METHODS AND APPARATUSES OF ESTIMATING THE POSITION OF A MOBILE USER IN A SYSTEM OF SATELLITE DIFFERENTIAL NAVIGATION

Description

FIELD OF THE INVENTION

[0001] The present invention relates to methods of information processing in satellite navigation systems with differential positioning of a mobile user.

BACKGROUND OF THE INVENTION

[0002] Satellite navigation systems, such as GPS (USA) and GLONASS (Russia), are intended for high accuracy self-positioning of different users possessing special navigation receivers. A navigation receiver receives and processes radio signals broadcasted by satellites located within line-of-sight distance. The satellite signals comprise carrier signals that are modulated by pseudo-random binary codes, which are then used to measure the delay relative to local reference clock or oscillator. These measurements enable one to determine the so-called pseudo-ranges (γ) between the receiver and the satellites. The pseudo-ranges are different from true ranges (D, distances) between the receiver and the satellites due to variations in the time scales of the satellites and receiver and various noise sources. To produce these time scales, each satellite has its own on-board atomic clock, and the receiver has its own on-board clock, which usually comprises a quartz crystal. If the number of satellites is large enough (more than four), then the measured pseudo-ranges can be processed to determine the user location (e.g., X, Y, and Z coordinates) and to reconcile the variations in the time scales. Finding the user location by this process is often referred to as solving a navigational problem or task.

[0003] The necessity to guarantee the solution of navigational tasks with accuracy better than 10 meters, and the desire to raise the stability and reliability of measurements, have led to the development of the mode of "differential navigation ranging," also called "differential navigation" (DN). In the DN mode, the task of finding the user position is performed relative to a Base station (Base), the coordinates of which are known with the high accuracy and precision. The Base station has a navigation receiver that receives the signals of the satellites and processes them to generate measurements. The results of these measurements enable one to calculate corrections, which are then transmitted to the user that also uses a navigation receiver. By using these corrections, the user obtains the ability to compensate for the major part of the strongly correlated errors in the measured pseudo-ranges, and to substantially improve the accuracy of his or her positioning.

[0004] Usually, the Base station is immobile during measurements. The user may be either immobile or mobile. We will call such a user "the Rover." The location coordinates of a moving Rover are continuously changing, and should be referenced to a time scale.

[0005] Depending on the navigational tasks to be solved, different modes of operation may be used in the DN mode. They differ in the way in which the measurement results are transmitted from the Base to the Rover. In the Post-processing (PP) mode, these results are transmitted as digital recordings and go to the user after all the measurements have been finished. In the PP mode, the user reconstructs his or her location for definite time moments in the past.

[0006] Another mode is the Real-Time Processing (RTP) mode, and it provides for the positioning of the Rover receiver just during the measurements. The RTP mode uses a communication link (usually it is a radio communication link), through which all the necessary information is transmitted from the Base to the Rover receiver in digital form.

[0007] Further improvement of accuracy of differential navigation may be reached by supplementing the measurements of the pseudoranges with the measurements of the phases of the satellite carrier signals. If one measures the carrier phase of the signal received from a satellite in the Base receiver and compares it with the carrier phase of the same satellite measured in the Rover receiver, one can obtain measurement accuracy to within several percent of the carrier's wavelength, i.e. , to within several centimeters.

[0008] The practical implementation of those advantages, which might be guaranteed by the measurement of the carrier phases, runs into the problem of there being ambiguities in the phase measurements.

[0009] The ambiguities are caused by two factors. First, the difference of distances ΔD from any satellite to the Base and Rover is much greater than the carrier's wavelength λ. Therefore, the difference in the phase delays of a carrier signal Δϕ =ΔD/λ received by the Base and Rover receivers exceeds several cycles. Second, it is not possible to measure the integer number of cycles in Δϕ from the incoming satellite signals; one can only measure the fractional part of Δϕ. Therefore, it is necessary to determine the integer part of Δϕ, which is called the "ambiguity".

[0010] More precisely, we need to determine the set of all such integer parts for all the satellites being tracked, one integer part for each satellite. One has to determine this set along with other unknown values, which include the Rover's coordinates and the variations in the time scales.

[0011] In a general way, the task of generating highly-accurate navigation measurements is formulated as follows: one determines the state vector of a system, with the vector containing n_Σ unknown components. Those include three Rover coordinates (usually along Cartesian axes X, Y, Z) in a given coordinate system (sometimes time derivatives of coordinates are added too); the variations of the time scales which is caused by the phase drift of the local main reference oscillator; and n integer unknown values associated with the ambiguities of the phase measurements of the carrier frequencies. The value of n is determined by the number of different carrier signals being processed, and accordingly coincides with the number of satellite channels actively functioning in the receiver. At least one satellite channel is used for each satellite whose broadcast signals are being received and processed by the receiver. Some satellites broadcast more than one code-modulated carrier signal, such as a GPS satellite that broadcasts a carrier in the L₁ frequency band and a carrier in the L₂ frequency band. If the receiver processes the carrier signals in both of the L₁ and L₂ bands, the number of satellite channels (n) increases correspondingly.

[0012] Two sets of navigation parameters are measured by the Base and Rover receivers, respectively, and are used to determine the unknown state vector. Each set of parameters includes the pseudo-range of each satellite to the receiver, and the full (complete) phase of each satellite carrier signal, the latter of which may contain ambiguities. Each pseudorange is obtained by measuring the time delay of a code modulation signal of the corresponding satellite. The code modulation signal is tracked by a delay-lock loop (DLL) circuit in each satellite-tracking channel. The full phase of a satellite's carrier signal is tracked by phase counter (as described below) with input from a phase-lock-loop (PLL) in the corresponding satellite tracking channel (an example of which is described below in greater detail). An observation vector is generated as the collection of the measured navigation parameters for specific (definite) moments of time.

[0013] The relationship between the state vector and the observation vector is defined by a well-known system of navigation equations. Given an observation vector, the system of equations may be solved to find the state vector if the number of equations equals or exceeds the number of unknowns in the state vector. In the latter case, conventional statistical methods are used to solve the system: the least-squares method, the method of dynamic Kalman filtering, and various modifications of these methods.

[0014] Practical implementations of these methods in digital form may vary widely. In implementing or developing such a method on a processor, one usually must find a compromise between the accuracy of the results and speed of obtaining results for a given amount of processor capability, while not exceeding a certain amount of loading on the processor. The present invention is directed to novel methods and apparatuses for accelerating the obtaining of reliable estimates for the integer ambiguities at an acceptable processor load.

[0015] More particularly, the present invention is directed to novel methods and apparatuses for more quickly obtaining such estimates in floating-point form (non-integer form) which are close to the integer values. With these floating-point forms, which we call floating ambiguities, conventional methods may be used to derive the corresponding integer ambiguities.

[0016] US patent no. US 6,268,824 describes methods and apparatuses of positioning a mobile user in a system of satellite differential navigation. Satellites broadcast code signals, which are modulated onto respective carrier signals, and which are received by two receivers on Earth. The first receiver is situated at a point with known coordinates. The results of its measurements are transmitted to a user at a second receiver through a connection link, the user being interested in knowing his or her positioning. The second receiver is similar to the first receiver and it receives the signals of the same satellites. By processing data from the measurements of code delays and carrier phase shifts in the satellite signals received by the two receivers, the described methods and apparatuses determine the location of the user with high precision and make the time indications of receiver clocks more accurate and exact. In one arrangement, the procedure of resolution of phase measurement ambiguities comprises the preliminary estimation of floating ambiguities by a recurrent (e.g., iterative) procedure including the simultaneous processing of code and phase measurements for all satellites for each processing time interval, and the gradual improvement of the result as the information is accumulated.

BRIEF DESCRIPTION OF THE DRAWINGS

[0017] 

FIG. 1 is schmatic diagram of an exemplary receiver which may be used in practicing the present invention.

FIG. 2 is a perspective view of a rover station and a base station in a first exemplary coinfiguration according to the present invention.

FIG. 3 is a perspective view of a rover station and a base station in a second exemplary coinfiguration according to the present invention.

FIG. 4 is a flow diagram illustrating a general set of exemplary embodiments according to the present invention.

FIG. 5 is a schematic diagram of an exemplary apparatus according to the present invention.

FIG. 6 is a diagram of an exemplary computer program product according to the present invention.

FIG. 7 is a flow diagram illustrating another general set of exemplary embodiments according to the present invention.

FIG. 8 is a diagram of another exemplary computer program product according to the present invention.

SUMMARY OF THE INVENTION

[0018] Broadly stated, the present invention encompasses methods and apparatuses for estimating the floating ambiguities associated with the measurement of the carrier signals of a plurality of global positioning satellites as recited in the accompanying claims.

[0019] The floating ambiguities are associated with a set of phase measurements of a plurality n of satellite carrier signals made by a first navigation receiver (B) and a second navigation receiver (R) separated by a distance, wherein a baseline vector (x^o,y^o,z^o) relates the position of the second receiver to the first receiver. Each satellite carrier signal is transmitted by a satellite and has a wavelength, and each receiver has a time clock for referencing its measurements. Any difference between the time clocks may be represented by an offset. Methods and apparatuses according to the present invention receive, for a plurality of two or more time moments j, the following inputs: a vector

representative of a plurality of pseudo-ranges measured by the first navigation receiver (B) and corresponding to the plurality of satellite carrier signals, a vector

representative of a plurality of pseudo-ranges measured by the second navigation receiver (R) and corresponding to the plurality of satellite carrier signals, a vector

representative of a plurality of estimated distances between the satellites and the first navigation receiver (B), a vector

representative of a plurality of estimated distances between the satellites and the second navigation receiver (R), a vector

representative of a plurality of full phase measurements of the satellite carrier signals measured by the first navigation receiver (B), a vector

representative of a plurality of full phase measurements of the satellite carrier signals measured by the second navigation receiver (R), and a geometric Jacobian matrix

whose matrix elements are representative of the changes in the distances between the satellites and one of the receivers that would be caused by changes in that receiver's position and time clock offset. (As used herein, the term "representative of," as used for example used when indicating that a first entity is representative of a second entity, includes cases where the first entity is equal to the second entity, where the first entity is proportional to the second entity, and where the first entity is otherwise related to the second entity.)

[0020] The present invention may be practiced in real time, where estimates of the floating ambiguities are generated as the above satellite information is being received. The present invention may also be practiced in post-processing mode, where the floating ambiguities are estimated after all the above satellite information has been received. In the latter case, block processing according to the present invention may be done.

[0021] Preferred methods and apparatuses according to the present invention generate, for each time moment j, a vector Δγ_j of a plurality of range residuals of pseudo-range measurements made by the first and second navigation receivers in the form of:

and also generate, for each time moment j, a vector Δϕ_j of a plurality of phase residuals of full phase measurements made by the first and second navigation receivers in the form of:

where Λ^-1 is a diagonal matrix comprising the inverse wavelengths of the satellites. In the case of real-time processing, an LU-factorization of a matrix M₁, or a matrix inverse of matrix M₁, is generated for a first time moment (denoted as j = 1), with the matrix M₁ being a function of at least Λ^-1 and

Also for this initial time moment, an initial vector N₁ of floating ambiguities is generated as a function of at least Δγ₁, Δϕ₁, and the LU-factorization of matrix M₁ or the matrix inverse of matrix M₁. For an additional time moment j, an LU-factorization of a matrix M_j, or a matrix inverse of matrix M_j, is generated, with the matrix M_j being a function of at least Λ^-1 and

Also for an additional time moment j, a vector N_j of estimated floating ambiguities is generated as a function of at least Δγ_j, Δϕ_j, and the LU-factorization or matrix M_j or the matrix inverse of matrix M_j. Exemplary forms of matrices M_j and vectors N_j are provided below. In this manner, a set of successively more accurate estimates of the floating ambiguities are generated in real-time with the vectors N_j. This method, of course, may also be practiced in a post-processing environment, where the data has been previously recorded and then processed according to the above steps.

[0022] In the post-processing environment, the following block processing approach according to the present invention may be practiced. As with above-described embodiments for real-time processing, the vectors of pseudo-range residuals Δγ_k and vectors of phase residuals Δϕ_k, k=1,... j, are generated. Thereafter, a general matrix M is generated from the data, with M being a function of at least Λ^-1 and

for index k of

covering at least two of the time moments j, and an LU-factorization of matrix M or a matrix inverse of matrix M is generated. Thereafter, a vector N of estimated floating ambiguities is generated as a function of at least the set of range residuals Δγ_k, the set of phase residuals Δϕ_k, and the LU-factorization of matrix M or the matrix inverse of matrix M.

[0023] As an advantage of the present invention, the nature of matrix M, as described in greater detail below, enables a compact way of accumulating the measured data in order to resolve the floating ambiguities. As a further advantage, forms of matrix M provide a stable manner of factorizing the matrix using previous information. In preferred embodiments, matrix M is substantially positive definite, and also preferably symmetric.

[0024] Accordingly, it is an objective of the present invention to improve the stability of generating estimates of the floating ambiguities, and a further objective to reduce the amount of computations required to generate estimates of the floating ambiguities.

[0025] This and other advantages and objectives of the present invention will become apparent to those of ordinary skill in the art in view of the following description.

DETAILED DESCRIPTION OF THE PRESENT INVENTION

NOMENCLATURE

[0026] All vectors presented herein are in column form. The transpose of a vector or matrix is indicated by the conventional superscript "^T" affixed to the end of the name of the vector or matrix. The inverse of a matrix is indicated by the conventional superscript "^-1" affixed to the end of the name of the matrix. For the convenience of the reader, we provide a summary notation symbols below:

_γ - denotes a pseudorange measurement, also called a code measurement; a pseudorange is the approximate distance between a satellite and a receiver is equal to the speed of light c multiplied by the time it takes the satellite signal to reach the receiver.

ϕ - denotes a phase measurement.

n - denotes the number of satellites from which measurements are collected

^S - is a superscript index that is attached to a measurement quantity (e.g., γ , ϕ) or other data to identify the satellite to which the quantity corresponds.

_j and _k - are subscript indices that are attached to a measurement quantity (e.g., γ , ϕ ) or other data to identify the moment in time (e.g., epoch) to which the measure the quantity corresponds.

c - speed of light in air.

λ^S - denotes a wavelength of a carrier signal emitted by the S-th satellite. The wavelength may denote either the L1-band carrier or the L2-band carrier, as indicated by the context of the discussion.

Λ - denotes a diagonal matrix of wavelengths: Λ = diag(λ^l, . . ., λⁿ).

O_l,mor O_lxm - denotes an l × m zero matrix (all matrix elements are zero). The indices l and m take on the appropriate values in the context that the matrix is used. In other words, specific numerical values or other indices may appear in the places of l and m.

I_l or I_l×l - denotes a square identity matrix of size l × l. The index l takes on the appropriate value in the context that the square identity matrix is used. In other words, specific numerical values or other indices may appear in place of l.

µ - is a vector of observable equations, preferably linearized observable equations. µ = (Δγ¹,..., Δγⁿ, Δϕ¹,..., Δϕⁿ)^T, where each element of µ is associated with a corresponding satellite, where the superscript indices affixed to the elements identify the satellites.

- denotes the observation matrix H associates with the pseudo-ranges γ. In the art, is also called the Jacobian matrix, the Jacobi matrix, the geometric function matrix, the Jacobian geometric matrix, the matrix of directional cosines, and the directional cosine matrix.

has dimensions of n × 4.

Q_k - denotes a compound matrix formed by

and Λ as follows:

having dimensions of 2n × 4.

G - denotes a compound matrix

formed by the zero matrix O_n,n and the identity matrix I_n, having dimensions of 2n × n . The matrix G has many purposes. One purpose is to select the lower-right n x n sub-matrix of a larger 2n × 2n matrix as follows: G^T C G = C₂₂, where

- denotes a compound matrix

which is a 2n × (4+n) matrix, ∥·∥_F-1 denotes F^-1 - weighted norm,

Brief Background on the Structure of the Satellite Signals

[0027] Before describing the present invention, we briefly describe the structure of the satellite signals and of a typical receiver suitable for differential navigation applications. Each of the satellites radiates signals in two frequency bands: the L1 band and the L2 band. Two carrier signals are simultaneously transmitted in the L1-band; both carrier signals have the same frequency, but are shifted in phase by π/2 (90°). The first L1 carrier signal is modulated by the clear acquisition C/A-code signal and the second L1 carrier signal is modulated by the precision P-code signal. One carrier signal is transmitted in the L2 band, and uses a different frequency than the L1 carrier signals. The L2 carrier signal is modulated by the same P-code signal used to modulate the second L1 carrier signal. These carrier frequencies are between 1 GHz and 2 GHz in value. Each C/A-code signal and P-code signal comprises a repeating sequence of segments, or "chips", where each chip is of a predetermined time period (Δ) and has a pre-selected value, which is either +1 or -1. The segment values follow a pseudo-random pattern, and thus the C/A-codes and the P-codes are called pseudo-random code signals, or PR-code signals. Additionally, before each C/A-code signal and P-code signal is modulated onto its respective carrier signal, each code signal is modulated by a low frequency (50 Hz) information signal (so-called information symbols).
The approximate distance between a satellite and a receiver is equal to the speed of light c multiplied by the transmission time it takes the satellite signal to reach the receiver. This approximate distance is called the pseudorange γ, and it can be corrected for certain errors to find a corrected distance D between the satellite and the receiver. There is a pseudorage between each visible satellite and the receiver. The transmission time from satellite to receiver is measured with the aid of clocks in the receiver and the satellite, and with the aid of several time scales (i.e., timing marks) present within the received satellite signal. The clocks in the satellites and the receiver are set to substantially the same time, but it is assumed that the receiver clock has a time offset τ because the receiver clock is based upon a quartz-crystal whereas each satellite clock is based upon a more accurate atomic reference clock. The receiver has the orbital patterns of the satellites stored in a memory, and it can determine the orbital position of the satellites based on the time of its clock. The receiver reads the timing marks on the satellite's signal, and compares them against it own clock to determine the transmission time from satellite to receiver. The satellite's low-frequency (50Hz) information signal provides the least precise timing information, the C/A-code signal provides the next most precise timing information, and the P-code signal provides the most precise timing information. The pseudorange is determined from the low-frequency information si gnal and the C/A-code signal for civilian users and some military users, and is determined from the low-frequency information signal and the P-code signal for most military users. Accurate use of the P-code signal requires knowledge of a certain code signal that is only known to military users. Precision better than that provided by the P-code signal can be obtained by measuring the phase of the satellite carrier signal in a differential navigation mode using two receivers.

[0028] Referring to FIG. 1, a typical receiver for differential navigation applications has an antenna, an amplifier/filter unit that receives the antenna's output, a frequency down-conversion unit that receives the output of the amplifier/filter unit, and several individual tracking channels of the coherent type, each of which receives the down-converted satellite signals. The receiver also comprises a receiver clock, which provides base timing signals to the local oscillator of the down-conversion unit, and to components within each individual tracking channel. The down-conversion unit provides down-converted and quantized versions of the satellite si gnals to the channels, with the down-converted signals having frequencies generally in the range of 10 MHz to 20 MHz. Each channel tracks a selected one of the satellite signals. Each tracking channel measures the delay of one PR-code signal within the satellite signal (e.g., C/A-code or P-code signal), and also the phase of the down-converter version of the satellite's carrier signal. A typical tracking channel comprises a Delay-Lock Loop (DLL) circuit for tracking the delay, of the PR-code, a Phase-Lock Loop (PLL) circuit for tracking the phase of the satellite's carrier signal, and three correlators which generate the input signals for the DLL and PLL circuits.

[0029] Referring to FIG. 1, the DLL circuit has a reference code generator that generates a set of reference code signals, each of which tracks the PR-code of the satellite signal, and each of which is provided as an input to a respective correlator. Each correlator output is representative of the degree to which the reference code signals are tracking the satellite code signal (i.e., the amount by which the reference signals are advanced or retarded in time with respect to the satellite signal). The DLL circuit also has a DLL discriminator and a DLL filter. The DLL discriminator receives inputs from two correlators, a DLL correlator that generates a base signal for controlling the DLL circuit and a main correlator that generates a signal useful for normalizing the base signal from the DLL correlator. The DLL discriminator generates an error control signal from these inputs; this error signal is filtered by the DLL filter before being provided to the DLL reference code generator as a control signal. The value of the DLL error signal varies by the amount that the reference code signals of the DLL generator are delayed or advanced with respect to the satellite code signal being tracked, and causes the code generator to adjust the timing of the reference code signals to better match the satellite's code signal. In this manner, tracking is achieved. The pseudorange γ may be generated by methods known to the art from the receiver's clock, the 50 Hz information signal, and any one of the reference code signals generated by the DLL. This is indicated in the "γ_j(T_j)" box.

[0030] In a similar manner, the PLL has a reference carrier generator that generates a reference carrier signal that tracks the down-converter version of the satellite's carrier signal. We denote the frequency of the reference carrier signal as f_NCO since the reference carrier frequency is often generated by a numerically-controlled oscillator (NCO) within the reference carrier generator. Referring to FIG. 1, the PLL circuit also has a PLL discriminator and a PLL filter. The PLL discriminator receives inputs from two correlators, a PLL correlator that generates a base signal for controlling the PLL circuit, and the main correlator that generates a signal useful for normalizing the base signal from the PLL correlator. Each correlator output is representative of the degree to which the reference carrier signal is tracking the satellite's carrier signal (i.e., the amount by which the reference carirer signal is advanced or retarded in time with respect to the satellite's carrier signal). The PLL discriminator generates an error control signal from these inputs; this error control signal is filtered by the PLL filter before being provided to the PLL reference carrier generator as a control signal. The value of the PLL error signal varies by the amount that the reference carrier signal of the PLL reference carrier generator is delayed or advanced with respect to the satellite carrier signal being tracked, and causes the reference carriergenerator to adjust the timing of the reference carrier signal to better match the satellite's carrier signal. The reference carrier generator provides a phase signal φ_jNCO(T_j) representative of the phase (integrated frequency) of the PLL's reference oscillator. This signal may be combined with other information, as described below, to provide a signal representative of the full phase of the satellite signal (but possibly with ambiguities in it). This is indicated in the "φ_j(T_j)" box.

[0031] Finally, each individual tracking channel usually comprises a search system which initially varies the reference signals to bring the PLL and DLL circuits into lock mode. The construction of this and the other above components is well know to the art, and a further detailed description thereof is not needed in order for one of ordinary skill.in the art to make and use the present invention.

Brief Background on the Navigation Parameters.

[0032] Because of the time offset τ and other error effects, the ps eudorange γ between a satellite and a receiver is usually not the actual distance between the satellite and the receiver. By looking at the pseudoranges of four or more satellites, there are well-known methods of estimating the time offset τ of the receiver and of accounting for some of the error effects to generate the computed distance D between the satellites and the receiver. The receiver's position may then be computed. However, because of various sources of noise and the relatively low resolution of the pseudo-random code signal, the true distances (i.e., true ranges), and receiver's position coordinates will not be exactly known, and will have errors.

[0033] In theory, more precise values for the receiver's position and clock offset could be obtained by measuring the number of carrier cycles that occur between each satellite and the receiver. The phase of the carrier of the satellite signal as transmitted by a satellite can be expressed as:

where

is an initial phase value, f_S is the satellite carrier frequency, and t is the satellite's time. Because the satellite carrier frequency is generated from a very precise time base, we may assume that f_S is a constant and not time dependent, and we may replace the above integral with f_S•t, as we have shown in the second line in the above equation. The phase of this signal when it reaches the receiver's antenna over the range distance D(t) is denoted as φ^S_A(t), and its value would be:

where c is the speed of light, and where τ_ATM(t) is a delay due to anomalous atmospheric effects which occur mostly in the upper atmosphere. The number of cycles f_s•τ_ATM(t) due to the atmospheric effects cannot be predicted or determined to within an accuracy of one carrier cycle by a stand-alone receiver (i.e., cannot be determined by absolute positioning). However, the atmospheric effect can be substantially eliminated in a differential GPS mode where the phase of the satellite is measured at a rover station and a base station, φ^S_A,R(t) and φ^S_A,B(t) respectively, and then subtracted from one another. Over the short baseline between the rover and base stations, the atmospheric delay τ_ATM(t) in both of these phases is equal for practical purposes, and the difference in phases is:

[0034] The terms

and f_S•t have also been cancelled from the difference. In FIG. 2, we show a base station, a rover station, and four satellites, in schematic form. The wave fronts of the satellite carrier signal are nearly planar by the time they reach the base and rover stations since the satellites are at least 22,000 km away. These wave fronts are illustrated in FIG. 2 for the first and fourth satellites (S1 and S4). Assuming that a wave front of the satellite carrier reaches one of the antennas first (either rover or base), the above difference is the number of additional wave fronts that the satellite carrier must travel before it reaches the second antenna (either base or rover). We have illustrated this for the first satellite S1 as f_S1•D_R(t)/c - f_S1•D_B(t)/c. Using the angle between the baserover baseline vector and the line-of-sight vector to the satellite from either of the stations, the number of carrier cycles that lie along the baseline can be determined by trigonometry, and thus the baseline can, in theory, be accurately determined from the differential phase.

