WO2013080183A9 - A quasi tightly coupled gnss-ins integration process - Google Patents

Abstract

A quasi tightly coupled (QTC) aided INS (AINS) process has an inertial navigator system with a loosely-coupled AINS Kalman filter that constructs INS-GNSS position measurements, a GNSS position engine that computes a position fix from observabies provided by a GNSS receiver and an externally provided a priori position and position VCV matrix. An INS position seeding process in which the externally provided a priori position to the GNSS position engine is an antenna position computed from the INS position and attitude solution. An observable subspace constraint (OSC) process computes an OCS matrix that has rows and columns that suppress the approximately uncorrected component of the GNSS position error due to a poor geometry of satellites in in the GNSS position solution in the 1G position measurement constructed by the AINS Kalman filter and that multiplies the OSC matrix and the IG position measurement and measurement model to suppress the approximately uncorrected component of the GNSS position error in the IG position measurement and measurement model.

Description

A Quasi tightly coupled GNSS-INS integration Process

This invention claims priority from provisional patent application for A QUASI-TIGHTLY- COUPLED GNSS-INS INTEGRATION PROCESS filed on 11/30/2011 and having serial number 61/565,466, Confirmation No. 1028 and docket number A3089P.

Field of the Invention

This invention is related to the field of Inertial Navigation Systems aided with data from a Global Navigation Satellite System referred to as I S-GNSS systems. This application teaches how to make an INS-GNSS system that includes a quasi-tightly-coupled INS-GNSS integration process or method (QTC). The GNSS legend stands for Global Navigation Satellite System. This is the generic term for any satellite system used for positioning and navigation. Currently there are 4 GNSS's systems: the GPS (U.S.), GLONASS (Russia), Galileo (European Union) and the Compass system from (China).

Field of the Invention

This invention is related to the field of Inertial Navigation Systems aided with data from a Global Navigation Satellite System referred to as INS-GNSS systems. This application teaches how to make an INS-GNSS system that includes a quasi-tightly-coupled INS-GNSS integration process or method (QTC). The GNSS legend stands for Global Navigation Satellite System. This is the generic term for any satellite system used for positioning and navigation. Currently there are 4 GNSS's systems: the GPS (U.S.), GLONASS (Russia), Galileo (European Union) and the Compass system from (China).

A GNSS positioning algorithm being executed in a GNSS receiver computes a position fix using a trilateration of measured ranges in the form of pseduoranges and carrier phases from the receiver antenna to each tracked satellite. Its use in a GNSS-aided INS requires in principle a loosely-coupled (LC) integration in which the aided INS (AINS) Kalman filter

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RECTIFIED SHEET (RULE 91.1) processes the GNSS position fix as a measurement. The known disadvantage of a LC integration is that the GNSS receiver cannot compute a position fix with fewer than 4 satellites and consequently the AINS Kalman filter cannot construct an INS-GNSS (IG) position measurement The complete loss of aiding data during a such a partial outage of satellites allows the AINS errors to grow without the constraints imposed by GNSS aiding data.

By constrast, a tightly-coupled (TC) integration uses an AINS Kalman filter that processes the pseudoranges and possibly carrier phases generated by the rover GNSS receiver and possibly one or more base receivers at fixed and known locations as measurements. The salient advantage of a TC integration is that during a partial GNSS outage the AINS Kalman filLer processes Ihe available observables from fewer than the minimum 4 satellites that the GNSS receiver requires to compute a fully-constrained position fix. Consequently the AINS errors are partially constrained.

A QTC integration is a LC integration that exhibits the salient characteristics of a TC integration, these being continued aiding and partial position-time error regulation with fewer than 4 satellites, and rapid RTK/KAR recovery after recovery of 4 or more satellites. The purpose of pursuing a QTC integration is to allow the integration of a GNSS positioning algorithm into a GNSS AINS with TC integration behavior with as few modifications as possible to the GNSS positioning algorithm. This is often desirable if the algorithm implementation in software is complex, mature and hence not easily modifiable. Such algorithms are often found on high performance GNSS receivers used for precision positioning and survey.

A typical GNSS positioning algorithm is a type of statistical position estimator. Its starting data comprises an a priori antenna position and a position variance-covariance ( VCV) matrix. It performs a statistical estimation operation on successive epoch observables comprising pseudoranges and possibly carrier phases, and thereby computes an updated antenna position and VCV that is optimal according to an estimation cost function. The

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RECTIFIED SHEET (RULE 91.1) simplest example of such an estimator is a Gauss-Markov least-squares adjustment (LSA).

More sophisticated examples include the Kalman filter and its many variants (extended

Kalman filter, unscented Kalman filter], and Bayesian estimators.

QTC integration is achieved via two key mechanisms added to a LC integration:

INS position seeding of the GNSS positioning algorithm. Observable subspace constraint on

INS-GNSS position aiding.

The INS position seeding mechanization requires the a priori antenna position and position VCV in the GNSS positioning algorithm to be computed from the INS position and attitude (roll, pitch, heading) solution. This can be done by simple assignment, i.e. the a priori antenna position and VCV are set equal to the INS-derived antenna position and VCV. Alternatively the GNSS positioning algorithm treats the INS-derived antenna position and VCV as epoch measurements along with the GNSS observables. This causes the GNSS position estimator to adopt the INS-derived antenna position and VCV at a given epoch, which is approximately the same as assignment. Either approach is usually an easy modification to a GNSS positioning algorithm.

The observable subspace constraint on INS-GNSS position aiding comprises two steps. The basis vectors of the observable subspace in a 3-dimensioned position error space are computed. Then the LC IG position measurement in an AINS Kalman filter is constrained to lie in the observable subspace spanned by these vectors. The observable subspace basis vectors are obtained from the singular value decomposition (SVD) of the satellite- differenced range measurement model matrix for the available observables. Satellite differenced range measurements are the differences between pseudoranges from different satellites so as to cancel the receiver clock error. The range measurement model matrix thus maps the satellite-differenced range measurements onto the 3-dimensioned antenna position error space. It is typically used in a least-squares adjustment (LSA) that computes GNSS antenna position from these satellite-differenced range measurements. It is exactly column rank deficient, i.e. has column rank less than 3, and will have a kernel (also called a null space) if fewer than 4 (i.e. 2 or 3) satellite observables are used to construct fewer than

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RECTIFIED SHEET (RULE 91.1) 3 (i.e. 1 or 2) satellite-differenced range measurements. The orthogonal complement of the kernel is the observable subspace for these measurements.

The SVD is a particular method of decomposing a matrix into its SVD components comprising the left singular vectors, the right singular vectors and the singular values. These components provide useful information about the properties of the matrix, such as its image space, kernel or null space, its rank and nearness to lower rank. The SVD is used in the QTC integration process to compute the right singular vectors of the SVD of the satellite-differenced range measurement model matrix. These right singular vectors are orthonormal and divide into basis vectors for unobservable and observable subspaces corresponding to the kernel and kernel orthogonal complement of the satellite-differenced range measurement model matrix. Orthonormal vectors are vectors that have unit length and that are orthogonal to each other.

If 4 or more satellites are available but the geometry formed by these satellites is poor as expressed in a large dilution of precision (DOP), then the satellite-differenced range measurement model matrix is close to being column rank deficient and is said to be approximately column rank deficient. The SVD will in this case yield one or more singular values close to zero. The approximate kernel is spanned by the right singular vectors corresponding to the small singular values. Its orthogonal complement is the strongly observable subspace for these measurements, spanned by the right singular vectors corresponding to its large singular values.

The basis vectors obtained from the SVD for the exact or strongly observable subspace are assembled into a transformation matrix that multiplies the LC IG position measurement and measurement model in an AINS Kalman filter to implement the observable subspace constraint This is again an easy modification of a LC integration. The consequence is that the AINS Kalman filter position measurement model is constructed only in the observable subspace defined by the available satellite-differenced range measurements, and is thereby consistent with the actual position measurement.

