Fault detection and exclusion for global position systems

[0001] 1. Field of the Invention

[0002] The present invention relates to global position systems, GPS, and more particularly to an improvement for detecting and excluding erroneous information from a transmitting satellite in order to assure greater accuracy and integrity.

[0003] 2. Description of the Prior Art

[0004] U.S. Pat. No. 5,760,737 (herein the '737 patent) entitled Navigation System with Solution Separation Apparatus for Detecting Accuracy Failures issued Jun. 3, 1998 to Mats Brenner, and assigned to the applicant of the present invention, discloses a GPS system for determining when an error occurs in one of a plurality of transmitting satellites, and for establishing an accuracy or error bound. This is achieved using a position solution obtained from at least five transmitting satellites and comparing that with a plurality of position sub-solutions, each of which is derived from a different subset of the at least five transmitting satellites. Specifically, a filter that excludes a different one of the transmitting satellites is employed to obtain the sub-solutions. This yields a plurality of solution separation parameters each based on the statistics of a separation between the position solution and respective position sub-solution. A plurality of discriminators, each based on the distance between a position solution and a respective sub-solution, is obtained and an indicator indicates a fault when one of the discriminators is greater than one of the solution separation parameters.

[0005] Unfortunately, the determination that a fault exists does not provide information as to which of the transmitting satellites is in error.

[0006] The present invention employs a plurality of sub-sub solutions for each of the sub-solutions. Each sub-sub-solution excludes a different satellite in addition to the satellite excluded by the parent sub-solution. As was the case with the parent solution, the separation between each sub-sub-solution and its parent sub-solution is compared to a detection threshold. If the separation for a particular sub-solution and any of its sub-sub-solutions exceeds the detection threshold then the satellite represented by the particular sub-solution cannot be the failed satellite. If one and only one sub-solution is found for which the separation between it and all of its sub-sub-solutions are less than the detection threshold, that the satellite represented by that sub-solution is the failed satellite.

[0007] FIG. 1 shows the first navigation system of the '737 patent and the present invention;

[0008] FIG. 2 shows the first method of operating the first navigation system of the '737 patent with the addition of the present invention;

[0009] FIGS. 2A, 2B, and 2C show details of steps 280, 300 and 310 of FIG. 2;

[0010] FIG. 3 shows the second navigation system of the '737 patent and the present invention; and

[0011] FIG. 4 shows the second method of operating the second navigation system of the '737 patent with the addition of the present invention.

[0012] In the above referred to '737 patent, herein incorporated by reference, the system utilizes a number, N, of transmitting satellites to determine a fall solution for the position of the receiving aircraft and then by taking N groups of satellites, each group omitting a different one of the transmitting satellites, determining a plurality of sub-solutions from which it could be found if one of the satellites was transmitting an erroneous signal greater than a calculated protection limit or error bound and alerting the pilot or flight management system that the integrity of the position system was unacceptable. More particularly, if it is assumed that six transmitting satellites are in view and the full solution is generated by analyzing their signals. The first sub-group might include satellites 2, 3, 4, 5, and 6 (omitting 1), the second sub-group might include satellites 1, 3, 4, 5, and 6 (omitting 2), the third subgroup might include satellites 1, 2, 4, 5, and 6 (omitting 3) etc. to the sixth sub-group which would include satellites 1, 2, 3, 4, and 5 (omitting 6). From each sub-group, a different sub-solution is obtained and from the sub-solutions, it may be determined if there is a satellite that should be considered faulty. However, determining which of the six satellites is the faulty one is not determinable from the system of the '737 patent.

