Lecture 04 Principles of GPS - Home-Precision …...Read Chapter 2. (Pages 9-15) of the TEXT book: Precision farming A guide for Agriculturists. By John Deere. PRINCIPLE OF GLOBAL

Principles of GPS. Fall Semester 2015. Raj Khosla Page 1 9/3/2015

Lecture 4-5

Principles of Global Positioning Systems Suggested Reading for Students: Read Chapter 2. (Pages 9-15) of the TEXT book: Precision farming A guide for

Agriculturists. By John Deere. PRINCIPLE OF GLOBAL POSITIONING SYSTEM:

How the Global Positioning System works is, conceptually, really very simple. The GPS system is based on a distance measuring (satellite ranging) system. That means that we find ‘our position’ or ‘position of an object’ on earth by measuring our distance from a group of satellites in space. The satellites act as precise reference points for us. The GPS satellites transmit signals in all directions, with a preferential orientation toward the Earth.

You might ask: "How do we measure exactly how far we are from a satellite way out in space? And how do we know exactly where a moving satellite is?" These are a couple of the details we will deal with later.

Lets start with a simple example of triangulation: Here's a story problem similar to the ones we find in high school trigonometry textbooks:

Assume that Sam's boat travels at a speed of 6 miles per hour. Sam has determined that his favorite fishing spot in the Lake is: 10 minutes from dock A, 5 minutes from dock B, and 15 minutes from dock C Can you locate the fishing spot on a map of the lake that shows the dock locations? (Sam doesn't have a compass, but he can see each of the three docks from the fishing hole.)

The first step is to calculate how far the fishing spot is from each of the docks. We do this by multiplying his speed of 6 miles per hour (or ____ mile per minute) by the time required to reach the fishing spot from each dock.

DISTANCE = SPEED X TIME Distance to dock A:

0.1 miles/minute X 10 minutes = ______ miles

Distance to dock B:


0.1 miles/minute X 5 minutes = ______ miles

Distance to dock C: 0.1 miles/minute X 15 minutes = ______ miles

Let’s draw a circle around dock A on the map with a radius of ____ mile. Now

we know that Sam's fishing spot is somewhere on this circle. Let’s draw another circle around dock B with a radius of _____ miles. Notice these two circles intersect, or cross, at two points. This means the fishing spot is at one of these two intersecting points. If we draw a third circle around dock C with a ______ mile radius, all three circles cross at a single point. This point marks the fishing spot. From this example, we can define the steps required to locate the fishing spot as follows: 1. Determine the time required to reach the fishing spot from each of three boating

docks on the lake 2. Assume that the speed of the boat is constant at 6 miles per hour. 3. Calculate the distance, in miles, from each of the three docks. 4. Locate each of the boating docks already marked on the map. 5. Determine the fishing spot location by drawing circles on the map around the three

docks, which were equivalent to calculated distances.

These are essentially the same steps a GPS receiver uses to determine its position. In the example above, the “ranging method” was used to determine the location of the fishing spot in TWO dimensions based on distances from three reference points, by intersecting three circles at a single point. Likewise, GPS provides positions information in THREE dimensions by using signals from four (or more) satellites.

Let us have a look how GPS works. Let's say we are lost on planet earth and we are trying to locate ourselves on earth. If we know that we are a certain distance from satellite A, say 11,000 miles, that really narrows down where in the whole universe we can be. It tells us we must be somewhere on an imaginary sphere that is centered on the satellite and that has a radius of 11,000 miles. Now if at the same time we also know that we are 12,000 miles from another satellite, satellite B that further narrows down where we can be. Because the only place in the universe where we can be is on the intersection of the two spheres (11,000 miles from satellite A and 12,000 miles from satellite B). Then if we make a measurement from a third satellite we can really pinpoint ourselves. Because if we know that at the same time we are 13,000 miles from satellite C, there are only two points in space where that can be true. Those two points are where the 13,000-mile sphere cuts through the circle that's the inter-section of the 11,000-mile sphere and the 12,000-mile sphere.

That's right. By ranging from three satellites we can narrow down where we are to just two points in space. But how do we decide which one of those two points is our true location? We do that by making a fourth measurement from another satellite. However there are other ways, for example, we can make an intelligent assumption,


because one of the two points is a ridiculous answer. Reason: the incorrect point may not be close to the earth. Or it may have an impossibly high velocity. Or if you're sure of your altitude, like mariners are (they know they're at sea level), you can eliminate one of the satellite measurements. However, the computers in the GPS receivers have various techniques for distinguishing the correct point from the incorrect one.

That is the basic principle behind GPS: using satellites as reference points for

triangulating your position somewhere on earth. Technically the process is called trilateration. In trilateration, the position of an unknown point (GPS receiver) is determined by measuring the lengths (distance) of the sides of a triangle between the unknown point (the receiver) and two or more known points (i.e., the satellites). Whereas in triangulation, a position is determined by taking angular bearings from two points a known distance apart and computing the unknown point's position from the resultant triangle. In essence, the GPS operates on the principle of trilateration. In simple terms: the steps required in finding position on earth are as follows: 1. Determine the time required for satellite signal to reach the GPS unit/receiver. 2. Assume that the speed of light is constant at 186,000 miles per second. 3. Receiver calculates the distance, from each of the four or more satellites. 4. Locate each of the satellites on specified orbits. 5. Make corrections about the position and read out the coordinates. MEASURING YOUR DISTANCE FROM SATELLITES.

