ABSTRACT
The feasibility of improving airborne GPS/INS integrated navigation by
use of accurate deflection of the vertical (DOV) information is investigated in
this paper. Based on the available 2’´2’ DOV grid, tests were run
on GPS/INS navigation data, and the results were compared with the
corresponding results where the DOV compensation was not used. The issue of
stochastic modeling of the anomalous gravity field in the state vector of the
integrated filter is also addressed here. The tests were conducted using the
Airborne Integrated Mapping System (AIMSÔ), whose positioning
component is based on a tightly integrated GPS/INS. The system is designed to
provide orientation and position of an aerial platform, with estimated accuracy
of 4-7 centimeters in position, and ~ 10 arcsec in orientation in post-mission
mode. A brief overview of the system is also given, including a discussion of
current error modeling and airborne test results.
INTRODUCTION
Integrated systems based on Global Positioning System (GPS) and Inertial Navigation System (INS) generated increased interest in the airborne survey and remote sensing community over the past few years as a tool for direct georeferencing of aerial imagery [17, 25, 20, 22, 1, 28, 7]. At the same time, the use of GPS/INS systems for airborne gravimetry has been researched by the geodetic and navigation community [16, 3, 26, 6, 12, 13, 10, 31]. With full operational GPS capability, it has been recognized that an optimal combination of GPS with inertial navigation brings a number of advantages over stand-alone inertial or GPS navigation. GPS contributes its high accuracy and stability over time, enabling continuous monitoring of inertial sensor errors. Implementation of closed-loop INS error calibration allows continuous, on-the-fly error update that bounds INS errors, leading to increased estimation accuracy. On the other hand, INS contributes immunity to GPS outages, continuous attitude solution, and reduction of the GPS ambiguity search volume/time. Using a GPS-calibrated, high to medium accuracy inertial system, attitude accuracy in the range of 10-30 arcsec can be achieved [27, 1, 4, 7, 8]. Conversely, an array of GPS antennas mounted on a mobile platform can also provide attitude estimates, complementary to those, determined by the calibrated gyro observable. The GPS-derived attitude components, available usually at 1-10 Hz rate, can be good to 1-2 arcmin [18, 31], depending mainly on the GPS antenna baseline length and orientation with respect to line-of-sight, and multipath level. The issue of ambiguity resolution is not addressed here; it is assumed that some ambiguity resolution method is applied before the attitude components are estimated. For example, the Ashtech GPS 3DF-ADU multi-antenna system is capable of providing 2-Hz attitude information, with accuracy estimated at 0.2° RMS for heading, and 0.4° RMS for pitch and roll, based a 1-meter square antenna array (3DF-ADU Manual), and the Trimble TANS Vector delivers 10-Hz attitude information with an accuracy of 0.3° for 1-meter baseline (TANS Vector product description). In order to obtain 0.1° pointing accuracy, however, the baselines connecting the electrical centers of GPS antennas must be known to within 2 mm on a 1 m baseline [32]. This requires baseline estimation by the attitude determination filter, as the baseline separation can change from the pre-surveyed value during the airborne or spaceborne mission. In general, the stand-alone multi-antenna GPS provides less accurate, and at much lower frequency, attitude information as compared to integrated GPS/INS with a high accuracy inertial system supported by differential GPS (DGPS). Using quality DGPS it’s possible to bound the gyro drifts and obtain high accuracy attitude estimates even with a single roving GPS antenna. The implementation of a multi-array GPS system, however, may become more important for the integrated systems that utilize low accuracy INS, rendering more robust system. The gyros can estimate the phase differences related to the airborne GPS baselines, forming a class of external updates to the integrated filter. In such a case, special care should be exercised towards proper parameterization and dynamic modeling of the filter states, since attitude and baseline errors (that need to be added to the state vector) are closely coupled, and some observability issues can arise. Moreover, an attitude motion is needed to separate baseline and attitude errors in the combined filter. In addition, since attitude is usually needed with respect to some frame different from the body frame defined by the GPS antennas, a transformation between the frame defined by the GPS antenna array, and the frame of interest is needed. Some external measurement is required to resolve the rotation between these two frames. However, if any baseline flexure is present, this relationship will not be constant, and would require external information to resolve this rotation.
