CN107271956A - The localization method based on arrival time of unknown initial time in nlos environment - Google Patents

Abstract

The invention discloses a time-of-arrival positioning method based on an unknown start time in a non-line-of-sight environment, which first establishes a model of the signal transmission distance between a target source and each sensor; then obtains a description of measurement noise according to the model; Then, according to the description of the measurement noise, and using the worst-case robust weighted least squares method, the positioning problem in an environment with unknown signal transmission start time and non-line-of-sight errors is obtained, and then transformed into a non-convex positioning problem; then by introducing auxiliary variables and using the second-order cone relaxation method, the second-order cone programming problem is obtained; finally, the interior point method is used to solve the second-order cone programming problem, and the final coordinate position of the target source in the reference coordinate system is obtained Estimated value; the advantage is that it can solve the existing problems of unknown signal transmission start time and non-line-of-sight error at the same time, thereby improving positioning accuracy.

Description

Time-of-arrival-based localization method with unknown origin time in non-line-of-sight environment

technical field

The invention relates to a target positioning method, in particular to a time-of-arrival positioning method with unknown starting time in a non-line-of-sight environment.

Background technique

In recent years, wireless sensor network positioning technology has been widely used in many fields, which can conveniently and quickly realize positioning navigation, environmental monitoring, smart home, industrial control, etc. With the development of technology and the progress of society, high-precision positioning technology has shown a wide range of application prospects in various fields. Therefore, it is very necessary to research on high-precision target localization methods in wireless sensor networks.

At present, there are many basic methods to achieve target positioning. The most widely used is the time measurement method based on Time of Arrival (TimeofArrival). The basic network structure is shown in Figure 1. The advantage of this method is that the complexity of the time measurement system is low, and it can achieve high-precision target positioning. Therefore, most studies on object localization methods are performed based on time-of-arrival measurements.

The precise location of the target is the key to the design of the target location method. However, there are many factors that affect the positioning accuracy in the actual wireless sensor network. The main problems are the clock asynchronous problem and the non-line-of-sight error problem in the wireless sensor network, as shown in Figure 1. The so-called clock asynchronous problem mainly refers to the accurate start time of the unknown target transmitting signal, resulting in inaccurate measurement of the transmission time of the signal between the target and the sensor, causing a deviation between the measured distance and the real distance, and thus affecting the positioning accuracy. The so-called non-line-of-sight error means that the transmission of the signal is blocked between the target and the sensor, resulting in a propagation delay, which makes the measurement time longer, produces a large distance measurement error, and reduces the positioning accuracy. In fact, if the problems of unknown signal transmission start time and non-line-of-sight error cannot be effectively dealt with, the positioning accuracy will not be greatly improved effectively.

In order to solve the problem of unknown signal transmission start time and non-line-of-sight error in the time-of-arrival positioning method, it is necessary to design a method to jointly eliminate the adverse effects of clock asynchrony and non-line-of-sight error on positioning accuracy. At present, there are few methods to jointly deal with the two kinds of errors, but the research on the methods for unknown start time and non-line-of-sight errors alone has been very extensive. For example, use convex relaxation techniques for efficient non-line-of-sight error processing; use some methods to jointly estimate the unknown start time and target position of signal emission. However, in practical applications, the existence of unknown start time and non-line-of-sight errors at the same time is very common, and it is difficult to apply individual error processing to real life. Therefore, joint processing of multiple errors will be an inevitable trend.

Contents of the invention

The technical problem to be solved by the present invention is to provide a positioning method based on time of arrival, which can simultaneously solve the existing problems of unknown signal transmission start time and non-line-of-sight error, thereby improving positioning accuracy.

The technical scheme adopted by the present invention to solve the above-mentioned technical problems is: a positioning method based on arrival time with unknown start time in a non-line-of-sight environment, characterized in that it includes the following steps:

① Establish a plane coordinate system or space coordinate system as a reference coordinate system in the wireless sensor network; set a target source for transmitting measurement signals and N sensors for receiving measurement signals in the wireless sensor network, and set The clocks of the N sensors are synchronized, but the clocks of the target source are not synchronized with the clocks of the sensors; the coordinate positions of the N sensors in the reference coordinate system are correspondingly recorded as s ₁ ,…,s _N , and the target source is in the reference coordinate system The coordinate position of is denoted as x; among them, N≥3, s ₁ represents the coordinate position of the first sensor in the reference coordinate system, and s _N represents the coordinate position of the Nth sensor in the reference coordinate system;

