﻿ 数学project代做 Math 5610代写 Term Project代写 - 作业代写, 数学代写

# 数学project代做 Math 5610代写 Term Project代写

2022-03-23 14:43 星期三 所属： 作业代写 浏览：238

## Math 5610 Fall 2020 Term Project1

数学project代做 However, beyond that the project illustrates a recent technologicaldevelopment which I hope you will find interesting and motivating

### 1 Introduction

The primary numerical task in this semester project is the solution of certain systems ofnonlinear equations. However, beyond that the project illustrates a recent technologicaldevelopment which I hope you will find interesting and motivating. To get everything towork in this project we will have to put many pieces together, and they have to work justright, as in most real life problems. The results of your work, a couple of programs, will need to work with the programs of your classmates and my own. When we reach that stage we will be able to say with assurance that we truly understand the problem and its solution.   数学project代做

The original US Global Positioning System (GPS) consists of 24 satellites that orbit the earth and constantly transmit their position and the precise time of that transmission,in addition to other data. Commercially available receivers, (Global Positioning Devices)process that information and, using the differences in the runtimes of the satellite signals, compute their position and the precise time. The position can be expressed, e.g., as longitude, latitude, and altitude. Its accuracy depends on the type and quality of the device and its software, and other circumstances. It ranges from less than one centimeter to about 100 meters2.

#### In this project, we will write two programs that simulate a much simplified version of this system.

GPS was developed originally for the US armed forces, but there are of course also civilian and scientific applications, many of which are still being discovered. As far as volume and resources involved, civilian applications now vastly outstrip military applications. The following is an incomplete list of applications I have read or heard about (or, in the case of the first three applications, actually practiced):

• Navigating in a car. This may be the largest civilian application at present. You can buy devices that will tell you where you are and what’s near you, and give you oral turn by turn instructions on how to get to your destination. Many high end models now come with built-in GPS based navigation systems.
• Finding your way in the

1by Peter Alfeld, [email protected]

#### 2A meter is a little more than 3 feet. Most of the literature on GPS uses the metric system, and we follow that convention.

• Geocaching is a high-tech treasure hunting game played throughout the world by adventure seekers equipped with GPS devices. This is a highly popular past time where people search for a cache established by others, given it’s coordinates, remove a token item, and replace it with another. The basic idea is to locate hidden containers, called geocaches, outdoors and then share your experiences online. The official web site is http://www.geocaching.com/ .
• Electronically marking a spot (a good fishing place, a buried treasure or mine, man or item overboard) in a featureless area such as the ocean, a lake, a desert, or the Salt Flats.     数学project代做
• Aerial navigation, including precision approach and
• Measuring continental drift and
• Telling blind people (via Braille or Audio) where they
• Steeringa tractor planting vine plants along a suitable
• Surveying with high
• ##### Controllinga fleet of vehicles (e.g., Police, Taxi, Truck or Bus companies).
• Transfer of extremely accurate time
• Installation of several GPS devices in a vehicle, like a ship or plane, and determina- tion not just of position but also of attitude, roll, pitch, and yaw.
• By coupling a cell phone with a GPS device, in an emergency it can tell your location to the dispatcher even if you don’t know it yourself. If the phone is installed in your car, and the car is stolen, you might be able to call your car and ask where it is.
• GPS is available anywhere on or near earth, 24 hours a day, in fog, space, or dark- ness. By making several measurements over time one can compute and display such quantities as speed, bearing, or estimated time of arrival. It’s eerie to have your car give you detailed instructions on how to get where you are going, as you are going. Or, for example, you can transform a simple rental boat into a sophisticated yacht with compass and speedometer simply by placing a GPS device next to the wheel.

### 2 Sources ofInformation   数学project代做

If you are interested in GPS and like to learn more, here are some places to get started:

A standard text on GPS is B. Hofmann-Wellenhof, H. Lichtenegger, and J. Collins: GPS, Theory and Practice, Springer-Verlag Wien New York, 5th edition, 2001, ISBN 3211835342. This excellent but very technical book gives lots of details and references.

Another standard detailed and technical text is Elliott D. Kaplan (ed.), Understand- ing GPS Principles and Applications, Artech House, Boston, London, 1996, ISBN 0890067937.

#### There’s a great deal of information on the web. The following Table shows the number of results found by a Google search for GPS.

 May 2004 17, 000, 000 May 2005 58, 600, 000 August 2007 257, 000, 000 May 2008 407, 000, 000 August 2009 244, 000, 000 May 2012 1, 330, 000, 000 May 2013   数学project代做 720, 000, 000 May 2014 298, 000, 000 May 2015 620, 000, 000 July 2017 258, 000, 000

Table: Number of Google results for GPS

• There is excellent information in a concise and readable form about the shape of the earth and its gravitational field in the Encyclopedia Britannica, particularly in the article The Earth in the macropedia.
• A useful text giving all kinds of mathematical formulas and information is the VNR Concise Encyclopedia of Mathematics. It’s published by Van Nostrand Reinhold Company. My copy was published in 1975. The ISBN is 0442226462.

