Location determination using signals from fewer than four satellites

Method and apparatus for determining the location of a point on a rotating body, using location determination (LD) signals received from as few as one satellite, preferably non-geosynchronous. Where signals from two or more satellites are received, one may be geosynchronous. Pseudoranges are measured from one or more satellites at two or more selected, spaced apart observation times, and the simultaneous rotations of the body and the satellite(s) relative to each other result in different body-satellite constellations for which the initial location coordinates (and, optionally, signal receiver time offset) of the selected point are determined exactly, without approximation or iteration. The selected point may be motionless or may be allowed to move with known coordinate differences between the initial unknown location and the present location at each observation time. Pseudoranges from different satellites, or even from different satellite systems (GPS, GLONASS, LEO, etc.) can be measured and used in this procedure.

FIELD OF THE INVENTION 
This invention relates to determination of location coordinates and/or time 
coordinate for a user that receives timed signals from fewer than four 
satellites. 
BACKGROUND OF THE INVENTION 
Timed signals received by a receiver/processor from satellites are now 
routinely used to determine location coordinates and/or receiver time 
offset of the receiver. Two examples of this are the signals received as 
part of a Global Positioning System (GPS), signals received as part of a 
Global Orbiting Navigational Satellite System (GLONASS), and signals 
received as part of a Low Earth Orbit (LEO) system, such as the 
66-satellite Iridium constellation proposed by Motorola. 
One potential problem here is that the minimum number of satellites needed 
for determination of location coordinates alone (usually two or three) or 
for determination of location coordinates and receiver time offset (four) 
may not be in view simultaneously. A related problem occurs for an 
extraterrestrial exploration vehicle, such as the landing vehicle for the 
Mars Explorer that touched down on that planet's surface on or around Jul. 
3, 1997, wherein only a single satellite, the mother ship orbiting the 
planet, is available to provide timed signals for location determination. 
The U.S. Navy Navigation Satellite System, also known as "Transit," used 
triangulation from signals received from a plurality of satellites with 
orbits in polar planes to estimate a user's location. Development of the 
Transit system began at Johns Hopkins in 1958, became partly operational 
in 1963 and became fully operational in 1968. The Transit system included 
at least six satellites, arranged in polar orbits (angle of inclination 
relative to the equator=90.degree.), with altitudes of about 1075 
kilometers (km) above the Earth's surface with orbit time intervals of 107 
minutes. The six orbits formed a "bird cage" constellation around the 
Earth. The average time between usable satellite passes was about 90 
minutes, but the actual time can vary between 30 minutes and several 
hours, depending upon the observer's location. 
Each satellite transmitted signals at each of two frequencies, 150 MHz and 
400 MHz, with a frequency stability of about 1 part in 10.sup.11. The 
transmitted signal includes a navigation message that is repeated every 
two minutes. The navigation message itself was updated every 12 hours but 
could run for up to 16 hours without requiring an update. The Transit 
ground support system included four tracking stations within the U.S., 
plus two signal injection stations to inject navigation message updates 
and a computer center that created the message update. The navigation 
message included a fixed part with geometrical parameters that describe a 
perfectly smooth elliptical orbit and a variable part that provides 
corrections to the elliptical orbit parameters; every two minutes, a new 
orbital correction is added and an older correction is deleted. 
The signal frequency transmitted at 400 MHz (nominal) by each Transit 
satellite was offset by 32 kHz to provide a relatively low frequency 
difference (32 kHz.+-.8 kHz) after mixing the received signal (f=F.sub.G) 
with a 400 MHz signal (f=F.sub.R) at a ground station. The distance or 
slant range of a given Transit satellite from a ground-based user was 
determined using a carrier phase count of the number of full and partial 
cycles N received between two selected time markers t1 and t2, viz. 
EQU N=(F.sub.G -F.sub.R)dt=(F.sub.G -F.sub.R)(t2-t1)+F.sub.G (R2-R1)/c, (1) 
where R1 and R2 are the as-yet-unknown distances of the satellite from the 
user at the beginning and end of the integration interval, respectively. 
