WO2002086642A1

WO2002086642A1 - Method for determining 3-dimensional attitude of object using satellite vehicle and decrease of integer ambiguity to be searched

Info

Publication number: WO2002086642A1
Application number: PCT/KR2002/000676
Authority: WO
Inventors: Sangjeong Lee; Chansik Park; Seokbo Son; Youngbaek Kim
Original assignee: Navicom Co., Ltd.
Priority date: 2001-04-13
Filing date: 2002-04-12
Publication date: 2002-10-31
Also published as: KR20020079125A; KR100401154B1

Abstract

The present invention is related to a method for determining fast the integer number causing an integer ambiguity in obtaining the three-dimensional attitude of a vehicle by using a GPS attitude determination system. In particular, the present invention provides a method for reducing the number of independent integer numbers to be searched by using known lengths of at least two baseline vectors, each baseline vector being defined by two selected antennas, and an angle between two baseline vectors in determining the integer number included in measured values of carrier wave phase of GPS signals. The number of the independent integer number to be searched can be reduced by at least one, and thus time required for resolving the integer ambiguity and for determining the attitude of the vehicle can be reduced, calculation volume and the required memory of the calculator also can be reduced.

Description

Method for Determining 3-dimensional Attitude of Object Using Satellite Vehicle and Decrease of Integer Ambiguity to be Searched

Field of the Invention

The present invention relates to a method for providing fast and precise determination of a 3-dimensional attitude of a vehicle by using a Global Navigation Satellite System (GNSS) such as a Global Positioning System (GPS) , and more particularly, to a method for decreasing the range of integer numbers to be searched by using a length and a relative disposition of at least two baseline vectors between GPS antennas, and to a 3-dimensional attitude determination method using the same.

Description of the Prior Art

A Global Positioning System (hereinafter GPS' ) is a kind of a satellite navigation system enabling for user to obtain information on a position, velocity and time of the user everywhere in the world.

A GPS attitude determination system provides attitude information of an object as well as position information by using at least two antennae and a carrier phase measurement. In case that two antennas set in a progression direction of a vehicle, the attitude may be determined by using a pitch and a yaw.

Furthermore, when at least three antennas are provided, the 3-dimentional attitude may be determined.

An error of the attitude information determined by the GPS attitude determination system is not affected by the velocity of the vehicle, but depends on a length of baseline vector which defines a vector between antennas and an intensity of a receiver noise. The invention related to attitude determination method by using the GPS attitude determination system is disclosed in Korea patent application No. 1997-057696 filed by the same applicant of this application. An integer ambiguity included in a double-differenced carrier wave should be resolved in order to determine the attitude of the vehicle -by the GPS attitude determination system. The integer ambiguity is an initial bias of the carrier phase observed as a random cycle number. An initial satellite measurement is obtained when a GPS receiver gains initial capture of GPS signal. As of the initial capture, an ambiguity for a cycle integer may occur due to difficulty in fixing the accurate cycle number, which is called "integer ambiguity." That is, the integer ambiguity means the number of sine waves (or the phase number) existing between the satellite and the GPS receiver. Since the initial integer ambiguity may not be eliminated even by a double differencing, the integer ambiguity has to be resolved precisely in order to improve an accuracy of the position determination. Since the initial integer ambiguity occurs only as of the initial capture of the GPS signal, it is not necessary to resolve it again during consecutive receiving of the GPS signal from then on once the integer ambiguity is resolved. That is, the integer ambiguity is a random constant which changes values whenever the receiver initially captures the GPS signal but remains constant thereafter.

In summary, it is required to determine "an integer number" of wavelengths of the carrier wave component for a GPS attitude determination, and this determination of "the integer number" is called integer ambiguity resolution.

In a geodesy field requiring relatively low level of real-time positioning, various methods using the carrier phase have been researched. In order to use the carrier phase in determining an attitude of a vehicle by using the GPS, it is required a real-time determination for the integer number included in the carrier phase.

However, since the number of unknown integer numbers is more than that of measured number and the integer ambiguity should be an integer, it is substantially impossible to determine the integer numbers causing the integer ambiguity by an analytical method. Moreover, since a convexity in an integer domain may not be guaranteed, the integer ambiguity can not be resolved even bya nonlinear programming (NLP) .

Therefore, a method for searching a search region including the integer number is mainly used in resolving the integer ambiguity. In the integer ambiguity resolution for the attitude determination requiring a real-time resolution of the integer ambiguity rather than the geodesy where accurate locating is the key concern, research has been focused on determining the integer number by a single measurement .

The integer number causing the integer ambiguity may be determined by searching and should be an integer. The measurements of multiple epochs should be collectively used, since various On-the-fly (OTF) methods based on an integer least square method use only carrier phase and need an estimation value for the integer number. Therefore, if a measuring period is short, a search range of the integer number and a calculation amount for searching may increase. On the other hand, if the measuring period becomes longer, a calculation amount at each searching point may increase though the search range decreases. Moreover, it is not adequate for real-time application due to ignorance of a change of involving satellites during a searching process. In particular, if the transmission of satellite signals is interrupted for any reason even after resolving the integer ambiguity, all integer numbers should be determined again.