[0035] However, the task of measuring carrier phase is not as easy as it appears. In practice, we must use non-ideal receivers to measure the phases φ^S_A,R(t) and φ^S_A,B(t), with each receiver having a different clock offset with respect to the GPS time, and with each receiver having phase errors occurring during the measurement process. In addition, at the present time, it is not practical to individually count the carrier cycles as they are received by the receiver's antenna since the frequency of the carrier signal is over 1 GHz. However, the PLL loop can easily track the Doppler-shift frequency f_D of the carrier signal, which is in the kHz range. With a few assumptions, the phases φ^S_A,R(t) and φ^S_A,B(t) can be related to their respective Doppler-shift frequencies. As is known in the art, the satellite transmits at a fixed frequency of f_S, but the relative motion between the satellite and receiver causes the frequency seen by the receiver to be slightly shifted from the value of f_S by the Doppler frequency f_D. We may write the frequency seen by the receiver's antenna as f_S + f_D, where f_D has a positive value when the distance between the satellite and receiver's antenna is shrinking, and a negative value when the distance is increasing. Each receiver can then assume that the received phase is proportional to the predictable amount of f_S•t, minus the amount f_D•t due to the Doppler-shift. The Doppler amount f_D•t is subtracted from f_S•t because the Doppler frequency increases as the distance between the satellite and receiver's antenna decreases. The predictable amount f_S•t will be the same for each receiver, but the Doppler frequencies will usually be different.

[0036] As previously mentioned with reference to FIG. 1, the reference oscillator (e.g., NCO) of the PLL circuit tracks the frequency of a selected one of the down-converted satellite signals. As a result, it inherently tracks the Doppler-shift frequency of the satellite's carrier signal. Before being provided to the PLL circuit, the carrier signal is down-converted from the GHz range by a local oscillator having a frequency f_L. The frequency seen by the PLL circuit is (f_S+f_D)-f_L, which can be rearranged as: (f_S-f_L) + f_D. The quantity (f_S-f_L) is known as the pedestal frequency f_p, which is typically around 10 MHz to 20 MHz. The PLL's reference oscillator tracks the down-converted frequency of f_p + f_D. We would like to integrate the frequency of the reference oscillator to obtain a phase observable which is proportional to - f_D•t. Starting at a time moment T_p when the PLL circuit initially locks onto the down-converted satellite signal, with the time moment T_p being measured by the receiver's clock, we will generate a phase observable φ_j(T_j) at discrete time moments T_j of the receiver's clock T as follows:

where f_p,nom is the nominal value of the pedestal frequency, and where φ_jNCO(T_j) is the phase (integrated frequency) of the PLL's reference oscillator (e.g., NCO). The time moments T_j are spaced apart from each other by a time interval ΔT_j, as measured by the receiver's clock, and may be express as T_j = j·ΔT_j, where j is an integer. φ_j(T_j) is in units of cycles, and is proportional to the negative of the integrated Doppler-shift frequency. This is because φj_NCO(T_j) changes value in proportional to the quantity (f_p + f_D)•(T_j - T_p).

[0037] While it may be possible to set φ_jNCO(T_p) to any value at the initial lock moment Tp, it is preferably to set φ_jNCO(T_p) to a value substantially equal to (f_p,nom• T_p - f_S• D̃p/c), where Tp is measured from the start of GPS time, and where D̃p is the approximate distance between the satellite and the receiver, as found by the pseudorange measured by the DLL, or as found by a single point positioning solution. This setting of φ_jNCO(T_p) is conventional and provides values of φ_j(T_j) for the base and rover stations which are referenced from the same starting time point.

[0038] In U.S. Patent No. 6,268,824, which is commonly assigned with the present application, it is shown how the observable φ_j(T_j) is related to the distance D between the receiver and the satellite. We summarize the relationship here, neglecting the atmospheric delay τ_ATM(t_j) since this delay will be canceled out when we take the difference between receivers, and refer the reader to the patent for the derivation:

where

• f_S is the satellite carrier frequency,
• D_j is the distance between satellite and receiver at time moment T_j,
• c is speed of light in the atmosphere,
• T_j is the offset between the satellite time clock t and the receiver time clock T (T_j = t_j +'τ_j ), τ_j varies with time,
• Tp is the time moment when the PLL circuit initially locks onto the down-converted satellite signal,
• φ'₀ = Φ_LO + f_L,nom•T_p, where Φ_LO is the phase offset of the receiver's local oscillator and f_L,nom is the nominal frequency of the receiver's local oscillator,
• 
  
  is an initial phase value of the satellite,
• N_nco is an unknown integer number of carrier cycles due to the PLL not knowing which carrier cycle it first locks onto, N_nco may be positive or negative, and
• ξ_φj is a small tracking error due to the operation of the PLL circuit.

As it turns out, if we chose the time increments ΔT_j for T_j (T_j = j•ΔT_j) to be equal to ΔT_j = 2 ms or an integer multiple of 2 ms, then the term - f_S•T_p is an integer number which can be consolidated with the integer ambiguity -N_nco into a single ambiguity +N.

Thus, our Doppler phase observable φ_j(T_j) has been related to the distance D_j between the satellite and receiver, the time offset τ_j of the receiver, an integer ambiguity N, and some initial phase offsets

which can be readily determined.

[0039] We will now write equation (7) for the base and rover stations, adding superscripts "B" and "R" for the base and rover stations, and subscript "m" to indicate the m-th satellite signal and the m-th individual tracking channel.



For the differential navigation mode, the difference of these phases is formed:

Using N^m = (N_m^B - N_m^R) to represent the difference between the ambiguities, and using the well-known relationship f_S/c =1/λ_m, where λ_m is the wavelength of the satellite carrier signal, we have:

The values for

and

can be readily determined. The values of

and

cannot be determined, but they have zero mean values (and should average to zero over time).

[0040] Now, we relate this back to the initial objective and problem that we expressed with respect to FIG. 2. As indicated, we wanted to find the phase difference quantity (f_S•D_R(t)/C-f_S•D_B(t)/c) associated with each satellite in order to determine the number of carrier cycles that lie along the baseline, and thus provide a better estimate of the coordinates of the baseline. By examination of the phase difference quantity and equations (9) and (10), we seek that (f_S•D_R(t)/c-f_S•D_B(t)/c) is equal to

,_m(T_j) , which we can measure, minus the cycle ambiguity N^m, minus the error in the clocks

minus the phase offsets

and minus the errors



the last of which can be removed by averaging. Thus, estimating the cycle ambiguities N will be important in using the phase information to improve the estimated coordinates of the baseline.

Our Inventions related to Ambiguity Resolution - Estimating Floating Ambiguities

[0041] Our inventions can be applied to a number of application areas. In general application areas, the rover and base stations can move with respect to one another. In one set of specific application areas, the base and rover stations are fixed to a body or vehicle (e.g., planes and ships) and separated by a fixed distance L, as shown in FIG. 3. We refer to these applications as the constant distance constraint applications. The movement of the vehicle may be arbitrary while the distance between two antennas is held constant due to the rigidity of the vehicle body. This distance L_RB is known with a certain degree of confidence is considered as an additional constraint or measurement available at every time instant, because it depends on both Base and Rover positions. The constraint can be mathematically expressed as:

where x_R, y_R, and z_R, represent the rover station coordinates and where x_B, y_B, and z_B represent the base station coordinates. The information from the constraint can be used to better estimate the Rover position relative to the Base. For this, it is preferably to form a cost penalty function mathematically equivalent to the form:

where, in preferred embodiments, the cost function F(*) will be minimized such that and its the first derivatives with respect to the rover coordinates (or alternatively the base coordinates) are reduced to values near zero:

In the preferred methods described below, the amount of weighting given to minimizing the cost function F(*) will set by a weighting parameter q (e.g., q•F(*)) , with no weighting being given when q=0, and increasing degrees of weighting being given by selecting q>0. The case of setting q=0 is also equivalent to removing the constraint that the rover and base stations are separated by a fixed distance L.

[0042] The ambiguity resolution task generally comprises the following three main parts:

1. Generation of the floating ambiguity estimations (estimating the floating ambiguities),

2. Generation of the integer ambiguities, and

3. Formation of a signal of integer ambiguities resolution.

[0043] The present invention pertains to the first part, that of generating the floating ambiguity estimates. The second and third parts may be accomplished by apparatuses and methods known in the art. The apparatuses and methods of our present invention use as an input data:

(a) the pseudo-ranges and full phase measurements obtained in the Base and Rover receivers;

(b) the satellites coordinates (which are highly predictable can be readily determined from the GPS time and information provided in the 50 Hz low-frequency information signal by methods well known to the art);

(c) the estimated coordinates of the Base and Rover stations (as measured at their antennas); which may be generated from then pseudo-range data by single point solutions, and oftentimes the coordinates of the base station are precisely known;

(d) optionally, a set of weight coefficients characterizing the accuracy of the measurements; and

(e) optionally, an estimate r_j' of the baseline coordinates r_j and the value of L_RB if the cost function F(*) is used.

[0044] According to the input data, a vector of observations and a covariance matrix of measurements are formed. After that a state vector is generated, components of which are floating ambiguities, the number of which is equal to the number of satellites. On the basis of the floating ambiguity values, a search of the integer ambiguities is performed with use of a least-squares method that is modified for integer estimations.

[0045] Improvement of the floating ambiguity estimations can take place step-by-step, and the probability of correct ambiguity resolution increases step-by-step as information is accumulated. Preferred finishing of this process is registered by appearance of the signal of integer ambiguities resolution, which indicates that ambiguity resolution was performed sufficiently safely. After that, the integer ambiguities together with other input ) data are used for accurate determination of the base line vector. The tasks of determining the integer ambiguities and of generating the signal of integer ambiguities resolution are known to the art and do not form part of our invention. These tasks, therefore, are not described in greater detail.

[0046] Our invention is suitable for both the RTP mode and for the PP mode under movable Rover, where the Rover coordinates may be random and independent in adjacent clock moments. Our invention is also suitable for both the RTP mode and for the PP mode when both the base and rover stations are moving, where the coordinates of the stations may be random, but constrained by fixed distance L_RB, in adjacent clock moments.

[0047] We start by rewriting equation (10), which has the integer ambiguities N^m in vector form:

where Λ^-1 is a diagonal matrix of inverse wavelengths. We note that the components of

can be selected to be integer constants, and can therefore be incorporated into the integer ambiguities N, which may be called the modified integer ambiguities N. Thus, we may simplify the above equation:

After this modified N are found, the "true" N may be found by subtracting

from the modified N.

[0048] As an alternative equivalent to the above, the vector

can be computed and carried over to the left-hand side of equation (14) and subtracted from

As such, the N that is determined will be the "true" N. Once the "true" N is found, then the underlying unknown integer number of carrier cycles N_nco may be found by negating the "true" N (i.e., multipling by-1) and then subtracting f_S•T_p (N_nco = - "true" N - f_S•T_p). However, for the ultimate goal of computing more accurate coordinates of the baseline, the modified ambiguities N in equation (14) can be used, and they greatly simplify the computations. Nonetheless, it will be appreciated that the present invention may be practiced by using the "true" N or N_nco in equation (14) by suitable modification of the left-hand side of equation (14), and that the appended claims cover such practices.

[0049] Our approach comprises solving the above equations (14) at a plurality of time moments jointly to find an estimation for vector N. We cannot readily find measured values for vectors

and

and we have some errors in the range vectors

and

Our approach is to represent (model) the errors in the range vectors

and

and the terms

by the following error vector: Λ^-1.

where

is the Jacobian matrix (e.g., directional cosine matrix), and where [Δx, Δy, Δz, c•Δτ ]^T are corrections to the baseline coordinates and clock offsets of the receivers. Thus, we will model the above equation as:

The pseudoranges will be used to estimate the vector

as follows:

This will be done through the formation of observation vectors, state vectors, and observation matrices, and an estimation process over a plurality of time moments, as described below in greater detail. Vector N will be jointly estimated in this process.

Generation of the Vector of Observations.

[0050] A vector of observations µ_j is generated at each clock time moment t_j=j•ΔT_J and comprises 2•n components, where n is the number of the satellite channels. The first n components are the residuals of the single differences of the Base and Rover pseudoranges, which we denote in vector form as Δγ_j:

where

and

are vectors containing the pseudo-ranges of the satellites, measured in the Base and Rover receivers, respectively; and

and

are vectors of the estimated ranges of the satellites from Base and Rover stations at the moment of j-th signal radiation

and

are computed from the known positions of the satellites, the estimated position of the rover, and the known or estimated position of the base station by standard geometry.

[0051] The next n components of the observation vector µ_j are the residuals of the single differences of the Base and Rover full phases (Δφ_j), which we indicate here in vector form:

where

and

are vectors of the full phases of the given satellite signal, measured at the Base and Rover receivers, respectively (the phases are measured in cycles); and

Λ^-1 is a diagonal matrix of inverse wavelengths, where each diagonal component corresponds to a channel and is equal to 1/λ, where λ is the wavelength of the carrier signal in the given channel. The full phase vectors ϕ^B_j and ϕ^R_j may be constructed in the form provided by equation (5):



or may be constructed in the form which includes the phase offsets of the base receiver:

In either case, we will use the convention practice and have the base receiver correct its clock so that the base time T_j will be equivalent to the GPS time t_j for the purposes of implementing our inventions (the time offset has already been accounted for in the above equations). For all practical purposes, times T_j and t_j will refer to the same processing time increment, and can be interchanged in the above equations.

State Vector Representation.

[0052] The forms represented by Equations (15)-(18) maybe represented in matrix form equivalent to:

where vector A_j is a state vector related to observation matrix µ_j and where matrix

is an observation matrix that specifies the relationship between the components of the observations vector µ_j and the state vector A_j. State vector vector A_j comprises (4+n) components. The first three components are increments (Δx, Δy, Δz) to the coordinates (x°, y°, z°) of the baseline vector unknown at the j-th clock moment, the fourth component is the unknown increment of the reference oscillator phases (c•Δτ). The remaining n components are the unknown floating ambiguities, different in different channels (N¹, N²... Nⁿ). Matrix

comprises 2n rows and (4+n) columns, and may be divided into the following 4 parts (sub-matrices):

The first part, the left upper corner of this matrix (the first four columns by the first n rows), is occupied by the observation matrix

relating to the pseudo-range measurements, each row corresponding to one channel (from the 1-st to the n-th). For the n-th channel, the corresponding row appears like this:

where α_jn , β_jn, h_jn - the directional cosines of the range vector to the n-th satellite from Rover for the j-th time moment. Methods of computing directional cosines are well known to the art and a description thereof herein is not needed for one of ordinary skill in the art to make and use the present invention.

[0053] The second part of matrix

the left lower comer (the first four columns by the last n rows), is occupied by the matrix product

relating to the full phase measurements, each row corresponding to one channel (from the 1-st to the n-th). For the n-th channel, the corresponding row appears like this:

where λ_n is the wavelength of the carrier signal in the n-th channel.

[0054] The third part of matrix

the right upper corner (the last n columns by the first n rows) is occupied by zeroes. And the fourth part, the right lower corner (the last n rows by the last n columns) is occupied by the elements relating to the floating ambiguities. This part represents the identity matrix I_n with dimensions of nxn. For the discussion that follows below, it will be convenient to identified sub-blocks of matrix

as follows

where Q_j is a compound matrix formed by

and Λ as follows:

and where G is a compound matrix the zero matrix O_n×n and the identity matrix I_n as follows:

Equation (18) has 2n equations (according to the number of components of the observation vector), and may be used to determine the state vector A_j at the j-th clock moment (i.e., to determine 4+n unknown values). Solution of such a system of equations at n≥4 may be performed by the method of least squares. However, our invention relates to solving for the ambiguity vector N, which is a component of A_j, over a plurality of time moments. Before we describe that process, however, we want to first describe the groundwork of how we can integrate the minimization of the cost function F(*) for constrained distance between receivers, if used, with the solution of Equation (19), and to then describe some covariance matrices that characterize the accuracy of the measured data used to generate µ_j. These covariance matrices are helpful to our invention, but not strictly necessary.

[0055] The cost function F(*) may be expressed in the following form:

where:

is a vector of the first four components of A_j. and

is a vector of the coordinates of the baseline vector, with a zero as the four vector component. Using a second-order truncation of the Taylor series expansion, the cost function F(*) may be approximated as:

where



Because forms (21)-(23) share common variables with equation (18), the minimization of q•F(*) can be integrated with the solution of equation (18), as described below in greater detail.

Generation of the measurements covariance matrix.

[0056] Measurements covariance matrix R_J is preferably formed in the following way on the basis of weight coefficients obtained in Base and Rover receivers:

where

and

The weight coefficients

characterize the accuracy of the measurements of the residuals Δγ_j of the pseudo-range single differences for the corresponding satellite channels (1-st, 2-nd, ..., n-th). The generation of the weight coefficients is not a critical part of the present invention, and one may use his particular method of weighting. We present here one of our ways, where each of these coefficients maybe determined according to the weight coefficients measured in each channel by Base and Rover receivers for the pseudo-ranges, i.e., by values

and

[0057] Thus, for example, for the n-th channel:

where

and

are determined taking into account the measured signal-to-noise ratio in the receivers and the satellite elevation angles in the n-th channel (of Base and Rover, respectively). Specifically, for each of the receivers (no superscript used),



where

is the signal strength of the m-th satellite carrier signal as received by the receiver (it has been normalized to a maximum value and made dimensionless), where ξ_k,m is the elevation angle of the m-th satellite as seen by the receiver, where a minimum elevation angle ξ^B _min at which the signal becomes visible at the receiver, where σ_γ² is the variance of the code measurements (σγ ≈1 m). The factor

is dimensionless.

[0058] Weight coefficients

characterize the accuracy of the measurements of the residuals Δφ_j of the phase single differences, and are determined similarly. Here the same input data is used: the signal-to-noise ratio and the angle of elevation, but another scale for the phase measurements is considered (e.g., σ_ϕ² instead of σ_γ²). Specifically, for each of the receivers (no superscript used),

and

where

and ξ^B_min are as they are above, and where σ_ϕ² is the variance of the code measurements (σ_ϕ ≈ 1 mm).

[0059] When the magnitudes of either of weight coefficients

or

is less than a first selected small threshold value, the value of

is generated as a first small number which is less than the first threshold value. This is equivalent to setting

to a large number equal to the inverse of the first small number. Similarly, when the magnitudes of either of weight coefficients

or

is less than a second selected small threshold value, the value of

is generated as a second small number which is less than the second threshold value. This is equivalent to setting

to a large number equal to the inverse of the second small number.

[0060] In some instances, satellite signals are blocked from view and should be excluded from the ambiguity resolution process. The elements of the covariance matrix R_j corresponding to these satellite signals are replaced by a very large number. The very large number is selected in advance, and has a value which exceeds by several orders of magnitude the nominal values of the covariance matrix components

or

encountered during operation. Consequently, in further computations, the weights of all measurements relating to these channels become so small that they do not influence the result.

[0061] As a more simple approach, but currently less preferred, one can use the following form of covariance matrix R_j, or a scaled version thereof:

This form gives an equal weighting of 1/σ_γ² of to the rows of pseudorange data provided by Δγ_j and an equal weighting of 1/σ_ϕ² to the rows of phase data provided by Δ ϕ_j. When a satellite is blocked from view or not visible, the elements of the covariance matrix R_j corresponding to the satellite are preferably replaced by a very large number, as described above.

Central Aspects of the Present Invention

[0062] Summarizing equation (19), the linearized measurement model at the j-th epoch takes the form:

We expect that the vector of floating ambiguities N_j will be constant in time, and therefore we will drop the subscript index j. Instead, we will denote N_j the evaluation of N obtained through the epoch from 1 to j.

Generation of the floating ambiguity estimations

[0063] We create the following scalar quantity J in equation (27) which integrates the minimization of equation (26) with the minimization of the truncated cost functions F_k for the constrained distance condition (if used) according to equations (21) - (23):

As mentioned above, q is a scalar weighting factor (penalty parameter) which is equal to or greater than 0 (q ≥0). We seek to minimize J in value in order to obtain a_k, k = 1,..., j and N . The minimization of scalar J essentially seeks to minimize the errors in the individual forms of

and conditions (11), if applicable, for data of all of the j epochs being considered, but with the constraint that floating ambiguity vector N in each individual state vector A_k be the same. The individual vectors a_k are allowed to have different values. If the fixed distance constraint between rover and base stations is not considered to be present, then cost function F_k is omitted from equation (27), and all following equations based on equation (27). This may be simply done by setting weighting parameter q=0. If the fixed distance constraint is used, values for the weighting parameter q can range from approximately 0.5 to approximately 4, with a typical range being between approximately 1 and approximately 3. The best value of q often depends upon the amount of noise in the signals and the receivers, and can be found by trying several values within the above ranges (i.e., fine tuning). The inventors have found a value of q=2 to be useful for their test applications. In the case where the distance between the receivers is constrained to a fixed value, we emphasize that the use of the cost function qF is optional, and that one is not required to use it. Using the methods of the present invention without the cost function qF will still provide estimates of the floating ambiguities. The inclusion of the cost function qF generally enables more accurate estimates.

[0064] The inventors have discovered that a set of N which minimizes scalar J can be found by solving the following block linear system for the set N, which is a vector.

To the inventors' knowledge, the form of scalar J provide by equation (27) and form of the linear system provided by equation (28) are not found in the prior art. While it is preferred to use the weighting matrices R and their inverse matrices, it may be appreciated that the present invention can be practiced without them. In the latter case, each weighting matrix R may be replaced by an identity matrix of similar dimension in Equation (28) and the following equations; each inverse matrix R^-1 is similarly replaced by an identity matrix.

[0065] The inventors have further constructed an inverse matrix for the block matrix on the left-hand side of the equation (28), and from this inverse matrix have found a form of N which satisfies equation (28) as follows:

where this form comprises an n x n inverse matrix multiplying an n x 1 vector. In the discussion that follows, we will identify the n x n inverse matrix as M^-1 and the n x 1 vector as B, with N = M^-1 B. The above form may be more simply expressed if we form a matrix matrix P_k for the data of the k-th time moment in the following form:

and a vector:

[0066] With matrix P_k, equation (29) may now be written as:

We refer to P_k as a projection-like matrix for q=0 and a quasi-projection matrix for q>0. Each component matrix

is symmetric positive definite and may be inverted. Moreover, a matrix comprised of a summation of symmetric positive definite matrices is also symmetric positive definite. Thus, matrix M is symmetric positive definite and can be inverted. In preferred implementations of the present invention, the inverse of M is not directly computed. Instead, a factorization of M into a lower triangular matrix L and an upper triangular matrix U is produced as follows: M = LU. Several different factorizations are possible, and L and U are not unique for a given matrix M. The LU factorization of matrix M enables us to compute the floating ambiguities N through a sequence of forward and reverse substitutions. These substitutions are well known to the art (cf. any basic text on numerical analysis or matrix computations). With symmetric positive definite matrices, one may choose L and U such that U=L^T, which gives a factorization of M = LL^T. This is known as the Cholesky factorization, and it generally has low error due to numerical rounding than other factorizations.

[0067] With N being generated from this form, the inventors have further found that each individual vector a_k can be generated according to the following form:

The generation of the individual vectors a_k is optional to the process of generating a set of floating ambiguities N. However, when using cost function F(*) and equations (22) and (23), one can generate a_k in order to update r_k. The forms of N and a_k which the inventors have discovered enable one to generate a vector N without having to first generate the individual vectors a_k.

[0068] When using the cost function F(*), S_k and h_k (and g_k which is derived from h_k and S_k), an estimate of the baseline vector r_k at time moment "k" has to be generated. As indicated by equations (22) and (23), both S_k and h_k depend upon L_RB, which does not normally change, and upon r_k, which can change and often does change. For generating S_k and h_k, it is usually sufficient to use an estimate of r_k, which we denote at r_k' and which may be provided by the user or general process that is using the present invention. As part of generating the calculated distances D_k^R and D_k^B, the user or general process uses the estimated position of the rover and the known or estimated position of the base station. An estimate r_k' can be generated by subtracting the position of the base station from the estimated position of the rover. If desired, one can refine the estimate by generating a_k from equation (31), and then using the combination (rk'+a_k) as a more refined estimate of r_k in generating S_k and h_k. To do this, one may first generate an estimate N' to N by using equation (30) with S_k=0 and g_k=0. Then equation (31) can be evaluated using N=N', S_k=0, and g_k=0 to generate refined baseline vectors rk =(r_k'+a_k), and initial values of S_k', h_k', and g_k'. Equation (30) is then again evaluated using the initial values S_k', h_k', and g_k' for S_k, h_k, and g_k, respectively. The process may be repeated again. To speed convergence, one can consider using prior values of S_k' and g_k' in generating a_k for k>1 as follows:

First Exemplary Set of Method Implementations of the Present Invention

[0069] The inventors have discovered a number of ways to employ the form of N provided by Equation (30). For post-processing application, the satellite data may be collected for a plurality j of time moments, the matrices R_k and Q_k (and optionally S_k) and the vectors µ_k (and optionally h_k and g_k) for each k-th time moment may then be computed, and the n x n inverse matrix M^-1 and the n x 1 vector B may then be generated and thereafter multiplied together to generate the vector N. The plurality of time moments may be spaced apart from one another by equal amounts of time or by unequal amounts of time. As indicated above, instead of generating the inverse matrix M^-1; one may generate the LU factorization of M and perform the forward and reverse substitutions. The factorization-substitution process is faster than generating the inverse matrix, and more numerically stable than most matrix inversion processes. If necessary, Equations (30) and (31) may be iterated as described above until convergence for significantly nonlinear dependency of quantities in equations (22) and (23) on the Rover position.