[0013] The present invention takes the procedure one step further. In the present invention, each sub-group is split into N1 sub-sub-groups with each sub-sub-group excluding a different one of the satellites in its group in addition the satellite omitted in the first sub-group. More particularly, using the example above, with the first sub-group consisting of satellites 2, 3, 4, 5, and 6 (omitting 1), five sub-sub-groups would be formed. The first sub-sub-group might consist of satellites 3, 4, 5, and 6 (omitting 1 and 2), the second sub-sub-group might consist of satellites 2, 4, 5, and 6 (omitting 1 and 3), the third sub-sub-group might consist of satellites 2, 3, 5, and 6 (omitting 1 and 4) etc. to the fifth sub-sub-group consisting of 2, 3, 4, and 5 (omitting 1 and 6). In similar fashion the second sub-group consisting of satellites 1, 3, 4, 5, and 6 (omitting 2) might be divided into another five sub-sub-groups with the first consisting of satellites 3, 4, 5, and 6 (omitting 2 and 1) the second consisting of satellites 1, 4, 5, and 6 (omitting 2 and 3) etc. to the fifth sub-sub-group consisting of satellites 1, 3, 4, and 5 (omitting 2 and 6). This sub-sub-grouping continues for each of the sub-groups until there are a total of 30 sub-sub-groups, each consisting of 4 satellites with 2 of the satellites omitted.

[0014] The sub-sub-groups are processed to produce sub-sub-solutions in substantially the same way that the sub-solutions were obtained from the sub-groups in the '737 patent. Furthermore, the sub-sub-solutions are processed in substantially the same way that the sub-solutions were processed in the '737 patent, and, as will be shown, a result is obtained that not only determines the integrity limit (which bounds the error due to both an undetected fault and a rare normal excursion) but also determines an exclusion limit (which bounds the error due to a fault which has not yet been isolated and excluded.).

[0015] FIG. 1 is the same as FIG. 1 of the above referred to '737 patent and performs the same functions as was described in the '737 patent. It will not be further described here.

[0016] In FIG. 2, those portions which are joined by double line arrows are the same as FIG. 2 of the '737 patent and those portions joined by single line arrows represent the additions of the present invention. The portions shown by double arrows operate in the same manner as in the '737 patent and will not be further described here. The original reference numerals have been retained for those elements of the '737 patent, and thus steps 30, 50, 60, 70, 80, 90, 100, 110, and 120 operate the same as FIG. 2 of the '737 patent, with a few exceptions which will be discussed hereinafter.

[0017] Steps shown by boxes 260, 270, 280 290, 300, and 310 are the steps of the present invention and in most cases these steps are similar to the steps 60, 70, 80, 90, 100 and 110 respectively. Accordingly, for ease in following the description of the present invention, the new reference numerals have been chosen to be the same as those in the '737 patent except that they are increased by 200. Thus, step 260 is similar to step 60, etc.

[0018] As in the '737 patent, normal operation begins with processor obtaining pseudorange measurements from receiver 14 in Step 30. In this step, the processor also determines the number of satellite transmitters N. The processor then forms an N-element vector _meas that contains the pseudorange measurements. The process then linearizes the non-linear equations governing the position solution around an initial estimate of the position solution and an initial receiver clock offset estimate. This requires defining _meas, an N-element vector of pseudorange residuals according to

[0019] where _meas is the N-element column vector of measured pseudoranges and _est is an N-element vector for the estimated pseudoranges derived from the initial position estimate and the initial clock offset estimate.

[0020] In step 50, the processor determines a full position solution r₀, using pseudorange residuals derived from the N signals and defined as

[0021] where S₀ is a 4N least squares solution matrix.

[0022] In step 60, the '737 patent calculates several sub-solutions r₁, r₂ . . . r_N, each based on a respective subset of the N pseudorange measurements. The nth sub-solution in the '737 patent is defined as

[0023] Step 260 of the present invention is like step 60, except that a plurality of sub-sub-solutions, r_nm(n1, 2 . . . N; m1, 2 . . . N; nm) are calculated with each based on a respective sub-subset of the N sub-solutions. As explained above, each sub-sub-group contains the same satellites as each sub-group (N1) except that it also excludes one other additional satellite. The mth sub-sub-solution of sub-solution n, is defined as

[0024] where S_nm is a 4N least-square sub-sub-solution matrix with a zero nth column and a zero mth column. Note that where an equation applicable to the present invention is similar to an equation applicable to the '737 patent, the equation is numbered the same except that an s is added thereto.

[0025] In the preferred embodiment, the sub-sub-solutions r_nm, are, as in the '737 patent, 4-element vector quantities with three elements representing respective north, east, and down coordinates and a fourth element ct where c is the speed of light and t is the receiver clock offset.