In the condition that the length of the baseline vector is 1 meter and LI frequency is used, the number of integer number candidates within the search range may reach several hundreds. Therefore, calculation amount for searching may increase, and an error probability of the integer number determination may increase.

In a method for determining 2-dimensional attitude of a vehicle, it is necessary to determine two independent integer numbers for one baseline vector between two antennas. As similar, for the 3-dimensional attitude determination, three 'independent integer numbers must be determined to define each baseline vector.

In case of multiple GPS satellite vehicle (SV) , integer numbers as many as number of satellite minus 1 (# of SV-1) should be determined for each baseline vector. However, by using a method, so called "Ambiguity- Resolution with Constraint Equation (ARCE)", only three independent integer numbers should be determined, and remaining integer numbers of (# of SV -l)-3 may be determined dependently. Such integer numbers of (# of SV -l)-3 are called "dependent integer numbers." The detail of the ARCE method is disclosed in Korean patent application no. 1997-576^'96.

In the conventional method, in order to determine three independent integer numbers used for defining one 3- dimensional baseline vector, the search range for each integer number is determined respectively from a double differencing (between satellites and between antennas) code signal and a carrier phase measurement for at least 4 satellites. Subsequently, an objective function may be calculated by using the search range for each independent integer number, and then a candidate integer number which makes a ^' value of the objective function to the minimum may^" be determined as a true integer number. As mentioned above, the search processes for three search range should be performed to define one baseline vector, which requires much process time and memory space since the calculation of the objective function should be performed for all candidates. To reduce calculation volume, there has been provided a method in which the search range for only two independent integer numbers may be searched, and remaining one independent integer number may be derived using the length of the baseline vector on the assumption that the length of the baseline vector- is known.

For the 3-dimensional attitude determination, however, even by using the state-of-the-art method, search for at least 2 (N - 1) integer numbers is necessary since N(>3) antennas should be used and at least two baseline vectors should be determined. Summary of the Invention

An object of the present invention is to provide a fast and easy method for resolving the integer ambiguity within the carrier phase in the 3-dimensional attitude determination by using a attitude determination GPS.

Another object of the present invention is to provide a method for reducing the number of the independent integer numbers to be searched by using the length of the at least two baseline vectors and relative position thereof.

Yet another object of the present invention is to provide a method for determining a 3-dimensional attitude of a vehicle by using the ambiguity resolution method.

According to an aspect of the present invention, there is provided a method for determining 3-dimensional attitude of a vehicle by using a Global Positioning System (GPS) satellites and an attitude determination GPS system comprising the steps of : (i) receiving GPS signals from at least 4 satellites (i, j, k, 1,...; represented by subscript) by using the attitude determination GPS system including one reference antenna (A ; represented by superscript) and at least two target antennas (B, C, ... ; represented by superscript) , and measuring a code signal measurement and a carrier phase measurement which are double differenced between antennas and between satellites; (ii) determining three independent integer numbers for each antenna combination (AB, AC, AD,...) of the reference antenna (A) and any of the at least two target antennas (B, C,...) by using at least one the carrier phase measurement, the code signal measurement, double differencing value for each line-of- sight vector between any of the satellites and any of the antennas, the ^"known lengths of the at least 2 baseline vectors, and the known angle between two selected baseline vectors; and (iii) determining the 3-dimensional attitude of the vehicle by defining at least two baseline vectors for two of the antenna combinations by using the determined three independent integer numbers.

In the step (ii) , particularly, in determining three independent integer numbers for each antenna combinations, a method for reducing the number of the independent integer numbers to be searched is used, which comprises the sub- steps of : (ii-1) measuring code signal measurements and carrier phase measurements which are double-differenced between -at least 4 satellites and between the reference antenna (A) and a first target antenna (B) , and calculating at least three first double-differencing Equations; (ii-2) determining a search range for two independent integer numbers (n_AB,ι, n_AB,2) among three independent integer numbers included in the at least three first double-differencing Equations, determining two true integer numbers by selecting candidate integers within the search range, each of the two true integer number being a candidate integer making the value of an objective function to the minimum, and determining third independent integer number (ΠAB,3) by using the two true integer numbers and a lengths of first baseline vector {b__) between the reference antenna ^"and the first target antenna; (ii-3) measuring code signal measurements and carrier phase measurements which are^" double-differenced between the at least 4 satellites and between the reference antenna (A) and a second target antenna (C) , and calculating at least three second double- differencing Equations; (ii-4) determining a search range for one independent integer number (ΠA_CI) among three integer numbers included in the three second double- differencing Equations, determining one true integer number by selecting a candidate integer within the search range, the true integer number being the candidate integer making a value of an objective- function to the minimum, and determining remaining two independent integer numbers (n_Ac,2, nac-,3) by using the one true integer number, the length of the first baseline vector {b ) between reference antenna and the first target antenna, a length of second baseline vector (bac) between the reference vector and a second target antenna, and an angle (θ) between the first baseline vector and the second baseline vector; and (ii-5) in case that there are at least 4 antennas, performing the sub-steps (ii-3) and (ii-4) for each of additional target antennas (D,E,...) , and determining three independent integer numbers (n_AD,ir n_AD/2, n_AD,3, n_AE,ι, n_AE,2, •••) for each baseline vector (r_Ao, r__f ...) between the reference antenna and any of the additional target antennas (D,E,...) . Moreover, in case that more than 4 satellites is used, dependent integer numbers can be determined by using at least "two baseline vectors determined, as above..

