zotero/storage/G7B599DJ/.zotero-ft-cache

Computing GPS Satellite Velocity and Acceleration from the
Broadcast Navigation Message
Blair F. Thompson , Steven W. Lewis , Steven A. Brown
Lt. Colonel, 42d Combat Training Squadron, Peterson Air Force Base, Colorado
Todd M. Scott
Command Chief Master Sergeant, 310th Space Wing, Schriever Air Force Base, Colorado
ABSTRACT We present an extension to the Global Positioning System (GPS) broadcast navigation message user equations for
computing GPS space vehicle (SV) velocity and acceleration. Although similar extensions have been published (e.g., Remondi,1 Zhang J.,2 Zhang W.3), the extension presented herein includes a distinct kinematic method for computing SV acceleration which signiﬁcantly reduces the complexity of the equations and improves the mean magnitude results by approximately one order of magnitude by including oblate Earth perturbation effects. Additionally, detailed analyses and validation results using multiple days of precise ephemeris data and multiple broadcast navigation messages are presented. Improvements in the equations for computing SV position are also included, removing ambiguity and redundancy in the existing user equations. The recommended changes make the user equations more complete and more suitable for implementation in a wide variety of programming languages employed by GPS users. Furthermore, relativistic SV clock error rate computation is enabled by the recommended equations. A complete, stand-alone table of the equations in the format and notation of the GPS interface speciﬁcation4 is provided, along with benchmark test cases to simplify implementation and veriﬁcation.
1 | INTRODUCTION Basic positioning of a Global Positioning System (GPS) receiver requires accurate modeling of the location of the
antenna phase center of four or more orbiting space vehicles (SV) in view. The SV orbit is nominally determined and predicted in four-hour arcs, with two hours of overlap, by the Master Control Station (MCS) at Schriever Air Force Base, Colorado. The predicted orbit is parameterized and, along with other information, becomes the broadcast navigation message. The message is periodically uploaded to each SV where it is modulated onto a carrier signal, along with a unique pseudorandom noise (PRN) navigation code, and broadcast to GPS user segment receivers. Users can then apply equations like those prescribed in Table 20-IV of the GPS interface speciﬁcation (IS)4 to accurately
1
DISTRIBUTION A: Approved for public release; distribution unlimited.

compute the position of each SV antenna phase center in the WGS-84 earth-centered earth-ﬁxed (ECEF) rotating coordinate system.5 (For brevity, we refer to the position, velocity, and acceleration of the SV antenna phase center as simply the position, velocity, and acceleration of the SV.) Accuracy estimates of SV position computed from the broadcast navigation message are on the order of 1.5 meters rms.6
The published broadcast navigation user equations were formulated for computing SV position in near real time. Users may also require SV velocity and acceleration for more complex, near real-time navigation purposes such as receiver velocity determination, GPS/INS (inertial navigation system) integration, etc. SV velocity and acceleration can be computed by extension of the broadcast navigation user equations. This has been done by several researchers including Remondi,1 Zhang J.,2 Zhang W.3 and others, undoubtedly. We present a comparable extension for SV velocity with thorough validation against multiple days of precise ephemeris data and multiple broadcast navigation messages for vehicle PRN 11, which, at the time of this writing, has the greatest off-nominal eccentricity and inclination of all SV in the GPS constellation. (SV number and PRN number are generally not the same, but we follow the common practice of using the PRN number to uniquely identify a particular SV at a particular time.) For SV acceleration we use a kinematic approach that signiﬁcantly reduces the complexity of the equations and simpliﬁes inclusion of oblate Earth (J2) gravity perturbations. Including J2 perturbations reduces the mean acceleration magnitude difference by approximately one order of magnitude compared to the more commonly used derivative method. The acceleration equations were also validated against multiple days of precise ephemeris and multiple broadcast navigation messages for PRN 11. The effects of other potential error sources were also analyzed including polar motion, Earth precession, and higher-order gravity.
In the sections that follow, we present improvements to the SV position equations and the derivation of the SV velocity and acceleration equations. Detailed validation results are presented along with a complete table of the recommended user equations in the format and notation of the public interface speciﬁcation. Benchmark test cases are provided to assist with implementation and veriﬁcation of the entire set of equations.
2 | SV POSITION – ECCENTRIC AND TRUE ANOMALY
The broadcast navigation user equation tables correctly state that Kepler’s equation (M = E−e sin E) can be solved for eccentric anomaly (E) by iteration. There are many known methods of various complexity for doing this,7,8 but no particular method is speciﬁed or recommended by the IS.4 Because SV orbits are near-circular (maximum valid eccentricity e = 0.03, according to Table 20-III in the IS), simple methods requiring a limited number of iterations can be used. We evaluated two such methods for use with the broadcast navigation user equations: 1) successive substitutions,9 and 2) Newton iteration.10 For successive substitutions, the initial estimate of eccentric anomaly is set equal to the mean anomaly (M ), and the ﬁnal value of E is converged upon by iteration of a simple variation of
2