The Ohio State University Center for Mapping has developed an integrated GPS/INS system as a part of a fully digital Airborne Integrated Mapping System (AIMS™), designed for large-scale mapping and other precise mapping applications. The AIMS™ positioning module is based on a tightly integrated, dual frequency differential GPS (single roving antenna) and the Litton LN100 inertial navigation system. The project’s goal is to acquire the position and orientation of an aerial platform with an accuracy of 4-7 centimeters and below 10 arcsec, respectively, in post‑processing mode. This level of accuracy offers a possibility to reduce or even eliminate the need for aerotriangulation, translating to substantial savings in data processing time and cost in aerial surveying. The performance of AIMSÔ positioning has been repeatedly evaluated during several tests with different flight conditions and varying base-rover separation [9, 29]. The system’s accuracy estimates derived from the covariance matrix were not yet validated by an independent, high‑accuracy ground truth test; the major reason for that being the limited accuracy of the aerotriangulation solution, as presented by [29]. However, assuming reliable sensor error modeling [21], the internally estimated position and attitude error terms can be considered as a dependable measure of accuracy. It is our belief that the sensor error models implemented in the integrated filter are reliable (see Table 2), since they are based on several years of testing by the LN100 manufacturer; therefore, the RMS analysis should provide a good indication of system accuracy. Further tests against the ground truth are currently in progress. Table 1 shows the typical errors internally estimated from several test flights [9]. It should be mentioned that carefully planned maneuvers performed between the steady portions of the test flights were able to keep the platform orientation at less than 10 arcsec level even for the long base-rover separation.
|
Estimated Error
Components (RMS) |
Base-rover
separation |
Units |
|
|||||||||
|
50 |
100 |
200 |
350 |
[km] |
|
|||||||
|
North coordinate |
10 |
9 |
9 |
15 |
[mm] |
|
||||||
|
East Coordinate |
9 |
8 |
8 |
15 |
[mm] |
||||||
|
Vertical
Coordinate |
20 |
21 |
18 |
31 |
[mm] |
|
||||||
|
Heading |
6.0 |
6.3 |
7.5 |
7.0 |
[arcsec] |
|
||||||
|
Pitch |
3.0 |
3.7 |
5.1 |
5.1 |
[arcsec] |
|
||||||
|
Roll |
3.3 |
4.1 |
5.1 |
5.0 |
[arcsec] |
|
||||||
Table 1. Estimated standard deviations in position and orientation as a function of base-rover separation.
This paper presents the ongoing effort to upgrade the system to provide
even more accurate estimates of position, and especially, orientation. In the
integrated system during the GPS outages, the quality of free-inertial
navigation with the high- to medium-accuracy INS currently may well depend on
the knowledge of the local gravity field. Therefore, gravity anomaly and
deflection maps given on dense grids, based on gravimetric surveys, are
becoming more significant to precise navigation. High-accuracy vertical
deflection compensation should refine not only the positioning accuracy, but
also the attitude determination of the platform. On one hand, the method of
improving attitude estimates by accurate vertical deflection information should
be labeled as “low altitude technique” due to the rapid attenuation of DOV with
altitude. On the other hand, the AIMS™
system is designed particularly for large-scale mapping, and therefore, is
naturally confined to low altitudes, where the method is specifically
applicable. For the high altitude flights supported by integrated GPS/INS, the
multi-antenna array may become an optimal solution, providing attitude
determination support independently of the platform’s altitude. In this paper
the implementation of the National Imagery and Mapping Agency (NIMA) 2’´2’ deflection of the vertical (DOV) data, and
the resulting performance evaluation of the platform position and orientation,
are presented.
AIMS™: CURRENT CONFIGURATION
The AIMS™ positioning component currently comprises
two dual-frequency, single-antenna Trimble 4000SSI GPS receivers, and a
medium-accuracy, and highly reliable, strapdown Litton LN-100 inertial
navigation system, based on the Zero-lockTM Laser Gyro (ZLGTM)
and an A-4 accelerometer triad (0.8 nmi/h CEP, gyro bias – 0.003°/h,
accelerometer bias – 25mg). The LN-100 firmware,
modified for the AIMSTM project, allows for access to the raw IMU
data, updated at 256 Hz [21]. Estimation of errors in position, velocity, and
attitude, as well as errors in inertial and GPS measurements, is accomplished
by a centralized Kalman filter that processes GPS L1/L2 phase observables in
double-differenced mode, together with the INS strapdown navigation solution
[4, 7]. The state vector components and
their stochastic model representation are presented in Table 2. Figure 1
illustrates the current hardware configuration.
