**Autonomous Time-based**

**Precision Global Positioning**

M. Böhm

Telphykas, D-70499 Stuttgart, Solitudestr. 389, Germany

Tel.: 49.711.831131

Fax: 49.711.831124

e-mail: Dr.Boehm-Telphykas@t-online.de

**Abstract**

Three-dimensional time-based global positioning by exclu-sive autonomous
precision time measurements is achieved by using astronomical data and a
precision clock. The clock is providing the continuously changing
universal time t~x~ (UT), while fixed points in time t~i~ provide
\"inertial\" references. A local position coordinate, say meridian, is
determined by using the equation L = 360° (t~x~ - t~i~ )/T~i~ , with
T~i~ being 24 hours, i.e. the time of a daily rotation of earth.

**Introduction**

Measurement of physical quantities generally re-quires a reference
quantity and a quantity to be measured of the same kind, for instance
two distances, two angles, two forces, two frequencies or two points in
time. For the autonomous, or self-contained, measurement of two
quantities, take angles like geographic longitudes, the reference
quantity is to be available at the same location as the quantity to be
measured, and it must be independent of the local position. Therefore it
has to be ensured that any position change results in a change of the
quantity to be measured, but does not effect the reference quantity.
Inertially viewed this quantity must be independent of local position,
while it changes apparently if the local position is used as the
reference. A positioning result, however, is not influenced by the
choice of the respective reference frame.

Autonomous positioning, i.e. positioning without any link to the
\"outside\" world, is at present available only via inertial platforms
comprising gyros and accelerometers Such platforms provide reference
frames calibrated at known positions, while position changes with
respect to these positions are measured via three-axis accelerations and
twice the integration of them over time^1^.

Another well-known example for dynamic effects is the pendulum of
Foucault, against which earth rotates at 15°/h. The local observer may
interprete this rotation as that of the oscillation plane of the
pendulum against earth considered to be fixed, i.e. not rotating. The
main disadvantage of such dynamic approaches, i.e. those using forces,
are the many error sources, which cause drift errors growing with
mission duration. This results in quite a technical effort to limit
these errors, while accuracies available still cannot match those
achievable via kinematic approaches like GPS.

GPS is based upon differential travel time mea-surements of signals
transmitted from satellites, the positions of which are exactly known at
any time and at any receiver via the received signals. GPS signals for
public use do not provide the highest possible accuracies which are
available to military users only.

Higher accuracies for civil applications are provided by utilization of
DGPS, where a ground station transmits corrective signals which enable a
civil GPS receiver to reduce the errors of its determined GPS position
fix ^2,3,4^.

For autonomous precision positioning utilization of time signals, which
are the most accurate means available in physics, is a must^13^. Time
signals for reference and measurement also ensure GPS accuracy, although
not self-contained or autono-mously.

The objective of this paper is to offer a new time-based kinematic
approach for autonomous global positioning without any radio signals and
without any gyros and accelerometers, while providing equal and better
accuracies than the best GPS or - perhaps in the future - Galileo system
can provide for, at a fraction of the expenses required for
satellite-based positioning systems. Main issue besides unmatch-able
accuracy is always the autonomy of the positioning device, i.e.
reference and measuring signals are to be available at the same local
position. Time-based autonomous positioning, which does not require -
exept for calibration at a known position before a mission begins - any
external means, is exclusively utilizing differences in time between
known fixed points in time and a local precision clock. However, it has
to be recognized that the positioning results achieved autonomously a
priori do not involve inertial attitude or vectorial information and may
be compared in this respect with GPS fixes which also lack such kind of
information. In addition, it has to be kept in mind, that also precision
clocks are subject to drift^12^. However, the drift error effect can be
easily overcome, as will be outlined later.

**Physical Background**

Just the time signal (UT) of a clock operated autonomously cannot
indicate a position. Classic positioning on ships therefore required, in
addition to UT, the momentaneous position of the sun, i.e. an inertial
reference to determine local time (LT), i.e. the local meridian.

An autonomous positioning approach can substitute the external -
inertial - sun by an internal sun, which is, however, no more inertial.
The position of this internal sun can be defined with respect to any
surface point on earth as a specific time difference, which changes with
the change of position of this surface point. Utilization of an internal
sun via given known points in local time, accurately related to known
points on earth as references, and the variable local time of the own,
still unknown position makes it possible, by means of the local
precision clock, to continuously determine the local position via local
measurements of time differences. This approach applies to the
determination of the local meridian, the local parallel, and the local
altitude, as will be discussed in more detail later.

