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

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.

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.

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.

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.

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.

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.

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.

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".

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.

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.

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.

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.

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.

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.

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.

• 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.

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.

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 ${\gamma }_j^B$ 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 ${\gamma }_j^R$ 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 $D_j^B$ representative of a plurality of estimated distances between the satellites and the first navigation receiver (B), a vector $D_j^R$ representative of a plurality of estimated distances between the satellites and the second navigation receiver (R), a vector ${\varphi }_j^{\mathrm{B}}$ representative of a plurality of full phase measurements of the satellite carrier signals measured by the first navigation receiver (B), a vector ${\varphi }_j^{\mathrm{R}}$ 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 $H_j^{\gamma }$ 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.)

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.

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: $$\mathrm{\Delta }{\gamma }_j=\left({\gamma }_j^R-{\gamma }_j^B\right)-\left(D_j^R-D_j^B\right);$$ and also generate, 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: ${\mathit{\Delta \varphi }}_j=\left({\varphi }_j^{\mathrm{R}\prime }-{\varphi }_j^{\mathrm{B}}\right)-{\mathrm{\Lambda }}^{-1}\cdot \left(D_j^R-D_j^B\right),$ 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 $H_l^{\gamma }\mathrm{.}$ 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 $H_j^{\gamma }\mathrm{.}$ 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.

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 $H_k^{\gamma },$ for index k of $H_k^{\gamma }$ 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.

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.

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.

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.

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.
• ${\mathrm{H}}_k^{\gamma }$ - 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. ${\mathrm{H}}_k^{\gamma }$ has dimensions of n × 4.
• Q_k - denotes a compound matrix formed by ${\mathrm{H}}_k^{\gamma }$ and Λ as follows: ${\mathrm{Q}}_k=\left[\underset{{\mathrm{\Lambda }}^{-1}{\mathrm{H}}_k^{\gamma }}{\stackrel{{\mathrm{H}}_k^{\gamma }}{---}}\right],$ having dimensions of 2n × 4.
• G - denotes a compound matrix $\mathrm{G}=\left[\begin{array}{c}{\mathrm{O}}_{\mathit{n},\mathit{n}}\\ {\mathrm{I}}_{\mathit{n}}\end{array}\right]$ 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 ${\mathbf{G}}^{\mathbf{T}}\mathbf{CG}={\mathbf{C}}_{\mathbf{22}},\mathrm{where}\mathbf{C}=\left[\begin{array}{cc}{\mathbf{C}}_{11}& {\mathbf{C}}_{12}\\ {\mathbf{C}}_{21}& {\mathbf{C}}_{22}\end{array}\right]\mathrm{.}$
• ${\mathrm{H}}_j^{\mathit{\mu }}$ - denotes a compound matrix ${\mathrm{H}}_j^{\mathit{\mu }}=\left[{\mathrm{Q}}_{\mathrm{j}}|\mathrm{G}\right],$ which is a 2n × (4+n) matrix, ∥·∥_F-1 denotes F^-1 - weighted norm, ${\Vert \cdot \Vert }_{F^{-1}}\mathrm{denotes}{\mathbf{F}}^{-1}-\mathrm{weighted\; norm},{\Vert \mathbf{X}\Vert }_{F^{-1}}^2={\mathbf{X}}^T{\mathbf{F}}^{-1}\mathbf{X}\mathrm{.}$

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.

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.

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.

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.

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.

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.

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: $$\begin{array}{ll}{\phi }^S\left(t\right)& ={\mathrm{\Phi }}_{\mathit{0}}^S+\underset{0}{\overset{t}{\int }}{f}_S\cdot \mathit{dt}\\ & ={\mathrm{\Phi }}_{\mathit{0}}^S+f_S\cdot \mathit{dt}\end{array}$$ where ${\mathrm{\Phi }}_{\mathit{0}}^S$ 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: $$\begin{array}{ll}{{\phi }^S}_A\left(t\right)& ={\phi }^S\left(t-D\left(t\right)-{\tau }_{\mathit{ATM}}\left(t\right)\right)\\ & ={\mathrm{\Phi }}_{\mathit{0}}^S+f_S\cdot \left(t-D\left(t\right)-{\tau }_{\mathit{ATM}}\left(t\right)\right)\\ & ={\mathrm{\Phi }}_{\mathit{0}}^S+f_S\cdot \mathit{t}-f_S\cdot D\left(t\right)/\mathit{c}-f_S\cdot {\tau }_{\mathit{ATM}}\left(t\right),\end{array}$$ 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: $${{\phi }^S}_{A,R}\left(t\right)-{{\phi }^S}_{A,B}\left(t\right)=f_S\cdot D_R\left(t\right)/c-f_S\cdot D_B\left(t\right)/c\mathrm{.}$$

The terms ${\mathrm{\Phi }}_{\mathit{0}}^S$ 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.

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.

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: $${\phi }_j\left(T_j\right)=d_{p,\mathit{nom}}\cdot \left(T_j-T_p\right)-{\phi }_{\mathit{jNCO}}\left(T_j\right)$$ 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).

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.

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: $${\phi }_j\left(T_j\right)=f_S\cdot D_j/c+f_S\cdot \left({\tau }_j-T_p\right)+\left({\phi }_0^{ʹ}-{\mathrm{\Phi }}_{\mathit{0}}^S\right)-N_{\mathit{nco}}-{\zeta }_{\mathit{\phi j}}$$ 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,
• ${\mathrm{\Phi }}_{\mathit{0}}^S$ 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.

$${\phi }_j\left(T_j\right)=f_S\cdot D_j/c+f_S\cdot {\tau }_j+\left({\phi }_0^{ʹ}-{\mathrm{\Phi }}_{\mathit{0}}^S\right)+N-{\zeta }_{\mathit{\phi j}}$$