② Calculate the signal transmission distance between the target source and each sensor, record the signal transmission distance between the target source and the i-th sensor as d _i , d _i =c×t _i , where, 1≤i≤N, c represents the speed of light, t _i represents the time elapsed from the measurement signal being sent from the target source to the i-th sensor receiving;

③Describe the signal transmission distance between the target source and each sensor in a model, and express the model of d _i as: d _i =d ₀ +||xs _i ||+e _i +n _i , where d ₀ represents the signal transmission distance deviation caused by the unknown signal transmission start time, d ₀ =c×Δt, d ₀ ≥ 0, Δt represents the difference between the clock of the target source and the clock of the sensor, and the symbol "|| ||" is Euclidean norm symbol, s _i represents the coordinate position of the i-th sensor in the reference coordinate system, e _i represents the non-line-of-sight error existing on the path from the target source to the i-th sensor receiving the measurement signal , n _i represents the measurement noise that exists on the path from the target source to the i-th sensor receiving the measurement signal, and n _i obeys a Gaussian distribution with zero mean Indicates the power of n _i , 0≤|n _i |<<e _i ≤ρ _i , the symbol "||" is the absolute value symbol, and ρ _i represents the path that the measurement signal goes through from the target source to the i-th sensor receiving The upper limit of the non-line-of-sight error existing on ;

④ Transform d _i =d ₀ +||xs _i ||+e _i +n _i into d _i -e _i =(d ₀ +||xs _i ||)+n _i ; then d _i -e _i ＝(d ₀ +||xs _i ||)+n _i is squared on both sides to get then ignore middle Get (d _i -e _i ) ² ≈(d ₀ +||xs _i ||) ² +2n _i (d ₀ +||xs _i ||); then (d _i -e _i )2≈(d ₀ +||xs _i ||) ² +2n _i (d ₀ +||xs _i ||) becomes

⑤ According to And using the worst-case robust weighted least squares method, the positioning problem in an environment with unknown signal transmission start time and non-line-of-sight errors is obtained, which is described as: Then order Will Into Next will Into Then determine the maximum value of f(e _i ) in the interval [0,ρ _i ], when d _i ≤ρ _i has When d _i >ρ _i , there is Finally will with substitute In , the non-convex localization problem is obtained,

described as: if d _i ≤ ρ _i ;

If d _i >ρ _i

Among them, min() is the minimum value function, max() is the maximum value function,

"st" means "subject to";

⑥Introduce auxiliary variables η ₁ , η ₂ ,…,η _i ,…η _N in the non-convex positioning problem, the

If d _i ≤ ρ _i is equivalently described as:

If d _i >ρ _i

; Then substitute f(0), f(ρ _i ), f(d _i ) into

If d _i ≤ ρ _i

And introduce auxiliary variables b ₁ , b ₂ ,…b _i ,…b _N , r and y to get

If d _i ≤ ρ _i

If d _i ≤ ρ _i ; then use second-order cone relaxation

A[x,y,d ₀ ,r] ^T ≤ f

b _i =2d ₀ ||xs _i ||

method will

If d _i ≤ ρ _i

A[x,y,d ₀ ,r] ^T ≤ f

b _i =2d ₀ ||xs _i ||

The y=||x|| ² in slack is ||x|| ² ≤y, relax to The second-order cone programming problem is obtained, which is described as:

if d _i ≤ ρ _i ;

A[x,y,d ₀ ,r] ^T ≤ f

Among them, η ₁ , η ₂ ,...,η _i ,...η _N correspond to the first auxiliary variable introduced in the non-convex positioning problem, the second auxiliary variable, ..., the i-th auxiliary variable, ..., the Nth Auxiliary variables, b ₁ ,b ₂ ,… _bi ,…b _N correspond to

The first auxiliary variable, the second auxiliary variable, ..., the first auxiliary variable introduced in If d _i ≤ ρ _i

The i auxiliary variable, ..., the Nth auxiliary variable, r and y correspond to represent

The additional two auxiliary variables introduced in if d _i ≤ ρ _i , is the transpose of _si , is the transpose of s ₁ , is the transpose of s _N , d ₁ represents the signal transmission distance between the target source and the first sensor, d _N represents the signal transmission distance between the target source and the Nth sensor, [x,y,d ₀ ,r ] ^T is the transpose of [x,y,d ₀ ,r],

⑦ Use the interior point method to solve the second-order cone programming problem, and obtain the global optimal solution, denoted as x ^* , x ^* is the final estimated value of the coordinate position of the target source in the reference coordinate system.

