Wong's angles to determine trajectories of objects

A set of three geometric anagles is initiated to track the trajectories of moving objects. The objects to be tracked include but not limit to: satellites, space shuttles, airplanes, missiles moving in aerospace; submarines, ships, fish groups and other moving objects in hydrospace; cars, trucks, trains, human-beings and anmals moving on lands; planet motions in the solar system; particle motions in a controlled experiment in our national labories; contaminants, and hazardous materials in a variety of environmental conditions. All these objects can be detected and monitored by means of various sensing technologies which include seismic, acoustic, electro-magnetic, thermal, chemical, electro-optical and infrared. Mapping and monitoring systems could be deployed in unattended arrays or aboard ground, air, or maritime vehicles.

FIELD OF THE INVENTION 
The present invention provides a unique algorithm to determine the real 
trajectories of moving objects that are under the actions of various force 
fields. The moving objects include but are not limit to satellites, space 
shuttles, airplanes, missiles in aerospace; submarines, ships, fish in 
hydrospace; cars, trucks, trains, tanks on land; planet motions in the 
solar system; particle motions in a controlled experiment in our national 
laboratories; human-being and animals and materials in forensic analyses; 
pollutants, contaminants, and hazardous materials in a variety of 
environmental conditions; etc. All these objects can be detected and 
monitored by means of various sensing technologies which include seismic, 
acoustic, electro-magnetic, thermal, chemical, electro-optical and 
infrared. Mapping and monitoring systems could be deployed in unattended 
arrays or aboard ground, air, or maritime vehicles. 
BACKGROUND OF THE INVENTION 
The tracking of a moving object and determining its trajectory has long 
been regarded as a part of the famous previously unsolved P.sub.2 
targeting problem which was proclaimed solved in October 1974 and it 
became a U.S. Pat. No.: 5,084,232 entitled "TRAJECTORY SOLID ANGLE'S 
IMTS TO PHYSICS AND HIGH TECHNOLOGIES". Recently, the U.S. Army issued 
a solicitation under the Small Business Innovation Research (SBIR) Program 
topic No. A93-308 seeking for the solution of a topic entitled "Global 
Positioning System (GPS) Error Modeling for incorporation into 
Post-Mission Trajectory Estimation" which is in reality as a part of the 
P.sub.2 targeting problem. 
According to abstracts and summaries of reports by relevant scientists, 
engineers and mathematicians who have been doing research work in (GPS) 
questions on the accuracy of the (GPS) have been raised. 
SUMMARY OF THE INVENTION 
The errors, contributed from all previous models as of to date, can be 
summarized from repeatedly using the following techniques and methods for 
measurements: 
(1) The measurement of the range (distance) between the (GPS) ground 
station and the (GPS) satellite is by means of either using radars and/or 
lasers. 
(2) The measurement of the range is done from only one (GPS) ground station 
at a given time and that the geometric and trigonometric laws among the 
same measurements from the other two stations at the same time have been 
overlooked. As a result of this, it produces numerous incompatible data 
and that none of them can be chosen as a standard base for calculation of 
errors. 
It is obvious that the range (distance) measurement by method (1) is 
affected by all the uncertainties of all physical parameters between the 
space from the point where the (GPS) satellite is located and to the point 
where the (GPS) ground station is located. It is also obvious that the 
measurement by method (2) is not sufficient to describe precisely the 
trajectory of the (GPS) satellite with respect to the center of the earth, 
therefore, the true altitude of the (GPS) satellite has always been in 
question as having been shown by reports in the prior art. Thus, the 
successive usage of measurement by method (1) and then followed by method 
(2) will compound all the errors and uncertainties. 