At sub-relativistic velocities the two terms universal time (UT) and
local time (LT), or local angle (longitude, latitude), are linked by the
third term angular velocity ω. For mathematical treatment of different
points on earth consideration of the relativity of simultaneity is
sufficient. Local time is an analogy to space time, known from General
Relativity. Its unit is the second, if position is determined by angles
as fractions of 2π, i.e. without metric units.

Exclusive utilization of time for autonomous global positioning in
combination with known positions, lines, or planes recommends a careful
consideration of what is meant by the term \"time\". There are many
publications on this item ^5,6,7,8,9,10^. Of major importance is the
term relativity of simultaneity. Also consideration of the Sagnac effect
is helpful. In general, it has to be analyzed how different terms like
time, clock time, local time, differential time, space time etc. apply
to the task of autonomous positioning on earth, and how they can be
utilized.

Definitions:

*Time*: Basic term to describe matter motion

*Clock time*: Point in time

*Point in time*: Momentaneous clock display

*Differential time*: Difference between two points

in time

*Space time*: Desciption term for events

*Local time*: point in time of a surface point on earth,

referred to sun position

*Solar time*: Based upon a day with 86400 s

*Siderial time*: Unit is siderial day between two upper

culmination points of the spring point V

*Time equation*: True solar time minus mean solar

time

*Relativity of simultaneity*:

> ct~2~ = (ct~1~ - (v/c)x~1~) (1- v^2^/c^2^)^-1/2^

For autonomous positioning on earth the relativistic term of the
equation above can be neglected since v^2^/c^2^ ≈ 10^-8^. Thus ct~2~ =
ct~1~ - (v/c)x~1~. The approach described in this paper utilizes time
and position differences measured and computed autonomously within a
moving object.

It has to be checked which differences result when the measurements are
referred to the resting inertial frame instead of to the moving object,
and vice versa.

P. Mittelstaedt^3^ writes: \"If one considers two events E~1~(x~1~,t~1~)
and E~2~(x~2~,t~2~), then their space-time (four-dimensional) distance
∆(x,t)^2^ = c^2^(t~2~-t~1~)^2^ - (x~2~-x~1~)^2^ is invariant towards
Lorentz transformations and therefore equal within any inertial frame,
the time difference ∆t = t~2~ - t~1~ itself, however, is not
Lorentz-invariant\". From this, under certain conditions, the relativity
of simultaneity results. It governs the synchronisation of clocks within
an inertial system via electromagnetic signals from the spatial origin
according to the equation t = (∆s/c) + t~0~ , or t-t~0~ = ∆s/c ^8^. ∆s
is the spatial Euklidian distance to the spatial origin, t~0~ is the
point in time of transmission of a synchronising signal, and c is the
velocity of light. It becomes apparent that this equation also results
from the application of the Sagnac effect to the same situation.

For the Sagnac effect can be formally applied to the rotating earth
which transports any fixed electronic clock with its known angular
velocity. The well known formula for this effect is ∆Φ = (4A/λr)β =
(4A/λc)ω. With ∆Φ = ω∆t it can be derived for earth - the angular
velocity ω of which is practically constant and exactly known - ∆t =
(r/c) ∆Φ. The phase ∆Φ of any point on a parallel is, in addition,
ω∆t~LT~ = ∆Φ. This leads to ∆t⋅c/r =∆t⋅Ω = ω ∆t~LT~. The time shift ∆t
is then the fraction ω/Ω of the local time difference ∆t~LT~. The factor
ω/Ω is about 1,55^.^ 10^-6^. That means travel time for a full
circulation of a signal around earth from and to its origin is 0,133673
seconds, compared to the 24 hours of a daily rotation of earth.

These considerations result in a new approach for autonomous time-based
positioning on the rotating earth. Fixed points on earth could be
determined - if their angular velocities ω~i~ were known - by simple
differential time measurements. For objects moving relative to earth or
selected points on it, however, the angular velocity is not known. In
these cases one has to use a virtual angular velocity Ω = c/r =47
s^-\ 1^, which results from the diameter of earth 2r and light velocity
c. There is no signal which can travel around earth faster than about
133 milliseconds. Compared to the real angular velocity of the daily
earth rotation which is ω = 7,272^.^10^-5^ s^-1^ this means an increase
of angular velocity by the factor of about 648 000. The meaning of this
factor will be outlined later. Working with the virtual angular velocity
Ω is always required for autonomous time-based positioning when the
motion of an object with respect to earth surface changes the inertial
angular velocity of the object from ω of earth to ω±∆ω, with ∆ω a priori
unknown. One then has to continuously utilize the macroscopic Sagnac
effect by relying upon the virtual angular velocity Ω. Of course, this
approach can also be used for positioning of fixed objects. The
introduction of r, however, provides an additional parameter for the
respective calculations, which have to consider the variations of r.