Kalman Filter States1
|
Number of States
|
Initial Conditions (1 sigma) |
|
Navigation
Parameters Position
errors Velocity
errors Attitude
errors Heading |
3 3 2 1 |
1.0 m 0.5 m/s 0.01° 0.05° |
|
Accelerometer
Errors Biases Scale factor errors |
3 3 |
30.0mg 50 ppm, (5.0 mg*Ös) |
|
Gyro Errors Drifts |
3 |
0.010 deg/h (0.001deg/Öh) |
|
Gravity
Deflection
(Gauss-Markov, T = 20 nmi) Anomaly (Gauss-Markov, T = 20 nmi) |
2 1 |
25 mgal 35 mgal (5mgal)2 |
|
GPS Errors Ionospheric delay (random walk) |
Number of DD |
2 cm 1 mm*Ös |
Table 2. Filter state vector characteristics.
A compact form of the state equation of the filter model can be presented by the following matrix equation:
where 
, 
, 
,
, 
, and 
are, respectively,
the error vectors of the inertial navigation solution, the accelerometer
measurement error, the gyro measurement error, the gravity anomaly and
deflections, the antenna lever arm errors, and the GPS ionospheric errors; 
, 
, 
, 
, and 
are all zero-mean
Gaussian white noise vectors. More detailed expressions of the matrices F1j (j=1,2,3,4) and F22
are provided in Appendix A.
Figure 1. AIMS™ hardware
configuration – INS and digital camera mount.
GRAVITY ERRORS
IN INERTIAL NAVIGATION
Inertial navigation is
based on Newton’s second law of motion defined in the inertial (non-rotating)
frame. Thus, the basic mathematical model is expressed as the following system
of second order differential equations:
Where:
and`r is the total acceleration
vector, `a is
the acceleration sensed by the
accelerometer, `g
is the total gravitational acceleration vector, `gm is the gravity model, D`g is
the difference between the actual gravity and the gravity model used, go
is the nominal value of gravity, x and h are vertical deflections of
the vertical, and Dg is the gravity anomaly. Since the
accelerometer triad measures the difference between kinematic inertial
acceleration and mass gravitation, errors in the measurements are affected by
errors in knowledge of the gravity, leading ultimately to positioning errors.
The uncertainty in gravity (deflections of the vertical and the gravity
anomaly) is defined as the difference between the actual gravity and the
gravity model implemented in the inertial navigation algorithm. Deflections of
the vertical and the gravity anomaly contain the information mostly in the
medium- to short‑frequency range whereas the model usually represents the
long wavelength part of the spectrum.
Different models, ranging from the normal gravity to high order
spherical harmonic expansion, can be used to approximate the Earth’s gravity
field. The normal gravity model approximates the Earth’s gravitation with an
accuracy of 1 part in 104, whereas detailed global gravitational
models are good to 1 part in 105 [13]. Historically, normal gravity
was satisfactory for free inertial navigation, but today’s high-accuracy
requirements for GPS-aided inertial systems bring even more demands for
knowledge of the Earth’s gravity field. During GPS outages, the quality of
free-inertial navigation with the high- to medium-accuracy INS currently may
well depend on the knowledge of the local gravity field. The gravity anomaly
and deflection maps given on dense grids, based on gravimetric surveys, are
becoming of more importance to applications that require precise navigation.