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RECTIFIED SHEET (RULE 91.1) The following variations on when to compute and apply the observable subspace constraint can be used in any embodiment.

One approach has the AINS Kalman filter applying an observable subspacc constraint to the IG position measurement only if fewer than 4 satellites are used to compute the GNSS position solution or DOP is large. In this case the satellite differenced model matrices are exactly or approximately column rank deficient, and the IG position measurement must be constrained to the exact or approximate observable subspace. When 4 or more satellites are available, then the AINS Kalman filter computes the regular, i.e. unconstrained IG position measurement.

The second approach has the AINS Kalman filter applying the observable subspace constraint to the IG position measurement at every epoch. If 4 or more satellites are used In the GNSS position estimator, then the transformation is a non-singular rotation from the navigation frame in which the IG position measurement is normally resolved into a basis spanned by the right singular vectors of the model matrix having a full column rank of 3. The information in the measurement is preserved since the transformation is orthonormal. The preferred embodiment uses the satellite differenced range measurement model matrix and its SVD to compute the observable subspace constraint based on zero or small singular values. It applies the observable subspace constraint to the IG position measurement every epoch to maintain simplicity and consistency.

Notation

The description that follows provides a number of equations in which, by way of example, the following notation. denotes a vector with no specific reference frame of resolution.

The notation *" denotes a vector resolved in a coordinate frame called the a-frame. AH coordinate frames are right-handed orthogonal frames. The X-Y-Z axes form an orthogonal triad in the forward, right and down directions. Typical coordinate frames of interest are the geographic frame (g-frame) whose principal axes coincide with the North, East and Down directions, and the inertial sensor body frame (b-frame), whose principal axes coincide with the input axes of the inertial sensors of an inertial guidance system.

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RECTIFIED SHEET (RULE 91.1) Subscripts on vectors are used to indicate a particular property or identification of the vector. For example, ^« is a lever arm vector from an INS sensor frame (s-frame) origin to the GNSS antenna frame (a-frame) resolved in the INS body (b-frame) frame coordinates.

Γ*

Capital letters, when used as in , denotes a direction cosine matrix (DCM) that

— « x" - C^hx" transforms the vector ^x from the a-frame to the b-frame, as in the equation, .

Time dependency of a quantity is indicated with round brackets around a time variable or

C" (i ) t

index. For example, ^{0 v 1} ' denotes the value of the DCM at time ¹ .

An increment of a variable is indicated with the symbol Δ . For example, &x denotes the increment of the vector ^x over a predefined time interval.

An error in a variable is indicated with the symbol ^ . For example, denotes the error in the vector Six denotes the error in the increment of vector * over a predefined time interval.

The small hat above the vector * implies that the vector * is approximately but not exactkly known. It can be an estimate of ^x generated by an estimation process such as a least-squares adjustment (LSA) or Kalman filter. A superscript - on a vector * to yield * indicates an a priori estimate of ^x , which is the estimate or best guess of * before the occurrence of an estimation process involving new of timely information. A superscript on a vector ^x to yield ^x+ indicates an a posteriori estimate of ^x, which is output of an estimation process involving new of timely information.

Background and Related Art

Inertial Navigation System

An inertial navigation system (INS) is a navigation instrument that computes its navigation solution by propagating Newton's equations of motion using as inputs measured specific

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RECTIFIED SHEET (RULE 91.1) forces or incremental velocities from a triad of accelerometers and measured angular rates or incremental angles from a triad of gyros. A terrestrial INS is designed to navigate on the earth where it is subjected to gravity and earth rate. A celestial INS is designed to navigate in space where in is subjected to smaller gravitational forces from multiple celestial bodies. This disclosure is concerned only with a terrestrial INS. The qualifier "terrestrial" is hereafter implied but not cited explicitly.

An INS can navigate with a specified accuracy after an initialization of the inertial navigator mechanization during which it determines its initial position, initial velocity and North and down directions to a specified accuracy that is commensurate with its inertial sensor errors. The term "alignment" is used to describe this initialization and any ongoing corrections of the inertial navigator mechanization. A free-inertial INS performs an initial alignment and then propagates its navigation solution with no further corrections. See the text by George Siouris, Aerospace Avionics Systems, A Modern Synthesis, Academic Press 1993 for an overview of an INS.

A BRIEF DESCRIPTION OF THE DRAWINGS

Figure 1 is a data flow diagram that shows a generic closed-loop AINS architecture.

Figure 2 is a data flow diagram that shows a generic feedforward aided INS solution.

Figure 3 is a data flow diagram that shows an LC (loosly coupled) integration architecture.

Figure 4 is a data flow diagram that shows the QTC GNSS AINS architecture and data flow. It comprises a combination of the generic LC integration in Figure 3 with the following additions:

Figure 5 is a process flow chart that shows the the initial and final steps in the QTC process and

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RECTIFIED SHEET (RULE 91.1) Figure 6 is a process flow chart that shows the steps in the OSC process extending from Figure 5. 28 is applied to the IG position measurement as described in Figure 6.

Figure 7 is a process flow chart that shows the process as the OSC process concludes its steps and exits Figure 6.

GNSS Receiver

A Global Navigation Satellite System (GNSS) is one of several satellite-based navigation systems. Current GNSS's are the United States deployed Global Positioning System (GPS), the Russian Federation deployed GLONASS, the European Community deployed GALILEO and the Peoples Republic of China deployed COMPASS. A GNSS receiver uses signals received from satellites to compute a position fix and possibly a velocity every receiver epoch. A typical epoch is one second, but can be shorter depending on the receiver design. A GNSS receiver outputs navigation data comprising time, position and possibly velocity every epoch. Some GNSS receivers also output channel data for each tracked satellite comprising pseudoranges, carrier phases and possibly Doppler frequencies for each frequency and modulation protocol broadcast by the satellite. These are often called observables data or simply observables. Reference [7] provides descriptions about GPS and GPS receivers.

Aided INS

An aided INS (AINS) undergoes ongoing corrections to its inertial navigator mechanization or its computed navigation solution. The traditional AINS uses a Kalman filter to estimate INS errors and some means of INS error control to correct the INS errors. A closed-loop AINS uses the estimated INS errors from the Kalman filter to correct the inertial navigator mechanization integrators. This causes the INS alignment to be continuously corrected, and as such is a method for achieving mobile alignment. A feeedforward AINS corrects the

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RECTIFIED SHEET (RULE 91.1) computed INS navigation solution but leaves the inertial navigator mechanization uncorrected.

Figure 1 shows a generic closed-loop AINS architecture. The inertial measurement unit (IMU) 1 generates incremental velocities and incremental angles at the IMU sampling rate, typically 50 to 500 samples per second. The corresponding IMU sampling time interval is the inverse of the IMU sampling rate, typicallyl/50 to 1/1000 seconds. The incremental velocities are the specific forces from the IMU accelerometers integrated over the IMU sampling time interval. The incremental angles are the angular rates from the IMU gyros integrated over the IMU sampling time interval. See Reference [2] for information on inertial sensors and IMU mechanizations. The inertial navigator 2 receives the inertial data from the IMU and computes the current IMU position (typically latitude, longitude, altitude), velocity (typically North, East and Down components) and orientation (roll, pitch and heading) at the IMU sampling rate.

The aiding sensors 5 are any sensors that provide navigation information that is statistically independent of the inertial navigation solution that the INS generates. A GNSS receiver is the most widely used aiding sensor, and is the aiding sensor to which this invention applies.