The determination of the search range for the independent integer number and the determination of the true integer number may be performed by 'using a plurality of the numerical Equations as described below.

Brief Description of the Drawings

The above and other objects and features of the present invention will become apparent from the following ^'description of a preferred embodiment given in conjunction with the accompanying drawings, in which:

Fig. 1 shows a process for determining an attitude of a vehicle using a conventional GPS for 3-dimentional attitude determination;

Fig. 2 illustrates a flow chart of the integer ambiguity resolution method and the attitude determination method according to the present invention; Fig. 3 describes a detailed flow chart illustrating a method for determining three independent integer numbers for each of the all antenna combination of the reference antenna and a target antenna according to the present invention; and

Fig. 4 exemplifies an example of a vehicle including three antennas (A, B and C) .

Detailed Description of the Invention

Hereinafter, a preferred embodiment of the present -invention will be described in detail with reference to the accompanying drawings.

Fig. 1 shows a process for determining an attitude of a vehicle using a conventional GPS for - 3-dimentional attitude determination.

In general attitude determination GPS system acquires a vehicle attitude by obtaining double-differenced measurements between satellites and between antennas by using the GPS signals from at least 3 antennas disposed separately on a vehicle. Subsequently, the. system obtains baseline vector (s) between two selected antennas by determining integer numbers included in the measurements, ^•and determines the vehicle attitude by using the baseline vectors.

In Fig. 1, the attitude determination GPS system tracks and receives satellite signals from at least 4 satellites by using at least 3 antennas disposed spatially on a vehicle, (Step Sll) and generates data streams by way of, e.g., a demodulation of the satellite signals (Step S12) . Subsequently, based on the data stream, the attitude determination GPS system calculates inter-satellites and inter-antennas double-differenced measurements respectively for a code signal and a carrier phase, (Step S13, 14) and determines the search range for two independent integer numbers (Step S15) . Then,, the attitude determination GPS system calculates an objective function for candidate integers within the search range, and determines a candidate integer making the value of the objective function minimum as a true integer number (Step SI6) . Another independent integer number may be determined by using two true integer numbers already determined and known length of a baseline vector between two antennas corresponding thereto, and the baseline vector may be defined based on the determined 3 independent integer numbers (Step S17) .

It is possible to determine at least two baseline' vectors by performing the above processes for all antenna combinations (a reference antenna + one target antenna) , and to determine the 3-dimensional attitude of the vehicle on the basis of the baseline vectors (Step S18) .

Though the flow charts in Fig. 1 describes the method for determining a third independent integer number by using two integer numbers and the length of the baseline vector, it should be noted that the three independent integer numbers can be attained by, e.g., determining a search range for all of three independent integer numbers and then determining the independent integer numbers by searching.

The above-described procedure for the attitude determination will be described in more detail with reference to^'some numerical Equations.

An order of involving satellites for the independent integer numbers can be determined by selecting a satellite group making a covariance of the terms of dependent integer numbers minimum and assigning priorities for the involving satellites by the order of an elevation angle of the satellite.

At least four satellites should be involved for terms of the independent integer numbers, and measurements of the satellite signal are expressed by Equation 1 and Equation 2 as follows.

[Equation 1]

PAB,I ⁼ HAB,I TAB,I + VAB,I, wherein, PAB,I represents double-differenced (between antennas) code signal for the independent integer number, H_AB,I represents a part of the HAB (wherein, H is a line-of- sight vector between satellite and antenna) divided by a term of the independent integer number, r_AB,ι represents a baseline vector between antennas A and B for the term of the independent integer number, and VAB,_I represents double- differenced measurement noise for the term of the independent integer number. [Equation 2] ./?,/ = ^AB,i^rAB,l + ANAB,I + ^WAB,I i wherein, 1AB,I represents double-differenced carrier phase measurement, N_AB,I represents the term _ of the independent integer number, λ represents carrier wavelength

(about 19cm for the LI carrier wave) and w_AB,ι represents a double-differenced measurement noise for the independent integer number term.^"

An estimated value for the term of the independent integer number can be expressed by Equation 3 as follows. [Equation 3]

wherein,

(DD»DD^T) if there is no correlation between the code signal and the carrier phase, and characteristics of receiver channels are identical. Therefore, an estimated value for a covariance- of the independent integer number may be expressed by Equation 4 as follows.

