GPS Solutions: Closed Forms, Critical and SpecialConfigurations of P4P

ERIK W. GRAFAREND Department of GeoUesy ami Geoinfm<matics, Stuttgart University, Geschwlster-Scholl-Str. 240,0-70174 Stuttgart, Germany

JEFFREY SHAN Geomatlcs Engineering, School of Civil Engineering, Purdue University, West Lafayette, IN 47907

P4P is the pseudo-mnging 4-point problem as it appears

as the basic configuration of satellite positioning with

pseudo-ranges as observables. In order to determine the

ground receiver/ satellite receiver (LEO networks) position

from four positions of satellite transmitters given, a system

of four nonlinear (algebraic) equations has to be solved.

The solution point is the intersection of four spherical

cones if the ground receiver /satellite receiver clock bias is

implemented as an unknown. Here we determine the crit-

ical configuration manifold (Determinantal Loci, Inverse

Function Theorem, Jacobi map) where no solution ofP4P

exists. Four examples demonstrate the critical linear man-

ifold. The algorithm GS solves in a closed form P4P in a

manner similar to Groebner bases: The algebraic nonlin-

ear observational equations are reduced in the forward

step to one quadratic equation in the clock bias unknown.

In the backward step two solutions of the position

unknowns are generated in closed form. Prior information

in P4P has to be implemented in order to decide which

solution is acceptable. Finally, the main target of our con-

tribution is formulated: Can we identify a special configu-

ration of satellite transmitters and ground receiver/satel-

lite receiver where the two solutions are reduced to one. A

special geometric analysis of the discriminant solves this

problem. © 2002 Wiley Periodicals, Inc.

INTRODUCTIONhe solution of nonlinear observational equa-

tions in closed form has been a key issue of GPS

positioning and navigation (see, e.g., Abel & Chaffee,

GPS Solutions, Vol. 5, No. 3, pp. 29-41 (2002)© 2002 Wiley Periodicals, Inc.

1991; Bancroft, 1985; Chaffee & Abel, 1994; Grafarend &

Shan, 1997a, 1997b; Kleusberg, 1994; Lichtenegger, 1995).

Here we are aiming at a closed form solution for the pseu-

do-ranging 4-point problem (P4P), mainly to analyze two

key problems.

First, we want to study critical configurations of P4P,

namely those configurations of satellite transmitters and

ground receivers/satellite receivers, where no point posi-

tioning is possible. Such an analysis of critical P4P

extends results obtained for satellite laser ranging (Killian

& Meissl, 1969; Tsimis, 1973; Grafarend & Mueller, 1985)

and for satellite Doppler positioning (Killian & Meissl,

1980; Mueller, 1983; Schatz, 1980; Tsimis, 1973; Grafarend

& Mueller, 1985) to pseudo-ranging satellite networks.

Second, we try to find the bifurcation manifold,

namely, the special configuration where the two solutions

of nonlinear P4P reduce to one solution. Such a problem

has already been introduced by Abel and Chaffee (1991),

Bancroft (1985), Chaffee and Abel (1994), and Grafarend

and Shan (1997a, 1997b) as an unsolved problem.

Our contribution is organized as follows. In Section

1, we generate in the nonlinear P4P observational equa-

tions. They are interpreted as the intersection of four

spherical cones as long as we consider the ground receiv-

er/satellite receiver clock bias as an unknown. In Section

2 we derive the critical configuration of P4P, namely, the

linear manifold built up in the four-dimensional space of

unknown point positioning coordinates (x, y, z) and the

ground receiver /satellite receiver clock bias p. Explicitly

we determine the point set of Determinantal Loci which

are illustrated by four examples. Section 3 is devoted to

an explicit solution of the P4P nonlinear observational

equations. A type of Groebner basis (Becker & Weispfen-

Closed Forms, Critical and Special Configurations of P4P 28

ning, 1993; Cox, Little, & O'Shea, 1996) is constructed byreducing the system of nonlinear algebraic equations toone quadratic equation in the unknown ground receiver/satellite receiver clock bias. A backward substitution of

the two solutions for clock bias into the positioning equa-tions leads us to two solutions (x+, y+, z+), (x-, y~, z~) for

the Cartesian coordinates of the placement of the groundreceiver/satellite receiver. Finally, in Section 4 we identi-fy the configuration of satellite transmitters and groundreceiver/satellite receiver that guarantees one uniquesolution of P4P. The analysis of such a special configura-

tion is illustrated by a numerical example.