The Kalman filter 4 is a recursive minimum-variance estimation algorithm that computes an estimate of a state vector based on constructed measurements. The measurements typically comprise computed differences between the inertial navigation solution elements and corresponding data elements from the aiding sensors. For example, an inertial-GNSS position measurement comprises the differences in the latitudes, longitudes and altitudes respectively computed by the inertial navigator and a GNSS receiver. The true positions cancel in the differences, so that the differences in the position errors remain. A Kalman filter designed for integration of an INS and aiding sensors will typically estimate the errors in the I S and aiding sensors. The INS errors typically comprise the following:

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RECTIFIED SHEET (RULE 91.1) • inertial North, East and Down position errors

• inertial North, East and Down velocity errors

• inertial platform misalignment errors

• accelerometer biases

• gyro biases

GNSS errors can include the following:

• North, East and Down position errors

• Carrier phase ambiguities

• Atmospheric range errors

Reference [3] is a definitive and comprehensive treatment of Kalman filtering. It also contains the aided INS as an example application. Reference [4] provides a detailed analysis of different INS error models that can be used in an AINS Kalman filter.

The error controller 3 computes a vector of resets from the INS error estimates generated by the Kalman filter and applies these to the inertial navigator integration processes, thereby regulating the inertial navigator errors in a closed-loop error control loop. This causes the inertial navigator errors to be continuously regulated and hence maintained at significantly smaller magnitudes that an uncontrolled or free-inertial navigator would be capable of.

Figure 2 shows a generic feedforward AINS. The IMU 1, inertial navigator 2, Kalman filter 4 and GNSS receiver and other aiding sensors 5 are the same as those shown in the closed- loop AINS in Figure 1. The inertial navigator 2 is assumed to be aligned and operating free-incrtially with a position error rate that is typical of an INS and commensurate with its inertial sensors. The error controller 6 computes corrections to the free-inertial INS solution. The summing junction 7 performs the corrections.

The state-of-the-art in aided inertial navigation is mature. The technology originated in the late 1960's, and found application on several military navigation systems. Since then, much research has been conducted and much literature has been generated on the subject Types of GNSS-Aided INS Integrations

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RECTIFIED SHEET (RULE 91.1) Tightly-Coupled Integration

A tightly-coupled (TC) GNSS-AINS integration uses the observables from the GNSS receiver to construct Kalman filter range measurements, typically one for each receiver channel that tracks a satellite. The commonly known advantage of a TC integration is its optimal use of any and all information in the observables regardless of the number of satellites, including fewer than 4 satellites. An additional advantage reported in [11] is rapid fixed integer ambiguity recovery following a GNSS outage. This is a consequence of the inertial coast of position accuracy through the outage that accelerates the convergence of estimated ambiguities and subsequent ambiguity resolution.

l.l.l LOOSELY-COUPLED INTEGRATION

A loosely-coupled (LC) GNSS-AINS integration uses the position fixes and possibly the velocity fixes computed by the GNSS receiver. Figure 3 shows a LC integration architecture. The AINS Kalman filter 13 (same as or similar to the AINS Kalman filter 4 in Figure 1) constructs an INS-GNSS (IG) position measurement that differences the respective position fixes from the INS and the GNSS receiver.

ζισ ^{~~ r}ms ~ ¹GNSS

where

r^{£ s} is the predicted GNSS antenna position computed from the INS navigation solution

GNSS i_s GNSS antenna position measured by the GNSS receiver.

The IG position measurement can be constructed in any Cartesian coordinate frame of convenience. The following is an example of the IG position measurement constructed in the ECEF coordinate frame.

(2) where

'INS ^rs ^ ^g^b 'sa (

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RECTIFIED SHEET (RULE 91.1) where

is the INS position resolved in the ECEF coordinate frame,

'GNSS j_S e GNSS antenna position computed by the GNSS receiver (same as in (1)), /*

*^» is the lever arm vector from the INS to the GNSS antenna resolved in the INS body frame,

* is a direction cosine matrix (DCM) from the INS body frame to the local geographic frame (North-East-Down coordinates) computed from the INS-generated roll, pitch and heading,

C

g is a DCM from the geographic frame to the ECEF frame computed from the INS- generated latitude and longitude.

The following is an example of the IG position measurement constructed in the local geographic or North-East-Down coordinate frame.

- ^L„)l^r.∞* K + C^gl

- +

(4) where (4"^"^) and (4"^^<"Ό are respectively the INS and GNSS antenna geographic coordinates (latitude, longitude, altitude) and ^e is the mean earth radius.

The AINS Kalman filter 4 implements a measurement model that has the generic form

^Z10 ^~~ A_{l l} ^XAINS ⁺ *llG [5) x A

where ^Am is the AINS Kalman filter state vector, ⁷⁰ is the IG position measurement model matrix and is the measurement noise model with covariance . The measurement model is expanded as follows.

The components of the measurement (1) comprise the true antenna position ^r" plus INS and GNSS errors as follows.

'ws ^~ l .\

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RECTIFIED SHEET (RULE 91.1) *GNSS ^~ , ^^~ ^ONSS f 7) where

^Sr'^NS is the error in ^r«* ,

S^faⁱ Nss i_s the error in ^r°^Nss.

The true antenna position cancels in measurement (1) leaving a difference of the INS and GNSS errors.

The measurement model [5] is expressed in terms of the components of the state vector x

^{A s} t errors.

^XAINX 9) where

' is a sub-vector of ^MNS comprisinf the INS position errors resolved in the INS wander angle navigation frame, ^G is a sub-vector of -^ws comprisinf the GNSS position errors resolved in the local geographic frame, is the measurement noise model with covariance ^^{0 = E}[^?7ro^ ] .

The measurement model (9) assumes the INS and GNSS position errors are statistically uncorrelated. Consequently a Kalman filter update will yield an INS position VCV that is constrained by the IG measurement in all three axes of the Cartesian navigation frame in which ^Z,G is constructed.

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RECTIFIED SHEET (RULE 91.1) GNSS Positioning Algorithm

The rover GNSS receiver 10 generates a set of observables per receiver channel that tracks a satellite signal every epoch. The epoch duration is typically 1 second but can be shorter depending on the receiver design and on its configuration set by the user. The observables for a GPS satellite comprise some or all of the following depending the design of the receiver:

LI, L2, L2C, L2M, L5 pseudoranges

LI, L2, L2C, L2M, L5 carrier phases

Comparable observables may be obtained for GLONASS, GALILEO and COMPASS satellites, again depending on the receiver design. The optional reference GNSS receiver 11 is located at a known stationary position. It generates some or all of the observables generated by the rover receiver that are typically required to compute one of several types of differential GNSS position fixes. The GNSS position engine 12 in a typical receiver implements a GNSS positioning algorithm. It receives the rover observables and possibly the reference observables, and with these computes an antenna position fix and possibly an estimation variance-covariance (VCV) matrix. The GNSS positioning algorithm can range from a simple trilateration of pseudoranges to a sophisticated precision position algorithm that computes a kinematic ambiguity resolution (KAR) solution having centimeter-level accuracy relative to the reference receiver position.

GNSS Positioning Using Least Squares Adjustment

The following is a summary of the single point position (SPP) solution in any Cartesian coordinate frame such as the earth-centered-earth-fixed (ECEF) frame. This is likely the simplest GNSS positioning algorithm, and can be cast as representative of any other positioning algorithm that computes an antenna position fix from GNSS range measurements.

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RECTIFIED SHEET (RULE 91.1) Given an a priori (before estimation) rover antenna position r , the a posteriori SPP solution is the corrected a priori position

where Sf is the estimated error in the a priori position solution obtained from a least squares adjustment (LSA) of range measurements. The following two least squares adjustments using undifferenced and satellite differenced range measurement models can be used tp compute δΨ .

1.1.2 UNDIFFERENCED RANGE MEASUREMENT MODEL

The linearized measurement model relating a GNSS range measurement to the antenna position and receiver clock error is the following.

η -_Ρι = -ΰ^ -δΤ (11) where r_t = ji*; - r_g j is the predicted range between satellite position f_t and the computed antenna position r , p_t is the i-th GNSS range measurement such as the i-th pseudorange or range- equivalent carrier phase,

δτ is the antenna position error,

ST is the receiver clock error.

m satellite observables are used to construct m measurements of the form (11) with /^' = 1, m to form the vector measurement equation of the form

2 = m (12) where (12) has the following components.