[Equation 4]

cov(N_AB ) = i l Ξil _DD . _DD^τ) ;

A wherein, N_AB,I is defined as [nAB,i nAB,2, nAB,3j^T, and Equation 4 can be rewritten as Equation 5. Furthermore, a search range may be defined as Equation 6.

[Equation 5]

[Equation 6]

^• on_t ≤ n, ≤ m + δ , i = 1,2,3. ,

wherein, δm is ?—-y ^_o ^+σψ) (β = significant level).

Meanwhile, a baseline vector may be expressed by Equation 7 using Equation 2, and Equation 8 may be derived from Equation 7 that the length between antennas is already known.

[Equation 7]

[Equation 8] t>AB,I ~ AB,I - ΛNAB,I HABJHABJ QAB,! ~ ^ ABj) •

Equation 8 may be rewritten as Equation 9 since H_AB,_I and H^T _AB,i may be expressed as the LL^T, a multiplication of lower triangular matrices, by using the Cholesky factorization.

[Equation 9] h\_B = l_AB, - AN_AB f[L-^l(l_AB>, -AN_AB )] , wherein, l_AB,ι is defined as [1AB,I, 1AB,2, 1AB,3.^T. Equation 11 may be derived when L^"1 is defined as Equation 10, and a constraint for length may be expressed by Equation 12

[Equation 10]

[Equation 11]

[Equation 12]

"°AB ~~ ^AB,l + ^AB,2 + ^AB,3 ^•

Equation 13 may be derived from Equation 12, which, in turn, Equations 14 and 15 may be expressed with respect to the integer number.

[Equation 13]

— b ^JA_ΛB_R

AB '"ABA <λ ^■AB,2 < -ft ^':AB ^■λ" ΑBΛ

[Equation 14] ~b >A_ΔB_{R .} , / -AB,\ _{< <} b_AB l_AB λκ_n ^{+ _} λ ^~~~^{≤ n}AB,I ^≤~ λκ ⁺ ~ λT^".

[Equation 15]

^■ AB ϋ?,l _μ ^bA~B~^λAB,_ _μ λK. ~*~bl - ⁿAB,2 — _T __ ^+C=\ ' λK 2-2

wherein, ξi is ^21 VΑB,\ ^AB ) ^"*^{" K}22*AB,2

Arc- ₂₂ The integer number may be searched by a measurement noise covariance centering around a position calculated by using a code information in Equation 6, and searched by a geometrical condition centering around a position of the reference antenna in Equation 14. The search range of nβ,ι may be expressed by Equation 16. [Equation 16]

max(—<5w,,—^-)<n_AS, < in( —δn_χ, ^AB ) . λκ_n " λκ_n

It will now be described in detail the procedure for searching integer number using Equation 16. At first, the range of nβ,2 is calculated for an independent- integer number n_AB,ι in the range defined in Equation 16. The range of n_AB2 can be determined by Equation 16 by also using Equations 6 and 15. Subsequently, an objective function for candidates (n_AB,ι and nω,2) of the independent integer numbers is calculated. All of the candidates within the search range are substituted into the objective function, and candidate (s) making the value of the objective function minimum is determined as the true integer number (s). Detailed description oh the method for determining the true integer number from the candidates of the integer number and the objective function is disclosed in Korean patent application no. 1997-57696. The objective function may be determined by using Equation 16-1 on the basis of the double-differenced carrier phase during n number of epochs for the m number of satellites. [Equation 16-1]

Ω* ^~~ ∑Ω_ε(0 = ∑ [(l_E{t) - E(t)^τN)^τQ_E(tr^l{l_E( ) - λE(t)^τN)] t=\ f=l

λ- cNl(t){E_D(t)Q- t)E_D ^τ(t)}δN_D(t) , t=\ wherein, E(t) is null (H (t) ^τ) , N is [nAB,ι, n_ω,2, n_ffi,3, nAB,D, n_AB,D5,... n_AB,D(m-i)]^T, a measurement noise w_E(t) is approximately N(0, E^τ (t) Q_Dψ (t) E (t) s N(0, Q_E(t)), a dependent integer number in the true number domain No ≡N_D - δN_D, and E^τ(t) is a part of the E(t) divided on the basis of the

definition of No .

Once two independent integer numbers nAB,ι, nAB,2 are determined, a value n_As,3 in a true number domain for a candidate AB,3 of the remaining independent integer number can be calculated by using Equation 17. A value n_AB,3 in an integral number domain of the remaining independent integer number can be determined by Equation 18.

