A Tri-Star Time Difference of Arrival (TDOA) Location Method under Non-Convergent Conditions

Abstract

The present invention discloses a three-star ground-to-space time difference positioning method under non-convergent conditions, comprising the following steps: respectively obtaining the times when the main satellite, the first deputy satellite, and the second deputy satellite receive the same signal from the radiation source, as well as the coordinates of the three satellites at the corresponding moments, and constructing two hyperboloids and an ellipsoid with the time difference of the signal reaching the satellites and the Earth's surface; selecting the sub-satellite point T0 of the main satellite as the initial target point, and respectively searching for the point P1 closest to the target point on the first hyperboloid and the point P2 closest to the target point on the second hyperboloid, and then searching for the point T on the ellipsoid with the minimum sum of the squares of the distances from the point P1 and the point P2 m as the new target point; repeating the above search process until the coordinate values of the point T m converge; outputting the coordinate values of the point T m as the positioning result of the radiation source. The present invention solves the problem of three-star ground-to-space time difference positioning under non-convergent conditions and provides a reliable initial value of the target position for the iterative algorithm.

Description

Technical Field

[0001] The present invention relates to satellite positioning technology, and particularly to a three-star ground-to-space time difference positioning method under non-convergent conditions Background Art

[0002] Common positioning methods of space-based passive positioning systems include single-star direction finding positioning, two-star time difference / frequency difference positioning, and multi-star time difference positioning, etc. Among them, the multi-star time difference positioning method only needs to measure the signal arrival time to achieve instantaneous positioning of the radiation source (positioning based on a single pulse), and has high positioning accuracy, so it has been widely used. The current three-star ground-to-space time difference positioning method is generally as follows: By assuming the target height, an equation of the earth ellipsoid surface where the target is located is constructed, and then two hyperboloid equations based on time difference are constructed by combining the signal capture times measured by three satellites, and the target positioning is achieved by solving the intersection point of the above three equations. The solving methods include analytical calculation method, spherical iterative method, Newton iterative method, etc. No matter which solving method is adopted for the current three-star ground-to-space time difference positioning method, it is required that the two hyperboloids constructed by the three-star time difference intersect with the earth ellipsoid surface at one point. If this intersection point does not exist, problems such as the solution result being a complex number will occur, thus causing the positioning method to fail

[0003] In practical applications, due to reasons such as the unknown actual height of the target, deviation of the real-time coordinates of the satellite, and time difference measurement deviation, there will be a situation where the two hyperboloids constructed by the three-star time difference cannot intersect with the earth ellipsoid surface, resulting in the non-existence of the analytical solution of the intersection point of the three surfaces. In addition, in iterative algorithms such as the Newton iterative method and the spherical iterative method, it is necessary to set the initial value of the target position as the starting point of iteration. This point usually uses the analytical solution of the intersection point of the three surfaces under the assumption that the target elevation is 0, or the sub-satellite point of the master station satellite as the initial value. When the analytical solution of the intersection point of the three surfaces does not exist, or the sub-satellite point is far from the actual target, the iterative algorithm cannot proceed or the iterative convergence performance is affected Summary of the Invention

[0004] The technical problem to be solved by the present invention: Aiming at the above problems of the prior art, a three-star ground-to-space time difference positioning method under non-convergent conditions is provided. By adopting the method of approximately estimating the target position, the restriction that the two hyperboloids formed by the three-star time difference must intersect with the earth ellipsoid surface is removed, and instead, the point on the earth ellipsoid surface with the closest distance to the two hyperboloids is solved as the target position estimation result to solve the three-star ground-to-space time difference positioning problem under non-convergent conditions, and a reliable target position initial value for the three-star time difference positioning iterative algorithm is also provided

[0005] In order to solve the above technical problems, the technical solution adopted by the present invention is as follows

[0006] A three-star ground-to-space time difference positioning method under non-convergent conditions, comprising the following steps

[0007] S1) Obtain the times when the main satellite, the first deputy satellite, and the second deputy satellite receive the same signal from the radiation source, as well as the coordinates of the main satellite, the first deputy satellite, and the second deputy satellite when receiving the signal, and construct the first hyperboloid and the second hyperboloid based on these;

[0008] S2) Select the sub-satellite point T0 of the main satellite on the earth as the initial target point, and search for the point P1 on the first hyperboloid that is closest to the initial target point and the point P2 on the second hyperboloid that is closest to the initial target point respectively;

[0009] S3) Search for the point T on the earth with the minimum sum of the squares of the distances to the point P1 and the point P2 m As the new target point, search for the point P1 on the first hyperboloid that is closest to this target point and the point P2 on the second hyperboloid that is closest to this target point again, and repeat this step until the coordinate values of the point T m Converge;

[0010] S4) Output the coordinate values of the point T m As the positioning result of the radiation source.