[0070] For real-time applications, M^-1 and B may be initially computed at a first time moment from a first set of data (e.g., R₁, Q₁, µ_1, R₂, Q₂, µ₂) and then recomputed at subsequent time moments when additional data becomes available. For instance, we can initially compute M and B based on l time moments k=1 to k=l as follows:



where the subscript "l" has been used with M_l and B_l to indicate that they are based on the l time moments k=1 to k=l. The floating ambiguity may then be computed from N_l= [Ml^-1] B_l, using matrix inversion or LU-factorization and substitution. Data from the next time moment l+1 can then be used to generate the updated quantities M_l+1 and B_l+1 as follows using the previously computed quantities M_l and B_l:



With an updated set of floating ambiguities being computed from N_l+1 = [M_l+1]^-1 B_l+1, using matrix inversion or LU-factorization and substitution. It may be appreciated that the above forms of M_l+1 and B_l+1 may be employed recursively (e.g., iteratively) in time to compute updated floating ambiguities N_l+1 from previously-computed values of the quantities M_l and B_l as satellite data is collected. The recursion may be done with each recursive step adding data from one epoch, or with each recursive step adding data from multiple epochs, such as provided by the following forms:



where M_l and B_l are based upon j time moments, and M_m and B_m are based on these time moments plus the time moments l+1 through to m, where m > l+1. While these recursion methods are preferably applied to real-time applications, it may be appreciated that they may be equally used in post processing applications. Furthermore, while one typically arranges the time moments such that each time moment k+1 occurs after time moment k, it may be appreciated that other ordering of the data may be used, particularly for post processing applications.

[0071] The steps of the above method are generally illustrated in a flow diagram 40 shown in FIG. 4. Initially, the data is received (e.g.,

and optionally r_j' and L_RB) as indicated at reference number 41. At step 42, the method generates, for each time moment j, a vector Δγ_j of a plurality of range residuals of pseudo-range measurements made by the first and second navigation receivers in the form of:

At step 43, the method generates, for each time moment j, a vector Δϕ_j of a plurality of phase residuals of full phase measurements made by the first and second navigation receivers in the form of:

-

where Λ^-1 is a diagonal matrix comprising the inverse wavelengths of the satellites, the set of phase residuals being denoted as Δϕ_k , k=1, .., j. At step 44, the method generates an LU-factorization of a matrix M or a matrix inverse of matrix M, the matrix M being a function of at least Λ^-1 and H_k^γ, for index k of H_k^γ covering at least two of the time moments j. Exemplary forms of matrix M have been provided above. At step 45, the method generates a vector N of estimated floating ambiguities as a function of at least the set of range residuals Δγ_k, the set of phase residuals Δϕ_k, and the LU-factorization of matrix M or the matrix inverse of matrix M. Exemplary forms of vector N have been provided above. If the cost function F(*) is included, then steps 44 and 45 may be reiterated, along with a step of generating a_k for refined estimates of r_k, as indicated above.

[0072] It may be appreciated that the above steps may be performed other sequences than that specifically illustrated in FIG. 4, as long as the data needed to perform a specific step is available. For example, parts of steps 41, 42, and 43 may be performed as matrix M is being assembled in step 44.

[0073] The above method maybe carried out on the exemplary apparatus 100 shown in FIG. 5. Apparatus 100 comprises a data processor 110, an instruction memory 112 and data memory 114 for data processor 110, an optional keyboard/display 115 for interfacing between data processor 110 and a human user, and a generalized data portal 120 for receiving the measured data

and optionally r_j'and L_RB, each data having been described in detail above. Memories 112 and 114 may be separate, or difference sections of the same memory bank. Generalized data portal 120 may take any number of conventional forms, such as one or more input/output ports, or one or more files stored on disk, tape, non-volatile memory, volatile memory, or other forms of computer readable medium. The data can be placed in data portal 120 by any number of means, such as by a user of the apparatus, or by a more general apparatus that utilizes the apparatus of the present invention in carrying out its functions (such as, for example, computing precise estimates of the position of the rover or the coordinates of the baseline vector). In the former case, the keyboard/display 115 may be used to receive information from the user as to when new data has been provided to data portal 120 (this may be useful in post-processing applications). In the latter case, a more general apparatus may comprise the rover station (including receiver channels such as that shown in FIG. 1) and radio transceiver for receiving data from the base station. In the case that both the base and rover stations are located on the same vessel, the more general apparatus may comprise the base and rover stations.

[0074] Data processor 110 may be configured to implement the above-described method embodiments, such as exemplified by the steps in FIG. 4, by running under the direction of a computer product program present within instruction memory 112. An exemplary computer program product 60 is illustrated in FIG. 6. Computer program product 60 may be stored on any computer-readable medium and then loaded into instruction memory 112 as needed. Instruction memory 112 may comprises a non-volatile memory, a programmable ROM, and hard-wired ROM, or a volatile memory.

[0075] Computer program produce 60 comprises five instruction sets. Instruction Set #1 directs data processor 110 to receive the measured data from data portal 120. The measured data from portal 120 may be loaded into data memory 114 by Instruction Set #1. As another implementation, the data may be loaded into memory 114 by subsequent instruction sets as the data is needed. In the latter case, Instruction Set #1 can take the form of a low-level I/O routine that is called by the other instruction sets as needed. As such, data portal 120 and data processor 110 under the direction of instruction set #1 provide means for receiving the measured data for apparatus 100. Instruction Set #2 directs data processor to generate, for each time moment j, a vector Δγ_j of a plurality of range residuals of pseudo-range measurements made by the first and second navigation receivers in the form of:

As such, data processor 110 under the direction of instruction set #2 provides means for generating the range residuals Δγ_j for apparatus 100. Instruction Set #3 directs data processor 110 to generate, for each time moment j, a vector Δϕ_j of a plurality of phase residuals of full phase measurements made by the first and second navigation receivers in the form of:



where Λ^-1 is a diagonal matrix comprising the inverse wavelengths of the satellites. As such, data processor 110 under the direction of instruction set #3 provides apparatus 100 with means for generating the phase residuals Δϕ_j. The residuals may be stored in data memory 114.

[0076] Instruction Set #4 directs data processor 110 to generate an LU-factorization of matrix M or a matrix inverse of matrix M, the matrix M being a function of at least Λ^-1 and

for index k of

covering at least two of the time moments j. Exemplary forms of matrix M have been provided above. Matrix M and its LU-factorization or inverse may be stored in data memory 114. As such, data processor 110 under the direction of instruction set #4 provide apparatus 100 with means for generating matrix M and its LU-factorization or inverse. Finally, instruction Set #5 directs data processor 110 to generate a vector N of estimated floating ambiguities as a function of at least the set of range residuals Δγ_k, the set of phase residuals Δϕ_k, and the LU-factorization of matrix M or the matrix inverse of matrix M. Exemplary forms of vector N have been provided above. Vector N may be stored in data memory 114. As such, data processor 110 under the direction of instruction set #5 provide apparatus 100 with means for generating an vector N, which is an estimate of the floating ambiguities.

[0077] The resulting estimates provided by vector N may be outputted on keyboard/display 115, or may be provided to the more general process through data portal 120 or by other transfer means (such as by memory 114 if data processor 110 is also used to implement the more general process).

Second Exemplary Set of Method Implementations of the Present Invention

[0078] The inventors have developed additional recursive methods that are generally better suited to real-time applications. The second exemplary set of methods generally facilitate implementations which require less memory storage space and fewer computations. We previously defined a projection-like matrix P_k for the data of the k-th time moment as follows:

Equation (32) was then applied to the form of equation (30) to provide the form:

where M and B are identified in the forms of:

The forms of equations (33B) can be expressed in the following recursion form:

Then

noting that N_j = [M_j]^-1 B_j for the data from time moments 1 through j, we can write N_j-1 = [M_j-1]^-1 B_j-1 for the data from time moments 1 through j-1. The latter can be rearranged as M_j-1 N_j-1 = B_j-1 and used to substitute for the term B_j-1 in equation (35A) to provide:

To equation (35B), we now add and subtract the term G^TP_j G N_j-1 from the second. bracketed quantity to obtain:

Noting that the first two terms of the second bracketed quantity can be factored as (M_j-1+G^TP_jG) N_j-1 and that the factor (M_j-1+G^TP_jG) is the inverse of the first bracketed quantity in equation (35C), it can be seen that the first bracketed quantity multiplied onto (M_j-1+G^TP_jG) N_j-1 is simply N_j-1. Therefore, equation (35C) can be simplified as:

The second bracketed quantity of equation (35D) can be further simplified as:

Using equation (34) this becomes:

With this, the following recursive method may be used in a real time application:

(1) Generate initial values of M₀ and N₀, set epoch counter k to zero ( k = 0).

(2) Increment the epoch counter by one; obtain the data needed to generate the matrices R_k and Q_k, and the vector µ_k.

(3) Generate R_k and Q_k, and µ_k. If the constant distance constraint is to be used, then also compute S_k, h_k, and g_k, and select a value for q greater than zero; otherwise, set qS_k =0 and qg_k=0 in steps (4)-(7) below.

(4) Generate P_k in a form equivalent to:

(5A) Generate an LU-factorization of M_k, where M_k is in a form equivalent to M_k = M_k-1 + G^TP_k G. Exemplary ways of generating LU factorizations are described in greater detail below.
-- OR --

(5B) Generate an inverse matrix M_k^-1 of M_k, where M_k is in a form equivalent to M_k = M_k-1 + G^T P_k G. In general, this approach requires more computation than the approach of step (5A), and is currently the less preferred approach.

(6) Generate N_k = N_k-1 + M_k^-1(G^TP_k (µ_k - G N_k-1) + qg_k). If the inverse matrix M_k^-1 has been generated according to step (5B), this form may be generated by conventional multiplication of M_k^-1 onto a vector F_k, where F_k = (G^TP_k (µ_k - G N_k-1) + qg_k). N_k-1) + qg_k). If the LU factorization for M_k has been generated according to step (5A), then the quantity X_k = M_k^-1 (G^TP_k (µ_k - G N_k-1) + qg_k) may be generated from a forward and reverse substitution on the form: L_k U_k X_k = F_k, where L_k is the lower triangular matrix and U_k is the upper triangular matrix of the factorization. The quantity X_k is then added to N_k-1 to form N_k.

(7) Optionally generate a_k=(Q_k^TR_k^-1Q_k)^-1 (Q_k^TR_k^-1(µ_k-GN_k)-qg_k)

(8) Reiterate steps (2)-(7).

For initial values it is convenient to use M₀ = 0 and N₀ = 0. One can also use satellite data from a single epoch to generate values for M₀ and N₀ using forms (4), (5A), (7A). This is in fact what is done with the above steps (2)-(6) are performed on the data for the first epoch with M₀= 0 and N₀= 0. In step (3), if the distance constraint is to be used, one may use r_k' to generate S_k, h_k, and g_k , or one may generate more refined estimate as (r_k'+a_k), with a_k being generated as:

or as

[0079] During the first few recursion steps, matrix M_k may be ill-conditioned, and thus the generation of the LU factorization or inverse of M_k may incur some rounding errors. This problem can be mitigated by using R_k = I during the few recursions, or using an R_k with more equal weightings between the psuedorange and phase data, and then switching a desired form for R_k.

[0080] The steps of the above method are generally illustrated in a flow diagram 70 shown in FIG. 7. Initially, the data is received (e.g.,

and optionally r_j' and L_RB) as indicated at reference number 71. At step 72, the method generates, for each time moment j, a vector Δγ_j of a plurality of range residuals of pseudo-range measurements made by the first and second navigation receivers in the form of: Δγ_j

At step 73, the method generates, for each time moment j, a vector Δγ_j of a plurality of phase residuals of full phase measurements made by the first and second navigation receivers in the form of:

where Λ^-1 is a diagonal matrix comprising the inverse wavelengths of the satellites, the set of phase residuals being denoted as Δϕ_k, k=1, .., j.

[0081] At step 74, the method generates, for time moment j = 1, an LU-factorization of a matrix M₁ or a matrix inverse of matrix M₁, the matrix M₁ being a function of at least Λ^-1 and

Any of the forms for M described above may be used. At step 75, the method generates, for time moment j = 1, a vector N₁ as a function of at least Δγ₁, Δϕ₁, and the LU-factorization of matrix M₁ or the matrix inverse of matrix M₁. At step 76, the method generates, for an additional time moment j ≠ 1, an LU-factorization of a matrix M_j or a matrix inverse of matrix M_j, the matrix M_j being a function of at least Λ^-1 and

At step 77, the method generates, for an additional time moment j ≠ 1, a vector N_j as a function of at least Δγ_j, Δϕ_j and the LU-factorization or matrix M_j or the matrix inverse of matrix M_j. At step 78, the estimated ambiguity vector N_j is reported. If the estimated ambiguity vector has not achieved sufficient accuracy, as set by the user, steps 76 and 77 are repeated, with steps 76-78 forming a loop. If the estimated ambiguity vector has achieved a sufficient degree of accuracy, or if steps 76-78 have been repeated for a maximum number of times set by the user, the process is ended. If the cost function F(*) is included, then step 76 may include the generation of a_k for refined estimates of r_k, as indicated above.

[0082] It may be appreciated that the above steps may be performed other sequences than that specifically illustrated in FIG. 7, as long as the data needed to perform a specific step is available. For example, parts of steps 71-73 may be performed as matrix M₁ is being assembled in step 74, and as matrix M_j is being assembled in step 76. It may also be appreciated that the data set provided to the process may span several tens of time moments to several thousands of time moments, and that the time moment j=1 illustrated above may correspond to any of the time moments in the data set, not just the earliest time moment or the first received time moment. It may also be appreciated that steps 74-75 may generated their respective M matrices and N vectors from data measured over two or more time moments as discussed above with regard to equations (31A)-(31F), as well as just one time moment, and that steps 76-77 may generated their respective M matrices and N vectors from data measured over two or more time moments as discussed above with regard to equations (31A)-(31F), as well as just one time moment. It may be further appreciated that approproate ones of the appended claims are intended to cover these possibilities.

[0083] The above method embodiments may be carried out on the exemplary apparatus 100 shown in FIG. 5, using a different set of instructions in memory 112. Specifically, data processor 110 may be configured to implement the above-described method embodiments exemplified by the steps in FIG. 7, by running under the direction of a computer product program present within instruction memory 112. An exemplary computer program product 80 is illustrated in FIG. 8. Computer program product 80 may be stored on any computer-readable medium and then loaded into instruction memory 112 as needed. Instruction memory 112 may comprises a non-volatile memory, a programmable ROM, and hard-wired ROM, or a volatile memory.

[0084] Computer program produce 80 comprises eight instruction sets. Instruction Set #1 directs data processor 110 to receive the measured data from data portal 120. The measured data from portal 120 may be loaded into data memory 114 by Instruction Set #1. As another implementation, the data may be loaded into memory 114 by subsequent instruction sets as the data is needed. In the latter case, Instruction Set #1 can take the form of a low-level I/O routine that is called by the other instruction sets as needed. As such, data portal 120 and data processor 110 under the direction of instruction set #1 provide means for receiving the measured data for apparatus 100. Instruction Set #2 directs data processor to generate, for each time moment j, a vector Δγ_j of a plurality of range residuals of pseudo-range measurements made by the first and second navigation receivers in the form of:

As such, data processor 110 under the direction of instruction set #2 provides means for generating the range residuals Δγ_j for apparatus 100. Instruction Set #3 directs data processor 110 to generate, for each time moment j, a vector Δϕ_j of a plurality of phase residuals of full phase measurements made by the first and second navigation receivers in the form of:



where Λ^-1 is a diagonal matrix comprising the inverse wavelengths of the satellites. As such, data processor 110 under the direction of instruction set #3 provides apparatus 100 with means for generating the phase residuals Δϕ_j. The residuals may be stored in data memory 114.

[0085] Instruction Set #4 directs data processor 110 to generate, for time moment j = 1, an LU-factorization of a matrix M₁ or a matrix inverse of matrix M₁, the matrix M₁ being a function of at least Λ^-1 and

Exemplary forms of matrix M₁ have been provided above. Matrix M₁ and its LU-factorization or inverse may be stored in data memory 114. As such, data processor 110 under the direction of instruction set #4 provide apparatus 100 with means for generating matrix M₁ and its LU-factorization or inverse. Instruction Set #5 directs the data processor 110 to generate, for time moment j = 1, an estimated ambiguity vector N₁ as a function of at least Δγ₁, Δϕ₁, and the LU-factorization of matrix M₁ or the matrix inverse of matrix M₁. Exemplary forms of vector N₁ have been provided above. Vector N₁ may be stored in data memory 114. As such, data processor 110 under the direction of instruction set #5 provide apparatus 100 with means for generating an vector N₁, which is an estimate of the floating ambiguities.

[0086] Instruction Set #6 directs data processor 110 to generate, for one or more additional time moments j ≠1, an LU-factorization of a matrix M_j or a matrix inverse of matrix M_j, the matrix M_j being a function of at least Λ^-1 and

Exemplary forms of matrix M_j have been provided above. Matrix M_j and its LU-factorization or inverse may be stored in data memory 114. As such, data processor 110 under the direction of instruction set #6 provide apparatus 100 with means for generating matrix M_j and its LU-factorization or inverse. Because of their similar operations, Instruction Set #6 may share or duplicate portions of Instruction Set #4. Instruction Set #7 directs the data processor 110 to generate, for one or more additional time moments j ≠1, a vector N_j as a function of at least Δγ_j, Δϕ_j , and the LU-factorization or matrix M_j or the matrix inverse of matrix M_j. Exemplary forms of vector N_j have been provided above. Vector N_j may be stored in data memory 114. As such, data processor 110 under the direction of instruction set #7 provide apparatus 100 with means for generating an vector N_j, which is an estimate of the floating ambiguities. Because of their similar operations, Instruction Set #7 may share or duplicate portions of Instruction Set #5.

[0087] Instruction Set #8 directs the data processor 110 to report vector N_j as having estimates of the floating ambiguities, and to repeat Instruction Sets #6 and #7 if vector does not have sufficient (or desired) accuracy, or if it is desired to keep the process going even through sufficient accuracy has been reached. The resulting estimates provided by vector N may be outputted on keyboard/display 115, or may be provided to the more general process through data portal 120 or by other transfer means (such as by memory 114 if data processor 110 is also used to implement the more general process).

[0088] Give the detailed description of the present Specification, it is well within the ability of one skilled in the GPS art to construct all of the above Instruction Sets without undue experimentation, and a detailed code listing thereof is not needed for one of ordinary skill in the art to make and use the present invention.

Methods of LU-factorization of Matrix M

First method

[0089] We now discuss methods of LU-factorization for matrix M. The factorization methods may be used in any of the above steps or computer instruction sets where an LU-factorization is generated or where an inverse matrix is generated, such as in step (5A) described above, and also used in step (5B) to construct an inverse matrix for M, although such is computationally costly. In one approach of LU-factorization, the matrix M_k-1 from the previous iteration is retained for the current iteration, and G^T P_k G from the current iteration is added to it to form the matrix M_k for the current iteration of step (5B). Then, any LU-factorization method may be used to find a lower triangular matrix L_k and an upper triangular matrix U_k which satisfies M_k = L_k U_k . Since M_k is symmetric positive definite, the Cholesky method may be used. This method has good numerical stability, and generates upper triangular matrix U_k such that it equals the transpose of lower triangular matrix. L_k: U_k = L_k^T. With this factorization, M_k = L_k L_k^T. The standard Cholesky method requires a square-root operation for each row of the matrix (n rows requires n square-root operations). Such operations may be difficult or time consuming to perform on mid-range processor chips. A modification of the Cholesky method developed by Wilkinson and Reinsch may be used to avoid these square-root operations. In this method, the factorization of M_k = L̃DL̃^T is used, where D is a diagonal matrix. We refer readers who are not familiar with this area of the art to the following references for further information:

1. Kendall E. Atkinson, An Introduction to Numerical Analysis, publisher: John Wiley & Sons, 1978, pages 450-454;

2. J. Wilkinson and C. Reinsch, Linear Algebra, Handbook for Automatic Computation, Vol. 2, publisher: Springer-Verlag, New York, 1971, pages 10-30;

3. Gene Golub and Charles Van Loan, Matrix Computations, publisher: The Johns Hopkins University Press, Baltimore, Maryland, pages 81-86.

The above described main instruction sets that generate LU factorizations of matrix M can be constructed to include instructions that direct data processor 110 to carry out the above forms of Cholesky factorization of matrix M under this first method. The combination of these instructions and data processor 110 provides apparatus 100 with means for performing the Cholesky factorizations.

Second method.

[0090] Instead of using the modified Cholesky method, the following method developed by the inventors may be used. The inventors currently prefer this method. Given that we have generated the previous factorization M_k-1 = L_k-1 L_k-1^T, we generate a factorization T_k T_k^T for the quantitiy G^T P_k G as follows: T_k T_k^T = G^T P_k G. Later, we will describe how this factorization T_k T_k^T may be generated. Using T_k T_k^T = G^T P_k G, the factorization L_k L_k^Tof M_k may be written as:

It is known in the matrix computation art that the product of two n x n matrices X and Y is equal to the sum of the outer products of the columns of these matrices, as specified below:

where {x₁, x₂ , ..., x_n } are the columns of matrix X, and where {y₁, y₂, ..., y_n } are the columns of matrix Y. The inventors have applied this general knowledge to their development of the invention to recognize that

where {t_k,1, t_k,2,..., t_k,n} are the columns of matrix T_k. Each outer product

is an n x n matrix of rank one. It is known from the article entitled "Methods for Modifying Matrix Factorizations" by P.E. Gill, G.H. Golub, W. Murray, and M.A. Saunders (Mathematics of Computation, Vol. 28, No. 126, April 1974, pp. 505-535) that when a rank-one matrix of the form z z^T is added to a symmetric positive definite matrix A, a previously computed Cholesky factorization of matrix A may be modified with a relatively few number of computations (much less than the number of computations need to generate a factorization of A+z z^T). The inventors have further recognized that performing n such rank-one modifications on the previous factorization L_k-1 L_k-1^T for M_k-1 using

for s = 1 to s = n would also require less computations and would have better accuracy that a new factorization for M_k.

[0091] Let us illustrate this approach by first defining a set of intermediate factorization matrices L̃₁ , L̃₂, ..., L̃_n , each of which has the same dimensions as L_k-1 and L_k. We now go through a sequence of n rank-one modifications which will sequentially generate the matrices L̃₁ through L̃_n, with the last matrix L̃_n being in a form which is substantially equal to the desired factorization L_k. The first rank-one modification starts with any of the matrices

Without loss of generality, we will start with s=1 in order to simplify the presentation. We then write:

which can be factored as:

where vectors c₁ and t_k,1 are related to one another as follows: L_k-1 ^Tc₁ = t_k,1. Vector c₁ is readily obtained from vector t_k,1 with a forward substitution process with the previously computed matrix L_k-1, since L_k-1 is lower triangular. Then, the above reference by Gill, et al. teaches how to readily obtain a Cholesky factorization of (I +

), which we denote here as

The reader is referred to that reference, as well any other references teaching such factorizations, for the implementation details. From this it can be seen that the form of equation (35) becomes:

and thus L̃₁= L_k-1 L₁ . The multiplication of two lower triangular matrices is relatively easy to perform and computationally inexpensive.

[0092] We now perform the second rank-one modification using rank-one modification the matrix

in a similar manner by writing:

which can be factored as:

where vector c₂ is generated from t_k,2 by forward substitution according to: L̃₁ c₂ = t_k,2. A Cholesky factorization of

is then obtained, which we denote here as

Thus, L̃₂=L̃₁ L₂ . The recursion process continues in this manner until L̃_n is computed. The following recursion sequence can be used:

(1) Allocate matrices L̃₀, L̃₁,L̃₂, ..., L̃_n, and L₁ , L̃₂, ..., L_n , and vectors c₁,c₂, ..., c_n.

(2) Set an index j =1, and set L₀ = L_k-1.

(3) Repeat the following steps (4) - (7) n times:

(4) Generate vector c_j from vector t_k,j by forward substitution according to:

(5) Generate matrix L_j as a Cholesky factorization of the quantity

(6) Generate matrix L̃_j as the matrix multiplication of L̃_j-1 and L̃_j :

(7) Increment index j.

(8) Provide L̃ as L_k.

Third method.

[0093] Close examination of the second method shows that the following more compact recursion sequence may be used:

(1) Allocate matrices L̃ and L and a vectors c.

(2) Set an index j =1, and set L̃ = L_k-1.

