CN103634903A - Noise canceling and positioning method of low-power-consumption small-scale equipment - Google Patents

Abstract

The invention discloses a method for anti-noise positioning of small devices with low power consumption. If the calculation result is not in the applicable area of the Bancroft method, the Chan method is used to determine the position of the node to be located, and the calculation result is used as the final positioning result. The positioning method of the present invention overcomes the problem of pathological regions existing in the existing methods, and innovatively combines the advantages of the two methods to complement each other. The method is simple in calculation, strong in noise resistance, and high in positioning accuracy, and is especially suitable for low power consumption. Positioning of small devices.

Description

Anti-noise localization method for small devices with low power consumption

technical field

The invention belongs to the technical field of object positioning, in particular to an anti-noise positioning method for small devices with low power consumption based on distance difference.

Background technique

As the key basic information of modern society, location information plays an increasingly important role in national economy, national defense technology and daily life. The satellite-based positioning system represented by GPS has enabled many applications to obtain convenient and fast location services, but in systems such as wireless sensor networks, mobile Internet, and tactical Internet, satellite-based positioning systems are sometimes difficult to provide effective location services. For example: in wireless sensor networks, it is not feasible economically or technically to add GPS receivers to numerous miniature low-power nodes; in mobile Internet, users are usually in urban or indoor environments, due to the Weak, often difficult to function due to building occlusion; in the tactical Internet, satellite-based navigation systems are likely to be unusable due to enemy interference. At this time, it is necessary to rely on the self-location technology of these systems to provide position information. At the same time, the ranging noise in these systems is usually large, posing new challenges for providing precise location services.

Positioning techniques are generally divided into two categories: ranging-based positioning and non-ranging positioning. Non-ranging positioning relies on connection information between nodes to infer node position information, and the positioning accuracy is not high. Positioning based on distance measurement is generally based on the measurement of certain distance information to determine the position, and usually has high positioning accuracy, so it is more suitable for providing high-precision location services.

For ranging-based positioning systems, the positioning process can be divided into two stages: ranging and position calculation.

The ranging stage is to obtain information such as distance, distance difference or azimuth angle according to certain characteristics of the radio signal (such as amplitude, phase, time of arrival, etc.). The ranging method includes time of arrival ranging (TOA), time difference of arrival ranging ( TDOA), Azimuth Ranging (AOA), Received Signal Strength Ranging (RSSI), and Radio Interference Ranging (RIR), etc. The position calculation stage is the process of running a certain positioning algorithm according to the ranging information to determine the final position of the target to be located. The choice of positioning algorithm in the position solution stage plays a vital role in the final positioning accuracy.

Positioning algorithms can be divided into iterative algorithms and closed solution algorithms. Among them, iterative algorithms have high computational complexity and local convergence problems of iterative results. They are not suitable for applications with low power consumption, weak computing power or high real-time requirements. scenario; while the closed solution algorithm has an explicit solution, low computational complexity, and no local convergence problem. Among the closed solution positioning algorithms based on distance difference, the two most commonly used and relatively good performance are the Chan method [see literature Chan Y.T., Ho K.C., "A simple and efficient estimator for hyperbolic location". IEEETransactions on Signal Processing, vol .42, no.8, pp.1905-1915, August1994] and Bancroft method [see literature Bancroft S., "An Algebraic Solution of the GPSEquations". IEEE Transactions on Aerospace and Electronic Systems, vol.AES-21, no. 1, pp.56-59, January1985 and Mellen G.II, Pachter M., Raquet J.,"Closed-form solution for determining emitter location using time difference of arrival measurements". IEEE Transactions on Aerospace and Electronic Systems, vol.39 , no.3, pp.1056-1058, July2003]. The Chan method first regards the redundant variables introduced in the linearization process and the variables to be estimated as independent variables to solve the first step of weighted least squares, and then uses the relationship between the redundant variables and the variables to be estimated to perform the second step of weighted least squares Multiplied to further improve the accuracy of positioning results. The Bancroft method was originally proposed to solve the GPS equation, but its idea can also be used for positioning solutions based on distance differences. This method first uses the least squares to express the variables to be estimated with formulas containing redundant variables, and then uses The relationship between them constitutes a quadratic equation about the redundant variables, and the estimated value of the redundant variable is obtained by solving the quadratic equation, and then the estimated value of the variable to be estimated is obtained.