Kepler’s equation: E0 = M (1) Ej = M + e sin Ej−1
where j is the iteration index. The Newton iteration method begins the same way. The iterative equation for E is
slightly more complex, but no more complex than other broadcast navigation user equations.

E0 = M

Ej

=

Ej−1

+

M

− Ej−1 + e sin Ej−1 1 − e cos Ej−1

(2)

At the nominal GPS altitude (r ≈ 26, 560 km), an angular variation of 1 × 10−10 radians corresponds to an in-track positional difference on the order of 3mm, well within the expected positional accuracy of the broadcast navigation equations (≈ 1.5 meters rms6). Fig. 1 shows the number of iterations required for both methods of solving Kepler’s equation to converge on eccentric anomaly to a precision of 1 × 10−10 radians, for the maximum valid eccentricity of e = 0.03. The ﬁgure shows that iterating three times on eq. (2) for any value of mean anomaly will guarantee

Figure 1. Iterations required for two methods of solving Kepler’s equation for eccentric anomaly (e = 0.03).
convergence within ≈ 3mm in-track (or the level of precision supported by the computer being used) of the broadcast navigation message for any valid eccentricity. The Newton iteration method of eq. (2) is simple to implement in a wide range of programming languages, and guaranteed to quickly converge for eccentricity e < 0.03. Including it
3

explicitly in the IS tables, along with the recommended number of iterations (three), enables users less familiar with astrodynamics and Kepler’s equation to directly apply the equations without the need to ﬁnd a suitable solution by consulting an external reference, thus saving time and effort and reducing risk.

True anomaly (ν) is computed from eccentric anomaly and orbital eccentricity using inverse trigonometry functions. The equations currently listed in the IS result in quadrant ambiguity. No information is given for resolving the quadrant, requiring the user to invoke a function such as atan2 (which may not be available in all programming languages) or to resolve the ambiguity by some other, unspeciﬁed means. We recommend deleting from the tables the current, ambiguous equations for true anomaly and replacing them with the unambiguous form derived directly from the geometric relationship between the eccentric and true anomalies:9–12

ν = 2 tan−1

1+e E tan

(3)

1−e 2

We note that eq. (3) results in an unambiguous value of true anomaly in the range −π ≤ ν ≤ π, which is different from the more customary range 0 ≤ ν < 2π. However, this poses no issue for the user equations because of the periodic nature of the trigonometric functions that use true anomaly as the independent variable, either directly or indirectly (see Table 3).

We further recommend deleting the equation for eccentric anomaly that appears after the equations for true anomaly in the IS user equation table. Eccentric anomaly is computed prior to computing true anomaly. The redundant equation for eccentric anomaly is unnecessary and could lead to confusion when the equations are being implemented by a programmer with little or no knowledge of astrodynamics.

The upper panel of Fig. 2 shows the WGS-84 position coordinates and RSS (root sum squared) magnitude of an example 72-hour series (approximately six GPS orbital periods) of precise ephemeris for PRN 11. The precise ephemeris data are publicly available from the National Geospatial-Intelligence Agency (NGA).13 The lower panel shows the position coordinates and magnitude differences between the precise ephemeris and the broadcast navigation message results using eqs. (2) and (3) over the same time period. The differences are shown in radial, in-track, and cross-track (RIC) coordinates in the upper panel of Fig. 3. The relatively consistent differences in the radial direction and magnitude are indicative of the orbit prediction efforts of the Master Control Station. The lower panel of Fig. 3 shows the position differences without fully converging on eccentric anomaly. As expected, large positioning errors arise, primarily in the in-track direction, by simply not allowing proper convergence of Kepler’s equation. We therefore recommend the minimum number of iterations (three) be speciﬁed explicitly along with the user equations in the IS (e.g., see Table 3). Again, experienced users would not be required to use these equations or number of iterations. They are simply baseline recommendations for users and programmers with limited knowledge of astrodynamics.