The determination process of the constraint condition A[x,y,d ₀ ,r] ^T ≤ f in the step ⑥ is as follows: According to the conclusion that the measurement noise is much smaller than the non-line-of-sight error in general, it is obtained that d _i ≥d ₀ + ||xs _i || _; then transform d _{i ≥d 0} +||xs _i || into d _i -d ₀ ≥|| _{xs i ||} Square expansion on both sides, we get Then according to y=||x|| ² and Will Into Finally will Described in vector form, described as: A[x,y,d ₀ ,r] ^T ≤f,

The constraints in the step ⑥ The determination process is:

⑥_1. According to the constraints before relaxation b _i =2d ₀ ||xs _i ||, use the square sum inequality to get Next will Expanding the square term in , we get Then y=||x|| ² and substitute , get the constraints on the clock error Then according to the known condition d ₀ ≥0, get b _i ≥0;

⑥_2. For the constraint condition before relaxation, b _i =2d ₀ ||xs _i || Then substitute y=||x|| ² into in, get Next will Relax to Convex Constraints then substitute in, get

Compared with the prior art, the present invention has the advantages of:

1) In the process of establishing the model of the signal transmission distance between the target source and each sensor, the method of the present invention fully considers the non-line-of-sight error and the error caused by clock asynchrony, so as to jointly deal with the existence of the non-line-of-sight error The problem that the clock of the problem and the target source are not synchronized is closer to the actual application.

2) According to the measurement noise, the method of the present invention adopts the worst-case robust weighted least squares method to obtain the positioning problem in the environment where the unknown signal transmission start time and the non-line-of-sight error exist, and then converts it into a non-line-of-sight error. Convex positioning problem, robust processing of non-line-of-sight errors, making full use of the upper limit of non-line-of-sight errors to improve positioning performance, even in harsh conditions with many non-line-of-sight paths, it has sufficient performance advantages, making The method of the present invention is closer to practical application.

3) the method of the present invention obtains the second-order cone programming problem by introducing auxiliary variables and adopting the second-order cone relaxation method, because the reasonable constraints about the clock error are added in the second-order cone programming problem, the method of the present invention can be compared Accurately estimating the clock error reduces the influence of the clock error on the positioning performance, thus effectively increasing the robustness against the clock error of the method of the present invention and having stronger anti-interference ability.

Description of drawings

FIG. 1 is a schematic diagram of a time-of-arrival (TOA)-based positioning environment under the condition of an unknown start time in a non-line-of-sight environment;

Fig. 2 is the overall flow chart of the inventive method;

Fig. 3 is the root mean square error between the estimated value of the coordinates and the real value of the coordinates of the method of the present invention, the existing robust positive semi-definite relaxation method and the existing robust second-order cone relaxation method in the case of many non-line-of-sight paths Variation plot with increasing measurement noise;

Fig. 4 shows the root mean square error between the estimated value of the coordinates and the real value of the coordinates of the method of the present invention, the existing robust positive semi-definite relaxation method and the existing robust second-order cone relaxation method in the case of fewer non-line-of-sight paths Variation plot with increasing measurement noise;

Fig. 5 is the change graph of the root mean square error between the estimated value of the coordinates and the real value of the coordinates of the method of the present invention, the existing robust positive semi-definite relaxation method and the existing robust second-order cone relaxation method as the number of non-line-of-sight paths increases .

detailed description

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

A time-of-arrival positioning method based on an unknown start time in a non-line-of-sight environment proposed by the present invention, its overall flow diagram is shown in Figure 2, and it includes the following steps:

① Establish a plane coordinate system or space coordinate system as a reference coordinate system in the wireless sensor network; set a target source for transmitting measurement signals and N sensors for receiving measurement signals in the wireless sensor network, and set The clocks of the N sensors are synchronized, but the clocks of the target source are not synchronized with the clocks of the sensors; the coordinate positions of the N sensors in the reference coordinate system are correspondingly recorded as s ₁ ,…,s _N , and the target source is in the reference coordinate system The coordinate position of is denoted as x; wherein, N≥3, N=8 in this embodiment, s ₁ represents the coordinate position of the first sensor in the reference coordinate system, s _N represents the Nth sensor in the reference coordinate system The coordinate position in .