[Equation 17]

wherein, ξ₂ is ^K 1 VAB_.I ~~ ^ⁿA ,\ ) ^{+ K}31 -AB,2 ^ AB,2 ) ^{+ K}33^AB,3 λκ₃₃ [Equation 18]

ⁿAB,3 ⁼ round (HAB, ) . Subsequently, (m-l)-3 number of dependent integer numbers (N_D) for m number of satellites may be determined by following procedure. The constraint Equation as Equation 19 may be derived by the double-differenced carrier phase measurement in Equation 2.

[Equation 19] β,E ^~~ ΛEAB^N' AB + ^WAB,E r wherein, EΛB is null(HAB), 1AB,E is E^T _AB1AB, and WM^B is

E^TABW_AB, and AB is approximately N(0, E^TABQDΨEAB)≡ N(0, Q__) . When the candidates for the integer numbers are divided into three independent terms and remaining dependent terms, Equation 19 can be rewritten as Equation 20. [Equation 20]

I_AB,E ~ E_AB,I N_AB,I + E_AB,D^ NAB,D+ >_{AB E} ,

wherein, NAB,I represents the terms of three independent integer numbers among NAB, NAB,D- represents, the terms of (m-l)-3 number of dependent integer numbers in the true number domain, E^TAB,I represents a part of the ^'E_AB divided according to the definition of NAB,I, E^T _AB,D represents a part of the EAB divided according to the definition of

NAB,D . The dependent integer numbers in the true number domain may be calculated by Equation 21 since E_AB,D is a square matrix. The dependent integer numbers may be finally determined by using Equation 21. [Equation 21] ND - E_ABβ(l_AB - E_AB λN_AB ) ,

[Equation 22]

N_D = round(No) .

In accordance with the method described above, baseline vectors for all antenna combinations can be defined^" by determining three independent integer numbers and consecutively (m-l)-3 number of dependent integer numbers, provided that m number of visible satellites are involved.

Hereinafter, an integer ambiguity resolution method capable of reducing the number of independent integer numbers to be searched, and a method for determining attitude of a vehicle in accordance with the present invention will now be described in detail.

A 3-dimensional attitude of the vehicle can be determined by using a GPS system with at least 3 antennas spatially disposed on the vehicle. A baseline vector is defined as a vector from one antenna to another antenna, therefore at least 2 baseline vectors are may be defined by at least 3 antenna. The 3-dimensional attitude of the vehicle may be determined if two baseline vectors are determined.

When a GPS system employing number of antennas is used, total (α-1) baseline vectors may be defined, and an integer ambiguity should be resolved for each baseline vector according to the above-described process, respectively. In other word, it is required to search at least 2 (α-1) number of independent integer number terms based on the conventional method. Such searching and ^" determination for the integer numbers may require prodigious calculation volume and time, which hamper a real-time attitude determination.

Meanwhile, a vehicle of which attitude is sought is usually a rigid body such as an automobile, an aircraft etc. The assumption that the vehicle^' is a rigid body may provides a big advantage for ease and effective an integer ambiguity resolution. That is, if the vehicle is a rigid body, a relative dispositions of the baseline vectors remain unchanged despite the movement of a vehicle, ^'which helps make independent factors minimal in searching and determining the integer numbers .

In summary, the present invention provides a method for reducing the number of the integer numbers to be searched by at least one by using lengths of the baseline vectors and an angle between the baseline vectors for use in an attitude determination method employing an attitude determination GPS system.

According to the prior art, wherein the GPS system with α number of antennas is used, it is required to search for at least 2 (α-1) integer numbers. In contrast, in accordance with the present invention, search of only α number of independent integer numbers should be conducted. Therefore, the number of the independent integer numbers to be searched can be dramatically reduced by amount of (α-2). Fig. 2 is a flow chart explaining a method for resolving an integer -ambiguity and determining an attitude of a vehicle.

At first, the attitude determination GPS system acquires GPS signals from at least 4 satellites (satellite indices being superscripts i, j, k,...) by using at least 3 antennas (antenna indices being subscript A, B, C,...), and measures code signal measurements and carrier phase measurements which are double-differenced between antennas and between satellites (Steps S21, S22) . In this process, maximum of (number of satellites involved minus one; that is m-1) double-differencing Equations are formed per baseline vector (Step S23) .

Subsequently, three independent integer numbers (n_AB,ι, n_B,2, n_AB,3, ΠAC,I n_Ac,2, n_Ac,3, n_AD,ι,...) for each of all combinations of a reference antenna (A) ^■ and a target antenna (B, C, D, ...) are determined by using at least one measurements • of a double-differenced carrier phase, a double-differenced line-of-sight vector between a satellite and an antenna, and a length of each baseline vector and an angle between the baseline vectors (Step S24) . At least two baseline vectors (r_AB, r_ACf rω,...) are defined by using three independent integer numbers already determined (Step S25) , and thus a 3-dimensional attitude of the vehicle can be determined (Step S27).