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Figure 2. WGS-84 precise ephemeris position (upper panel) and differences with broadcast navigation messages (lower panel).
Figure 3. RIC position differences using fully converged Kepler equation (upper panel) and partially converged Kepler equation (lower panel).
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3 | SV VELOCITY The ECEF velocity of the SV can be computed by taking the time-derivative of the SV position equations in the
interface speciﬁcation.4 The resulting ECEF velocity equations are:

x˙ = −x Ω˙ k sin Ω − y (Ω˙ k cos Ω cos i − (di/dt) sin Ω sin i) + x˙ cos Ω − y˙ sin Ω cos i

y˙ = x Ω˙ k cos Ω − y (Ω˙ k sin Ω cos i + (di/dt) cos Ω sin i) + x˙ sin Ω + y˙ cos Ω cos i

(4)

z˙ = y (di/dt) cos i + y˙ sin i

where x and y are the orbit-plane position coordinates prior to transformation to the ECEF system, and Ω˙ k is the rate of the corrected longitude of the ascending node. To fully implement these equations, several additional derivatives are required (Table 1). These derivatives are functions of the parameters of the broadcast navigation message and parameters computed by the SV position equations. Note that the term (di/dt) is not the same as IDOT . The term IDOT is the “Rate of Inclination Angle” from the broadcast navigation message (not to be confused with IODE or IODC), while (di/dt) is the inclination rate corrected by the equation in Table 1. The E˙ equation results from taking
Table 1. Velocity ancillary equations.

E˙ = n/(1 − e cos E) ν˙ = E˙ √1 − e2/(1 − e cos E) (di/dt) = (IDOT ) + 2ν˙ (cis cos 2Φ − cic sin 2Φ) u˙ = ν˙ + 2ν˙ (cus cos 2Φ − cuc sin 2Φ) r˙ = eAE˙ sin E + 2ν˙ (crs cos 2Φ − crc sin 2Φ) Ω˙ k = Ω˙ − Ω˙ e x˙ = r˙ cos u − ru˙ sin u y˙ = r˙ sin u + ru˙ cos u

the time-derivative of Kepler’s equation (M = E − e sin E) and substituting mean motion (n) for the derivative of

mean anomaly6 (i.e., M˙ = n). The rate of true anomaly (ν˙ ) results from taking the time-derivative of an alternate

form of the true anomaly equation10

√

1 − e2 sin E sin ν =

(5)

1 − e cos E

and substituting for cos ν from

cos E − e

cos ν =

(6)

1 − e cos E

Furthermore, the argument of perigee is treated as constant (i.e., ω˙ = 0) for the derivation of the equations in Table 1. Fig. 4 (upper panel) shows 72-hours of precise ephemeris velocity magnitude along with the absolute value of SV

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latitude. As expected, the inverse correlation between velocity magnitude and SV latitude (absolute value) is apparent
Figure 4. WGS-84 precision ephemeris velocity magnitude and absolute value SV latitude (upper panel) and velocity differences with broadcast navigation message and polynomial derivative (lower panel).
in the ﬁgure and indicates that the periodic variations in velocity magnitude are due primarily to the motion of the SV in its inclined orbit relative to the WGS-84 coordinate system. The differences between the precise ephemeris velocity and velocity computed by eqs. (4) are shown in the lower panel by component and RSS magnitude. Also shown in Fig. 4 are the velocity magnitude differences of the time-derivative of a 12th order Chebyshev polynomial ﬁt to the precision position data. This was included to show that the achievable accuracy of the polynomial derivative is commensurate with the differences between the precise and computed velocity. Because precise acceleration data are not publicly available from the NGA, 12th order Chebyshev polynomials14 were ﬁt to the precise position and velocity data in three-hour segments (approximately 1/4 GPS orbital period) with constrained end points. To assess the accuracy of computing precise acceleration by this method (next section), the polynomial derived velocity was compared to the precise ephemeris velocity in Fig. 4. 4 | SV ACCELERATION
SV acceleration can be computed by taking the time-derivative of the velocity equations, as shown by Zhang J.2 and Zhang W.3 However, with the WGS-84 position and velocity known (i.e., computed previously), Newton’s second law of motion (for constant mass) and the kinematic equation of acceleration10 can be employed instead, resulting in
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a much simpler formulation.