② Calculate the signal transmission distance between the target source and each sensor, record the signal transmission distance between the target source and the i-th sensor as d _i , d _i =c×t _i , where, 1≤i≤N, c represents the speed of light, and t _i represents the time elapsed from the measurement signal being sent from the target source to the i-th sensor receiving it.

③Describe the signal transmission distance between the target source and each sensor in a model, and express the model of d _i as: d _i =d ₀ +||xs _i ||+e _i +n _i , where d ₀ represents the signal transmission distance deviation caused by the unknown signal transmission start time, d ₀ =c×Δt, d ₀ ≥ 0, Δt represents the difference between the clock of the target source and the clock of the sensor, and the symbol "||||" is the Euclidean norm symbol, s _i represents the coordinate position of the i-th sensor in the reference coordinate system, e _i represents the non-line-of-sight error existing on the path from the target source to the i-th sensor receiving the measurement signal , n _i represents the measurement noise that exists on the path from the target source to the i-th sensor receiving the measurement signal, and n _i obeys a Gaussian distribution with zero mean Indicates the power of n _i , in general, 0≤|n _i |<<e _i ≤ρ _i , the symbol "||" is an absolute value symbol, and ρ _i indicates that the measurement signal is sent from the target source to the i-th sensor to receive The upper limit of the non-line-of-sight error existing on the path experienced, the value of ρ _i has been measured before target positioning, and ρ _i is a constant.

④ Transform d _i =d ₀ +||xs _i ||+e _i +n _i into d _i -e _i =(d ₀ +||xs _i ||)+n _i ; then d _i -e _i ＝(d ₀ +||xs _i ||)+n _i is squared on both sides to get then ignore middle Get (d _i -e _i ) ² ≈(d ₀ +||xs _i ||) ² +2n _i (d ₀ +||xs _i ||); then (d _i -e _i )2≈(d ₀ +||xs _i ||) ² +2n _i (d ₀ +||xs _i ||) becomes

⑤ According to And using the worst-case robust weighted least squares method, the positioning problem in an environment with unknown signal transmission start time and non-line-of-sight errors is obtained, which is described as: Then order Will Into Since f(e _i ) is always positive, then the Into Then determine the maximum value of f(e _i ) in the interval [0,ρ _i ], when d _i ≤ρ _i has When d _i >ρ _i , there is Finally will with substitute In , a non-convex positioning problem is obtained, which is described as:

if d _i ≤ ρ _i ;

If d _i >ρ _i

Among them, min() is the minimum value function, max() is the maximum value function,

"st" means "subject to...".

⑥Introduce auxiliary variables η ₁ , η ₂ ,…,η _i ,…η _N in the non-convex positioning problem, the

If d _i ≤ ρ _i is equivalently described as:

If d _i >ρ _i

Then substitute f(0), f(ρ _i ), f(d _i ) into

If d _i ≤ ρ _i

And introduce auxiliary variables b ₁ , b ₂ ,…b _i ,…b _N , r and y to get If d _i ≤ ρ _i

If d _i ≤ ρ _i ; then use second-order cone relaxation

A[x,y,d ₀ ,r] ^T ≤ f

b _i =2d ₀ ||xs _i ||

method will

If d _i ≤ ρ _i

A[x,y,d ₀ ,r] ^T ≤ f

b _i =2d ₀ ||xs _i ||

The y=||x|| ² in slack is ||x|| ² ≤y, relax to The second-order cone programming problem is obtained, which is described as:

if d _i ≤ ρ _i ;

A[x,y,d ₀ ,r] ^T ≤ f

Among them, η ₁ , η ₂ ,...,η _i ,...η _N correspond to the first auxiliary variable introduced in the non-convex positioning problem, the second auxiliary variable, ..., the i-th auxiliary variable, ..., the Nth Auxiliary variables, b ₁ ,b ₂ ,… _bi ,…b _N correspond to

The first auxiliary variable, the second auxiliary variable, ..., the first auxiliary variable introduced in If d _i ≤ ρ _i

The i auxiliary variable, ..., the Nth auxiliary variable, r and y correspond to represent The additional two auxiliary variables introduced in if d _i ≤ ρ _i , is the transpose of _si , is the transpose of s ₁ , is the transpose of s _N , d ₁ represents the signal transmission distance between the target source and the first sensor, d _N represents the signal transmission distance between the target source and the Nth sensor, [x,y,d ₀ ,r ] ^T is the transpose of [x,y,d ₀ ,r],