### 3 Note onNotation

Notation can get quite cumbersome in this project. I use boldface letters to denote vectors in cartesian coordinates. The coordinates of these vectors themselves are lower case Roman letters that may be subscripted. For example, the vector x is usually denoted by

(1)

but sometimes, for example in the discussion of Newton’s method, by

(2)

Subscripts are often used with an S to denote correspondence to a satellite. for example,xSi denotes correspondence to the i-th of several satellites.

### 4 The SpaceSegment

The number and configuration of the orbiting satellites is evolving. We will consider a setof 24 satellites orbiting in six planes at an inclination of 55 to the equator at an altitude of about 20,200 km.  数学project代做

To help you get going on the project, sprinkled throughout this assignment are a totalof 15 exercises. You will have to solve these exercises or closely related problems during the work on the project, so they collectively form our first home work in this class.

### 5 TheModel   数学project代做

The model will consist of three modules, i.e., programs, the vehicle, the satellite, and the receiver. The vehicle generates positional data, the satellite converts those into data of the kind processed by GPS receivers, and the receiver converts these back into positional data. The three modules are piped together in the standard Unix fashion:

vehicle | satellite | receiver (3)

If all goes well the standard output of receiver will equal (almost) the standard output of vehicle. Your job is to write satellite and receiver (and perhaps a rudi- mentary vehicle for testing purposes).

The model depends on a set of data provided in a file called

data.dat

which is read by satellite. The beginning part of that file is also read by receiver. Figure 1 summarizes the model. Arrows indicate data flow.

Following are some of the assumptions we’ll make to construct our model. Numerical values are given here for illustration. They will be actually read from data.dat and theprecise numerical values are subject to change (with appropriate notification). You or I may also change those parameters for the purpose of testing our programs.

#### 1.TheEarth is perfectly spherical and has a radius Rof

R = 6, 367, 444.50 m. (4)

1. The Earth turns at a constant rate and completes one revolution in one sidereal day3of

s = 86, 164.09 seconds. (5)

1. Thesatellites move at a constant speed in perfectly circular orbits at an altitude of

h = 20, 200, 000 m (6)

and with an orbital period

(7)

of exactly half a sidereal day.

1. The satellites are evenly spaced in groups of four in six planes, each of which is inclined 55to the equator. According to Hofmann-Wellenhof et al this provides global coverage of four to eight satellites being more than 15 above the horizon at all times and in all places on earth.

### 6 CoordinateSystems

We use two coordinate systems: longitude, latitude and altitude to express positions on and near earth, and a cartesian coordinate system to describe the space in which the satellites orbit and the earth rotates. Like commercial global positioning devices, weexpress angles in degrees, minutes of degree, and seconds of degree. The two coordinate systems are linked by the following conventions:

The North Pole is located at (0, 0, R), the South Pole at (0, 0, R), (assuming both have altitude 0, which is definitely wrong for the South Pole). At time 0 the point O of zero longitude, latitude, and altitude, is at (R,0,0), and the Earth rotates (west to east) once in a sidereal day.   数学project代做

3A sidereal day is the time required for a complete rotation of the earth in reference to a fixed star. It differs from a standard solar day of 24 hours because in the course of a day the earth not only rotates, but also changes position in its orbit around the Sun. Hence the sun changes its position in the sky and the earth needs to rotate an additional 4 minutes or so to catch up with the sun before it is noon again.

Exercise 1: Find a formula that describes the trajectory of the point O in cartesian coordinates as a function of time.

### 7 ThePrograms   数学project代做

As stated above, our model will consist of three programs:

1. The This program, which you can download from our Canvas folder, producesa stream of data sets (to standard output) each of which has the form:

tV ψd ψm ψs NS λd λm λs EW h (8)

where tV denotes Universal time4 in seconds, ψd, ψm, ψs is latitude in degrees, minutes, and seconds, and, similarly, λd, λm, λs is longitude in degrees, minutes, and seconds. More specifically,

• tV is a real number given to an accuracy of 102 seconds ranging from 0 to 106. This is the time at which the vehicle is at the specified position.
• ψd is an integer ranging from 0 (i.e., the Equator) to +90 (i.e., the North or South Pole).     数学project代做
• ψmis an integer ranging from 0 to 59 minutes of
• ψs is a real number ranging from 0 to 59.9999 seconds of degree. It should be given to an accuracy of 102 (which corresponds to an accuracy of about a foot).
• NS is an integer that is +1 North of the equator and 1 South of the
• λd is an integer ranging from 0 (i.e., the meridian of Greenwich) to 180 (i.e., 180 degrees east, or west, the date line).
• λmis an integer ranging from 0 to 59 minutes of degree.

#### 4The mean solar time of the Greenwich meridian (0◦longitude).

Universal time replaced the desig- nation Greenwich mean time in 1928; it is now used to denote the solar time when an accuracy of about 1 second suffices. In 1955 the International Astronomical Union defined several categories of Universal Time of successively increasing accuracy. UT0 represents the initial values of Universal Time obtained by optical observations of star transits at various astronomical observatories. These values differ slightly from each other because of the effects of polar motion. UT1, which gives the precise angular coordinate of the Earth about its spin axis, is obtained by correcting UT0 for the effects of polar motion. Finally, an empirical correction to take account of annual changes in the Earth’s speed of rotation is added to UT1 to convert it into UT2.

