elays ' 25 an ' 45 for the USBL 254 system. Next we calculate the absolute values of the following ifferences: 1= ' = ' = ' an 4= ' (the values an 45 are obtaine through the measurement). Now we calculate the geometric mean 1 of the an 4 ( 1=( ) 0.5 ). The same proceure is repeate with opposite sign esignation for the coorinate Z 123. The geometric mean in this case will be 2. The coorinates corresponing to a less geometric mean are taken as correct. The same proceure is applie to the USBL 234 USBL 341 USBL 412 systems. The next stage of the algorithm is solving the problem of the low precision of coorinate etermination in cases when an object is foun in the plane (or near to the plane) of the receiving horizontal bases. For that we calculate the spherical coorinates: ς q = ; ξ ς ξ ς T ( R ψ ϕ ) ς ξ ς ξ ς T q = ( R ψ ϕ ) ; ς q = ; ξ ς ξ ς T ( R ψ ϕ ) ς ξ ς ξ ς T q = ( R ψ ϕ ) ; (13) If the values of the polar angle of some USBL systems lie within the iapason [ ] the calculate values are iscare. If the values of the polar angle of the analyze USBL systems lay outsie of the iapason [ ] the corresponing Cartesian coorinates of these systems are consiering as reliable an the values of the object coorinates ( p p p an p ) USBL 123 USBL 234 USBL 341 USBL 412 in the carrier coorinate system are calculate accoring to the formulas (9). In the thir stage of the algorithm the object coorinates are calculate using the time elays obtaine by the incline measuring systems (USBL163 USBL274 USBL381 an USBL492). We consier the USBL 163 system in orer to escribe this proceure (for the other incline USBL systems this part of the algorithm functions in the same way). First we calculate the coorinates of the object base on the elays measure with the USBL 163 system (p 163=[X 163 Y 163 Z 163] T ) an then the sign of the Z 163 coorinate is etermine. The sign of the Z 163 coorinate just as in the case of the horizontal systems is obtaine using the values of time elays of the USBL 153 system an the values of the time elays of the USBL 254 system. For the incline USBL systems we initially also assume that the value of the Z coorinate (for the USBL 163 system it is the Z 163 coorinate) is positive. Just as in the first an the secon stages of the algorithm the values of the components of the vector p 163=[X 163 Y 163 Z 163] T allow us to calculate the values of the elays ' 15 an ' 35 for the USBL 153 system an the values of elays ' 25 an ' 45 for the USBL 254 system. Next we calculate the absolute values of the following ifferences: 1= ' = ' = ' an 4= ' (the values an 45 are obtaine through the measurement). Now we calculate the geometric mean 1 of the an 4. The same proceure is repeate with opposite sign esignation for the coorinate Z 163. The geometric mean in this case will be 2. The coorinates corresponing to a less geometric mean are taken as correct. The same proceure is applie to the USBL274 USBL381 an USBL492 systems. The next stage of the algorithm is the calculation of the spherical coorinates of the object for the elemental incline USBL systems (for solving the accuracy problem when the object is foun in the plane or near the plane of some incline array). The coorinate vectors in the spherical coorinate can be expresse as follows: ς q = ; ξ ς ξ ς T ( R ψ ϕ ) ς ξ ς ξ ς T q = ( R ψ ϕ ) ; ς q = ; ξ ς ξ ς T ( R ψ ϕ ) ς ξ ς ξ ς T q = ( R ψ ϕ ) (14) If the values of the polar angle of some USBL systems lie within the iapason [ ] the calculate values are iscare. If the values of the polar angle of the analyze USBL systems lay outsie of the iapason [ ] the corresponing Cartesian coorinates of these systems are consiering as reliable an the values of the object coorinates ( p p p p ) in USBL 163 USBL 274 USBL 381 USBL 492 the carrier coorinate system are calculate accoring to the formulas (11). The last step of the algorithm implies the calculation of the means of the object coorinates in the carrier coorinate system with reliable ata obtaine by the elemental USBL systems: p=(xyz) T = mean ( p p p p p p USBL 153 p USBL 163 USBL 254 p USBL 274 USBL 123 p USBL 381 USBL 234 p ). USBL 492 USBL 341 USBL 412 ISSN: Issue 8 Volume 8 August 2009