f = ao + aR + 2(ω × vR) + (ω˙ × rR) + ω × (ω × rR)

(7)

All vectors are in the ECEF (WGS-84) coordinate system – the earth-centered inertial (ECI) system is not used. The vectors in eq. (7) are deﬁned below:
f speciﬁc force, i.e. net external force per (constant) mass ao acceleration of the origin of the rotating frame aR acceleration with respect to the rotating frame ω angular velocity of the rotating frame rR position with respect to the rotating frame vR velocity with respect to the rotating frame
Note that ao = 0 (null vector) because the origin of WGS-84 is ﬁxed at the center of mass of Earth, co-located with origin of the inertial frame. Let ω = [0, 0, Ω˙ e]T and ω˙ = 0, where Ω˙ e is the WGS-84 Earth rotation rate. Solve for aR, the desired acceleration vector with respect to the ECEF frame,

aR = f − 2(ω × vR) − ω × (ω × rR)

(8)

Let f be the two-body gravitational force-per-mass expressed in ECEF coordinates. Carrying out the vector cross products in eq. (8), the equations for ECEF acceleration in component form become

ax

=

µx − r3

+

2y˙Ω˙ e

+

xΩ˙ 2e

ay

=

µy − r3

−

2x˙ Ω˙ e

+

yΩ˙ 2e

(9)

µz az = − r3

where r = x2 + y2 + z2, and µ is the Earth gravitational parameter (µ = GM ). These equations are much less complex than those of the derivative method and produce the same level of accuracy. For comparison and completeness, the derivative method equations of acceleration are included here without derivation.

µx

x¨ = − r3

(10)

µy

y¨ = − r3

8

x¨ = − x Ω˙ 2 cos Ωk + x¨ cos Ωk − y¨ sin Ωk cos ik

+ y (Ω˙ 2k + (dik/dt)2) sin Ωk cos ik + 2Ω˙ k(dik/dt) cos Ωk sin ik

(11)

− 2x˙ Ω˙ k sin Ωk − 2y˙ (Ω˙ k cos Ωk cos ik − (dik/dt) sin Ωk sin ik)

y¨ = − x Ω˙ 2 sin Ωk + x¨ sin Ωk + y¨ cos Ωk cos ik

− y (Ω˙ 2k + (dik/dt)2) cos Ωk cos ik − 2Ω˙ k(dik/dt) sin Ωk sin ik

(12)

+ 2x˙ Ω˙ k cos Ωk − 2y˙ (Ω˙ k sin Ωk cos ik + (dik/dt) cos Ωk sin ik)

z¨ = −y (dik/dt)2 sin ik + 2y˙ (dik/dt) cos ik + y¨ sin ik

(13)

Note the difference in complexity between eqs. (9) and (10) - (13). Also note that eq. (10) is the two-body gravitational acceleration in the orbit-plane coordinate system of the broadcast navigation position equations (see Table 3), where x points to the ascending node of the SV orbit, and y points to 90◦ in-track in the orbital plane. (There is no z position component because the SV position vector lies entirely in the orbital plane, z = 0 always). The ﬁnal three broadcast navigation position equations are, in fact, the orbit-plane SV position coordinates transformed to ECEF by an orthogonal transformation comprising orbital inclination (ik) and longitude of the ascending node (Ωk). Incorporating non-spherical Earth gravity effects in eq. (10) would ﬁrst require transforming the gravitational acceleration from ECEF to orbit-plane coordinates. This would also introduce a non-zero z¨ acceleration component. Including nonspherical gravity in the derivative eqs. (10) - (13) is possible, but unnecessarily complex. It is much simpler to modify the kinematic eqs. (9) to include non-spherical gravity, as shown below in eq. (14). Earth oblateness (J2) is the dominant perturbing force acting on the SV orbits. The third-body gravity of the Sun and Moon are approximately one order of magnitude less than J2, and solar radiation pressure is approximately two orders of magnitude less.6,8 The broadcast navigation orbit model includes correction terms to account for J2 and other perturbations, nominally over the span of four hours. It follows that computing acceleration with the dominant J2 effects is commensurate with the accuracy of the broadcast navigation position and velocity.