[0011] Further, the steps of constructing the first hyperboloid and the second hyperboloid in step S1) include:

[0012] Convert the coordinates of the main satellite, the first deputy satellite, and the second deputy satellite from the geodetic coordinate system to the earth-fixed coordinate system respectively;

[0013] Establish a first hyperboloid equation based on the time difference between the main satellite and the first deputy satellite receiving the same signal and the coordinates of the main satellite and the first deputy satellite when receiving the signal, and establish a second hyperboloid equation based on the time difference between the main satellite and the second deputy satellite receiving the same signal and the coordinates of the main satellite and the second deputy satellite when receiving the signal.

[0014] Further, both step S2) and step S3) include the step of searching for the point on the current hyperboloid that is closest to the target point, specifically including:

[0015] S21) Calculate the intersection points of the line connecting the main satellite of the current hyperboloid and the earth's center with the current hyperboloid, and the intersection points of the line connecting the first / second deputy satellite of the current hyperboloid and the earth's center with the current hyperboloid respectively, and take the intersection point that is closest to the target point as the initial estimation point;

[0016] S22) Based on the initial estimation point, use the Newton iteration algorithm to determine the coordinate estimation value of the point on the current hyperboloid that is closest to the target point.

[0017] Further, when the current hyperboloid is the first hyperboloid, step S22) specifically includes:

[0018] S221) Establish the system of equations that the point P1 satisfies F(P1 k) = [f1 f2 f3] T ;

[0019] S222) Derive the partial differential equation system F′(P1 k ) by differentiating the equation system F(P1 k );

[0020] S223) Establish an iterative formula and start iterating from the initial estimation point until the distance between the estimation points obtained from two consecutive calculations is less than a preset first threshold. The expression of the iterative formula is as follows:

[0021]

[0022] In the above formula, is the estimated point of point P1 obtained in the k-th iteration.

[0023] Furthermore, in step S221), the expressions of the equations f1, f2, and f3 in the equation system F(P1 k ) are as follows:

[0024] f1 = r0k1 + r1k2

[0025] f2 = r0k1′ + r1k2′

[0026]

[0027] In the above formula, where r0 is the distance from point P1 to the main satellite, Δr1 is the distance difference between the radiation source and the main satellite and the first deputy satellite, r1 is the distance from point P1 to the first deputy satellite, x t , y t , z t are the coordinate values of the radiation source respectively, x s0 , y s0 , z s0 are the coordinate values of the main satellite respectively, x s1 , y s1 , z s1 are the coordinate values of the first deputy satellite respectively, x p1 , y p1 , z p1 are the estimated coordinate values of point P1 respectively.

[0028] Furthermore, in step S3), search for the point T on the earth with the minimum sum of the squares of the distances to point P1 and point P2 m Specifically includes:

[0029] S31) Set the conditions satisfied by point T m including:

[0030] Condition 1:

[0031] Condition 2:

[0032] wherein, are the coordinate values of point T m respectively, d i , i = 1, 2 are the distances from point T m to P1 and P2; N is the radius of curvature of the prime vertical circle of the earth at the point where point T m is located, and e 2 is the square of the first eccentricity of the earth;

[0033] S32) Convert the distances from point T m to P1 and P2 into the projections of the vectors from T m to P1 and P2 on the normal lines at points P1 and P2, to obtain:

[0034]

[0035] In the above formula, x p1 , y p1 , z p1 are the estimated coordinate values of point P1 respectively, and x p2 , y p2 , z p2 are the estimated coordinate values of point P2 respectively, n1 is the normal vector of the first hyperboloid, and n2 is the normal vector of the second hyperboloid;

[0036] S33) Based on the formula d1 2 + d2 2 take partial derivatives with respect to respectively and make them equal to 0, to obtain the equation of with respect to , and then substitute the equation of with respect to into Condition 2, to obtain the estimated result of ;

[0037] S34) Remove the ambiguous values less than or equal to 0 from the estimated result of , and then substitute the estimated result of into the equation of with respect to , to obtain the estimated result of .