In accord with the invention, a unique and original method of measuring the 
position vector, the velocity vector and the acceleration vector for 
moving objects including but not limiting the (GPS) satellite has been 
obtained. This method will be able to reduce the number of uncertainties 
of both physical and geometric parameters from numerous into only three 
angles to determine the position vector of the moving object/(GPS) 
satellite. They are called the Wong's Angles in order to simplify the 
title of this application and to distinguish from the well-known Euler's 
Angles which are used to define the position of a rigid body rotating 
about a fixed point. The combined use of both the Wong's Angles and the 
Euler's Angles that are set for a spinning satellite will determine not 
only the trajectory of the center of mass of the satellite but also the 
trajectory of any point in/on the satellite in rotation about a certain 
point. 
In summary, the advantages of the invention over all other methods in the 
past to determine the trajectories of moving objects are: 
(1) The invention of the Wong's Angles provides the most precise (GPS) 
satellite position vector that has never been obtained by all other 
methods as of to date. 
(2) All the parameters of uncertainties affecting the position vector of 
the (GPS) satellite are reduced into the minimum of only 3 Wong's Angles 
which are measurable with much less uncertainties from the (GPS) ground 
stations. 
(3) It follows that the velocity vector of the (GPS) satellite can be 
determined from only six parameters which are the 3 Wong's Angles and 
their 3 first derivatives with respect to time all of which are measurable 
with much less uncertainties from the (GPS) ground stations. 
(4) It follows that the acceleration vector of the (GPS) satellite can also 
be determined from only 9 parameters which include 3 Wong's Angles; 3 
first order time derivatives of the Wong's Angles and 3 second order time 
derivatives of the Wong's Angles. As a result, the forces acting on the 
(GPS) satellite can also be determined while the gravitational force 
acting on the (GPS) satellite is already determined from (2) at any time. 
(5) Applying the Wong's Angles to determine the precise position vectors of 
3 (GPS) satellites in the space which are formed as a new triangular base 
to determine other moving objects relative to the 3 (GPS) satellites such 
that the trajectories of other moving objects can also be determined. This 
particular application is important to measure objects moving far away 
from our solar system because the 3 (GPS) ground stations position vectors 
to the far-distance moving object (for example 1,000 time the radius of 
the earth) can no longer be distinguished. It is important to choose the 
proper distance scales for the ground stations as the bases of measurement 
of the Wong's Angles depending on the type of moving objects which can be 
airplanes approaching to the airport; submarines moving under-neath the 
ocean; base ball moving in the ball park . . . etc. Thus, the Wong's 
Angles are applicable for determination of the trajectories of all moving 
objects that can be detected and pointed by means of sensing technologies 
which include seismic, acoustic, electro-magnetic, thermal, chemical, 
electro-optical and infrared. Its application to determine the precise 
position vector of the (GPS) satellite is merely one of its numerous 
applications. 
(6) The invention of this set of Wong's Angles can be used to confirm the 
truth TRAJECTORY SOLID ANGLE (TSA) in U.S. Pat. No. 5,084,232: 
(TSA) provides the precise definition of the probability functions for 
targeting problems in theory. 
The Wong's Angles provides a precise method to determine the real 
trajectories of any objects under the action of many-force fields and to 
guide the using of the appropriate instruments for measurements in 
experiments. 
Putting both the (TSA) and the Wong's Angles together provides a complete 
solution of the targeting problem. 
(7) Like (TSA), the invention of the Wong's Angles provides the most 
precise method with the least number of uncertainties (3 angles) for 
measurements to determine the trajectories of any moving objects relative 
to the earth. While the use of other methods will compound the errors of 
measurements. 
(8) The definition of the Wong's Angles is explicitly defined with all 
parameters implicitly contained within the definition while all the other 
methods do not. 
(9) The Wong's Angles appeared in the position vector, the velocity vector 
and the acceleration vector of the moving object are closed-form 
analytical solution that provides a good cenceptual and optimal analysis 
for the trajectory problems to be solved. 
(10) Algorithms developed from using the Wong's Angles provide simplified 
parametric study of computing outputs. 
(11) The computing time of the algorithms developed from using the Wong's 
Angles and their costs will be drastically reduced. 
(12) The accuracy and the precision of the numerical values from the 
algorithms developed from using the Wong's Angles are thus far more better 
than other approaches.