(3) Repeat the following steps (4) - (7) n times:

(4) Generate vector c from t_k,j by forward substitution according to: L̃ c_j = t_k,j.

(5) Generate matrix L as a Cholesky factorization of the quantity (I + cc^T).

(6) Generate the matrix multiplication product L̃ L, and store the result as L̃ (e.g., overwrite the storage location of L̃ with the product L̃ L ).

(7) Increment index j.

(8) Provide L̃ as L_k.

The above-described n sequential rank-one modifications require approximately half as many operations as direct re-factorization. In addition, low rank modifications to a matrix factorization are generally more numerically stable than direct re-factorization.

[0094] The previously-described main instruction sets that generate LU factorizations of matrix M can be constructed to include instructions that direct data processor 110 to carry out the above forms steps of factorizing matrix M under the above second and third methods. Specifically, there would be a first subset of instructions that direct the data processor to generate an LU-factorization of matrix M_j-1 in a form equivalent to L_j-1 L_j-1^T wherein L_j-1 is a low-triangular matrix and L_j-1^T is the transpose of L_j-1. In addition, there would be a second subset of instructions that direct the data processor to generate a factorization of G^T P_j G in a form equivalent to T_j T_j^T = G^T P_j G, where Tj^T is the transpose of T_j (examplary methods for this are described below). There would also be a third subset of instructions that direct the data processor to generate an LU-factorization of matrix M_j in a form equivalent to L_j L_j^T from a plurality n of rank-one modifications of matrix L_j-1, as described above, each rank-one modification being based on a respective column of matrix T_j, where n is the number of rows in matrix M_j. The combination of these subsets of instructions and data processor 110 provides apparatus 100 with means for performing the above tasks.

Generation of matrix T_k

[0095] The last two factorization methods requires finding T_k T_k^T = G^T P_k G so that the column vectors t_k,s may be used. One can use the Cholesky factorization method to generate T_k since G^TP_k G is symmetric positive definite (we call this the first method of generating matrix T_k). The above second subset of instructions be constructed to include further instructions that direct data processor 110 to carry out the any form of Cholesky factorization of G^T P_k G.to generate T_k,. The combination of these further instructions and data processor 110 provides apparatus 100 with means for generate matrix T_j from a Cholesky factorization of G^T P_j G. However, the inventors have developed the following more efficient second method of generation of matrix T_k. It must be noted that this second method of generation of matrix T_k is applicable if the penalty parameter q related to the constant distance constraints defined by equation (11), and appearing starting with the Equation (27), is set to zero. In other words, the second method of generating matrix T_k described below is applicable if the constant distant constraints are not used.

The second method of generating matrix T_k

[0096] The second method is based on a novel block Householder transformation of the matrix G^T P_k G which the inventors have developed. This method is based on constraining the weighting matrices R^γ and R^ϕ to the following forms that are based on a common weighing matrix W:

where σ_γ and σ_ϕ are scalar parameters selected by the user, and where

is either the L1-band wavelength or the L2-band wavelength of the GPS system. If all the wavelengths in Λ are the same, the above forms reduce to:

[0097] With W, σ_γ σ_ϕ,

R^γ and R^ϕ selected, the following steps are employed to generate matrix T_k

1. Generate a scalar b in a form equivalent to:

= (in the case of dual band measurements

2. Generate a matrix H̃ in a form equivalent to H̃ = W^1/2H_k.

3. Generate a Householder matrix S_HH for matrix H̃.

4. Generate matrix T_k in a form equivalent to:

or in the case of dual band measurements:

where sub-matrixces A11, A21, and A22 are as follows:



and

[0098] The derivation of this method is explained in Appendix A.

[0099] The above second subset of instructions be constructed to include further instructions that direct data processor 110 to carry out the above four general step under this second method of generating matrix T_k. The combination of these further instructions and data processor 110 provides apparatus 100 with means for generate matrix T_j .

[0100] While the present invention has been particularly described with respect to the illustrated embodiments, it will be appreciated that various alterations, modifications and adaptations may be made based on the present disclosure, and are intended to be within the scope of the present invention. While the invention has been described in connection with what is presently considered to be the most practical and preferred embodiments, it is to be understood that the present invention is not limited to the disclosed embodiments but, on the contrary, is intended to cover various modifications and equivalent arrangements included within the scope of the appended claims.

APPENDIX A - Derivation of the Second Method of Generating Matrix T.

[0101] To simplify the presentation of this derivation, we take the case of where the measurements are provided in only one GPS frequency band (e.g., L1-band). Generalization to two-band data and multiband data (GLONASS) is straight-forward matter. First, let us relate the range weighting matrix (R^γ)^-1 and the phase weighting matrix (R^ϕ)^-1 to a common weight matrix W and two scalar parameters σ_γ and σ_ϕ as follows:

where W is a diagonal positive definite n × n matrix. We now want to look at the block factorization of the component

which is part of

To simplify the presentation, we are going to omit the time-moment subscript "k" and the pseudorange superscript "γ" from, our notation for

and simply use the notation


in the following discussion. With the above, we can write the matrix component

as follows:

With the condition that the measurement data is within one signal band, all of the wavelengths are the same (for the purposes of determining the floating ambiguity), and the diagonal wavelength matrix Λ may be replaced by the identity matrix I multiplied by the scalar wavelength value λ: Λ = λI This leads to the simplification:

[0102] The quantity

may be represented a single scalar value

and the above may be further simplified as:

[0103] The matrix W is diagonal positive definite so we can write W = W^1/2 W^1/2. Then we have

[0104] Denoting the matrix product W ^1/2H more simply as H̃ (i.e., W^1/2H = H̃), the above form can be rewritten as:

[0105] We now generate reduced forms for H̃ and H̃^T . We apply the Householder transformation to matrix H̃^T, which is indicated by Householder matrix S_HH:

where matrix V is a lower diagonal 4 x 4 matrix. Matrix S is an. n x n matrix, and is an orthogonal (so that S_HHS_HH^T= I_n) since it is a Householder matrix. Multiplying both sides of the above equation by S_HH^T, and using the fact that S_HHS_HH^T = I_n , we can find that:

[0106] Then, by the transpose rule, we can find H̃ = S_HHṼ^T. We now substitute these reduced forms for H̃ and H^T in the prior equation for G^TPG, and perform the following sequence of substitutions, expansions, and regrouping:

[0107] Finally, we obtain

[0108] In the case when we use both GPS and GLONASS measurements, take matrices (R^γ)^-1 and (R^ϕ)^-1 in the form

where λ is the GPS wave length, then all the above reasoning hold including the formula for T . However the value

is close to 1 (but always less than 1), so

is a small value, the operation of taking a square root does not reduce accuracy.

[0109] This method of generating the matrix T may be extended to L1 and L2 band measurements. Let

where λ₁ and λ₂ are the wavelengths of GPS L1 and L2 bands, respectively, and W is the weighing matrix, as above. Then

Matrix Q thus takes the form

so that

and

where b is defined as

Note that in this case the matrix G takes the form

[0110] As in the case of L1, measurements, denote

and, implementing Householder transformation: H̃^TS_HH=Ṽ = [V | O_4×(n-4)] that is

we obtain:

[0111] Matrix K has the following structure:

[0112] The Cholesky factorization of matrix K: K = K̃K̃^T may be obtained using finite formula:

[0113] Thus, for matrix T, where GPG^T = TT^T , we may write

where sub-matrixces A11, A21, and A22 are as follows:



and

Claims

1. A method of estimating a set of floating ambiguities associated with a set of phase measurements of a plurality n of satellite carrier signals made by a first navigation receiver (B) and a second navigation receiver (R) separated by a distance, wherein a baseline vector (x^o,y^o,z^o) relates the position of the second receiver to the first receiver, each satellite carrier signal being transmitted by a satellite and having a wavelength, wherein each receiver has a time clock for referencing its measurements and wherein any difference between the time clocks may be represented by an offset, said method receiving (71), for a plurality of two or more time moments j, the following inputs:

a vector

representative of a plurality of pseudo-ranges measured by the first navigation receiver (B) and corresponding to the plurality of satellite carrier signals,

a vector

representative of a plurality of pseudo-ranges measured by the second navigation receiver (R) and corresponding to the plurality of satellite carrier signals,

a vector

representative of a plurality of estimated distances between the satellites and the first navigation receiver (B),

a vector

representative of a plurality of estimated distances between the satellites and the second navigation receiver (R),

a vector

representative of a plurality of full phase measurements of the satellite carrier signals measured by the first navigation receiver (B),

a vector

representative of a plurality of full phase measurements of the satellite carrier signals measured by the second navigation receiver (R),

a geometric Jacobian matrix

whose matrix elements are representative of the changes in the distances between the satellites and one of the receivers that would be caused by changes in that receiver's position and time clock offset, said method characterized by:

(a) generating (72), for each time moment j, a vector Δγ_j of a plurality of range residuals of pseudo-range measurements made by the first and second navigation receivers in the form of:

(b) generating (73), for each time moment j, a vector Δϕ_j of a plurality of phase residuals of full phase measurements made by the first and second navigation receivers in the form of:

where Λ^-1 is a diagonal matrix comprising the inverse wavelengths of the satellites;

(c) generating (74), for time moment j = 1, an LU-factorization of a matrix M₁ or a matrix inverse of matrix M₁, the matrix M₁ being a function of at least Λ^-1 and

(d) generating (75), for time moment j = 1, a vector N₁ as a function of at least Δγ_j , Δϕ_j, and the LU-factorization of matrix M₁ or the matrix inverse of matrix M₁;

(e) generating (76), for an additional time moment j ≠ 1, an LU-factorization of a matrix M_j or a matrix inverse of matrix M_j, the matrix M_j being a function of at least Λ^-1 ,

and an instance of matrix M generated for a different time moment;

(f) generating (77), for an additional time moment j ≠ 1, a vector N_j as a function of at least Δγ_j , Δϕ_j , and the LU-factorization or matrix M_j or the matrix inverse of matrix M_j, the vector N_j having estimates of the floating ambiguities; and

(g) determining (78) if Nj has sufficient accuracy, wherein steps (e) and (f) are repeated if Nj does not have sufficient accuracy.

2. The method of Claim 1 wherein step (c) comprises generating an LU-factorization for a matrix comprising a form equivalent to (G^TP₁G), where:

• the matrix G has 2n rows, n columns, an upper sub-matrix, and a lower sub-matrix, one of the sub-matrices comprises an n x n zero matrix and the other sub-matrix comprising an n x n identity matrix,

• the matrix G^T comprises the transpose matrix of matrix G, and
the matrix P₁ has 2n rows, 2n columns, and a form which comprises a matrix equivalent to

where the matrix R₁ is a weighting matrix, where the matrix R₁^-1 comprises an inverse of matrix R₁, where the matrix Q₁ has an upper sub-matrix and a lower sub-matrix, one of the sub-matrices of Q₁ comprising matrix

and the other of the sub-matrices of Q₁ comprising the matrix product

and wherein the matrix Q₁^T comprises the transpose of matrix Q₁, and where the quantity qS_k is a zero matrix when the distance
between the first and second navigation receivers is unconstrained and where q may be a non-zero weighting parameter and S_k may be a non-zero matrix when the distance between the first and second navigation receivers is constrained.

3. The method of Claim 2 wherein step (d) comprises the step of generating vector N₁ to comprise a vector having a form equivalent to

where the matrix M₁^-1 comprises an inverse of matrix of matrix M₁, and where the vector µ₁ comprises the vector [Δγ₁, Δϕ₁ ]^T, and where the quantity qg_k is a zero vector when the distance between the first and second navigation receivers is unconstrained and where q may be non-zero and g_k may be a non-zero vector when the distance between the first and second navigation receivers is constrained.

4. The method of Claim 1 wherein step (e) comprises generating an LU-factorization for a matrix comprising a form equivalent to M_j = M _j-1 + G^TP_jG , where:

• M_j-1 comprises the matrix M₁ of step (c) when j = 2 and comprises the matrix M_j of step (e) for the j-1 time moment when j > 2,

• the matrix G has 2n rows, n columns, an upper sub-matrix, and a lower sub-matrix, one of the sub-matrices comprising an n x n zero matrix and the other sub-matrix comprising an n x n identity matrix,

• the matrix G^T comprises the transpose matrix of matrix G, and

• the matrix P_j has 2n rows, 2n columns, and a form which comprise a matrix equivalent to

where the matrix R_j is a weighting matrix, where the matrix R_j^-1 comprises an inverse of matrix R_j, where the matrix Q_j has an upper sub-matrix and a lower sub-matrix, one of the sub-matrices of Q_j comprising matrix

and the other of the sub-matrices of Q_j comprising the matrix product

and wherein the matrix Q_j^T comprises the transpose of matrix Q_j, and where the quantity qS_j is a zero matrix when the distance between the first and second navigation receivers is unconstrained and where q may be a non-zero weighting parameter and S_j may be a non-zero matrix when the distance between the first and second navigation receivers is constrained.

5. The method of Claim 4 wherein step (f) comprises the step of generating vector N_j to comprise a vector having a form equivalent to

where the matrix M_j^-1 comprises an inverse of matrix of matrix M_j, where the vector µ_j, comprises the vector [Δγ_j, Δϕ_j ]^T, and where the vector N_j-1 comprises the vector N₁ generated by step (d) when j = 2 and comprises the vector N_j-1 generated by step (f) for the j-1 time moment when j > 2, and where the quantity qg_j is a zero vector when the distance between the first and second navigation receivers is unconstrained and where q may be non-zero and g_j may be a non-zero vector when the distance between the first and second navigation receivers is constrained.

6. The method of claim 3 wherein the distance between the first and second navigation receivers is constrained to a distance L_RB, wherein step (c) generates matrix S₁ in a form equivalent to:

where r₁ is a vector comprising estimates of the three coordinates of the baseline vector for the time moment j=1 and a zero as fourth component, where r₁^T is the vector transpose of r₁, where I₃ is the 3-by-3 identity matrix, where O_1x3 is a row vector of three zeros, and where O_3x1 is a column vector of three zeros; and
wherein step (d) generates vector g₁ for the time moment j=1 in a form equivalent to:

where:

7. The method of Claim 3 wherein weighting matrix R₁ comprises an identity matrix multiplied by a scalar quantity.

8. The method of claim 5 wherein the distance between the first and second navigation receivers is constrained to a distance L_RB, wherein step (e) generates matrix S_j in a form equivalent to:

where r_j is a vector comprising estimates of the three coordinates of the baseline vector for the j-th time moment and a zero as fourth vector component, where r_j^T is the vector transpose of r_j, where I₃ is the 3-by-3 identity matrix, where O_1x3 is a row vector of three zeros, and where O_3x1 is a column vector of three zeros; and
wherein step (f) generates vector g_j for the j-th time moment in a form equivalent to:

where:

9. The method of Claim 5 wherein the weighting matrix R_j comprises an identity matrix multiplied by a scalar quantity for at least one time moment j.

10. The method of Claim 4 wherein the generation of the LU-factorization in step (e) comprises the steps of:

(g) generating an LU-factorization of matrix M_j-1 in a form equivalent to L_j-1 L_j-1^T wherein L_j-1 is a low-triangular matrix and L_j-1^T is the transpose of L_j-1;

(h) generating a factorization of G^T P_j G in a form equivalent to T_j T_j^T = G^T P_j G, where T_j^T is the transpose of T_j;

(i) generating an LU-factorization of matrix M_j in a form equivalent to L_j L_j^T from a plurality n of rank-one modifications of matrix L_j-1, each rank-one modification being based on a respective column of matrix T_j, where n is the number of rows in matrix M_j .

11. The method of Claim 10 wherein step (h) generates matrix T_j from a Cholesky factorization of G^T P_j G.

12. The method of Claim 10 wherein weighting matrix R_j has a form equivalent to:


where R^γ and R^ϕ are weighting matrices; wherein R^γ and R^ϕ are related to a common weighting matrix W and scaling parameters σ_γ and σ_ϕ as follows

and

wherein step (h) of generating matrix T_j comprises the steps of:

generating a scalar b in a form equivalent to:

where λ_GPS is the wavelength of the satellite signals,

generating a matrix H̃ in a form equivalent to

= generating a Householder matrix S_HH for matrix H̃ , and

generating matrix T_j in a form equivalent to:

13. The method of Claim 10 wherein weighting matrix R_j is applied to a case where there is a first group of satellite signals having carrier signals in a first wavelength band and a second group of satellite signals having a carrier signals in a second wavelength band, the weighting frequency having a form equivalent to:


where R^γ and R^ϕ are weighting matrices; wherein R^γ and R^ϕ are related to a common weighting matrix W, the carrier wavelengths of the first group of signals as represented by matrix Λ⁽¹⁾, the carrier wavelengths of the second group of signals as represented by matrix Λ⁽²⁾, the center wavelength of the first band as represented by λ₁ the center wavelength of the first band as represented by λ₂ and scaling parameters σ_γ and σ_ϕ, as follows:



where


wherein step (h) of generating matrix T_j comprises the steps of:
generating a scalar b in a form equivalent to:

where λ₁ is the wavelength of a first group of satellite signals and λ₂ is the wavelength of a second group of satellite signals,
generating a matrix H̃ in a form equivalent to

generating a Householder matrix S_HH for matrix H̃, and generating matrix T_j in a form equivalent to:

where sub-matrixces A11, A21, and A22 are as follows:



and

14. The method of Claim 4 wherein step (d) comprises generating a vector B₁ to comprise a vector having a form equivalent to G^T P₁ µ₁+qg₁, where the vector µ₁ comprises the vector [Δγ₁, Δϕ₁]^T, and where the quantity qg_j is a zero vector when the distance between the first and second navigation receivers is unconstrained and where q may be non-zero and g_j may be a non-zero vector when the distance between the first and second navigation receivers is constrained; and
wherein step (f) further comprises generating, for each time moment j ≠ 1, a vector B_j to comprise a matrix having a form equivalent to B_j-1 + G^T P_jµ_j +qg_j, where the vector µ_j comprises the vector [Δγ_j, Δϕ_j]^T, and where the vector B_j-1 is the vector B₁ generated by step (d) when j = 2 and comprises the vector generated by step (f) for the j-1 time moment when j > 2, and where the quantity qg_j is a zero vector when the distance between the first and second navigation receivers is unconstrained and where q may be non-zero and g_j may be a non-zero vector when the distance between the first and second navigation receivers is constrained; and
wherein step (f) further comprises generating vector N_j to comprise a vector having a form equivalent to N_j = [M_j]^-1 B_j, where the matrix M_j^-1 comprises an inverse of matrix of matrix M_j.

15. The method of claim 14 wherein the distance between the first and second navigation receivers is constrained to a distance L_RB, wherein step (c) generates matrix S₁ in a form equivalent to:

where r₁ is a vector comprising estimates of the three coordinates of the baseline vector for the time moment j=1 and a zero as fourth component, where r₁^T is the vector transpose of r₁, where I₃ is the 3-by-3 identity matrix, where O_1x3 is a row vector of three zeros, and where O_3x1 is a column vector of three zeros; and
wherein step (d) generates vector g₁ for the time moment j=1 in a form equivalent to:

where:

16. The method of claim 14 wherein the distance between the first and second navigation receivers is constrained to a distance L_RB, wherein step (e) generates matrix S_j in a form equivalent to:

where r_j is a vector comprising estimates of the three coordinates of the baseline vector for the j-th time moment and a zero as fourth vector component, where r_j^T is the vector transpose of r_j, where I₃ is the 3-by-3 identity matrix, where O_1x3 is a row vector of three zeros, and where O_3x1 is a column vector of three zeros; and
wherein step (f) generates vector g_j for the j-th time moment in a form equivalent to:

where:

17. The method of Claim 14 wherein the weighting matrix R_j comprises an identity matrix multiplied by a scalar quantity for at least one time moment j.

18. A method of estimating a set of floating ambiguities associated with a set of phase measurements of a plurality n of satellite carrier signals made by a first navigation receiver (B) and a second navigation receiver (R), wherein a baseline vector (x^o, y^o, z^o) relates the position of the second receiver to the first receiver, each satellite carrier signal being transmitted by a satellite and having a wavelength, wherein each receiver has a time clock for referencing its measurements and wherein any difference between the time clocks may be represented by an offset, said method receiving (41), for a plurality of two or more time moments j, the following inputs for each time moment j:

a vector

representative of a plurality of pseudo-ranges measured by the first navigation receiver (B) and corresponding to the plurality of satellite carrier signals,

a vector

representative of a plurality of pseudo-ranges measured by the second navigation receiver (R) and corresponding to the plurality of satellite carrier signals,

a vector

representative of a plurality of estimated distances between the satellites and the first navigation receiver (B),

a vector

representative of a plurality of estimated distances between the satellites and the second navigation receiver (R),

a vector

representative of a plurality of full phase measurements of the satellite carrier signals measured by the first navigation receiver (B),

a vector

representative of a plurality of full phase measurements of the satellite carrier signals measured by the second navigation receiver (R),

a geometric Jacobian matrix

whose matrix elements are representative of the changes in the distances between the satellites and one of the receivers that would be caused by changes in that receiver's position and time clock offset, said method characterised by:

(a) generating (42), for each time moment j, a vector Δγ_j of a plurality of range residuals of pseudo-range measurements made by the first and second navigation receivers in the form of:

said step generating a set of range residuals Δγ_k, k=1, .., j;

(b) generating (43), for each time moment j, a vector Δϕ_j of a plurality of phase residuals of full phase measurements made by the first and second navigation receivers in the form of:

where Λ^-1 is a diagonal matrix comprising the inverse wavelengths of the satellites, said step generating a set of phase residuals Δϕ_k k=1, .., j;

(c) generating (44) an LU-factorization of a matrix M or a matrix inverse of matrix M, the matrix M being a function of at least Λ^-1 and

for index k of

covering at least two of the time moments j ;

(d) generating (45) a vector N of estimated floating ambiguities as a function of at least the set of range residuals Δγ_k, the set of phase residuals Δ ϕ_k, and the LU-factorization of matrix M or the matrix inverse of matrix M.

19. The method of Claim 18 wherein step (c) comprises generating matrix M in a form equivalent to the summation

where:

• the matrix G has 2n rows, n columns, an upper sub-matrix, and a lower sub-matrix, one of the sub-matrices comprises an n x n zero matrix and the other sub-matrix comprising an n x n identity matrix,

• the matrix G^T comprises the transpose matrix of matrix G, and

• the matrix P_k has 2n rows, 2n columns, and a form which comprises a matrix equivalent to

where the matrix R_k is a weighting matrix, where the matrix R_k^-1 comprises an inverse of matrix R_k, where the matrix Q_k has an upper sub-matrix and a lower sub-matrix, one of the sub-matrices of Q_k comprising matrix

and the other of the sub-matrices of Q_k comprising the matrix product

and wherein the matrix Q_k^T comprises the transpose of matrix Q_k, and where the quantity qS_k is a zero matrix when the distance between the first and second navigation receivers is unconstrained and where q may be a non-zero weighting parameter and S_k may be a non-zero matrix when the distance between the first and second navigation receivers is constrained.

20. The method of Claim 19 wherein step (d) comprises generating matrix N in a form equivalent to:

where the matrix M^-1 comprises an inverse of matrix of matrix M, where the vector µ_k comprises the vector [Δγ_k, Δϕ_k ]^T, and where the quantity qg_k is a zero vector when the distance between the first and second navigation receivers is unconstrained and where q may be non-zero and g_k may be a non-zero vector when the distance between the first and second navigation receivers is constrained.

21. The method of claim 20 wherein the distance between the first and second navigation receivers is constrained to a distance L_RB, wherein step (c) generates matrix S_k for the k-th time moment in a form equivalent to:

where r_k is a vector comprising estimates of the three coordinates of the baseline vector at the k-th time moment, and a zero as fourth component, where r_k^T is the vector transpose of r_k, where I₃ is the 3-by-3 identity matrix, where O_1x3 is a row vector of three zeros, and where O_3x1 is a column vector of three zeros; and
wherein step (d) generates vector g_k for the k-th time moment in a form equivalent to:

where:

22. The method of Claim 19 wherein at least one of the weighting matrices R_k comprises an identity matrix multiplied by a scalar quantity.