[0026] In step 270, like in step 70 of the '737 patent, the processor computes discriminators d_nm(n1, 2 . . . N; m1, 2 . . . N; nm) for the respective sub-sub-solutions r_nm(n1, 2 . . . N; m1, 2 . . . N; nm). The discriminator d_nm may be any mathematical quantity based on a mathematical distance or separation between the mth sub-sub-solution r_nm and its parent sub-solution r_n. The preferred embodiment defines discriminators d_nm as the actual horizontal distance between each sub-solution r_n and the respective sub-sub-solutions r_nm. In the horizontal, or north-east coordinate plane, the distance d_nm between sub-sub-solution r_nm, and sub-solution r_n, is $\begin{array}{cc}{d}_{\mathrm{nm}}=\sqrt{[{\left(r_n\left(N\right)-r_{\mathrm{nm}}\left(N\right)\right]}^2+[{\left(r_n\left(E\right)-r_{\mathrm{nm}}\left(E\right)\right]}^2}& \left(\text{4s}\right)\end{array}$

[0027] where (N) and (E) denote the north and east components of the nth sub-solution and its mth sub-sub-solution.

[0028] In step 280, like in step 80 of the '737 patent, processor 16 computes a set of solution separation parameters D_nm(n1, 2 . . . N; m1, 2 . . . N; nm) based on the separation statistics of the sub-solution from the respective sub-sub-solutions. Computing the solution separation parameters entails sub-steps 280A, 280B, and 280C shown in FIG. 2A.

[0029] Step 280A requires computing covariance matrices dP_nm(n1, 2 . . . N; m1, 2 . . . N; nm) which describe the statistics of the respective separations between each sub-solution r_n and its sub-sub-solutions r_nm. The covariance matrix dP_mn is defined as

[0030] where E denotes expectation and superscript T denotes transposition. In the preferred implementation, processor 16 computes the covariance matrices, using the equivalent relation

[0031] where _pr² is the variance of the pseudorange measurements.

[0032] In step 280B, the processor determines the position-domain eigenvalues of covariance matrices dP_nm in the same way and for the same purpose as in the '737 patent and again, the maximum eigenvalue (_dPnm)_max of each covariance matrix determines the maximum separation variance.

[0033] Thus, in step 280C processor 16 uses the maximum eigenvalues (_dPnm)_max to determine the maximum separations, or solution separation parameters D_nm according to

[0034] where P_FD is the probability of false detection and, Q¹ denotes the inverse of $\begin{array}{cc}Q\left(x\right)=\frac{1}{\sqrt{2}}{}_x^{}{}^{-\frac{r^2}{2}}r& \left(\text{8s}\right)\end{array}$

[0035] the well-known normalized cumulative distribution function.

[0036] Having determined the solution separation parameters, processor 16 executes step 90 in the '737 patent and determines if a failure has occurred. If a failure occurs, this information is transferred to step 290 as shown by arrow 292. Step 290 involves a determination of which of the satellites is the one that failed. The logic for exclusion is as follows: Any transmitter, r, will be excluded as the failed transmitter if, and only if,

[0037] where d_rm is the horizontal distance between sub-sub-solution X_rm and sub-solution X_r.

[0038] If sub-solution n is separated from at least one of its sub-sub-solutions by more than the exclusion threshold, then transmitter n cannot be the failed transmitter because the failed transmitter is still included in sub-solution n. On the other hand, if the separation of sub-solution r from each of its sub-sub-solutions is less than the exclusion threshold, then transmitter r is the failed transmitter (providing each of the other sub-solutions is separated from at least one of its sub-sub-solutions by more than the exclusion threshold).

[0039] Having determined a failed transmitter, a signal indicative of the failed transmitter is returned to step 30 via line 294 so that at the next determination of the number of satellites N, step 30 will exclude the failed satellite.

[0040] Step 300 is like step 100 in which the error parameters A_nm(n1, 2 . . . N; m1, 2 . . . N; mn) are determined based on the noise induced errors in respective sub-sub-solutions r_nm. Step 300 comprises three sub-steps 300A, 300B and 300C which are shown in FIG. 2B.