The change in slant range, R2-R1, was determined from the Doppler shift 
change, represented by the last term in Eq. (1). Each full count 
(.DELTA.N=1) of a carrier phase signal represents about 0.75 meters. Two 
frequencies, 150 MHz and 400 MHz, were used to estimate the time delay due 
to signal propagation in the ionosphere. Time delay due to signal 
propagation in the troposphere and due to refraction was estimated as a 
unit. For this dual frequency set, the estimated maximum radial error and 
rms radial error for location using the Transit system are about 77 meters 
and 32 meters, respectively. A computer uses a least square error approach 
to estimate the best fit of location, based upon the Doppler shift signals 
determined for each of the two or three visible satellites. If the user 
moves during the time interval of receipt of the signals transmitted by 
the satellites, the system requires accurate specification of the velocity 
vector during this interval and usually relies upon dead reckoning between 
location fixes. 
The Transit system is discussed by Gueir and Weiffenbach, "A Satellite 
Doppler Navigation System", Proc. I.R.E. (1960) pp. 507-516, and by 
Williams, "Marine Satellite Navigation Systems", SERT Journal, vol. 13 
(1977) pp. 50-54. 
The Transit system required fixed inclination angles for each of a 
collection of satellites, used a least square error approach, rather than 
an exact analytical approach, to determine user location, did not 
explicitly account for the rotation of the Earth, relying instead on 
simultaneous visibility of two or three satellites, and relied upon dead 
reckoning to determine user location between location fixes. 
What is needed is an approach that allows receipt and analysis of fewer (as 
few as one) than the theoretical minimum number of (simultaneous) 
satellite signals needed, to determine the location coordinates and/or 
receiver time offset (referred to collectively here as "location fix 
coordinates") for a receiver. Preferably, the approach should be flexible 
and should allow (1) receipt and analysis of signals from as few as a 
single satellite and (2) receipt and analysis of signals from satellites 
that are part of different location determination (LD) systems ("mixed 
system signals"), such as a mixture of GPS and LEO signals or a mixture of 
GPS and GLONASS signals. 
SUMMARY OF THE INVENTION 
The invention meets these needs by providing methods that can receive and 
analyze signals from a single satellite, or from more than one satellite, 
at spaced apart times, by taking advantage of the rotation of a planet or 
planetary satellite (referred to collectively here as a "planet") and of 
the separate rotation of the signal-transmitting satellite(s) relative to 
the planet. The rotational axis and angular rotation for the planet and 
the rotational axis and angular rotation for the satellite are assumed to 
be determinable or known. Movement of the satellite relative to the 
receiver (which is also spinning below, on or near the planet's surface) 
is described parametrically in time, where the only unknowns are the 
location coordinates of the receiver at a selected time. An equation 
describing the orbit of each satellite used for LD is assumed to be known 
or determinable. An equation describing the distance or pseudorange 
between the receiver and each of one or more satellites used for LD is 
nonlinear in the unknown location fix coordinates. Assuming that 
associated errors such as ionospheric propagation time delay, tropospheric 
propagation time delay, receiver noise, satellite transmitter time offset 
and multipath propagation can be measured or modeled and compensated for, 
as is done in conventional LD systems, these nonlinear equations are 
analyzed exactly (no approximations or iterations) and are reduced to a 
sequence of exact equations linear in the unknown location fix 
coordinates. These linear equations, plus one nonlinear equation, are used 
to determine the user's location coordinates, for a stationary user or for 
a user that moves along a known path relative to an unknown initial 
location. 
The LD signals may be received and analyzed from one or more satellites 
belonging to a single LD system, such as GPS, GLONASS or LEO. 
Alternatively, the LD signals may be received and analyzed from two or 
more different LD systems, as long as (1) the orbit of each satellite from 
which LD signals are received is known and (2) timed LD signals received 
from a satellite can be distinguished and assigned to the source of those 
signals. One advantage of use of timed signals from a single satellite for 
LD is that no special orthogonality or frequency distinction need be built 
into these signals to distinguish one signal source from another signal 
source, as is required, for example, in a CDMA system. 
The receiver may be assumed to be stationary while the LD signals are being 
received. Alternatively, the receiver may be allowed to move along a known 
path relative to its initial unknown location.