However, the above-mentioned closed solution localization algorithm has weak anti-noise ability, and the research finds that there will be pathological regions with poor performance in this kind of method. As shown in Figure 2 and Figure 3, in ill-conditioned areas, the existing closed solution localization algorithms often cannot reach the lower bound of performance - Cramer-Rao Lower Bound (CRLB, Cramer-Rao Lower Bound), or the noise threshold that can reach CRLB is very low , the anti-noise ability is extremely weak, which seriously affects the accuracy of positioning results.

Contents of the invention

The object of the present invention is to provide an anti-noise positioning method for low-power small equipment, which has small calculation amount, strong anti-noise ability, and high positioning accuracy for low-power small equipment.

The technical solution to realize the purpose of the present invention is: a low-power consumption small-sized device anti-noise positioning method, characterized in that it includes the following steps:

10) Acquisition of distance difference: obtain the distance difference between the node to be positioned and the reference anchor node and the node to be positioned to no less than two other anchor nodes;

20) Initial position calculation: According to the distance difference between the node to be positioned and the reference anchor node and the node to be positioned to other anchor nodes, use the Bancroft method to determine the initial position of the node to be positioned;

30) Judging whether the initial position calculation is successful: judge whether the initial position calculation is successful according to the quadratic equation solution of the Bancroft method, if not, go to step (50);

40) Judging whether the node to be located is in the applicable area of the Bancroft method: determining whether the initial position of the node to be located is in the applicable area of the Bancroft method, if so, go to step (60);

50) Use the Chan method to determine the position of the node to be located;

60) Output the position of the node to be located.

Compared with the prior art, the present invention has significant advantages: because the technical scheme makes full use of the advantages of explicit solutions, low computational complexity, and no problem of local convergence with the Chan method and the Bancroft method, and the ill-conditioned regions of the two are the same as The non-pathological area has certain complementary characteristics, which overcomes the problem of the ill-conditioned area existing in the existing closed solution positioning method, innovatively uses the performance advantages of the two methods, and effectively improves the anti-noise ability and positioning of the closed solution positioning method. Accuracy, small amount of calculation, strong anti-noise ability, high positioning accuracy for small devices with low power consumption.

The present invention will be described in further detail below in conjunction with the accompanying drawings.

Description of drawings

FIG. 1 is a flow chart of the anti-noise positioning method for low-power small equipment of the present invention.

Fig. 2 is a comparison diagram of MSE-SNR when the node to be located is p ₁ =[-50,500] ^T.

Fig. 3 is a comparison diagram of MSE-SNR when the node to be located is p ₂ =[1500,-200] ^T.

Figure 4 is a comparison chart of the mean MSE of the positioning results.

In the figure: CRLB is Cramer-Rao Lower Bound (CRLB, Cramer-Rao Lower Bound), Chan is the result obtained by using the Chan method, Bancroft is the result obtained by using the Bancroft method, and the new method is the anti-noise positioning of the low-power small equipment of the present invention The results obtained by the method, the abscissa indicates the size of the ranging noise (dB), and the ordinate indicates the mean square error (MSE, Mean Squared Errors) of the positioning results.

Detailed ways

As shown in Figure 1, the anti-noise positioning method for low-power small equipment of the present invention includes the following steps:

10) Acquisition of distance difference: Obtain the distance difference between the node to be positioned to the reference anchor node and the node to be positioned to no less than two other anchor nodes;

This step is prior art.

The distance difference can be obtained by means of TDOA. First, clock synchronization is performed between each anchor node, and the synchronization is completed through the FTSP clock synchronization protocol; then the node to be positioned transmits the source signal, and each anchor node records the time t ₁ , t when the source signal is received ₂ ,...,t _M (where M is the number of anchor nodes); finally, according to the signal receiving time of each anchor node, calculate the distance difference between the node to be positioned and each anchor node and No. 1 anchor node (reference anchor node), namely r _i1 =c×(t _i -t ₁ ), where c is the propagation velocity of the source signal. In the cellular network, the measured value of the distance difference is obtained by observing the time difference of arrival (OTDOA, Observed Time Difference of Arrival). Propagation time difference, and then the distance difference between the mobile station and different base stations.