[0041] In step 300A, processor 16 calculates error covariance matrices P_nm describing the statistics of the noise-induced errors for corresponding sub-sub-solutions r_nm. The error covariance matrix P_nm is

[0042] where r_nm represents the effect of noise on the mth sub-sub-solution of sub-solution n.

[0043] where w is the N-dimensional measurement noise vector and E(ww) is _pr². In the preferred embodiment, the processor calculates matrices P_nm from

[0044] In step 300B, the processor determines the eigenvalues of error covariance matrices P_nm. The eigenvalues define the variance, that is, the spread of the errors around the average error, and thus control of the magnitude of the maximum error in sub-sub-solution r_nm depends on the square root of the maximum eigenvalue (_Pnm)_max of the corresponding error covariance matrix.

[0045] In step 300C, the processor computes error parameters A_nm using

[0046] where P_MD is the probability of missed detection.

[0047] The next step in FIG. 2 is step 310. In this step, as with determining the protection limit in step 110 of the '737 patent, the processor determines the exclusion limit, using the solution separation parameters D_nm and the sub-solution error parameters A_nm. This preferably entails determig the maximum sum of D_nm and A_nm, over n1, 2 . . . N; m1, 2 . . . N; nm. FIG. 2C shows this as two steps: step 310A determines M parameters sums (D_nmA_nm), over n1, 2 . . . N; m1, 2 . . . N; mn and step 310B selects the maximum of these parameter sums as the exclusion limit. Functionally, this is expressed as

[0048] After calculating the integrity limit in step 110 and the exclusion limit in step 310, processor 16 executes step 120 by outputting the full position solution, the protection limit and the exclusion limit as well as the detection flag to the flight management system 20 which indicates if a failure has been detected but not yet isolated. After executing the output step 120, the processor returns to step 30 to get new data from the receiver and to determine N, excluding any satellite that failed, and to calculate a new position solution, protection limit, new detection flag and identify which satellite failed, if applicable.

[0049] It is seen that the first embodiment accomplishes all that the '737 patent accomplished and, in addition, provides information not only when a failure occurs but also which satellite failed. Although the first embodiment explicitly operates only on the present measurement data, in so-called snapshot fashion, its advantage is in the unique suitability for Kalman filter implementations. Through the use of Kalman filters, the processor incorporates not only present measurement data, but also past measurement data into the integrity monitoring process. Moreover, Kalman filters enable the processor to incorporate inertial sensor data into the determination of satellite transmitter failures, as evidenced in the second embodiment.

[0050] As shown in FIG. 3, the second embodiment extends the radio navigation system of FIG. 1 with the addition of an inertial reference unit 22 for providing inertial data to the processor 16 and pressure altitude sensor 27 for providing altitude data to processor 16. FIG. 3 is the same in form and operation as FIG. 3 of the '737 patent and uses the same reference numerals. Accordingly, it will not be described further herein.

[0051] As was the case with respect to FIG. 2, those portions of FIG. 4 which are joined by double line arrows are the same as FIG. 4 of the '737 patent and those portions joined by single line arrows represent the additions of the present invention. The portions shown by double arrows operate in the same manner as in the '737 patent and will not be further described here. The original reference numerals are retained for those elements of the '737 patent and thus steps 30, 40, 50, 60, 70, 80, 90, 100, 110, 120 and 130 operate the same as FIG. 4 of the '737 patent with a few exceptions which will be discussed hereinafter. One significant exception, to be described, arises in that the present invention provides a new way of computing the separation covariance parameters (see step 380 of the present invention below). This is accomplished by using the error covariance parameters (which are obtained in step 375, below, in a manner similar to step 100 of the '737 patent). Furthermore, step 80 in the '737 patent could also be computed the same way as step 380 of the present invention, if desired. In any event, for purposes of the present invention, the step corresponding to step 100 of the '737 patent (which would have been step 400 of the present invention) is removed and inserted after step 370 in the present invention with reference numeral 375 for convenience.

[0052] As was the case in the '737 patent, inertial processor 25, determines an inertial position solution r_i, preferably a three-element vector in an earth-fixed reference frame. Inertial process or 26, also preferably converts the acceleration data into raw acceleration vector a_raw, and attitude data into raw angular velocity vector _raw. Inertial processor 26 also determines a transformation matrix C, for transforming body frame coordinates to local vertical frame L, a three element rotation vector _IE, which describes rotation of the earth based frame E versus inertial frame I transformed to L frame, and rotation vector _IE, which describes rotation of the L frame versus the earth fixed frame E transformed to L frame.