[0038] Furthermore, the equation expressions of with respect to in steps S33) and S34) are as follows:

[0039]

[0040] wherein,

[0041] x p1 、y p1 、z p1 are respectively the estimated coordinate values of point P1, x p2 、y p2 、z p2 are respectively the estimated coordinate values of point P2, n1 is the normal vector of the first hyperboloid, n2 is the normal vector of the second hyperboloid, and k i , i = 1, 2, 3 are the coordinate values of vector n1, k i ′, i = 1, 2, 3 are the coordinate values of vector n2.

[0042] Further, the convergence of the coordinate value of T in step S3) specifically means that: the distance between the points T m calculated continuously twice and the point T m is less than a preset second threshold. m-1

[0043] The present invention also provides a computer, which is programmed or configured to execute any one of the three-star ground time difference positioning methods under non-convergence conditions.

[0044] The present invention also provides a computer-readable storage medium, in which a computer program is stored that is programmed or configured to execute any one of the three-star ground time difference positioning methods under non-convergence conditions.

[0045] Compared with the prior art, the present invention has the following advantages:

[0046] 1. The present invention performs coordinate search based on the distance minimum criterion, adopts an iterative method, and alternately iteratively searches for the point on the earth ellipsoid with the minimum sum of the squares of the distances from the time difference hyperboloid, and the point on the hyperboloid closest to the target estimated point, and approaches the final solution through repeated iteration. It releases the constraint that the three-star time difference hyperboloid and the earth ellipsoid must intersect, realizes three-star positioning under non-convergence conditions, and expands the applicable range of the positioning method.

[0047] 2. The present invention performs iterative solution based on the geometric relationship between the two-star time difference surface and the earth ellipsoid. Through the distance minimum principle, a unique real solution can be determined, that is, there is no complex solution situation common in traditional three-star positioning algorithms, nor is there a problem of removing false targets from multiple estimated values. BRIEF DESCRIPTION OF THE DRAWINGS

[0048] Figure 1 is the geometric schematic diagram of the embodiment of the present invention.

[0049] Figure 2 is the flowchart of the embodiment of the present invention.​

[0050] Figure 3 This is the positioning error distribution diagram of the embodiment of the present invention. Detailed implementation manners

[0051] The present invention will be further described below in conjunction with the accompanying drawings of the specification and specific preferred embodiments, but the protection scope of the present invention is not limited thereby.

[0052] The time difference positioning of three stars relative to the ground requires three satellites, one of which is called the main satellite, and the other two are called sub-satellites. For easy distinction, these two sub-satellites are numbered 1 and 2 respectively, and can also be called the first sub-satellite and the second sub-satellite. The main satellite and the first sub-satellite form the foci of a hyperboloid, and this hyperboloid is determined by the time difference of the signal arriving at the main satellite and the first sub-satellite; at the same time, the main satellite and the second sub-satellite form the foci of another hyperboloid, and this hyperboloid is determined by the time difference of the signal arriving at the main satellite and the second sub-satellite; these two hyperboloids are also numbered 1 and 2 for distinction, and can also be called the first hyperboloid and the second hyperboloid. As Figure 1 shown, under non-convergent conditions, there are no intersection points between the first hyperboloid and the second hyperboloid and the Earth ellipsoid, resulting in the non-existence of the analytical solution of the system of equations formed by the first hyperboloid, the second hyperboloid and the Earth ellipsoid. Therefore, we consider using an iterative algorithm to alternately search for the point on the Earth ellipsoid with the minimum sum of the squares of the distances from the time difference hyperboloids, and the point on the hyperboloid closest to the target estimation point, and approach the final solution through repeated iterations.

[0053] Based on the above idea, this embodiment proposes a three-star time difference positioning method relative to the ground under non-convergent conditions, as Figure 2 shown, including the following steps:

[0054] S1) Respectively obtain the time when the main satellite, the first sub-satellite, and the second sub-satellite receive the same signal from the radiation source, and the coordinates of the main satellite, the first sub-satellite, and the second sub-satellite when receiving the signal, and construct the first hyperboloid and the second hyperboloid therefrom;

[0055] S2) Select the sub-satellite point T0 of the main satellite on the Earth as the initial target point, and respectively search for the point P1 on the first hyperboloid closest to the initial target point and the point P2 on the second hyperboloid closest to the initial target point;

[0056] S3) Search for the point T on the Earth with the minimum sum of the squares of the distances from the point P1 and the point P2 m as the new target point, and again respectively search for the point P1 on the first hyperboloid closest to this target point and the point P2 on the second hyperboloid closest to this target point, and repeat this step until the coordinate value of the point T m converges;

[0057] S4) Output the coordinate value of point T m as the positioning result of the radiation source.