23. A computer program product for directing a data processor to perform the method of any preceding claim.

24. An apparatus (100) for estimating a set of floating ambiguities associated with a set of phase measurements of a plurality n of satellite carrier signals made by a first navigation receiver (B) and a second navigation receiver (R) separated by a distance, wherein a baseline vector (x^o,y^o,z^o) relates the position of the second receiver to the first receiver, each satellite carrier signal being transmitted by a satellite and having a wavelength, wherein each receiver has a time clock for referencing its measurements and wherein any difference between the time clocks may be represented by an offset, said apparatus (100) for receiving (71), for a plurality of two or more time moments j, the following inputs:

a vector

representative of a plurality of pseudo-ranges measured by the first navigation receiver (B) and corresponding to the plurality of satellite carrier signals,

a vector

representative of a plurality of pseudo-ranges measured by the second navigation receiver (R) and corresponding to the plurality of satellite carrier signals,

a vector

representative of a plurality of estimated distances between the satellites and the first navigation receiver (B),

a vector

representative of a plurality of estimated distances between the satellites and the second navigation receiver (R),

a vector

representative of a plurality of full phase measurements of the satellite carrier signals measured by the first navigation receiver (B),

a vector

representative of a plurality of full phase measurements of the satellite carrier signals measured by the second navigation receiver (R),

a geometric Jacobian matrix

whose matrix elements are representative of the changes in the distances between the satellites and one of the receivers that would be caused by changes in that receiver's position and time clock offset,

said apparatus (100) characterised by:

(a) means for generating (72), for each time moment j, a vector Δγ_j a plurality of range residuals of pseudo-range measurements made by the first and second navigation receivers in the form of:

(b) means for generating (73), for each time moment j, a vector Δϕ_j of a plurality of phase residuals of full phase measurements made by the first and second navigation receivers in the form of:

where Λ^-1 is a diagonal matrix comprising the inverse wavelengths of the satellites;

(c) means for generating (74), for time moment j = 1, an LU-factorization of a matrix M₁ or a matrix inverse of matrix M₁, the matrix M₁ being a function of at least Λ^-1 and

(d) means for generating (75), for time moment j = 1, a vector N₁ as a function of at least Δγ₁, Δϕ₁, and the LU-factorization of matrix M₁ or the matrix inverse of matrix M₁;

(e) means for generating (76), for an additional time moment j ≠ 1, an LU-factorization of a matrix M_j or a matrix inverse of matrix M_j, the matrix M_j being a function of at least Λ^-1 and

(f) means for generating (77), for an additional time moment j ≠ 1, a vector N_j as a function of at least Δγ_j, Δϕ_j, and the LU-factorization or matrix M_j or the matrix inverse of matrix M_j, the vector N_j having estimates of the floating ambiguities; and

(g) means for determining (78) if Nj has sufficient accuracy, wherein steps (e) and (f) are repeated if Nj does not have sufficient accuracy.

25. The apparatus of Claim 24 wherein means (c) comprises means for generating an LU-factorization for a matrix comprising a form equivalent to (G^TP₁G), where:

• the matrix G has 2n rows, n columns, an upper sub-matrix, and a lower sub-matrix, one of the sub-matrices comprises an n x n zero matrix and the other sub-matrix comprising an n x n identity matrix,

• the matrix G^T comprises the transpose matrix of matrix G, and

• the matrix P₁ has 2n rows, 2n columns, and a form which comprises a matrix equivalent to

where the matrix R₁ is a weighting matrix, where the matrix R₁^-1 comprises an inverse of matrix R₁ where the matrix Q₁has an upper sub-matrix and a lower sub-matrix, one of the sub-matrices of Q₁ comprising matrix

and the other of the sub-matrices of Q₁ comprising the matrix product

and wherein the matrix Q₁^T comprises the transpose of matrix Q₁, and where the quantity qS_k is a zero matrix when the distance between the first and second navigation receivers is unconstrained and where q may be a non-zero weighting parameter and S_k may be a non-zero matrix when the distance between the first and second navigation receivers is constrained.

26. The apparatus of Claim 25 wherein means (d) comprises means for generating vector N₁ to comprise a vector having a form equivalent to

where the matrix M₁^-1 comprises an inverse of matrix of matrix M₁, and where the vector µ₁ comprises the vector [Δγ₁, Δϕ₁]^T, and where the quantity qg_k is a zero vector when the distance between the first and second navigation receivers is unconstrained and where q may be non-zero and g_k may be a non-zero vector when the distance between the first and second navigation receivers is constrained.

27. The apparatus of Claim 24 wherein means (e) comprises means for generating an LU-factorization for a matrix comprising a form equivalent to

where:

• M_j-1 comprises the matrix M₁ of generated by means (c) when j = 2 and comprises the matrix M_j generated by means (e) for the j-1 time moment when j > 2,

• the matrix G has 2n rows, n columns, an upper sub-matrix, and a lower sub-matrix, one of the sub-matrices comprising an n x n zero matrix and the other sub-matrix comprising an n x n identity matrix,

• the matrix G^T comprises the transpose matrix of matrix G, and

• the matrix P_j has 2n rows, 2n columns, and a form which comprise a matrix equivalent to

where the matrix R_j is a weighting matrix, where the matrix R_j^-1 comprises an inverse of matrix R_j, where the matrix Q_j has an upper sub-matrix and a lower sub-matrix, one of the sub-matrices of Q_j comprising matrix

and the other of the sub-matrices of Q_j comprising the matrix product

and wherein the matrix Q_j^T comprises the transpose of matrix Q_j, and where the quantity qS_j is a zero matrix when the distance between the first and second navigation receivers is unconstrained and where q may be a non-zero weighting parameter and S_j may be a non-zero matrix when the distance between the first and second navigation receivers is constrained.

28. The apparatus of Claim 27 wherein means (f) comprises means for generating vector N_j to comprise a vector having a form equivalent to

where the matrix

comprises an inverse of matrix of matrix M_j, where the vector µ_j comprises the vector [Δγ_j, Δϕ_j]^T, and where the vector N_j-1 comprises the vector N₁ generated by means (d) when j = 2 and comprises the vector N_j-1 generated by means (f) for the j-1 time moment when j > 2, and where the quantity qg_j is a zero vector when the distance between the first and second navigation receivers is unconstrained and where q may be non-zero and g_j may be a non-zero vector when the distance between the first and second navigation receivers is constrained.

29. The apparatus of claim 26 wherein the distance between the first and second navigation receivers is constrained to a distance L_RB, wherein means (c) comprises means for generating matrix S₁ in a form equivalent to:

where r₁ is a vector comprising estimates of the three coordinates of the baseline vector for the time moment j=1 and a zero as fourth component, where r₁^T is the vector transpose of r₁, where I₃ is the 3-by-3 identity matrix, where O_1x3 is a row vector of three zeros, and where O_3x1 is a column vector of three zeros; and
wherein means (d) comprises means for generating vector g₁ for the time moment j=1 in a form equivalent to:

where:

30. The apparatus of Claim 28 wherein the distance between the first and second navigation receivers is constrained to a distance L_RB, wherein means (e) comprises means for generating matrix S_j in a form equivalent to:

where r_j is a vector comprising estimates of the three coordinates of the baseline vector for the j-th time moment and a zero as fourth vector component, where r_j^T is the vector transpose of r_j, where I₃ is the 3-by-3 identity matrix, where O_1x3 is a row vector of three zeros, and where O_3x1 is a column vector of three zeros; and
wherein means (f) comprises means for generating vector g_j for the j-th time moment in a form equivalent to:

where:

31. The apparatus of Claim 27 wherein means (e) for generating the LU-factorization of M_j comprises:

(g) means for generating an LU-factorization of matrix M_j-1 in a form equivalent to L_j-1 L_j-1^T wherein L_j-1 is a low-triangular matrix and L_j-1^T is the transpose of L_j-1;

(h) means for generating a factorization of G^T P_j G in a form equivalent to

where

is the transpose of T_j; and

(i) means for generating an LU-factorization of matrix M_j in a form equivalent to

from a plurality n of rank-one modifications of matrix L_j-1, each rank-one modification being based on a respective column of matrix T_j, where n is the number of rows in matrix M_j.

32. The apparatus of Claim 31 wherein means (h) generates matrix T_j from a Cholesky factorization of G^T P_j G.

33. The apparatus of Claim 31 wherein weighting matrix R_j has a form equivalent to:


where R^γ and R^ϕ are weighting matrices; wherein R^γ and R^ϕ are related to a common weighting matrix W and scaling parameters σ_γ and σ_ϕ as follows

and

wherein means (h) of generating matrix T_j comprises:

means for generating a scalar b in a form equivalent to:

where

is the wavelength of the satellite signals,

means for generating a matrix H̃ in a form equivalent to

means for generating a Householder matrix S_HH for matrix H̃, and

means for generating matrix T_j in a form equivalent to:

34. The apparatus of Claim 31 wherein weighting matrix R_j is applied to a case where there is a first group of satellite signals having carrier signals in a first wavelength band and a second group of satellite signals having a carrier signals in a second wavelength band, the weighting frequency having a form equivalent to:

where R^γ and R^ϕ are weighting matrices; wherein R^γ and R^ϕ are related to a common weighting matrix W, the carrier wavelengths of the first group of signals as represented by matrix Λ⁽¹⁾, the carrier wavelengths of the second group of signals as represented by matrix Λ⁽²⁾, the center wavelength of the first band as represented by λ₁ the center wavelength of the first band as represented by λ₂, and scaling parameters σ_γ and σ_ϕ, as follows:



where

wherein step (h) of generating matrix T_j comprises the steps of:

generating a scalar b in a form equivalent to:

where λ₁ is the wavelength of a first group of satellite signals and λ₂ is the wavelength of a second group of satellite signals,

generating a matrix H̃ in a form equivalent to

generating a Householder matrix S_HH for matrix H̃, and

generating matrix T_j in a form equivalent to:

where sub-matrixces A11, A21, and A22 are as follows:



and

35. An apparatus (100) for estimating a set of floating ambiguities associated with a set of phase measurements of a plurality n of satellite carrier signals made by a first navigation receiver (B) and a second navigation receiver (R) separated by a distance, wherein a baseline vector (x^o,y^o,z^o) relates the position of the second receiver to the first receiver, each satellite carrier signal being transmitted by a satellite and having a wavelength, wherein each receiver has a time clock for referencing its measurements and wherein any difference between the time clocks may be represented by an offset, said apparatus (100) for receiving (41), for a plurality of two or more time moments j, the following inputs:

a vector

representative of a plurality of pseudo-ranges measured by the first navigation receiver (B) and corresponding to the plurality of satellite carrier signals,

a vector

representative of a plurality of pseudo-ranges measured by the second navigation receiver (R) and corresponding to the plurality of satellite carrier signals,

a vector

representative of a plurality of estimated distances between the satellites and the first navigation receiver (B),

a vector

representative of a plurality of estimated distances between the satellites and the second navigation receiver (R),

a vector

representative of a plurality of full phase measurements of the satellite carrier signals measured by the first navigation receiver (B),

a vector

representative of a plurality of full phase measurements of the satellite carrier signals measured by the second navigation receiver (R),

a geometric Jacobian matrix

whose matrix elements are representative of the changes in the distances between the satellites and one of the receivers that would be caused by changes in that receiver's position and time clock offset,

said apparatus (100) characterised by:

(a) means for generating (42), for each time moment j, a vector Δγ_j of a plurality of range residuals of pseudo-range measurements made by the first and second navigation receivers in the form of:

said means generating a set of range residuals Δγ_k, k=1, .., j;

(b) means for generating (43), for each time moment j, a vector Δϕ_j of a plurality of phase residuals of full phase measurements made by the first and second navigation receivers in the form of:

where Λ^-1 is a diagonal matrix comprising the inverse wavelengths of the satellites, said means generating a set of phase residuals Δϕ_k, k=1, .., j;

(c) means for generating (44) an LU-factorization of a matrix M or a matrix inverse of matrix M, the matrix M being a function of at least Λ^-1 and

for index k of

covering at least two of the time moments j ;

(d) means for generating (45) a vector N of estimated floating ambiguities as a function of at least the set of range residuals Δγ_k, the set of phase residuals Δϕ_k, and the LU-factorization of matrix M or the matrix inverse of matrix M.

36. The apparatus of Claim 35 wherein means (c) comprises means for generating matrix M in a form equivalent to the summation

where:

• the matrix G has 2n rows, n columns, an upper sub-matrix, and a lower sub-matrix, one of the sub-matrices comprises an n x n zero matrix and the other sub-matrix comprising an n x n identity matrix,

• the matrix G^T comprises the transpose matrix of matrix G, and

• the matrix P_k has 2n rows, 2n columns, and a form which comprises a matrix equivalent to

where the matrix R_k is a weighting matrix, where the matrix R_k^-1 comprises an inverse of matrix R_k, where the matrix Q_k has an upper sub-matrix and a lower sub-matrix, one of the sub-matrices of Q_k comprising matrix

and the other of the sub-matlices of Q_k comprising the matrix product

and wherein the matrix Q_k^T comprises the transpose of matrix Q_k, and where the quantity qS_k is a zero matrix when the distance between the first and second navigation receivers is unconstrained and where q may be a non-zero weighting parameter and S_k may be a non-zero matrix when the distance between the first and second navigation receivers is constrained.

37. The apparatus of Claim 36 wherein means (d) comprises means for generating matrix N in a form equivalent to:

where the matrix M^-1 comprises an inverse of matrix of matrix M, where the vector µ_k comprises the vector [Δγ_k , Δϕ_k]^T, and where the quantity qg_k is a zero vector when the distance between the first and second navigation receivers is unconstrained and where q may be non-zero and g_k may be a non-zero vector when the distance between the first and second navigation receivers is constrained.

38. The apparatus of claim 37 wherein the distance between the first and second navigation receivers is constrained to a distance L_RB, wherein means (c) comprises means for generating matrix S_k for the k-th time moment in a form equivalent to:

where r_k is a vector comprising estimates of the three coordinates of the baseline vector at the k-th time moment, and a zero as fourth component, where r_k^T is the vector transpose of r_k, where I₃ is the 3-by-3 identity matrix, where O_1x3 is a row vector of three zeros,
and where O_3x1 is a column vector of three zeros; and
wherein means (d) generates vector g_k for the k-th time moment in a form equivalent to:

where:

Ansprüche

1. Verfahren zum Schätzen eines Satzes von gleitenden Mehrdeutigkeiten, die einem Satz von Phasenmessungen einer Mehrzahl n von Satellitenträgersignalen zugeordnet sind, die von einem ersten Navigationsempfänger (B) und einem zweiten Navigationsempfänger (R) erzeugt werden, die durch einen Abstand voneinander getrennt sind, wobei ein Grundlinienvektor (x⁰, y⁰, z⁰) auf die Position des zweiten Empfängers zum ersten Empfänger bezogen ist, wobei jedes Satellitenträgersignal von einem Satelliten gesendet wird und eine Wellenlänge aufweist, wobei jeder Empfänger einen Zeittakt zum Referenzieren seiner Messungen aufweist, und wobei eine etwaige Differenz zwischen den Zeittakten durch einen Versatz repräsentiert werden kann, wobei das Verfahren, für eine Mehrzahl von zwei oder mehr Zeitmomenten j, die folgenden Eingaben empfängt (71):

einen Vektor

der eine Mehrzahl von Pseudobereichen repräsentiert, die von dem ersten Navigationsempfänger (B) gemessen werden und der Mehrzahl von Satellitenträgersignalen entsprechen,

einen Vektor

der eine Mehrzahl von Pseudobereichen repräsentiert, die von dem zweiten Navigationsempfänger (R) gemessen werden und der Mehrzahl von Satellitenträgersignalen entsprechen,

einen Vektor

der eine Mehrzahl von geschätzten Abständen zwischen den Satelliten und dem ersten Navigationsempfänger (B) repräsentiert,

einen Vektor

der eine Mehrzahl von geschätzten Abständen zwischen den Satelliten und dem zweiten Navigationsempfänger (R) repräsentiert,

einen Vektor

der eine Mehrzahl von Vollphasenmessungen der Satellitenträgersignale repräsentiert, die von dem ersten Navigationsempfänger (B) gemessen werden,

einen Vektor

der eine Mehrzahl von Vollphasenmessungen der Satellitenträgersignale repräsentiert, die von dem zweiten Navigationsempfänger (R) gemessen werden,

eine geometrische Jacobi-Matrix

deren Matrixelemente die Änderungen in den Abständen zwischen dem Satelliten und einem der Empfänger repräsentieren, die durch Änderungen der Position des Empfängers und dem Zeittaktversatz verursacht würden, wobei das Verfahren gekennzeichnet ist durch:

(a) Erzeugen (72), für jeden Zeitmoment j, eines Vektors Δγ_j einer Mehrzahl von Bereichsresten von Pseudobereichmessungen, die von den ersten und zweiten Navigationsempfängern vorgenommen werden, in der Form von:

(b) Erzeugen (73), für jeden Zeitmoment j eines Vektors Δϕ_j einer Mehrzahl von Phasenresten von Vollphasenmessungen, die von den ersten und zweiten Navigationsempfängern vorgenommen werden, in der Form von:

wobei Λ^-1 eine diagonale Matrix ist, die zwei inverse Wellenlängen der Satelliten aufweist;

(c) Erzeugen (74), für den Zeitmoment j = 1, eine LU-Faktorisierung einer Matrix M₁ oder einer Inversmatrix der Matrix M₁, wobei die Matrix M₁ eine Funktion von zumindest Λ^-1 and

ist;

(d) Erzeugen (75), für den Zeitmoment j = 1, eines Vektors N₁ als Funktion von zumindest Δγ₁, Δϕ₁, und der LU-Faktorisierung der Matrix M₁ oder der Inversmatrix der Matrix M₁;

(e) Erzeugen (76), für einen zusätzlichen Zeitmoment j ≠ 1, einer LU-Faktorisierung der Matrix M_j oder einer Inversmatrix der Matrix M_f, wobei die Matrix M_j eine Funktion von zumindest Λ^-1,

und eines Exemplars der Matrix M, die für ein anderes Zeitmoment erzeugt wurde, ist;

(f) Erzeugen (77), für einen zusätzlichen Zeitmoment j ≠ 1, eines Vektors N_j als Funktion von zumindest Δγ_j, Δϕ_j, der LU-Faktorisierung oder Matrix M_j oder der Inversmatrix der Matrix M_j, wobei der Vektor N_j Schätzungen der gleitenden Mehrdeutigkeiten aufweist; und

(g) Bestimmen (78), ob N_j eine ausreichende Genauigkeit hat, wobei die Schritte (e) und (f) wiederholt werden, wenn N_j keine ausreichende Genauigkeit hat.

2. Das Verfahren von Anspruch 1, wobei Schritt (c) aufweist, eine LU-Faktorisierung für eine Matrix zu erzeugen, die eine Form äquivalent zu (G^TP₁G) hat, wobei:

• die Matrix G 2n Zeilen , n Spalten, eine obere Submatrix und eine untere Submatrix aufweist, wobei eine der Submatrices eine n x n Nullmatrix aufweist und die andere Submatrix eine n x n Identitätsmatrix aufweist,

• die Matrix G^T die transponierte Matrix der Matrix G aufweist, und
die Matrix P₁ 2n Zeilen, 2n Spalten, und eine Form aufweist, die eine Matrix aufweist, äquivalent zu

wobei die Matrix R₁ eine Gewichtungsmatrix ist, wobei die Matrix R₁^-1 eine Inversmatrix R₁ aufweist, wobei die Matrix Q₁ eine obere Submatrix und eine untere Submatrix aufweist, wobei eine der Submatrices von Q₁ die Matrix

aufweist und die andere der Submatrices von Q₁ das Matrixprodukt

aufweist, und wobei die Matrix Q₁^T die Transponierte der Matrix Q₁ aufweist, und wobei die Größe qS_k eine Nullmatrix ist, wenn der Abstand zwischen den ersten und zweiten Navigationsempfängern unbeschränkt ist, und wobei q ein Nicht-null-Gewichtungsparameter sein kann, und S_k eine Nicht-null-Matrix sein kann, wenn der Abstand zwischen den ersten und zweiten Navigationsempfängern beschränkt ist.

3. Das Verfahren von Anspruch 2, wobei Schritt (d) den Schritt des Erzeugen vom Vektor N₁ aufweist, dass er einen Vektor aufweist, der eine Form äquivalent zu

hat, wobei die Matrix M₁^-1 eine Inversmatrix der Matrix M₁ aufweist, und wobei der Vektor µ₁ den Vektor [Δγ₁, Δϕ₁]^T aufweist, und wobei die Größe qg_k ein Null-Vektor ist, wenn der Abstand zwischen ersten und zweiten Navigationsempfängern unbeschränkt ist, und wobei q nicht null sein kann und g_k ein Nicht-null-Vektor sein kann, wenn der Abstand zwischen ersten und zweiten Navigationsempfängern beschränkt ist.

4. Das Verfahren von Anspruch 1, wobei Schritt (e) aufweist, eine LU-Faktorisierung für eine Matrix zu erzeugen, die eine Form ä quivalent zu M_j = M_j-1 +G^TP_jG aufweist, wobei:

• M_j-1 die Matrix M₁ von Schritt (c) aufweist, wenn j = 2, und die Matrix M_j von Schritt (e) für das j-1 Zeitmoment aufweist, wenn j > 2,

• die Matrix G 2n Zeilen, n Spalten, eine obere Sub-matrix, und eine untere Submatrix aufweist, wobei eine der Sub-Matrices eine n x n Null-Matrix aufweist und die andere Submatrix eine n x n Identitäts-matrix aufweist,

• die Matrix G^T die transponierte Matrix der Matrix G aufweist, und

• die Matrix P_j 2n Zeilen, 2n Spalten und eine Form aufweist, welche eine Matrix äquivalent zu

aufweist, wobei die Matrix R_j eine Gewichtungsmatrix ist, wobei die Matrix

eine Inverse der Matrix R_j aufweist, wobei die Matrix Q_j eine obere Sub-Matrix und eine untere Sub-Matrix aufweist, wobei eine der Sub-Matrices von Q_j eine Matrix

aufweist und die andere der Sub-Matrices von Q_j das Matrixprodukt

aufweist und wobei die Matrix

die Transponierte der Matrix Q_j aufweist, und wobei die Größe qS_j eine Null-Matrix ist, wenn der Abstand zwischen ersten und zweiten Navigationsempfängern unbeschränkt ist und wobei q ein Nicht-null-Gewichtungsparameter sein kann und S_j eine Nicht-null-Matrix sein kann, wenn der Abstand zwischen ersten und zweiten Navigationsempfiängern beschränkt ist.

5. Das Verfahren von Anspruch 4, wobei Schritt (f) den Schritt des Ereugens eines Vektors N_j aufweist, so dass er einen Vektor aufweist, der eine Form äquivalent zu

hat, wobei die Matrix

eine Inversmatrix der Matrix M_j aufweist, und wobei der Vektor µj den Vektor [Δϕ_j, Δϕ_j]^T aufweist, und wobei der Vektor N_j-1 den in Schritt (d) erzeugten Vektor N₁ aufweist, wenn j = 2, und den in Schritt (f) für den j-1 Zeitmoment erzeugten Vektor in N_j-1, aufweist, wenn j > 2, und wobei die Größe qg_j ein Null-Vektor ist, wenn der Abstand zwischen den ersten und zweiten Navigationsempfängern unbeschränkt ist, und wobei q nicht null sein kann und g_j ein Nicht-null-Vektor sein kann, wenn der Abstand zwischen ersten und zweiten Navigationsempfängern beschränkt ist.

6. Das Verfahren von Anspruch 3, wobei der Abstand zwischen ersten und zweiten Navigationsempfängern auf einen Abstand L_RB beschränkt ist, wobei der Schritt (c) die Matrix S₁ in einer Form erzeugt, die äquivalent ist zu:

wobei r₁ ein Vektor ist, der Schätzungen der drei Koordinaten des Grundlinienvektors für den Zeitmoment j=1 und eine Null als vierte Komponente aufweist, wobei r₁^T die Vektortransponierte von r₁ ist, wobei I₃ is the 3-x-3-Identitätsmatrix ist, wobei O_1x3 ein Zeilenvektor von drei Nullen ist und wobei O_3x1 ein Spaltenvektor von drei Nullen ist; und wobei Schritt (d) den Vektor g₁ für das Zeitmoment j=1 in einer Form erzeugt, die äquivalent ist zu:

wobei

7. Das Verfahren von Anspruch 3, wobei die Gewichtungsmatrix R₁ eine Identitätsmatrix aufweist, die mit einer skalaren Größe multipliziert wird.

8. Das Verfahren von Anspruch 5, wobei der Abstand zwischen den ersten und zweiten Navigationsempfängern auf einen Abstand L_RB beschränkt ist, wobei Schritt (e) die Matrix S_j in einer Form erzeugt, die äquivalent ist zu:

wobei r_j ein Vektor ist, der Schätzungen der drei Koordinaten des Grundlinienvektors für den j-ten Zeitmoment und eine Null als vierte Vektorkomponente aufweist, wobei

die Vektortransponierte von r_j ist, und wobei I₃ die 3x3-Identitätsmatrix ist, wobei O_1x3 ein Zeilenvektor von drei Nullen ist und wobei O_3x1 ein Spaltenvektor von drei Nullen ist; und
wobei Schritt (f) den Vektor g_j für den j-te Zeitmoment in einer Form erzeugt, die äquivalent ist zu:

wobei:

9. Das Verfahren von Anspruch 5, wobei die Gewichtungsmatrix R_j eine Identitätsmatrix aufweist, die mit einer skalaren Größe für zumindest einen Zeitmoment j multipliziert ist.

