BACKGROUND ART The velocity V of interest in navigating a vehicle relative to the earth is defined by the equation where 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), asf is the specific-force acceleration experienced by the inertial navigation system on board the vehicle, g is gravity, Q 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 io is the rotation rate of the local-level frame relative to the inertial frame. In order to

integrate and obtain V, an accurate expression for g is required.

The so-called normal gravity potential (the most accurate gravity model presently available) is given in terms of ellipsoidal coordinates (u,,) as and (D = m- arctan 0'a'sin') 6 - - Cos (2) c u 2qo 3 2 gA = ° (5) c'=a'-(6) v2 = u2 + c2 (7) 2 q=201+ 2 tarctans (9) 2 c u c z b 'ctan - 3b (10) 2 c b c q = 2 1 - -arctan c 1 (11) c c u where G,"is the earth's gravitational constant, a 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 is the earth rotation rate.

Given the vehicle's location in geodetic coordinates (0,,h), the normal gravity vector expressed in the ECEF (earth-centered, earth-fixed) frame is given by -*jE'''i\ where

is the transformation matrix from the ellipsoidal (U) frame to the ECEF (E) frame, and gu, the normal gravity vector, is expressed in ellipsoidal coordinates as Note that u=b, q=q0, v=a, and g =0 when h=0.

The normal gravity vector expressed in NAV (N) frame coordinates is given by where 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 x = (N+h)cos#cos# y = N + hcos sin , (17) z = (Nb²/a² + h)sin# where The transformation to ellipsoidal coordinates is defined by the equations i = arctan (19) X = arctan(Y) xJ

where rz = x2 + yz + z2 (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 J2 gravity model is utilized in present-day inertial navigation systems. The J2 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.

DISCLOSURE OF 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 Lu, LnS Hu, and Hn from the position coordinates, L and Ln being predetermined functions of geodetic latitude and Hu and Hn being predetermined functions of geodetic altitude, (c) determining the vertical component of gravity by substituting either or both L and Hu in a first polynomial expression, (d) determining the north-south component of gravity by substituting either or both Ln and Hn 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.

BRIEF DESCRIPTION OF DRAWINGS FIG. 1 shows a simplified flow diagram which describes the steps performed by the digital processor in an inertial navigation system in obtaining a vehicle's velocity and position.

BEST MODE FOR CARRYING OUT THE INVENTION In the preferred embodiment of the invention, the gravity vector is an approximation to the normal gravity model defined by the equations gn = D" h sin(20) <BR> <BR> gu = (C22sin4 # + C12 sin² # + C02)h² <BR> <BR> <BR> <BR> + (C21 sin4 # + C11 sin² # +C01)h (21)<BR> <BR> <BR> <BR> + (C20 sin4 # + C10 sin² # + C00) The quantity is the eccentric latitude and is related to the geodetic latitude by the equation tan r = b tan 0 (22) a The coefficients D", C0O, C,0, C20, C0,, C", C2" C02, C,2, and C22 are determined by fitting the above equations to the normal gravity model in the least square error sense. If 18, 281 points are used (: 1° increments from-90° to 90°;h: 1000 ft increments from 0 ft to 100, 000 ft), the following values for the coefficients are obtained : D"= -2. 475 925 058 626 642 x 10-9 C. = -9. 780 326 582 929 618 C0, = 9.411 353 888 873 278 x 10-7 Co2=-6.685 260 859 851881 x 10-'4 C, o=-5. 197 841 463 945 455 x 10-2 C, 1 = -1. 347 079 301 177 616 x 10-9 C,2 = 1.878 969 973 008 548 x 10-16 C20 = 1.188 523 953 283 804 x 10-4 C2, = 3.034 117 526 395 185 x 10-12 C22 = 1. 271727 412 728 199 x 10-"

A comparison of the accuracies of the invention and the use of the J2 gravit model in approximating the normal gravity model for the 18, 281 (101 x 181) grid points is shown below in units of g's (1 g = 980.6194 cm/s2 x 10 6).

North-South Component Invention J2 gravity model RMS Error 0. 00767 3. 79994 MAX Error 0. 02540 6. 36700 MIN Error -0. 02540 -6. 36700 Vertical Component Invention J2 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 Fig. 1 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.