![](media/image1.wmf)

The space-time distance ∆(x,t) between two events can be expressed,
under certain conditions, by time differences, if one eliminates the
constant factors like light velocity c for the spatial terms. It is then
possible to determine a spatial coordinate by the local measurement of
points in time (UT), while other required points in time can be taken
from a local data bank, which contains well-kown astronomical data of
the orbits of all points on earth in form of rectascension and
declination values.

When determining the local meridian of an object P, a reference point
P~0~ on a reference meridian which has the angle Φ~x~ referred to the
position of the sun or its local time Φ~x~/ω with respect to the line
sun-earth center (S-M) at any UT point in time t~x~ , is always exactly
known as a means to determine the still unknown local meridian of object
P. The local meridian L (ϑ) results, if P~0~ is defined as a point on
the Greenwich meridan (0°). Astronomical data of earth on its orbit are
needed and are available as published tables^11^ of rectascension and
declination values.

![](media/image1.wmf)

Decisive for the feasibility of the described approach is the
utilization of accurate table values in combination with a precision
clock, which should not only provide accurate universal time, but also
high phase stability and little drift. In addition, for each mission the
system is to be calibrated at a known position, for instance a depart
point with also known altitude. Another key element is a motion line
which represents the angular velocity Ω and directly connects both the
unknown local point P und the reference point P~0~ .

**Measuring Principles**

The proposed positioning method is described using twelve figures. In
fig.1 the basic situation for autonomous time-based positioning on the
rotating earth is outlined. There is sketched a parallel 1 with the

Greenwich meridian 0° at 20.00 hours local time and the two meridians
120° with the local time 04.00 hours and 240° with the local time 12.00
hours. Universal time t~x~ be 10.30 hours. For this configuration there
are - sequentially - always changing points in universal time t~xi~
which can be taken from available tables^11^. Clock 2 is located at the
Greenwich meridian, with t~x~ = 10.30 hours, clock 3 is located at the
meridian 120°, with the same point in UT time 10.30 hours, however with
a different local time, namely shifted by 8 hours to 04.00 hours. Here
it becomes clear that t~x~ is valid for all meridians on earth, while
local time is specific for each meridian. It is also apparent that local
time and local angle are quantities with identical information
concerning position. They are linked by the equation ∆t~LT~ ω = Φ~L~. On
earth eight hours difference in local time mean an angle difference of
120°. If now the meridian P of clock 3 is unknown and only t~x~ (10.30)
is available, then this is not sufficient for positioning, but
additionally local time has to be introduced. Inertially viewed, the
tables^11^ already mentioned contain the local time for every meridian
at every point in universal time. Locally, however, this does not help
directly, and initially one does not know at P that the local meridian
is 120°. If one wants to determine the local meridian L (or ϑ) then two
quantities are required locally, namely one which does not vary with
position, i.e. a reference, and another one which is correctly related
to the local time of P. Dynamic methods, operating with gyros and
accelerometers, offer these informations. Purely kinematic autonomous
methods, however, cause concerns since it is argued that any position
change from P~0~ to P by the angle ϑ does change the reference time
signal in the same way as the measurement signal, and that the angle ϑ
of P, which is to be measured with respect to P~0~, is thus eliminated.
These concerns referring to traditional considerations are
understandable and principally justified. However, they do not apply to
the time-based positioning method described in this paper, since this
approach utilizes fixed points t~ai~ in universal time, which define
certain reference meridians with known properties like 00.00 or 12.00
hours local time, and which are neither time- nor position-dependent,
i.e. valid for any unknown point P.

During a measurement the time difference between the fixed reference
point in time t~a~ (S) and the continuously changing point in time t~x~
(UT) of the local position (P) is related to the real position of the
sun (S). The time difference measured between a fixed point in time t~a~
and t~x~ is a measure for the angular distance of the local meridian to
the inertial reference point, the sun. The local time t~LT~ = Φ~L~/ω of
every point on earth surface follows an exactly definable specific
function of time t~UT~. The differences in local time of these
practically linear functions can be interpreted as phase differences
with respect to the inertial reference, the line sun-earth (S-M). The
task is to determine a specific phase difference between P~0~ and P.