[0058] In step S1) of this embodiment, assume that the times when the main satellite, the first deputy satellite, and the second deputy satellite receive the same signal from the radiation source are t i , i = 0, 1, 2, and the satellite coordinates at this time are O i (B si , L si , H si ), i = 0, 1, 2, where B, L, and H respectively refer to latitude, longitude, and altitude in the geodetic coordinate system; i is the satellite serial number, the main satellite serial number is 0, the first deputy satellite serial number is 1, and the second deputy satellite serial number is 2; the steps of constructing the first hyperboloid and the second hyperboloid at this time include:

[0059] First, convert the coordinates of the main satellite, the first deputy satellite, and the second deputy satellite from the geodetic coordinate system to the Earth-fixed coordinate system respectively to obtain the corresponding coordinate values O i (x si , y si , z si ), i = 0, 1, 2. Converting from the geodetic coordinate system to the Earth-fixed coordinate system is a common method for those skilled in the art, and the conversion process will not be elaborated here.

[0060] Then, according to the time difference between the main satellite and the first deputy satellite receiving the same signal and the conversion results of the coordinates of the main satellite and the first deputy satellite when receiving the signal, establish the first hyperboloid equation, and according to the time difference between the main satellite and the second deputy satellite receiving the same signal and the conversion results of the coordinates of the main satellite and the second deputy satellite when receiving the signal, establish the second hyperboloid equation.

[0061] Specifically, the time difference Δt1 between the main satellite and the first deputy satellite receiving the same signal is Δt1 = t1 - t0, and the time difference Δt2 between the main satellite and the second deputy satellite receiving the same signal is Δt2 = t2 - t0. These two time differences correspond to the distance difference between the radiation source and the two satellites: Δr i = c·Δt i , i = 1, 2, where c is the speed of light.

[0062] Obviously, the points with two satellites as foci and the distance difference to the two satellites equal to Δr i , i = 1, 2 form a hyperboloid C i , i = 1, 2. The radiation source T(x t , y t , z t ) should be located on this hyperboloid. Therefore, the equations of the first hyperboloid and the second hyperboloid can be expressed as

[0063]

[0064] In the above formula, x t , y t , z t are the coordinate values of the radiation source respectively, and x s0 , y s0 , z s0 are the coordinate values of the main satellite respectively, and x s1 , y s1 , z s1 are the coordinate values of the first deputy satellite respectively, and x s2 , y s2 , z s2 are the coordinate values of the second deputy satellite respectively. Δr1 is the distance difference between the radiation source and the main satellite and the first deputy satellite, and Δr2 is the distance difference between the radiation source and the main satellite and the second deputy satellite.

[0065] In step S2) of this embodiment, point T0 can be any point on the earth ellipsoid surface close to the main satellite, but usually the sub-satellite point of the main satellite is selected. The coordinate value of point T0 in the geodetic coordinate system is T0(B s0 , L s0 , 0). Similarly, it needs to be converted to the coordinate value in the earth-fixed coordinate system as

[0066] After obtaining the coordinate value of the initial target point T0, it is necessary to search for the point closest to the target point on the first hyperboloid and the second hyperboloid respectively, including the following steps:

[0067] S21) Calculate the intersection points of the connection lines between the main satellite and the earth center and the current hyperboloid, and the connection lines between the first / second deputy satellites and the earth center and the current hyperboloid respectively, and use the intersection point closest to the target point among them as the initial estimation point;

[0068] S22) Based on the initial estimation point, use the Newton iteration algorithm to determine the coordinate estimation value of the point closest to the target point on the current hyperboloid.

