CN106483501A - One kind is based on DOP value analytical acoustics alignment system multiple-answering machine optimal distribution method - Google Patents

Abstract

The invention discloses a kind of be based on DOP value analytical acoustics alignment system multiple-answering machine optimal distribution method.A kind of method and technology field the invention belongs to acoustic positioning system multiple-answering machine is structured the formation, and in particular to transponder space geometry optimal distribution method suitable for multiple-answering machine acoustic positioning system.This method establishes a kind of idea and method good and bad from space geometry angle quantitative analysis multiple-answering machine layout；The theory can be directly used for the calibration laid with transponder location of transponder arrays.And by taking trisponder as an example, labor is carried out to space geometry optimal location thinking, the optimal location scheme of trisponder battle array in planar rondure region has been obtained using the analysis of our standard measure.

Description

One kind is based on DOP value analytical acoustics alignment system multiple-answering machine optimal distribution method

Technical field

The method and technology field the invention belongs to acoustic positioning system multiple-answering machine is structured the formation, and in particular to a kind of applicable Transponder space geometry optimal distribution method in multiple-answering machine acoustic positioning system.

Background technology

For acoustic positioning system, when having three transponders even more multiple-answering machine can be used, should Answer relative geometry position relation between the space geometry formation arrangement of device and transponder and basic matrix all and positioning accurate Degree has certain relation.Particularly with the acoustic positioning system crossed as positioning principle with sphere, multiple-answering machine Space geometry optimal distribution conceptual design is significant to improving overall precision.

With the acoustic localization technique development of navigation field and application under water, have developed including long base both at home and abroad Line, short baseline and ultra-short baseline are in interior multiple acoustic positioning system typical products, acoustic positioning system development Unit is absorbed in the product for developing high-performance (sensor error is little), but still lacks one kind and quantitatively can divide The good and bad method of analysis multiple-answering machine space geometry layout, fails, in the case that sensor error is certain, to pass through The formation optimization of multiple-answering machine is come the method that reduces acoustic positioning system systematic error.

Therefore, need badly and develop a kind of transponder space geometry optimum suitable for multiple-answering machine acoustic positioning system Structure the formation method such that it is able to which quantitative analysis multiple-answering machine space geometry layout is good and bad,

Content of the invention

The technical problem to be solved in the present invention is to provide a kind of acoustic positioning system multiple-answering machine space geometry optimum Structure the formation method, from the quality of the quantitative analysis multiple-answering machine layout of space geometry angle, and then it is many to obtain one kind The optimized design that transponder formation lays, reduces acoustic positioning system systematic error to a certain extent.

In order to realize this purpose, the present invention is adopted the technical scheme that：

One kind is based on DOP value analytical acoustics alignment system multiple-answering machine optimal distribution method, comprises the following steps：

(1) using azimuth angle alpha and the height angle beta of ultra short baseline locating system USBL output, DOP is set up Observing matrix A：

Wherein, α i and β i represent azimuth and the elevation angle of i-th transponder, i=1～n, n=3 respectively；

Do not consider the time difference of USBL measurement, set up the site error least square solution of 3 transponders：

Wherein Δ ρ is observation noise vector, and its variance is σ²；

Site error variance after simplification is：

(2) the weight coefficient matrix Q of 3 transponder DOP observational equations is set up：

(3) computing formula for defining three kinds of DOP values of multiple-answering machine USBL positioning according to formula 0 is as follows：

Horizontal geometric dilution of precision (Horizontal Dilution of Precision, HDOP)：

Vertical dilution of precision (Vertical Dilution of Precision, VDOP)：

Three-dimensional geometry dilution of precision (Positional Dilution of Precision, PDOP)：

(4) determine the relative position relation of trisponder in planar rondure region

(4.1) elevation angle and azimuth in observing matrix are determined in planar rondure region

Radius is height angle beta in observing matrix in the planar rondure region of R₁,β₂,β₃All it is set as 0, if three Individual transponder is followed successively by α with respect to the azimuth of basic matrix₁,α₂,α₃；

DOP value observing matrix is：

Weight coefficient matrix is：

Wherein：

(4.2) horizontal geometric dilution of precision HDOP value in analysis planar rondure region, by non-zero in 0 Take out, constitute in plane and simplify HDOP weight coefficient matrix Q_H：

Wherein：

Then HDOP is

Optimal location needs to set A=Amax, α₁+α₂+α₃=360 °, obtain differing all when three azimuths During for 120 °, radius can be obtained for the optimal location in the planar rondure region of R, so that it is determined that three responses Device in the length of side is respectivelyThree summits of equilateral triangle on；