The undifferenced range measurement vector is the followin .

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RECTIFIED SHEET (RULE 91.1) The undifferenced range measurement model matrix is the following.

-T - 1

H = (14)

-Hi -1

The LSA state vector is x - (15) δτ where the unit line-of-sight (LOS) vector from the rover position r_g to the i-th satellite position η is the following.

The LSA that minimizes the ob ective

IS

where W is a positive definite symmetric weight matrix. The LSA solution is unique if H has full column rank, which occurs with near certainty if m≥ 4 . Column rank deficiency with m≥ 4 is a consequence of two LOS vectors coinciding, which is a rare occurrence.

In a system of m equations and n unknowns described by (12), the null space or kernel of the m χ n matrix H comprises all n-dimensioned vectors x such that Hx = 0. The kernel of H is written as Ker (// ) and is non-trivial if H is rank deficient, which happens if m is less than n, as in the case where there are fewer equations to be solved than the number of variables in each equation.

Only column rank deficient matrices have kernels. Column rank deficiency implies the matrix does not have linearly independent columns. If the columns of a matrix // are Λ„...,Λ„ then linear dependence implies at least one column can be cast as a linear combination of the other columns, i.e. h_t + - - - + s_i__{l i}__l +ί_Ι-,Λ-_ι1 +■··+ s_nh_n . An m x n matrix is column rank deficient ifm is less than n or if n is greater than m and its columns are not linearly independent

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RECTIFIED SHEET (RULE 91.1) 1.1.3 SATELLITE-DIFFERENCED MEASUREMENT MODEL

The receiver clock error is removed by computing differences between GNSS range measurements in (11) as follows using Δχ,, = x, - x_} and ΔΜ₍ = H, - w_y

(19) m satellites observables are used to construct m-1 satellite differenced measurements where without loss of generality /=l,.„,m-l and j-m is suppressed for clarity. The satellite differences are generated by pre-multiplying both sides of (12) by an mxm differencing matrix D given by

The satellite differenced measurement model is give

^ZD H_Dx (21) where z_D = Dz = (22)

I *. J

(23)

— M where (21) has the following components.

The satellite differenced range measurement vector is given as follows.

where the m-th vindifferenced range measurement vector is given as follows.

(25) The satellite differenced range measurement model matrix is given as follows.

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RECTIFIED SHEET (RULE 91 . 1) H, (26)

-toil

The LSA that minimizes the objective min ((¾ - H_Dxf W (z_D - H_Dx†j = mm ((z - Hxf D^TWD(z - Hi)) (27) is = (H H_rf )^~' H ¾ (28) δΤ = -ζ„^~ ΰ„^τδ? (29) where W is the upper (m-l)x(m-l) submatrix of W. The LSA solution is unique if H_d has column rank, which occurs with near certainty if m > 4.

The satellite differencing operation thus separates the least-squares estimations of position error and clock error. (28) is the LSA solution of the satellite differenced measurement model

(30) and (29) is the LSA estimate of the receiver clock error given an estimated position error (28). This is not in general the same solution as (18) in which both are estimated together. The LSA solution (28) using the satellite differenced measurement model (30) does however provide a means of estimating only the position error if the receiver clock error is not required.

1.1.4 GENERALIZED SATELLITE DIFFERENCING

The satellite-differenced range measurement model matrix (26) can be generalized as follows. The undifferenced range measurement model matrix (14) is cast as follows,

H = I (31) wh«

An mxm nonsingular transformation E that effects receiver clock error cancellation is given as follows

18

RECTIFIED SHEET (RULE 91 . 1 )

so that

Η δΤ

EHx = (34)

H_2]H_} r +HJT

(33) implies that F. - 0 which in turn implies that the sum of each row of Ei is zero. This describes one or more satellite differences per row. £,Z,⁷ = H, must has column rank one less than the column rank of H for any H, consequently

must have full row rank. This implies that each row of Ei must describe a unique satellite difference. Ei is any 1 xm matrix so that £ is nonsingular.

There exists a nonsingular mxm matrix T such that£ = TD . This implies that the differencing transformation D in (20) is representative of all possible transformations E that can be used in the invention. The differencing transformation D in (20) is used hereafter without loss of generality.

1.1.5 COLUMN RANK DEFICIENT MEASUREMENT MODEL

The range measurement model H is column rank deficient when it has fewer than 4 rows. It has a kernel or null space labeled as Ker ( /) and defined as follows: X e Ker (H) if and only if Hx = 0.

Similarly the measurement model H_d is column rank deficient range when it has fewer than 3 rows. Both column rank deficiencies occur when fewer than 4 satellites are available for computing a position solution. The LSA as given in (18) cannot be computed because H^RWH is singular. Likewise the LSA as given in (28) cannot be computed because H_D ^TW_DH_D is singular.

The range measurement model H can be column rank deficient when it has 4 or more rows and one or more rows are linear combinations of the remaining rows so that there are in fact fewer than 4 linearly independent rows. This occurs in a range measurement model matrix only if two or more satellites are exactly coincident For example the following H constructed with 4 satellites is a 4x4 matrix

where hj is the i-th row of H represented as a transposed vector. If for example Λ₄ ^Γ = O because the satellites corresponding to rows 1 and 4 are exactly coincident, then H is row and column rank deficient because it has only 3 independent rows and hence only 3 independent columns. A similar

19

RECTIFIED SHEET (RULE 91.1) example can be constructed for a satellite differenced measurement model matrix H_d . This cannot occur within a single GNSS such as GPS because the satellite orbital radii are the same. This might happen ifH describes range measurements from two different GNSS's such as GPS and GLONASS since their respective satellites have different orbit radii. This is a pathological occurance, i.e. has miniscule probability of happening. Satellites can however be close to each other and hence close to being coincident In this case H or H_d can become close to being column rank deficient

The following analysis of the undifferenced range measurement model H is equally applicable to the satellite differenced range measurement model H_d .

The singular value decomposition (SVD) of an mxn matrix H is the following,

o^"

H = U∑V^T = [U₁ U_z ] (36)

0 0

where

V is an nxn orthonormal matrix that spans the n-dimensioned state space 91" ,

U is an mxm orthonormal matrix that spans the m-dimensioned measurement space

∑ is an mxn trapezoidal matrix with∑, a diagonal matrix of non-zero singular values.

The SVD is a method of decomposing a matrix H into its SVD components U, V and∑ that have useful properties for characterizing a matrix and understanding its properties. The following SVD properties are relevant to this disclosure:

1. The columns of V form an orthonormal basis for Ker(H)¹ the orthogonal complement of the null space or kernel of H.

2. The columns of V₂ form an orthonormal basis for er (H) the kernel of//.

The following properties result from the SVD (36) when H is column rank deficient

Subspace Components

The columns of V = [V_] V₂ form an orthonormal basis for . Then for any x e ¾" there exists such that

A, (37)

20

RECTIFIED SHEET (RULE 91.1) where x, = Vfy and x₂ - V₂b₂.

Furthermore V^Tx = b im lies

Thus V is a rotation matrix that transforms x into a canonical form comprising partitioned sub- vectors in Ker(H) and Ker(H)¹ . The sub-vectors are generated as follows. v?i = v vfi +v?v₂b₂ =b, (39)

V₂V₂ x = V₂b₂ = x₂ (42) LSA Solution

The measurement equation (12) becomes the following.

(43)

The complete set of solutions that minimize the least squares objective function (17) are characterized as follows.

x - x_l + ₂ (44) where

x^ V^U^Tz (45) and x₂ = V₂ ^Tb is any linear combination of the columns of V₂. The LSA solution (44) is therefore not unique, comprising the sum of x, e Ker(/ )^± and any vector x₂ e Ker(H) . The solution with the minimum vector norm is x = x. .

Kernel Equivalence e Ker(H) is equivalent to Sr e Ker(/ _d ) and S_m ^TSr (46)

ST

21

RECTIFIED SHEET (RULE 91.1) This is shown as follows.