$\left({\phi }_0^{\prime }-{\mathrm{\Phi }}_{\mathit{0}}^S\right)$

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. $${\phi }_{j,m}^B\left(T_j\right)=f_S\cdot D_{j,m}^B/c+f_S\cdot {\tau }_j^B+\left({\phi }_0^{\mathit{ʹB}}-{\mathrm{\Phi }}_{\mathit{0}}^S\right)+{N_m}^B-{\zeta }_{\mathit{\phi j},m}^B$$ $${\phi }_{j,m}^R\left(T_j\right)=f_S\cdot D_{j,m}^R/c+f_S\cdot {\tau }_j^R+\left({\phi }_0^{\mathit{ʹR}}-{\mathrm{\Phi }}_{\mathit{0}}^S\right)+{N_m}^R-{\zeta }_{\mathit{\phi j},m}^R$$ For the differential navigation mode, the difference of these phases is formed: $${\phi }_{j,m}^B\left(T_j\right)-{\phi }_{j,m}^R\left(T_j\right)=f_S\cdot D_{j,m}^B/c-f_S\cdot D_{j,m}^R/c+f_S\cdot \left({\tau }_j^B-{\tau }_j^R\right)+\left({N_m}^B-{N_m}^R\right)+\left({\phi }_0^{\mathit{ʹB}}-{\phi }_0^{\mathit{ʹR}}\right)-\left({\zeta }_{\mathit{\phi j},m}^B-{\zeta }_{\mathit{\phi j},m}^R\right)$$ 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: $${\phi }_{j,m}^B\left(T_j\right)-{\phi }_{j,m}^R\left(T_j\right)=\left(1/{\lambda }_m\right)\cdot \left(D_{j,m}^B-D_{j,m}^R\right)+f_S\cdot \left({\tau }_j^B-{\tau }_j^R\right)+N^m+\left({\phi }_0^{\mathit{ʹB}}-{\phi }_0^{\mathit{ʹR}}\right)-\left({\zeta }_{\mathit{\phi j},m}^B-{\zeta }_{\mathit{\phi j},m}^R\right)$$ The values for ${\phi }_{\mathit{0}}^{\prime B}$ and ${\phi }_{\mathit{0}}^{\prime R}$ can be readily determined. The values of ${\zeta }_{\mathit{\phi j},}^B$and ${\zeta }_{\mathit{\phi j},m}^R$ cannot be determined, but they have zero mean values (and should average to zero over time).

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 ${\phi }_{j,}^B\left(T_j\right)-{\phi }_j^R$ ,_m(T_j) , which we can measure, minus the cycle ambiguity N^m, minus the error in the clocks $f_s\cdot \left({\tau }_j^B-{\tau }_j^R\right),$ minus the phase offsets $\left({\phi }_{\mathit{0}}^{\prime B}-{\phi }_{\mathit{0}}^{\prime R}\right),$ and minus the errors $({\zeta }_{\mathit{\phi j},m}^B-$ ${\zeta }_{\mathit{\phi j},m}^R),$ 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 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: $${\left({\left(x_R-x_B\right)}^2+{\left(y_R-y_B\right)}^2+{\left(z_R-z_B\right)}^2\right)}^{1/2}=L_{\mathit{RB}}$$ 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: $$F\left(x_R-x_B,y_R-y_B,z_R-z_B\right)=\frac{1}{2}{\left({\left({\left(x_R-x_B\right)}^2+{\left(y_R-y_B\right)}^2+{\left(z_R-z_B\right)}^2\right)}^{1/2}-L_{\mathit{RB}}\right)}^2$$ 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: $$F\left(*\right)\approx 0,\frac{\partial F}{\partial x_R}\approx 0,\frac{\partial F}{\partial y_R}\approx 0,\frac{\partial F}{\partial z_R}\approx 0$$ 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.

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

1. 1. Generation of the floating ambiguity estimations (estimating the floating ambiguities),
2. 2. Generation of the integer ambiguities, and
3. 3. Formation of a signal of integer ambiguities resolution.

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:

1. (a) the pseudo-ranges and full phase measurements obtained in the Base and Rover receivers;
2. (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);
3. (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;
4. (d) optionally, a set of weight coefficients characterizing the accuracy of the measurements; and
5. (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.

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.

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.

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.

We start by rewriting equation (10), which has the integer ambiguities N^m in vector form: $${\mathit{\phi }}_{\mathit{j}}^{\mathit{B}}\left(T_i\right)-{\mathit{\phi }}_{\mathit{j}}^{\mathit{R}}\left(T_i\right)={\mathbf{\Lambda }}^{-1}\cdot \left({\mathit{D}}_{\mathit{j}}^{\mathit{B}}-{\mathit{D}}_{\mathit{j}}^{\mathit{R}}\right)+f_S\cdot \left({\mathit{\tau }}_{\mathit{j}}^{\mathit{B}}-{\mathit{\tau }}_{\mathit{j}}^{\mathit{R}}\right)+N+\left({\mathit{\phi }}_{\mathit{0}}^{\mathit{ʹB}}-{\mathit{\phi }}_{\mathit{0}}^{\mathit{ʹR}}\right)-\left({\mathit{\zeta }}_{\mathit{\phi j}}^{\mathit{B}}-{\mathit{\zeta }}_{\mathit{\phi j}}^{\mathit{R}}\right),$$ where Λ^-1 is a diagonal matrix of inverse wavelengths. We note that the components of $\left({\phi }_{\mathit{0}}^{\prime B}-{\phi }_{\mathit{0}}^{\prime R}\right)$ 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: $${\mathit{\phi }}_{\mathit{j}}^{\mathit{B}}\left(T_j\right)-{\mathit{\phi }}_{\mathit{j}}^{\mathit{R}}\left(T_j\right)={\mathbf{\Lambda }}^{-1}\cdot \left({\mathit{D}}_{\mathit{j}}^{\mathit{B}}-{\mathit{D}}_{\mathit{j}}^{\mathit{R}}\right)+f_S\cdot \left({\mathit{\tau }}_{\mathit{j}}^{\mathit{B}}-{\mathit{\tau }}_{\mathit{j}}^{\mathit{R}}\right)+N-\left({\mathit{\zeta }}_{\mathit{\phi j}}^{\mathit{B}}-{\mathit{\zeta }}_{\mathit{\phi j}}^{\mathit{R}}\right)$$ After this modified N are found, the "true" N may be found by subtracting $\left({\phi }_{\mathit{0}}^{\prime B}-{\phi }_{\mathit{0}}^{\prime R}\right)$ from the modified N.

As an alternative equivalent to the above, the vector $\left({\phi }_{\mathit{0}}^{\prime B}-{\phi }_{\mathit{0}}^{\prime R}\right)$ can be computed and carried over to the left-hand side of equation (14) and subtracted from $\left({\phi }_j^B\left(T_i\right)-{\phi }_j^R\left(T_i\right)\right)\mathrm{.}$ 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.