Determine that the azimuth of three in same level transponder is followed successively by conjunction with 0 and 0： + 240 ° of+120 ° of α, α, α, in the case that in three transponders of guarantee, azimuth difference is 120 ° between any two, α is unspecified angle, and HDOP value is unaffected；

(5) determine the optimal solution of border circular areas inside radius and depth relationship

On the premise of optimal distribution scheme in step (4) planar rondure region, USBL absolute fix is solved Three-dimensional geometry dilution of precision optimum point；

Three summits for taking the inscribe equilateral triangle of the circle that radius is R are the placement of transponder, three Azimuth is followed successively by α₁=0 °, α₂=120 °, α₃=240 °, calculate from mathematics geometrical relationship and determine, optimal solution is round On axial direction, the height h which arrives a transponder in each planar rondure region is equal；Make elevation angle β₁=β₂=β₃=β；Wherein,

DOP value observing matrix is：

Weight coefficient matrix is：

Plane positioning dilution of precision is：

Vertical dilution of precision is：

Three-dimensional localization dilution of precision is：

Optimal location needs to set PDOP value minimum, abbreviation above formula：

Can obtain that and if only ifI.e.When, PDOP value obtains minimum of a value, now for determining circle Shape region inside radius and the optimal solution of depth relationship.

Further, as above a kind of based on DOP value analytical acoustics alignment system multiple-answering machine optimum cloth Battle array method, when number of transponders is more than 3, takes n ＞ 3, is solved using the method for step (1)～(5) Optimal distribution.

The beneficial effects of the present invention is, one kind is established from space geometry angle quantitative analysis multiple-answering machine cloth The good and bad idea and method of office；The theory can be directly used for the calibration laid with transponder location of transponder arrays. And by taking trisponder as an example, labor is carried out to space geometry optimal location thinking, using our standard measure Analysis obtained the optimal location scheme of trisponder battle array in planar rondure region.Additionally, the method also has Help improve the absolute position stated accuracy of transponder.

Specific embodiment

DOP value analytical acoustics alignment system multiple-answering machine is based on reference to specific embodiment is a kind of to the present invention Optimal distribution method is described in detail.The method is comprised the following steps：

(1) using azimuth angle alpha and the height angle beta of ultra short baseline locating system USBL output, DOP is set up Observing matrix A：

Wherein, α i and β i represent azimuth and the elevation angle of i-th transponder, i=1～n, n=3 respectively；

Do not consider the time difference of USBL measurement, set up the site error least square solution of 3 transponders：

Wherein Δ ρ is observation noise vector, and its variance is σ²；

Site error variance after simplification is：

(2) the weight coefficient matrix Q of 3 transponder DOP observational equations is set up：

(3) computing formula for defining three kinds of DOP values of multiple-answering machine USBL positioning according to formula 0 is as follows：

Horizontal geometric dilution of precision (Horizontal Dilution of Precision, HDOP)：

Vertical dilution of precision (Vertical Dilution of Precision, VDOP)：

Three-dimensional geometry dilution of precision (Positional Dilution of Precision, PDOP)：

(4) determine the relative position relation of trisponder in planar rondure region

(4.1) elevation angle and azimuth in observing matrix are determined in planar rondure region

Radius is height angle beta in observing matrix in the planar rondure region of R₁,β₂,β₃All it is set as 0, if three Individual transponder is followed successively by α with respect to the azimuth of basic matrix₁,α₂,α₃；

DOP value observing matrix is：

Weight coefficient matrix is：

Wherein：

(4.2) horizontal geometric dilution of precision HDOP value in analysis planar rondure region, by non-zero in 0 Take out, constitute in plane and simplify HDOP weight coefficient matrix Q_H：

Wherein：

Then HDOP is

Optimal location needs to set A=Amax, α₁+α₂+α₃=360 °, obtain differing all when three azimuths During for 120 °, radius can be obtained for the optimal location in the planar rondure region of R, so that it is determined that three responses Device in the length of side is respectivelyThree summits of equilateral triangle on；

Determine that the azimuth of three in same level transponder is followed successively by conjunction with 0 and 0： + 240 ° of+120 ° of α, α, α, in the case that in three transponders of guarantee, azimuth difference is 120 ° between any two, α is unspecified angle, and HDOP value is unaffected；

(5) determine the optimal solution of border circular areas inside radius and depth relationship, in step (4) planar rondure area In domain on the premise of optimal distribution scheme, USBL absolute fix three-dimensional geometry dilution of precision optimum point is solved；

The position that i.e. analysis basic matrix positional precision optimal solution is located, is favorably improved the absolute position of each transponder Stated accuracy.On the premise of optimal distribution scheme in planar rondure region, USBL absolute fix three is solved Dimension geometric dilution of precision optimum point.This optimum point analysis can to USBL transponder lay using with And demarcation is provided fundamental basis, the such as high-precision calibration of transponder initial position, or be R in radii fixus Border circular areas in transponder should lay many deep (h) relatively small PDOP value could be obtained.