DESCRIPTION OF BEST MODES OF THE INVENTION 
In FIG. 1, an LD signal receiver/processor 11 and associated LD signal 
antenna 13, which are part of a LD system, receive LD signals from one, 
two or more LD signal sources 15A, 15B, 15C, 15D that are visible (by 
line-of-sight view) from the LD antenna 13. Each LD signal received by the 
LD receiver/processor 13 from an LD signal source 15s (s=A, B, C, D) is a 
timed signal, preferably but not necessarily a CDMA signal or other signal 
that changes in a determinable manner with time (a "time-determined" 
signal). When a particular portion of an LD signal sequence is received at 
the receiver/processor 11, this portion is analyzed, and the time at which 
this portion of the LD signal was transmitted by the LD signal source 15s 
is determined. Two signals transmitted by two different LD signal sources 
15s and 15s' are distinguishable. Thus, the time .DELTA.t(15s;rcvr) 
required for propagation of an LD signal, transmitted from an identified 
LD source 15s and received by the LD antenna 13 or the LD 
receiver/processor 11 is readily determined. This invention does not 
require that the propagation times .DELTA.t(15s;rcvr) (s=A, B, C and D) 
from four or more distinct LD signal sources be simultaneously available. 
The situation where LD signals from only a single satellite are received 
at a stationary receiver is considered first. 
Assume that a single satellite orbits about a rotating planetary body in an 
orbital plane with an associated azimuthal angle .PHI. and an associated 
polar angle .theta., as illustrated in FIG. 2. The orbital plane is 
described in an (x,y,z) Cartesian coordinate system, with origin at the 
body center, by the parametric equation 
EQU x cos.PHI. cos.theta.+y sin.PHI. cos.theta.+z sin.theta.=d, (2) 
where d is the perpendicular distance from the orbital plane to the center 
C of the planetary body. No loss of generality occurs by assuming that 
d=0, in which event the orbital plane contains the center C of the 
planetary body. The parametric equations for motion in time (t) of the 
satellite in the orbital plane are 
##EQU1## 
where a is the radius of the orbit (assumed to be circular here, but this 
is extended to elliptical orbits in the following), .omega. is the 
observed angular velocity associated with the orbit of the satellite, and 
t=.tau.0 is a selected time for the satellite orbit. 
The satellite is assumed to transmit a time-determined message or sequence 
of digital values, but this message need not be a CDMA format signal, such 
as used for GPS signals, unless more than one satellite is present and 
transmitting such signals at the same carrier frequency and at 
substantially the same time. The planetary body has a nominal radius a0 
and rotates around a known axis with angular velocity .omega.0. A digital 
signal receiver positioned on or near the surface of the planetary body is 
assumed to have a replica of the satellite transmitted message and is 
assumed to be provided with the values of the parameters .PHI., .theta., 
a, .omega., a0 and .omega.0. A point with as-yet unknown Cartesian 
coordinates (x0,y0,z0) that is fixed on or near the surface of the 
rotating planetary body is thus describable by the parametric equations 
EQU x(t)=x0 cos.omega.0(t-t0)+y0 sin.omega.0(t-t0), (6) 
EQU y(t)=-x0 sin.omega.0(t-t0)+y0 cos.omega. (t-t0), (7) 
EQU z(t)=z0, (8) 
where the time t=t0 is an arbitrary system initialization time. 
It is assumed here that the ionospheric time delay, tropospheric time 
delay, multipath time deviation and receiver noise errors are modeled and 
removed or otherwise compensated for, as is done in a standard GPS signal 
analysis. The compensated pseudorange PR(t;t0;j) measured at the receiver, 
based on the received satellite signal (j), is represented mathematically 
as 
EQU PR(t;t0;j)=b+{(x.sub.s (t;j)-x(t)).sup.2 +(y.sub.s (t;j)-y(t)).sup.2 
+(z.sub.s (t;j)-z(t)).sup.2 }.sup.1/2, (9) 
where b=c..DELTA.t is a length equivalent of an unknown time offset of the 
receiver and c is the velocity of light in the medium. Pseudorange is 
measured at each of four or more distinct observation times t=t.sub.m and 
t=t.sub.n (m, n=1, 2, 3, 4; m&lt;n) for satellite numbers j1 and j2, 
respectively, with t1&lt;t2&lt;t3&lt;t4, and the following differences are formed. 