[0069] Taking the first hyperboloid as an example of the current hyperboloid, in step S21), first use the intersection point of the connection line between the main satellite O0 or the first deputy satellite O1 and the earth center and the first hyperboloid C1 as the initial point for estimating point P1 Obviously, the connection lines between the two satellites and the earth center must intersect with the first hyperboloid C1, but which intersection point is used as the initial point is related to the spatial characteristics of the hyperboloid. In practical applications, the intersection points of the connection lines between these two satellites and the earth center and the first hyperboloid C1 can be calculated respectively, and the point closest to T0 among all the solutions can be selected as the initial point

[0070] Taking the main satellite O0 as an example, the analytical solution of the intersection point is analyzed as follows: Since the geocentric coordinates in the geocentric fixed coordinate system are (0, 0, 0), the coordinates of any point on the line connecting the main satellite O0 and the geocenter are (k0·x s0 , k0·y s0 , k0·z s0 ), where 0 < k0 < 1. Substituting this point into the equation of the first hyperboloid C1 in Equation (1), the intersection point equation is obtained as follows:

[0071]

[0072] The analytical solution of this equation is:

[0073]

[0074] In the above formula, 1 is the straight-line distance from the satellite O i , i = 0, 1 to the geocenter. Furthermore, the intersection point of the line connecting the main satellite O0 and the geocenter and the first hyperboloid C1 can be obtained as

[0075] Using the same algorithm, the intersection point of the line connecting the first secondary satellite O1 and the geocenter and the first hyperboloid C1 can be obtained Then That is, the point closest to T0 is selected as the initial point Here, L(J i , T0) represents the distance between the intersection point J i and T0.

[0076] Taking the first hyperboloid as an example for the current hyperboloid, step S22) specifically includes:

[0077] S221) Establish the system of equations satisfied by point P1 The expressions of the equations f1, f2, and f3 in the system of equations are as follows:

[0078] f1 = r0k1 + r1k2

[0079] f2 = r0k1′ + r1k2′

[0080]

[0081] In the above formula, where r0 is the distance from point P1 to the main satellite, Δr1 is the distance difference between the radiation source and the main satellite and the first secondary satellite, r1 is the distance from point P1 to the first secondary satellite, x t , y t , z t are the coordinate values of the radiation source respectively, and x s0 , y s0 , z s0The coordinate values of the main satellite are x s1 , y s1 , z s1 The coordinate values of the first deputy satellite are x p1 , y p1 , z p1 They are respectively the coordinate estimated values of point P1;

[0082] S222) Differentiate the system of equations F(P1 k ) to obtain the system of partial differential equations F′(P1 k ), and the expression is as follows:

[0083]

[0084] Among them, r0 is the distance from point P1 to the main satellite, Δr1 is the distance difference between the radiation source and the main satellite and the first deputy satellite, r1 is the distance from point P1 to the first deputy satellite, x t , y t , z t They are respectively the coordinate values of the radiation source, x s0 , y s0 , z s0 They are respectively the coordinate values of the main satellite, x s1 , y s1 , z s1 They are respectively the coordinate values of the first deputy satellite, x p1 , y p1 , z p1 They are respectively the coordinate estimated values of point P1;

[0085] S223) Establish an iterative formula and start iteration from the initial estimated point , and the expression of the iterative formula is as follows:

[0086]

[0087] In the above formula, is the estimated point of point P1 obtained by the k-th iteration;

[0088] Until the distance between the estimated points obtained by two consecutive calculations is less than the preset first threshold, that is, the distance d1 between the P1 estimated points obtained by two consecutive calculations satisfies the following formula:

[0089]

[0090] Among them, λ1 is the given first threshold, k represents the iteration number in step S223), is the coordinate estimated value of point P1 obtained by the k-th iteration, x k+1 p1 , y k+1p1 , z k+1 p1 is the coordinate estimation value of point P1 obtained in the (k + 1)-th iteration.

[0091] Obviously, the equation f3 in step S221) is derived from the hyperbolic function, which will not be elaborated here; the derivation processes of equations f1 and f2 are as follows: Since point P1 is the point on the first hyperboloid C1 closest to point T0, the line connecting point P1 and point T0 must be perpendicular to the first hyperboloid C1. The normal vector n1 of the first hyperboloid C1 can be obtained by taking the partial derivative of the hyperboloid function, that is:

[0092]

[0093] where

[0094] Since both point P1 and radiation source T are on this normal line, there is Here This equation forms two equations, namely f1 and f2.