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 ${\tau }_j^B,{\tau }_j^R,{\zeta }_{\mathit{\phi j}}^B,$ and ${\zeta }_{\mathit{\phi j}}^R,$ and we have some errors in the range vectors $D_j^B$ and $D_j^R\mathrm{.}$ Our approach is to represent (model) the errors in the range vectors $D_j^B$ and $D_j^R$ and the terms $f_S\cdot \left({\tau }_j^B-{\tau }_j^R\right)-\left({\zeta }_{\mathit{\phi j}}^B-{\zeta }_{\mathit{\phi j}}^R\right)$ by the following error vector: Λ^-1. $H_j^{\gamma }\cdot {\left[\mathit{\Delta x},\mathit{\Delta y},\mathit{\Delta z},c\cdot \mathit{\Delta \tau }\right]}^{\mathrm{T}},$ where $H_j^{\gamma }$ 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: $$\left({\varphi }_j^{\mathrm{R}}-{\varphi }_j^{\mathrm{B}}\right)-{\mathbf{\Lambda }}^{-1}\left({\mathit{D}}_{\mathit{j}}^{\mathit{R}}-{\mathit{D}}_{\mathit{j}}^{\mathit{R}}\right)=\mathit{N}+{\mathbf{\Lambda }}^{-1}\cdot {\mathit{H}}_j^{\gamma }\cdot {\left[\mathit{\Delta x},\mathit{\Delta y},\mathit{\Delta z},c\cdot \mathit{\Delta \tau }\right]}^{\mathrm{T}}$$ The pseudoranges will be used to estimate the vector $H_j^{\gamma }\cdot {\left[\mathit{\Delta x},\mathit{\Delta y},\mathit{\Delta z},c\cdot \mathit{\Delta \tau }\right]}^{\mathrm{T}}$ as follows: $$\left({\gamma }_j^R-{\gamma }_j^B\right)-{\mathbf{\Lambda }}^{-1}\left({\mathit{D}}_{\mathit{j}}^{\mathit{R}}-{\mathit{D}}_{\mathit{j}}^{\mathit{B}}\right)={\mathit{H}}_j^{\mathbf{\gamma }}\cdot {\left[\mathit{\Delta x},\mathit{\Delta y},\mathit{\Delta z},c\cdot \mathit{\Delta \tau }\right]}^{\mathrm{T}}$$ 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.

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: $$\begin{array}{l}\mathbf{\Delta }{\mathit{\gamma }}_{\mathit{j}}={\mathit{\gamma }}_{\mathit{j}}-{\mathit{D}}_{\mathit{j}}\\ \mathrm{with}{\mathit{\gamma }}_{\mathit{j}}={\mathit{\gamma }}_{\mathit{j}}^{\mathit{R}}-{\mathit{\gamma }}_{\mathit{j}}^{\mathit{B}},\mathrm{and}{\mathit{D}}_{\mathit{j}}={\mathit{D}}_{\mathit{j}}^{\mathit{R}}-{\mathit{D}}_{\mathit{j}}^{\mathit{B}};\end{array}$$ where ${\gamma }_j^R$ and ${\gamma }_j^B$are vectors containing the pseudo-ranges of the satellites, measured in the Base and Rover receivers, respectively; and $D_j^R$ and $D_j^B$ are vectors of the estimated ranges of the satellites from Base and Rover stations at the moment of j-th signal radiation $D_j^R$ and $D_j^B$ 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.

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: $$\mathit{\Delta }{\mathit{\varphi }}_{\mathit{j}}={\mathit{\varphi }}_{\mathit{j}}-{\mathbf{\Lambda }}^{-1}\cdot {\mathit{D}}_{\mathit{j}}$$ where

• ${\varphi }_j={{\varphi }_j}^{\mathrm{R}}-{{\varphi }_j}^{\mathrm{B}};$
• ${\varphi }_j^{\mathrm{R}}$ and ${{\varphi }_j}^{\mathrm{B}}$ 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): $${{\varphi }^B}_j\left(T_j\right)=f_{p,\mathit{nom}}\cdot \left(T_j-{T_p}^B\right)-{{\phi }^B}_{j,\mathit{NCO}}\left(T_j\right),$$ $${{\varphi }^B}_j\left(T_j\right)=f_{p,\mathit{nom}}\cdot \left(T_j-{T_p}^B\right)-{{\phi }^B}_{j,\mathit{NCO}}\left(T_j\right),$$ or may be constructed in the form which includes the phase offsets of the base receiver: $${{\varphi }^B}_j\left(T_j\right)=\left[f_{p,\mathit{nom}}\cdot \left(T_j-{T_p}^B\right)-{{\phi }^B}_{j,\mathit{NCO}}\left(T_j\right)\right]-{\mathit{\phi }}_{\mathit{0}}^{\mathit{ʹB}},$$ $${{\varphi }^R}_j\left(T_j\right)=\left[f_{p,\mathit{nom}}\cdot \left(T_j-{T_p}^R\right)-{{\phi }^R}_{j,\mathit{NCO}}\left(T_j\right)\right]-{\mathit{\phi }}_{\mathit{0}}^{\mathit{ʹR}}\mathrm{.}$$

The forms represented by Equations (15)-(18) maybe represented in matrix form equivalent to: $${\mu }_j={\mathit{H}}_j^{\mu }\cdot {\mathit{A}}_j,$$ where vector A_j is a state vector related to observation matrix µ_j and where matrix ${\mathit{H}}_{\mathit{j}}^{\mathit{\mu }}$ 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 ${\mathit{H}}_{\mathit{j}}^{\mathit{\mu }}$ comprises 2n rows and (4+n) columns, and may be divided into the following 4 parts (sub-matrices): $${\mathit{H}}_j^{\mu }=\left[\begin{array}{cc}{\mathit{H}}_j^{\gamma }& {\mathbf{0}}_{n\times n}\\ {\mathit{\Lambda }}^{-1}\cdot {\mathit{H}}_j^{\mathrm{\gamma }}& {\mathit{I}}_n\end{array}\right]\mathrm{.}$$ 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 ${\mathit{H}}_{\mathit{j}}^{\gamma }$ 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: $$\left[{\mathit{\alpha }}_{\mathit{jn}},{\mathit{\beta }}_{\mathit{jn}},{\mathit{h}}_{\mathit{jn}},\mathit{1}\right],$$ 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.