Furthermore, when there are m number of visible satellites involved, there may be provided an additional step for determining (m-l)-3 number of remaining dependent integer numbers on the basis of the independent integer numbers and/or the baseline vector already determined.

Fig. 3 describes a flow chart illustrating the detailed steps for determining three independent integer numbers for each of all antenna combinations (the reference antenna and one target antenna) .

At first, the GPS system measures a code signal PAB,I and a carrier phase 1AB,_I which are double-differenced between each of 4 satellites and between a reference antenna (A) and a first target antenna (B) , and acquires first three double-differencing Equations (Step S31) . Subsequently, the GPS system determines a search range for- two independent integer numbers (nAB,ι, n_AB,2) among integer numbers within the first three- double-differencing Equations by using the above-mentioned numerical Equations (Step S32) . Then, the GPS system determines two true integer numbers by selecting two candidates in the search range, the candidate making the value of an objective function minimum (Step S33) , and determines third independent integer number (n_AB,3) by using determined two integer numbers and a length of a first baseline vector (DAB) between the reference antenna A and the first target antenna B (Step S34) . The system measures a code signal

Pac,ι and a carrier phase 1AC,I which are double-differenced between each of the 4 satellites and between the reference antenna (A)"- and a second target antenna (C) , and acquires a second three double-differencing ^'Equations (Step S35) . Subsequently, the GPS system determines ^"a search range for" one independent integer number ( Ac,ι) among integer numbers within the second three double-differencing Equations (Step S36) . Then, the GPS system determines one true integer number (nAc,ι) by selecting a candidate in the search range, the candidate making the value of an objective function minimum (Step S37), and determines two remaining independent integer numbers (nac,2, nac,3) by using determined integer number (n_Ac,ι) , a length of the first baseline vector (ba_B) , a length of a second baseline vector

(b_AC) between antenna A and antenna C, and an angle (θ) between the first baseline vector and the second baseline vector (Step S38) . In addition, when more than -3 antennas are employed in the GPS system according ^'the present invention, the GPS system performs the same processes for acquiring the three independent integer numbers n_AC,i (i=l,2,3) as described above with respect to each additional antenna (D,E,...), and determines three independent integer numbers n_AD,i ΠAE,! (i=l,2,3) for each additional antenna (Step S39) . Furthermore, when it is required to determine an attitude of a vehicle where at more than 4 visible satellites are involved, the system may determine dependent integer numbers N_D by using at least two baseline vectors determined according to the above processes and the above- described numerical Equations.

Hereinafter,, an integer ambiguity resolution method using a GPS system with three antennas (one reference antenna and two target antennas) will be described in detail with reference to the following numerical Equations.

Fig. 4 shows an example of a vehicle with 3 GPS antennas (A,B and C) .

In Fig.4, "A" represents a reference antenna, "B" and "C" represent a first and a second target antenna respectively, "r_AB" and "r_Ac" represent a first and a second baseline vector, "θ" represents an angle between two baseline vectors, and "b__." _. and "b_AC" represent a length of the first and the second baseline vector respectively.

Given that the vehicle is a rigid body, a length and a relative disposition keep constant despite the movement of the vehicle. The above rigid-body assumption may be expressed by Equations 23, 24 and 25. Equations 23 and 24 represent a fixed length condition, and Equation 25 represents a constant angle condition, respectively. [Equation 23]

T _ ι 2 ^rAB ^{' r}AB — °AB I [Equation 24] r _ r 2

^TAC ^{' r}AC — ° C r

[Equation 25] rju_t ^• r_AC ^ b^c cosθ . In case that two ^' baseline vectors raβ" and "r_Ac" are used, two integer number terms should be searched per baseline vector, i.e. total' four integer number- terms by using Equation 23 and Equation 24. However, when Equation 25 is used along with Equations 23 and 24, only three integer number terms may be searched, which will now be further described.

At first, two integer number terms (ΠAB,I and n_AB,2) for the first baseline vector (rω) are searched and determined in accordance with the conventional searching method. Once two independent integer numbers are determined, remaining one independent integer number and dependent integer numbers (including nAB,3, No) can be determined by using Equations 18, 22 and 23.

At least five common visible satellites are required for each of three antennas in order to determine integer numbers for an additional second baseline vector r_Ac- Arranging measurements from these satellites in same order results in Equation 26, and then the lower triangular matrix L^-1 in Equation 10 has an identical value for two baseline vectors rAB and r_Ac, respectively. Therefore, Equation 14 can be rewritten as Equation 27 by adjusting Equation 14 with respect to first independent integer number for the second baseline vector.

Furthermore, provided that performance of the GPS System is identical, a search range _ calculated by a covariance can be expressed by Equation 6, and the search range can finally be expressed as Equation 28. [Equation 26]

[Equation 27]

^' [Equation 28]

— h h maxC-^ ,,— -*£-) <n_ACl≤ min^,-^-) .