Errors in inertial navigation can be divided into two major groups –
system related errors, and modeling errors. Measurement noise, calibration and
alignment errors comprise the first group, whereas the second group contains
approximation, linearization and lumping errors [2]. The uncertainties in the
gravity vector fall in the second category, and are an important error source
for high-quality inertial navigation systems, especially during long missions,
where the gravity errors tend to build up. The uncertainties in knowledge of
the gravity field produce errors that cannot be eliminated by pure inertial
system. In the aided navigation systems, however, sensor errors and the errors
caused by gravity can be bounded. For example, an optimal combination of the
inertial sensor with GPS allows continuous estimation of the errors in the inertial
sensors and the gravity model, which can be fed back to the navigation
algorithm, leading to a precise inertial navigation solution. Subsequently, an
error-calibrated inertial system can be used for navigation during the GPS
outages, and fills in between the GPS discrete trajectory estimates. A detailed
expression of the INS system error model depends on the type of mathematical
equations chosen for describing the gyro and accelerometer measurement errors,
as well as for the gravity anomaly and deflections. Clearly, the filter design
object is to achieve the desired modeling accuracy without unnecessarily
increasing the complexity of the models and, thereby, the filter [11].
A vast array of literature exists discussing the effects of gravity
uncertainties on inertial navigation, and stochastic gravity modeling. For a
more detail discussion, and closed expressions for the various covariance
models of anomalous gravity and deflections, the reader is referred to [19, 2,
14, 15, 23, 5, 6, 12, 13]. It is worth mentioning, however, that generally all
the authors agree that the Gauss-Markov process can be used as the stochastic
representation of the anomalous gravity field in the state vector estimation
process. This implies that, by definition, the anomalous gravity field is
stationary and its probability density is Gaussian. Medium-order (second to
fourth) Gauss-Markov processes are considered as self-consistent in
representing the anomalous gravity field, and therefore better serve the
purpose than the first order process. Gauss-Markov models for the gravity field
are, in general, designed to reflect the omission of the long-wavelength
component in the estimation process, and thus assume that the compensation
models (or basic approximation) used are significantly higher than the normal
gravity model. This implies that the unmodeled effects become more and more
local in character.
In general, the higher the order of the spherical harmonic expansion
used for computing the gravity intensity for the vertical axis in the
navigation algorithm, the better the accuracy of the vertical velocity and
altitude determination. Subsequently, since the velocity error couples from the
vertical to horizontal axes (and vice versa) through the error in the Coriolis
term, these effects are reduced as the velocity error is reduced due to less
error from the gravity anomaly vector. Moreover, when the deflections of the
vertical are compensated, there is less tilt error, and consequently, less
coupling of the horizontal accelerations into the vertical axis. Thus, it can
be expected that (high-accuracy) deflection of the vertical compensation should
improve not only the positioning accuracy, but also the attitude determination
of the platform. This can be verified by close examination of the dynamics
matrix of the system, which shows the interrelation among the estimated errors.
The errors in deflections of the vertical enter directly into the horizontal
velocity errors in linear combination with the attitude errors. This, generally
speaking, complicates full separation in the estimation procedure.
Consequently, if one of the components, in this case DOV, becomes (partially)
known, an improvement in estimation of the attitude should ultimately be
achieved.
The following sections present the NIMA deflection of the vertical
(DOV) data used in the airborne tests with the GPS/INS measurements and the
system performance analysis with and without the partial deflection
compensation.
NIMA DOV: PRODUCT DESCRIPTION
In the tests presented here,
the unclassified NIMA Standard Inertial Navigation Product (INP) was used over
the designated test areas. INP is a three-dimensional, gridded database of
deflections of the vertical and their errors at 2’´2’ separation ranging from 0 ft to 90K ft
above the geoid level. It was derived from 1’´1’ free-air gravity
anomalies, selected from a NIMA point gravity anomaly (PGA) files using the
Variable Interval selection program. The accuracy of the gravity anomaly ranges
between 1 and 5 mgal (5 mgal corresponds roughly to 1 arcsec in deflection
accuracy). A short description of the INP development process is given below
[Steven Kenyon, NIMA, personal communication]. The GRAVSOFT software package as
described in [30], was used for INP development purpose. The designated
reference system is WGS 84.
As a first gravity reduction
step, the EGM96 (360, 360) gravity model was used to remove the long-wavelength
effects from the original point gravity data. Subsequently, the high frequency
components were also removed using Residual Terrain Models (RTM), implementing
a prism integration technique. NIMA's most detailed set of 1’´1’ Digital Terrain Elevation Data (DTED),
along with 5’ and 30’ mean elevations, were used in the RTM computations
process. After the EGM96 and RTM reductions, the point gravity was gridded to a
2’´2’ mesh, using a local least-squares
collocation procedure. The next step was to derive a 2’´2’ grid of elevations from the original 1’´1’ DTED, and use them to downward continue
the gravity anomalies to the surface of the geoid. A two-kilometer attenuation
factor was used.