The distance difference can also be obtained by radio interferometric ranging (RIR, Radio Interferometric Ranging), where the most basic ranging unit is a quadruple composed of two transmitting nodes and two receiving nodes. First, the clock is synchronized, and then the two transmitting nodes transmit sine wave signals with close frequencies, and the two receiving nodes measure the amplitude, phase and other information of the received beat frequency signals, and the two receiving nodes obtained at multiple frequencies The measured phase difference of the beat frequency signal can obtain the distance constraint relationship between the four nodes, and the distance constraint relationship can be further converted into a distance difference.

20) Initial position calculation: According to the distance difference between the node to be positioned and the reference anchor node and the node to be positioned to other anchor nodes, use the Bancroft method to determine the initial position of the node to be positioned;

The step of using the Bancroft method to determine the initial position of the node to be positioned (20) is specifically:

21) Solving quadratic equations

(g ^T g-1)r ₁ ² -2g ^T hr ₁ +h ^T h=0

get an estimate of _r1 ;

In the formula:

h=(A ^T A) ^-1 A ^T b,g=(A ^T A) ^-1 A ^T R, where A=[x ₂ -x ₁ ,x ₃ -x ₁ ,…,x _M -x ₁ ; y ₂ -y ₁ ,y ₃ -y ₁ ,…,y _M -y ₁ ] ^T ，b=-0.5[r ₂ , ₁ ² -||A ₂ || ² +||A ₁ || ² ,r ₃ , ₁ ² －||A ₃ || ² ＋||A ₁ || ² ,…,r _M , ₁ ² －||A _M || ² ＋||A ₁ || ² ] ^T ，R=[ r ₂ , ₁ ,r ₃ , ₁ ,…,r _M , ₁ ] ^T ;

The anchor node is A _i =[x _i ,y _i ] ^T , i=1,2,…,M (M is the number of anchor nodes, M≥3), and the node to be positioned is p=[x,y] ^T , r _i =||p–A _i || represents the distance between the node p to be positioned and the anchor node A _i , r _i , _j represents the distance difference between the node p to be positioned and the anchor nodes A _i and A _j ;

22) Judging the situation of the solution of the quadratic equation:

Let Δ=(g ^T h) ² -(g ^T g-1)h ^T h, then

221) If g ^T g-1=0, the equation has a unique solution r ₁ =h ^T h/(2g ^T h);

222) If Δ<0, set r ₁ =-1, p=A ₁ , and the solution of Bancroft method ends;

223) If Δ=0, the equation has a unique solution r ₁ =g ^T h/(g ^T g-1);

224) If Δ>0, continue to make the following judgments:

2241) If (g ^T h-Δ ^0.5 )/(g ^T g-1)>0 and (g ^T h+Δ ^0.5 )/(g ^T g-1)<0, the correct solution of the equation is r ₁ =( g ^T h-Δ ^0.5 )/(g ^T g-1);

2242) If (g ^T h-Δ ^0.5 )/(g ^T g-1)<0 and (g ^T h+Δ ^0.5 )/(g ^T g-1)>0, the correct solution of the equation is r ₁ =( g ^T h+Δ ^0.5 )/(g ^T g-1);

2243) If (g ^T h-Δ ^0.5 )/(g ^T g-1)<0 and (g ^T h+Δ ^0.5 )/(g ^T g-1)<0, then let r ¹ =-1, p =A ¹ , the solution of Bancroft method ends;

判断正确解，2244) If (g ^T h-Δ ^0.5 )/(g ^T g-1)>0 and (g ^T h+Δ ^0.5 )/(g ^T g-1)>0, the equation has two valid solutions r ₁ = (g ^T h-Δ ^0.5 )/(g ^T g-1) and r ₁ =(g ^T h+Δ ^0.5 )/(g ^T g-1), using least squares metric

judge the correct solution,

表示根据二次方程的解得到位置估计值之后计算出的距离差，待定位节点的位置根据p=h-gr₁进行计算，

表示距离差测量值，即与测量值的残差平方和最小的为正确解；in

Indicates the distance difference calculated after the position estimate is obtained according to the solution of the quadratic equation, and the position of the node to be located is calculated according to p=h-gr ₁ ,

Indicates the measured value of the distance difference, that is, the correct solution is the smallest sum of squared residuals with the measured value;

23) After solving the quadratic equation to obtain the estimated value of r ₁ , substitute it into the formula p=h-gr ₁ to obtain the unweighted estimated value p _u of the position of the node to be located:

Calculate the distance r _i from the node to be located to each anchor node Ai by p _u ,

Let h=(A ^T WA) ^-1 A ^T Wb, g=(A ^T WA) ^-1 A ^T WR, where the definitions of A, b and R are the same as above, W=(DQD) ^-1 , D=diag{r ₂ ,r ₃ ,…,r _M }, Q=I _M-1 +1, solve the quadratic equation (g ^T g-1)r ₁ ² -2g ^T hr ₁ +h ^T h=0 to get r ₁ The estimated value of , the solution of the quadratic equation will encounter several situations in the same step (22), and adopt the same processing method

24) After obtaining the estimated value of r ₁ , substitute it into the formula p=h-gr ₁ to obtain the weighted estimated value p _w of the position of the node to be located:

在p_u和p_w中选择残差最小的作为待定位节点初始位置p。25) Calculate the distance difference between p _u and each anchor node and the distance difference between p _w and each anchor node, and then according to the least square index

Select the one with the smallest residual error among p _u and p _w as the initial position p of the node to be located.

30) Judging whether the initial position calculation is successful: judge whether the initial position calculation is successful according to the quadratic equation solution of the Bancroft method, if not, go to step (50);

The step of judging whether the initial position calculation is successful (30) is specifically:

If the quadratic equation solution r ₁ =-1, the initial position solution is unsuccessful, otherwise it is successful.

40) Judging whether the node to be located is in the applicable area of the Bancroft method: determining whether the initial position of the node to be located is in the applicable area of the Bancroft method, if so, go to step (60);

Divide the area to be located into two parts: the outer area of the anchor node and the inner area of the anchor node, and use the Chan method and the Bancroft method to solve the problem respectively.

A method of the step of judging whether the initial position of the node to be positioned is in the applicable area of the Bancroft method (40) is specifically:

41) Take the centroid C of the anchor node,

42) For all adjacent anchor nodes A _i and A _j that follow the direction of the polygon composition, check whether they are in the Bancroft direction

Judgment on the area of application of the law;

For all adjacent anchor nodes A _i and A _j , the step of judging whether they are in the applicable area of the Bancroft method (42) is as follows:

421) Take the midpoint M of the anchor nodes A _i and A _j , make an extension line from C to point M, and take a point P on the extension line whose distance is half the length of the line segment A _i A _j ;

422) Judging whether the node p to be positioned and the centroid C are substituted into the A _i A _j straight line equation whether the positive or negative is the same;

423) If different, the node to be located is not in the applicable area of the Bancroft method;

424) If they are the same, judge whether the positive and negative are the same when substituting the node to be positioned and the centroid point C into the linear equations of PA _i and PA _j ;

425) If different, the node to be located is not in the applicable area of the Bancroft method.

43) If the node to be located is not in the applicable area of the Bancroft method after the judgment of all adjacent anchor nodes is completed, the node to be located is in the applicable area of the Bancroft method.

The step of judging whether the initial position of the node to be positioned is located in the applicable area of the Bancroft method (40) may also include:

61) Determine whether the node p to be located is inside the triangle formed by any three anchor nodes, if it is inside at least one of the triangles, then the node p to be located is in the applicable area of the Bancroft method, and the judgment ends; otherwise, proceed to the following steps;

The methods for judging whether the node p to be located is inside the triangle include the area judgment method, the vector cross product judgment method, the same-side judgment method and the directed area judgment method.

62) Take the centroid C of the anchor node,

63) For all adjacent anchor nodes A _i and A _j that follow the direction of the polygon composition, take the midpoint M of the anchor nodes A _i and A _j , make an extension line from C to point M, and take a distance of l on the extension line Point P, where 0≤l≤50*A _i A _j ; if the node to be positioned p is located inside the triangle PA _i A _j , the node to be positioned is in the applicable area of the Bancroft method, and the judgment is over;

64) If none of the above conditions for terminating the judgment occurs, the node to be located is not located in the applicable area of the Bancroft method.

The step of judging whether the initial position of the node to be positioned is located in the applicable area of the Bancroft method (40) may also include:

(71) Determine whether the node p to be located is located inside the polygon composed of all anchor nodes, if so, the node p to be located is in the applicable area of the Bancroft method, and the judgment is over; otherwise, proceed to the following steps;

The methods for judging whether the node p to be located is inside the polygon include horizontal/vertical intersection number discrimination method, cross product discrimination method, angle sum discrimination method, and directed area judgment method.