A constraint condition of Equation 25 can be rewritten as Equation 30 from Equation 7 and Equation 30. [Equation 29] _

Λ

TACJ =H_AC1(l_AC1 -AN_ACI) .

[Equation 30]

_jH tr_A-_c1^l _>I(l_AC -λN_AC )

=b_ABb_AC cosθ

Equation 30 may be rewritten as Equation 33 by using the definition of Equations 31 and 32 since the integer numbers (NAB,I) for the first baseline vector (r_M) are already determined.

[Equation 31] [L^~x (l_AB - λN_AB )f ¹ = [μ_x , μ₂, μ₃] .

[Equation 32^'] di - ^l _AB,ι - Λn_AC , ^' = 1,2,3.

[Equation 33] b_ABb_AC cos6> = μ_xd_x + μ₂d₂ + μ₃d₃ . A constraint formula for the length of the ^'baseline vector r_Ac in Equation 24 can be rewritten as Equation 34 from Equations 11, 12 and 32. In addition, L^-1 = is a matrix acquired by the Cholesky factorization for T H_AB,I, H _B,I, H_AB,I is [n_AB,ι, niiB,2(

, 1AB,I=[ 1_AB,I, 1AB,2/ 1AB,3 ] is a carrier phase measurement double-differenced between satellites and between antenna A and B, n_AB,i is a i-th independent integer number, and λ is a carrier wave wavelength.

[Equation 34] b_AB = (K_ud_x)- + (κ₂ d_λ + r ₂₂d₂)² + (κ_3Λd_x + κ₃₂d₂ + f ₃d₃)²

Therefore, once one true independent integer number n_Ac,ι for the baseline vector r_Ac may be determined by searching for one independent integer number term as represented in Equation 28, remaining two integer numbers n_Ac,2 and nc,3 may be determined directly by using Equation 33 and 34. Though two solutions are acquired from the quadratic Equations 33 and 34, one solution may be determined as the true integer number by using a range defined in Equation 6.

However, when one baseline vector is in parallel with the other baseline vector, the inventive method may not be- applicable since the angle θ is zero, and, therefore, Equation .33 cannot provide no additional information. This event is, however, exceptional and further, te 3-dimensional attitude cannot be determined even by using two parallel baseline vectors.

While the embodiments illustrated in the figures and described above are presently preferred, it should be understood that these embodiments are offered by way of example only. The invention is not limited to a particular embodiment, but extends to various modifications, combinations, and permutations that nevertheless fall within the scope and spirit of the appended claims .

In accordance with the present invention, in order to determine integer numbers of carrier wave causing an integer ambiguity for use in determining 3-dimensional attitude of a vehicle by using the GPS system, the number of the independent integer numbers can be reduced by the amount of number of antennas minus 2 (α-2) since, provided α number of antennas are used in the GPS system, minimum α number of integer numbers should be searched, while the prior art method requires 2 (α-1) number of the integer numbers to be searched.

Since the number of the integer numbers to be searched can be reduced, the integer number causing the integer ambiguity can be fast determined, and finally, a calculation volume, ^'time and memory required can be greatly reduced accordingly.

Claims

What is claimed is:

1. A method for determining a 3-dimensional attitude of a vehicle for use with a satellite navigation system by receiving satellite signals" transmitted from at least 4 satellites with at least 3 antennas spatially disposed on the vehicle and defining at least 2 baseline vectors, each baseline vector being defined by each combination of two antennas, comprising the procedure of : reducing the number of integer numbers to be searched by at least one by using known lengths of at least 2 baseline vectors and a known angle between two selected baseline vectors in determining the integer numbers causing an integer ambiguity, the integer numbers being included in a carrier phase.

2. The method as claimed in claim 1, wherein the satellites are Global Positioning System (GPS) satellites and the satellite navigation system is an attitude determination GPS system, and the procedure comprises the steps of :

(i) receiving GPS signals from at least 4 satellites (i, j, k, 1,...; represented by subscript) by using the attitude determination GPS system including one reference antenna (A ; represented by superscript) and at least two target antennas (B, C, ... ; represented by superscript) , and measuring a code signal measurement and a carrier phase measurement which are double differenced between antennas and between satellites;

(ii) determining three independent integer numbers for each antenna combination (AB, AC, AD,...) of the reference ^" antenna (A) and any of the at least two target antennas (B, C,...) by using at least one the carrier phase measurement, the ^"code signal measurement, double differencing value for each line-of-sight vector between any of the satellites and any of the antennas, the known lengths of the at least 2 baseline vectors, and the known angle between two selected baseline vectors; and

(iii) determining the 3-dimensional attitude of the vehicle by defining at least two baseline vectors for two of the antenna combinations by using the determined three independent integer numbers.