##EQU2## 
The notation adopted in Eqs. (10)-(16) applies to signals received from 
two different satellites (j1.noteq.j2) and to signals received from the 
same satellite (j1=j2). One or both of the observation times may, if 
desired, coincide with the time t0. The satellite coordinates (x.sub.s 
(t),y.sub.s (t),z.sub.s (t)) for the times t=t.sub.m and t=t.sub.n need 
not refer to the same satellite, or even to two or more satellites within 
the same system of satellites (e.g., all GPS or all GLONASS or all LEO), 
as long as the satellite coordinates are accurately known. Thus, different 
satellites with known coordinates can be used to determine the 
coefficients Dmn, Emn and Fmn for different indices n. 
First consider the situation where signals are received from a single 
satellite (j1=j2), at two different observation times. Three difference 
equations (10), formed using the four times t.sub.m,t.sub.n =t1, t2, t3, 
and t4 (t.sub.m .noteq.t.sub.n), produce three linear equations in the 
four unknowns x0, y0, z0 and b. These three linear difference equations 
are expressible in the form 
##EQU3## 
The matrix entries Dmn, Emn and Fmn are interpretable as x, y and z 
location coordinate differences resulting from rotating the respective 
locations given by (x.sub.s (t.sub.s,n),y.sub.s (t.sub.s,n),z.sub.s 
(t.sub.s,n)) and (x.sub.s (t.sub.s,m),y.sub.s (t.sub.s,m),z.sub.s 
(t.sub.s,m)) around the planet axis according to the respective rotation 
matrices 
##EQU4## 
Requiring that the determinant of the matrix H(1;2,3,4) be non-zero is 
equivalent to requiring that the four sets of rotated location coordinates 
##EQU5## 
be non-coplanar. The satellite is acted upon primarily by a central force 
potential that depends only upon the radial distance between the planet 
center and the satellite. The non-coplanarity condition is satisfied for 
the four sets of rotated location coordinates for most choices of the 
distinct observation times t=t1, t2, t3 and t4. An exception occurs if the 
satellite orbit lies in a plane that is parallel to the equatorial plane 
of the planetary body, or if all satellites are geosynchronous. 
The determinant of the matrix H(1;2,3,4) is thus non-zero, and the inverse 
matrix H(1;2,3,4).sup.-1 exists and is computable by well known 
procedures. Equation (16) is thus invertible and yields the partial 
solution 
EQU X=H(1;2,3,4).sup.-1 {A-C+B.multidot.b} (24) 
expressing the spatial location coordinates x0, y0 and z0 in terms of the 
time offset b. This approach also works where one or more location fix 
coordinates x0, y0, z0 and b is known, by substituting known values. A 
fourth equation, expressible in terms of only the remaining unknown (here, 
b) is obtained by squaring Eq. (9) for one of the selected observation 
times to obtain a quadratic equation in the remaining unknown, such as b, 
viz. 
EQU (PR(t.sub.n ;t0)-b).sup.2 =(x.sub.s (t.sub.s,n)-x(t.sub.n)).sup.2 +(y.sub.s 
(t.sub.s,n)-y(t.sub.n)).sup.2 +(z.sub.s (t.sub.s,n)-z(t.sub.n)).sup.2. 
(25) 
Here, three of the four unknown parameters x0, y0, z0 and b are replaced by 
their linear relationship in terms of the fourth unknown parameter, using 
Eq. (17) or (24). 
The quadratic equation (25) in the fourth unknown (for example, b) has two 
roots, one of which will have a physically realistic small magnitude (of 
the order of microseconds or milliseconds). This physically realistic 
value of b is then substituted into the three difference equations (10) or 
the matrix equation (24), and the values for the remaining unknowns x0, y0 
and z0 are determined. Because measurements at only four times are used, 
the solutions for the unknowns x0, y0, z0 and b should be internally 
consistent for this set of measurements. Any one of the four location fix 
coordinates x0, y0, z0 and b can be used as the "fourth" unknown, although 
it is most convenient to use b because the physically realistic value of b 
is relatively small, of the order of 10-10,000 cm or less. 
Alternatively, four difference equations (10) can be formed using five 
distinct times t.sub.m, t.sub.n =t1, t2, t3, t4 and t5 (t.sub.m 
.noteq.t.sub.n), and the solutions for the four unknowns x0, y0, z0 and b 
can be determined from the resulting linear equations. An advantage of 
this alternative is that only linear equations are analyzed. Possible 
disadvantages of this alternative are that (1) five measurements instead 
of four must be made and (2) the solutions for the four unknowns may not 
be internally consistent among the measurements at the five distinct 
measurement times. 