[0053] The focal point of the Kalman filter implementation is the hybrid state vectors X₀X_N, with each having 18 error states and stemming from the exclusion of the nth satellite transmitter. In other words, X₀, excludes no satellite transmitters and X_N, excludes the nth satellite transmitter. The nth state vector is defined as

[0054] where

[0055] is a three-element attitude error vector;

[0056] v is a three-element velocity vector;

[0057] r is a three-element hybrid position solution excluding the nth pseudorange measurement;

[0058] rtc is a one-element receiver clock phase error expressed as a distance (analogous to ct in the first embodiment);

[0059] vfc is a one-element receiver clock frequency error expressed as a velocity;

[0060] ₀ is a three element vector of gyro biases modeled as first order Markov processes with a 100 hour time constant;

[0061] a₀ is a three element vector of accelerometer biases modeled as first order Markov processes with a 100 hour time constant; and

[0062] alt is a one-element pressure altitude bias.

[0063] (For clarity, the n subscripts are not shown on each state variable in X_n.)

[0064] Steps shown by boxes 360, 370, 375, 380 390 and 410 are the steps of the present invention and in most cases these steps are similar to the steps 60, 70, 80, 90, 100 and 110 respectively except, as noted above, step 375 of the present invention is similar to step 100 of the '737 patent but to provide a basis for a new calculation of step 380, has been moved to a position prior to step 380 and has been renumbered 375. Other than this one substitution, in the following description of the present invention, the new reference numerals are the same as those in the '737 patent except that they are increased by 300.

[0065] Again, the method begins in step 30 when the processor retrieves the following inertial data from the inertial reference unit 22:

[0066] inertial position solution r_i,

[0067] raw acceleration vector a_raw,

[0068] raw angular velocity vector _raw,

[0069] transformation matrix C,

[0070] rotation vector _IE, and

[0071] rotation vector _EL.

[0072] After retrieving this inertial data, the processor models the inertial error dynamics based on motion of the aircraft. This occurs within the context of a linearized inertial error model that is referenced to local vertical coordinate frame L. This inertial error model defines an altitude error vector , a velocity error vector v, and a hybrid position solution r, each stemming from the solution of the following three differential equations:

[0073] where

[0074] ₀ is gyro bias modeled as first order Markov processes with a 100-hour time constant;

[0075] _IB is ₀scale factor and misalignment errors;

[0076] a₀ is accelerometer bias modeled as first order Markov processes with a 100 hour time constant;

[0077] a_IB is ₀scale factor and misalignment errors;

[0078] g gravity deflections and anomaly (v_xg, v_yg, g_c)

[0079] G is 33 matrix defined as $\begin{array}{cc}G=\left[\begin{array}{ccc}-g\left(U\right)/R^{\left(1\right)}& 0& 0\\ 0& -g\left(U\right)/R^{\left(1\right)}& 0\\ 0& 0& 2g\left(U\right)/R^{\left(2\right)}\end{array}\right]& \left(18\right)\end{array}$

[0080] where R denotes the radius of the earth and g(0) denotes gravity at zero height. (1) and (2) identify the specific terms affecting integrity as is discussed in the '737 patent.

[0081] Step 30 further requires the processor to retrieve pseudorange measurements from receiver 14 and determines the number of pseudorange measurements N. After retrieving the pseudoranges, the processor determines N-element pseudorange residual vectors ₀(k)_N(k), with the nth pseudorange residual >_N(k) defined as

[0082] In this equation, _n,meas are the actual pseudorange measurements and _n,est are the estimated pseudoranges based on the previous nth absolute hybrid sub-solution r_n(k) and the receiver clock offset rtc_n, with the subscript, n, denoting exclusion of the nth pseudorange from the vector. The nth absolute hybrid sub-solution, r_n(k), refers to

[0083] where the superscript E denotes that the relative hybrid position solution r_n has been transformed from the L frame to the earth fixed frame for addition to the inertial position solution r_i.