72) Take the centroid C of the anchor node,

73) For all adjacent anchor nodes A _i and A _j following the composition direction of the polygon, take the midpoint M of the anchor nodes A _i and A _j , make an extension line from C to point M, and take a distance of l on the extension line Point P, where 0≤l≤50*A _i A _j ; if the node to be positioned p is located inside the triangle PA _i A _j , the node to be positioned is in the applicable area of the Bancroft method, and the judgment is over;

74) If none of the above conditions for terminating the judgment occurs, the node to be located is not located in the applicable area of the Bancroft method.

50) Use the Chan method to determine the position of the node to be located;

If the node to be located is located in the applicable area of the Chan method, or the Bancroft method cannot be successfully solved, the Chan method is used for the solution.

The step of using the Chan method to determine the position of the node to be located (50) is specifically:

51) The preliminary estimated value pe of the node to be located _is

p _e =(G ₁ ^T Q ^-1 G ₁ ) ^-1 G ₁ ^T Q ^-1 h ₁ ,

Where: Q=I _M-1 +1, G ₁ =[x ₂ -x ₁ ,x ₃ -x ₁ ,…,x _M -x ₁ ;y ₂ -y ₁ ,y ₃ -y ₁ ,…,y _M -y ₁ ;r ₂₁ ,r ₃ , ₁ ,…,r _M , ₁ ] ^T ，p _e =[x,y,r ₁ ] ^T ，h ₁ =-0.5[r ₂ , ₁ ² －||A ₂ || ² ＋||A ₁ || ² ,r ₃ , ₁ ² －||A ₃ || ² ＋||A ₁ || ² ,…,r _M , ₁ ² －||A _M || ² ＋||A ₁ || ² ] ^T ;

52) Calculate the distance r _i from the node p to be located to each anchor node A _i by using the preliminary estimated value p _e of the position of the node to be located;

53) Estimated value of the position of the node to be located

p _e =( _G 1 ^T W ₁ G ₁ ) ^-1 G ₁ ^T W ₁ h ₁ ,

Where: W ₁ =(DQD) ^-1 ,D=diag{r ₂ ,r ₃ ,…,r _M };

54) The position reference value of the node to be positioned

p _a =(G ₂ ^T W ₂ G ₂ ) ^-1 G ₂ ^T W ₂ h ² ,

Where: G ₂ =[1,0;0,1;1,1;],p _a =[(xx ₁ ) ² ;(yy ₁ ) ² ],h ₂ =[(p _e (1)-x ₁ ) ² ;(p _e (2)-y ₁ ) ² ;(p _e (3)) ² ], W ₂ =(D′(G ₁ ^T W ₁ G ₁ ) ^-1 D′) ^-1 ,D′ =diag{x–x ₁ ,y–y ₁ ,r ₁ };

Thus, the constraint relationship between r ₁ and x, y is used to further improve the estimation accuracy.

55) Determine the position of the node to be located

p=p _a ^0.5 ⊙sgn(p _e (1,2)-A ₁ )+A ₁ ,

Among them: sgn() is a symbolic function;

Therefore, the ambiguity solution is eliminated by using the estimated value _pe of the node to be located.

If the elements in p _a have negative values under high noise, then

p = p _e (1,2).

60) Output the position of the node to be located.

Output the position coordinates of the node to be positioned to the display;

Since this technical solution makes full use of the advantages of explicit solutions, low computational complexity, and no local convergence problems with the Chan method and Bancroft method, and there is a certain complementarity between the ill-conditioned area and the non-ill-conditioned area of the two, it overcomes the existing There are ill-conditioned area problems in the closed solution positioning method, which effectively improves the anti-noise ability of the closed solution positioning method and the accuracy of positioning results.

The present invention can verify the validity of the method through simulation. The anchor node coordinates used in the simulation are A ₁ =[0,0] ^T , A ₂ =[600,-100] ^T , A ₃ =[1225,50] ^T , A ₄ =[1300,1350] ^T , A ₅ =[650,1550] ^T ,A ₆ =[-150,1200] ^T .

Fig. 2 shows the MSE-SNR diagram of the positioning result when the node to be positioned is p ₁ =[-50,500] ^T.