The solution obtained above uses compensated pseudorange values measured at 
four different times, preferably separated by several minutes to a few 
hours to obtain a more favorable constellation or configuration of the 
satellite at the four observation points. As the satellite moves in a 
first orbit across the sky, the receiver location (x0,y0,z0) will move in 
another orbit around the body's axis. This joint motion of the satellite 
location and the receiver location should produce four non-coplanar 
locations in space for the four pseudorange observations. An exception 
occurs when the orbital plane for the satellite is substantially parallel 
to, or coincides with, the equatorial plane of the planetary body. This 
exception does not occur for the GPS satellites orbiting the Earth and is 
very unlikely to occur for a "mother ship" orbiting a planetary body, such 
as Mars. 
The above development allows determination of the location of a fixed 
receiver on the surface of a planetary body, such as Mars or Earth, where 
the orbit parameters of a single satellite are known and the compensated 
pseudorange for the satellite can be measured at four distinct times. If 
the receiver time offset is unimportant, only three distinct observation 
times are used. 
The above formalism extends to observation of compensated pseudoranges for 
one, two, three or four distinct satellites at distinct times. This 
extends the GPS analysis to situations where fewer than four satellites 
are observable; this also allows a pseudorange measurement for a first 
satellite to be used with a second pseudorange measurement for a second 
satellite at a second time, with a third pseudorange measurement for a 
third satellite at a third time, and/or with a fourth pseudorange 
measurement for a fourth satellite at a fourth time, if the same satellite 
cannot be observed at all four times. Further, LD signals from different 
LD systems can be used to determine location according to the invention, 
using a "mixed system." For example, one or more LD signals received from 
GPS signal sources can be used together with one or more LD signals 
received from GLONASS or LEO signal sources. 
Consider next the situation where signals are received from two different 
satellites (j1.noteq.j2) at the same time t.sub.m =t.sub.n (=t0 
optionally). The formalism set forth in Eqs. (10)-(16) is applicable here 
as well, with t.sub.m =t.sub.n =t0. Equations (10)-(13) and (16) are 
unchanged, and Eqs. (14) and (15) simplify to 
EQU Dmn=2{x.sub.s (t.sub.s,n ;j2)-x.sub.s (t.sub.s,m ;j1)}, (14') 
EQU Emn=2{y.sub.s (t.sub.s,n ;j2)-y.sub.s (t.sub.s,m ;j1)}. (15') 
The resulting equations (17)-(25) are formally unchanged, although the 
coefficients Dmn and Emn for the difference in Eq. (10) will computed 
differently for two signals from the same satellite measured at different 
times (j1=j2; t.sub.m .noteq.t.sub.n) and for two signals measured at the 
same time from different satellites (j1.noteq.j2; t.sub.m =t.sub.n =t0). 
In the remaining situation, one satellite signal (j=j1) in the difference 
formed in Eq. (10) is measured at t=t.sub.m =t0 and the other satellite 
signal (j=j2.noteq.j1) in the difference is measured at a time t=t.sub.n, 
which may be the same as, or different from, the time t=t0. In this 
situation, Eqs. (10)-(13) and (16) are unchanged and Eqs. (14) and (15) 
simplify to 
##EQU6## 
Again, the resulting equations (17)-(25) are formally unchanged. 
The preceding discussions cover the physically possible situations for 
formation of the difference of the pseudorange squares in Eq. (10): (1) 
observation of a single satellite pseudorange at two, three or four 
distinct times; (2) observation of a first satellite pseudorange at one, 
two or three distinct times and observation of a second satellite 
pseudorange at a selected time, where the second satellite observation 
time may, but need not, coincide with one of the observation times for the 
first satellite; (3) observation of a first satellite pseudorange at N 
distinct times and observation of a second satellite pseudorange at M-N 
distinct times (M.gtoreq.N+1; N=1,2,3), where any one of the second 
satellite observation times may, but need not coincide with any one of the 
observation times for the first satellite; (4) observation of a first 
satellite pseudorange at first and second distinct times and observation 
of a second satellite pseudorange at third and fourth distinct times, 
where one of the third and fourth times may, but need not, coincide with 
at least one of the first and second times; (5) observation of a first 
satellite pseudorange at a first time, observation of a second satellite 
pseudorange at a second time and observation of a third satellite 
pseudorange at third and fourth distinct times, where the first time 
and/or the second time may, but need not, coincide with at least one of 
the third time and fourth times; and (6) observation of a first satellite 
pseudorange at a first time, observation of a second satellite pseudorange 
at a second time, observation of a third satellite pseudorange at a third 
time and observation of a fourth satellite pseudorange at a fourth time, 
where two or more of the first, second, third and fourth times may, but 
need not coincide. 