In this specific embodiment, the determination process of the constraint condition A[x,y,d ₀ ,r] ^T ≤ f in step ⑥ is as follows: According to the conclusion that the measurement noise is much smaller than the non-line-of-sight error in general cases, d _i ≥d ₀ +||xs _i || _; then transform d _{i ≥d 0} +||xs _i || into d _i -d ₀ ≥||xs _i || xs _i || carry out square expansion on both sides to get Then according to y=||x|| ² and Will Into Finally will Described in vector form, described as: A[x,y,d ₀ ,r] ^T ≤f,

In this specific embodiment, the constraints in step ⑥

The determination process is:

⑥_1. According to the constraint condition before relaxation b _i =2d ₀ ||xs _i ||, it can be seen that it is difficult to relax this non-convex constraint into a convex constraint, but the corresponding conversion can be used, so the sum of squares inequality can be used to obtain Next will Expanding the square term in , we get Then y=||x|| ² and substitute , get the constraints on the clock error Then according to the known condition d ₀ ≥0, it is obtained that b _i ≥0.

⑥_2. For the constraint condition before relaxation, b _i =2d ₀ ||xs _i || Then substitute y=||x|| ² into in, get due to constraints is non-convex, so then the Relax to Convex Constraints then substitute in, get

⑦ Use the interior point method to solve the second-order cone programming problem, and obtain the global optimal solution, denoted as x ^* , x ^* is the final estimated value of the coordinate position of the target source in the reference coordinate system.

In order to verify the feasibility and effectiveness of the method of the present invention, the method of the present invention is simulated.

Assume that there are N=8 sensors uniformly distributed on the edge of a square of 10×10m ² centered on the origin (0,0), and the position of the target source is randomly selected within the area of 15×15m ² . Assuming that the power (variance) of the measurement noise of all sensors is the same, that is, The upper bound of the non-line-of-sight error is the same, that is, ρ ₁ =ρ ₂ =...=ρ _N =ρ.

The variation of the performance of the method of the present invention with the increase of measurement noise is tested in the case of different numbers of non-line-of-sight paths. Fig. 3 has provided in the environment that exists non-line-of-sight, under the condition of unknown starting time, when the number of non-line-of-sight paths is 6, the method of the present invention and existing robust semi-definite relaxation method and existing The positioning performance of the robust second-order cone relaxation method changes with the increase of measurement noise; Figure 4 shows that in an environment with non-line-of-sight, the number of non-line-of-sight paths is 2 under the condition of unknown start time When, the positioning performance of the method of the present invention, the existing robust positive semi-definite relaxation method and the existing robust second-order cone relaxation method varies with the increase of measurement noise. It can be seen from Fig. 3 and Fig. 4 that no matter when there are many or few non-line-of-sight paths, the performance of the method of the present invention is always better than that of the existing robust positive semi-definite relaxation method and the existing robust second-order cone relaxation The method is enough to show that the method of the present invention has sufficient advantages in terms of positioning accuracy.

The performance of the method of the present invention was tested as a function of the number of non-line-of-sight paths. Figure 5 shows that in an environment with non-line-of-sight, under the condition of unknown start time, the standard deviation σ of the measurement noise is randomly selected in the interval [0.2,1]. The method of the present invention and the existing robust positive semi-definite Localization performance of the relaxation method and the existing robust second-order cone relaxation method as a function of the number of non-line-of-sight paths. It can be seen from Figure 5 that the performance of the method of the present invention is always better than the existing robust semi-definite relaxation method and the existing robust second-order cone relaxation method, which is enough to show that the method of the present invention has sufficient advantages in positioning accuracy .

In the simulation figures 3, 4 and 5, "robust positive semi-definite relaxation method" means the algorithm based on maximum likelihood positive semi-definite relaxation under the premise of time synchronization and non-line-of-sight error; "robust second-order Cone relaxation method" means under the premise of time synchronization and non-line-of-sight error, a robust second-order cone relaxation algorithm based on least squares; the method of the present invention is under the condition of clock asynchrony and non-line-of-sight error method.