![](media/image1.wmf)**Traditional Considerations**

The sketches in fig. 2 indicate the equator as their horizontal center
lines, while the respective left ordinates are the reference meridians
defined by t~a~. The sketches show both principal ways to describe local
times, namely as inertial (1) or local (2) situation. One may use either
a simulated quasi-inertial display, for which a selected fixed point
like the sun is shown fixed also on the display (1), while

the simulated earth rotates. Or one choses a fixed earth, with the sun
rotating around it (2). If one assumes (in 1) that the (internal) sun is
the fixed reference then points P~0~, P~E~ and P move time-dependent
with respect to the fixed point in time t~a~, which defines a reference
meridian pointing to the sun. Or one selects (in 2) a point of the
respective three ones as a (local) reference, against which, if this
point is moving, the sun seems to move. If there is an observer at point
P, the reference point P~0~ at the point in time t~a~ moves, together
with the sun, to a different place than if the observer would be at
point P~E~. The spatial transition from P to P~E~ would mean a position
transformation, which would have to include a respective transformation
of local time.

![](media/image1.wmf)Staying at one of the moving points, the reference
point in time t~a~ can be interpreted as inertial point in the time
domain which defines the total local time domain. If locally observed
from P, this local time domain - comprising 24 hours or 86400 seconds -
would pass this point because of the motion of P with earth. The
respective inertial position change of P can be described by a
respective change of time or time difference. Of course, there is no
true vectorial inertial reference, as will be outlined later. However,
the angle difference between P and the selected reference, sun or
Greenwhich meridian, exactly fits.

In fig. 3 time dependence of the different local times of points P~0~,
P~1~, P~2~ and P~3~ is shown. The local times are expressed as local
angles. The four points may be located on the same parallels or on
different ones. P~0~ be the 0°-meridian, which at the point in time t~a~
exactly points to the sun (12.00). During 24 hours, the period T~L~,
this meridian rotates by the angle 2π, which is covered at the point in
time t~b~ = t~a~ +T~L~. The meridians of the three other points P~1~,
P~2~ und P~3~ also cover 2π during T~L~ = t~b~-t~a~. With respect to the
sun, and to P~0~, however, these points have the different angle
components ϑ~1~, ϑ~2~, and ϑ~3~, which are identical with the respective
inertial position angles. If both a local reference signal as well as
the measuring signal would contain the respective ϑ, then this quasi-dc
component would be eliminated and one could not measure locally and
autonomously any change of position. A position change can only be
measured if the local reference signal would not contain any
position-dependent \"dc\" angle component.

The motion lines of the four points are parallel to each other, and they
intersect with the UT axis at specific points in time. For P~0~ the
intersection point in time is t~a~.

The motion lines, the slope of which corresponds to the angular velocity
ω of the daily earth rotation, can be considered as inertial motion
paths of the points and comprise the elements universal time and local
time. A clock in Berlin and another one in Bonn

![](media/image1.wmf)show identical universal times, but their locations
correlate to different local times, of course. The respective local
times are usually not available via these clocks, though.

**New Approach**

Fig.4 shows the general relation of time and meridian or local time of
three selected fixed points on earth, the reference point P~0~, the
calibration point P~E~, and the local position P. The figure illustrates
the inertial angular change of the three points as a function of time
(UT). On earth this is determined by the angular velocity ω of the daily
rotation of earth which is practically constant.

The rotation angle is given by the equation ∆Φ = ω∆t. This equation
determines the relation between time and inertial position, i.e. local
time, which then is defined by Φ/ω = t~LT~. This means, that the
universal time difference ∆t~UT~ and the local time difference ∆t~LT~
are equal, and also that the angular change of any fixed point on earth
over time leads to the constant angular velocity ω. However, if one
locally connects the unknown point P with a reference point P~0~ then
the angular velocity of this motion line is not known.That is also valid
for points moving relative to earth surface. One has to utilize then, in
analogy to the \"microscopic\" Sagnac effect, the \"macroscopic\" Sagnac
effect with the ultimate angular velocity Ω = c/r. That was already
discussed in the introduction section. How the macroscopic Sagnac effect
is applied to the determination of local meridians, is dealt with using
figures 5 and 6. The same basic procedure is then applied to the
autonomous determination of latitude and altitude.