Figure 3 shows the MSE-SNR diagram of the positioning results when the node to be positioned is p ₂ =[1500,-200] ^T , the abscissa in the figure represents the size of the ranging noise (-10dB～25dB), and the vertical axis represents the mean value of the positioning results. The size of the variance (MSE, Mean Squared Errors).

Figure 4 shows the mean MSE value of the positioning results under -10dB ~ 25dB noise when the working range of the positioning system is [-500, 1700; -500, 1700] and the positioning points are taken within the working range at intervals of 10×10.

It can be seen from the simulation results that the Chan method seriously fails at the position of p1, and can only reach CRLB when the noise is less than -5dB, while the new method can still reach CRLB when the noise level is 25dB, which is 30dB higher than the Chan method ; Bancroft method seriously fails at the p2 position, even when the noise level is -10dB, it still cannot reach the CRLB, while the new method can also reach the CRLB when the noise level is 25dB. It can be seen from Figure 4 that the average MSE of the Bancroft method is very poor, much higher than that of CRLB, indicating that there are many pathological regions in the method, which seriously affects the overall performance of the method; while the Chan method can reach CRLB under low noise, However, when the noise increases, the MSE of the positioning result will be higher than that of CRLB; at the same time, the new method can be closer to CRLB than the previous two methods. The average MSE value is 38.91% of the Chan method), and the performance improvement compared to the Bancroft method is even greater.

The simulation results show that the noise threshold is increased from -5dB to 25dB at p1 compared with the Chan method, and the noise threshold is increased from -10dB to 25dB at p2 compared with the Bancroft method; in the best case, the average MSE value can be reduced It is 38.91% of the Chan method, and the performance improvement compared to the Bancroft method is even greater. The technical proposal significantly improves the anti-noise capability and positioning accuracy of the method with low computational complexity.

Claims (8)

1. A method for anti-noise positioning of low-power small-sized equipment, is characterized in that, comprises the steps: 10) Acquisition of distance difference: obtain the distance difference between the node to be positioned and the reference anchor node and the node to be positioned to no less than two other anchor nodes; 20) Initial position calculation: According to the distance difference between the node to be positioned and the reference anchor node and the node to be positioned to other anchor nodes, use the Bancroft method to determine the initial position of the node to be positioned; 30) Judging whether the initial position calculation is successful: judge whether the initial position calculation is successful according to the quadratic equation solution of the Bancroft method, if not, go to step (50); 40) Judging whether the node to be located is in the applicable area of the Bancroft method: determining whether the initial position of the node to be located is in the applicable area of the Bancroft method, if so, go to step (60); 50) Use the Chan method to determine the position of the node to be located; 60) Output the position of the node to be located. 2. The anti-noise positioning method for low-power small equipment according to claim 1, wherein the initial position calculation (20) step includes: 21) Solving quadratic equations (g ^T g-1)r ₁ ² -2g ^T hr ₁ +h ^T h=0 get an estimate of _r1 ; In the formula: h=(A ^T A) ^-1 A ^T b,g=(A ^T A) ^-1 A ^T R, where A=[x ₂ -x ₁ ,x ₃ -x ₁ ,…,x _M -x ₁ ; y ₂ -y ₁ ,y ₃ -y ₁ ,…,y _M -y ₁ ] ^T ，b=-0.5[r ₂ , ₁ ² -||A ₂ || ² +||A ₁ || ² ,r ₃ , ₁ ² －||A ₃ || ² ＋||A ₁ || ² ,…,r _M , ₁ ² －||A _M || ² ＋||A ₁ || ² ] ^T ，R=[ r ₂ , ₁ ,r ₃ , ₁ ,…,r _M , ₁ ] ^T ; The anchor node is A _i =[x _i ,y _i ] ^T , i=1,2,…,M (M is the number of anchor nodes, M≥3), and the node to be positioned is p=[x,y] ^T , r _i =||p–A _i || represents the distance between the node p to be positioned and the anchor node A _i , r _i , _j represents the distance difference between the node p to be positioned and the anchor nodes A _i and A _j ; 22) Judging the situation of the solution of the quadratic equation: Let Δ=(g ^T h) ² -(g ^T g-1)h ^T h, then 221) If g ^T g-1=0, the equation has a unique solution r ₁ =h ^T h/(2g ^T h); 222) If Δ<0, set r ₁ =-1, p=A ₁ , and the solution of Bancroft method ends; 223) If Δ=0, the equation has a unique solution r ₁ =g ^T h/(g ^T g-1); 224) If Δ>0, continue to make the following judgments: 2241) If (g ^T h-Δ ^0.5 )/(g ^T g-1)>0 and (g ^T h+Δ ^0.5 )/(g ^T g-1)<0, the correct solution of the equation is r ₁ =( g ^T h-Δ ^0.5 )/(g ^T g-1); 2242) If (g ^T h-Δ ^0.5 )/(g ^T g-1)<0 and (g ^T h+Δ ^0.5 )/(g ^T g-1)>0, the correct solution of the equation is r ₁ =( g ^T h+Δ ^0.5 )/(g ^T g-1); 2243) If (g ^T h-Δ ^0.5 )/(g ^T g-1)<0 and (g ^T h+Δ ^0.5 )/(g ^T g-1)<0, then let r ₁ =-1, p =A ₁ , the solution of Bancroft method ends; 2244) If (g ^T h-Δ ^0.5 )/(g ^T g-1)>0 and (g ^T h+Δ ^0.5 )/(g ^T g-1)>0, the equation has two valid solutions r ₁ = (g ^T h-Δ ^0.5 )/(g ^T g-1) and r ₁ =(g ^T h+Δ ^0.5 )/(g ^T g-1), using least squares metric