##### Coordinated Universal Time, the international basis of civil and scientific time, is obtained from an atomic clock that is adjusted so as to remain close to UT1;

in this way, the solar time that is indicated by Universal time is kept in close coordination with atomic time. [Encyclopedia Britannica, 1992]. We will assume that all the times described here are identical and call them Universal Time.     数学project代做

• λsis a real number ranging from 0 to 9999 seconds of degree, given to the same accuracy as ψs.
• EWis an integer that is +1 east of Greenwich and 1 west of Greenwich.
• his a real number giving the altitude in meters, to an accuracy of 1cm.

For example, according to my Magellan Trailblazer the street light labeled B125 in front of the South Window of my office (at time t) is located at:

t 40 45 55.0 1 111 50 58.0  1 1372.00 (9)

i.e., at latitude 40 45’ 55” North, longitude 111 50’ 58” West, and an altitude of 1372 m.

#### Exercise 2: Write a program that converts angles from degrees, minutes, and seconds to radians, and vice versa. Make sure your program does what it’s supposed to do.

For the following four exercises assume that tV equals true Universal time and denote it by t.

Exercise 3: Find a formula that converts position as given in (8) at time t = 0 into cartesian coordinates.

Exercise 4: Find a formula that converts position and general time t as given in

(8) into cartesian coordinates.

Exercise 5: Find a formula that converts a position given in cartesian coordinates at time t = 0 into a position of the form (8).   数学project代做

Exercise 6: Find a formula that converts general time t and a position given in cartesian coordinates into a position of the form (8).

Exercise 7: Find a formula that describes the trajectory of lamp post B12 in cartesian coordinates as a function of time.

5If you look around on campus you’ll see that all street lights have  been labeled with a letter and   a number. On some lights, the labels have disappeared, or, as in the case of B12, obstructed by an attachment.

The vehicle should not produce impossible positions (like a latitude of 90 59’ 59”). For testing purposes we will use vehicles that produce a stream of data corresponding, for example, to a walk from JWB to the Marriott Library, a hike up to Mount Olympus, or a flight from Salt Lake City to the North Pole. The data sets should be spaced at least 1 second (of time) apart.

In addition vehicle writes a copy of the standard input and the standard output into the file

vehicle.log.

#### 2.The Satellite. This program reads data from dat and then reads the data generated by a vehicle and processes those data as follows:

A.Itcomputes the position xV of the vehicle in cartesian coordinates (measured in meters) at the time tV .

B.For each satellite S that is above the horizon6at the vehicle’s position it com- putes the time tS at which a signal needs to be sent to reach the vehicle at time tV and position xV , assuming the signal moves at the speed c of light in vacuum where

c = 2.99792458 × 108m/s. (10)

The satellite needs to be above the horizon at time tS, The satellite also computes the position xS of the satellite at the time tS It then writes the following data to standard output:

iS tS xS (11)   数学project代做

where iS is the reference number of the satellite (ranging from 0 to 23). The time tS should be given to an accuracy of 1011 seconds, and the position xS should be given as a triple of cartesian coordinates with an accuracy of 1cm.

C.In addition, satellite writes a copy of the standard input and the standard output into thefile

satellite.log.

It should terminate gracefully when the data stream from the satellite program is terminated.

Exercise 8: Given a point x on earth and a point s in space, both in cartesian coordinates, find a condition that tells you whether s as viewed from x is above the horizon.

Exercise 9: Discuss how to compute tS and xS.

6My own receiver will check this condition. I figured that assuming a transparent earth would be too much of a simplification. However, calculations in the actual GPS system are usually based on satellites at least 15 above the horizon.

#### 3.The Receiver. This program reads the data written by satellite and converts them into position and time data in the same form as the data generated by the vehicle.

If all goes well those data should closely approximate the data sent by vehicle. Note that satellite does not transfer tV. This has to be reconstructed by receiver, thus simulating the transfer of precise  In addition the program writes a copy of the standard input and the standard output into the file

It should terminate gracefully when the data stream from the satellite program is terminated.

Letting xv denote the position at which the vehicle receives the satellite signal, tV the time that it receives the signal, and, similarly, xS and tS the position and time at which the satellite sends the signal, the basic equation for our system is

||xV xS|| = c(tV  tS).   (12)

In this equation we know tS and xS, and we don’t know tV and xV . We have one such equation for each satellite from which we receive a signal.  数学project代做

The time tV enters these equations linearly, and can be eliminated. It also plays a different role than the coordinates of xV . It needs to be computed to a much higher accuracy than the location. So it is indeed best to eliminate it. Suppose we have equations of the form (12) for two satellites S1 and S2. Taking the difference gives the equation

||xV  xS||  ||xV  xS|| c(tS tS1 ). (13)

##### Exercise10:Suppose you have data of the from (11) from 4 satellites. Write down a setof four equations whose solutions are the position of the vehicle in cartesian coordinates, and tV.

Usually you will have data from more than four satellites. To use the above approach one would have to choose four specific ones of those satellites. This is in itself a non- trivial problem. A better way is to use all observations and to think of the (possibly overdetermined) nonlinear system of equations

F (x) = 0 (14)

as a nonlinear Least Squares problem

f (x) = F (x)T F (x) = min . (15)

The least squares problem can be solved by setting the gradient of f in (15) to zero, which gives rise to another system of four equations in four unknowns.