Two-dimensional FFT
transform of Vening Meinesz formulas were then employed to calculate DOV from the
reduced free-air gravity anomalies. The entire area of conterminous US (CONUS)
was segmented into 10°´10° blocks with 3-degree
overlap. Based on the local covariance modeling technique, developed by
Forsberg [5], least-squares collocation was used to obtain the accuracy
estimates for the DOV. First the empirical covariance models were developed;
next the analytical models with proper variance and correlation length were
fit. Subsequently, the DOV were upward continued to 10K, 20K, and 30K ft
elevations. Subsequently, the DOV due to RTM and EGM96 (360, 360) were computed
at 0K, 10K, 20K, and 30K, and restored to the final predictions.
The bilinear interpolation
to the three-dimensional locations in space that uses two neighboring layers of
the gridded DOV data was applied to provide the final deflections to the
strapdown navigation algorithm. The platform position at discrete 1-second
locations was passed to the interpolating algorithm, and the current
deflections were used for the entire second, for the 64 Hz navigation
solution. The quality of the DOV data provided for the tests presented here was
at the level (1 sigma) of 0.5 arcsec [Steven Kenyon, NIMA, personal
communication]. However, a somewhat conservative standard deviation of 1 arcsec
was used in the data processing.
NUMERICAL RESULTS
Effects of DOV compensation on the positioning and attitude estimates
The model most commonly used for the effects of the anomalous gravity
field in the state vector is the time correlated noise process, as mentioned in
the previous sections. The second- to fourth-order Gauss-Markov process
combined with the high-order reference field is recommended by [15, 23, 12].
However, the first-order Gauss-Markov process is often used for its simplicity,
and was also applied together with normal gravity as a basic gravity field
approximation, in the data reduction for the tests presented in this section.
It should be mentioned that the first order Gauss-Markov process, combined with
normal gravity as a basic approximation, can understate the actual errors in
the horizontal gravity components, primarily due to the fact that the very long
wavelength cannot be estimated [12].
The test flights analyzed in this paper were conducted in 1997 and
early 1998 with St. Louis-based Image America (formerly OMNI Solutions
International Ltd) in the St. Louis area. A total of two missions comprising
four test flights (see Table 3) were flown to evaluate the performance of the
integrated positioning/orientation component. The data collected during these
tests were processed with two different data reduction modes provided by the
AIMS™ software — mode 1, where no prior knowledge of DOV was assumed, and mode
2, where the NIMA 2’´2’ DOV data were used — with
RMS (1 sigma) of 1 arcsec. The first-order Gauss-Markov process was used as a
stochastic representation of the anomalous gravity field as indicated above
(see Table 2).
|
Flight Number |
Date |
Altitude [m] |
Speed [m/s] |
|
1 |
4/23/98 |
4500 |
125 |
|
2 |
4/24/98 |
4800 |
120 |
|
3 |
3/6/97 |
4800 |
120 |
|
4 |
3/6/97 |
4800 |
125 |
Table 3. Test flight schedule.
The position and orientation
estimates from both processing modes were compared, and typical results are
presented in Figures 2-5 and Table 4. Figure 2 illustrates the attitude
difference between the solutions, i.e., with and without DOV data, for the
steady-state portion of test flight #2. The estimated attitude standard
deviations (square roots of the diagonal terms of the filter covariance matrix)
for the same steady-state portion of test flight 2, for DOV-compensated case
are shown in Figures 3 and 4, whereas their counterparts from the
DOV-uncompensated solution are shown in Figure 5. The comparison of Figures 3,
4, and 5 indicates improvement in the estimated standard deviations with
respect to mode 1 by 1.5-2.5 arcsec. This represents the average gain due
to the DOV compensation, and is observed at a similar level for all the
analyzed test flights. In addition, some improvement in the order of 1 arcsec
(average) for the heading standard deviation was observed. Generally, the
DOV-compensated solution shows faster convergence and steadier spectrum of the
standard deviation, as compared to the uncompensated mode. Table 4 presents the
summary of the comparison between both processing modes for the test flights
listed in Table 3. It shows that the RMS of the attitude differences ranges
from 3 to 5.6 arcsec for pitch, and 3.1 to 17.2 for roll, with the largest
differences reaching 14 and 21.4 arcsec, respectively.