Table 1 sets forth non-equivalent possibilities for use of LD signals 
received from one, two, three or four distinct satellites, indicated as a, 
b, c, d in column three, for four time values t1, t2, t3, t4 satisfying 
t1.ltoreq.t2.ltoreq.t3.ltoreq.t4. 
TABLE 1 
______________________________________ 
Satellite signal assignments 
No. of 
Time seguence 
satellites 
Satellite signals received 
______________________________________ 
t1 = t2 = t3 = t4 
4 a,b,c,d (at times t1, t2, t3, t4, respectively) 
t1 &lt; t2 = t3 = t4 
3, 4 a,a,b,c a,b,c,d 
t1 = t2 &lt; t3 = t4 
2, 3, 4 a,b,a,b a,b,b,c a,b,a,c a,b,c,d 
t1 = t2 = t3 &lt; t4 
3, 4 a,b,c,a a,b,c,d 
t1 &lt; t2 &lt; t3 = t4 
2, 3, 4 a,a,a,b a,b,a,b a,a,b,c a,b,a,c a,b,c,a 
a,b,c,d 
t1 &lt; t2 = t3 &lt; t4 
2, 3, 4 a,a,b,a a,a,b,b a,a,b,c a,b,c,a a,b,c,b 
a,b,c,c a,b,c,d 
t1 = t2 &lt; t3 &lt; t4 
2, 3, 4 a,b,a,a a,b,a,b a,b,b,a a,b,b,b a,b,a,c 
a,b,b,c a,b,c,d 
t1 &lt; t2 &lt; t3 &lt; t4 
1, 2, 3, 4 
a,a,a,a a,a,a,b a,a,b,a a,b,a,a a,a,b,b 
a,b,a,b a,b,b,a a,b,a,c a,b,c,a a,b,b,c 
a,b,c,b a,b,c,c a,b,b,b a,b,c,d 
______________________________________ 
The eight time sequences in column 1 divide naturally into four subgroups 
of sizes one, three, three and one, as indicated, although the number of 
possibilities for satellite signals received in column 3 is not 
necessarily the same for each member of a subgroup. The time-ordered sets 
of satellite signals set forth in column 3 of Table 1 may not exhaust the 
possibilities but are included to illustrate the possibilities of use of 
signals received from one, two, three or four distinct satellites. 
Receipt of signals from two distinct satellites at "simultaneous" times, 
indicated by a partial time sequence such as t1=t2, is to be interpreted 
here as covering receipt of these signals from the two satellites "a" and 
"b" within a few hundred milliseconds, or less, of each other. In theory, 
receipt of signals, from either the same satellite or from two distinct 
satellites, at two non-simultaneous times, indicated by a partial time 
sequence such as t1&lt;t2, can theoretically occur at two times t1 and t2 
that are at least a few hundred milliseconds apart. In practice, however, 
two non-simultaneous times t1 and t2 are preferably at least a few minutes 
apart in order to provide an adequate constellation for the two, three or 
four locations of the satellite(s) used to provide the pseudorange values 
for the computations set forth in the preceding development. 
The number of pseudorange measurements required here will depend upon the 
number of location fix coordinates (x0,y0,z0,b) that are required for the 
situation. If, for example, K of these location fix coordinates (K=1,2,3) 
are already known, 4-K independent pseudorange measurements are needed to 
determine the remaining 4-K unknown location fix coordinates. These 4-K 
pseudorange measurements may be made using a single satellite or any 
larger number of satellites, up to 4-K. 