If m is the number of satellites being received, I recommend that you form m  1 equations of the form (13) and apply the Least Squares approach to find xV . Once you have xV you can compute tV .   数学project代做

Exercise 11: Suppose you have data of the form (11) from more than 4 satellites. Write down a least squares problem whose solution the position of the vehicle in cartesian coordinates, and tV .

Exercise 12: Think about the ground track of satellite 1, i.e., the position in geographic coordinates directly underneath the satellite on the surface of the earth, as a function of time. Do you notice anything particular? What is the significance of the orbital period being exactly one half sidereal day?

### 8 MathematicalApproach   数学project代做

Mathematically, we have four unknowns: three coordinates of the location, and time. Each satellite signal provides a range c(tS tV ) where we don’t know tV . Given four data we should be able to compute position and time. So receiver has to solve a system of four nonlinear equations for the four unknowns. In addition the programs will have to do some coordinate conversions.   数学project代做

The natural way (why?) to solve the nonlinear equations arising in this project is by Newton’s Method (for systems of nonlinear equations). We will discuss it in detail in class.  However, for reference here is a preliminary description: Let F : Rn Rn be a given function, and suppose we want to solve the nonlinear system of equations

F (x) = 0. (16)

Let J denote the Jacobian Matrix of F . Thus the (i, j) entry of J is the partial derivative of the i-th component of F with respect to the j-th component of x, i.e.,

(17)

#### Given a starting point x(0), Newton’s method proceeds iteratively by defining

for k = 0, 1, 2, · · · , until satisfied

x(k+1)  = x(k)  J  x(k) (1) F  x(k)  . (18)

Of course, we never invert a matrix. Instead, your program should do something like this For k = 0, 1, 2, · · · until satisfied :

Solve J x(k) s(k) = F x(k)

Let x(k+1) = x(k) + s(k)       (19)   数学project代做

The performance of Newton’s method is very sensitive to the choice of the starting point x(0). As always, the precise choice of that point depends on the problem. For ourpurposes you may assume that at least initially you are operating your GPS device in the vicinity of Salt Lake City, i.e., you can use (9) as a starting point.

Exercise 13: Find a precise description of Newton’s method as it is applied to the nonlinear system obtained by processing data from 4 satellites, as derived in an earlierexercise. Your answer should include an explicit specification of the derivatives involved.

#### Exercise14:Similarly, find Newton’s method for the nonlinear system obtained fromthe least squares approach. Again, your answer should include an explicit specification of the derivatives involved.

A common problem with nonlinear equations is non-existence of a solution, or existence of multiple solutions. For our project, for example, if the difference in run times of two satellite signals exceeds the time corresponding to the maximum distance of the satellitesthen there is no solution. On the other hand, the way we set up the problem there should always be a solution since we start with a well defined position. Non-existence of a solution would therefore indicate a programming error of some sort. It is also possible to haveseveral solutions.  数学project代做

For simplicity, consider a situation where you are given the distances (i.e., the receiver has an exact clock) from three satellites at known positions. So yourlocation is on the intersection of three spheres (with the satellites as their centers and the known distances as their radii). Two spheres intersect in a circle, and the third sphere intersects the circle in two points. There are thus two solutions of the nonlinear equations.Which should we decide on as our current location? In practice, global positioning devicesuse the following criteria: closeness to the previously computed position, closeness to the surface of the earth, and slow motion.

Exercise 15: Think about the number of solutions obtained by analyzing four satellitesignals with an unknown vehicle time tV . This is an open ended question that will not be graded!

##### Here are some technical comments on writing the programs:

Use the highest available accuracy throughout, e.g., use the double data type in java. Keep in mind that roughly speaking a meter makes a difference in the 7th or 8th digit of the distances involved. Light takes about 3 109 seconds to pass that distance, and since we contemplate times ranging up to one million seconds we need to process time information accurately to at least 15 digits.

• Makesure your standard output carries enough
• Since your programs will have to work with those written by others you have to follow carefully the conventions laid out here. vehicle and satellite can’t write anything to standard output other than what’s specified (and for better compara- bility, receiver also should not write anything else).
• ###### Your satellite should read the data file in the form given in the appendix and should not be deterred by the embedded comments.
• You satellite will send data for all visible satellites. receiver does not know before- hand the number of those satellites. It will therefore have to decide when a group of signals starts and ends, and it will have to take reasonable action if the number of satellite signals is different from 4. If it’s less it could write a message saying that there isn’t enough information, if there are more it could select a subset of 4 signals, or it could solve a least squares problem. My own inclination is to set up a least squares problem and solve it whenever the number of satellite signals is 4 or greater.  数学project代做
• You should have your receiver check whether or not your satellite sends only signals from satellites that are above the horizon. Otherwise you may be simulating a technical breakthrough that renders the earth transparent, but your satellite will not work with my less advanced receiver.
• Both receiver and satellite should terminate gracefully when they receive no more input. Thus they should shut down without irrelevant system messages, and they should close their log file before terminating.