3. The method as claimed in claim 2, wherein the step (ii) comprises the sub-steps of :

(ii-1) measuring code signal measurements and carrier ^• phase measurements which are double-differenced between at least 4 satellites and between the reference antenna (A) and a first target antenna (B) , and calculating at least three first double-differencing Equations;

(ii-2) determining a search range for two independent integer numbers (nAB,i/ nAB,2) among three independent integer numbers included in the at least three first double- differencing Equations, determining two true integer numbers by selecting candidate integers within the search range, each of the two true integer number being a candidate integer making the value of an objective function to the minimum, and determining third independent integer number (-A_B,₃) by using the two true integer numbers and a lengths of first baseline vector (bω) between the" reference antenna and the first target antenna;

(ii-3) measuring code signal measurements and carrier phase measurements which are double-differenced between the at least 4 satellites and between the reference antenna (A) and a second target antenna (C) , and calculating at least three second double-differencing Equations;

(ii-4) determining a search range for one independent integer number (n_Ac,ι) among three integer numbers included in the three second double-differencing Equations, determining one true integer number by selecting a candidate integer within the search range, the true integer number being the candidate integer making a value of an objective function to the minimum, and determining remaining two independent integer numbers ^' (n_Ac,2, nAc,3) by using the one true integer number, the length of the first baseline vector

(b_M) between reference antenna and the first target antenna, a length of second baseline vector (b_C) between the reference vector and a second target antenna, and an angle (θ) between the first baseline vector and the second baseline vector; and

(ii-5) in case that there are at least 4 antennas, performing the sub-steps (ii-3) and (ii-4) for each of additional target antennas (D,E,...), and determining ^' three independent integer numbers (n_ω,ι, n_AD,2, nao,3, n__, ι, ΠAE,2, ...) for each baseline vector (r_AD, r_AE, ...) between the reference antenna and any of the additional target antennas (D,E,...) .

4. The method as claimed in claim 2, wherein the step (ii) comprises the sub-steps of :^■

(ii-a) measuring code signal measurements and carrier phase measurements which are double-differenced between at least 4 satellites and between the reference antenna (A) and a first target antenna (B) , and calculating at least three first double-differencing Equations;

(ii-b) determining a search range for three independent integer numbers (n_ω,ι, nm_.2, n_ω,3) among integer numbers- included in the three first double-differencing Equations, determining three true integer numbers by selecting candidate integers within the search range, each of the three true integer number being a candidate integer making the value of an objective function to the minimum;

(ii-c) measuring code signal measurements and carrier phase measurements which are double-differenced between the at least 4 satellites and between the reference antenna (A) and a second target antenna (C) , and calculating at least three second double-differencing Equations;

(ii-d) determining a search range for one independent integer number (n_Ac,ι) among three integer numbers included in the three second double-differencing Equations, determining^"-one true integer number by selecting a candidate integer within the search range, the ^'one true integer number being the candidate integer making a value of an objective function to the minimum, and determining remaining two independent integer numbers (n_Ac,2, n_AC,3) by using the one true integer number, the length of the first baseline vector

( aβ) between reference antenna and the first target antenna, a length of second baseline vector (b_Ac) between the reference vector and a second target antenna, and an angle

(θ) between the first baseline vector and the second baseline vector; and

(ii-e) in case that there are at least 4 antennas, performing the sub-steps (ii-c) and (ii-d) for each of additional target antennas (D,E,...), and determining three independent integer numbers (nAD,ι, n_AD,2, nω,3, Ώ-ΆZ. I , n_AE,2, ...) for each baseline vector (r_AD, r_ω, .-..) between the reference antenna and any of the additional target antennas (D,E,...).

5. The method as claimed in claim 3 or 4, wherein in the sub-steps (ii-4) and (ii-d), the procedure for determining the remaining two independent integer numbers (n_AC,2, n_Ac,3) by using the one true integer number (n_Ac,ι) , the length of the first baseline vector (b_AB) , the length of the second baseline vector (b_Ac) / and an angle (θ) between the first baseline vector and the second baseline vector is performed by Equation as follow : ^bAB^bAC ^C0SΘ = MA + 2 ^bAB = (^K_Af + (*2l +

+ *32^rf2 + *lA? > wherein, [L^~ l_AB - λN_{AB l})fl7^l ≡ [μ_x,μ₂,μ₃] .

6. The method as claimed in claim 5, in case of more than 4 satellites, further comprising an additional sub-step for determining (# of satellites - l)-3 number of dependent integer numbers (N_D) by using (# of satellites -1) number of baseline vectors and Equation as follow : d_t ≡ l_AB i - λn_AC ,. , i = 1,2,3 ,

wherein, L ¹ is a matrix acquired by the

Cholesky factorization for H_AB',ι and H^TAB,I, HAB,I is [nAB,ι, naB,2, nAs,3]^Tf 1AB,I=[ 1AB,I, 1AB,2, 1AB,3,]^T is the carrier phase measurement double-differenced between any of the at least 4 satellite and between the reference antenna (A) and the first target antenna (B) , nAB,i is a i-th independent integer number, and λ is a carrier wave wavelength.