The second part of matrix ${\mathit{H}}_{\mathit{j}}^{\mathit{\mu }},$ the left lower comer (the first four columns by the last n rows), is occupied by the matrix product ${\mathrm{\Lambda }}^{-1}\cdot {\mathit{H}}_{\mathit{j}}^{\mathit{\gamma }}$ 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: $$\left[{\mathit{\alpha }}_{\mathit{jn}}/{\lambda }_n,{\mathit{\beta }}_{\mathit{jn}}/{\lambda }_n,{\mathit{h}}_{\mathit{jn}}/{\lambda }_n,\mathit{1}/{\lambda }_n\right],$$ where λ_n is the wavelength of the carrier signal in the n-th channel.

The third part of matrix ${\mathit{H}}_{\mathit{j}}^{\mathit{\mu }},$ 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 ${\mathit{H}}_{\mathit{j}}^{\mathit{\mu }}$ as follows ${\mathrm{H}}_{\mathit{j}}^{\mathit{\mu }}=\left[{\mathrm{Q}}_{\mathrm{j}}|\mathrm{G}\right],$ where Q_j is a compound matrix formed by ${\mathrm{H}}_{\mathit{j}}^{\gamma }$ and Λ as follows: $${\mathbf{Q}}_j=\left[\begin{array}{c}{\mathbf{H}}_j^{\gamma }\\ {\mathrm{\Lambda }}^{-1}{\mathbf{H}}_j^{\gamma }\end{array}\right]\left(\mathrm{having\; dimensions\; of}2n\times 4\right),$$ and where G is a compound matrix the zero matrix O_n×n and the identity matrix I_n as follows: $$\mathbf{G}=\left[\begin{array}{c}{\mathbf{O}}_{n\times n}\\ {\mathbf{I}}_n\end{array}\right]\left(\mathrm{having\; dimensions\; of}2n\times n\right)\mathrm{.}$$ 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.

The cost function F(*) may be expressed in the following form: $$F\left({\mathbf{r}}_j+{\mathbf{a}}_j\right)=\frac{1}{2}{\left({\left({\left(x_j°+\mathrm{\Delta }{x}_j\right)}^2+{\left(y_j°+\mathrm{\Delta }{y}_j\right)}^2+{\left(z_j°-\mathrm{\Delta }{z}_j\right)}^2\right)}^{1/2}-L_{\mathit{RB}}\right)}^2$$ where: ${\mathbf{a}}_j=\left(\begin{array}{c}{\mathrm{\Delta }\mathit{x}}_j\\ {\mathrm{\Delta }\mathit{y}}_j\\ {\mathrm{\Delta }\mathit{z}}_j\\ c\cdot {\mathrm{\Delta }\mathit{\tau }}_j\end{array}\right)$ is a vector of the first four components of A_j. and ${\mathbf{r}}_j=\left(\begin{array}{c}{x_j}^o\\ {y_j}^o\\ {z_j}^o\\ 0\end{array}\right)$ 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: $$F\left({\mathbf{r}}_j+{\mathbf{a}}_j\right)-F\left({\mathbf{r}}_j\right)\approx {{\mathbf{a}}_j}^T\frac{\partial F}{\partial r}+\frac{1}{2}{{\mathbf{a}}_j}^T\frac{{\partial }^{2}F}{\partial r^2}{\mathbf{a}}_j={{\mathbf{a}}_j}^T{\mathbf{h}}_j+\frac{1}{2}{{\mathbf{a}}_j}^T{\mathbf{S}}_j{\mathbf{a}}_j$$ where $${\mathbf{h}}_j=\left(1-\frac{L_{\mathit{RB}}}{\Vert {\mathbf{r}}_j\Vert }\right){\mathbf{r}}_j,$$ $${\mathbf{S}}_j=\left(1-\frac{L_{\mathit{RB}}}{\Vert {\mathbf{r}}_j\Vert }\right)\left(\begin{array}{cc}{\mathbf{I}}_3& {\mathbf{O}}_{3\times 1}\\ {\mathbf{O}}_{1\times 3}& 0\end{array}\right)+\frac{L_{\mathit{RB}}}{\Vert {\mathbf{r}}_j\Vert }{\mathbf{r}}_j{{\mathbf{r}}_j}^T\mathrm{.}$$ 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.

Measurements covariance matrix R_J is preferably formed in the following way on the basis of weight coefficients obtained in Base and Rover receivers: $${\mathit{R}}_{\mathit{j}}=\left[\begin{array}{cc}{\mathit{R}}_{\mathit{j}}^{\mathit{\gamma }}& {\mathbf{0}}_{n\times n}\\ {\mathbf{0}}_{n\times n}& {\mathit{R}}_{\mathit{j}}^{\mathit{\varphi }}\end{array}\right],$$ where $${\mathit{R}}_{\mathit{j}}^{\mathit{\gamma }}=\left[\begin{array}{cccc}{\left(K_{j,1}^{\gamma }\right)}^{-1}& 0& \cdots & 0\\ 0& {\left(K_{j,2}^{\gamma }\right)}^{-1}& \cdots & 0\\ ⋮& ⋮& \ddots & ⋮\\ 0& 0& \cdots & {\left(K_{j,n}^{\gamma }\right)}^{-1}\end{array}\right],$$ and $${\mathit{R}}_{\mathit{j}}^{\mathit{\varphi }}=\left[\begin{array}{cccc}{\left(K_{j,1}^{\mathit{\varphi }}\right)}^{-1}& 0& \cdots & 0\\ 0& {\left(K_{j,2}^{\mathit{\varphi }}\right)}^{-1}& \cdots & 0\\ ⋮& ⋮& \ddots & ⋮\\ 0& 0& \cdots & {\left(K_{j,n}^{\mathit{\varphi }}\right)}^{-1}\end{array}\right]\mathrm{.}$$ The weight coefficients $\left(K_{j1}^{\gamma },K_{j2}^{\gamma },\dots ,K_{\mathit{jn}}^{\gamma }\right)$ 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 $K_j^{\mathit{\gamma B}}$ and $K_j^{\mathit{\gamma R}}\mathrm{.}$