![](media/image1.wmf)In fig.5 the relation between universal time t~UT~
and local time t~LT~ is explained in more detail. This figure shows the
fundamental procedure to determine the unknown local meridian of any
fixed point P on earth. In order to make this \"inertial\" illustration
possible the meridian of the reference point P~0~ at the point in time
t~a~ has to be known. P~0~ changes its inertial position with respect to
the inertial reference point sun in an exactly known manner as a
function of time on the sketched motion line ω. At the point in time
t~a~ the real P~0~ meridian points exactly to the sun, and P~0~ at the
display, quasi as a sun substitute, may be considered to be an internal
sun. In an inertial representation both points P~0~ and P - if they are
fixed points - change their inertial positions as exactly known
functions of time, since they must move with the angular velocity of the
daily earth rotation and with respect to the inertial reference point in
time t~a~. However, the initial location of P at t~a~ is not known.
Therefore only the inertial meridian of P~0~ is defined exactly at every
point in time. The spatial attitude of any display does not matter at
all, since the relative positions are only functions of time. This is
valid also, if the local observer at P does not know the local meridian.
The time-based determination of a coordinate like a meridian can be
based upon the measurement of the time difference

∆t = (t~x~ - t~a~), provided the motion line ω\* is used which directly
connects P and P~0~. ∆t⋅ω\* delivers the angular distance between the
reference point in time (the internal sun) and the local meridian at any
local point in time. For a correct utilization of this basic
configuration the following procedure has to be considered. Each point
on earth moves on its own individual inertial path. All motion paths are
parallel to each other. If one keeps the fixed (inertial) time frame
t~a~ and t~a~ +T, then the meridian of P~0~ changes its position within
this time frame or window continuously, while the meridian of P with
respect to P~0~ does not change. The total meridan collective moves
through this fixed time window once within T (24 hours). Such
observation cannot be influenced by the attitude of the used display.
But, of course, such a display has no true inertial meaning (which
applies to any GPS measurement result, too).

The continuous measurement of the unknown local meridian of P via the
clock at this position principally is achieved as follows.

With P~0~, the meridian of which - at the point in time t~a~ - has the
local time 12.00 hours and points exactly to the sun, one defines a
reference coordinate system, the ordinate of which is the universal time
axis and the horizontal axis of which is the local time axis. The
inertial motion path ω of P~0~ does not meet P, but the motion path ω\*
does and connects P~0~ and P. If one reflects P at the t~a~-axis, one
gets point P\' which is mirror-symmetric to the t~a~-axis. P\' sits on a
motion line of P~0~ which corresponds to -ω\*. The local meridian of P
basically results from the equation

L = 360° (t~x~ - t~a~)/T provided one knows ω\*.

Physically one may interprete this procedure as an expanded Sagnac
approach. Starting at t~a~ (UT) and 12.00 (LT) from P~0~ (internal sun),
the stable electronic oscillation within a clock is transported in
reality (counterclockwise) to P, which at the moment of measurement t~x~
(UT) has the local time t~x~ - 12.00, and additionally a virtual
electronic oscillation is transported clockwise to point P\'. The time
distance (difference) between these two points is twice the time
difference t~x~-t~a~, and unambiguously corresponds to the local
meridian of P. Of course, the virtual point P\' is not a must for the
procedure. But it helps to reduce clock drift.

Since the angular velocity ω\* of P is not known, the procedure sketched
in fig. 5 has to be modified. Fig. 6 shows it for fixed as well as for
moving objects for meridian measurement. The essential difference to the
procedure shown in fig. 5 is the introduction of the known angular
velocity Ω = c/r. It is required, since the inertial angular velocity
ω\*= ω+∆ω of an object on earth with respect to an inertial reference
point is not known a priori. ∆ω can have any value, except one which
is - because of the relativity of simultanenity - beyond c/r.

If one therefore uses this limit of angular velocities on earth, then
one don\'t need to know ω\* = ω+∆ω. The price to pay is r, the radius of
earth which is a variable term and introduces an additional difficulty
in autonomous time-based positioning.

Neglecting this, however, determination of the local meridian of a
resting or moving P on earth is achieved as follows.

If point P moves relative to earth, then there are various motion lines,
which means that for identical points in universal time t~x~ there are
undefined points in local time for P, i.e. local meridians. The direct
utilization of ∆t = t~x~ - t~a~ would lead to an error, since the
rotation period of P changes from T (86400 seconds) to T\', which may be
shorter or longer than T, coresponding to the velocity vector of P
relative to earth surface.

In order to avoid this effect, the real but unknown angular velocity ω\*
of P with respect to P~0~ is substituted by the ultimate angular
velocity Ω = c/r, as discussed already before. Since for both
approaches - Sagnac effect and relativity of simultanenity - the same
local meridan must result it follows Φ~x~ = (t~x~ - t~a~) ω\' = (t~x~ -
t~a~\*) Ω. In this equation t~a~\* can be derived from t~a~, as will be
discussed later. The time difference ∆t\* = t~x~ - t~a~\* leads to the
looked-for local meridian in connection with the universal time period
T\* by multiplication with Ω as follows.