ΗβΤ

Hx - 0 for any x e Ker(//) implies DHx = = 0 and hence 5f e er(/ _</) and

-ΰ_^τδΤ-δ

uJ^T5r - -ST . Since D is nonsingular, the reverse implication DHx - 0 => Hx = 0 also holds.

1.2 DILUTION OF PRECISION (DOP]

1.2.1 DOP FOR UNDIFFERENCED RANGE MEASUREMENTS

The position dilution of precision {PDOP) and time dilution of precision {TDOP) for the undifferenced range measurement model (13) are defined as follows.

G=(H^TH)^~1 (47)

PDOP = ∑G{i,i) TDOP = {A,A) (48)

The dilution of precision {DOP) is the root-sum-square of PDOP and TDOP.

DOP = J PDOP² + TDOP² = ^ G (, /) (49)

The SVD representation H =U∑V^T applied to (47) yields

G=(V∑U^TU∑V^T)^~] =(K∑V^r)^"' =V∑-²V^T (50) since V^T = ^"' . Then (50) becomes the following

where { v, } is the th element of the /-th right singular vector. (49) then becomes the following.

22

RECTIFIED SHEET (RULE 91.1)

(52) relates DOP to the singular values of H. It also shows how an ill-conditioned H having at least one singular value that is near zero generates a large DOP.

1.2.2 DOP FOR SATELLITE DIFFERENCED RANGE MEASUREMENTS

PDOP for a satellite differenced range measurement model is defined as follows.

^G _d = ( ^] (53)

[15] provides the following bounds on PDOP_d for m > 4.

-)=PDOP≤ PDOP_d < PDOP (55) A development similar to (50) to (52) results in

^PDOP_d = ∑\ (56) c,. where ς for /^" = 1, 2, 3 are the singular values of H_d . (56) relates PDOP for satellite differenced range measurements to singular values of H_d , and in particular points to a method for identifying nearness to rank deficiency from PDOP.

1.2.3 NEARNESS TO COLUMN RANK DEFICIENCY

If all of the singular values of H_d are non-zero, then H_d technically has rank 3 which is full column rank. It may however be close to being column rank deficient because one or more of its singular values are small. This can happen if two or more satellites are close to each other and hence close to being coincident (56) shows that PDOP_a will be large, and (55) shows that PDOP will also be large.

The specification of a small singular value of H_d is c_; < c_mm (57) where c_mm is a specified minimum value of a large singular value. If the n» smallest singular values c, for = 4-n₅, 3 and n_s = 1 or 2 are small according to (57), then

23

RECTIFIED SHEET (RULE 91.1) (58] shows that

which in turn is a lower bound on PDOP from (55).

A specified maximum PDOP d^ is used to determine when H_d is considered nearly column rank deficient Setting - P_d > c _max if n_s singular values are smaller than c_{a n} . To avoid dep

so that (58) becomes DOP_d > d_m^≥d^ (59)

Thus a specification d_m3x translates to a specification on a small singular value threshold c_min = l/i/_max for the purpose of identifying small singular values and H_d being nearnly column rank deficient

24

RECTIFIED SHEET (RULE 91.1) 2 INVENTION DESCRI PTIO N

2.1 SCOPE

Section 2.2 provides a theoretical development of the key process steps that are described at the beginning of the section, i.e. INS position seeding and OSC. Section 2.2.2 describes INS position seeding of the GNSS position algorithm. It describes how the unobservable component of the estimated position error is a consequence of I NS position seeding when not enough satellites are available and how this presents a problem to the GNSS-AINS without the OSC. Section 0 describes the OSC as the solution to this problem and how it is constructed using the SVD of tf_d. Section 2.3 presents the preferred embodiment of the QTC algorithm as a step-by-step process with flowcharts. It references the key equations that were developed in Sections 0 and 2.2 simply to not be repetitive.

2.2 QTC INTEGRATION ALGORITHM

2.2.1 ARCHITECTURE AND DATA FLOW

QTC integration is achieved via two key mechanisms added to a standard LC integration:

• INS position seeding of the GNSS position fix algorithm.

• Observable subspace constraint (OSC) on INS-GNSS position measurement in the AINS Kalman filter.

Figure 4 shows the QTC GNSS AINS architecture and data flow. It comprises the generic LC integration in Figure 3 with the following additions:

• The a priori antenna position 15 derived from the INS navigation solution and position VCV matrix 16 derived from the AINS Kalman filter estimation VCV is fed back to the GNSS position engine 12 that uses this data to implement INS position seeding.

• A modified IG position measurement in the AINS Kalman filter 13 implements the OSC. 2.2.2 INS POSITION SEEDING

The following describes the a priori antenna position 15 computed from INS data resolved in an earth-centered-earth-fixed (ECEF) Cartesian coordinate frame.

where

25

RECTIFIED SHEET (RULE 91.1) r is the a priori INS position vector in the ECEF coordinate frame,

/,* is the lever arm vector from the INS to the GNSS antenna resolved in the INS body coordinate frame,

C is the direction cosine matrix (DCM) from the INS body frame to the INS navigation frame (typically the wander angle frame) computed using the INS roll, pitch and platform heading,

C^* is the DCM from the INS navigation frame to the ECEF frame using the INS latitude, longitude and wander angle.

The corresponding position VCV matrix 16 is computed as follows.

^ = (61) where

P*_r is the 3x3 sub-matrix of the AINS Kalman filter estimation VCV matrix corresponding to the INS position error states.

The GNSS position engine 12 receives the following necessary input data every GNSS epoch for computing a position solution as part of the QTC integration:

• Rover GNSS observables comprising one set of pseudoranges and carrier phases

corresponding to one or more carrier frequencies for each tracked satellite

• A priori antenna position 15 from the INS and corresponding position error VCV matrix 16 from the AINS Kalman filter.

The GNSS position engine 12 may receive the following optional input data every GNSS epoch for computing a differential GNSS position solution:

• GNSS observables comprising one set of pseudoranges and carrier phases corresponding to one or more carrier frequencies for each tracked satellite from one or more reference receivers 11.

The GNSS position engine 12 computes an antenna position and position VCV matrix from these data. As part of this position fix computation, it does one of the following:

a. It uses the antenna position 15 computed from INS data as its a priori position and the INS position error VCV 16 computed from the AINS Kalman filter as its a priori position VCV. An example is the LSA SPP algorithm (17) and (18) that uses the INS-derived antenna position as an a priori position.

26

RECTIFIED SHEET (RULE 91.1) b. It uses the antenna position 15 computed from INS data as a position measurement in its estimation process. It uses the INS position error VCV 16 computed from the AINS alman filter to compute the position measurement covariance. An example of such a GNSS position engine is a GNSS Kalman filter that estimates the antenna position using measurements constructed from GNSS observables and the INS-derived antenn position.

The actual GNSS positioning algorithm thatthe GNSS position engine 12 implements is not important The actual algorithm is approximated by a simple GNSS positioning algorithm comprising the following sequence of operations:

1. The a priori antenna position is set to the antenna position 15 computed from INS data.

2. The a priori position VCV is set to the INS position error VCV 16.

3. A SPP algorithm uses a range measurement model (12) to (16) and a set of pseudo ranges to the m satellites used by the GNSS positioning algorithm. It computes the antenna position error as in (18) and from this a corrected antenna position as in (10). The pseudo ranges p i = l,... , m are consistent with the range error model (11) and have accuracies consistent with the actual position error generated by the GNSS position engine 12.

The range measurement model matrix (14) contains the satellite geometry that every GNSS positioning algorithm has to work with. PDOP described by (48) parametrizes the geometry- dependent position error that every GNSS positioning algorithm typically generates.