8 3.4 Simulation results For the algorithm simulation a special computer program was esigne. During the simulation of the algorithm it was assume that the istance to the object an the pitch an roll angles were being measure precisely. We assume that the measurement of the time elays is provie by utilizing the binary counters an the signal-to-noise ratio (SNR) on the inputs of receiving elements an signal reception conitions allow us to measure the time elays without errors. It is also suppose that the accuracy of measurement of the time elays is limite by the clock rive frequency of the time elay counters. The ifferent values of the horizontal istance to the object the epth of the object the azimuth angle an the pitch an roll angles were utilize for moeling the ifficult conitions to measure the object coorinates with high accuracy (cases when an object is foun in the plane of the horizontal receiving bases or near to the plane of the horizontal receiving bases). The computer simulation of algorithm will estimate the instrumental precision of the USBL system. Let X Y Z be the true values of the coorinates of the object in the carrier coorinate system =(0xyz). Let R be the true incline istance to the object. Let X USBL Y USBL Z USBL be the values of the coorinates obtaine by applying of the evelope algorithm (the coorinates of the object are calculate utilizing the expression (9) (10) an (11) for USBL 153 USBL 254 USBL 123 USBL 234 USBL 341 USBL 412 USBL163 USBL274 USBL381 an USBL492 systems). The values of the true errors of etermination of the coorinates are: X=X USBL -X; Y= Y USBL -Y Z= Z USBL -Z. The values of the relative true errors of the coorinates are: X/R; Y/R; Z/R. In process of the simulation the azimuth angle is changing clockwise (if looking own on the horizontal plane see Figs.13) from 0 to 360 in the (xy) coorinate plane (zero reaing is coincie with x-axis of the =(0xyz) carrier coorinate system see Fig.1). The other parameters of algorithm simulation have following values: the spee of the soun in the water c=1500m/s; the size of receiving bases =0.056m; transponer pulse carrier frequency f=11khz (operating frequency of USBL system); the frequency of the time elay counter f c =25MHz. The results of esigne algorithm simulation are shown in Figs.4-8. First we will consier the case when the receiving antenna oes not have any inclination an the object is locate in the horizontal istance of 100 meters an the relative epth is 5 meters. The angular position of the object (azimuth angle ) is changing with the step of 1. The results of the algorithm simulation for the examining case (R=100m; Z=5 m; =0 ; = 0 ) are shown in Fig.4. In the absence of the pitch an roll the moulus of the latitue angle to the object (in graphs this angle is esignate as ) shoul be invariable an the value of the latitue angle is approximately It means that the horizontal USBL systems are not participate in the calculation of coorinate means an the maximum number of systems that are taken into account in this case is 6 (N=6). In the relatively lengthy iapasons of azimuth angle changing the number of utilize elemental USBL is 6 (see Fig.4). In other azimuth angle iapasons the number of utilize USBL systems varies from 3 to 5. We will consier in etail the behavior of the function N=N( ) only in the iapason of changing of the angle from 0 to 90. In the other iapasons of ([ ] [ ] [ ]) the behavior of the function N=N( ) will be analogous. In the azimuth angle range from 0 to 10 the number of the utilize USBL system is 3 (N=3) an the systems that are utilize in the calculation of the coorinate means are: USBL 153 USBL 163 an USBL 381 (see Figs.34). For the =11 the number of use systems is 4 (N=4 utilize systems are: USBL 153 USBL 163 USBL 381 an USBL 452 ). The system USBL 452 is utilize in the calculation of coorinate means because the azimuth angle is more than 10. For the USBL 452 system (the case: = 0 ) the azimuth angle also can be interprete as the altitue angle to the object relative to the plane of the 10 4 ( X/R) 10 4 ( Y/R) 10 4 ( Z/R) N N 10 4 ( Y/R) ( X/R) 10 4 ( Z/R) eg. Fig.4. Variation of the relative errors of the object coorinates number of the elemental USBL systems N that are use for calculation of coorinate means an moulus of the latitue angle to the transponer (object) relative to the USBL 1234 plane; R=100m; Z=5 m; =0 ; = 0. ISSN: Issue 8 Volume 8 August 2009