Beginning at the - unknown - local position P one draws a motion line Ω
with known slope at every point in time t~x~ until it intersects with
the t~a~\* - axis. This intersection corresponds to P~0~, or the
internal sun. The t~a~\* - axis is derived from the t~a~-axis of fig. 5
by applying the factor ω/Ω to the time difference t~x~ - t~a~, measured
continuously. That means one derives t~a~\* = ( t~x~ - t~a~) ω/Ω and
thus can divide (t~a~ + T) - t~a~, i.e. T, into Ω/ω (≈ 648 000 ) ranges
of T\*. t~a~\* is that constant local time axis, which comes closest to
the local time axis defined by t~a~ if one starts at t~x~.

Once P~0~\* is determined, one can draw also the motion line -Ω and
define point P\' which is reflected at the t~a~\*-axis. The distance
P-P\' , which corresponds to the time difference 2(t~x~-t~a~\*),
unambiguoulsy determines the local meridian if P~0~ is defined as a
point located on the Greenwich meridian, which means that t~a~ has to be
that point in time, at which the Greenwich meridian exactly points to
the sun. The local meridian of P is given by L = K (t~x~ -
t~a~\*)/T~L~\*. K is a known factor, which determines the measuring
range, i.e. 360° or a derivation from this if one uses \"eastern
longitude\" or \"longitude west\".

Time resolution of a 24 h period, i.e. one full daily rotation of earth,
also determines the achievable angle or local time resolution. If one
divides this full rotation cycle or period into 10^6^ parts, then one
part has the duration of 86,4 ms. During this time any point at the
equator moves tangentially with respect to the earth center by about 40
m. There is no need to also consider orbit velocity.

Within the local time period T~LT~ one has the assumed 10^6^ time
intervals of the universal time range as also local time or angle
segments. One angle segment corresponds to 2π/10^6^ = 1,59^.^10^-5^ or
20,63\'\'. Since Ω is employed, which has a period duration of T\* =
0,133 s, the assumed resolution of 10^6^ causes clock impulse intervals
of 0,133⋅10^-6^ s. This corresponds to a clock frequency of about 7 Mhz.
Since the present clock rate state of art is three orders of magnitude
higher, resolution can be increased respectively, which means to go from
the assumed 40 m (10^6^) resolution to 0,04 m (10^9^). Potential
resolution limits are not imposed by the achievable clock rate, but by
the drift rate of the applied clock. Also atomic clocks are subject to
drift^12^. It is, however, possible to neutralize any clock drift by
employing measuring time intervals of more than one period, for instance
by using instead of one daily rotation a yearly rotation which means 365
daily rotations.

Points in time as well as angles usually are already coded as digital
information. While local time is periodic with 24 hours, universal time
changes unidirectionally. Since t~a~ and the meridian pointing to the
sun at this point in time are very acurately known, as well as t~a~ ± nT
(n may be any integer), and also t~x~ can be measured with utmost
accuracy, the local meridian derived from these accurate terms - which
practically can\'t be distorted - is also highly accurate.

The reference point in time t~a~ - rectascension - is very accurately
known from literature^11^. Despite being an inertial angle, it is given
as an accurate time for every day of a year. The values of all meridians
could be stored for all days of a year autonomously, and be used as
needed for positioning computations.

The point in time t~a~\* is a numerical value for an inertial reference.
But as a number it only defines which angle Φ~x~ a chosen reference
meridian like that of Greewich has at any point in time t~x~ from the
selected inertial reference (sun). Locally there is no true, vectorial
inertial information connected to these numbers. The local meridian of P
is correctly determined, but the inertial attidude of this meridian in
space remains unknown from only this information.

![](media/image1.wmf)The option to shift the fixed point in time t~a~\*
by n periods - with n being any integer, for instance 365 for going to
the yearly instead of to the daily rotation of earth - opens two ways to
improved performance. The first one leads to the reduction of clock
rate, and the second one allows for neutralizing drift rates of clocks,
which means that simpler clocks can substitute high performance atomic
clocks.

When using the numerical time for one yearly earth rotation, the
required clock rate for the chosen position resolution can be reduced by
the factor 365. And the position error caused by the clock drift is
reduced then by the same factor.