1. SETUP OF THE PSEUDO-RANGING P4P PROBLEM:NONLINEAR OBSERVATIONAL EQUATIONSThe modeling of GPS measurements is conventionallydivided into two parts: In the peripheric model thoseeffects are taken care of which can be computed from prior

information of the placement of the satellite transmitterand the ground receiver with a sufficient precision, accu-racy and reliability. Those computations are labeled as

reduction of measurements. Let us refer to two examples:GPS measurements are performed in spacetime,

which is nowadays considered as a four-dimensional

pseudo-Riemann manifold of signature (+ + + -). The GPScontrol station has already implemented a general rela-

tivistic spacetime model that reduces measured propertime differences to coordinate time differences. Thetransformation from proper time to geocentric coordi-

nate time is based on the exterior Schwarzschild metricand a circular GPS orbit. In a special reduction GPSmeasurements are further transformed by means of theellipticity of a realistic GPS orbit. Such an effect is of theorder of 5 m (Ludvigsen, 2000).

While the relativistic reductions lead us from thecurved spacetime (3,1)-pseudo-Riemann manifold to

their flat coordinate charts, the second reductionaccounts for the action of the refraction index in theatmospheric environment (troposphere, ionosphere) on

GPS measurements. The atmospheric medium betweenthe satellite transmitter and the ground receiver is dis-persive. Accordingly the wavelength dependence of the

refractive index is used by multifrequency GPS measure-ments to determine the influential parameters of therefractive index. The reduction of ionosphere and tropo-sphere refraction on GPS relativistically reduced meas-

urements accounts for an effect on the distance of thesatellite transmitter-ground receiver in the order of 150 m

(Hristov, 2000).As a result of these two reductions we end up with a

set of reduced GPS measurements in a flat Euclidean

space, namely a three-dimensional space R3 equippedwith a Euclidean metric. In its canonical form, the coor-dinates of the metric tensor are elements of a unit matrixIs. Now we are prepared to set up the core model of GPS

observational equations.The pseudo-ranging 4-point problem (P4P) as it

appears in its basic configuration in Figure 1 is defined as

FIGURE 1. Configuration of 4P pseudo-ringing.

30 Gralarend and Shan

Closed Forms, Critical and Special Configurations of P4P 31

Grafarend and Shan

Corollary: The affine variety characterizing P4P is theintersection of four spherical cones.

Corollary: For the special cases (no receiver bias fi) the

affine variety reduces to the intersection ofthree spheres of type

2. CRITICAL CONFIGURATIONS OF P4PTo better understand the internal singularities of satellite

networks, critical configurations have been studied.

• For satellite laser ranging, we mention the basic workby Killian and Meissl (1969), Tsimis (1973), and Gra-

farend and Mueller (1985).• For satellite Doppler measurements, critical configu-

rations had been identified by Killian and Meissl(1980), Mueller (1963), Schatz (1983), Tsimis (1973),and Grafarend and Mueller (1985).

• For GPS networks, first critical configurations

excluding the receiver bias have been identified byWunderlich (1993).

These references document that practically nothinghas been done in GPS pseudo-ranging and carrier beatphase networks to identify critical configurations. Here

we shall focus on the three-dimensional critical manifold

for P4P.What are the critical configurations of P4P? To answer

this question we must first determine how to find them!

FIGURE 4. Critical configuration ol reduced P4F (zero receiver Was).

Closed Forms, Critical and Special Configurations of P4P 33

FIGURE §. Critical configuration of reduced P4P.

34 Orafarend and Shan

J is twice the area of the triangle . As soon as the areagoes to zero we end up with the critical configuration.

Example 2.2 (reduced P4P, critical configuration):

Let us consider the special P4P case where and holdsand P4P is reduced to two algebraic equations of type

satellite points 1 and 2, respectively and the pseudo-ranges d\ (first observation) and d,2 (second observation).As soon as the area approaches zero we end up with thecritical configuration.

Example2.3 (zero bias solution, critical configuration):If there is no receiver bias (3, P4P is reduced to three

algebraic equations of type

The Determinantal Loci generated by |J| = 0 are just

points on the plane FI2 e P3, in particular on the straight

line A2 e P3 if x - y. We have performed the algebraicmanipulations in order to demonstrate that the trilinearterms xyz and the bilinear terms xy, yz, zx vanish within

|J|. What is left is the equation of a linear manifold.

Closed Forms, Critical and Special Configurations of P4P 35

Finally there is the standard geometric interpreta-tion of the Jacobian |J| = 0.