judge the correct solution, where r

Indicates the distance difference calculated after the position estimate is obtained according to the solution of the quadratic equation, and the position of the node to be located is calculated according to p=h-gr ₁ ,

Indicates the measured value of the distance difference, that is, the correct solution is the smallest sum of squared residuals with the measured value; 23) After solving the quadratic equation to obtain the estimated value of r ₁ , substitute it into the formula p=h-gr ₁ to obtain the unweighted estimated value p _u of the position of the node to be located: Calculate the distance r _i from the node to be located to each anchor node A _i by p _u , Let h=(A ^T WA) ^-1 A ^T Wb, g=(A ^T WA) ^-1 A ^T WR, where the definitions of A, b and R are the same as above, W=(DQD) ^-1 , D=diag{r ₂ ,r ₃ ,…,r _M }, Q=I _M-1 +1, Solve the quadratic equation (g ^T g-1)r ₁ ² -2g ^T hr ₁ +h ^T h=0 to get the estimated value of r ₁ , the solution of the quadratic equation will encounter several situations in the same step (22), and take the same approach; 24) After obtaining the estimated value of r ₁ , substitute it into the formula p=h-gr ₁ to obtain the weighted estimated value p _w of the position of the node to be located: 25) Calculate the distance difference between p _u and each anchor node and the distance difference between p _w and each anchor node, and then according to the least square index

Select the one with the smallest residual error among p _u and p _w as the initial position p of the node to be located. 3. The anti-noise positioning method for low-power small equipment according to claim 2, characterized in that: the step of judging whether the initial position calculation is successful (30) is specifically: If the quadratic equation solution r ₁ =-1, the initial position solution is unsuccessful, otherwise it is successful. 4. The anti-noise positioning method for small low-power equipment according to claim 2, wherein the step of judging whether the initial position of the node to be positioned is located in the applicable area of the Bancroft method (40) includes: 41) Take the centroid C of the anchor node,

42) For all adjacent anchor nodes A _i and A _j that follow the direction of the polygon, judge whether they are in the applicable area of the Bancroft method; 43) If the node to be located is not in the applicable area of the Bancroft method after the judgment of all adjacent anchor nodes is completed, the node to be located is in the applicable area of the Bancroft method. 5. The anti-noise positioning method for small low-power equipment according to claim 4, wherein, for all adjacent anchor nodes Ai and Aj that follow the direction of the polygon, it is judged whether it is in the applicable area of the Bancroft method ( 42) Steps include: 421) Take the midpoint M of the anchor nodes A _i and A _j , make an extension line from C to point M, and take a point P on the extension line whose distance is l, where 0≤l≤50*A _i A _j ; 422) Judging whether the node p to be positioned and the centroid C are substituted into the A _i A _j straight line equation whether the positive or negative is the same; 423) If different, the node to be located is not in the applicable area of the Bancroft method; 424) If they are the same, judge whether the positive and negative are the same when substituting the node to be positioned and the centroid point C into the linear equations of PA _i and PA _j ; 425) If different, the node to be located is not in the applicable area of the Bancroft method. 6. The anti-noise positioning method for small low-power equipment according to claim 2, characterized in that: the step of judging whether the initial position of the node to be positioned is in the applicable area of the Bancroft method (40) includes: 61) Determine whether the node p to be located is inside the triangle formed by any three anchor nodes, if it is inside at least one of the triangles, then the node p to be located is in the applicable area of the Bancroft method, and the judgment ends; otherwise, proceed to the following steps; 62) Take the centroid C of the anchor node,