### 9 What ToDo  数学project代做

2. Writethe programs satellite and receiver. You will need a rudimentary vehicle for testing  You can download one from the Canvas folder or you can write your own. Your programs must run on all or some of our Unix systems. They may be written in any language, but of course they must accommodate the Unix standard input and output concepts. E-mail your programs and a pdf file with a written report of your work to me,  by  the deadline of this project.  The report should  be suitable for distribution among your class mates. Describe your mathematical approach, your software, how to compile and use it, the lessons you learned in this project, and anything else you may find worth mentioning.
3. After I test and process your programs we will have a discussion of this project. I willask some (or all if practical) teams or individuals to present part of their work to the whole class. Details will be announced as we approach the deadline.

### 10 Note on TeamWork

As mentioned in class, I recommend you work with one or two partners, to study together,and to work on the term project and the home works together. If you do work in a team hand in just one set of answers. Each of you will get the same score on all parts of an assignment.

### 11 SimplifyingAssumptions

As additional information this sections contains an incomplete list of phenomena that haveto be incorporated to make GPS viable in practice. This information is taken mostly from Hofmann-Wellenhof et al which gives detailed mathematical descriptions of most of these effects.   数学project代做

• Signal Acquisition. The satellite signals are weak and the signal to noise ratio is low. As a consequence the receivers work with a very narrow bandwidth. Due to the Doppler effect the frequency at which the signal is sent varies across a spectrum significantly wider than the bandwidth. Thus the receiver doesn’t know the precise frequency at which the satellite is transmitting unless it knows the satellite’s position and velocity. As a consequence, if starting from scratch, signal acquisition may take quite a while. The problem is alleviated by the fact that each satellite sends information on the current location of all other satellites (this information is called ephemerides).   数学project代做
• Positioning with Doppler data or Carrier Phases. Our programs model positioning with range data. More accurate positioning can be accomplished by taking into account Doppler data and carrier phases. The latter determine distance only within an integer multiple of the wavelength (which is 19.0cm or 24.4 cm). However, phases can be measured to better than 1% of wavelength, which potentially gives GPS an accuracy of about 1mm or less.
• ##### RelativePositioning.

In many applications the important item to be measured is the vector between two points, rather than the location of a point with respect to a global coordinate system. This can usually be done more accurately than the determination of a position.

• Selected Availability and Anti Spoofing. In the interest of national security the Department of Defense used to limit availability and accuracy of GPS to civilian users by encrypting the high accuracy version of the satellite signal (selected avail- ability) and by dithering (distorting) the public signal (anti-spoofing). On May 2, 2000, these measures were turned off and the full accuracy of GPS is now available to the public. Interestingly, while selected availability or anti-spoofing were still in effect, they were turned off during crises like the first gulf war or the invasion of Haiti, to make it possible for the armed forces to make full use of commercially available GPS devices.
• #### Spherical Earth. The Earth is of course not spherical. It is more accurately described as an ellipsoid with a semimajor axis of 6,378,137 m and a semiminor axis of 6,356,752 m.

When contemplating the determination of altitude within a cm or so that definition is also inadequate. What’s really involved here is a precise definition of the term “sea level”. The resulting shape of the earth is called the “datum” underlying a map or other navigational tool. For example, the Magellan Trailblazer can use any of 12 built-in datums.  数学project代做

• Constant axis of rotation. The North and South Poles move (at the Chandler period of about 430 days) within an area that has a diameter of roughly 6 m.
• Circular Orbits. The satellites do not move along circular or even elliptical orbits. The orbits are disturbed by a number of factors: non-spherical earth, gravitational anomalies, tidal forces, solar radiation pressure, and even air drag. The problem is overcome by the satellites broadcasting their actual position (rather than one based on an orbit calculation), but this needs to be determined using a sophisticated ground and space based infrastructure.
• Time Dilation. Time passes more slowly for an object that is moving at a high speed. This affects the accuracy of the satellite clocks.
• Time of transmission. We have to define precisely what we mean by the time of transmission of a signal. Each relevant signal consists of 1023 bits that are separated by about 106 seconds. Thus at the speed of light the distance between two bits  is about 300m. This is called the “chip length” (and the bits are called “chips”). Current timing accuracy is between 0.1% and 1% of the chip length.
• Ionospheric Refraction. This is a major cause of inaccuracies. Its precise effects depend for example on current sunspot activity.
• ##### TroposphericRefraction.This depends largely on the amount of water in the various parts of the troposphere, i.e., on the weather.
• Multipath Effects. A receiver may receive more than one version of the signal due to reflection (e.g., on walls or cliffs).     数学project代做
• Position of what? GPS receivers are larger than the potential accuracy with which they can be used. Thus one has to figure out exactly which point (called the phase center, and located usually within the antenna) corresponds to the computed solution. The location of the phase center of course depends on the geometry of the antenna and receiver, but also on the frequency and the intensity of the signal.
• Physical Rigor. Often GPS devices are employed in demanding conditions (e.g., rain, temperature extremes, dirt, vibrations). They need to be physically rugged.
• Software. The quality of a GPS device depends very much on its software. The cost of software development constitutes a very significant portion of the total purchase price.
• Utility. The utility of a device depends much on what it can do beyond computing position (and perhaps speed and direction). For example a device might contain tables of magnetic deviations from true North (like the Magellan Trail Blazer) or a database of (almost) all streets in the United States (like the Magellan GPS Map 7000) and so be able to give magnetic directions or a map with your current location.