The orbit(s) of the satellite(s), assumed to be circular in Eqs. (3)-(5), 
may be extended to an elliptical orbit, illustrated in FIG. 3, by the 
following development. The ellipse has a semi-major axis (x') of length a, 
a semi-minor axis (parallel to y') of length b, and an eccentricity given 
by 
EQU .epsilon.={1-b.sup.2 /a.sup.2 }.sup.1/2, (26) 
where b&lt;a. The radius length p of a line segment from an ellipse focus F0 
to its intersection with the ellipse curve is given by 
EQU .rho.(.phi.)=a(1-.epsilon..sup.2)/(1+.epsilon. cos(.phi.+.pi./2)), (27) 
where .phi. is the angle the line segment makes with the y'-axis in FIG. 3. 
(K. R. Symon, Mechanics, Addison Wesley, Cambridge, Mass., 1953, pp. 
111-114). Note that the azimuthal angle .phi. is defined with reference to 
the x'-axis rather than to the more conventional y'-axis. The analogues of 
Eqs. (3)-(5) for an elliptical orbit, instead of a circular orbit, are 
verified to be 
EQU x.sub.s (t;j)=.rho.(.phi.) cos.PHI. cos.omega.(t-.tau.0)+.rho.(.phi.) 
sin.PHI. sin.omega.(t-.tau.0), (28) 
EQU y.sub.s (t,j)=-.rho.(.phi.) sin.PHI. cos.theta. 
cos.omega.(t-.tau.0)+.rho.(.phi.) cos.PHI. cos.theta. 
sin.omega.(t-.tau.0)+z sin .theta., (29) 
EQU z.sub.s (t;j)=.rho.(.phi.) sin.PHI. sin.theta. 
cos.omega.(t-.tau.0)-.rho.(.phi.) cos.PHI. sin.theta. 
sin.omega.(t-.tau.0)+z cos.theta., (30) 
where the planetary body is located at an ellipse focus F0 (FIG. 3). 
Thus far, the LD signal receiver is assumed to be motionless, or nearly so, 
during a continuous time interval including the observation times t=t1, 
t2, t3 and t4. The LD signal receiver is now allowed to move along an 
arbitrary path P, where the initial spatial location coordinates 
(x0,y0,z0) are unknown but the differential location coordinates 
(.DELTA..times.0'(t.sub.n),.DELTA.y0'(t.sub.n),.DELTA.z0'(t.sub.n)) of the 
receiver relative to this initial location at an earlier or later time 
t=t.sub.n are known (n=1, 2, 3, 4). If Eqs. (6), (7) and (8) are 
redeveloped for the time t=t.sub.n, it is easily verified that these 
relations take the alternate forms 
EQU x(t.sub.n)=(x0+.DELTA.x0'(t.sub.n)) cos.omega.0(t.sub.n 
-t0)+(y0+.DELTA.y0'(t.sub.n)) sin.omega.0(t.sub.n -t0), (31) 
EQU y(t.sub.n)=-(x0+.DELTA.x0'(t.sub.n)) sin.omega.0(t.sub.n 
-t0)+(y0+.DELTA.y0'(t.sub.n)) cos.omega.0(t.sub.n -t0), (32) 
EQU z(t.sub.n)=z0+.DELTA.z0'(t.sub.n), (33) 
where .DELTA.x0'(t.sub.n), .DELTA.y0'(t.sub.n) and .DELTA.z0'(t.sub.n) are 
known. One verifies that, with the indicated changes made in the observed 
location coordinates (x0+.DELTA.x0'(t.sub.n), y0+.DELTA.y0'(t.sub.n), 
z0+.DELTA.z0'(t.sub.n)), Eq. (17) assumes the alternate form 
##EQU7## 
The analogue of Eqs. (17) and (24) become, respectively, 
EQU H(1,2,3,4)X=A-C-.DELTA.X0'+B.multidot.b}, (37) 
EQU X=H(1;2,3,4).sup.-1 {A-C-.DELTA.X0'+B.multidot.b}, (38) 
and Eq. (25) is unchanged, except that the location coordinates 
(x(t),y(t),z(t)) now become (x0+.DELTA.x0'(t), y0+.DELTA.y0'(t), 
z0+.DELTA.z0'(t)) in Eq. (25). 
Allowing the initial unknown receiver coordinates (x0,y0,z0) to change by 
known amounts (.DELTA.x0'(t), .DELTA.y0'(t), .DELTA.z0'(t)) relative to an 
initially unknown location at subsequent times will be useful in surveying 
activities, or where a particular satellite cannot be observed at a 
particular time because of blockage by canyon walls, trees, buildings or 
other structures. Note that the path P by which the receiver arrived at a 
subsequent location with coordinates (x0+.DELTA.x0'(t), y0+.DELTA.y0'(t), 
z0+.DELTA.z0'(t)) is not important: only the coordinates at the 
observation times t=t1, t2, t3, t4 are involved in the solution of Eqs. 