Claims (3)

1. a time-of-arrival positioning method based on an unknown starting time in a non-line-of-sight environment, characterized in that it may further comprise the steps: ① Establish a plane coordinate system or space coordinate system as a reference coordinate system in the wireless sensor network; set a target source for transmitting measurement signals and N sensors for receiving measurement signals in the wireless sensor network, and set The clocks of the N sensors are synchronized, but the clocks of the target source are not synchronized with the clocks of the sensors; the coordinate positions of the N sensors in the reference coordinate system are correspondingly recorded as s ₁ ,…,s _N , and the target source is in the reference coordinate system The coordinate position of is denoted as x; among them, N≥3, s ₁ represents the coordinate position of the first sensor in the reference coordinate system, and s _N represents the coordinate position of the Nth sensor in the reference coordinate system; ② Calculate the signal transmission distance between the target source and each sensor, record the signal transmission distance between the target source and the i-th sensor as d _i , d _i =c×t _i , where, 1≤i≤N, c represents the speed of light, t _i represents the time elapsed from the measurement signal being sent from the target source to the i-th sensor receiving; ③Describe the signal transmission distance between the target source and each sensor in a model, and express the model of d _i as: d _i =d ₀ +||xs _i ||+e _i +n _i , where d ₀ represents the signal transmission distance deviation caused by the unknown signal transmission start time, d ₀ =c×Δt, d ₀ ≥ 0, Δt represents the difference between the clock of the target source and the clock of the sensor, and the symbol "|| ||" is Euclidean norm symbol, s _i represents the coordinate position of the i-th sensor in the reference coordinate system, e _i represents the non-line-of-sight error existing on the path from the target source to the i-th sensor receiving the measurement signal , n _i represents the measurement noise that exists on the path from the target source to the i-th sensor receiving the measurement signal, and n _i obeys a Gaussian distribution with zero mean Indicates the power of n _i , 0≤|n _i |<<e _i ≤ρ _i , the symbol "| |" is the absolute value symbol, and ρ _i represents the path that the measurement signal goes through from the target source to the i-th sensor receiving The upper limit of the non-line-of-sight error existing on ; ④ Transform d _i =d ₀ +||xs _i ||+e _i +n _i into d _i -e _i =(d ₀ +||xs _i ||)+n _i ; then d _i -e _i ＝(d ₀ +||xs _i ||)+n _i is squared on both sides to get then ignore middle get (d _i -e _i ) ² ≈(d ₀ +||xs _i ||) ² +2n _i (d ₀ +||xs _i ||); (d _i -e _i ) ² ≈(d ₀ +||xs _i ||) ² +2n _i (d ₀ +||xs _i ||) becomes ⑤ According to And using the worst-case robust weighted least squares method, the positioning problem in an environment with unknown signal transmission start time and non-line-of-sight errors is obtained, which is described as: Then order Will Into Next will Into Then determine the maximum value of f(e _i ) in the interval [0,ρ _i ], when d _i ≤ρ _i has When d _i >ρ _i , there is Finally will with substitute In , a non-convex positioning problem is obtained, which is described as: Among them, min() is the minimum value function, max() is the maximum value