### 12 Appendix: The DataFile   数学project代做

The whole project depends on a few parameters, like the speed of light, the radius of earth, and the orbits of the satellites. Rather than hardwiring these data into the programs we’ll put them into a file that can then be easily modified. For example, if we wanted to run GPS on Jupiter or the Sun we would only have to modify that file.

We express the (perfectly circular) orbit of a satellite as

(20)

where:

• t is time measured inseconds
• x(t) is the location of a satellite at time t expressed in cartesian coordinates with meters being the unit of length
• Ris the radius of the earth
• his the altitude of the satellite
• u is a unitvector
• vis a unit vector that is orthogonal to u
• pis the periodicity of the orbit (assumed to be half a sidereal day)
• θis the phase of the

#### Crucial to the project is the data file listed in Table 1. (The leading line numbers are not part of the file.) You can download it from the Canvas folder.

3.141592653589793116E+00 /= pi
2.997924580000000000E+08 /= c, speed of light, [m/s]
6.367444500000000000E+06 /= R, radius of earth, [m]
8.616408999999999651E+04 /= s, length of a sidereal day, [s]
1.000000000000000000E+00 /= u1 of Sat. 0
0.000000000000000000E+00 /= u2 of Sat. 0
0.000000000000000000E+00 /= u3 of Sat. 0
0.000000000000000000E+00 /= v1 of Sat. 0
5.735764363510461594E-01 /= v2 of Sat. 0
8.191520442889917986E-01 /= v3 of Sat. 0
4.308204499999999825E+04 /= periodicity of Sat. 0 [s]
2.020000000000000000E+07 /= altitude  of  Sat. 0 [m]
0.000000000000000000E+00 /= phase of Sat. 0 [rad]
1.000000000000000000E+00 /= u1 of Sat. 1
0.000000000000000000E+00 /= u2 of Sat. 1
0.000000000000000000E+00 /= u3 of Sat. 1
0.000000000000000000E+00 /= v1 of Sat. 1
5.735764363510461594E-01 /= v2 of Sat. 1
8.191520442889917986E-01 /= v3 of Sat. 1
4.308204499999999825E+04 /= periodicity of Sat. 1 [s]
2.020000000000000000E+07 /= altitude  of  Sat. 1 [m]
1.570796326794896558E+00 /= phase of Sat. 1 [rad]
1.000000000000000000E+00 /= u1 of Sat. 2
0.000000000000000000E+00 /= u2 of Sat. 2
0.000000000000000000E+00 /= u3 of Sat. 2
0.000000000000000000E+00 /= v1 of Sat. 2
5.735764363510461594E-01 /= v2 of Sat. 2
8.191520442889917986E-01 /= v3 of Sat. 2
##### 4.308204499999999825E+04
/= periodicity of Sat. 2 [s]
2.020000000000000000E+07 /= altitude  of  Sat. 2 [m]
3.141592653589793116E+00 /= phase of Sat. 2 [rad]
1.000000000000000000E+00 /= u1 of Sat. 3
0.000000000000000000E+00 /= u2 of Sat. 3
0.000000000000000000E+00 /= u3 of Sat. 3
0.000000000000000000E+00 /= v1 of Sat. 3
5.735764363510461594E-01 /= v2 of Sat. 3
8.191520442889917986E-01 /= v3 of Sat. 3
4.308204499999999825E+04 /= periodicity of Sat. 3 [s]
2.020000000000000000E+07 /= altitude  of  Sat. 3 [m]
4.712388980384689674E+00 /= phase of Sat. 3 [rad]
5.000000000000000000E-01 /= u1 of Sat. 4
8.660254037844385966E-01 /= u2 of Sat. 4

#### 0.000000000000000000E+00

/= u3 of Sat. 4
-4.967317648921540929E-01 /= v1 of Sat. 4
2.867882181755230797E-01 /= v2 of Sat. 4
8.191520442889917986E-01 /= v3 of Sat. 4
4.308204499999999825E+04 /= periodicity of Sat. 4 [s]
2.020000000000000000E+07   数学project代做 /= altitude  of  Sat. 4 [m]
1.000000000000000000E+00 /= phase of Sat. 4 [rad]
5.000000000000000000E-01 /= u1 of Sat. 5
8.660254037844385966E-01 /= u2 of Sat. 5
0.000000000000000000E+00 /= u3 of Sat. 5
-4.967317648921540929E-01 /= v1 of Sat. 5
2.867882181755230797E-01 /= v2 of Sat. 5
8.191520442889917986E-01 /= v3 of Sat. 5
4.308204499999999825E+04 /= periodicity of Sat. 5 [s]
##### 2.020000000000000000E+07
/= altitude  of  Sat. 5 [m]
2.570796326794896558E+00 /= phase of Sat. 5 [rad]
5.000000000000000000E-01 /= u1 of Sat. 6
8.660254037844385966E-01 /= u2 of Sat. 6
0.000000000000000000E+00  数学project代做 /= u3 of Sat. 6
-4.967317648921540929E-01 /= v1 of Sat. 6
2.867882181755230797E-01 /= v2 of Sat. 6
8.191520442889917986E-01 /= v3 of Sat. 6
4.308204499999999825E+04 /= periodicity of Sat. 6 [s]
2.020000000000000000E+07 /= altitude  of  Sat. 6 [m]
4.141592653589793116E+00 /= phase of Sat. 6 [rad]
5.000000000000000000E-01 /= u1 of Sat. 7
8.660254037844385966E-01 /= u2 of Sat. 7
0.000000000000000000E+00 /= u3 of Sat. 7
-4.967317648921540929E-01 /= v1 of Sat. 7
2.867882181755230797E-01 /= v2 of Sat. 7
8.191520442889917986E-01 /= v3 of Sat. 7
4.308204499999999825E+04   数学project代做 /= periodicity of Sat. 7 [s]
2.020000000000000000E+07 /= altitude  of  Sat. 7 [m]
5.712388980384689674E+00 /= phase of Sat. 7 [rad]
-4.999999999999998890E-01 /= u1 of Sat. 8
8.660254037844385966E-01 /= u2 of Sat. 8
0.000000000000000000E+00 /= u3 of Sat. 8
-4.967317648921540929E-01 /= v1 of Sat. 8
-2.867882181755230242E-01 /= v2 of Sat. 8
8.191520442889917986E-01 /= v3 of Sat. 8
4.308204499999999825E+04 /= periodicity of Sat. 8 [s]
2.020000000000000000E+07 /= altitude  of  Sat. 8 [m]