Thus, for example, for the n-th channel: $${\left(K_{\mathit{jn}}^{\gamma }\right)}^{-1}={\left(K_{\mathit{jn}}^{\mathit{\gamma B}}\right)}^{-1}+{\left(K_{\mathit{jn}}^{\mathit{\gamma B}}\right)}^{-1},$$ where $K_j^{\mathit{\gamma B}}$ and $K_j^{\mathit{\gamma R}}$ 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), $$K_{\mathit{j},\mathit{m}}^{\gamma }=Z_{k,m}^{\mathit{2}}\cdot \sin \left({\xi }_{k,m}-{\xi }_{\mathit{min}}\right)\cdot {{\mathrm{\sigma }}_{\mathrm{\gamma }}}^2\mathrm{when}{\xi }_{k,m}>{\xi }_{\mathit{min}},$$ $$K_{\mathit{j},\mathit{m}}^{\gamma }=0\mathrm{when}{\xi }_{k,m}\le {\xi }_{\mathit{min}},$$ where $Z_{k,m}^2$ 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 $Z_{k,m}^2\cdot \sin \left({\xi }_{k,m}-{\xi }_{\mathit{min}}\right)$ is dimensionless.

Weight coefficients $K_{j1}^{\varphi },K_{j2}^{\varphi },\dots ,K_{\mathit{jn}}^{\varphi }$ 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), $$K_{\mathit{j},\mathit{m}}^{\varphi }=Z_{k,m}^{\mathit{2}}\cdot \sin \left({\xi }_{k,m}-{\xi }_{\mathit{min}}\right)\cdot {{\mathrm{\sigma }}_{\mathrm{\varphi }}}^2\mathrm{when}{\xi }_{k,m}>{\xi }_{\mathit{min}},$$ and $$K_{\mathit{j},\mathit{m}}^{\varphi }=0\mathrm{when}{\xi }_{k,m}\le {\xi }_{\mathit{min}},$$ where $Z_{k,m}^2,{\xi }_{k,m}$ and ξ^B_min are as they are above, and where σ_ϕ² is the variance of the code measurements (σ_ϕ ≈ 1 mm).

When the magnitudes of either of weight coefficients $K_{j,m}^{\mathit{\gamma B}}$ or $K_{j,m}^{\mathit{\gamma R}}$ is less than a first selected small threshold value, the value of $K_{j,m}^{\gamma }$ is generated as a first small number which is less than the first threshold value. This is equivalent to setting ${\left({K^{\mathit{\gamma }}}_{j,m}\right)}^{-1}$ to a large number equal to the inverse of the first small number. Similarly, when the magnitudes of either of weight coefficients $K_{j,m}^{\mathit{\varphi B}}$ or $K_{j,m}^{\mathit{\varphi R}}$ is less than a second selected small threshold value, the value of $K_{j,m}^{\varphi }$ is generated as a second small number which is less than the second threshold value. This is equivalent to setting ${\left(K_{j,m}^{\varphi }\right)}^{-1}$ to a large number equal to the inverse of the second small number.

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 ${\left(K_j^{\gamma }\right)}^{-1}$ or ${\left(K_j^{\varphi }\right)}^{-1}$ 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.

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: $${\mathit{R}}_{\mathit{j}}=\left[\begin{array}{cc}\frac{1}{{\sigma }_{\gamma }^2}\cdot {\mathbf{I}}_{n\times n}& {\mathbf{0}}_{n\times n}\\ {\mathbf{0}}_{n\times n}& \frac{1}{{\sigma }_{\varphi }^2}\cdot {\mathbf{I}}_{n\times n}\end{array}\right]\mathrm{.}$$ 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.

Summarizing equation (19), the linearized measurement model at the j-th epoch takes the form: $${\mathbf{H}}_{\mathbf{j}}^{\mathbf{\mu }}{\mathbf{A}}_{\mathbf{j}}={\mathbf{\mu }}_{\mathbf{j}},{\mathbf{A}}_{\mathbf{j}}=\left(\begin{array}{c}\begin{array}{c}\mathrm{\Delta }{x}_j\\ \mathrm{\Delta }{y}_j\\ \mathrm{\Delta }{z}_j\\ c\cdot \mathrm{\Delta }{\tau }_j\end{array}\\ {\mathbf{N}}_{\mathbf{j}}\end{array}\right)=\left(\begin{array}{c}{\mathbf{a}}_{\mathbf{j}}\\ {\mathbf{N}}_{\mathbf{j}}\end{array}\right)$$ 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.

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): $$\begin{array}{l}J=\frac{1}{2}{\displaystyle \sum _{k=1}^j}{\Vert {\mathbf{H}}_k^{\mu }{\mathbf{A}}_k-{\mathbf{\mu }}_k\Vert }_{{\mathbf{R}}_{\mathbf{k}}^{-\mathbf{1}}}^2+q{\displaystyle \sum _{k=1}^j}{F}_k\\ =\frac{1}{2}{\displaystyle \sum _{k=1}^j}{\Vert {\mathbf{Q}}_k{\mathbf{a}}_k+\mathbf{G}\cdot \mathbf{N}-{\mathbf{\mu }}_k\Vert }_{{\mathbf{R}}_{\mathbf{k}}^{-\mathbf{1}}}^2+q{\displaystyle \sum _{k=1}^j}\left({{\mathbf{a}}_k}^T{\mathbf{h}}_{\mathbf{k}}+\frac{1}{2}{{\mathbf{a}}_k}^T{\mathbf{S}}_{\mathbf{k}}{\mathbf{a}}_k\right)\to \min \end{array}$$ 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 ${\Vert {{\mathrm{H}}_k}^{\mathit{\mu }}\cdot {\mathrm{A}}_k-{\mathit{\mu }}_k\Vert }^2$ 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.