63) For all adjacent anchor nodes A _i and A _j that follow the direction of the polygon composition, take the midpoint M of the anchor nodes A _i and A _j , make an extension line from C to point M, and take a distance of l on the extension line Point P, where 0≤l≤50*A _i A _j ; if the node to be positioned p is located inside the triangle PA _i A _j , the node to be positioned is in the applicable area of the Bancroft method, and the judgment is over; 64) If none of the above conditions for terminating the judgment occurs, the node to be located is not located in the applicable area of the Bancroft method. 7. The anti-noise positioning method for small low-power equipment according to claim 2, wherein the step of judging whether the initial position of the node to be positioned is located in the applicable area of the Bancroft method (40) includes: (71) Judging whether the node p to be located is located inside the polygon formed by all anchor nodes, if yes, the node p to be located is in the applicable area of the Bancroft method, and the judgment ends, otherwise, proceed to the following steps; 72) Take the centroid C of the anchor node,

73) For all adjacent anchor nodes A _i and A _j following the composition direction of the polygon, take the midpoint M of the anchor nodes A _i and A _j , make an extension line from C to point M, and take a distance of l on the extension line Point P, where 0≤l≤50*A _i A _j ; if the node to be positioned p is located inside the triangle PA _i A _j , the node to be positioned is in the applicable area of the Bancroft method, and the judgment is over; 74) If none of the above conditions for terminating the judgment occurs, the node to be located is not located in the applicable area of the Bancroft method. 8. The anti-noise positioning method for small low-power equipment according to claim 1, wherein the step of using the Chan method to determine the position of the node to be positioned (50) includes: 51) The preliminary estimated value pe of the node to be located _is p _e =(G ₁ ^T Q ^-1 G ₁ ) ^-1 G ₁ ^T Q ^-1 h ₁ , Where: Q=I _M-1 +1, G ₁ =[x ₂ -x ₁ ,x ₃ -x ₁ ,…,x _M -x ₁ ;y ₂ -y ₁ ,y ₃ -y ₁ ,…,y _M -y ₁ ;r ₂₁ ,r ₃ , ₁ ,…,r _M , _1] ^T ，p _e =[x,y,r ₁ ] ^T ，h ₁ =-0.5[r ₂ , ₁ ^2－ ||A ₂ || ² ＋||A ₁ || ² ,r ₃ , ₁ ² －||A ₃ || ² ＋||A ₁ || ² ,…,r _M , ₁ ² －||A _M || ² ＋||A ₁ || ² ] ^T ; 52) Calculate the distance r _i from the node p to be located to each anchor node A _i by using the preliminary estimated value p _e of the position of the node to be located; 53) Estimated value of the position of the node to be located p _e =(G ₁ ^T W ₁ G ₁ ) ^-1 G ₁ ^T W ₁ h ₁ , Where: W ₁ =(DQD) ^-1 ,D=diag{r ₂ ,r ₃ ,…,r _M }; 54) The position reference value of the node to be positioned p _a =(G ₂ ^T W ₂ G ₂ ) ^-1 G ₂ ^T W ₂ h ² , Where: G ₂ =[1,0;0,1;1,1;],p _a =[(xx ₁ ) ² ;(yy ₁ ) ² ],h ₂ =[(p _e (1)-x ₁ ) ² ;(p _e (2)-y ₁ ) ² ;(p _e (3)) ² ], W ₂ =(D′(G ₁ ^T W ₁ G ₁ ) ^-1 D′) ^-1 ,D′ =diag{x–x ₁ ,y–y ₁ ,r ₁ }; 55) Determine the position of the node to be located p=p _a ^0.5 ⊙sgn(p _e (1,2)-A ₁ )+A ₁ , Among them: sgn() is a symbolic function; If the elements in p _a have negative values, then p=p _e (1,2).

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