##### 2.000000000000000000E+00
/= phase of Sat. 8 [rad]
-4.999999999999998890E-01 /= u1 of Sat. 9
8.660254037844385966E-01 /= u2 of Sat. 9
0.000000000000000000E+00 /= u3 of Sat. 9
-4.967317648921540929E-01 /= v1 of Sat. 9
-2.867882181755230242E-01 /= v2 of Sat. 9
8.191520442889917986E-01 /= v3 of Sat. 9
4.308204499999999825E+04 /= periodicity of Sat. 9 [s]
2.020000000000000000E+07 /= altitude  of  Sat. 9 [m]
3.570796326794896558E+00 /= phase of Sat. 9 [rad]
-4.999999999999998890E-01 /= u1 of Sat. 10
8.660254037844385966E-01 /= u2 of Sat. 10
0.000000000000000000E+00 /= u3 of Sat. 10
-4.967317648921540929E-01 /= v1 of Sat. 10
-2.867882181755230242E-01 /= v2 of Sat. 10
8.191520442889917986E-01 /= v3 of Sat. 10
4.308204499999999825E+04 /= periodicity of Sat. 10 [s]
2.020000000000000000E+07 /= altitude of Sat. 10 [m]
5.141592653589793116E+00 /= phase of Sat. 10 [rad]
-4.999999999999998890E-01 /= u1 of Sat. 11
8.660254037844385966E-01 /= u2 of Sat. 11
0.000000000000000000E+00 /= u3 of Sat. 11
-4.967317648921540929E-01 /= v1 of Sat. 11
-2.867882181755230242E-01 /= v2 of Sat. 11
8.191520442889917986E-01 /= v3 of Sat. 11
4.308204499999999825E+04 /= periodicity of Sat. 11 [s]
##### 2.020000000000000000E+07
/= altitude of Sat. 11 [m]
6.712388980384689674E+00 /= phase of Sat. 11 [rad]
-9.999999999999997780E-01 /= u1 of Sat. 12
1.110223024625156540E-16 /= u2 of Sat. 12
0.000000000000000000E+00 /= u3 of Sat. 12
-5.551115123125782702E-17 /= v1 of Sat. 12
-5.735764363510460484E-01  数学project代做 /= v2 of Sat. 12
8.191520442889917986E-01 /= v3 of Sat. 12
4.308204499999999825E+04 /= periodicity of Sat. 12 [s]
2.020000000000000000E+07 /= altitude of Sat. 12 [m]
3.000000000000000000E+00 /= phase of Sat. 12 [rad]
-9.999999999999997780E-01 /= u1 of Sat. 13
1.110223024625156540E-16 /= u2 of Sat. 13
0.000000000000000000E+00 /= u3 of Sat. 13
-5.551115123125782702E-17 /= v1 of Sat. 13
-5.735764363510460484E-01 /= v2 of Sat. 13