An example 72-hour acceleration comparison is shown in Fig. 5. The upper panel shows acceleration magnitude and SV latitude (absolute value), revealing strong correlation due to the motion of the SV in the WGS-84 coordinate frame. The lower panel shows the RSS magnitude differences compared to precise values (i.e., time-derivative of the Chebyshev velocity polynomials) for the derivative method and the kinematic method. Also shown in the lower panel is the effect of including oblate Earth gravity (J2) in the kinematic method, which improves the mean acceleration magnitude difference by approximately one order of magnitude. The numerical value for J2 was derived from the fully-normalized coefﬁcient C¯20 of the EGM-2008 gravity model15 (by convention: J2 = −√5C¯20). The kinematic
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Figure 5. WGS-84 acceleration magnitude and absolute value SV latitude (upper panel) and acceleration differences computed using various methods (lower panel).

acceleration equations including J2 perturbations are16

µx x¨ = − r3 + F

1 − 5(z/r)2

(x/r) + 2y˙Ω˙ e + xΩ˙ 2e

µy y¨ = − r3 + F

1 − 5(z/r)2

(y/r) − 2x˙ Ω˙ e + yΩ˙ 2e

(14)

µz z¨ = − r3 + F

3 − 5(z/r)2

(z/r)

where F = −(3/2)J2(µ/r2)(RE/r)2 and RE is the equatorial radius of Earth. With the J2 terms included, these equations are still less complex than the less accurate, two-body form of the derivative-based equations (10) - (13), and the mean magnitude difference results are improved by approximately one order of magnitude.
Errors introduced by simplifying assumptions (i.e., ω = [0, 0, Ω˙ e]T and ω˙ = 0) and other effects on the acceleration equations were analyzed, including the difference between the geodetic constants of the WGS-84 system and the EGM2008 gravity ﬁeld (Table 2). Because EGM-2008 is a tide-free gravity ﬁeld, the effect of the solid earth permanent tide should be included in the oblate gravity term for high precision applications.17

<

∆J2

>=

−

<

∆C¯20

>

√ 5

=

9.3324

×

10−9

(15)

Including the permanent tide effect on J2 was found to have no signiﬁcant impact on the accuracy of the kinematic 10

Table 2. Geodetic parameters. The WGS-84 value for µ comes from the broadcast navigation equations in the IS, not the separately published value.

Parameter µ (m3/s2) RE (meters) C¯2,0 C¯2,1 S¯2,1

WGS-84 3.986005 × 1014
6378137.0

EGM-2008 3.986004415 × 1014
6378136.3 −0.484165143790815 × 10−3 −0.206615509074176 × 10−9 +0.138441389137979 × 10−8

acceleration equations (see Fig. 6). The ﬁnal value of J2 used for evaluation was J2 = 0.0010826262.

Figure 6. WGS-84 acceleration differences with various corrections applied for comparison.

Also considered were the effects of polar motion – the location of the true Earth rotation axis with respect to the conventional pole. Because the J2 oblate gravity term is dominant over the higher order terms, the approximate mean pole location, or Earth Orientation Parameters (EOP), can be derived from the gravity ﬁeld coefﬁcients,17

x¯p

≈

√C3¯C2¯,12,0

6.48 × 105 π

=

0.051

y¯p

≈

√−3S¯C¯22,1,0

6.48 × 105 π

=

0.341

arcsec arcsec

(16)

which include the conversion from radians to arc-seconds. Polar motion EOP data from the International Earth Rotation and Reference Systems Service (IERS)18 from 1962 to 2018 are plotted in Fig. 7, the mean values from eqs.

11

(16) indicated by dashed lines. From the ﬁgure, the “worst case” maximum polar motion EOP for this ≈ 60 year data set is seen to be approximately xp = 0.05 arcsec and yp = 0.6 arcsec. The mean and maximum polar motion had no signiﬁcant effect on the computed SV acceleration (Fig. 6). The effects of omitting higher order gravity perturbations (up to spherical harmonic degree and order twenty – 20x20) were also found to be insigniﬁcant at the precision level of the kinematic acceleration equations, as shown by Fig. 6. Using the precise ephemeris values for position and velocity in lieu of the less accurate computed values in the kinematic acceleration equations was also found to have insigniﬁcant impact. Finally, Earth precession (≈ 50 arcsec per year) and polar motion rate were analyzed and found to have no signiﬁcant effect on SV acceleration. This analysis shows that the accuracy of the kinematic equations for SV acceleration is not signiﬁcantly affected by the simplifying assumptions and omission of perturbations when deriving the equations.