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. $$\left[\begin{array}{cc}\begin{array}{cccc}{\mathbf{Q}}_1^{\mathbf{T}}{\mathbf{R}}_1^{-1}{\mathbf{Q}}_1+q{\mathbf{S}}_1& \mathbf{O}& \cdots & \mathbf{O}\\ \mathbf{O}& {\mathbf{Q}}_2^{\mathbf{T}}{\mathbf{R}}_2^{-1}{\mathbf{Q}}_2+q{\mathbf{S}}_2& \cdots & \mathbf{O}\\ ⋮& ⋮& \ddots & ⋮\\ \mathbf{O}& \mathbf{O}& \cdots & {\mathbf{Q}}_{\mathrm{j}}^{\mathbf{T}}{\mathbf{R}}_{\mathrm{j}}^{-1}{\mathbf{Q}}_{\mathrm{j}}+q{\mathbf{S}}_{\mathrm{j}}\end{array}& \begin{array}{c}{\mathbf{Q}}_1^{\mathbf{T}}{\mathbf{R}}_2^{-1}\mathbf{G}\\ {\mathbf{Q}}_2^{\mathbf{T}}{\mathbf{R}}_2^{-1}\mathbf{G}\\ ⋮\\ {\mathbf{Q}}_{\mathrm{j}}^{\mathbf{T}}{\mathbf{R}}_{\mathrm{j}}^{-1}\mathbf{G}\end{array}\\ \begin{array}{cccc}{\mathbf{G}}^{\mathbf{T}}{\mathbf{R}}_1^{-1}{\mathbf{Q}}_1& {\mathbf{G}}^{\mathbf{T}}{\mathbf{R}}_1^{-1}{\mathbf{Q}}_1& \dots & {\mathbf{G}}^{\mathbf{T}}{\mathbf{R}}_1^{-1}{\mathbf{Q}}_1\end{array}& {\displaystyle \sum _{k=1}^j}{\mathbf{G}}^{\mathbf{T}}{\mathbf{R}}_1^{-1}\mathbf{G}\end{array}\right]\left[\begin{array}{c}\begin{array}{c}{\mathbf{a}}_1\\ {\mathbf{a}}_2\\ ⋮\\ {\mathbf{a}}_{\mathrm{j}}\end{array}\\ \mathbf{N}\end{array}\right]=\left[\begin{array}{c}\begin{array}{c}{\mathbf{Q}}_2^{\mathbf{T}}{\mathbf{R}}_2^{-1}{\mathbf{\mu }}_1-q{\mathbf{h}}_1\\ {\mathbf{Q}}_2^{\mathbf{T}}{\mathbf{R}}_2^{-1}{\mathbf{\mu }}_2-q{\mathbf{h}}_2\\ ⋮\\ {\mathbf{Q}}_{\mathrm{j}}^{\mathbf{T}}{\mathbf{R}}_{\mathrm{j}}^{-1}{\mathbf{\mu }}_k-q{\mathbf{h}}_{\mathrm{j}}\end{array}\\ {\displaystyle \sum _{k=1}^j}{\mathbf{G}}^{\mathbf{T}}{\mathbf{R}}_k^{-1}{\mathbf{\mu }}_k\end{array}\right]$$ 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.

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: $$\mathbf{N}={\left[{\displaystyle \sum _{k=1}^j}\left({\mathbf{G}}^T{\mathbf{R}}_k^{-1}\mathbf{G}-{\mathbf{G}}^T{\mathbf{R}}_k^{-1}{\mathbf{Q}}_k{\left({\mathbf{Q}}_k^{\mathbf{T}}{\mathbf{R}}_k^{-1}{\mathbf{Q}}_k+q{\mathbf{S}}_k\right)}^{-\mathbf{1}}{\mathbf{Q}}_k^{\mathbf{T}}{\mathbf{R}}_k^{-1}\mathbf{G}\right)\right]}^{-1}\times \left[{\displaystyle \sum _{k=1}^j}\left({\mathbf{G}}^T{\mathbf{R}}_k^{-1}{\mathbf{\mu }}_k-{\mathbf{G}}^T{\mathbf{R}}_k^{-1}{\mathbf{Q}}_k{\left({\mathbf{Q}}_k^{\mathbf{T}}{\mathbf{R}}_k^{-1}{\mathbf{Q}}_k+q{\mathbf{S}}_k\right)}^{-\mathbf{1}}\left({\mathbf{Q}}_k^{\mathbf{T}}{\mathbf{R}}_k^{-1}{\mathbf{\mu }}_k-q{\mathbf{h}}_k\right)\right)\right],$$ 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: $${\mathbf{P}}_k={\mathbf{R}}_k^{-1}-{\mathbf{R}}_k^{-1}{\mathbf{Q}}_k{\left({\mathbf{Q}}_k^{\mathbf{T}}{\mathbf{R}}_k^{-1}{\mathbf{Q}}_k+q{\mathbf{S}}_k\right)}^{-\mathbf{1}}{\mathbf{Q}}_k^{\mathbf{T}}{\mathbf{R}}_k^{-1}$$ and a vector: $${\mathbf{g}}_k={\mathbf{G}}^T{\mathbf{R}}_k^{-1}{\mathbf{Q}}_k{\left({\mathbf{Q}}_k^{\mathbf{T}}{\mathbf{R}}_k^{-1}{\mathbf{Q}}_k+q{\mathbf{S}}_k\right)}^{-\mathbf{1}}{\mathbf{h}}_k$$

With matrix P_k, equation (29) may now be written as: $$\mathbf{N}={\left[{\displaystyle \sum _{k=1}^j}\left({\mathbf{G}}^T{\mathbf{P}}_k\mathbf{G}\right)\right]}^{-1}\times \left[{\displaystyle \sum _{k=1}^j}\left({\mathbf{G}}^{\mathbf{T}}{\mathbf{P}}_k{\mathbf{\mu }}_k+q{\mathbf{g}}_k\right)\right]$$ We refer to P_k as a projection-like matrix for q=0 and a quasi-projection matrix for q>0. Each component matrix ${\mathrm{G}}^{\mathrm{T}}{P}_k\mathrm{G\; of\; M}=\left[{\displaystyle \sum _{k=1}^j}\left({\mathbf{G}}^{\mathbf{T}}{\mathrm{P}}_k\mathrm{G}\right)\right]$ 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.

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: $${\mathbf{a}}_{\mathbf{k}}={\left({{\mathbf{Q}}_{\mathbf{k}}}^{\mathbf{T}}{{\mathbf{R}}_{\mathbf{k}}}^{-1}{\mathbf{Q}}_{\mathbf{k}}+q{\mathbf{S}}_k\right)}^{-\mathbf{1}}\left({{\mathbf{Q}}_{\mathbf{k}}}^{\mathbf{T}}{{\mathbf{R}}_{\mathbf{k}}}^{-1}\left({\mathbf{\mu }}_{\mathbf{k}}-\mathbf{GN}\right)-q{\mathbf{g}}_k\right)\mathrm{.}$$ 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.