**Latitude and Altitude**

Fig. 7 shows principally how latitude and altitude can be measured
autonomously. Fg. 7a shows time range 1 and local time range 2, which is
drawn thicker. At a point in time t~x~ (UT) the point P with its locally
unknown position is assumed to be located at a southern parallel of
about 10°. For measurement the local view is used and compared with the
simulated inertial view, similarly to what was described already for
meridian measurement using figures 5 and 6. Also for latitude
measurement an inertial reference point P~B~ has to be introduced, which
is determined by a fixed point in time t~c~. The difference in time
t~x~ - t~c~ = ∆t~B~ corresponds to the parallel of P within the latitude
range of ± 90° and is given by B = ±90° (∆t~B~/T~B~). For defining the
inertial reference point P~B~ declination can be used which is available
very accurately from literature^11^ and which changes daily during the
yearly orbit of earth around the sun.

Fig. 7b illustrates the principle of the autonomous time-based
measurement of altitude. As inertial reference sea level NN is chosen,
although the real sea level locally changes as a function of tides and
weather. It is possible to use a special internal scanning frequency for
the altitude range 6 from NN to H. Because of moving objects like
aircraft again it is required to introduce the highest possible virtual
scanning velocity c/(H-NN) = Ω\* or Ω\* = c/H\*. The altitude of P then
can be derived, in analogy to longitude and latitude determination, from
h = H\*(t~x~-t~e~)/T~H~, as is discussed in more detail referring to
fig. 8.

This additional discussion is useful since it discloses that autonomous
time-based determination of a quantity does not require to consider
earth rotation. And it clarifies that, like altitude, any distance to
selected time-based reference points can be measured autonomously. It
should be emphasized again, however, that the true inertial attitude of
the altitude range H\* cannot be taken from the scalar measurement
results. One would have to get the plumb vector from a different sensor.

Fig. 8 explains hwo the time-based autonomous altitude determination of
P within a linear range H\*

![](media/image1.wmf)is achieved. 1 is the time range, H\* the altitude
range from sea level NN to maximum altitude H, t~c~ the reference point
in time when the scanning frequency - which simulates a relative motion
between an inertial reference system and the actual location of P -
starts at sea level (NN).

![](media/image1.wmf)It is assumed that the reference level P~N~
periodically scans the altitude range H\* as a function of universal
time t~UT~ with the angular velocity Ω\*\* = c/H\*. Thus a common
virtual motion line Ω\*\* is allocated to P~0~ and P.

At P one measures universal time t~x~ and thus has the time difference
t~x~ - t~N~ at every point in universal time. Because of the selected
angular velocity this time difference is directly proportional to the
altitude of P. It helps in understanding to consider this approach as a
kind of autonomous on-way-DME where a reference point in time t~N~ is
allocated to the \"inertial\" sea level. If the altitude of P remains
unchanged, then also the respective time difference as a phase
difference between the periodic reference signal with the period T~H~
and universal time t~x~ remains unchanged. Relativity of simultaneity
makes this approach possible.

Once again it is emphasized that the measured altitude has no vectorial
character.

Measurement of altitude h within the altitude range H\* is achieved as
follows. Beginning at sea level which is marked with the fixed reference
point in time t~N~ P~N~ scans virtually the altitude range H\*
periodically with the angular velocity Ω\* = c/H\*.

During the scan of H\* the reference signal P~B~ (NN) coincides at the
point in time t~x~ with the local point P = H\* (t~x~ - t~N~) /T~H~. If
P is at the low end of H\*, i.e. at sea level and at the start of a scan
period T~H~ then t~x~ equals t~N~, i.e. t~x~ - t~N~ = 0, and therefore P
= NN. One could say that the phase difference of P referred to sea level
is zero. If P is located in the middle of H\*, then t~x~ - t~N~ = T~H~/2
and therefore P = H\*/2. One could say, the phase difference between P
and NN is π. If P is at the high end of H\* then t~x~ - t~N~ = T~H~ ,
and thus P = H. The phase difference between P and NN is then 2π.

This procedure is feasible since the position of P within H\* can be
changed only at the expense of local time, i.e. differential phase,
while the reference point in time t~N~ is not effected by any positional
change of P. Therefore a change of P altitude results in a change of
local time and thus a change of phase between P and sea level which
allows for measuring such altitude change. Graphically that was
explained already in figures 5 and 6 for meridian measure-ment.

The result of these explanations are the given equations which allow for
continuous rapid computations of the respective 3D-position of P at any
point in time. The autonomous altitude measurement (and any other
distance measurement) can be achieved locally via a precise clock 3 and
a computer 4.