10. Das Verfahren von Anspruch 4, wobei die Erzeugung der LU-Faktorisierung in Schritt (e) die Schritte aufweist:

(g) Erzeugen einer LU-Faktorisierung der Matrix M_j-1 in einer Form äquivalent zu L_j-1 L_j-1^T, wobei L_j-1 eine niedrige dreieckige Matrix ist und L_j-1^T die Transponierte von L_j-1 ist;

(h) Erzeugen einer Faktorisierung von G^TP_jG in einer Form äquivalent zu

wobei

die Transponierte von T_j ist;

(i) Erzeugen einer LU-Faktorisierung der Matrix M_j in einer Form äquivalent zu

aus einer Mehrzahl n von erstrangigen Modifizierungen der Matrix L_j-1, wobei jede erstrangige Modifizierung auf einer erstrangigen Spalte der Matrix T_j beruht, wobei n die Anzahl von Zeilen in der Matrix M_j ist.

11. Das Verfahren von Anspruch 10, wobei Schritt (h) die Matrix T_j aus einer Cholesky-Faktorisierung von G^TP_jG erzeugt.

12. Das Verfahren von Anspruch 10, wobei die Gewichtungsmatrix R_j eine Form aufweist, die äquivalent ist zu:


wobei R^γ und R^φ Gewichtungsmatrices sind, wobei R^γ und R^ϕ auf eine gemeinsame Gewichtungsmatrix W und Skalierungsparameter σ_γ und σ_ϕ wie folgt

und

bezogen sind,
wobei Schritt (h) der Erzeugung der Matrix T_j die Schritte aufweist:

Erzeugen eines Skalars b in einer Form äquivalent zu:

wobei λ_GPS die Wellenlänge der Satellitensignale ist,

Erzeugen einer Matrix H̃ in einer Form äquivalent zu

Erzeugen einer Haushältermatrix S_HH für die Matrix H̃, und

Erzeugen der Matrix T_j in einer Form äquivalent zu:

13. Das Verfahren von Anspruch 10, wobei die Gewichtungsmatrix R_j auf einen Fall angewendet wird, wo es eine erste Gruppe von Satellitensignalen, die Trägersignale in einem ersten Wellenlängenband aufweisen, und eine zweite Gruppe von Satellitensignalen, die Trägersignale in einem zweiten Wellenlängenband aufweisen, gibt, wobei die Gewichtungfrequenz eine Form aufweist, die äquivalent ist zu:


wobei R^γ und R^ϕ Gewichtungsmatrices sind; wobei R^γ und R^ϕ auf eine gemeinsame Gewichtungsmatrix W, wobei die Trägerwellenlängen der ersten Signalgruppe repräsentiert durch die Matrix Λ⁽¹⁾, die Trägerwellenlängen der zweiten Signalgruppe repräsentiert durch die Matrix Λ⁽²⁾, die Mittelwellenlänge des ersten Bands repräsentiert durch λ₁ die Mittelwellenlänge des ersten Bands repräsentiert durch λ₂, und Skalierungsparameter σ_γ und σ_ϕ bezogen sind, wie folgt:





wobei Schritt (h) vom Erzeugen der Matrix T_j die Schritte aufweist: Erzeugen eines Skalars b in einer Form äquivalent zu:

wobei λ₁ die Wellenlänge einer ersten Gruppe von Satellitensignalen ist und λ₂ die Wellenlänge einer zweiten Gruppen von Satelliten-signalen ist,
Erzeugen einer Matrix H̃ in einer form äquivalent zu H̃ = W^1/2 H^γ_j, Erzeugen einer Haushältermatrix S_HH für die Matrix H̃, und Erzeugen der Matrix T_J in einer Form äquivalent zu:

wobei die Sub-Matrixces A11, A21, und A22 wie folgt sind:



und

14. Das Verfahren von Anspruch 4, wobei Schritt (d) das Erzeugen eines Vektors B₁ aufweist, der einen Vektor aufweist, der eine Form äquivalent zu G^T P₁ µ₁+1+qg₁ hat, wobei der Vektor µ₁ den Vektor [Δγ₁, Δϕ₁]^T aufweist, und wobei die Größe qg_j ein Null-Vektor ist, wenn der Abstand zwischen den ersten und zweiten Navigationsempfängern unbeschränkt ist, und wobei q nicht null sein kann und g_j ein Nicht-null-Vektor sein kann, wenn der Abstand zwischen den ersten und zweiten Navigationsempfängern beschränkt ist; und
wobei Schritt (f) ferner aufweist, für jeden Zeitmoment j ≠ 1, einen Vektor B_j zu erzeugen, so dass er eine Matrix mit einer Form äquivalent zu B_J-1 + G^T P_jµ_j +qg_j hat, wobei der Vektor µ_j den Vektor [Δγ_j, Δϕ_j]^T aufweist und wobei der Vektor B_j-1 der in Schritt (d) erzeugte Vektor B₁ ist, wenn j = 2, und den in Schritt (f) für den j-1 Zeitmoment erzeugten Vektor aufweist, wenn j > 2, und wobei die Größe qg_j ein Null-Vektor ist, wenn der Abstand zwischen den ersten und zweiten Navigationsempfängern unbeschränkt ist, und wobei q nicht null sein kann und g_j ein Nicht-null-Vektor sein kann, wenn der Abstand zwischen den ersten und zweiten Navigationsempfängern beschränkt ist; und
wobei Schritt (f) ferner aufweist, Vektor N_j zu erzeugen, so dass er einen Vektor mit einer Form äquivalent zu N_j = [M_j]^-1 B_j, aufweist, wobei die Matrix

eine Inversmatrix der Matrix M_j aufweist.

15. Das Verfahren von Anspruch 14, wobei der Abstand zwischen den ersten und zweiten Navigationsempfängern auf einen Abstand L_RB beschränkt ist, und wobei der Schritt (c) eine Matrix S₁ in einer Form äquivalent zu erzeugt:

wobei r₁ ein Vektor ist, der Schätzungen der drei Koordinaten des Grundlinienvektors für den Zeitmoment j=1 und eine Null als vierte Komponente aufweist, wobei r₁^T die Vektortransponierte von r₁ ist, und wobei I₃ die 3x3-Identitätsmatrix ist, wobei O_1x3 ein Zeilenvektor von drei Nullen ist, und wobei O_3x1 ein Spaltenvektor von drei Nullen ist; und
wobei Schritt (d) den Vektor g₁ für den Zeitmoment j=1 in einer Form erzeugt, die äquivalent ist zu:

wobei:

16. Das Verfahren von Anspruch 14, wobei der Abstand zwischen den ersten und zweiten Navigationsempfängern auf einen Abstand L_RB beschränkt ist, wobei der Schritt (e) die Matrix S_j in einer Form äquivalent zu erzeugt:

wobei r_j ein Vektor ist, der Schätzungen der drei Koordinate des Grund-linienvektors für den j-ten Zeitmoment und eine Null als vierte Vektorkomponente aufweist, wobei r_j^T die Vektortransponierte von r_j ist, wobei l₃ die 3x3-Identitätsmatrix ist, wobei O_1x3 ein Zeilenvektor von drei Nullen ist und wobei O_3x1 ein Spaltenvektor von drei Nullen ist; und
wobei Schritt (f) den Vektor g_j für den j-ten Zeitmoment in einer Form erzeugt, die äquivalent ist zu:

wobei:

17. Das Verfahren von Anspruch 14, wobei die Gewichtungsmatrix R_j eine Identitätsmatrix aufweist, die mit einer skalaren Größe für zumindest einen Zeitmoment j multipliziert wird.

18. Verfahren zum Schätzen eines Satzes von gleitenden Mehrdeutigkeiten, die einem Satz von Phasenmessungen einer Mehrzahl n von Satellitenträgersignalen zugeordnet sind, die von einem ersten Navigationsempfänger (B) erzeugt werden, wobei ein Grundlinienvektor (x°, y°, z°) auf die Position des zweiten Empfängers zum ersten Empfänger bezogen ist, wobei jedes Satellitenträgersignal von einem Satelliten gesendet wird und eine Wellenlänge aufweist, wobei jeder Empfänger einen Zeittakt zum Referenzieren seiner Messungen aufweist, und wobei eine etwaige Differenz zwischen den Zeittakten durch einen Versatz repräsentiert werden kann, wobei das Verfahren, für eine Mehrzahl von zwei oder mehr Zeitmomenten j, die folgenden Eingaben für jeden Zeitmoment j empfängt (41):

einen Vektor

der eine Mehrzahl von Pseudobereichen repräsentiert, die von dem ersten Navigationsempfänger (B) gemessen werden und der Mehrzahl von Satellitenträgersignalen entsprechen,

einen Vektor

der eine Mehrzahl von Pseudobereichen repräsentiert, die von dem zweiten Navigationsempfänger (R) gemessen werden und der Mehrzahl von Satellitenträgersignalen entsprechen,

einen Vektor

der eine Mehrzahl von geschätzten Abständen zwischen den Satelliten und dem ersten Navigationsempfänger (B) repräsentiert,

einen Vektor

der eine Mehrzahl von geschätzten Abständen zwischen den Satelliten und dem zweiten Navigationsempfänger (R) repräsentiert,

einen Vektor

der eine Mehrzahl von Vollphasenmessungen der Satellitenträgersignale repräsentiert, die von dem ersten Navigationsempfänger (B) gemessen werden,

einen Vektor

der eine Mehrzahl von Vollphasenmessungen der Satellitenträgersignale repräsentiert,die von dem zweiten Navigationsempfänger (R) gemessen werden,

eine geometrische Jacobi-Matrix

deren Matrixelemente die Änderungen in den Abständen zwischen dem Satelliten und einem der Empfänger repräsentieren, die durch Änderungen der Position des Empfängers und dem Zeittaktversatz verursacht würden, wobei das Verfahren gekennzeichnet ist durch:

(a) Erzeugen (42), für jeden Zeitmoment j, eines Vektors Δγ_j einer Mehrzahl von Bereichsresten von Pseudobereichmessungen, die von den ersten und zweiten Navigationsempfängern durchgeführt werden, in der Form von:

wobei der Schritt einen Satz von Bereichsresten Δγ_k , k=1, .., j, erzeugt;

(b) Erzeugen (43), für jeden Zeitmoment j, eines Vektors Δϕ_j einer Mehrzahl von Phasenresten von Vollphasenmessungen, die von den ersten und zweiten Navigationsempfängern vorgenommen werden, in der Form von:

wobei Λ^-1 eine diagonale Matrix ist, die zwei inverse Wellenlängen der Satelliten aufweist, wo-bei der Schritt einen Satz von Phasenresten Δϕ_k , k=1, .., j; erzeugt;

(c) Erzeugen (44) einer LU-Faktorisierung einer Matrix M oder einer Inversmatrix der Matrix M, wobei die Matrix M eine Funktion von zumindest Λ^-1 and

ist, für Index k von

die zumindest zwei der Zeitmomente j abdeckt

(d) Erzeugen (45) eines Vektors N von geschätzten gleitenden Mehrdeutigkeiten als Funktion von zumindest dem Satz von Bereichsresten Δγ_k, des Satzes von Phasenresten Δϕ_k, und der LU-Faktorisierung der Matrix M oder der Inversmatrix der Matrix M.

19. Das Verfahren von Anspruch 18, wobei der Schritt (c) aufweist, die Matrix M in einer Form zu erzeugen, die äquivalent ist zu der Summierung

wobei die Matrix G 2n Zeilen, n Spalten, eine obere Sub-Matrix, und eine untere Sub-Matrix aufweist, wobei eine der Sub-Matrices eine n xn-Nullmatrix aufweist und die andere Sub-Matrix eine nxn-Identitätsmatrix aufweist,

• die Matrix G^T die transponierte Matrix der Matrix G aufweist, und

• die Matrix P_k 2n Zeilen, 2n Spalten und eine Form aufweist, die eine Matrix aufweist, die äquivalent ist zu

wobei die Matrix R_k eine Gewichtungsmatrix ist, wobei die Matrix R_k^-1 eine Inverse der Matrix R_k aufweist, wobei die Matrix Q_k eine obere Sub-Matrix und eine untere Sub-Matrix aufweist, wobei eine der Submatrices von Q_k die Matrix H_k^γ aufweist und die andere der Sub-Matrices von Q_k das Matrixprodukt Λ^-1H_k^γ aufweist, und wobei die Matrix Q_k^T die Transponierte der Matrix Q_k aufweist, und wobei die Größe qS_k eine Null-Matrix ist, wenn der Abstand zwischen ersten und zweiten Navigationsempfängern unbeschränkt ist, und wobei q ein Nicht-null-Gewichtungsparameter sein kann und S_k eine Nicht-null-Matrix sein kann, wenn der Abstand zwischen den ersten und zweiten Navigationsempfängern beschränkt ist.

20. Das Verfahren von Anspruch 19, wobei der Schritt (d) aufweist, die Matrix N in einer Form zu erzeugen, die äquivalent ist zu:

wobei die Matrix M^-1 eine Inversmatrix der Matrix M aufweist, wobei der Vektor µ_k den Vektor [Δγ_k, Δϕ_k]^T aufweist und wobei die Größe qg_k ein Null-Vektor ist, wenn der Abstand zwischen den ersten und zweiten Navigationsempfängern unbeschränkt ist, und wobei q nicht null sein kann und g_k ein Nicht-null-Vektor sein kann, wenn der Abstand zwischen den ersten und zweiten Navigationsempfängem beschränkt ist.

21. Das Verfahren von Anspruch 20, wobei der Abstand zwischen den ersten und zweiten Navigationsempfängern auf einen Abstand L_RB beschränkt ist, wobei der Schritt (c) die Matrix S_k für den k-ten Zeitmoment in einer Form zu erzeugen, die äquivalent ist zu:

wobei r_k ein Vektor ist, der Schätzungen der drei Koordinaten des Grundlinienvektors zum k-ten Zeitmoment sowie eine Null als vierte Komponente aufweist, wobei r_k^T die Vektortransponierte von r_k ist, wobei I₃ die 3x3-Identitätsmatrix ist, wobei O_1x3 ein Zeilenvektor von drei Nullen ist, und wobei O_3x1 ein Spaltenvektor von drei Nullen ist; und wobei Schritt (d) einen Vektor g_k für den k-ten Zeitmoment in einer Form erzeugt, die äquivalent ist zu:

wobei:

22. Das Verfahren von Anspruch 19, wobei zumindest eine der Gewichtungsmatrices R_k eine Identitätsmatrix aufweist, die mit einer skalaren Größe multipliziert wird.

23. Computerprogrammprodukt zum Anweisen eines Datenprozessors, das Verfahren von einem vorangehenden Anspruch durchzuführen.

24. Vorrichtung (100) zum Schätzen eines Satzes von gleitenden Mehrdeutigkeiten, die einem Satz von Phasenmessungen einer Mehrzahl n von Satellitenträgersignalen zugeordnet sind, die von einem ersten Navigationsempfänger (B) und einem zweiten Navigationsempfänger (R) erzeugt werden, die durch einen Abstand voneinander getrennt sind, wobei ein Grundlinienvektor (x°, y°, z°) auf die Position des zweiten Empfängers zum ersten Empfänger bezogen ist, wobei jedes Satellitenträgersignal von einem Satelliten gesendet wird und eine Wellenlänge aufweist, wobei jeder Empfänger einen Zeittakt zum Referenzieren seiner Messungen aufweist, und wobei eine etwaige Differenz zwischen den Zeittakten durch einen Versatz repräsentiert werden kann, wobei die Vorrichtung (100) dazu dient, für eine Mehrzahl von zwei oder mehr Zeitmomenten j, die folgenden Eingaben zu empfangen (71):

einen Vektor

der eine Mehrzahl von Pseudobereichen repräsentiert, die von dem ersten Navigationsempfänger (B) gemessen werden und der Mehrzahl von Satellitenträgersignalen entsprechen,

einen Vektor

der eine Mehrzahl von Pseudobereichen repräsentiert, die von dem zweiten Navigationsempfänger (R) gemessen werden und der Mehrzahl von Satellitenträgersignalen entsprechen,

einen Vektor

der eine Mehrzahl von geschätzten Abständen zwischen den Satelliten und dem ersten Navigationsempfänger (B) repräsentiert,

einen Vektor

der eine Mehrzahl von geschätzten Abständen zwischen den Satelliten und dem zweiten Navigationsempfänger (R) repräsentiert,

einen Vektor

der eine Mehrzahl von Vollphasenmessungen der Satellitenträgersignale repräsentiert, die von dem ersten Navigationsempfänger (B) gemessen werden,

einen Vektor

der eine Mehrzahl von Vollphasenmessungen der Satellitenträgersignale repräsentiert, die von dem zweiten Navigationsempfänger (R) gemessen werden,

eine geometrische Jacobi-Matrix

deren Matrixelemente die Änderungen in den Abständen zwischen dem Satelliten und einem der Empfänger repräsentieren, die durch Änderungen der Position des Empfängers und dem Zeittaktversatz verursacht würden,

wobei die Vorrichtung (100) gekennzeichnet ist durch:

(a) Mittel zum Erzeugen (72), für jeden Zeitmoment j, eines Vektors Δγ_j einer Mehrzahl von Bereichsresten von Pseudobereichmessungen, die von den ersten und zweiten Navigationsempfängern vorgenommen werden, in der Form von:

(b) Mittel zum Erzeugen (73), für jeden Zeitmoment j, eines Vektors Δϕ_j einer Mehrzahl von Phasenresten von Vollphasenmessungen, die von den ersten und zweiten Navigationsempfängern vorgenommen werden, in der Form von:

wobei Λ^-1 eine diagonale Matrix ist, die zwei inverse Wellenlängen der Satelliten aufweist;

(c) Mittel zum Erzeugen (74), für den Zeitmoment j = 1, eine LU-Faktorisierung einer Matrix M₁ oder einer Inversmatrix der Matrix M₁, wobei die Matrix M₁ eine Funktion von zumindest Λ^-1 and H₁^γ ist;

(d) Mittel zum Erzeugen (75), für den Zeitmoment j = 1, eines Vektors N₁ als Funktion von zumindest Δγ₁, Δϕ₁, und der LU-Faktorisierung der Matrix M₁ oder der Inversmatrix der Matrix M₁;

(e) Mittel zum Erzeugen (76), für einen zusätzlichen Zeitmoment j ≠ 1, einer LU-Faktorisierung der Matrix M_j oder einer Inversmatrix der Matrix M_j, wobei die Matrix M_j eine Funktion von zumindest Λ^-1,

ist;

(f) Mittel zum Erzeugen (77), für einen zusätzlichen Zeitmoment j ≠ 1, eines Vektors N_j als Funktion von zumindest Δγ_j, Δϕ_j, der LU-Faktorisierung oder Matrix M_j oder der Inversmatrix der Matrix M_j, wobei der Vektor N_j Schätzungen der gleitenden Mehrdeutigkeiten aufweist; und

(g) Mittel zum Bestimmen (78), ob N_j eine ausreichende Genauigkeit hat, wobei die Schritte (e) und (f) wiederholt werden, wenn N_j keine ausreichende Genauigkeit hat.

25. Die Vorrichtung von Anspruch 24, wobei das Mittel (c) ein Mittel zum Erzeugen einer LU-Faktorisierung für eine Matrix aufweist, die eine Form äquivalent zu (G^TP₁G) aufweist, wobei:

• die Matrix G 2n Zeilen, n Spalten, eine obere Sub-matrix, und eine untere Submatrix aufweist, wobei eine der Sub-Matrices eine n x n Null-Matrix aufweist und die andere Submatrix eine n x n Identitätsmatrix aufweist,

• die Matrix G^T die transponierte Matrix der Matrix G aufweist, und

• die Matrix P_j 2n Zeilen, 2n Spalten und eine Form aufweist, welche eine Matrix äquivalent zu

aufweist, wobei die Matrix R_j eine Gewichtungsmatrix ist, wobei die Matrix R₁^-1 eine Inverse der Matrix R_j aufweist, wobei die Matrix Q₁ eine obere Sub-Matrix und eine untere Sub-Matrix aufweist, wobei eine der Sub-Matrices von Q₁ eine Matrix H₁^γ aufweist und die andere der Sub-Matrices von Q₁ das Matrixproduct Λ^-1H_j^γ aufweist und wobei die Matrix Q_jT die Transponierte der Matrix Q_j aufweist, und wobei die Größe qS_k eine Null-Matrix ist, wenn der Abstand zwischen ersten und zweiten Navigationsempfängern unbeschränkt ist und wobei q ein Nicht-null-Gewichtungsparameter sein kann und S_k eine Nicht-null-Matrix sein kann, wenn der Abstand zwischen ersten und zweiten Navigations-empfängern beschränkt ist.

26. Die Vorrichtung von Anspruch 25, wobei das Mittel (d) ein Mittel aufweist, um den Vektor N₁ so zu erzeugen, dass er einen Vektor aufweist, der eine Form äquivalent zu

hat, wobei die Matrix M₁^-1 eine Inversmatrix der Matrix M₁ aufweist, und wobei der Vektor µ₁, den Vektor [Δγ₁, Δϕ₁]^T aufweist, und wobei die Größe qg_k ein Null-Vektor ist, wenn der Abstand zwischen ersten und zweiten Navigationsempfängem unbeschränkt ist, und wobei q nicht null sein kann und g_k ein Nicht-null-Vektor sein kann, wenn der Abstand zwischen ersten und zweiten Navigationsempfängern beschränkt ist.

27. Die Vorrichtung von Anspruch 24 wobei das Mittel (e) ein Mittel aufweist, um eine LU-Faktorisierung für eine Matrix für eine Form zu erzeugen, die äquivalent ist zu

wobei:

• M_j-1 die vom Mittel (c) erzeugte Matrix M₁ aufweist, wenn j = 2, und die vom Mittel (e) erzeugte Matrix M_j den j-1 Zeitmoment aufweist, wenn j > 2,

• die Matrix G 2n Zeilen, n Spalten, eine obere Sub-matrix, und eine untere Submatrix aufweist, wobei eine der Sub-Matrices eine n x n Null-Matrix aufweist und die andere Submatrix eine n x n Identitäts-matrix aufweist,

• die Matrix G^T die transponierte Matrix der Matrix G aufweist, und

• die Matrix P_j 2n Zeilen, 2n Spalten und eine Form aufweist, welche eine Matrix äquivalent zu

aufweist, wobei die Matrix R_j eine Gewichtungsmatrix ist, wobei die Matrix

eine Inverse der Matrix R_j aufweist, wobei die Matrix Q_j eine obere Sub-Matrix und eine untere Sub-Matrix aufweist, wobei eine der Sub-Matrices von Q_j eine Matrix H_j^γ aufweist und die andere der Sub-Matrices von Q_j das Matrixproduct

aufweist und wobei die Matrix Q_j^T die Transponierte der Matrix Q_j aufweist, und wobei die Größe qS_j eine Null-Matrix ist, wenn der Abstand zwischen ersten und zweiten Navigationsempfängern unbeschränkt ist und wobei q ein Nicht-null-Gewichtungsparameter sein kann und S_j eine Nicht-null-Matrix sein kann, wenn der Abstand zwischen ersten und zweiten Navigationsempfängern beschränkt ist.

28. Die Vorrichtung von Anspruch 27, wobei das Mittel (f) ein Mittel aufweist, um den Vektor N_j so zu erzeugen, dass er einen Vektor aufweist, der eine Form äquivalent zu

hat, wobei die Matrix

eine Inversmatrix der Matrix M_j aufweist, und wobei der Vektor µ_j den Vektor [Δγ_j, Δϕ_j]^T aufweist, und wobei der Vektor Nj_-1 den in Schritt (d) erzeugten Vektor N₁ aufweist, wenn j = 2, und den durch das Mittel (f) für den j-1 Zeitmoment erzeugten Vektor in N_j-1 aufweist, wenn j > 2, und wobei die Größe qg_j ein Null-Vektor ist, wenn der Abstand zwischen den ersten und zweiten Navigationsempfängern unbeschränkt ist, und wobei q nicht null sein kann und g_j ein Nicht-null-Vektor sein kann, wenn der Abstand zwischen ersten und zweiten Navigationsempfängern beschränkt ist.