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: $${\mathbf{a}}_{\mathbf{k}}={\left({\mathbf{Q}}_{\mathbf{k}}^{\mathbf{T}}{\mathbf{R}}_{\mathbf{k}}^{-\mathbf{1}}{\mathbf{Q}}_{\mathbf{k}}+q{\mathbf{S}}_{k-1}\right)}^{-\mathbf{1}}\left({\mathbf{Q}}_{\mathbf{k}}^{\mathbf{T}}{\mathbf{R}}_{\mathbf{k}}^{-\mathbf{1}}\left({\mathbf{\mu }}_{\mathbf{k}}-\mathbf{GNʹ}\right)-q{\mathbf{g}}_{k-1}\right)\mathrm{.}$$

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.

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: $${\mathbf{M}}_l=\left[{\displaystyle \sum _{k=1}^l}\left({\mathbf{G}}^{\mathbf{T}}{\mathbf{P}}_k\mathbf{G}\right)\right]$$ $${\mathbf{B}}_l=\left[{\displaystyle \sum _{k=1}^l}\left({\mathbf{G}}^{\mathbf{T}}{\mathbf{P}}_k{\mathbf{\mu }}_k+q{\mathbf{g}}_k\right)\right],$$ 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: $${\mathbf{M}}_{l+\mathit{1}}={\mathbf{M}}_l+{\mathbf{G}}^{\mathbf{T}}{\mathbf{P}}_{l+1}\mathbf{G}$$ $${\mathbf{B}}_{l+\mathit{1}}={\mathbf{B}}_l+{\mathbf{G}}^{\mathbf{T}}{\mathbf{P}}_{l+1}{\mathbf{\mu }}_{l+1}+q{\mathbf{g}}_{l+1},$$ 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: $${\mathbf{M}}_m={\mathbf{M}}_l+\left[{\displaystyle \sum _{k=l+1}^{k=m}}{\mathbf{G}}^{\mathbf{T}}{\mathbf{P}}_k\mathbf{G}\right]$$ $${\mathbf{B}}_m={\mathbf{B}}_l+\left[{\displaystyle \sum _{k=l+1}^{k=m}}\left({\mathbf{G}}^{\mathbf{T}}{\mathbf{P}}_k{\mathbf{\mu }}_k+q{\mathbf{g}}_k\right)\right],$$ 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.

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., ${\gamma }_j^B,{\gamma }_j^R,D_j^B,D_j^R,{\varphi }_j^{\mathrm{B}},{\varphi }_j^{\mathrm{R}},H_j^{\gamma }$ 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: ${\mathrm{\Delta }\mathit{\gamma }}_j=\left({\gamma }_j^R-{\gamma }_j^B\right)-\left(D_j^R-D_j^B\right)\mathrm{.}$ 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: ${\mathit{\Delta \varphi }}_j=\left({\varphi }_j^{\mathrm{R}}-{\varphi }_j^{\mathrm{B}}\right)-{\mathrm{\Lambda }}^{-1}\cdot (D_j^R$ - $-D_j^B),$ 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.

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.

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 ${\gamma }_j^B,{\gamma }_j^R,D_j^B,D_j^R,{\varphi }_j^{\mathrm{B}},{\varphi }_j^{\mathrm{R}},H_j^{\gamma },$ 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.

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.

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: ${\mathrm{\Delta }\mathit{\gamma }}_j=\left({\gamma }_j^R-{\gamma }_j^B\right)-\left({D_j}^R-{D_j}^B\right),$ 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: ${\mathrm{\Delta }\mathit{\varphi }}_j=\left({{\varphi }_j}^{\mathrm{R}}-{{\varphi }_j}^{\mathrm{B}}\right)-$ ${\mathrm{\Lambda }}^{-1}\cdot \left({D_j}^R-{D_j}^B\right),$ 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.

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 ${H_k}^{\gamma },$ for index k of ${H_k}^{\gamma }$ 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.

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).