**Autonomous v. Inertial**

Fig. 9 illustrates how the three-dimensional positio-ning results in
time differences 2 (L), 3 (B) and 4 (h) can be interpreted, which are
achieved by only measuring them at a point in time t~x~. Fig. 9a shows a
pseudo spatial sketch with the time intervals T~i~ arranged orthogonally
to each other for longitude, latitude and altitude measurements. These
time intervals are beginning with specific fixed points in universal
time t~L~ for longitude, t~B~ for latitude and t~N~ for altitude, and
end after the respective periods of time T~L~, T~B~ and T~H~. The
measured position is determined by the local point in time t~x~
delivered by the local clock. This clock impulse divides the periods
T~i~ of the three coordinate ranges (2π, ±π/2, H) by a ratio
corresponding to the respective coor-dinates of P. The orthogonal
arrangement of the coordinates is arbitrary, since the numerical values
of the coordinates don\'t involve any vectorial infor-mation. Therefore
they can be arranged, as shown in fig. 9b, also as parallel line
segments. In practical applications the coordinates may usually not be
displayed directly, but processed suitably within a more complex master
control system.

In fig. 10 it is shown how two positioning results, longitude and
latitude, can be displayed on an x-y-display.

The y-axis 3 shows the latitude of P, the x-axis 2 the longitude of it.
P is chosen as the fixed frame while equator and Greewich meridian move
with respect to it although in reality P moves. Of course, the available
data allow for dislaying a fixed earth with P moving on it as well.

Fig. 11 shows the difference between autonomous and inertial
measurements. I is the inertial system with sun S, point P on the real
parallel circle with its center M and the longitude ϑ at the point in
time t~a~, when the Greenwich meridian points exactly to the sun. This
inertial system I has to be discriminated from the autonomous system A,
the attitude of which in space is not identified. Only point P is common
to both systems. All points of system A are numerical, time-dependent
values which have no true inertial meaning. But their relative
coordinates at any point in time t~x~ are correct like within a true
inertial system.

If one needs the autonomous coordinates as inertial coordinates, then
additional information from vectorial sensors like plumb and north gyros
is required. The transformation of autonomous quanti-ties into inertial
ones is a familiar mathematical exercise. A principal problem for this
conversion may arise when the autonomous values are more accurate than
the inertial vectors available. In those cases more complex conversion
procedures (not discussed here) may help.

**Technical Realisation**

![](media/image1.wmf)Fig. 12 gives an idea of how to realize technically
the discussed positioning procedures.The block diagram shows the clock
1, the longitude processor 2, the latitude processor 3, the altitude
processor 4, the A/I converter 6, the data bank 5 with input 11 and
output 12, central control 7, control and display interface 8, and the
data bus 10, which connects all subsystems with each other, and the bus
control 9 with the remote control input 13. The device works as follows
(calibration assumed to have been done already).

The data bank 5 supplies the three coordinate processors 2 (L), 3 (B)
and 4 (h) with the specific fixed points in time t~L~ t~B~ and t~H~ and
the respective periodic time intervals T~L~, T~B~ und T~H~. Clock 1
supplies the three processors with the universal time t~x~ resolved as
required by the desired positioning resolution. Generally geodetic
resolution require-ments are much tougher than those for fast vehicles.

The coordinate processors continuously compute - based upon the data
input - the three coordinates longitude, latitude and altitude with the
refresh rate required by the system, and digitally delivers these values
to the master system via the data bus and the output interface 12, and
via the converter 6 to the control and display interface 8.

**Conclusions**

Summarizing, the new autonomous, time-based positioning approach
described in this paper for measuring longitude, latitude and altitude
combines the signal of a UT clock with very accurate data of fixed
points in time and coordinates belonging to it. Positioning is reduced
to the measurements of time differences.

The difference between a given fixed point in time t~a~ and the local
clock time t~x~ divides the respective period time T, and therefore the
respecctive measure-ment range, by the ratio ∆t~x~ /T, from which the
coordinate to be measured can be derived very accurately. The procedure
can be used for precise 3D positioning within microseconds independent
of the attitude of the object to be positioned.

The described autonomous positioning method offers a number of
advantages compared to the state of art, in particular achievable
maximum accuracy, full autonomy, high integrity depending mainly upon
the local clock and software, and the suitability for application in any
fixed and moving objects where a radio system like GPS can\'t be used.
Clock drift rates can be neutralized by selecting long time intervals,
like those for one year or more instead of for a day. Calibration
essentially is a test of the local clock and does not comprise special
expertise.

The described procedure can be supplemented by further functions, for
instance autonomous deriva-tion of velocity over ground of an airplane
via several position points sequentially measured. In addition, local
inertial vectors like north or lumb can be corrected and improved in
accuracy.

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