If the range measurement model matrix (14) is column rank deficient because fewer than 4 satellites are available, then Ker(H) defines an unobservable subspace in the 4-dimensioned position-time error space defined by the state vector (15). In this case the GNSS positioning algorithm uses the minimum norm SPP solution (45) to estimate position and receiver clock error in the observable subspace defined by the kernel orthogonal complement er(/ )¹ , and with this compute a corrected antenna position as in (10).

The a priori position 15 used by the SPP algorithm at a given epoch comprises the true position plus the INS position error.

r = r + 0f_INS (62)

The true a priori state vector (15) is partitioned as in (44) as follows

X = = χ + x₂ (63)

ST

27

RECTIFIED SHEET (RULE 91.1) δκ

where x. e Ker(H)' and x₂ = : Ker(H) .

δτ δτ:

The true measurement vector (12) is given as follows:

(64)

The SPP approximating the true GNSS positioning algorithm uses the minimum norm LSA solution (45) because it is unique in Ker(H)¹ . This is called the rank deficient SPP solution and is characterized as follows.

St GNSS) (65) δΤ,

The components Sr_GNSn and rST, are respectively used to correct the a priori GNSS position (which is the INS position) and the GNSS clock error. The updated error state that reflects these corrections is

where u^l GNSS I

δΤ^*

— X - X, (67)

The minus superscript"-" in equation (66) indicate that x is an a priori value of x, which is a best estimate of x that the SPP algorithm uses as its starting value. The hat "^Λ" in the vector J , in equations (66) and (67) indicates the quantity that the SPP algorithm is able to estimate because it is not in Ker (H) . The plus superscript "+"in equation (66) indicates that x^* is an a posteriori x , which is rafter being corrected by the estimated , generated by the SPP algorithm.

Equation (66) shows that the a priori state component x^~ e Ker(H) remains unchanged by the position-time error correction.

28

RECTIFIED SHEET (RULE 91.1) Observables Subspace Constraint

The AJNS Kalman filter uses the corrected GNSS position to construct the IG position measurement and corresponding IG position measurement model (9}. The measurement model assumes the INS and GNSS position errors are statistically uncorrected so that a Kalman filter update will yield an INS position VCV that is constrained by the IG measurement in all three axes of the Cartesian navigation frame in which ^Z,c is constructed.

The residual INS and GNSS errors in the IG position measurement (8) from the rank deficient SPP solution are the following.

The unobservable position error Sf_m._? remains in the updated GNSS position solution and hence cancels in the IG position measurement The IG measurement thus constrains only _m] , and δ κει >^s unconstrained and can grow with time.

The IG position measurement model (5) describes the assumed measurement (8) given by

which assumes the SPP solution is not rank deficient and has computed a statistically independent a posteriori value for 0 ^_NSS2. The IG position measurement model (5] does not account for the cancellation of Sf_INS2 in the IG position measurement and the consequent growing INS position error, which causes the AINS Kalman filler's estimated state to become pertured with a growing bias. Thus the IG position measurement model (5), (9) does not correctly describe the residual INS and GNSS position errors in the measurement [68) when H is column rank deficient This is a problem brought on by INS position seeding that requires a solution in order for QTC integration to work.

The solution to this problem is the observable subspace constraint (OSC) on the IG position measurement [1] and corresponding IG position measurement model (5), [9). The following two equations comprise the OSC.

The IG position measurement is transformed as follows,

29

RECTIFIED SHEET (RULE 91.1) ^∑ia ^~ *^{■ z}ia

(70)

The corres onding IG position measurement model is transformed as follows,

The transformed IG position measurement model (71) correctly describes the transformed IG position measurement (70) . This is the desired outcome of the OCS.

The OSC requires a 3x3 transformation matrix Γ to be determined so that

(72) (72) states that δΤ_ΙΝ82 e Ker (Γ) , which in turn implies 5f_1NS e Ker (Γ) ' .

Any Γ that fulfills the requirement (72) can be used to implement the OSC given by (70) and (71). There are several different ways of computing a suitable transformation matrix Γ. The transformation matrix Γ that implements the OSC for the preferred embodiment is constructed as follows.

The SVD of H_d given by (26) is used to generate the SVD components U_d , V_d and∑_d whose diagonal elements are the singular values of H_d . Thus H_d is described in terms of its SVD components as follows.

(73) which becomes the following when H_d is column rank deficient

u_a = [u_dl u_d2] (74) (75)

0

(76)

0 0

(73) the becomes the following.

0^"

H_d = [U_d U_d2] (77)

0 0 V^T

30

RECTIFIED SHEET (RULE 91.1) The kernel equivalence property (46] shows that x₂ e Ker(H) implies Sr_1KS2 e Ker( _j) spanned by the columns of V_d2 . Since V_d ^T5r₂ = 0 for any Sr₂ e Ker (H_d ) , Γ = is a suitable IG measurement transformation that implements the observable subspace constraint when // and hence H_d are column rank deficient Furthermore this transformation is nonsingular when 4 or more satellites are available and H_d has full column rank. It can therefore be constructed and applied at every epoch, and thereby automatically implement the observable subspace constraint when H is deemded to be column rank deficient

Column rank deficiency or nearness to column rank deficiency is determined from the singular values of H_d which are the diagonal elements of∑_d in (73). A threshold d_mm on PDOP is specified to the QTC integration algorithm to indicate that H _d is nearly rank deficient and should be treated as rank deficient if PDOP exceeds the threshold. From (59), this translates to a threshold c_miri - l/d_mm on the singular values of H_d .

Γ - j, is then assembled as follows using the singular values C_j,c₂,c₃ and right singular vectors that are the columns of V_d obtained from the SVD (73).

(78) where ν_ώ is the right singular vector corresponding to the i-th singular value c_i for / = 1,2,3. By construction, the largest singular value will pass this test, and hence V_d] will contain at least one column. If all singular values equal or exceed l/d_max , then the Γ = V_d ^T will be an orthonormal matrix and the IG position measurement vector transformation (71) becomes a rotation. If one or two singular values are smaller than

j will have less than 3 rows and so constrain the transformed IG position measurement vector to be in the observable subspace defined by Ker (H ¹ .

The observable subspace constraint (70) and (71) on the IG position measurement in the AINS Kalman filter becomes the following. The transformed IG position measurement is computed every epoch as follows. Let V_dX be an mx3 matrix assembled from the right singular vectors

corresponding to m large singular values equal to or larger than

where m equals 1, 2 or 3. The transformed IG position measurement that the AINS Kalman filter 13 processes comprise the IG position measurement premultiplied by Γ = V_dy .

31

RECTIFIED SHEET (RULE 91 . 1 ) The corresponding transformed IG position measurement model is the IG position measurement model (5) also premultiplied by Γ = ν] ,

where ή_!σ is an m-dimensioned vector of uncorrelated white noises with diagonal variance matrix R_ie . Equations (79) and (80) comprise the OSC for the preferred embodiment

2.3 PREFERRED EMBODIMENT DESCRIPTION

This section is concerned with an explicit step-by-step description with words and the flowcharts of the process implemented in the preferred embodiment This description references equations in the previous sections.

The QTC integration process flowcharts shown in Figure 5 and describe the sequence of processing operations in the preferred embodiment It is part of a closed-loop AINS process described by Figure 1 or a feedforward AINS process described by Figure 2. Consequently Figure 5 and Figure 6 show only the steps that are specific to the QTC integration. Figure 7 shows the AINS Kalman filter measurement update process that uses the output of the QTC integration process among other sources of aiding information.

The QTC integration process starts at a given GNSS epoch that corresponds to the AINS Kalman filter epoch. The AINS Kalman filter has at this point completed its extrapolation process 20 that generates the extrapolated state and VCV matrix valid at the current epoch,

The GNSS position engine 26 designed to do INS position seeding receives the following input data required to compute a GNSS position fix:

• GNSS antenna position 24 derived from the INS 21 solution using equation (60)

• GNSS antenna position VCV 23 from the AINS Kalman filter extrapolation process 20 using equation (61)

• GNSS observables 25 for m satellites from the rover receiver 22 and possibly one or more reference receivers (not shown), where m is at least 2 and can be as high as 40.