Figure 2. Differences in attitude between solutions with and without DOV
compensation (flight 2).
Figure 3. Pitch standard deviation
(square roots of the diagonal terms of the filter covariance matrix) for
solution with DOV compensation (flight 2).
Figure 4. Roll standard deviation (square roots of the
diagonal terms of the filter covariance matrix) for solution with DOV
compensation (flight 2).
Figure 5. Attitude standard deviation (square roots of the
diagonal terms of the filter covariance matrix) for solution without DOV
compensation (flight 2).
|
Tilt |
Flight # |
RMS [arcsec] |
Max [arcsec] |
Mean [arcsec] |
|
Pitch |
1 2 3 4 |
3.9 3.0 5.6 3.9 |
10.9 8.4 12.2 -14.0 |
1.7 -0.1 -1.6 -3.2 |
|
Roll |
1 2 3 4 |
3.3 3.1 4.0 17.2 |
-11.1 11.1 12.1 -21.4 |
-0.8 -0.4 2.7 -17.1 |
Table 4. Difference in attitude
between solutions with and without DOV compensation.
Figures 6 and 7
illustrate the deflection of the vertical compensation used in the flight #2
data reduction process. It shows, that north-south DOV, changes between 1 and 5
arcsec for the entire flight duration. The east-west component is significantly
smaller, ranging from 0.5 to –2.5 arcsec. The amount of compensation applied in
mode 2 is reflected in the order of magnitude of the attitude differences
between both solutions, as plotted in Figure 2.
Figure 6. North-south deflection signature, flight #2.
Figure 7. East-west deflection signature, flight #2.
One aspect worth analyzing is the behavior of the attitude standard deviations related to filter’s covariance matrix, as a function of the flight trajectory. For example, the portion of the flight between epochs ~3000-4000 in Figures 3-5 corresponds to a longer (~ 90 km) straight portion of the trajectory, where no maneuvers were performed. It appears that for the solution without DOV compensation the attitude standard deviations show an increase from 2 to 4 arcsec, whereas in the DOV-compensated case they show a rather flat spectrum at ~1.5 arcsec level, with only very minor growth at the specified interval. The aircraft maneuvers following this part of the trajectory brought the mode 1 estimated errors in attitude down to the 3 arcsec level.
Another test performed with the results from both solutions (i.e., DOV-compensated and uncompensated) is illustrated in Figures 8-11. They show typical horizontal error growth and positional standard deviations during the outage of the GPS signal. In both cases, the filter was run for approximately 15 minutes to allow for error calibration, and then an intentional gap of 60 s was introduced to both solutions at epoch marked 60 in Figures 8-11. Resulting trajectories were compared to the reference truth based on DGPS/INS post-mission solution. The results from the DOV-compensated case show an improvement of about 50% in the error level of the north component, and some minor improvement in the east component (Figure 10), as compared to Figure 8, which illustrates the DOV‑uncompensated results. The respective positional standard deviations (square roots of the diagonal terms of the filter covariance matrix) also show improvement of ~20%, as compared to the solution with no DOV compensation. Smaller improvement in the east component contrary to the north coordinate can be explained by smaller DOV signature observed in east-west direction of the analyzed flight. It can be observed from Figure 8 that the average error growth rate is about 1.2 mm/s for the north, and 0.4 mm/s for the east component, for the first 60 s of GPS total loss of lock. These results indicate that most of the correlated IMU errors have been correctly estimated by the Kalman filter, and removed from the IMU measurements by the feedback update loop. Thus, the system is able to provide accurate velocity and position information even when GPS lock is lost for a short period of time. Moreover, an introduction of gravity compensation leads to further improvement, lowering the rate growth for the north component to 0.5 mm/s. A high quality of free inertial navigation during the GPS outages is very important for fast ambiguity recovery after tracking is reestablished. It supports the GPS cycle-slip correction, and makes fast and robust On-The-Fly (OTF) ambiguity resolution possible. INS error growth in AIMSÔ is below 10 cm for the horizontal components, even for solution without DOV compensation, and below 20 cm in the vertical direction (not shown here), after a 60-second loss of GPS lock, which still enables instantaneous ambiguity recovery after the GPS signal is recovered (for details see Grejner-Brzezinska et al., 1998].