function, <mrow><mi>f</mi><mrow><mo>(</mo><mn>0</mn><mo>)</mo></mrow><mo>=</mo><mfrac><mrow><mo>|</mo><msup><mrow><mo>(</mo><msub><mi>d</mi><mi>i</mi></msub><mo>)</mo></mrow><mn>2</mn></msup><mo>-</mo><msup><mrow><mo>(</mo><msub><mi>d</mi><mn>0</mn></msub><mo>+</mo><mo>|</mo><mo>|</mo><mi>x</mi><mo>-</mo><msub><mi>s</mi><mi>i</mi></msub><mo>|</mo><mo>|</mo><mo>)</mo></mrow><mn>2</mn></msup><mo>|</mo></mrow><mrow><mo>(</mo><msub><mi>d</mi><mn>0</mn></msub><mo>+</mo><mo>|</mo><mo>|</mo><mi>x</mi><mo>-</mo><msub><mi>s</mi><mi>i</mi></msub><mo>|</mo><mo>|</mo><mo>)</mo></mrow></mfrac><mo>,</mo><mi>f</mi><mrow><mo>(</mo><msub><mi>d</mi><mi>i</mi></msub><mo>)</mo></mrow><mo>=</mo><mfrac><mrow><mo>|</mo><mo>-</mo><msup><mrow><mo>(</mo><msub><mi>d</mi><mn>0</mn></msub><mo>+</mo><mo>|</mo><mo>|</mo><mi>x</mi><mo>-</mo><msub><mi>s</mi><mi>i</mi></msub><mo>|</mo><mo>|</mo><mo>)</mo></mrow><mn>2</mn></msup><mo>|</mo></mrow><mrow><mo>(</mo><msub><mi>d</mi><mn>0</mn></msub><mo>+</mo><mo>|</mo><mo>|</mo><mi>x</mi><mo>-</mo><msub><mi>s</mi><mi>i</mi></msub><mo>|</mo><mo>|</mo><mo>)</mo></mrow></mfrac><mo>=</mo><mrow><mo>(</mo><msub><mi>d</mi><mn>0</mn></msub><mo>+</mo><mo>|</mo><mo>|</mo><mi>x</mi><mo>-</mo><msub><mi>s</mi><mi>i</mi></msub><mo>|</mo><mo>|</mo><mo>)</mo></mrow><mo>,</mo></mrow> "st" means "subject to"; ⑥Introduce auxiliary variables η ₁ , η ₂ ,…,η _i ,…η _N in the non-convex positioning problem, the Equivalently described as: Then substitute f(0), f(ρ _i ), f(d _i ) into And introduce auxiliary variables b ₁ , b ₂ ,…b _i ,…b _N , r and y to get Then, the second-order cone relaxation method is used to <mrow><munder><mrow><mi>m</mi><mi>i</mi><mi>n</mi></mrow><mrow><mi>x</mi><mo>,</mo><msub><mi>d</mi><mn>0</mn></msub><mo>,</mo><mi>y</mi><mo>,</mo><mi>r</mi><mo>,</mo><mo>{</mo><msub><mi>&amp;eta;</mi><mi>i</mi></msub><mo>,</mo><msub><mi>b</mi><mi>i</mi></msub><mo>}</mo></mrow></munder><mrow><mo>(</mo><munderover><mo>&amp;Sigma;</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>N</mi></munderover><msub><mi>&amp;eta;</mi><mi>i</mi></msub><mo>)</mo></mrow></mrow> <mfenced open = "" close = ""><mtable><mtr><mtd><mrow><mi>s</mi><mo>.</mo><mi>t</mi><mo>.</mo></mrow></mtd><mtd><mrow><mfrac><msup><mrow><mo>(</mo><msubsup><mi>d</mi><mi>i</mi><mn>2</mn></msubsup><mo>-</mo><mi>r</mi><mo>-</mo><mi>y</mi><mo>+</mo><mn>2</mn><msubsup><mi>s</mi><mi>i</mi><mi>T</mi></msubsup><mi>x</mi><mo>-</mo><mo>|</mo><mo>|</mo><msub><mi>s</mi><mi>i</mi></msub><mo>|</mo><msup><mo>|</mo><mn>2</mn></msup><mo>-</mo><msub><mi>b</mi><mi>i</mi></msub><mo>)</mo></mrow><mn>2</mn></msup><mrow><mi>r</mi><mo>+</mo><mi>y</mi><mo>-</mo><mn>2</mn><msubsup><mi>s</mi><mi>i</mi><mi>T</mi></msubsup><mi>x</mi><mo>+</mo><mo>|</mo><mo>|</mo><msub><mi>s</mi><mi>i</mi></msub><mo>|</mo><msup><mo>|</mo><mn>2</mn></msup><mo>+</mo><msub><mi>b</mi><mi>i</mi></msub></mrow></mfrac><mo>&amp;le;</mo><mn>4</mn><msubsup><mi>&amp;sigma;</mi><mi>i</mi><mn>2</mn></msubsup><msub><mi>&amp;eta;</mi><mi>i</mi></msub></mrow></mtd></mtr></mtable></mfenced> <mrow><mfrac><msup><mrow><mo>(</mo><msubsup><mi>&amp;rho;</mi><mi>i</mi><mn>2</mn></msubsup><mo>-</mo><mn>2</mn><msub><mi>d</mi><mi>i</mi></msub><msub><mi>&amp;rho;</mi><mi>i</mi></msub><mo>+</mo><msubsup><mi>d</mi><mi>i</mi><mn>2</mn></msubsup><mo>-</mo><mi>r</mi><mo>-</mo><mi>y</mi><mo>+</mo><mn>2</mn><msubsup><mi>s</mi><mi>i</mi><mi>T</mi></msubsup><mi>x</mi><mo>-</mo><mo>|</mo><mo>|</mo><msub><mi>s</mi><mi>i</mi></msub><mo>|</mo><msup><mo>|</mo><mn>2</mn></msup><mo>-</mo><msub><mi>b</mi><mi>i</mi></msub><mo>)</mo></mrow><mn>2</mn></msup><mrow><mi>r</mi><mo>+</mo><mi>y</mi><mo>-</mo><mn>2</mn><msubsup><mi>s</mi><mi>i</mi><mi>T</mi></msubsup><mi>x</mi><mo>+</mo><mo>|</mo><mo>|</mo><msub><mi>s</mi><mi>i</mi></msub><mo>|</mo><msup><mo>|</mo><mn>2</mn></msup><mo>+</mo><msub><mi>b</mi><mi>i</mi></msub></mrow></mfrac><mo>&amp;le;</mo><mn>4</mn><msubsup><mi>&amp;sigma;</mi><mi>i</mi><mn>2</mn></msubsup><msub><mi>&amp;eta;</mi><mi>i</mi></msub></mrow> 