[0084] In this embodiment, the Kalman filters include an error state for the altitude bias. Thus, in this step, the processor also determines a pressure altitude measurement residual based on the input pressure altitude from sensor 27 and the pressure altitude estimate based on the predicted altitude and the predicted pressure altitude bias. The predicted altitude stems from r_n(k) as already defined. The altitude residual is defined the same for all the state vectors.

[0085] In step 40, the processor establishes the initial values for position solution r₀(k) and sub-solutions r₁(k)r_N(k), separation covariance matrices dP₁(k)dP_N(k) and error covariance matrices P₁(k)P_N(k), where k signifies their temporal correspondence to the kth iteration.

[0086] In the preferred embodiment, step 40 also requires the processor to monitor changes in the number and composition of the satellite transmitters determining the position solution. If the number of satellites increases, the processor assigns a state vector (sub-filter) and related variables to that satellite for processing along with the existing ones. And, if a satellite transmitter moves out of view or otherwise becomes unavailable, the processor decides when it can safely release its corresponding state vector X_n, that is, cease computing a particular sub-solution X_n and its associated discriminator d_n, separation parameter D_n, and error parameter A_n. Sub-solution X_n becomes unnecessary, and thus eligible for release, when the nth satellite transmitter fails, moves out of view, or otherwise becomes irrelevant to the full position solution. To ensure that the remaining sub-solutions remain unaffected by this failure, the processor delays the release until the remaining sub-solutions have maintained their respective HDOPs (horizontal dilution of precision) below a threshold for a specific time period, such as five minutes.

[0087] As indicated in steps 50 and 60, the next steps entail computing the full solution r₀(k) and sub-solutions r₁(k)r_N(k). Similarly, steps 50 and 360 determine the sub-sub-solutions r_nm(n1, 2 . . . N, m1, 2 . . . N; nm), each based on a respective sub-subset of the N sub-solutions as explained above. As explained in the '737 patent, with the Kalman implementation, these solutions depend on the Kalman gain vectors using math detailed in the '737 patent and not repeated here.

[0088] Likewise, since steps 70, 80, 90, 100, 110 and 120 are performed in the same manner as in the '737 patent they will not be further discussed here except to note that if step 80 were performed in the same manner as step 380 of the present invention, step 100 would precede step 80 and that in step 90, when a check for failure is made, any indication of failure is presented via line 292 to step 390 of the present invention as will be discussed below.

[0089] In the present invention, steps 360 and 370 operate in the same manner as in the first embodiment. Step 360 calculates a plurality of sub-sub-solutions, r_nm(n1, 2 . . . N, m1, 2 . . . N; nm) with each based on a respective sub-subset of the N sub-solutions. As explained above, each sub-sub-group contains the same satellites as each sub-group (N1) and also excludes one other additional satellite. The mth sub-sub-solution of sub-solution n, is defined as in equation (3s).

[0090] In the preferred embodiment, the sub-sub-solutions r_nm, are, as in the '737 patent, 4-element vector quantities with three elements representing respective north, east, and down coordinates and a fourth element ct where c is the speed of light and t is the receiver clock offset.

[0091] In step 370, like in the first embodiment, the processor computes the sub-sub-solution discriminators d_nm(n1, 2 . . . N, m1, 2 . . . N; nm) for the respective sub-sub-solutions r_nm(n1, 2 . . . N, m1, 2 . . . N; nm).

[0092] Now, in step 375, which operates in the same manner as step 300 of the first embodiment, in which the error parameters A_nm(n1, 2 . . . N, m1, 2 . . . N; mn) are determined based on the noise induced errors in respective sub-sub-solutions r_nm. Step 375 comprises the same three substeps 300A, 300B and 300C s in the first embodiment and which are shown in FIG. 2B. In step 375, the processor computes error parameters A_nm.