9 USBL 452 three-element array. With the further increasing of the the altitue angle to object relative to the USBL274 array became more then 10 an this system (USBL274 system) is also begun to utilize for calculation of the coorinate means (on the graph of N this is the case when the azimuth angle belongs to the interval [12-17 ]). Thus if the angle belongs to the interval [12-17 ] an in the calculation of the coorinate means are utilize next USBL systems (N=5): USBL 153 USBL 452 USBL 163 USBL 274 an USBL 492. In the interval [18-72 ] both vertical (USBL 153 an USBL 452 ) systems an all incline (USBL 163 USBL 274 USBL 381 an USBL 492 ) systems are utilize. In iapason [73-78 ] the 5 systems (N=5) are utilize: USBL 153 USBL 452 USBL 274 USBL 381 an USBL 492. In ifference from the iapason [12-17 ] in the interval [73-78 ] the USBL 492 system is utilize instea of the USBL 163 system. For the angle =79 the following four (N=4) systems are use: USBL 153 USBL 452 USBL 274 an USBL 492. In the iapason [80-90 ] the next three (N=3) systems are utilize: USBL 452 USBL 274 an USBL 492. The behavior of the function N=N( ) is repeate in the other three 90 -sectors. The utilization of the ifferent elemental USBL arrays is efine accoring to algorithm escribe above. The relative errors of all three object coorinates ( X/R Y/R Z/R) have not exceee the threshol of 0.2% of incline istance to the object. The behavior of the relative errors is also similar within each 90 -sector. The greatest errors take place for the Z-coorinate of the object. The relative errors of Z-coorinate are reache the values of 0.15% whereas the errors of X- an Y-coorinates not exceee the values of 0.1%. Let that the USBL array has some inclination (pitch an roll angles are not zeros). The simulation results for these cases are shown in Figs.5-8. From Fig.5 (R=100m; Z=15m; =5 ; = 6 ) it is seen that the relative errors ( X/R Y/R Z/R) of all three coorinates have not exceee the threshol of 0.2% of incline istance to the object. The values of the relative errors X/R an Y/R are less than 0.1% in all examine iapason of. The number of the utilizing elemental systems (N) is varying from 3 to 10. From the graphs it is seen that if the the number of the utilize system is less or equal to 6. If the >10 the number of the utilize systems (N) is more than 6 an the exact number is efine by the values of altitue angle for the other three-element array planes. The graphs in Fig.6 illustrate the variation of relative errors of object coorinates for the case when R=100m; Z=40m; = 20 ; =10. The relative location of the measuring system an the object (with preetermine spatial orientation of receiving antenna) efines the case of significant inclination of receiving antenna an when the object can be foun in the plane of horizontal receiving bases of USBL system. It is seen that the relative errors o not excee the threshol of 0.15% of incline istance in all iapason of changing azimuth angle. Also it is seen that when the moulus of altitue angle is more than 10 the number of the utilize three-element systems is foun in interval between 7 an 10 (7 N 10) ( X/R) 10 4 ( Y/R) 10 4 ( Z/R) N N 10 4 ( Y/R) ( X/R) 10 4 ( Z/R) eg ( X/R) 10 4 ( Y/R) 10 4 ( Z/R) N N ( Y/R) 10 4 ( X/R) 10 4 ( Z/R) eg. Fig.5. Variation of the relative errors of the object coorinates number of the elemental USBL systems N that are use for calculation of coorinate means an moulus of the latitue angle to the transponer (object) relatively the USBL 1234 plane; R=100m; Z=15 m; =5 ; = -6. Fig.6. Variation of the relative errors of the object coorinates number of the elemental USBL systems N that are use for calculation of coorinate means an moulus of the latitue angle to the transponer (object) relatively the USBL 1234 plane; R=100m; Z=40 m; = -20 ; = 10. ISSN: Issue 8 Volume 8 August 2009

10 The results of the calculation of the relative errors for the significant inclination angles of the receiving antenna are shown in Fig.7 an Fig.8. In Fig.7 the pitch an roll angles have values: = 30 ; = 20 ; in the Fig.8 the pitch an roll angles are: = 40 ; = 40. From the presente curves we can note that as before the relative errors o not excee the threshol of 0.15% of incline istance in all iapason of changing azimuth angle. In the case of the big epth (Z=150m see Fig.8) the values of the relative errors o not excee the threshol of 0.12% of incline istance. It is necessary to notice that with the increasing of the relative epth (Z) the values of the relative errors Z/R are ecreasing (the mean of the relative errors is isplace from the negative values to zero see Figs.4-8). The number of utilizing USBL systems as before epens on the latitue angle moulus of the horizontal arrays an on the latitue angles of the other systems. If the number of the utilize systems is varies from 3 to 6 if >10 the number of the utilize systems is varies from 7 to 10. The