3 If d _i ≤ ρ _i A[x, y, d ₀ , r] ^T ≤ f y=||x|| ² , b _i =2d ₀ ||xs _i || The y=||x|| ² in slack is ||x|| ² ≤y, relax to The second-order cone programming problem is obtained, which is described as: Among them, η ₁ , η ₂ ,...,η _i ,...η _N correspond to the first auxiliary variable introduced in the non-convex positioning problem, the second auxiliary variable, ..., the i-th auxiliary variable, ..., the Nth Auxiliary variables, b ₁ ,b ₂ ,… _bi ,…b _N correspond to The first auxiliary variable introduced in , the second auxiliary variable, ..., the i-th auxiliary variable, ..., the N-th auxiliary variable, r and y correspond to the additional two auxiliary variables introduced in the representation, which are si Transpose, which is the transpose of s1, which is the transpose of sN, d1 represents the signal transmission distance between the target source and the first sensor, dN represents the signal transmission distance between the target source and the Nth sensor, [x, y,d0,r]T is the transpose of [x,y,d0,r], <mrow><mi>f</mi><mo>=</mo><mfenced open = "[" close = "]"><mtable><mtr><mtd><mrow><msubsup><mi>d</mi><mn>1</mn><mn>2</mn></msubsup><mo>-</mo><mo>|</mo><mo>|</mo><msub><mi>s</mi><mn>1</mn></msub><mo>|</mo><msup><mo>|</mo><mn>2</mn></msup></mrow></mtd></mtr><mtr><mtd><mo>.</mo></mtd></mtr><mtr><mtd><mo>.</mo></mtd></mtr><mtr><mtd><mo>.</mo></mtd></mtr>mtr><mtr><mtd><mrow><msubsup><mi>d</mi><mi>N</mi><mn>2</mn></msubsup><mo>-</mo><mo>|</mo><mo>|</mo><msub><mi>s</mi><mi>N</mi></msub><mo>|</mo><msup><mo>|</mo><mn>2</mn></msup></mrow></mtd></mtr></mtable></mfenced><mo>;</mo></mrow> ⑦ Use the interior point method to solve the second-order cone programming problem, and obtain the global optimal solution, denoted as x*, x* is the final estimated value of the coordinate position of the target source in the reference coordinate system. 2. The time-of-arrival positioning method based on the unknown starting time in the non-line-of-sight environment according to claim 1, characterized in that the constraint condition A[x,y,d ₀ ,r] ^T in the described step ⑥ The determination process of ≤f is: according to the conclusion that the measurement noise is much smaller than the non-line-of-sight error in general, it is obtained that d _i ≥d ₀ +||xs _i ||; and then d _i ≥d ₀ +||xs _i || Transform into d _i -d ₀ ≥||xs _i ||, and perform square expansion on both sides of d _i -d ₀ ≥||xs _i ||, get Then according to y=||x|| ² and Will Into Finally will Described in vector form, described as: A[x,y,d ₀ ,r] ^T ≤f, 3. The time-of-arrival positioning method based on the unknown starting time in the non-line-of-sight environment according to claim 1 or 2, characterized in that the constraints in the step ⑥ The determination process is: ⑥_1. According to the constraints before relaxation b _i =2d ₀ ||xs _i ||, use the square sum inequality to get Next will Expanding the square term in , we get Then y=||x|| ² and substitute , get the constraints on the clock error Then according to the known condition d ₀ ≥0, get b _i ≥0; ⑥_2. For the constraint condition before relaxation, b _i =2d ₀ ||xs _i || Then substitute y=||x|| ² into in, get Next will Relax to Convex Constraints then substitute in, get