[0093] In step 380, the processor determines the solution separation parameters D_nm over n1, 2 . . . N; m1, 2 . . . N; nm but does so using a different process. Whereas the '737 patent computed the separation covariance matrix as part of a dual (3636) partitionable covariance matrix, the preferred method in the present invention is to compute the separation covariance matrix describing the statistical separation between the nth sub-solution and its mth sub-sub-solution as follows:

[0094] where P_n is the 1818 error covariance matrix for the sub-solution X_n(k), P_nm(k) is the 1818 error covariance matrix for the mth sub-sub-solution of the nth sub-solution X_nm(k), and P_nm^cross(k) is the 1818 matrix describing the cross-covariance between the sub-solution X_n(k) and its mth sub-sub-solution X_nm(k). That is,

P_nm(k)EX_nm(k)X_nm^T(k)(23s) $\begin{array}{cc}{P}_{\mathrm{nm}}^{\mathrm{cross}}\left(k\right)=E\left[X_n\left(k\right)X_{\mathrm{nm}}^T\left(k\right)\right]& \left(\text{24s}\right)\end{array}$

[0095] The process updates the cross-covariance matrix to account for the measurement update of the sub-solution and the sub-solution for the ith satellite according to $\begin{array}{cc}{P}_{\mathrm{nm}}^{\mathrm{cross}}\left(k\right)=\left(I-g_n{h}_i^T\right)P_{\mathrm{nm}}^{\mathrm{cross}}\left(k\right)\left(I-h_i{g}_{\mathrm{nm}}^T\right)+g_n{\mathrm{rg}}_{\mathrm{nm}}^T& \left(\text{25s}\right)\end{array}$

[0096] where I is the 1818 identity matrix. Note the kth iteration notations have been omitted from the Kalman gains g_n and g_nm. Once the cross-covariance is updated, the separation covariance is computed per equation (21s).

[0097] In step 90, the processor checks for transmitter failures by comparing the actual separations d₁-d_N to respective solution separation parameters D₁-D_N. The solution separation parameters D₁-D_N function as failure detection thresholds for the individual sub-solutions. Thus, if the actual separation d_n between sub-solution r_n and the full solution r₀ exceeds its respective separation parameter D_n, the processor sets a failure flag indicating detection of a transmitter failure and sends a signal over line 392 to step 390 of the present invention.

[0098] Steps 100, 110 and 120 of the '737 patent and steps 390 and 410 of the present invention are the same as in the first embodiment except that if the new procedure for calculating the separation parameters is used, step 100 of the '737 patent would precede step 80.

[0099] After calculating the integrity limit in step 110 and the exclusion limit in step 410, processor 16 executes step 120 by outputting the full position solution, the protection limit and the exclusion limit as well as the failure flag to step 130 for projection ahead to the next iteration.

[0100] Step 130 not only involves the determining of the (k1)th separation covariance matrices dP₁(k1) through dP_N(k1) defining the statistical separations between the full solution and each of the N sub-solutions, but also entails determining the (k1)th separation covariance matrices dP_nm over n1, 2 . . . N; m1, 2 . . . N; mn defining the statistical separation between the nth sub-solution and its mth sub-sub-solution by first propagating the cross-covariance matrices P_nm^cross as follows: $\begin{array}{cc}{P}_{\mathrm{nm}}^{\mathrm{cross}}\left(k+1\right)=\left(k\right)P_{\mathrm{nm}}^{\mathrm{cross}}\left(k\right){\left(k\right)}^{}+Q\left(k\right)& \left(\text{26s}\right)\end{array}$

[0101] Again, once the cross-covariance is updated, the separation covariance is computed per equation (21s).

[0102] After completing step 130, the process returns to step 30 to repeat the process entire loop shown in FIG. 4.

[0103] It is thus seen that the second embodiment provides a simple Kalman-filter extension of the first embodiment that incorporates not only inertial data but also present and past measurement data into the position solution, protection limit calculations, and exclusion limit calculations as well as the transmitter failure detection and exclusion.

[0104] Although the present invention has been described with reference to preferred embodiments, workers skilled in the art will recognize that changes may be made in form and detail without departing from the spirit and scope of the invention. In particular, those in the art will recognize that a single processor could perform all of the operations for implementing the invention or that multiple processors could share these operations. Moreover, the method itself could be divided over distinct functional units other than those used for illustration here. Of course, other changes in form and detail are also with in the spirit and scope of the invention. Accordingly, we do not intend to be limited to the specific details used to describe the preferred embodiment. The following claims define the scope of the present invention.