##### 8.191520442889917986E-01
/= v3 of Sat. 13
4.308204499999999825E+04 /= periodicity of Sat. 13 [s]
2.020000000000000000E+07 /= altitude of Sat. 13 [m]
4.570796326794896558E+00 /= phase of Sat. 13 [rad]
-9.999999999999997780E-01 /= u1 of Sat. 14
1.110223024625156540E-16 /= u2 of Sat. 14
0.000000000000000000E+00 /= u3 of Sat. 14
-5.551115123125782702E-17 /= v1 of Sat. 14
-5.735764363510460484E-01 /= v2 of Sat. 14
8.191520442889917986E-01 /= v3 of Sat. 14
4.308204499999999825E+04 /= periodicity of Sat. 14 [s]
2.020000000000000000E+07 /= altitude of Sat. 14 [m]
6.141592653589793116E+00 /= phase of Sat. 14 [rad]
-9.999999999999997780E-01 /= u1 of Sat. 15
1.110223024625156540E-16 /= u2 of Sat. 15
0.000000000000000000E+00 /= u3 of Sat. 15
-5.551115123125782702E-17  数学project代做 /= v1 of Sat. 15
-5.735764363510460484E-01 /= v2 of Sat. 15
8.191520442889917986E-01 /= v3 of Sat. 15
4.308204499999999825E+04 /= periodicity of Sat. 15 [s]
2.020000000000000000E+07 /= altitude of Sat. 15 [m]
7.712388980384689674E+00 /= phase of Sat. 15 [rad]
-5.000000000000000000E-01 /= u1 of Sat. 16
-8.660254037844383745E-01 /= u2 of Sat. 16
##### 0.000000000000000000E+00
/= u3 of Sat. 16
4.967317648921539819E-01 /= v1 of Sat. 16
-2.867882181755230797E-01 /= v2 of Sat. 16
8.191520442889917986E-01 /= v3 of Sat. 16
4.308204499999999825E+04 /= periodicity of Sat. 16 [s]
2.020000000000000000E+07 /= altitude of Sat. 16 [m]
4.000000000000000000E+00 /= phase of Sat. 16 [rad]
-5.000000000000000000E-01 /= u1 of Sat. 17
-8.660254037844383745E-01 /= u2 of Sat. 17
0.000000000000000000E+00 /= u3 of Sat. 17
4.967317648921539819E-01 /= v1 of Sat. 17
-2.867882181755230797E-01 /= v2 of Sat. 17
8.191520442889917986E-01 /= v3 of Sat. 17
4.308204499999999825E+04 /= periodicity of Sat. 17 [s]
2.020000000000000000E+07 /= altitude of Sat. 17 [m]
5.570796326794896558E+00 /= phase of Sat. 17 [rad]
-5.000000000000000000E-01 /= u1 of Sat. 18
-8.660254037844383745E-01 /= u2 of Sat. 18

##### 0.000000000000000000E+00
/= u3 of Sat. 18
4.967317648921539819E-01 /= v1 of Sat. 18
-2.867882181755230797E-01 /= v2 of Sat. 18
8.191520442889917986E-01 /= v3 of Sat. 18
4.308204499999999825E+04 /= periodicity of Sat. 18 [s]
2.020000000000000000E+07 /= altitude of Sat. 18 [m]
7.141592653589793116E+00 /= phase of Sat. 18 [rad]
-5.000000000000000000E-01 /= u1 of Sat. 19
-8.660254037844383745E-01 /= u2 of Sat. 19
0.000000000000000000E+00 /= u3 of Sat. 19
4.967317648921539819E-01 /= v1 of Sat. 19
-2.867882181755230797E-01 /= v2 of Sat. 19
8.191520442889917986E-01 /= v3 of Sat. 19
4.308204499999999825E+04 /= periodicity of Sat. 19 [s]
2.020000000000000000E+07 /= altitude of Sat. 19 [m]
8.712388980384689674E+00 /= phase of Sat. 19 [rad]
4.999999999999996669E-01 /= u1 of Sat. 20
-8.660254037844384856E-01 /= u2 of Sat. 20
0.000000000000000000E+00 /= u3 of Sat. 20
4.967317648921540374E-01 /= v1 of Sat. 20
2.867882181755229132E-01 /= v2 of Sat. 20
8.191520442889917986E-01  数学project代做 /= v3 of Sat. 20
4.308204499999999825E+04 /= periodicity of Sat. 20 [s]
2.020000000000000000E+07 /= altitude of Sat. 20 [m]
##### 5.000000000000000000E+00
/= phase of Sat. 20 [rad]
4.999999999999996669E-01 /= u1 of Sat. 21
-8.660254037844384856E-01 /= u2 of Sat. 21
0.000000000000000000E+00 /= u3 of Sat. 21
4.967317648921540374E-01 /= v1 of Sat. 21
2.867882181755229132E-01 /= v2 of Sat. 21
8.191520442889917986E-01 /= v3 of Sat. 21
4.308204499999999825E+04 /= periodicity of Sat. 21 [s]
2.020000000000000000E+07 /= altitude of Sat. 21 [m]
6.570796326794896558E+00 /= phase of Sat. 21 [rad]
4.999999999999996669E-01 /= u1 of Sat. 22
-8.660254037844384856E-01 /= u2 of Sat. 22
0.000000000000000000E+00 /= u3 of Sat. 22
4.967317648921540374E-01 /= v1 of Sat. 22
2.867882181755229132E-01 /= v2 of Sat. 22
8.191520442889917986E-01 /= v3 of Sat. 22
4.308204499999999825E+04 /= periodicity of Sat. 22 [s]
2.020000000000000000E+07 /= altitude of Sat. 22 [m]

8.141592653589793116E+00 /= phase of Sat. 22 [rad]

4.999999999999996669E-01 /= u1 of Sat. 23

-8.660254037844384856E-01 /= u2 of Sat. 23

0.000000000000000000E+00 /=  u3  of  Sat. 23

4.967317648921540374E-01 /= v1 of Sat. 23

2.867882181755229132E-01 /= v2 of Sat. 23

8.191520442889917986E-01 /= v3 of Sat. 23

4.308204499999999825E+04 /= periodicity of Sat. 23 [s] 2.020000000000000000E+07 /= altitude of Sat. 23 [m]

9.712388980384689674E+00 /= phase of Sat. 23 [rad]

Table 1. The file data.dat.

You should call it data.dat and have your satellite read it and use the data. Your receiver should use the data contained in the first four lines of that file.