Method for determining gravity in an inertial navigation system

The invention is a method for determining gravity in an inertial navigation system which periodically produces and stores in memory position coordinates. The method comprises the steps of (a) retrieving the most recently determined position coordinates, (b) determining coordinates L.sub.u, L.sub.n, H.sub.u, and H.sub.n from the position coordinates, L.sub.u and L.sub.n being predetermined functions of geodetic latitude and H.sub.u and H.sub.n being predetermined functions of geodetic altitude, (c) determining the vertical component of gravity by substituting either or both L.sub.u and H.sub.u in a first polynomial expression, (d) determining the north-south component of gravity by substituting either or both L.sub.n and H.sub.n in a second polynomial expression, and (e) utilizing the components of gravity determined in steps (c) and (d) in the next determination of the position coordinates.

CROSS REFERENCE TO RELATED APPLICATIONS 
(Not applicable) 
STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH AND DEVELOPEMENT 
(Not applicable) 
BACKGROUND OF THE INVENTION 
This invention relates generally to inertial navigation systems and more 
specifically to determining gravity in such systems. 
The velocity V of interest in navigating a vehicle relative to the earth is 
defined by the equation 
##EQU1## 
where 
##EQU2## 
is the rate of change of the vehicle's velocity relative to the earth 
expressed in a NAV (N) frame of reference (local-level with origin fixed 
at the center of the earth), a.sub.s is the specific-force acceleration 
experienced by the inertial navigation system on board the vehicle, g is 
gravity, .PHI. is the rotation rate of an earth-fixed frame of reference 
relative to an inertial frame (i.e. earth's rotation rate vector), R is 
the position vector of the vehicle from the center of the earth, and 
.omega. is the rotation rate of the local-level frame relative to the 
inertial frame. In order to integrate 
##EQU3## 
and obtain V, an accurate expression for g is required. 
The so-called normal gravity potential .PHI. (the most accurate gravity 
model presently available) is given in terms of ellipsoidal coordinates 
(.mu.,.beta.,.lambda.) as 
##EQU4## 
and 
##EQU5## 
where G.sub.m is the earth's gravitational constant, .alpha. is the 
semi-major axis of WGS-84 ellipsoid (DMA Technical Report, Department of 
Defense WGS-84, TR 8350.2), b is the semi-minor axis of WGS-84 ellipsoid, 
(e,n,u) are the vehicle coordinates in the NAV (N) frame (e-east, n-north, 
u-up), and .OMEGA. is the earth rotation rate. 
Given the vehicle's location in geodetic coordinates (.phi., .lambda., h), 
the normal gravity vector expressed in the ECEF (earth-centered, 
earth-fixed) frame is given by 
##EQU6## 
where 
##EQU7## 
is the transformation matrix from the ellipsoidal (U) frame to the ECEF 
(E) frame, and g.sup.U, the normal gravity vector, is expressed in 
ellipsoidal coordinates as 
##EQU8## 
Note that u=b, q=q.sub.0, .nu.=a, and g.sub..beta. =0 when h=0. 
The normal gravity vector expressed in NAV (N) frame coordinates is given 
by 
##EQU9## 
where 
##EQU10## 
To determine the normal gravity vector at the vehicle's location expressed 
in NAV-frame coordinates, one first transforms to geodetic coordinates, 
then to ECEF coordinates, and finally to ellipsoidal coordinates. The 
geodetic-to-ECEF transformation is defined by the equations 
EQU x=(N+h) cos .phi. cos .lambda. 
EQU y=(N+h) cos .phi. sin .lambda. 
EQU z=(Nb.sup.2 /a.sup.2 +h) sin .phi. (17) 
where 
##EQU11## 
The transformation to ellipsoidal coordinates is defined by the equations 
##EQU12## 
where 
EQU r.sup.2 =x.sup.2 +y.sup.2 +z.sup.2 (20) 
When the vehicle's position in ellipsoidal coordinates has been determined, 
then the normal gravity vector in ellipsoidal coordinates can be 
calculated. The final step is to transform the normal gravity vector from 
ellipsoidal coordinates to NAV-frame coordinates using the equations 
presented above. 
To simplify the process of determining the gravity vector, the so-called 
J.sub.2 gravity model is utilized in present-day inertial navigation 
systems. The J.sub.2 gravity model can be expressed in rectangular ECEF 
coordinates and thereby greatly reduces the computational load associated 
with determining the gravity vector at a vehicle's location. 
Unfortunately, the cost of this reduction is a reduction in accuracy of 
the gravity vector. 
A need exists for a gravity-determining procedure which provides the 
accuracy of the normal model and can be implemented with 
currently-available inertial navigation system processors. 
BRIEF SUMMARY OF THE INVENTION 
The invention is a method for determining gravity in an inertial navigation 
system which periodically produces and stores in memory position 
coordinates. The method comprises the steps of (a) retrieving the most 
recently determined position coordinates, (b) determining expressions 
L.sub.u, L.sub.n, H.sub.u, and H.sub.n from the position coordinates, 
L.sub.u and L.sub.n being predetermined functions of geodetic latitude and 
H.sub.u and H.sub.n being predetermined functions of geodetic altitude, 
(c) determining the vertical component of gravity by substituting either 
or both L.sub.u and H.sub.u in a first polynomial expression, (d) 
determining the north-south component of gravity by substituting either or 
both L.sub.n and H.sub.n in a second polynomial expression, and (e) 
utilizing the components of gravity determined in steps (c) and (d) in the 
next iteration of position determination.