[0095] The process of determining point P2 through steps S21) and S22) is basically the same as the foregoing content, and only the relevant parameters of the first hyperboloid C1 and the first deputy satellite need to be replaced with the relevant parameters of the second hyperboloid C2 and the second deputy satellite, which will not be elaborated here.

[0096] After obtaining point P1(x1, y1, z1) and P2(x2, y2, z2) from steps S21) and S22), in step S3), the point T on the earth with the minimum sum of the squares of the distances to point P1 and point P2 is searched through multiple iterations m , and points P1 and P2 are updated correspondingly. m ≥ 1 represents the number of iterations. Searching for the point T on the earth with the minimum sum of the squares of the distances to point P1 and point P2 m includes the following steps:

[0097] S31) Set the conditions that point T m satisfies, including:

[0098] Condition 1:

[0099] Condition 2:

[0100] where are the coordinate values of point T m respectively, d i , i = 1, 2 are the distances from point T m to P1 and P2; N is the radius of curvature of the earth's prime vertical circle at the point where point T m is located, Here, R is the semi-major axis of the Earth, R = 6378137 m, and e 2 is the square of the first eccentricity of the Earth, e 2 = 0.00669437999013;

[0101] Obviously, condition two comes from point T m must satisfy the Earth ellipsoid equation, which will not be elaborated here. Next, the formula corresponding to condition one will be derived:

[0102] S32) The distance from point T m to P1 and P2 is equal to the projection of the vector from point T m to P1 and P2 on the normal vectors at points P1 and P2. Therefore, the distance from point T m to P1 and P2 is converted into the projection of the vector from T m to P1 and P2 on the normal vectors at points P1 and P2, resulting in:

[0103]

[0104] In the above formula, x p1 , y p1 , z p1 are the estimated coordinate values of point P1 respectively, and x p2 , y p2 , z p2 are the estimated coordinate values of point P2 respectively. n1 is the normal vector of the first hyperboloid, and n2 is the normal vector of the second hyperboloid. Both n1 and n2 can be obtained from the relevant content derived from equations f1 and f2 in step S221), which will not be elaborated here;

[0105] S33) Based on the formula of d1 2 +d2 2 take partial derivatives with respect to respectively, and set them to 0 to obtain the equation about , and the expression is as follows:

[0106]

[0107] Among them,

[0108]

[0109] and k i , i = 1, 2, 3 are the coordinate values of the vector n1, and k i′, where i = 1, 2, 3 are the coordinate values of the vector n2, that is, k1 = n1(1), k2 = n1(2), k3 = n1(3), k1′ = n2(1), k2′ = n2(2), k3′ = n2(3). The coordinate values of n1 and n2 can be obtained from the relevant content derived from the equations f1 and f2 in step S221), which will not be elaborated here;

[0110] Then substitute the equation about into condition two to obtain the estimation result of

[0111]

[0112] where A = b 2 + d 2 + 1 / (1 - e 2 ) 2 , B = 2ab + 2cd, C = a 2 + c 2 - N 2 ;

[0113] S34) Remove the fuzzy values less than or equal to 0 in the estimation result of because usually if all three satellites are in the Northern Hemisphere, the target must be in the Northern Hemisphere, that is, z t > 0. Then substitute the estimation result of into the equation of about in formula (12) to obtain the estimation result of m so as to obtain the coordinate estimation value of point T

[0114] Next, determine whether the coordinate values of T m converge, specifically:

[0115] If the distance between the points T m and T m-1 obtained from two consecutive calculations is less than the preset second threshold, that is, it satisfies d = ||T m-1 - T m || ≤ λ, where λ is the given second threshold, then jump to step S4);

[0116] If the above situation is not satisfied, then take point T m as the target point, and according to the coordinate values of point T m , execute step S21 and step S22 again, search for the points P1 and P2 closest to the target point on the first hyperboloid and the second hyperboloid respectively, and then continue to execute step S31) to step S34) to obtain point T m+1The coordinates of point T are estimated and the m With point T m+1 Whether the distance between them meets the above conditions.