DETAILED DESCRIPTION OF THE INVENTION 
FIG. 1 shows that a rectangular Cartesian coordinate system is set up at 
the center of the earth. The plane surface XOZ formed by the coordinate 
axis OX and OZ cuts through the center of the earth at 0 (0,0,0) and 
through Greenwich in England. OZ is the rotating axis of the earth and it 
is perpendicular to OX axis which is in the plane surface XOY that 
contains the equator of the earth. Thus, OX.perp.OY.perp.OZ and 
OX.andgate.OY.andgate.OZ=0 (0,0,0)=the center of the earth. P(x,y,z) 
represents a point at the center of mass of the moving object/(GPS) 
satellite relative to the center of the earth. OP is the position vector 
of the moving object/(GPS) satellite. Thus, OP=R=xe.sub.x +ye.sub.y 
+ze.sub.z =Re.sub.R and 
##EQU1## 
where .phi. is the longitudinal angle and .theta. is the latitudinal 
angle; e.sub.x , e.sub.y , e.sub.z and e.sub.R are the unit vectors of 
OX,OY,OZ and OP respectively. 
FIG. 2 shows how the Wong's Angles are defined with reference to the 
locations of the (GPS) ground stations A(x.sub.A,y.sub.A,z.sub.A); 
B(x.sub.B,y.sub.B,z.sub.B); C(x.sub.C,y.sub.C,z.sub.C) as a triangular 
base to locate the position of the moving object/(GPS) satellite at point 
P(x,y,z). Assuming that there is no earthquake at the moment of making 
measurements, the distances between the (GPS) ground stations and the 
distances from each (GPS) ground station to the center of the earth are 
constants without variation with respect to time. Thus, these distances 
can be determined uniquely from 3-D coordinate geometry as: 
##EQU2## 
The set of Wong's Angles determing the position P(x,y,z) of the moving 
object/(GPS) satellite from the FIG. 2 are clearly marked and defined as: 
.alpha.=.angle.PAB .beta.=.angle.PBA .gamma.=.angle.PBC 
It is obvious that the Wong's Angles vary with the moving object/(GPS) 
satellite. 
Certain details of the invention can be found with reference to U.S. Pat. 
No. 5,084,232, incorporated herein by reference, and with reference to 
Appendix A hereto. These details are summarized as follows: The range 
(distance) from the moving object/(GPS) satellite P to the (GPS) ground 
station B is 
##EQU3## 
The range (distance) from the moving object/(GPS) satellite P to the (GPS) 
ground station A is 
##EQU4## 
The range (distance)from the moving object/(GPS) satellite P to the (GPS) 
ground station C is 
##EQU5## 
The components of the position vector of the moving object/(GPS) satellite 
are: 
##EQU6## 
Notice that z.sub.o has two values for a given set of Wong's Angles. Taking 
the positive sign for the radical term means that the moving object/(GPS) 
satellite is above the geometric plane formed by the (GPS) ground stations 
A,B and C. Taking the negative sign for the radical term means that the 
moving object/(GPS) satellite is underneath the geometric-plane surface 
formed by the (GPS) ground stations A,B and C. In fact, the position 
vector obtained by taking the positive sign and that with the negative 
sign represent the object point and its image point reflected with 
symmetry about the geometric-plane surface formed by the (GPS) ground 
stations A,B and C. The perpendicular from the moving object/(GPS) 
satellite to the geometric-plane surface ABC is d which can be obtained 
as: 
##EQU7## 
Please note that this perpendicular d is entirely different from the 
altitude h of the moving object/GPS) satellite which is formally defined 
as the difference between R (distance from the center of the earth to the 
moving object) and R.sub.o (average or sea level radius of the earth=6,378 
km.). Thus h=R-R.sub.o. 
The gravitational acceleration of the moving object/(GPS) satellite in the 
direction of e.sub.R toward to the center of the earth is g which can be 
expressed as: 
##EQU8## 
As a result, both the velocity vector v and the acceleration vector a of 
the moving object/(GPS) satellite can also be obtained respectively as: 
##EQU9##