Three summits for taking the inscribe equilateral triangle of the circle that radius is R are the placement of transponder, three Azimuth is followed successively by α₁=0 °, α₂=120 °, α₃=240 °, calculate from mathematics geometrical relationship and determine, optimal solution is round On axial direction, the height h which arrives a transponder in each planar rondure region is equal；Make elevation angle β₁=β₂=β₃=β；Wherein,

DOP value observing matrix is：

Weight coefficient matrix is：

Plane positioning dilution of precision is：

Vertical dilution of precision is：

Three-dimensional localization dilution of precision is：

Optimal location needs to set PDOP value minimum, abbreviation above formula：

Can obtain that and if only ifI.e.When, PDOP value obtains minimum of a value, now for determining circle Shape region inside radius and the optimal solution of depth relationship.

When number of transponders is more than 3, n ＞ 3 is taken, solved most using the method for step (1)～(5) Excellent structure the formation so as to the range of application for extending this method, the theory can be directly used for laying and response for transponder arrays The calibration of device position.

Claims (2)

1. a kind of based on DOP value analytical acoustics alignment system multiple-answering machine optimal distribution method, its feature exists In comprising the following steps：

(1) using azimuth angle alpha and the height angle beta of ultra short baseline locating system USBL output, DOP is set up Observing matrix A：

$A=\left[\begin{array}{ccc}\cos \left({\mathrm{\β}}_1\right)\sin \left({\mathrm{\α}}_1\right)& \cos \left({\mathrm{\β}}_1\right)\cos \left({\mathrm{\α}}_1\right)& \sin \left({\mathrm{\β}}_1\right)\\ \cos \left({\mathrm{\β}}_2\right)\sin \left({\mathrm{\α}}_2\right)& \cos \left({\mathrm{\β}}_2\right)\cos \left({\mathrm{\α}}_2\right)& \sin \left({\mathrm{\β}}_2\right)\\ \·& \·& \·\\ \·& \·& \·\\ \·& \·& \·\\ \cos \left({\mathrm{\β}}_n\right)\sin \left({\mathrm{\α}}_n\right)& \cos \left({\mathrm{\β}}_n\right)\cos \left({\mathrm{\α}}_n\right)& \sin \left({\mathrm{\β}}_n\right)\end{array}\right]---\left(1\right)$

Wherein, α_iAnd β_iRepresent azimuth and the elevation angle of i-th transponder, i=1～n, n=3 respectively；No The time difference of consideration USBL measurement, set up the site error least square solution of 3 transponders：

$\begin{array}{c}\mathrm{\Δ}\hat{x}=A^{-1}\mathrm{\Δ}\mathrm{\ρ}\\ \mathrm{\Δ}\hat{x}={\left(A^{T}A\right)}^{-1}{A}^T\mathrm{\Δ}\mathrm{\ρ}\end{array}---\left(2\right)$

Wherein Δ ρ is observation noise vector, and its variance is σ²；

Site error variance after simplification is：

$E\[\mathrm{\Δ}\hat{x}\mathrm{\Δ}{\hat{x}}^T\]={\mathrm{\σ}}^2\·{\left(A^{T}A\right)}^{-1}---\left(3\right)$

(2) the weight coefficient matrix Q of 3 transponder DOP observational equations is set up：

$Q={\left(A^{T}A\right)}^{-1}=\left[\begin{array}{ccc}{Q}_{11}& Q_{12}& Q_{13}\\ Q_{21}& Q_{22}& Q_{23}\\ Q_{31}& Q_{32}& Q_{33}\end{array}\right]---\left(4\right)$

(3) computing formula for defining three kinds of DOP values of multiple-answering machine USBL positioning according to formula (4) is as follows：