[0117] The verification results of this scheme through simulation are as follows Figure 3 As shown in the figure, the simulation parameters are as follows: the coordinates of the main satellite in the geodetic coordinate system are: longitude 120°, latitude 45°, altitude 500Km; the coordinates of the secondary satellite 1 (i.e. the first secondary satellite) are longitude 115°, latitude 35°, altitude 500Km; the coordinates of the secondary satellite 2 (i.e. the second secondary satellite) are longitude 125°, latitude 35°, altitude 500Km, and the positioning error measurement standard is the straight-line distance between the estimated position and the actual position. Figure 3 As shown in the figure, when the true coordinate longitude of the radiation source is between [115°, 125°] and the latitude is distributed between [30°, 38°], the time difference measurement error is 100μs, the satellite longitude and latitude positioning error is 0.1°, and the height error is 1Km, the final positioning error is within 2Km, which meets the accuracy requirements of time difference positioning.

[0118] The present invention also provides a computer, which is programmed or configured to execute any of the three-star ground time difference positioning methods under non-convergence conditions.

[0119] The present invention also proposes a computer-readable storage medium, in which a computer program programmed or configured to execute any of the three-star ground time difference positioning methods under non-convergence conditions is stored.

[0120] In summary, the present invention has the following advantages:

[0121] (1) In order to solve the problem that the traditional three-star time difference positioning algorithm cannot converge due to large errors in time difference measurement and satellite coordinate measurement, a positioning algorithm based on the minimum distance criterion for coordinate search is proposed. This algorithm removes the constraint that the three-star time difference hyperbolic surface and the earth ellipsoid must intersect in the conventional algorithm, realizes three-star positioning under non-convergence conditions, and expands the scope of application of the positioning method.

[0122] (2) An iterative algorithm is used to perform alternating iterative searches for the point on the Earth's ellipsoid with the smallest sum of squared distances from the time difference hyperboloid and the point on the hyperboloid closest to the target estimated point, thereby reducing the computational difficulty and approaching the final solution through repeated iterations.

[0123] (3) An iterative solution is performed based on the geometric relationship between the binary star time difference surface and the earth's ellipsoid. The unique real number solution can be determined by the principle of minimum distance. That is, there is no complex number solution commonly seen in traditional three-star positioning algorithms, and there is no problem of removing false targets from multiple estimates.

[0124] (4) Relax the requirements for the measurement accuracy of the signal arrival time and the satellite positioning accuracy, and the implementation is simple, which is conducive to engineering implementation.

[0125] The above are only the preferred embodiments of the present invention, and do not impose any form of limitation on the present invention. Although the present invention has been disclosed above with the preferred embodiments, it is not intended to limit the present invention. Therefore, any simple modifications, equivalent changes and modifications made to the above embodiments based on the technical essence of the present invention without departing from the technical solution of the present invention shall fall within the scope of the protection of the technical solution of the present invention.

Claims

1. A three-star time difference of arrival (TDOA) positioning method under non-convergent conditions, characterized in that, Including the following steps: S1) Obtain the times when the main satellite, the first deputy satellite, and the second deputy satellite receive the same signal from the radiation source respectively, and the coordinates of the main satellite, the first deputy satellite, and the second deputy satellite when receiving the signal, and construct a first hyperboloid and a second hyperboloid therefrom; S2) Select the sub-satellite point T0 of the main satellite on the earth as the initial target point, and search for the point P1 on the first hyperboloid that is closest to the initial target point and the point P2 on the second hyperboloid that is closest to the initial target point respectively; S3) Search for the point T on the Earth with the minimum sum of the squares of the distances from point P1 and point P2 m As the new target point, search again for the point P1 closest to this target point on the first hyperboloid and the point P2 closest to this target point on the second hyperboloid, and repeat this step until the coordinate values of point T m converge; S4) Output the coordinate value of point T m as the positioning result of the radiation source.

2. The three-star time difference positioning method under non-convergence conditions according to claim 1, wherein, The steps of constructing the first hyperboloid and the second hyperboloid in step S1) include: Convert the coordinates of the main satellite, the first deputy satellite, and the second deputy satellite from the geodetic coordinate system to the earth-fixed coordinate system respectively; According to the time difference between the main satellite and the first deputy satellite receiving the same signal and the coordinates of the main satellite and the first deputy satellite when receiving the signal, establish a first hyperboloid equation, and according to the time difference between the main satellite and the second deputy satellite receiving the same signal and the coordinates of the main satellite and the second deputy satellite when receiving the signal, establish a second hyperboloid equation.