esigne algorithm has been examine for various relative locations of the measuring system an object (object location in the lower hemisphere was consiere the maximum horizontal istance was assume to be 100m). The values of pitch an roll angles ξ an ζ are assume to be in the range from 40 to +40. The computer simulation emonstrate the reliable operation of the esigne algorithm for all teste angular antenna positions an verifie locations of object. The results of the 10 4 ( X/R) 10 4 ( Y/R) 10 4 ( Z/R) N N ( Y/R) 10 4 ( X/R) 10 4 ( Z/R) eg. Fig.7. Variation of the relative errors of the object coorinates number of the elemental USBL systems N that are use for calculation of coorinate means an moulus of the latitue angle to the transponer (object) relatively the USBL 1234 plane; R=100m; Z=50 m; = -30 ; = ( X/R) 10 4 ( Y/R) 10 4 ( Z/R) N N 10 4 ( X/R) ( Y/R) 10 4 ( Z/R) eg. Fig.8. Variation of the relative errors of the object coorinates number of the elemental USBL systems N that are use for calculation of coorinate means an moulus of the latitue angle to the transponer (object) relatively the USBL 1234 plane; R=100m; Z=150 m; = -40 ; = -40. calculation of the errors of the etermination of the coorinates of the object show that the relative true errors of the coorinates have values less than 0.2% of the slant istance to the object. It is also necessary to mention that in a wie range of istances epths pitch an roll angles the values of true relative errors were less than 0.1%. 4 Conclusion In this article we have investigate the approach for esign of a multi-element USBL system. A case for the esign of a nine-element USBL system is presente. The paper focuse on the problem of improving the precision of coorinate etermination in conitions when the location of the object in the lower hemisphere is arbitrary an the receiving antenna can have significant inclinations. In this article this problem has been solve by means of increasing the number of elemental three-element USBL arrays an exploiting their ifferent spatial orientations. The propose algorithm allows us to accomplish the selection of reliable elemental USBL arrays utilizing the analysis of the values of latitue angles to the object for each elemental array. The presente algorithm is a significantly moifie an improve version of the algorithm esigne for the five-element USBL system [9]. In particular the logical part of algorithm has been notably moifie an the algorithm threshol epenence of the Z- coorinate sign etermination was eliminate. ISSN: Issue 8 Volume 8 August 2009

11 Algorithm simulation was realize for ifferent USBL systems an object mutual positions in wie range of pitch an roll angles of the receiving array (pitch an roll angles ξ an ζ are assume to be in the range from -40 to +40 ). For all teste angular receiving array positions an object locations the esigne algorithm showe reliable operation. The accuracy of coorinate etermination for the propose nine-element USBL system can be evaluate as the 0.2% of slant istance to the object. References: [1] P.H. Milne Unerwater Acoustic Positioning System Gulf Publishing Company [2] K. Vickery Acoustic positioning systems A practical overview of current systems. Proceeings of the 1998 Workshop on Autonomous Unerwater Vehicles 1998 p [3] M. Watson C. Loggins Y.T. Ochi A New High Accuracy Super Short Base Line (SSBL) System Unerwater Technology Proceeings of 1998 International Symposium 1998 pp [4] D. Thomson S. Elson New generation acoustic positioning systems Proceeings of Oceans '02 MTS/IEEE Vol pp [5] F.V.F. Lima C.M. Furukawa Development an Testing of an Acoustic Positioning System - Description an Signal Processing Ultrasonics Symposium Proceeings 2002 IEEE Vol pp [6] P.P.J. Beaujean A.I. Mohame R. Warin Acoustic Positioning Using a Tetraheral Ultrashort Baseline Array of an Acoustic Moem Source Transmitting Frequency-hoppe Sequences The Journal of the Acoustical Society of America Vol.121 No pp [7] D.R.C. Philip An Evaluation of USBL an SBL Acoustic Systems an the Optimisation of Methos of Calibration - Part 1 The Hyrographic Journal No pp [8] M. Arkhipov A Coorinate Determination Algorithm for USBL Systems. Proceeings of the 2n WSEAS International Conference on CIRCUITS SYSTEMS SIGNALS an TELECOMMUNICATIONS (CISST 08) Acapulco Mexico 2008 pp [9] M. Arkhipov An Algorithm for Improving the Accuracy of Z-Coorinate Determination for USBL Systems. WSEAS TRANSACTIONS on SYSTEMS Vol. 7 Num pp ISSN: Issue 8 Volume 8 August 2009