Figure 7. Polar motion Earth Orientation Parameters (EOP), 1962 - 2018. Mean values indicated by dashed lines.

5 | SV CLOCK RELATIVISTIC CORRECTION The SV clock error (or offset) relative to GPS Time at time t is modeled as

∆tSV = af0 + af1(t − toc) + af2(t − toc)2 + ∆tr

(17)

where toc is the clock data reference time, and af0, af1, and af2 are polynomial coefﬁcients unique to each SV clock and sent in the broadcast navigation message. The last term, ∆tr, is the unique relativistic correction arising from the
12

orbital altitude and speed of each SV. The interface speciﬁcation4 gives two different equations for computing ∆tr,

the ﬁrst being

√

∆tr = F e A sin E seconds

(18)

√ where e, A, and E are from the broadcast navigation message or computed by the user equations, and

F

=

−2√µ c2

=

−4.442807633

× 10−10

√s m

(19)

Equation (18) can be included in the broadcast navigation user equations. The second equation for ∆tr is

2r · v

∆tr = − c2 seconds

(20)

where r and v are the ECEF position and velocity vectors computed using the broadcast navigation user equations, i.e., rR and vR in eqs. (8). Equation (20) is utilized by the GPS Control Segment,4 but could alternatively be adopted by GPS users after computing r and v from the enhanced broadcast user equations.
The time-derivative of the relativistic clock correction is

d dt

(∆tr )

=

F e√AE˙

cos

E

=

2 − c2

|v|2 + r · a

(21)

where E˙ and the acceleration vector a (acceleration with respect to ECEF) are computed by the recommended user equations (see Table 3). The top panel in Fig. 8 shows 72 hours of clock error for PRN 28 computed from the broadcast navigation messages using eqs. (17) and (18) and a series of 12th order Chebyshev polynomials ﬁt to the clock error tabulated in the NGA precise ephemeris, corrected using eq. (20). The two data sets are practically indistinguishable at the scale of the ﬁgure, the maximum difference over the 72-hour period being 0.00903 microseconds. The bottom panel shows the time-derivative of the clock error including the relativistic correction computed by both forms in eq. (21). Again, the data are practically indistinguishable, the maximum difference being 0.0738 × 10−6 microsec/sec. Similar results occurred for all SV tested. The results in Fig. 8 validate eqs. (20) and (21) for use with the newly recommended user equations.

6 | RECOMMENDED EQUATIONS The recommended equations are shown as part of the complete set of user equations in Table 3, in the form and
notation of Table 20-IV in the interface speciﬁcation.4 Use of these equations would continue to be optional for all GPS users, as speciﬁed in section 20.3.3.4.3 User Algorithm for Ephemeris Determination of the IS: “The user shall compute the ECEF coordinates of position for the phase center of the SVs antennas utilizing a variation of the equations shown in Table 20-IV.” Note that we also suggest a more appropriate title for Table 20-IV. These changes

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Figure 8. SV clock error: broadcast navigation message vs. precise ephemeris polynomial (top panel), and clock drift rate (bottom panel).
and improvements are also applicable to Table 30-II in the IS,4 as well as Table 20-II in IS-GPS-705E,19 Table 3.5-2 in IS-GPS-800E,20 and other similar documents.
Table 3: Recommended updated Table 20-IV for Interface Speciﬁcation IS-GPS-200J.
Table 20-IV. Elements of Coordinate Systems Table 20-IV. Broadcast Navigation User Equations

µ = 3.986005 × 1014 meters3/sec2
Ω˙ e = 7.2921151467 × 10−5 rad/sec √2
A= A µ
n0 = A3 tk = t − t∗oe n = n0 + ∆n (table continues)

WGS 84 value of the earth’s gravitational constant for GPS user WGS 84 value of the earth’s rotation rate Semi-major axis Computed mean motion (rad/sec) Time from ephemeris reference epoch Corrected mean motion

14

Mk = M0 + ntk Mk = Ek − e sin Ek

E0 = Mk

Ej

=

Ej−1

+

Mk

− Ej−1 + e sin Ej−1 1 − e cos Ej−1

Ek = E3

Table 3 (continued) Mean anomaly Kepler’s Equation for Eccentric Anomaly (may be solved by iteration) (radians) Kepler’s equation (Mk = Ek − e sin Ek) solved for eccentric anomaly (Ek) by iteration: - Initial value (radians) - Reﬁned value, three iterations, (j = 1, 2, 3)
- Final value