3. The three-star time difference positioning method under non-convergent conditions according to claim 1, characterized in that Both step S2) and step S3) include the step of searching for the point on the current hyperboloid that is closest to the target point, specifically including: S21) Calculate the intersection points of the connection line between the main satellite of the current hyperboloid and the earth's center and the current hyperboloid respectively, and the intersection points of the connection line between the first / second deputy satellite of the current hyperboloid and the earth's center and the current hyperboloid, and take the intersection point that is closest to the target point as the initial estimation point; S22) Based on the initial estimation point, use the Newton iteration algorithm to determine the coordinate estimation value of the point on the current hyperboloid that is closest to the target point.

4. The three-star time difference positioning method under non-convergent conditions according to claim 3, wherein When the current hyperboloid is the first hyperboloid, step S22) specifically includes: S221) Establish the system of equations satisfied by point P1, F(P1 k ) = [f1 f2 f3] T ; S222) Differentiate the system of equations F(P1 k ) to obtain the system of partial differential equations F′(P1 k ); S223) Establish an iterative formula and start iteration from the initial estimation point until the distance between the estimation points obtained from two consecutive calculations is less than a preset first threshold. The expression of the iterative formula is as follows: In the above formula, is the estimated point of point P1 obtained in the k-th iteration.

5. The three-star time difference positioning method under non-convergent conditions according to claim 4, characterized in that In step S221), the expressions of equations f1, f2, and f3 in the system of equations F(P1 k ) are as follows: f1 = r0k1 + r1k2 f2 = r0k1′ + r1k2′ In the above formula, where r0 is the distance from point P1 to the main satellite, Δr1 is the distance difference between the radiation source and the main satellite and the first deputy satellite, r1 is the distance from point P1 to the first deputy satellite, x t , y t , z t are the coordinate values of the radiation source respectively, x s0 , y s0 , z s0 are the coordinate values of the main satellite respectively, x s1 , y s1 , z s1 are the coordinate values of the first deputy satellite respectively, x p1 , y p1 , z p1 are the estimated coordinate values of point P1 respectively.

6. The three-star time difference positioning method under non-convergence conditions according to claim 1, characterized in that In step S3), search for the point T on the earth with the minimum sum of the squares of the distances from point P1 and point P2 m Specifically include: S31) Set point T m The satisfied conditions include: Condition 1: Condition 2: Among them, are the coordinate values of point T m respectively, d i , i = 1, 2 are the distances from point T m to P1 and P2; N is the radius of curvature of the prime vertical circle of the earth at the point where point T m is located, and e 2 is the square of the first eccentricity of the earth. S32) Convert the distances from point T m to P1 and P2 into the projections of the vectors from T m to P1 and P2 onto the normal vectors at points P1 and P2, obtaining: In the above formula, x p1 y p1 z p1 are respectively the estimated coordinate values of point P1, and x p2 y p2 z p2 are respectively the estimated coordinate values of point P2, n1 is the normal vector of the first hyperboloid, and n2 is the normal vector of the second hyperboloid; S33) Based on d1 2 + d2 2 Take partial derivatives with respect to using the formula respectively, and set them to 0 to obtain the equation about Then substitute the equation about into the second condition to obtain the estimation result of ; S34) Remove the fuzzy values less than or equal to 0 in the estimation result of, and then substitute the estimation result of into the equation about to obtain the estimation result of .

7. The three-star time difference positioning method under non-convergent conditions according to claim 6, characterized in that In step S33) and step S34) Regarding The equation expression is as follows: Among them, x p1 , y p1 , z p1 are the estimated coordinate values of point P1, x p2 , y p2 , z p2 are the estimated coordinate values of point P2, n1 is the normal vector of the first hyperboloid, n2 is the normal vector of the second hyperboloid, and k i , i = 1, 2, 3 are the coordinate values of vector n1, k i ', i = 1, 2, 3 are the coordinate values of vector n2.

8. The method for three-star ground time difference positioning under non-convergent conditions according to claim 1, wherein In step S3), the coordinate value of T m converges specifically as follows: The distance between the point T m calculated twice in succession and the point T m-1 is less than a preset second threshold value.

9. A computer, characterized in that, The computer is programmed or configured to execute the three-satellite ground-to-ground time difference positioning method under non-convergent conditions according to any one of claims 1 to 8.

10. A computer-readable storage medium, characterized in that, The computer-readable storage medium stores a computer program that is programmed or configured to execute the three-satellite ground-to-ground time difference positioning method under non-convergent conditions according to any one of claims 1 to 8.

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