The GNSS position engine 26 uses the antenna position 23 and position VCV 24 to do INS position seeding as described in Section 2.2.2 either by setting the INS-derived antenna position 23 as the a priori position whose errors the GNSS position engine estimates and corrects, or by setting the INS- derived position 23 as a measurement in the GNSS position engine estimation process.

The GNSS position engine 26 outputs the following data:

32

RECTIFIED SHEET (RULE 91.1) List of satellites used in the GNSS position solution and their orbital positions 27

GNSS antenna position and position variances 28

The AINS Kalman filter measurement construction process 29 then computes the IG^'position measurement (1] and corresponding measurement model [5), (9) using the GNSS antenna position and position variances 28. The outputs 30 arc the IG position measurement z_lc and corresponding measurement model A_IG .

The OSC process then computes and applies the OSC to the IG position measurement and measurement model 30 as described by (79) and (80) as shown in the OSC flowchart in Figure 6.

The OSC process receives the following data 41:

• GNSS antenna position

• Satellites in the position solution

• Positions of the satellites computed from ephemeris data from the GNSS receiver (done as part of the GNSS position fix)

• IG position measurement and measurement model components

The OSC process block 42 constructs the satellite differenced range measurement model matrix H_d given by equation (26) from the GNSS antenna position and positions of the satellites used to compute the GNSS position fix. It then computes the SVD of H_d also in block 42. The SVD output data 43 comprises matrices U_d , V_d and singular values c, , c_?, c₃.

The OSC process then assembles V_d iteratively in phantom block 44 as follows. V_dx starts as an empty matrix having no columns. Then for index = 1, 2, 3 the following actions occur. If the i-th singular value c, is large according to the test 45 for a specified d_m then the corresponding right singular vector v_i in V_d is appended as a new column to V_iU in block 47. The assembly of V_dX is completed when all three singular values are processed according to the test 46. The output 48 of this iterative process is V_dl with 1, 2 or 3 columns.

The OSC process block 49 applies the transformation matrix V _x to the IG position measurement

(1) as described by equation (79) and to the IG position measurement model (5), (9) as described by equation (80).

The OSC process exits at block 50 and returns control to the AINS Kalman filter in . This is also the point at which the process specific to QTC integration ends.

The AINS Kalman filter in Figure 7 receives the transformed IG position measurement (79) and measurement model (80) 36. ft runs a measurement update 37 that processes the IG position measurement along with any other constructed measurements to generate an updated state estimate and estimation VCV matrix 38. The AINS Kalman filter exits at 39 and returns to the main

33

RECTIFIED SHEET (RULE 91.1) AINS processing algorithm described in Figure 1 or Figure 2. The QTC integration process awaits the next AINS Kalman filter epoch to repeat the above processing steps.

If the AINS is a closed-loop AINS as described in Figure 1, then ituses the updated state estimate and estimation VCV matrix to perform INS error correction as is typical of a closed loop AINS.

If the AINS is a feedforward AINS as described in Figure 2, then ituses the updated state estimate and estimation VCV matrix to correct the output INS solution as is typical of a feedforward AINS.

34

RECTIFIED SHEET (RULE 91.1) 2.4 ALTERNATIVE EMBODIMENTS

2.4.1 ALTERNATIVE OBSERVABLE SUBSPACE CONSTRAINT TRANSFORMATIONS

Alternative embodiments may use a different approach to construct the OSC transformation matrix Γ. The following are possible methods.

• An eigenvalues decomposition of H_d H_d by definition yields the singular values and right singular vectors of H_d .

• Another alternative is to perform an orthogonalization on the columns of H_d . This process generates a set of orthogonal basis vectors for Ker(H_d)^L . Possible methods for this are the Gram-Schmidt algorithm, Hausholder transformation or Givens rotation. Then assemble the transposed basis vectors as the rows of Γ. This works only for exactly rank deficient H_d when fewer than 4 satellites are used and m < 3 since this method does not provide any measure of nearness to rank deficiency.

• The Gaussian elimination method reduces a matrix to row echelon form and thereby

identifies the linearly independent rows of H_d . The kernel of H_d corresponds to the zero rows and its orthogonal complement is spanned by the non-zero rows.

• The QR algorithm is a numerically sound alternative to the Gausian elimination method.

A first independent claim is concerned with handling fewer than 4 satellites in the GNSS position solution, and the second concerned with handling 4 or more satellites but with poor geometry leading to a large DOP.

The independent claims describe an OSC matrix that suppresses uncorrected components of the GNSS position solution when defective or poor geometry occurs. The dependent claims qualify the OSC matrix as having an image equal to the orthogonal complement of the kernel of H that is further qualified in subsequent dependent claims.

The second independent claim, Independent claim 8, describes the invention in the scenario where there are a sufficient number of satellites, i.e. 4 or more, but PDOP is large. In this case, the Hd

35

RECTIFIED SHEET (RULE 91.1) matrix is approximately rank deficient and hence has one or more singular values that are nonzero but small. The observable subspace constraint is in this case approximate or barely observable, and approximated as unobservable in the OSC. The same algorithm described in Figs. 5 and 6 handles both cases. The singular value test in Fig. 6 decision box 42 identifies both zero and small singular values. VI is assembled from right singular vectors that correspond only to the singular values that pass the test

36

RECTIFIED SHEET (RULE 91.1)

Claims

What is claimed is:

A quasi tightly coupled aided INS (QTC AINS) comprising:

a. an inertial navigator 21 mechanization,

b. a loosely-coupled AINS with a Kalman filter 20 that constructs 1NS-GNSS position measurements,

c. a GNSS position engine 22 that computes a position fix from obsetvables provided by a GNSS receiver and an externally provided a priori position and position VCV matrix,

d. an INS position seeding process in which the externally provided a priori position to the GNSS position engine is an antenna position computed from the INS position and attitude solution,

e. an observable subspace constraint (OSC) process (FIG. 6) that

i. computes an OCS matrix that suppresses the uncorrected component of the GNSS position error due to an insufficient number of satellites in the GNSS position solution in the IG position measurement constructed by the AINS Kalman filter, and

ii. multiplies the OSC matrix and the IG position measurement and

measurement model to suppress the uncorrected component of the GNSS position error in the IG position measurement and measurement model.

2. The QTC AINS described in Claim 0 in which the AINS is a closed-loop configuration in which the estimated INS errors from the AINS Kalman filter are used to correct the inertial navigator mechanization,

3. The QTC AINS described in Claim 0 in which the AINS is a feedforward configuration in which the estimated INS errors from the AINS Kalman filter are used to correct the navigation solution output from the inertial navigator mechanization.

4. The QTC AINS described in Claim 0 in which the a priori position VCV matrix to the GNSS position engine is computed from the AINS Kalman filter estimation VCV matix.

5. The QTC AINS described in Claim 0 in which the kernel or null space of the OSC matrix is the kernel or null space of a satellite-differenced range measurement model matrix H_d given by (26).

6. The QTC AINS described in Claim 5 in which the rows of the OSC matrix are the transposed right singular vectors corresponding to the zero singular values obtained from a singular value decomposition (SVD) of H_d .

7. A quasi tightly coupled (QTC) aided INS (AINS) comprising

a. an inertial navigator mechanization.