Horizontal geometric dilution of precision HDOP：

$HDOP=\sqrt{Q_{11}+Q_{22}}---\left(5\right)$

Vertical dilution of precision VDOP：

$VDOP=\sqrt{Q_{33}}---\left(6\right)$

Three-dimensional geometry dilution of precision PDOP：

$PDOP=\sqrt{{\mathrm{HDOP}}^2+{\mathrm{VDOP}}^2}=\sqrt{Q_{11}+Q_{22}+Q_{33}}---\left(7\right)$

(4) determine the relative position relation of trisponder in planar rondure region

(4.1) elevation angle and azimuth in observing matrix are determined in planar rondure region

Radius is height angle beta in observing matrix in the planar rondure region of R₁,β₂,β₃All it is set as 0, if three Individual transponder is followed successively by α with respect to the azimuth of basic matrix₁,α₂,α₃；

DOP value observing matrix is：

$A=\left[\begin{array}{ccc}\cos \left({\mathrm{\β}}_1\right)\sin \left({\mathrm{\α}}_1\right)& \cos \left({\mathrm{\β}}_1\right)\cos \left({\mathrm{\α}}_1\right)& \sin \left({\mathrm{\β}}_1\right)\\ \cos \left({\mathrm{\β}}_2\right)\sin \left({\mathrm{\α}}_2\right)& \cos \left({\mathrm{\β}}_2\right)\cos \left({\mathrm{\α}}_2\right)& \sin \left({\mathrm{\β}}_2\right)\\ \cos \left({\mathrm{\β}}_3\right)\sin \left({\mathrm{\α}}_3\right)& \cos \left({\mathrm{\β}}_3\right)\cos \left({\mathrm{\α}}_3\right)& \sin \left({\mathrm{\β}}_3\right)\end{array}\right]=\left[\begin{array}{ccc}{\mathrm{sin\α}}_1& {\mathrm{cos\α}}_1& 0\\ {\mathrm{sin\α}}_2& {\mathrm{cos\α}}_2& 0\\ {\mathrm{sin\α}}_3& {\mathrm{cos\α}}_3& 0\end{array}\right]---\left(8\right)$

Weight coefficient matrix is：

$\begin{array}{c}Q={\left(A^{T}A\right)}^{-1}={\left(\left[\begin{array}{ccc}{\mathrm{sin\α}}_1& {\mathrm{sin\α}}_2& {\mathrm{sin\α}}_3\\ {\mathrm{cos\α}}_1& {\mathrm{cos\α}}_2& {\mathrm{cos\α}}_3\\ 0& 0& 0\end{array}\right]\left[\begin{array}{ccc}{\mathrm{sin\α}}_1& {\mathrm{cos\α}}_1& 0\\ {\mathrm{sin\α}}_2& {\mathrm{cos\α}}_2& 0\\ {\mathrm{sin\α}}_3& {\mathrm{cos\α}}_3& 0\end{array}\right]\right)}^{-1}\\ ={\left[\begin{array}{ccc}\underset{i=1}{\overset{3}{\Σ}}{\sin }^2{\mathrm{\α}}_i& \underset{i=1}{\overset{3}{\Σ}}{\mathrm{sin\α}}_i{\mathrm{cos\α}}_i& 0\\ \underset{i=1}{\overset{3}{\Σ}}{\mathrm{sin\α}}_i{\mathrm{cos\α}}_i& \underset{i=1}{\overset{3}{\Σ}}{\cos }^2{\mathrm{\α}}_i& 0\\ 0& 0& 0\end{array}\right]}^{-1}\end{array}---\left(9\right)$

Wherein： $\underset{i=1}{\overset{3}{\Σ}}{\sin }^2{\mathrm{\α}}_i={\sin }^2{\mathrm{\α}}_1+{\sin }^2{\mathrm{\α}}_2+{\sin }^2{\mathrm{\α}}_3,\underset{i=1}{\overset{3}{\Σ}}{\cos }^2{\mathrm{\α}}_i={\cos }^2{\mathrm{\α}}_1+{\cos }^2{\mathrm{\α}}_2+{\cos }^2{\mathrm{\α}}_3$

$\underset{i=1}{\overset{3}{\Σ}}{\mathrm{sin\α}}_i{\mathrm{cos\α}}_i={\mathrm{sin\α}}_1{\mathrm{cos\α}}_1+{\mathrm{sin\α}}_2{\mathrm{cos\α}}_2+{\mathrm{sin\α}}_3{\mathrm{cos\α}}_3$

(4.2) horizontal geometric dilution of precision HDOP value in analysis planar rondure region, by non-zero in (9) Take out, constitute in plane and simplify HDOP weight coefficient matrix Q_H：