29. Die Vorrichtung von Anspruch 26, wobei der Abstand zwischen ersten und zweiten Navigationsempfängern auf einen Abstand L_RB beschränkt ist, wobei das Mittel (c) die Matrix S₁ in einer Form erzeugt, die äquivalent ist zu:

wobei r₁ ein Vektor ist, der Schätzungen der drei Koordinaten des Grundlinienvektors für den Zeitmoment j=1 und eine Null als vierte Komponente aufweist, wobei r₁^T die Vektortransponierte von R₁ ist, wobei I₃ is the 3-x-3-Identitätsmatrix ist, wobei O_1x3 ein Zeilenvektor von drei Nullen ist und wobei O_3x1 ein Spaltenvektor von drei Nullen ist (d); und
wobei das Mittel (d) ein Mittel aufweist, um den Vektor g₁ für das Zeitmoment j=1 in einer Form zu erzeugen, die äquivalent ist zu:

wobei

30. Die Vorrichtung von Anspruch 28, wobei der Abstand zwischen den ersten und zweiten Navigationsempfängern auf einen Abstand L_RB beschränkt ist, wobei das Mittel (e) ein Mittel aufweist, um die Matrix S_j in einer Form zu erzeugen, die äquivalent ist zu:

wobei r_j ein Vektor ist, der Schätzungen der drei Koordinaten des Grundlinienvektors für den j-ten Zeitmoment und eine Null als vierte Vektorkomponente aufweist, wobei r_j^T die Vektortransponierte von r_j ist, und wobei I₃ die 3x3-Identitätsmatrix ist, wobei O_1x3 ein Zeilenvektor von drei Nullen ist und wobei O_3x1 ein Spaltenvektor von drei Nullen ist; und
wobei das Mittel (f) ein Mittel aufweist, um den Vektor g_j für den j-te Zeitmoment in einer Form zu erzeugen, die äquivalent ist zu:

wobei:

31. Die Vorrichtung von Anspruch 27, wobei das Mittel (e) zum Erzeugen der LU-Faktorisierung von M_j aufweist:

(g) Mittel zum Erzeugen einer LU-Faktorisierung der Matrix M_j-1 in einer Form äquivalent zu L_j-1 Lj_-1^T, wobei L_j-1 eine niedrige dreieckige Matrix ist und L_j-1^T die Transponierte von L_j-1 ist;

(h) Mittel zum Erzeugen einer Faktorisierung von G^TP_jG in einer Form äquivalent zu T_jT_j^T = G^TP_jG, wobei T_j^T die Transponierte von T_j ist;

(i) Mittel zum Erzeugen einer LU-Faktorisierung der Matrix M_j in einer Form äquivalent zu L_j L_j^T aus einer Mehrzahl n von erstrangigen Modifizierungen der Matrix L_j-1, wobei jede erstrangige Modifizierung auf einer erstrangigen Spalte der Matrix T_j beruht, wobei n die Anzahl von Zeilen in der Matrix M_j ist.

32. Die Vorrichtung von Anspruch 31, wobei das Mittel (h) die Matrix T_j aus einer Cholesky-Faktorisierung von G^TP_jG erzeugt.

33. Die Vorrichtung von Anspruch 31, wobei die Gewichtungsmatrix R_j eine Form äquivalent zu:


hat, wobei R^γ und R^ϕ Gewichtungsmatrices sind,
wobei R^γ und R^φ auf eine gemeinsame Gewichtungsmatrix W und Skalierungsparameter σ_γ und σ_φ wie folgt

und

bezogen sind, wobei das Mittel (h) der Erzeugung der Matrix T_j aufweist:

Mittel zum Erzeugen eines Skalars b in einer Form äquivalent zu:

wobei λ_GPS die Wellenlänge der Satellitensignale ist,

Mittel zum Erzeugen einer Matrix H̃ in einer Form äquivalent zu H̃ = W^1/2H^-γ_j,

Mittel zum Erzeugen einer hiaushältermatrix S_HH für die Matrix H̃, und Mittel zum Erzeugen der Matrix T_j in einer Form äquivalent zu:

34. Die Vorrichtung von Anspruch 31, wobei die Gewichtungsmatrix R_j auf einen Fall angewendet wird, wo es eine erste Gruppe von Satellitensignalen, die Trägersignale in einem ersten Wellenlängenband aufweisen, und eine zweite Gruppe von Satellitensignalen, die Trägersignale in einem zweiten Wellenlängenband aufweise, gibt, wobei die Gewichtungfrequenz eine Form aufweist, die äquivalent ist zu:


wobei R^γ und R^ϕ Gewichtungsmatrices sind; wobei R^γ und R^ϕ auf eine gemeinsame Gewichtungsmatrix W, wobei die Trägerwellenlängen der ersten Signalgruppe repräsentiert durch die Matrix Λ⁽¹⁾, die Trägerwellenlängen der zweiten Signalgruppe repräsentiert durch die Matrix Λ⁽²⁾, die Mittelwellenlänge des ersten Bands repräsentiert durch λ₁, die Mittelwellenlänge des ersten Bands repräsentiert durch λ₂, und Skalierungsparameter σ_γ und σ_ϕ bezogen sind, wie folgt:



wobei

wobei Schritt (h) zum Erzeugen der Matrix T_j die Schritte aufweist:

Erzeugen eines Skalars b in einer Form äquivalent zu:

wobei λ₁ die Wellenlänge einer ersten Gruppe von Satellitensigna - len ist und λ₂ Wellenlänge einer zweiten Gruppen von Satelliten-signalen ist,
Erzeugen einer Matrix H̃ in einer form äquivalent zu H̃ = W^1/2H^γ_j,
Erzeugen einer Haushältermatrix S_HH für die Matrix H̃, und

Erzeugen der Matrix T_j in einer Form äquivalent zu:

wobei die Sub-Matrixces A11, A21, und A22 wie folgt sind:



und

35. Vorrichtung (100) zum Schätzen eines Satzes von gleitenden Mehrdeutigkeiten, die einem Satz von Phasenmessungen einer Mehrzahl n von Satellitenträgersignalen zugeordnet sind, die von einem ersten Navigationsempfänger (B) und einem zweiten Navigationsempfänger (R) erzeugt werden, die durch einen Abstand voneinander getrennt sind, wobei ein Grundlinienvektor (x°, y°, z°) auf die Position des zweiten Empfängers zum ersten Empfänger bezogen ist, wobei jedes Satellitenträgersignal von einem Satelliten gesendet wird und eine Wellenlänge aufweist, wobei jeder Empfänger einen Zeittakt zum Referenzieren seiner Messungen aufweist, und wobei eine etwaige Differenz zwischen den Zeittakten durch einen Versatz repräsentiert werden kann, wobei die Vorrichtung (100) dazu dient, für eine Mehrzahl von zwei oder mehr Zeitmomenten j die folgenden Eingaben zu empfangen:

einen Vektor

der eine Mehrzahl von Pseudobereichen repräsentiert, die von dem ersten Navigationsempfänger (B) gemessen werden und der Mehrzahl von Satellitenträgersignalen entsprechen,

einen Vektor

der eine Mehrzahl von Pseudobereichen repräsentiert, gemessen von dem zweiten Navigationsempfänger (R) gemessen werden und der Mehrzahl von Satellitenträgersignalen entsprechen,

einen Vektor

der eine Mehrzahl von geschätzten Abständen zwischen den Satelliten und dem ersten Navigationsempfänger (B) repräsentiert,

einen Vektor

der eine Mehrzahl von geschätzten Abständen zwischen den Satelliten und dem zweiten Navigationsempfänger (R) repräsentiert,

einen Vektor

der eine Mehrzahl von Vollphasenmessungen der Satellitenträgersignale repräsentiert, die von dem ersten Navigationsempfänger (B) gemessen werden,

einen Vektor

der eine Mehrzahl von Vollphasenmessungen der Satellitenträgersignale repräsentiert, die von dem zweiten Navigati-onsempfänger (R) gemessen werden,

eine geometrische Jacobi-Matrix

deren Matrixelemente die Änderungen in den Abständen zwischen dem Satelliten und einem der Empfänger repräsentieren, die durch Änderungen der Position des Empfängers und dem Zeittaktversatz verursacht würden, wobei die Vorrichtung (100) gekennzeichnet ist durch:

(a) Mittel zum Erzeugen (42), für jeden Zeitmoment j, eines Vektors Δγ_j einer Mehrzahl von Bereichsresten von Pseudobereichmessun-gen, die von den ersten und zweiten Navigationsempfängern vorgenommen werden, in der Form von:

wobei der Schritt einen Satz von Bereichsresten Δγ_k , k=1, .., j, erzeugt;

(b) Mittel zum Erzeugen (43), für jeden Zeitmoment j, eines Vektors Δϕ_j einer Mehrzahl von Phasenresten von Vollphasenmessungen, die von den ersten und zweiten Navigationsempfängern vorgenommen werden, in der Form von:

wobei Λ^-1 eine diagonale Matrix ist, die zwei inverse Wellenlängen von Satelliten aufweist, wobei das Mittel einen Satz von Phasenresten Δϕ_k , k=1, .., j; erzeugt;

(c) Mittel zum Erzeugen (44) einer LU-Faktorisierung einer Matrix M oder eine Inversmatrix der Matrix M, wobei die Matrix M eine Funktion von zumindest Λ^-1 and

ist, für Index k von

die zumindest zwei der Zeitmomente j abdeckt;

(d) Mittel zum Erzeugen (45) eines Vektors N von geschätzten gleitenden Mehrdeutigkeiten als Funktion von zumindest dem Satz von Seu-eichsresten Δγ_k, des Satzes von Phasenresten Δϕ_k, und der LU-Faktorisierung der Matrix M oder der Inversmatrix der Matrix M.

36. Die Vorrichtung von Anspruch 35, wobei das Mittel (c) ein Mittel aufweist, um die Matrix M in einer Form zu erzeugen, die äquivalent ist zu der Summierung

wobei die Matrix G 2n Zeilen, n Spalten, eine obere Sub-Matrix, und eine untere Sub-Matrix aufweist, wobei eine der Sub-Matrices eine nxn-Nullmatrix aufweist und die andere Sub-Matrix eine nxn-Identitätsmatrix aufweist,

• die Matrix G^T die transponierte Matrix der Matrix G aufweist, und

• die Matrix P_k 2n Zeilen, 2n Spalten und eine Form aufweist, die eine Matrix aufweist, die äquivalent ist zu:

wobei die Matrix R_k eine Gewichtungsmatrix ist, wobei die Matrix R_k-1 eine Inverse der Matrix R_k aufweist, wobei die Matrix Q_k eine obere Sub-Matrix und eine untere Sub-Matrix aufweist, wobei eine der Submatrices von Q_k die Matrix H_k^γ aufweist und die andere der Sub-Matrices von Q_k das Matrixprodukt Λ^-1H_k^γ aufweist, und wobei die Matrix Q_k^T die Transponierte der Matrix Q_k aufweist, und wobei die Größe qS_k eine Null-Matrix ist, wenn der Abstand zwischen ersten und zweiten Navigationsempfängern unbeschränkt ist, und wobei q ein Nicht-null-Gewichtungsparameter sein kann und S_k eine Nicht-null-Matrix sein kann, wenn der Abstand zwischen den ersten und zweiten Navigationsempfängern beschränkt ist.

37. Die Vorrichtung von Anspruch 36, wobei das Mittel (d) ein Mittel aufweist, um die Matrix N in einer Form zu erzeugen, die äquivalent ist zu:

wobei die Matrix M^-1 eine Inversmatrix der Matrix M aufweist, wobei der Vektor µ_k den Vektor [Δγ_k, Δϕ_k]^T aufweist und wobei die Größe qg_k ein Null-Vektor ist, wenn der Abstand zwischen den ersten und zweiten Navigationsempfängern unbeschränkt ist, und wobei q nicht null sein kann und g_k ein Nicht-null-Vektor sein kann, wenn der Abstand zwischen den ersten und zweiten Navigationsempfängern beschränkt ist.

38. Die Vorrichtung von Anspruch 37, wobei der Abstand zwischen den ersten und zweiten Navigationsempfängern auf einen Abstand L_RB beschränkt ist, wobei das Mittel (c) die Matrix S_k für den k-ten Zeitmoment in einer Form erzeugt, die äquivalent ist zu:

wobei r_k ein Vektor ist, der Schätzungen der drei Koordinaten des Grundlinienvektors zum k-ten Zeitmoment sowie eine Null als vierte Komponente aufweist, wobei r_k^T die Vektortransponierte von r_k ist, wobei I₃ die 3x3-Identitätsmatrix ist, wobei O_1x3 ein Zeilenvektor von drei Nullen ist, und wobei O_3x1 ein Spaltenvektor von drei Nullen ist; und wobei das Mittel (d) einen Vektor g_k für den k-ten Zeitmoment in einer Form erzeugt, die äquivalent ist zu:

wobei:

Revendications

1. Procédé d'estimation d'un ensemble d'ambiguïtés flottantes associé à un ensemble de mesures de phases d'une pluralité n de signaux de porteuse de satellites réalisées par un premier récepteur de navigation (B) et un deuxième récepteur de navigation (R) séparés par une distance, où un vecteur de référence (x⁰, y⁰, z⁰) associe la position du deuxième récepteur au premier récepteur, chaque signal de porteuse de satellite étant émis par un satellite et ayant une longueur d'onde, où chaque récepteur a une horloge temporelle pour référencer ses mesures et où un différence quelconque entre les horloges temporelles peut être représentée par un décalage, ledit procédé recevant (71), pour une pluralité de deux instants j ou plus, les entrées suivantes :

un vecteur

représentant une pluralité de pseudo distances mesurées par le premier récepteur de navigation (B) et correspondant à la pluralité de signaux de porteuse de satellites,

un vecteur

représentant une pluralité de pseudo distances mesurées par le deuxième récepteur de navigation (R) et correspondant à la pluralité de signaux de porteuse de satellites,

un vecteur

représentant une pluralité de distances estimées entre les satellites et le premier récepteur de navigation (B),

un vecteur

représentant une pluralité de distances estimées entre les satellites et le deuxième récepteur de navigation (R),

un vecteur

i représentant une pluralité de mesures de phases totales des signaux de porteuse de satellites mesurées par le premier récepteur de navigation (B),

un vecteur

représentant une pluralité de mesures de phases totales des signaux de porteuse de satellites mesurées par le deuxième récepteur de navigation (R),

une matrice Jacobienne géométrique

dont les éléments matriciels représentent les variations des distances entre les satellites et l'un des récepteurs qui pourraient être provoquées par des variations de la position de ce récepteur et du décalage d'horloge temporelle, ledit procédé étant caractérisé par le fait :

(a) de générer (72), pour chaque instant j, un vecteur Δγ_j d'une pluralité de résidus de distance de mesures de pseudo distance réalisées par les premier et deuxième récepteurs de navigation sous forme de :

(b) de générer (73), pour chaque instant j, un vecteur Δϕ_j d'une pluralité de résidus de phase de mesures de phase totales réalisées par les premier et deuxième récepteurs de navigation sous forme de :

où Λ^-1 est une matrice diagonale comprenant les longueurs d'ondes inverses des satellites ;

(c) de générer (74), pour un instant j = 1, une factorisation LU d'une matrice M₁ ou d'un inverse de matrice de la matrice M₁, la matrice M₁ étant une fonction d'au moins Λ^-1 et

(d) de générer (75), pour un instant j = 1, un vecteur N₁ en tant que fonction d'au moins Δγ₁, Δϕ₁ et la factorisation LU de la matrice M₁ ou de l'inverse de matrice de la matrice M₁ ;

(e) de générer (76), pour un instant supplémentaire j ≠ 1, une factorisation LU d'une matrice M_j ou d'un inverse de matrice de la matrice M_j, la matrice M_j étant une fonction d'au moins Λ^-1,

et d'une instance de la matrice M générée pour un instant différent ;

(f) de générer (77), pour un instant supplémentaire j ≠ 1, un vecteur N_j en tant que fonction d'au moins Δγ_j, Δϕ_j, et la factorisation LU de la matrice M_j ou de l'inverse de matrice de la matrice M_j, le vecteur N_j ayant les estimations des ambiguïtés flottantes ; et

(g) de déterminer (78) si N_j a une précision suffisante, où les étapes (e) et (f) sont répétées si N_j n'a pas une précision suffisante.

2. Procédé de la revendication 1, dans lequel l'étape (c) comprend le fait de générer une factorisation LU pour une matrice comprenant une forme équivalente à (G^T P₁ G) où :

• la matrice G a 2n lignes, n colonnes, une sous-matrice supérieure et une sous-matrice inférieure, l'une des sous-matrices comprend une matrice nulle n x n et l'autre sous-matrice comprenant une matrice identité n x n,

• la matrice G^T comprend la matrice transposée de la matrice G, et
la matrice P₁ a 2n lignes, 2n colonnes, et une forme qui comprend une matrice équivalente à :

où la matrice R₁ est une matrice de pondération, où la matrice R₁^-1 comprend un inverse de la matrice R₁, où la matrice Q₁ a une sous-matrice supérieure et une sous-matrice inférieure, l'une des sous-matrices de Q₁ comprenant une matrice

et l'autre des sous-matrices de Q₁ comprenant le produit matriciel

et où la matrice Q₁^T comprend la transposée de la matrice Q₁, et où la quantité qS_k est une matrice nulle lorsque la distance entre les premier et deuxième récepteurs de navigation n'est pas limitée et où q peut être un paramètre de pondération non nul et S_k peut être une matrice non nulle lorsque la distance entre les premier et deuxième récepteurs de navigation est limitée.

3. Procédé de la revendication 2, dans lequel l'étape (d) comprend l'étape qui consiste à générer un vecteur N₁ pour comprendre un vecteur ayant une forme équivalente à :

où la matrice M₁^-1 comprend un inverse de matrice de la matrice M₁, et où le vecteur µ₁ comprend le vecteur [Δγ₁, Δϕ₁]^T , et où la quantité qg_k est un vecteur nul lorsque la distance entre les premier et deuxième récepteurs de navigation n'est pas limitée et où q peut être non nul et g_k peut être un vecteur non nul lorsque la distance entre les premier et deuxième récepteurs de navigation est limitée.

4. Procédé de la revendication 1, dans lequel l'étape (e) comprend le fait de générer une factorisation LU pour une matrice comprenant une forme équivalente à :

où

• M_j-₁ comprend la matrice M₁ de l'étape (c) lorsque j = 2 et comprend la matrice M_j de l'étape (e) pour l'instant j-1 lorsque j>2,

• la matrice G a 2n lignes, n colonnes, une sous-matrice supérieure et une sous matrice inférieure, l'une des sous-matrices comprenant une matrice nulle n x n et l'autre sous-matrice comprenant une matrice identité n × n.

• la matrice G^T comprend la matrice transposée de la matrice G, et

• la matrice P_j a 2n lignes, 2n colonnes, et une forme qui comprend une matrice équivalente à :

où la matrice R_j est une matrice de pondération, où la matrice R_j^-1 comprend un inverse de la matrice R_j,
où la matrice Q_j a une sous-matrice supérieure et une sous-matrice inférieure, l'une des sous-matrices de Q^j comprenant une matrice

et l'autre des sous-matrices de Qj comprenant le produit matriciel

et où la matrice Q_j^T comprend la transposée de la matrice Q_j, et où la quantité qS_j est une matrice nulle lorsque la distance entre les premier et deuxième récepteurs de navigation n'est pas limitée et où q peut être un paramètre de pondération non nul et S_j peut être une matrice non nulle lorsque la distance entre les premier et deuxième récepteurs de navigation est limitée.

5. Procédé de la revendication 4, dans lequel l'étape (f) comprend l'étape de génération d'un vecteur N_j pour comprendre un vecteur ayant une forme équivalente à :

où la matrice M_j^-1 comprend un inverse de matrice de la matrice M_j, où le vecteur µ_j comprend le vecteur [Δγ_j, Δϕ_j]^T, et où le vecteur N_j-1 comprend le vecteur N₁ généré par l'étape (d) lorsque j = 2 et comprend le vecteur N_j-1 généré par l'étape (f) pour l'instant j-1 lorsque j > 2, et où la quantité qg_j est un vecteur nul lorsque la distance entre les premier et deuxième récepteurs de navigation n'est pas limitée et où q peut être non nul et g_j peut être un vecteur non nul lorsque la distance entre les premier et deuxième récepteurs de navigation est limitée.

6. Procédé de la revendication 3, dans lequel la distance entre les premier et deuxième récepteurs de navigation est limitée à une distance L_RB, où l'étape (c) génère une matrice S₁ dans une forme équivalente à :

où r₁ est un vecteur comprenant des estimations des trois coordonnées du vecteur de référence pour l'instant j= 1 et un zéro en tant que quatrième composante, où r₁^T est la transposée de vecteur de r₁, où I₃ est la matrice identité 3 par 3, où O_1x3 est un vecteur ligne de trois zéros, et où O_3x1 est un vecteur colonne de trois zéros ; et
où l'étape (d) génère un vecteur g₁ pour l'instant j = 1 dans une forme équivalente à :

où :

7. Procédé de la revendication 3, dans lequel la matrice de pondération R₁ comprend une matrice identité multipliée par une grandeur scalaire.

8. Procédé de la revendication 5, dans lequel la distance entre les premier et deuxième récepteurs de navigation est limitée à une distance L_RB, où l'étape (e) génère une matrice S_j dans une forme équivalente à :

où r_j est un vecteur comprenant des estimations des trois coordonnées du vecteur de référence pour le j^ième instant et un zéro en tant que quatrième composante de vecteur, où r_j^T est la transposée de vecteur de r_j, où I₃; est la matrice identité 3 par 3, où O_1x3 est un vecteur ligne de 3 zéros, et où O_3x1 est un vecteur colonne de 3 zéros ; et
où l'étape (f) génère un vecteur g_j pour le j^ième instant dans une forme équivalente à :

où :

9. Procédé de la revendication 5, dans lequel la matrice de pondération R_j comprend une matrice identité multipliée par une grandeur scalaire pour au moins un instant j.

10. Procédé de la revendication 4, dans lequel la génération de la factorisation LU dans l'étape (e) comprend les étapes qui consistent :

(g) à générer une factorisation LU de la matrice M_j-1 dans une forme équivalente à L_j-1 L_j-1^T où L_j-1 est une matrice triangulaire inférieure et L_j-1^T est la transposée de L_j-1;

(h) à générer une factorisation de G^T P_j G dans une forme équivalente à T_jT_j^T = G^TP_jG, où T_j^T est la transposée de T_j;

(i) à générer une factorisation LU de la matrice M_j dans une forme équivalente à L_j L_j^T à partir d'une pluralité n de modifications de rang un de la matrice L_j-1, chaque modification de rang un étant basée sur une colonne respective de la matrice T_j, où n est le nombre de lignes dans la matrice M_j.

11. Procédé de la revendication 10, dans lequel l'étape (h) génère une matrice T_j à partir d'une factorisation de Cholesky de G^TP_jG.

12. Procédé de la revendication 10, dans lequel la matrice de pondération R_j a une forme équivalente à :

où R^γ et R^ϕ sont des matrices de pondération ;
où R^γ et R^ϕ sont liées à une matrice de pondération commune W et à des paramètres de mise à l'échelle σ_γ et σ_ϕ comme suit :

et

où l'étape (h) de génération de la matrice T_j comprend les étapes qui consistent :
à générer un scalaire b dans une forme équivalente à :

où λ_GPS est la longueur d'onde des signaux de satellites,
à générer une matrice H̃ dans une forme équivalente à

à générer une matrice de Householder S_HH pour la matrice H̃, et
à générer une matrice T_j dans une forme équivalente à :

13. Procédé de la revendication 10, dans lequel la matrice de pondération R_j est appliquée à un cas où il y a un premier groupe de signaux de satellites ayant des signaux de porteuse dans une première bande de longueurs d'ondes et un deuxième groupe de signaux de satellites ayant des signaux de porteuse dans une deuxième bande de longueurs d'ondes, la fréquence de pondération ayant une forme équivalente à :

où R^γ et R^ϕ sont des matrices de pondération ;
où R^γ et R^ϕ sont liées à une matrice de pondération commune W, aux longueurs d'ondes de porteuse du premier groupe de signaux comme représentées par la matrice Λ⁽¹⁾, aux longueurs d'ondes de porteuse du deuxième groupe de signaux comme représentées par la matrice Λ⁽²⁾, à la longueur d'onde centrale de la première bande comme représentée par λ₁, à la longueur d'onde centrale de la première bande comme représentée par λ₂, et à des paramètres de mise à l'échelle σ_γ et σ_ϕ comme suit :





où l'étape (h) de génération de la matrice T_j comprend les étapes qui consistent :

à générer un scalaire b dans une forme équivalente à :

où λ₁ est la longueur d'onde d'un premier groupe de signaux de satellites et λ₂ est la longueur d'onde d'un deuxième groupe de signaux de satellites,

à générer une matrice H̃ dans une forme équivalente à :

à générer une matrice de Householder S_HH pour la matrice H̃, et

à générer la matrice T_j dans une forme équivalente à :

où les sous-matrices A11, A21, et A22 sont comme suit :



et

14. Procédé de la revendication 4, dans lequel l'étape (d) comprend le fait de générer un vecteur B₁ pour comprendre un vecteur ayant une forme équivalente à G^TP₁µ₁ +qg₁, où le vecteur µ₁ comprend le vecteur [Δγ₁, Δϕ₁]^T, et où la quantité qg_j est un vecteur nul lorsque la distance entre les premier et deuxième récepteurs de navigation n'est limitée pas et où q peut être non nul et g_j peut être un vecteur non nul lorsque la distance entre les premier et deuxième récepteurs de navigation est limitée ; et
où l'étape (f) comprend en outre le fait de générer, pour chaque instant j ≠ 1, un vecteur B_j pour comprendre une matrice ayant une forme équivalente à B_j-1 + G^T P_jµ_j +qg_j, où le vecteur µ_j comprend le vecteur [Δγ_j, Δϕ_j]T, et où le vecteur B_j-1 est le vecteur B₁ généré par l'étape (d) lorsque j = 2 et comprend le vecteur généré par l'étape (f) pour l'instant j-1 lorsque j > 2, et où la quantité qg_j est un vecteur nul lorsque la distance entre les premier et deuxième récepteurs de navigation n'est pas limitée et où q peut être non nul et g_j peut être un vecteur non nul lorsque la distance entre les premier et deuxième récepteurs de navigation est limitée ; et
où l'étape (f) comprend en outre le fait de générer un vecteur N_j pour comprendre un vecteur ayant une forme équivalente à N_j = [M_j]^-1 B_j, où la matrice Mj^-1 comprend un inverse de matrice de la matrice M_j.