37

RECTIFIED SHEET (RULE 91.1) hh.. aa lloooosseellyy--ccoouupplleedd AAIINNSS KKaallmmaann ffiilltteerr tthhaatt ccoonnssttrruuccttss IINNSS--GGNNSSSS ppoossiittiioonn mmeeaassuurreemmeennttss,,

cc.. aa GGNNSSSS ppoossiittiioonn eennggiinnee tthhaatt ccoommppuutteess aa ppoossiittiioonn ffiixx ffrroomm oobbsseerrvvaabblleess pprroovviiddeedd bbyy aa GGNNSSSS rreecceeiivveerr aanndd aann eexxtteerrnnaallllyy pprroovviiddeedd aa pprriioorrii ppoossiittiioonn aanndd ppoossiittiioonn VVCCVV mmaattiixx,, dd.. aann IINNSS ppoossiittiioonn sseeeeddiinngg pprroocceessss iinn wwhhiicchh tthhee eexxtteerrnnaallllyy pprroovviiddeedd aa pprriioorrii ppoossiittiioonn ttoo tthhee GGNNSSSS ppoossiittiioonn eennggiinnee iiss aann aanntteennnnaa ppoossiittiioonn ccoommppuutteedd ffrroomm tthhee IINNSS ppoossiittiioonn aanndd aattttiittuuddee ssoolluuttiioonn,,

ee.. aann oobbsseerrvvaabbllee ssuubbssppaaccee ccoonnssttrraaiinntt ((OOSSCC)) pprroocceessss tthhaatt

ii.. ccoommppuutteess aann OOCCSS mmaattrriixx,, hhaavviinngg rroowwss aanndd hhaavviinngg ccoolluummnnss,, tthhaatt ssuupppprreesssseess tthhee aapppprrooxxiimmaatteellyy uunnccoorrrreecctteedd ccoommppoonneenntt ooff tthhee GGNNSSSS ppoossiittiioonn eerrrroorr dduuee ttoo aa ppoooorr ggeeoommeettrryy ooff ssaatteelllliitteess iinn iinn tthhee GGNNSSSS ppoossiittiioonn ssoolluuttiioonn iinn tthhee 1IGG ppoossiittiioonn mmeeaassuurreemmeenntt ccoonnssttrruucctteedd bbyy tthhee AAIINNSS KKaallmmaann ffiilltteerr,, aanndd iiii.. mmuullttiipplliieess tthhee OOSSCC mmaattrriixx aanndd tthhee IIGG ppoossiittiioonn mmeeaassuurreemmeenntt aanndd mmeeaassuurreemmeenntt mmooddeell ttoo ssuupppprreessss tthhee aapppprrooxxiimmaatteellyy uunnccoorrrreecctteedd ccoommppoonneenntt ooff tthhee GGNNSSSS ppoossiittiioonn eerrrroorr iinn tthhee IIGG ppoossiittiioonn mmeeaassuurreemmeenntt aanndd

mmeeaassuurreemmeenntt mmooddeell....

88.. TThhee QQTTCC AAIINNSS ddeessccrriibbeedd iinn CCllaaiimm 77 iinn wwhhiicchh tthhee AAIINNSS iiss aa cclloosseedd--lloooopp ccoonnffiigguurraattiioonn iinn wwhhiicchh tthhee eessttiimmaatteedd IINNSS eerrrroorrss ffrroomm tthhee AAIINNSS KKaallmmaann ffiilltteerr aarree uusseedd toto ccoorrrreecctt tthhee iinneerrttiiaall nnaavviiggaattoorr mmeecchhaanniizzaattiioonn..

99.. TThhee QQTTCC AAIINNSS ddeessccrriibbeedd iinn CCllaaiimm 77 iinn wwhhiicchh tthhee AAIINNSS iiss aa ffeeeeddffoorrwwaarrdd ccoonnffiigguurraattiioonn iinn wwhhiicchh tthhee eessttiimmaatteedd IINNSS eerrrroorrss ffrroomm tthhee AAIINNSS KKaallmmaann ffiilltteerr aarree uusseedd ttoo ccoorrrreecctt tthhee nnaavviiggaattiioonn ssoolluuttiioonn oouuttppuutt ffrroomm tthhee iinneerrttiiaall nnaavviiggaattoorr mmeecchhaanniizzaattiioonn..

1100.. TThhee QQTTCC AAIINNSS ddeessccrriibbeedd iinn CCllaaiimm 77 iinn wwhhiicchh tthhee aa pprriioorrii ppoossiittiioonn VVCCVV mmaattiixx ttoo tthhee GGNNSSSS ppoossiittiioonn eennggiinnee iiss ccoommppuutteedd ffrroomm tthhee AAIINNSS KKaallmmaann ffiilltteerr eessttiimmaattiioonn VVCCVV mmaattiixx..

1111.. TThhee QQTTCC AAIINNSS ddeessccrriibbeedd iinn CCllaaiimm 77 iinn wwhhiicchh tthhee kkeerrnneell oorr nnuullll ssppaaccee ooff tthhee OOCCSS mmaattrriixx iiss tthhee aapppprrooxxiimmaattee kkeerrnneell oorr nnuullll ssppaaccee ooff aa ssaatteelllliittee ddiiffffeerreenncceedd rraannggee mmeeaassuurreemmeenntt mmooddeell mmaattririxx HH_dd ggiivveenn bbyy ((2266J) wwhheenn iitt iiss cclloossee ttoo ccoolluummnn rraannkk ddeefificciieennccyy aass ddeetteerrmmiinneedd bbyy tthhee ssmmaalllleesstt ssiinngguullaarr vvaalluuee ooff HH_dd bbeeiinngg ssmmaalllleerr tthhaann aa ssppeecciiffiieedd tthhrreesshhoolldd ffoorr nneeaarrnneessss ttoo ccoolluummnn rraannkk ddeefificciieennccyy..

1122.. TThhee QQTTCC AAIINNSS ddeessccrriibbeedd iinn CCllaaiimm 1111 iinn wwhhiicchh HH_dd iiss ddeeffiinneedd ttoo bbee cclloossee ttoo rraannkk ddeeffiicciieennccyy wwhheenn tthhee ddiilluuttiioonn ooff pprreecciissiioonn ((DDOOPP)) ccoommppuutteedd aass tthhee ssqquuaarree rroooott ooff tthhee ssuumm ooff ddiiaaggoonnaall

13. The QTC AINS described in Claim 11 in which the rows of the OSC matrix are the transposed right singular vectors corresponding to the large singular values obtained from a singular value decomposition (SVD) of H_d.

38

RECTIFIED SHEET (RULE 91.1)

14. The QTC AINS described in Claim 13 in which a singular value is defined to be large if it equals or exceeds a large singular value threshold.

15. The QTC AINS described in Claims 12 and 13 in which the large singular value threshold for a large singular value is the inverse of the large DOP threshold,

16. A quasi tightly coupled GNSS-INS Integration Process (a QTC Process) for use in a GNSS-INS system for constraining the use of GNSS receiver data derived from observables from m satellites, as the receiver data quality falls to an acceptable but marginal quality, the QTC process selectively constrains the use of GNSS receiver data to data having a quality that is equal to or better than data with a level of quality that exceeds a predetermined threshold, the QTC process comprising:

A GNSS position engine 26 coupled to receive a-priori position VCV data from a Kalman Filter, a-priori position data from an INS, and observables from m satellites via the GNSS receiver, the GNSS position engine 26 implements a GNSS positioning algorithm to compute an antenna position fix and an estimation variance-covariance (VCV) matrix,

an AINS Kalman filter IG position measurement function 27 coupled to the GNSS position engine to calculate and receive the GNSS antenna position fix and an estimation variance- covariance (VCV matrix) to construct an INS-GNSS (IG) position measurement difference v

z_j(, data matrix and a measurement model data matrix I½ and transfers the INS-GNSS (IG) v

position measurement difference z_la data matrix and a measurement model data matrix Hm to the OSC process,

the OSC process uses the INS-GNSS (IG) position measurement difference /G data matrix and a measurement model data matrix Hm to compute the satellite differenced range measurement model matrix Hi, and the SVD of the H_& matrix 41,

the OSC constraint test process then uses a portion of the SVD to determine if a number of observables from m satellites were equal to or greater than a predetermined limit for acceptable quality, the constraint test process allowing the use of GNSS receiver data received within an epoch subject to the GNSS receiver data equal or exceeding a predetermined quality test limit.

39

RECTIFIED SHEET (RULE 91.1)