In order to test the influence of DOV compensation on the positional error for the extended period of the free navigation mode, the Kalman filter was calibrated for about 1000 s, and subsequently the GPS signal was intentionally cut off for 3000 s. This was repeated for two independent solutions – with and without DOV compensation. The resulting trajectories were again compared to the GPS/INS post-processed truth trajectory, and the differences in the north coordinate are plotted in Figure 12. It can be observed that the DOV-compensated solution shows positional error smaller by 20-50 m, as compared to DOV- uncompensated solution, depending on the error in compensation itself and the total duration of the free navigation mode. The positional errors due to gravity in the DOV-compensated case result from the finite grid resolution (2’), which might not capture the shorter wavelength signature, and the accuracy of DOV itself (assumed at 1 arcsec). It should also be mentioned that the east component (not shown here) presents similar behavior.
Figure 8. Horizontal
error growth during GPS outage introduced at epoch 60: solutions without DOV
compensation.
Figure 9. Estimated
standard deviations (square roots of the diagonal terms of the filter
covariance matrix) of the horizontal positioning during GPS outage introduced
at epoch 60: solution without DOV compensation.
.
Figure 10. Horizontal error growth during GPS outage introduced
at epoch 60: solution with DOV compensation.
Figure 11. Estimated
standard deviations (square roots of the diagonal terms of the filter
covariance matrix) of the horizontal positioning during GPS outage introduced
at epoch 60: solution with DOV compensation.
Figure 12. North coordinate error growth, free navigation mode
with DOV compensation.
Third order Gauss-Markov
model for gravity anomaly and vertical deflections
As discussed in the previous sections, the first order Gauss-Markov
process is not considered an optimal stochastic representation for the
anomalous gravity field, especially if the normal gravity is used as a basic
approximation for the gravity signature in the navigation algorithm. In order to test the effects of different
stochastic models for gravity anomaly and deflections on the filtering results,
a third-order Gauss-Markov model, which properly reflects the strong
attenuation of gravity signal with attitude, was initially implemented (1 sigma
for deflections = 25 mgal, 1 sigma
for gravity anomaly = 35 mgal,
T = 20 nmi for all three components). The positioning results from a
test flight over the Madison, OH test range, obtained with the third-order
Gauss-Markov model for gravity were compared to the corresponding first-order
Gauss-Markov processing results, as shown in Figure 13. The difference in horizontal coordinates
ranges between 2 and 6 cm, whereas vertical components differ by 10 cm at most,
being at the level of 2-3 cm for most of the flight. The results presented here
are preliminary; more tests have to be conducted towards estimating the best
correlation distances for gravity anomaly and deflections.
Figure 13. Positioning results difference between solutions
with 1st and 3rd order Gauss-Markov models for gravity.
CONCLUSIONS
The major objective of this paper was to test the feasibility of attitude estimation improvement by using high‑accuracy DOV information in the integrated GPS/INS navigation system. The covariance analysis results indicate that the estimability of the attitude components improves by adding partially compensated (1 arcsec RMS) gravity information. The Kalman filter converges faster and the standard deviation spectrum is less sensitive to the lack of maneuvers, which is normally observed during the straight portions of the flight. Free inertial navigation during the GPS outages in DOV-compensated mode shows slower error accumulation, especially in the horizontal components, as compared to the DOV-uncompensated solution. However, these results are preliminary and some improvements are needed, especially in the stochastic modeling of the gravity anomaly vector, as indicated in the previous section.
ACKNOWLEDGEMENTS
The AIMS™ project is supported by the NASA Stennis Space Center, MS, Commercial Remote Sensing Program grant #NAG13-42, and a grant from Litton Systems, Inc. Special thanks go to Richard Salman and Steven Kenyon from NIMA, St. Louis office, for providing the gravity data for the test flights analyzed here.
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