15. Procédé de la revendication 14, dans lequel la distance entre les premier et deuxième récepteurs de navigation est limitée à une distance L_RB, où l'étape (c) génère une matrice S₁ dans une forme équivalente à :

où r₁ est un vecteur comprenant des estimations des trois coordonnées du vecteur de référence pour l'instant j = 1 et un zéro en tant que quatrième composante, où r₁^T est la transposée de vecteur de r₁, où I₃ est la matrice identité 3 par 3, où O_1x3 est un vecteur ligne de trois zéros, et où O_3x1 est un vecteur colonne de trois zéros ; et
où l'étape (d) génère un vecteur g₁ pour l'instant j = 1 dans une forme équivalente à :

où

16. Procédé de la revendication 14, dans lequel la distance entre les premier et deuxième récepteurs de navigation est limitée à une distance L_RB, où l'étape (e) génère une matrice S_j dans une forme équivalente à :

où r_j est un vecteur comprenant des estimations des trois coordonnées du vecteur de référence pour le j^ième instant et un zéro en tant que quatrième composante de vecteur, où r_j^T est la transposée de vecteur de r_j, où I₃ est la matrice identité 3 par 3, où O_1x3 est un vecteur ligne de 3 zéros, et où O_3x1 est un vecteur colonne de 3 zéros ; et
où l'étape (f) génère un vecteur g_j pour le j^ième instant dans une forme équivalente à :

où

17. Procédé de la revendication 14, dans lequel la matrice de pondération R_j comprend une matrice identité multipliée par une grandeur scalaire pour au moins un instant j.

18. Procédé d'estimation d'un ensemble d'ambiguïtés flottantes associé à un ensemble de mesures de phases d'une pluralité n de signaux de porteuse de satellites réalisées par un premier récepteur de navigation (B) et un deuxième récepteur de navigation (R), où un vecteur de référence (x⁰, y^0, z⁰) associe la position du deuxième récepteur au premier récepteur, chaque signal de porteuse de satellite étant émis par un satellite et ayant une longueur d'onde, où chaque récepteur a une horloge temporelle pour référencer ses mesures et où une différence quelconque entre les horloges temporelles peut être représentée par un décalage, ledit procédé recevant (41), pour une pluralité de deux instants j ou plus, les entrées suivantes pour chaque instant j :

un vecteur

représentant une pluralité de pseudo distances mesurées par le premier récepteur de navigation (B) et correspondant à la pluralité de signaux de porteuse de satellites,

un vecteur

représentant une pluralité de pseudo distances mesurées par le deuxième récepteur de navigation (R) et correspondant à la pluralité de signaux de porteuse de satellites,

un vecteur

représentant une pluralité de distances estimées entre les satellites et le premier récepteur de navigation (B),

un vecteur

représentant une pluralité de distances estimées entre les satellites et le deuxième récepteur de navigation (R),

un vecteur

représentant une pluralité de mesures de phases totales des signaux de porteuse de satellites mesurées par le premier récepteur de navigation (B),

un vecteur

représentant une pluralité de mesures de phases totales des signaux de porteuse de satellites mesurées par le deuxième récepteur de navigation (R),

une matrice Jacobienne géométrique

dont les éléments matriciels représentent les variations des distances entre les satellites et l'un des récepteurs qui pourraient être provoquées par des variations de la position de ce récepteur et du décalage d'horloge temporelle, ledit procédé étant caractérisé par le fait :

(a) de générer (42), pour chaque instant j, un vecteur Δ_γj d'une pluralité de résidus de distance de mesures de pseudo distances réalisées par les premier et deuxième récepteurs de navigation sous forme de :

ladite étape de génération d'un ensemble de résidus de distance Δ_γk, k = 1, .., j ;

(b) de générer (43), pour chaque instant j, un vecteur Δ_ϕj d'une pluralité de résidus de phase de mesures de phases totales réalisées par les premier et deuxième récepteurs de navigation sous forme de :

où Λ^-1 est une matrice diagonale comprenant les longueurs d'ondes inverses des satellites, ladite étape de génération d'un ensemble de résidus de phase Δ_ϕk, k = 1, .., j ;

(c) de générer (44) une factorisation LU d'une matrice M ou d'un inverse de matrice de la matrice M, la matrice M étant une fonction d'au moins Λ^-1 et

pour un indice k de

couvrant au moins deux instants parmi les instants j ;

(d) de générer (45) un vecteur N d'ambigüités flottantes estimées en tant que fonction d'au moins l'ensemble de résidus de distance Δ_γk, l'ensemble de résidus de phase Δ_ϕk et la factorisation LU de la matrice M ou de l'inverse de matrice de la matrice M ;

19. Procédé de la revendication 18, dans lequel l'étape (c) comprend le fait de générer une matrice M dans une forme équivalente à la sommation

où

• la matrice G a 2n lignes, n colonnes, une sous-matrice supérieure et une sous-matrice inférieure, l'une des sous-matrices comprend une matrice nulle n × n et l'autre sous-matrice comprenant une matrice identité n × n,

• la matrice G^T comprend la matrice transposée de la matrice G, et

• la matrice P_k a 2n lignes, 2n colonnes, et une forme qui comprend une matrice équivalente à :

où la matrice R_k est une matrice de pondération, où la matrice R_k^-1 comprend un inverse de la matrice R_k, où la matrice Q_k a une sous-matrice supérieure et une sous-matrice inférieure, l'une des sous-matrices de Q_k comprenant une matrice

et l'autre des sous-matrices de Q_k comprenant le produit matriciel

et où la matrice Qk^T comprend la transposée de la matrice Q_k, et où la quantité qS_k est une matrice nulle lorsque la distance entre les premier et deuxième récepteurs de navigation n'est pas limitée et où q peut être un paramètre de pondération non nul et S_k peut être une matrice non nulle lorsque la distance entre les premier et deuxième récepteurs de navigation est limitée.

20. Procédé de la revendication 19, dans lequel l'étape (d) comprend le fait de générer une matrice N dans une forme équivalente à :

où la matrice M^-1 comprend un inverse de matrice de la matrice M, où le vecteur µ_k comprend le vecteur [Δγk, Δϕk]^T, et où la quantité qg_k est un vecteur nul lorsque la distance entre les premier et deuxième récepteurs de navigation n'est pas limitée et où q peut être non nul et g_k peut être un vecteur non nul lorsque la distance entre les premier et deuxième récepteurs de navigation est limitée.

21. Procédé de la revendication 20, dans lequel la distance entre les premier et deuxième récepteurs de navigation est limitée à une distance L_RB, où l'étape (c) génère une matrice S_k pour le k^ième instant dans une forme équivalente à :

où r_k est un vecteur comprenant des estimations des trois coordonnées du vecteur de référence au k^ième instant, et un zéro en tant que quatrième composante, où

est la transposée de vecteur de r_k, où I₃ est la matrice identité 3 par 3, où O_1x3 est un vecteur ligne de trois zéros, et où O_3x1 est un vecteur colonne de trois zéros ; et
où l'étape (d) génère un vecteur g_k pour le k^ieme instant dans une forme équivalente à :

où

22. Procédé de la revendication 19, dans lequel au moins l'une des matrices de pondération R_k comprend une matrice identité multipliée par une grandeur scalaire.

23. Produit de programme informatique pour diriger un processeur de données pour effectuer le procédé de l'une des revendications précédentes.

24. Appareil (100) pour estimer un ensemble d'ambiguïtés flottantes associé à un ensemble de mesures de phases d'une pluralité n de signaux de porteuse de satellites réalisées par un premier récepteur de navigation (B) et un deuxième récepteur de navigation (R) séparés par une distance, où un vecteur de référence (x⁰, y⁰, z⁰) associe la position du deuxième récepteur au premier récepteur, chaque signal de porteuse de satellite étant émis par un satellite et ayant une longueur d'onde, où chaque récepteur a une horloge temporelle pour référencer ses mesures et où une différence quelconque entre les horloges temporelles peut être représentée par un décalage, ledit appareil (100) pour recevoir (71), pour une pluralité de deux instants j ou plus, les entrées suivantes :

un vecteur

représentant une pluralité de pseudo distances mesurées par le premier récepteur de navigation (B) et correspondant à la pluralité de signaux de porteuse de satellites,

un vecteur

représentant une pluralité de pseudo distances mesurées par le deuxième récepteur de navigation (R) et correspondant à la pluralité de signaux de porteuse de satellites,

un vecteur

représentant une pluralité de distances estimées entre les satellites et le premier récepteur de navigation (B),

un vecteur

représentant une pluralité de distances estimées entre les satellites et le deuxième récepteur de navigation (R),

un vecteur

représentant une pluralité de mesures de phases totales des signaux de porteuse de satellites mesurées par le premier récepteur de navigation (B),

un vecteur

représentant une pluralité de mesures de phases totales des signaux de porteuse de satellites mesurées par le deuxième récepteur de navigation (R),

une matrice Jacobienne géométrique

dont les éléments matriciels représentent les variations des distances entre les satellites et l'un des récepteurs qui pourraient être provoquées par les variations de la position de ce récepteur et du décalage d'horloge temporelle,

ledit appareil (100) étant caractérisé par :

(a) un moyen pour générer (72), pour chaque instant j, un vecteur Δγ_j d'une pluralité de résidus de distance de mesures de pseudo distances réalisées par les premier et deuxième récepteurs de navigation sous forme de :

(b) un moyen pour générer (73), pour chaque instant j, un vecteur Δϕ_j d'une pluralité de résidus de phase de mesures de phases totales réalisées par les premier et deuxième récepteurs de navigation sous forme de :

où Λ^-1 est une matrice diagonale comprenant les longueurs d'ondes inverses des satellites ;

(c) un moyen pour générer (74), pour l'instant j = 1, une factorisation LU d'une matrice M₁ ou d'un inverse de matrice de la matrice M₁, la matrice M₁ étant une fonction d'au moins Λ^-1 et

(d) un moyen pour générer (75), pour l'instant j = 1, un vecteur N₁ en tant que fonction d'au moins Δγ₁, Δϕ₁, et la factorisation LU de la matrice M₁ ou de l'inverse de matrice de la matrice M₁;

(e) un moyen pour générer (76), pour un instant supplémentaire j ≠ 1, une factorisation LU d'une matrice M_j ou d'un inverse de matrice de la matrice M_j, la matrice M_j étant une fonction d'au moins A^-1 et

(f) un moyen pour générer (77), pour un instant supplémentaire j ≠ 1, un vecteur N_j en tant que fonction d'au moins Δγ_j, Δϕ_j, et la factorisation LU de la matrice M_j ou de l'inverse de matrice de la matrice M_j, le vecteur N_j ayant des estimations des ambiguïtés flottantes ; et

(g) un moyen pour déterminer (78) si Nj a une précision suffisante, où les étapes (e) et (f) sont répétées si Nj n'a pas une précision suffisante.

25. Appareil de la revendication 24, dans lequel un moyen (c) comprend un moyen pour générer une factorisation LU pour une matrice comprenant une forme équivalente à (G^T P₁G), où :

• la matrice G a 2n lignes, n colonnes, une sous-matrice supérieure, et une sous-matrice inférieure, l'une des sous-matrices comprend une matrice nulle n x n et l'autre sous-matrice comprenant une matrice identité n x n,

• la matrice G^T comprend la matrice transposée de la matrice G, et

• la matrice P₁ a 2n lignes, 2n colonnes, et une forme qui comprend une matrice équivalente à

où la matrice R₁ est une matrice de pondération, où la matrice

comprend un inverse de la matrice R₁, où la matrice Q₁ a une sous-matrice supérieure et une sous-matrice inférieure, l'une des sous-matrices de Q₁ comprenant la matrice

et l'autre des sous- matrices de Q₁ comprenant le produit matriciel

et où la matrice

comprend la transposée de matrice Q₁, et où la quantité qS_K est une matrice nulle lorsque la distance entre les premier et deuxième récepteurs de navigation n'est pas limitée et où q peut être un paramètre de pondération non nul et S_k peut être une matrice non nulle lorsque la distance entre les premier et deuxième récepteurs de navigation est limitée.

26. Appareil de la revendication 25, dans lequel le moyen (d) comprend un moyen pour générer un vecteur N₁ pour comprendre un vecteur ayant une forme équivalente à

où la matrice

comprend un inverse de matrice de la matrice M₁, et où le vecteur µ₁ comprend le vecteur [Δγ₁, Δϕ₁]T, et où la quantité qg_k est un vecteur nul lorsque la distance entre les premier et deuxième récepteurs de navigation n'est pas limitée et où q peut être non nul et g_k peut être un vecteur non nul lorsque la distance entre les premier et deuxième récepteurs de navigation est limitée.

27. Appareil de la revendication 24, dans lequel un moyen (e) comprend un moyen pour générer une factorisation LU pour une matrice comprenant une forme équivalente à

où :

• M_j-1 comprend la matrice M₁ générée par un moyen (c) lorsque j = 2 et comprend la matrice M_j générée par un moyen (e) pour l'instant j-1 lorsque j > 2,

• la matrice G a 2n lignes, n colonnes, une sous-matrice supérieure, et une sous-matrice inférieure, l'une des sous-matrices comprenant une matrice nulle n x n et l'autre sous-matrice comprenant une matrice identité n x n,

• la matrice G^T comprend la matrice transposée de la matrice G, et

• la matrice P_j a 2n lignes, 2n colonnes, et une forme qui comprend une matrice équivalente à

où la matrice R_j est une matrice de pondération, où la matrice

comprend un inverse de la matrice R_j, où la matrice Q_j a une sous-matrice supérieure et une sous-matrice inférieure, l'une des sous-matrices de Q_j comprenant une matrice

et l'autre des sous-matrices de Q_j comprenant le produit matriciel

et où la matrice

comprend la transposée de la matrice Q_j, et où la quantité qS_j est une matrice nulle lorsque la distance entre les premier et deuxième récepteurs de navigation n'est pas limitée et où q peut être un paramètre de pondération non nul et S_j peut être une matrice non nulle lorsque la distance entre les premier et deuxième récepteurs de navigation est limitée.

28. Appareil de la revendication 27, dans lequel un moyen (f) comprend un moyen pour générer un vecteur N_j pour comprendre un vecteur ayant une forme équivalente à

où la matrice

comprend un inverse de matrice de la matrice M_j, où le vecteur µ_j comprend le vecteur [Δγ_j, Δϕ_j]^T, et où le vecteur N_j-1 comprend le vecteur N₁ généré par un moyen (d) lorsque j = 2 et comprend le vecteur N_j-1 généré par un moyen (f) pour l'instant j-1 lorsque j > 2, et où la quantité qg_j est un vecteur nul lorsque la distance entre les premier et deuxième récepteurs de navigation n'est pas limitée et où q peut être non nul et g_j peut être un vecteur non nul lorsque la distance entre les premier et deuxième récepteurs de navigation est limitée.

29. Appareil de la revendication 26, dans lequel la distance entre les premier et deuxième récepteurs de navigation est limitée à une distance L_RB, où un moyen (c) comprend un moyen pour générer une matrice S₁ dans une forme équivalente à :

où r₁ est un vecteur comprenant des estimations des trois coordonnées du vecteur de référence pour l'instant j = 1 et un zéro en tant que quatrième composante, où

est la transposée de vecteur de r₁, où est la matrice identité 3 par 3, où O_1x3 est un vecteur ligne de trois zéros, et où O_3x1 est un vecteur colonne de trois zéros ; et
où un moyen (d) comprend un moyen pour générer un vecteur g₁ pour l'instant j = 1 dans une forme équivalente à :

où :

30. Appareil de la revendication 28, dans lequel la distance entre les premier et deuxième récepteurs de navigation est limitée à une distance L_RB, où un moyen (e) comprend un moyen pour générer une matrice S_j dans une forme équivalente à :

où r_j est un vecteur comprenant des estimations des trois coordonnées du vecteur de référence pour le j^ième instant et un zéro en tant que quatrième composante de vecteur, où

est la transposée de vecteur de r_j, où I₃ est la matrice identité 3 par 3, où O_1x3 est un vecteur ligne de trois zéros, et où O_3x1 est un vecteur colonne de trois zéros ; et
où un moyen (f) comprend un moyen pour générer un vecteur g_j pour le j^ième instant dans une forme équivalente à :


où :

31. Appareil de la revendication 27, dans lequel un moyen (e) pour générer la factorisation LU de M_j comprend :

(g) un moyen pour générer une factorisation LU de la matrice M_j-1 dans une forme équivalente à L_j-1 L_j-1^T où L_j-1 est une matrice triangulaire inférieure et L_j-1^T est la transposée de L_j-1 ;

(h) un moyen pour générer une factorisation de G^T P_j G dans une forme équivalente à T_j T_j^T = G^T P_j G, où

est la transposée de T_j ; et

(i) un moyen pour générer une factorisation LU de la matrice M_j dans une forme équivalente à

à partir d'une pluralité n de modifications de rang un de matrice L_j-1. chaque modification de rang un étant basée sur une colonne respective de la matrice T_j, où n est le nombre de lignes dans la matrice M_j.

32. Appareil de la revendication 31, dans lequel un moyen (h) génère une matrice T_j à partir d'une factorisation de Cholesky de G^T P_jG.

33. Appareil de la revendication 31, dans lequel la matrice de pondération R_j a une forme équivalente à :

où R^γ et R^ϕ sont des matrices de pondération ;
où R^γ et R^ϕ sont liées à une matrice de pondération commune W et à des paramètres de mise à l'échelle σ_γ et σ_ϕ comme suit

et

où un moyen (h) de génération de matrice T_j comprend :

un moyen pour générer un scalaire b dans une forme équivalente à :

où

est la longueur d'onde des signaux de satellites,

un moyen pour générer une matrice H̃ dans une forme équivalente à

un moyen pour générer une matrice de Householder SHH pour la matrice H̃, et

un moyen pour générer la matrice T_j dans une forme équivalente à :

34. Appareil de la revendication 31, dans lequel la matrice de pondération R_j est appliquée à un cas où il y a un premier groupe de signaux de satellites ayant des signaux de porteuse dans une première bande de longueurs d'ondes et un deuxième groupe de signaux de satellites ayant des signaux de porteuse dans une deuxième bande de longueurs d'ondes, la fréquence de pondération ayant une forme équivalente à :

où R^γ et R^ϕ sont des matrices de pondération ;
où R^γ et R^ϕ sont liées à une matrice de pondération commune W, aux longueurs d'ondes porteuses du premier groupe de signaux comme représentées par la matrice Λ⁽¹⁾, aux longueurs d'ondes porteuses du deuxième groupe de signaux comme représentées par la matrice Λ⁽²⁾, à la longueur d'onde centrale de la première bande comme représentée par λ₁, à la longueur d'onde centrale de la première bande comme représentée par λ₂, et à des paramètres de mise à l'échelle σ_y et σ_ϕ, comme suit :



où

où l'étape (h) de génération de matrice T_j comprend les étapes qui consistent :
à générer un scalaire b dans une forme équivalente à :

où λ₁ est la longueur d'onde d'un premier groupe de signaux de satellites et λ₂ est la longueur d'onde d'un deuxième groupe de signaux de satellites,
à générer une matrice H̃ dans une forme équivalente à

à générer une matrice de Householder S_HH pour la matrice H̃, et
à générer une matrice T_j dans une forme équivalente à :

où les sous-matrices A11, A21, et A22 sont comme suit :



et

35. Appareil (100) pour estimer un ensemble d'ambiguïtés flottantes associé à un ensemble de mesures de phases d'une pluralité n de signaux de porteuse de satellites réalisées par un premier récepteur de navigation (B) et un deuxième récepteur de navigation (R) séparés par une distance, où un vecteur de référence (x⁰, y⁰, z⁰) associe la position du deuxième récepteur au premier récepteur, chaque signal de porteuse de satellite étant émis par un satellite et ayant une longueur d'onde, où chaque récepteur a une horloge temporelle pour référencer ses mesures et où une différence quelconque entre les horloges temporelles peut être représentée par un décalage, ledit appareil (100) pour recevoir (41), pour une pluralité de deux instants j ou plus, les entrées suivantes :

un vecteur

représentant une pluralité de pseudo distances mesurées par le premier récepteur de navigation (B) et correspondant à la pluralité de signaux de porteuse de satellites,

un vecteur

représentant une pluralité de pseudo distances mesurées par le deuxième récepteur de navigation (R) et correspondant à la pluralité de signaux de porteuse de satellites,

un vecteur

représentant une pluralité de distances estimées entre les satellites et le premier récepteur de navigation (B),

un vecteur

représentant une pluralité de distances estimées entre les satellites et le deuxième récepteur de navigation (R),

un vecteur

représentant une pluralité de mesures de phases totales des signaux de porteuse de satellites mesurées par le premier récepteur de navigation (B),

un vecteur

représentant une pluralité de mesures de phases totales des signaux de porteuse de satellites mesurées par le deuxième récepteur de navigation (R),

une matrice Jacobienne géométrique

dont les éléments matriciels représentent les variations des distances entre les satellites et l'un des récepteurs qui pourraient être provoquées par les variations de la position de ce récepteur et du décalage d'horloge temporelle,

ledit appareil (100) étant caractérisé par :

(a) un moyen pour générer (42), pour chaque instant j, un vecteur Δγ_j d'une pluralité de résidus de distance de mesures de pseudo distances réalisées par les premier et deuxième récepteurs de navigation sous forme de :

ledit moyen générant un ensemble de résidus de distance Δγ_k, k = 1,..j ;

(b) un moyen pour générer (43), pour chaque instant j, un vecteur Δϕ_j d'une pluralité de résidus de phase de mesures de phases totales réalisées par les premier et deuxième récepteurs de navigation sous forme de :

où Λ^-1 est une matrice diagonale comprenant les longueurs d'ondes inverses des satellites, ledit moyen générant un ensemble de résidus de phase Δϕ_k, k = 1,..,_j ;

(c) un moyen pour générer (44) une factorisation LU d'une matrice M ou d'un inverse de matrice de la matrice M, la matrice M étant une fonction d'au moins Λ^-1 et

pour un indice k de

couvrant au moins deux instants parmi les instants j ;

(d) un moyen pour générer (45) un vecteur N d'ambiguïtés flottantes estimées en tant que fonction d'au moins l'ensemble de résidus de distance Δγ_k, l'ensemble de résidus de phase Δϕ_k, et la factorisation LU de la matrice M ou de l'inverse de matrice de la matrice M.

36. Appareil de la revendication 35, dans lequel un moyen (c) comprend un moyen pour générer une matrice M dans une forme équivalente à la sommation

où:

• la matrice G a 2n lignes, n colonnes, une sous-matrice supérieure, et une sous-matrice inférieure, l'une des sous-matrices comprenant une matrice nulle n x n et l'autre sous-matrice comprenant une matrice identité n x n,

• la matrice G^T comprend la matrice transposée de la matrice G, et

• la matrice P_k a 2n lignes, 2n colonnes, et une forme qui comprend une matrice équivalente à

où la matrice R_k est une matrice de pondération, où la matrice

comprend un inverse de la matrice R_k, où la matrice Q_k a une sous-matrice supérieure et une sous-matrice inférieure, l'une des sous-matrices de Q_k comprenant une matrice

et l'autre des sous-matrices de Q_k comprenant le produit matriciel

et où la matrice

comprend la transposée de la matrice Q_k, et où la quantité qS_k est une matrice nulle lorsque la distance entre les premier et deuxième récepteurs de navigation n'est pas limitée et où q peut être un paramètre de pondération non nul et S_k peut être une matrice non nulle lorsque la distance entre les premier et deuxième récepteurs de navigation est limitée.

37. Appareil de la revendication 36, dans lequel un moyen (d) comprend un moyen pour générer une matrice N dans une forme équivalente à :

où la matrice M^-1 comprend un inverse de matrice de la matrice M, où le vecteur µ_k comprend le vecteur [Δγ_k, Δϕ_k]^T, et où la quantité qg_k est un vecteur nul lorsque la distance entre les premier et deuxième récepteurs de navigation n'est pas limitée et où q peut être non nul et g_k peut être un vecteur non nul lorsque la distance entre les premier et deuxième récepteurs de navigation est limitée.

38. Appareil de la revendication 37, dans lequel la distance entre les premier et deuxième récepteurs de navigation est limitée à une distance L_RB, où un moyen (c) comprend un moyen pour générer la matrice S_k pour le k^ième instant dans une forme équivalente à :

où r_k est un vecteur comprenant des estimations des trois coordonnées du vecteur de référence au k^ième instant, et un zéro en tant que quatrième composante, où

est la transposée de vecteur de r_k, où I₃ est la matrice identité 3 par 3, où O_1x3 est un vecteur ligne de trois zéros, et où O_3x1 est un vecteur colonne de trois zéros ; et
où le moyen (d) génère un vecteur g_k pour le k^ième instant dans une forme équivalente à :

où :

Drawing