νk = tan−1 = tan−1

sin νk cos νk √
1−e2 sin Ek/(1−e cos Ek)
(cos Ek−e)/(1−e cos Ek)

νk = 2 tan−1

1 + e tan Ek 1−e 2

Ek = cos−1

e+cos νk 1+e cos νk

Φk = νk + ω

True Anomaly
True anomaly (unambiguous quadrant) Eccentric Anomaly Argument of Latitude

δuk = cus sin 2Φk + cuc cos 2Φk

Argument of Latitude Correction

δrk = crs sin 2Φk + crc cos 2Φk

Radius Correction

δik = cis sin 2Φk + cic cos 2Φk

Inclination Correction

uk = Φk + δuk

Corrected Argument of Latitude

rk = A(1 − e cos Ek) + δrk

Corrected Radius

ik = i0 + δik + (IDOT)tk

Corrected Inclination

xk = rk cos uk yk = rk sin uk Ωk = Ω0 + (Ω˙ − Ω˙ e)tk − Ω˙ etoe

Positions in Orbital Plane Corrected longitude of ascending node

xk = xk cos Ωk − yk cos ik sin Ωk

Earth-ﬁxed coordinates

yk = xk sin Ωk + yk cos ik cos Ωk

zk = yk sin ik

* t is GPS system time at time of transmission, i.e., GPS time corrected for transit time (range/speed of light).

Furthermore, tk shall be the actual total time difference between the time t and the epoch time toe, and must

account for beginning or end of week crossovers. That is, if tk is greater than 302,400 seconds, subtract

604,800 seconds from tk. If tk is less than -302,400 seconds, add 604,800 seconds to tk.

(table continues)

15

Table 3 (continued)

SV Velocity

E˙ k = n/(1 − e cos Ek)

ν˙k

=

E˙ k

√ 1

−

e2/(1

−

e cos Ek)

Eccentric anomaly rate True anomaly rate

(dik/dt) = (IDOT) + 2ν˙k(cis cos 2Φk − cic sin 2Φk) Corrected Inclination rate

u˙ k = ν˙k + 2ν˙k(cus cos 2Φk − cuc sin 2Φk) r˙k = eAE˙ k sin Ek + 2ν˙k(crs cos 2Φk − crc sin 2Φk) Ω˙ k = Ω˙ − Ω˙ e

Corrected Argument of Latitude rate Corrected Radius rate Longitude of ascending node rate

x˙ k = r˙k cos uk − rku˙ k sin uk

In-plane x velocity

y˙k = r˙k sin uk + rku˙ k cos uk

In-plane y velocity

x˙ k = −xkΩ˙ k sin Ωk + x˙ k cos Ωk − y˙k sin Ωk cos ik −yk(Ω˙ k cos Ωk cos ik − (dik/dt) sin Ωk sin ik)
y˙k = xkΩ˙ k cos Ωk + x˙ k sin Ωk + y˙k cos Ωk cos ik −yk(Ω˙ k sin Ωk cos ik + (dik/dt) cos Ωk sin ik)
z˙k = yk(dik/dt) cos ik + y˙k sin ik
SV Acceleration

Earth-ﬁxed x velocity (m/s) Earth-ﬁxed y velocity (m/s) Earth-ﬁxed z velocity (m/s)

RE = 6378137.0 meters

WGS 84 Earth equatorial radius

J2 = 0.0010826262

F = −(3/2)J2(µ/rk2)(RE/rk)2

x¨k

=

−µ

xk rk3

+F

(1 − 5(zk/rk)2)(xk/rk)

+2y˙kΩ˙ e + xkΩ˙ 2e

y¨k

=

−µ

yk rk3

+

F

(1 − 5(zk/rk)2)(yk/rk)

−2x˙ kΩ˙ e + ykΩ˙ 2e

z¨k

=

−µ

zk rk3

+F

(3 − 5(zk/rk)2)(zk/rk)

Oblate Earth gravity coefﬁcient Oblate Earth acceleration factor Earth-ﬁxed x acceleration (m/s2)
Earth-ﬁxed y acceleration (m/s2)
Earth-ﬁxed z acceleration (m/s2)

7 | IMPLEMENTATION AND BENCHMARK TESTS To assist with implementation and veriﬁcation of the recommended equations, we have included benchmark test
results for position, velocity, and acceleration. The associated broadcast navigation message parameters are shown in Table 4. The test results of the user equations are shown in Table 5 for two different times. The values of the precise ephemeris and broadcast navigation user equations are shown in component form along with the RSS magnitude. Note
16

that for acceleration the precise values were computed from the derivatives of Chebyshev polynomials ﬁt to the precise ephemeris velocity data, as described earlier.