DETAILED DESCRIPTION OF THE INVENTION 
In the preferred embodiment of the invention, the gravity vector is an 
approximation to the normal gravity model defined by the equations 
EQU g.sub.n =D.sub.11 h sin (2.phi.) 
EQU g.sub.n =(C.sub.22 sin.sup.4 .tau.+C.sub.12 sin.sup.2 
.tau.+C.sub.02)h.sup.2 +(C.sub.21 sin.sup.4 .tau.+C.sub.11 sin.sup.2 
.tau.+C.sub.01)h+(C.sub.20 sin.sup.4 .tau.+C.sub.10 sin.sup.2 
.tau.+C.sub.00) (21) 
The quantity .tau. is the eccentric latitude and is related to the geodetic 
latitude by the equation 
##EQU13## 
The coefficients D.sub.11, C.sub.00, C.sub.10, C.sub.20, C.sub.01, 
C.sub.11, C.sub.21, C.sub.02, C.sub.12, and C.sub.22 are determined by 
fitting the above equations to the normal gravity model in the least 
square error sense. If 18,281 points are used (.tau.: 1.degree. increments 
from -90.degree. to 90.degree.; h: 1000 ft increments from 0 ft to 100,000 
ft), the following values for the coefficients are obtained: 
EQU D.sub.11 =-2.475 925 058 626 642.times.10.sup.-9 C.sub.11 =-1.347 079 301 
177 616.times.10.sup.-9 
EQU C.sub.00 =-9.780 326 582 929 618 C.sub.12 =1.878 969 973 008 
548.times.10.sup.-16 
EQU C.sub.01 =9.411 353 888 873 278.times.10.sup.-7 C.sub.20 =1.188 523 953 283 
804.times.10.sup.-4 
EQU C.sub.02 =-6.685 260 859 851 881.times.10.sup.-14 C.sub.21 =3.034 117 526 
395 185.times.10.sup.-12 
EQU C.sub.10 =-5.197 841 463 945 455.times.10.sup.-2 C.sub.22 =1.271 727 412 
728 199.times.10.sup.-18 
A comparison of the accuracies of the invention and the use of the J.sub.2 
gravity model in approximating the normal gravity model for the 18,281 
(101.times.181) grid points is shown below in units of .mu.g's (1 
.mu.g=980.6194 cm/s.sup.2 .times.10.sup.-6). 
______________________________________ 
North-South Component 
Invention 
J.sub.2 gravity model 
______________________________________ 
RMS Error 0.00767 3.79994 
MAX Error 0.02540 6.36700 
MIN Error -0.02540 -6.36700 
______________________________________ 
______________________________________ 
Vertical Component 
Invention 
J.sub.2 gravity model 
______________________________________ 
RMS Error 0.00900 5.58837 
MAX Error 0.02770 5.13470 
MIN Error -0.02730 -12.07440 
______________________________________ 
The implementation of the invention is shown in the FIGURE which shows a 
simplified software flow diagram associated with the digital processor of 
an inertial navigation system. In step 11, the accelerometers and gyro 
outputs are read. In step 13, the components of gravity are determined 
using the polynomial expressions given above. In step 15, the acceleration 
of the vehicle is determined by adding to the components of gravity the 
specific-force acceleration components derived from the measurements 
supplied by the accelerometers. Finally, in step 17, the velocity and 
position of the vehicle are determined by updating the previous values. 
The process then repeats.