$\begin{array}{c}{Q}_H={\left[\begin{array}{cc}\underset{i=1}{\overset{3}{\Σ}}{\sin }^2{\mathrm{\α}}_i& \underset{i=1}{\overset{3}{\Σ}}{\mathrm{sin\α}}_i{\mathrm{cos\α}}_i\\ \underset{i=1}{\overset{3}{\Σ}}{\mathrm{sin\α}}_i{\mathrm{cos\α}}_i& \underset{i=1}{\overset{3}{\Σ}}{\cos }^2{\mathrm{\α}}_i\end{array}\right]}^{-1}\\ =\left[\begin{array}{cc}\frac{\underset{i=1}{\overset{3}{\Σ}}{\sin }^2{\mathrm{\α}}_i}{A}& \frac{-\underset{i=1}{\overset{3}{\Σ}}{\mathrm{sin\α}}_i{\mathrm{cos\α}}_i}{A}\\ \frac{-\underset{i=1}{\overset{3}{\Σ}}{\mathrm{sin\α}}_i{\mathrm{cos\α}}_i}{A}& \frac{\underset{i=1}{\overset{3}{\Σ}}{\cos }^2{\mathrm{\α}}_i}{A}\end{array}\right]\end{array}---\left(10\right)$

Wherein：

$\begin{array}{c}A=\underset{i=1}{\overset{3}{\Σ}}{\sin }^2{\mathrm{\α}}_i\underset{i=1}{\overset{3}{\Σ}}{\cos }^2{\mathrm{\α}}_i-\underset{i=1}{\overset{3}{\Σ}}{\sin }^2{\mathrm{\α}}_i{\cos }^2{\mathrm{\α}}_i\underset{i=1}{\overset{3}{\Σ}}{\sin }^2{\mathrm{\α}}_i{\cos }^2{\mathrm{\α}}_i\\ =({\sin }^2{\mathrm{\α}}_1+{\sin }^2{\mathrm{\α}}_2+{\sin }^2{\mathrm{\α}}_3)*({\cos }^2{\mathrm{\α}}_1+{\cos }^2{\mathrm{\α}}_2+{\cos }^2{\mathrm{\α}}_3)\\ -{({\mathrm{sin\α}}_1{\mathrm{cos\α}}_1+{\mathrm{sin\α}}_2{\mathrm{cos\α}}_2+{\mathrm{sin\α}}_3{\mathrm{cos\α}}_3)}^2\\ ={({\mathrm{sin\α}}_1{\mathrm{cos\α}}_2-{\mathrm{cos\α}}_1{\mathrm{sin\α}}_2)}^2+{({\mathrm{sin\α}}_2{\mathrm{cos\α}}_3-{\mathrm{cos\α}}_2{\mathrm{sin\α}}_3)}^2\\ +{({\mathrm{sin\α}}_3{\mathrm{cos\α}}_1-{\mathrm{cos\α}}_3{\mathrm{sin\α}}_1)}^2\\ ={\[\sin ({\mathrm{\α}}_1-{\mathrm{\α}}_2)\]}^2+{\[\sin ({\mathrm{\α}}_2-{\mathrm{\α}}_3)\]}^2+{\[\sin ({\mathrm{\α}}_3-{\mathrm{\α}}_1)\]}^2\end{array}---\left(11\right)$

Then HDOP is

$HDOP=\sqrt{Q_{H11}+Q_{H22}}=\sqrt{\frac{\underset{i=1}{\overset{3}{\Σ}}{\sin }^2{\mathrm{\α}}_i+\underset{i=1}{\overset{3}{\Σ}}{\cos }^2{\mathrm{\α}}_i}{A}}==\sqrt{\frac{3}{A}}---\left(12\right)$

Optimal location needs to set A=Amax, α₁+α₂+α₃=360 °, obtain differing all when three azimuths During for 120 °, radius can be obtained for the optimal location in the planar rondure region of R, so that it is determined that three responses Device in the length of side is respectivelyThree summits of equilateral triangle on；

Determine that the azimuth of three in same level transponder is followed successively by conjunction with (11) and (12)： + 240 ° of+120 ° of α, α, α, in the case that in three transponders of guarantee, azimuth difference is 120 ° between any two, α is unspecified angle, and HDOP value is unaffected；

(5) determine the optimal solution of border circular areas inside radius and depth relationship

On the premise of optimal distribution scheme in step (4) planar rondure region, USBL absolute fix is solved Three-dimensional geometry dilution of precision optimum point；