(34), (35), (36), (37) and (38). 
The LD receiver/processor 11 in FIG. 1 identifies the LD signal source and 
measures or otherwise determines a pseudorange from the LD signal source 
to the LD antenna 13, based on signal propagation time or another suitable 
measurement. A pseudorange is an approximate distance from an LD signal 
source, which transmits an LD signal, to an LD signal antenna, which 
subsequently receives the LD signal, and is usually based upon LD signal 
propagation time. For this purpose, the LD signal is often a 
time-determined signal, (1) with numerous time reference indicia included 
in the signal to allow quantitative determination of what portion of the 
signal is presently being received, and (2) with different indicia that 
are part of the signal being transmitted at specified times by the LD 
signal source. A "raw" pseudorange measurement, based on a signal 
transmitted from a satellite (j) and received by an LD receiver (i), often 
includes errors from one or more of the following artifacts: (1) time 
delay .tau..sub.I (t;i;j) (over and above the normal time delay due to 
signal propagation at the speed of light in vacuum) due to signal 
propagation in the ionosphere; (2) time delay .tau..sub.T (t;i;j) due to 
signal propagation in the troposphere; (3) satellite transmitter time 
offset SCB(t;j); (4) receiver noise .eta.(t;i,j); (5) receiver time offset 
RCB(t;i); and (6) receipt of multipath signals m(t;i;j) at the receiver 
antenna. It is assumed here that most or all of these errors, except 
possibly receiver time offset, are modeled and removed or otherwise 
compensated for from the measured pseudorange. 
The measured pseudorange signal is expressed in equivalent length units as 
EQU PR(t;i,j)=R(t;i,j)+SCB(t;j)+RCB(t;i)+.tau..sub.T (t;i,j)+.tau..sub.I 
(t;i,j)+m(t;i;j)+.eta.(t;i;j), (39 ) 
where R(t;i;j) represents the "true" range from the receiver number i to 
the satellite number j at the time t, as determined from navigation 
ephemeris (or almanac information) that is available at, or received by, 
receiver number i. 
FIG. 4 is a schematic view of LD signal apparatus 41 suitable for practice 
of the invention. An LD signal antenna 13 receives one or more LD signals 
from one or more LD signal sources (not shown) and passes this signal(s) 
to an LD signal receiver/processor 11 that is part of a computer 43. The 
computer 43 (or the receiver/processor 11) measures or otherwise 
determines a pseudorange associated with the presently received LD signal, 
for each of the one or more LD signals presently received. The computer 43 
then collects the pseudoranges measured at two or more selected (distinct 
and spaced apart) observation times and uses software that is temporarily 
or permanently stored in a computer memory 45 to perform the analysis 
discussed in the preceding development, using a computer CPU 45. 
Optionally, the computer 43 includes a visually perceptible or audibly 
perceptible display 47 that displays the initial location coordinates 
(x0,y0,z0) or one or more sets of the changing location coordinates 
(x0+.DELTA.x0'(t), y0+.DELTA.y0'(t), z0+.DELTA.z0'(t)) for the receiver, 
either graphically or in an alphanumeric format. 
The LD signal sources 15A, 15B, 15C, 15D in FIG. 1 may be GPS satellites or 
GLONASS satellites, which systems are discussed in U.S. Pat. No. 
5,563,917, issued to Sheynblat and incorporated by reference herein. 
Alternatively, these LD signal sources may be Low Earth Orbit (LEO) 
satellites, such as the Motorola Iridium system of approximately 66 
satellites that follow orbits that are usually no more than about 2000 
kilometers above the Earth's surface. Signals from satellites that are all 
geosynchronous cannot be used here, because the coordinate difference 
between a ground-based LD apparatus and a given satellite will change 
little or not at all with time. However, one or two LD signals from 
geosynchronous satellites might be used here (by setting .omega.=0 
formally for such a satellite) if the remaining LD signals are received 
from non-geosynchronous satellites whose locations relative to the LD 
signal apparatus change substantially with time.