Table 4. Broadcast navigation message parameters used for benchmark tests: GPST 7 Jan 2018, 00:00:00.0

Parameter PRN crs ∆n M0 cuc e
√cus A
toe

Value 11
−0.965625000000E + 01 0.583845748090E − 08
−0.286954703389E + 01 −0.379979610443E − 06
0.167867515702E − 01 0.277347862720E − 05 0.515375480270E + 04 0.000000000000E + 00

Parameter
cic Ω0 cis i0 crc Ω Ω˙
IDOT
GPS Week

Value 0.199303030968E − 06 −0.657960408566E + 00 0.173225998878E − 06 0.903782727230E + 00 0.293218750000E + 03 0.173129682312E + 01 −0.868929051526E − 08 0.789318592573E − 10 0.198300000000E + 04

Table 5. Benchmark test results.

GPST 7 Jan 2018 00:35:00.0

Position (m) Broadcast Nav Precise Ephemeris

xk

yk

zk

RSS

3166192.017 −21511945.818 −15899623.697 26936715.065

3166191.446 −21511947.161 −15899624.824 26936716.607

Velocity (m/s) Broadcast Nav Precise Ephemeris

x˙ k

y˙k

1533.973749 −1209.904136

1533.973891 −1209.904144

z˙k 2000.871636 2000.871617

Acceleration (m/s2) x¨k

Broadcast Nav

−0.224186

Precise Ephem. Poly. −0.224188

y¨k 0.100579 0.100577

z¨k 0.324295 0.324296

RSS 2796.503314 2796.503382
RSS 0.406870 0.406871

GPST 7 Jan 2018 01:50:00.0

Position (m) Broadcast Nav Precise Ephemeris

xk

yk

zk

7847635.362 −25169173.996 −4315772.358

7847635.584 −25169175.522 −4315773.249

RSS 26715137.871 26715139.518

Velocity (m/s) Broadcast Nav Precise Ephemeris

x˙ k 595.709009 595.708923

y˙k −259.303963 −259.304060

Acceleration (m/s2) x¨k

Broadcast Nav

−0.160162

Precise Ephem. Poly. −0.160162

y¨k 0.305506 0.305506

z˙k 2970.973426 2970.973219
z¨k 0.090248 0.090249

RSS 3041.182478 3041.182268
RSS 0.356554 0.356554

17

8 | SUMMARY Accurate SV position modeling is required for basic GPS navigation (positioning). SV velocity and acceleration
modeling are required for more advanced navigation and other purposes such as receiver velocity determination and GPS/INS integration. The current interface speciﬁcation includes equations for computing SV position only. We present equations for extending the broadcast navigation user equations to compute SV velocity and acceleration. Previous work in this area has been published, but the methods herein are validated over a longer period of time using multiple broadcast navigation messages. Additionally, the kinematic form of the acceleration equations are less complex and improve mean magnitude differences by approximately one order of magnitude when oblate Earth gravity effects are included. The new methods were validated using three days (6 GPS orbit periods) of precise ephemeris data from the NGA. For acceleration data, the derivative of a 12th order Chebyshev polynomial ﬁt to the velocity data was used. Also analyzed were the effects of the permanent tide on the J2 gravity coefﬁcient, mean and maximum polar motion effects, polar motion rate, precession, and higher order gravity. The recommended equations are provided in the format and notation of the existing user equations table in the interface speciﬁcation, and benchmark test results are provided to ease with implementation and validation.
ACKNOWLEDGMENTS We thank Air Force Space Command Chief Scientist Dr. Joel Mozer, Deputy Chief Scientist Dr. Michele Gau-
dreault, their staff, and Mr. Anthony Flores of SAIC for their support with this research and development effort.
DISCLAIMER The views expressed in this paper are those of the authors and do not reﬂect the ofﬁcial policy or position of the
United States Air Force, Department of Defense, or the U.S. Government.
ORCID Blair F. Thompson https://orcid.org/0000-0002-7541-6575 Steven W. Lewis https://orcid.org/0000-0002-5070-7691
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