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: $${\mathbf{P}}_k={\mathbf{R}}_k^{-1}-{\mathbf{R}}_k^{-1}{\mathbf{Q}}_{\mathbf{k}}{\left({\mathbf{Q}}_k^{\mathbf{T}}{\mathbf{R}}_k^{-1}{\mathbf{Q}}_{\mathbf{k}}+q{\mathbf{S}}_k\right)}^{-\mathbf{1}}{\mathbf{Q}}_k^{\mathbf{T}}{\mathbf{R}}_k^{-1}\mathrm{.}$$ Equation (32) was then applied to the form of equation (30) to provide the form: $${\mathbf{N}}_j={\left[{\displaystyle \sum _{k=1}^j}{\mathbf{G}}^T{\mathbf{P}}_k\mathbf{G}\right]}^{-1}\times \left[{\displaystyle \sum _{k=1}^j}\left({\mathbf{G}}^T{\mathbf{P}}_k{\mathbf{\mu }}_k+q{\mathbf{g}}_k\right)\right],$$ where M and B are identified in the forms of: $${\mathbf{M}}_j=\left[{\displaystyle \sum _{k=1}^j}{\mathbf{G}}^{\mathbf{T}}{\mathbf{P}}_k\mathbf{G}\right]\mathrm{and}{\mathbf{B}}_j=\left[{\displaystyle \sum _{k=1}^j}\left({\mathbf{G}}^{\mathbf{T}}{\mathbf{P}}_k{\mathbf{\mu }}_k+q{\mathbf{g}}_k\right)\right]\mathrm{.}$$ The forms of equations (33B) can be expressed in the following recursion form: $${\mathbf{M}}_j={\mathbf{M}}_{j-1}+{\mathbf{G}}^{\mathbf{T}}{\mathbf{P}}_j\mathbf{G}\mathrm{and}{\mathbf{B}}_j={\mathbf{B}}_{j-1}+{\mathbf{G}}^{\mathbf{T}}{\mathbf{P}}_j{\mathbf{\mu }}_j+q{\mathbf{g}}_j$$ Then $${\mathbf{N}}_j={\left({\mathbf{M}}_{j-1}+{\mathbf{G}}^{\mathbf{T}}{\mathbf{P}}_j\mathbf{G}\right)}^{-\mathbf{1}}\left({\mathbf{B}}_{j-1}+{\mathbf{G}}^{\mathbf{T}}{\mathbf{P}}_j{\mathbf{\mu }}_j+q{\mathbf{g}}_j\right)$$ 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: $${\mathbf{N}}_j={\left({\mathbf{M}}_{j-1}+{\mathbf{G}}^{\mathbf{T}}{\mathbf{P}}_j\mathbf{G}\right)}^{-1}\left({\mathbf{M}}_{j-1}{\mathbf{N}}_{j-1}+{\mathbf{G}}^{\mathbf{T}}{\mathbf{P}}_j{\mathbf{\mu }}_j+q{\mathbf{g}}_j\right)$$ To equation (35B), we now add and subtract the term G^TP_j G N_j-1 from the second. bracketed quantity to obtain: $${\mathbf{N}}_j={\left({\mathbf{M}}_{j-1}+{\mathbf{G}}^{\mathbf{T}}{\mathbf{P}}_j\mathbf{G}\right)}^{-\mathbf{1}}\left({\mathbf{M}}_{j-1}{\mathbf{N}}_{j-1}+{\mathbf{G}}^{\mathrm{T}}{\mathbf{P}}_j\mathbf{G}{\mathbf{N}}_{j-1}-{\mathbf{G}}^{\mathrm{T}}{\mathbf{P}}_j\mathbf{G}{\mathbf{N}}_{j-1}+{\mathbf{G}}^{\mathbf{T}}{\mathbf{P}}_j{\mathbf{\mu }}_j+q{\mathbf{g}}_j\right)$$ 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: $${\mathbf{N}}_j={\mathbf{N}}_{j-1}+{\left({\mathbf{M}}_{j-1}+{\mathbf{G}}^{\mathbf{T}}{\mathbf{P}}_j\mathbf{G}\right)}^{-\mathbf{1}}\left({\mathbf{G}}^{\mathrm{T}}{\mathbf{P}}_j\mathbf{G}{\mathbf{N}}_{j-1}+{\mathbf{G}}^{\mathbf{T}}{\mathbf{P}}_j{\mathbf{\mu }}_j+q{\mathbf{g}}_j\right)$$ The second bracketed quantity of equation (35D) can be further simplified as: $${\mathbf{N}}_j={\mathbf{N}}_{j-1}+{\left({\mathbf{M}}_{j-1}+{\mathbf{G}}^{\mathbf{T}}{\mathbf{P}}_j\mathbf{G}\right)}^{-1}\left({\mathbf{G}}^{\mathrm{T}}{\mathbf{P}}_j\left({\mathbf{\mu }}_j-\mathbf{G}{\mathbf{N}}_{j-1}\right)+q{\mathbf{g}}_j\right)$$ Using equation (34) this becomes: $${\mathbf{N}}_j={\mathbf{N}}_{j-1}+{\mathbf{M}}_j^{-1}\left({\mathbf{G}}^{\mathrm{T}}{\mathbf{P}}_j\left({\mathbf{\mu }}_j-\mathbf{G}{\mathbf{N}}_{j-1}\right)+q{\mathbf{g}}_j\right)$$ 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: $${\mathbf{P}}_k={\mathbf{R}}_k^{-1}-{\mathbf{R}}_k^{-1}{\mathbf{Q}}_k{\left({\mathbf{Q}}_k^{\mathbf{T}}{\mathbf{R}}_k^{-1}{\mathbf{Q}}_{\mathbf{k}}+q{\mathbf{S}}_k\right)}^{-\mathbf{1}}{\mathbf{Q}}_k^{\mathbf{T}}{\mathbf{R}}_k^{-1}\mathrm{.}$$
• (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).

$${\mathbf{a}}_{\mathbf{k}}={\left({{\mathbf{Q}}_{\mathbf{k}}}^{\mathbf{T}}{{\mathbf{R}}_{\mathbf{k}}}^{-1}{\mathbf{Q}}_{\mathbf{k}}\right)}^{-\mathbf{1}}\left({{\mathbf{Q}}_{\mathbf{k}}}^{\mathbf{T}}{{\mathbf{R}}_{\mathbf{k}}}^{-1}\left({\mathbf{\mu }}_{\mathbf{k}}-{\mathbf{GN}}_{k-1}\right)\right),$$

$${\mathbf{a}}_{\mathbf{k}}={\left({{\mathbf{Q}}_{\mathbf{k}}}^{\mathbf{T}}{{\mathbf{R}}_{\mathbf{k}}}^{-1}{\mathbf{Q}}_{\mathbf{k}}+q{\mathbf{S}}_{k-1}\right)}^{-\mathbf{1}}\left({{\mathbf{Q}}_{\mathbf{k}}}^{\mathbf{T}}{{\mathbf{R}}_{\mathbf{k}}}^{-1}\left({\mathbf{\mu }}_{\mathbf{k}}-{\mathbf{GN}}_{k-1}\right)-q{\mathbf{g}}_{k-1}\right)\mathrm{.}$$

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.

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., ${\gamma }_j^B,{\gamma }_j^R,{D_j}^B,{D_j}^R,{\varphi }_j^{\mathrm{B}},{\varphi }_j^{\mathrm{R}},H_j^{\gamma }$ 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 $=\left({\gamma }_j^R-{\gamma }_j^B\right)-\left({D_j}^R-{D_j}^B\right)\mathrm{.}$ 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: ${\mathrm{\Delta }\mathit{\varphi }}_j=\left({{\varphi }_j}^{\mathrm{R}}-{{\varphi }_j}^{\mathrm{B}}\right)-{\mathrm{\Lambda }}^{-1}\cdot \left({D_j}^R-{D_j}^B\right),$ 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 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 ${H_1}^{\gamma }\mathrm{.}$ 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 $H_j^{\gamma }\mathrm{.}$ 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.

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.

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.

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: ${\mathrm{\Delta }\mathit{\gamma }}_j=\left({\gamma }_j^R-{\gamma }_j^B\right)-\left({D_j}^R-{D_j}^B\right)\mathrm{.}$ 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: ${\mathrm{\Delta }\mathit{\varphi }}_j=\left({{\varphi }_j}^{\mathrm{R}}-{{\varphi }_j}^{\mathrm{B}}\right)-$ ${\mathrm{\Lambda }}^{-1}\cdot \left({D_j}^R-{D_j}^B\right),$ 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.

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 ${H_1}^{\gamma }\mathrm{.}$ 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.