Three summits for taking the inscribe equilateral triangle of the circle that radius is R are the placement of transponder, three Azimuth is followed successively by α₁=0 °, α₂=120 °, α₃=240 °, calculate from mathematics geometrical relationship and determine, optimal solution is round On axial direction, the height h which arrives a transponder in each planar rondure region is equal；Make elevation angle β₁=β₂=β₃=β；Wherein, $cos\mathrm{\β}=\frac{R}{\sqrt{R^2+h^2}},sin\mathrm{\β}=\frac{h}{\sqrt{R^2+h^2}};$

DOP value observing matrix is：

Weight coefficient matrix is：

$\begin{array}{c}Q={\left(A^{T}A\right)}^{-1}={\left(\left[\begin{array}{ccc}0& \frac{\sqrt{3}}{2}\cos \mathrm{\β}& -\frac{\sqrt{3}}{2}\cos \mathrm{\β}\\ \cos \mathrm{\β}& -\frac{1}{2}\cos \mathrm{\β}& \frac{1}{2}\cos \mathrm{\β}\\ \sin \mathrm{\β}& \sin \mathrm{\β}& \sin \mathrm{\β}\end{array}\right]\left[\begin{array}{ccc}0& \cos \mathrm{\β}& \sin \mathrm{\β}\\ \frac{\sqrt{3}}{2}\cos \mathrm{\β}& -\frac{1}{2}\cos \mathrm{\β}& \sin \mathrm{\β}\\ -\frac{\sqrt{3}}{2}\cos \mathrm{\β}& \frac{1}{2}\cos \mathrm{\β}& \sin \mathrm{\β}\end{array}\right]\right)}^{-1}\\ ={\left[\begin{array}{ccc}\frac{3}{2}{\cos }^2\mathrm{\β}& 0& 0\\ 0& \frac{3}{2}{\cos }^2\mathrm{\β}& 0\\ 0& 0& 3{\sin }^2\mathrm{\β}\end{array}\right]}^{-1}=\left[\begin{array}{ccc}\frac{2}{3{\cos }^2\mathrm{\β}}& 0& 0\\ 0& \frac{2}{3{\cos }^2\mathrm{\β}}& 0\\ 0& 0& \frac{1}{3{\sin }^2\mathrm{\β}}\end{array}\right]\end{array}---\left(14\right)$

Plane positioning dilution of precision is：

$HDOP=\sqrt{Q_{H11}+Q_{H22}}=\sqrt{\frac{4}{3{\cos }^2\mathrm{\β}}}=\frac{2\sqrt{3}}{3}\sqrt{\frac{R^2+h^2}{R^2}}---\left(15\right)$

Vertical dilution of precision is：

$VDOP=\sqrt{Q_{33}}=\sqrt{\frac{1}{3{\sin }^2\mathrm{\β}}}=\frac{\sqrt{3}}{3sin\mathrm{\β}}---\left(16\right)$

Three-dimensional localization dilution of precision is：

$PDOP=\sqrt{Q_{H11}+Q_{H22}+Q_{H33}}=\sqrt{\frac{4}{3{\cos }^2\mathrm{\β}}+\frac{1}{3{\sin }^2\mathrm{\β}}}=\sqrt{\frac{4(R^2+h^2)}{3R^2}+\frac{R^2+h^2}{3h^2}}---\left(17\right)$

Optimal location needs to set PDOP value minimum, abbreviation above formula：

$\begin{array}{c}PDOP=\sqrt{\frac{4(R^2+h^2)}{3R^2}+\frac{R^2+h^2}{3h^2}}=\sqrt{\frac{(R^2+h^2)(3R^2+12h^2)}{9R^2{h}^2}}\\ =\sqrt{\frac{3R^4+15R^2{h}^2+12h^4}{3R^2{h}^2}}=\sqrt{\frac{1}{3}(\frac{R^2}{h^2}+\frac{4h^2}{R^2}+5)}\\ >=\sqrt{\frac{1}{3}(2*\sqrt{\frac{R^2}{h^2}*\frac{4h^2}{R^2}}+5)}=\sqrt{3}\end{array}---\left(18\right)$

WhenWhen, PDOP value obtains minimum of a value, now for determining border circular areas inside radius and depth relationship Optimal solution.

2. as claimed in claim 1 a kind of based on DOP value analytical acoustics alignment system multiple-answering machine optimum cloth Battle array method, it is characterised in that when number of transponders is more than 3, take n ＞ 3, using step (1)～(5) Method solve optimal distribution.