[J| is six times the volume of the tetrahedron (P, P1,P2, P3). As soon as the volume goes to zero we end up withthe critical configuration.

manipulations in order to demonstrate that the trilinearterms xyz and the bilinear terms xy, yc/3, eft x vanish with-in | J|! What is left is the equation of a linear manifold.

Finally there is the standard geometric interpreta-

tion of the Jacobian |J| = 0.

J is six times the volume of the tetrahedron (x, y,

cft\xl, y1, di\x2, y2, d2\x3, y3, ds). As soon as the volumegoes to zero we end up with the critical configuration.

8. SOLVING THE P4P ALGEBRAIC EQUATIONSBy means of Equations 26-34 (see Section 3.1) we illus-trate the GS algorithm for a closed solution of the P4Pequations (5). The action is with the first forward step

where we compute from {/i, /a, fs, fa} the differences

36 Grafarend and Shan

Giosed Forms, Critical and Special Configurations of P4P 37

4. SPECIAL CONFIGURATION OF P4PThe previous section on solving the P4P algebraic equa-tions in closed form has documented that two solutions

of P4P exist. We had to solve one quadratic equation forground receiver/satellite receiver clock bias. Let us now

determine the bifurcation point, more accurately, themanifold where the solution (x\, x^) bifurcates (Rhein-boldt, 1978). Such a special configuration is triggered by

Figure 6 illustrates such special configurationsderived from the discriminant analysis by means of thefunction

4.1. Specific Configuration of P4P ("Discriminant")

FIGURE 8. Special configuration, discriminant analysis: the graph of the funotion 1/ssn2y.

Grafarend and Shan

(48)

(49)

(50)

Corollary: In terms of the inner product (a, b) and the

h' norms ||a|J2 and j|b|J2 of the vectors a and b, the discrim-inant A can be represented by

(51)

(52)

(53)

Obviously the (AAT) *-weighted norm ar(AAr) 1a

characterizes the dimension of the solution space. AAr

contains the configuration of the four satellites. In terms

of the three variables of the 1-array a = [di - (h, d\ - (fa, d\

- dn]T, aT(AAT)~la generates a triaxial ellipsoid. Indeedthe surface of the triaxial ellipsoid is also a critical sur-

face, since

Closed Forms, Critical and Special Configurations of P4P 39

• a point inside the triaxial ellipsoid generates twosolutions for X4,

• a point outside the triaxial ellipsoid generates no

solution at all,• a point on the triaxial ellipsoid corresponds to one

solution xl = xt.

Let us briefly discuss the results of this section: Wehave shown that the P4P solution is characterized by an

indicator ellipsoid which depends on the relative posi-tion of the 4 transmitter satellite and the measuredpseudo-ranges. In detail we have presented the discrim-inant analysis which makes the receiver clock bias

unique. If the discriminant describes a point on the sur-face of the indicator ellipsoid, we arrive at the uniquesolution. In contrast two solutions are computed if the

discriminant describes a point inside the indicator ellip-

soid. No solution is obtained for a discriminant thatgenerates a point outside the indicator ellipsoid. In theideal scenario, the receiver is equally distanced to the

four satellite transmitters, generating one unique solu-tion of P4P.

40 Grafarend and Shan

GPS satellite positions, pseudo-ranges and measures of configuration

BPS satellite number x(m) y(m) zlm) Aim)

1 14,832,308.660 -20,466,715.890 -7,428,634.750 24,310,764.0642 -15,799,854.050 -13,301,129.170 17,133,838.240 22,914,600.7843 1,984,818.910 -11,867,672.960 23,716,920.130 20,628,809.4054 -12,480,273.190 -23,382,560.530 3,278,472.680 23,422,377.972

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ÜQGRAPHIESErik W. Grafarend, Ph.D., is Professor and Head of the

Department of Geodetic Science, University of Stuttgart

(Germany). He is interested in a broad ränge of theories

underlying geodesy.

Jeffrey Shan, Ph.D., is assistant professor at Purdue

University, Department of Civil Engineering (area of Geo-

matics Engineering). He graduated from Wuhan Univer-

sity (P.R.C). His current research interests are geometry in

geodetic science, feature extraction, 3-D geographic

information Systems and visualization.

Peer Review: This paper has been peer-reviewed and

accepted for publication according to the guidelines pro-

vided in the Information for Contributors.

Ciosed Forms, Cntical and Special Configurations of P4P 41