Since GPS is based on knowing your distance to the satellites in space, we need a method for figuring out how far we are from those satellites. Surprisingly, the basic idea behind measuring a distance to a satellite is just the old "velocity times travel-time" equation we all learned in high school.

DISTANCE = VELOCITY X TIME

You remember those word problems: "If a car goes 60 miles an hour for two hours, how far has it gone?" Well, it's velocity (60-miles/hour) times travel time (2 hours) equals distance (120 miles).

The GPS system works by timing how long it takes for a radio signal travelling at the speed of light to reach us from a satellite and then calculating the distance from that time. So if we can figure out exactly when the GPS satellite started sending its radio message and when we received it, we'll know how long it took to reach us. We just multiply that time in seconds by 186,000 miles per second and that's our range to the satellite. (And remember we need four ranges to four different satellites and we've got our accurate position).


Since timing is a very important component of our GPS system, our clocks must be very accurate in measuring even miniscule period of time because light moves awfully fast. In fact, if a GPS satellites were right overhead it would only take about 6/l00ths of a second for the radio message to get to us. The kind of timing accuracy it demands is only possible with highly precise atomic clocks.

An atomic clock is a clock that keeps time using natural characteristic frequencies of atoms, such as cesium, hydrogen, or rubidium. Atomic clocks do not run on atomic energy. They get the name because they use the oscillations of a particular atom as their "metronome". Atomic clocks are extremely stable because the frequencies are almost unaffected by factors like temperature, pressure, or humidity. This tool for measuring time is a very good one, because two different atomic clocks will give you the same answer, even if they are far apart, oriented in different directions, and at different temperatures. So, we know how to measure the passage of time in a very consistent way.

The big trick to measuring the travel time of the radio signal is to figure out exactly when the signal left the satellite. To do that the designers of the GPS system came up with a clever idea: Synchronize the satellites and receivers so that they are generating the same code at exactly the same time. Then all we have to do is receive the codes from a satellite and then look back and see how long ago our receiver generated the same code. The difference in the time is how long the signal took to get down to us.

To picture how this works, imagine you and a friend were standing at opposite ends of a football stadium. Now suppose there was a way to make sure that you both started counting to ten at exactly the same moment. And you both yelled the numbers out as you counted. What you would hear at your end of the football stadium would be yourself saying: "One... two...three..." and then, a bit later, you would hear your friend's voice saying "one... two..." and so on. You might already be up to "three" by the time you heard him saying "one." That is because it takes a while for the sound of his voice to get all the way across the stadium to you.

Since you both started yelling at the same time you could measure the time

between when you said "one" and when you heard your friend say "one". That time would be the travel time for sound to cross the stadium. That's basically how the GPS system works.

The advantage of using a set of codes, or in the case of our analogy, a string of

numbers, is that you can make the time measurement any time you want. You don't necessarily have to measure between when you said "one" and when you hear your friend say "one." You could do the same measurement between any pair of numbers, like when you say "eight" and when you hear your friend say "eight" so you can jump in at any time.

The GPS system does not use numbers however. Both the satellites and the receivers actually generate a very complicated set of digital codes. The codes are made complicated on purpose so that they can be compared easily and unambiguously.


Anyway, the codes are so complicated they almost look like a long string of random pulses. They are not really random though; they are carefully chosen and are therefore termed as "pseudo-random" sequences that actually repeat every millisecond. GETTING PERFECT TIMING:

We know that light travels at 186,000 miles a second. If the satellite and our receiver were out of sync by even 1/l00th of a second, our distance measurement could be off by 1,860 miles! How do we know both our receiver and the satellite are really generating their codes at exactly the same time? Well, at least one side of the clock sync problem is easy to explain, the satellites have atomic clocks on board. They're unbelievably precise and unbelievably expensive. They cost about one hundred thousand dollars ($$100,000) a piece and each satellite has four atomic clocks, just to be sure one is always working. Atomic clock is the most stable and accurate time reference man has ever developed. So you can bet that when they think it is twelve noon, it is exactly twelve noon.

That is fine for the satellites, but what about the receiver here on earth. If we had to have a hundred thousand-dollar atomic clock in every GPS receiver only Donald Trump or Bill Gate’s yacht would have one.

Fortunately there's a way to get by with only moderately accurate clocks in our

receivers—and the secret is to make four satellite range measurement instead of only three.

Here's how it works: Suppose our receiver's clock is not perfect like an atomic clock. It is consistent like a quartz watch but it's not perfectly synced with universal time. Say it's a little fast, so that when it thinks it's twelve noon, it's really 11:59.59 AM. Let's look at what that would do to our position calculations.

Let us say that, in reality, were four seconds from satellite A, and six seconds

from satellite B. In two dimensions, those two ranges would be enough to locate us at a point. Let's call it "X" (Remember, it takes three measurements to locate a point in three dimensions). So "X" is where we really are and is the position we would get if all the clocks were working perfectly. But now what if we used our "imperfect" receiver, which is one second fast? It would read the distance to satellite A, as five seconds and the distance to satellite B, as seven seconds. And that causes the two circles to intersect at a different point: "XX". So XX is the point where our imperfect receiver would put us. And it would seem like a perfectly correct answer to us, since we have no way of knowing that our receiver was a little fast. But it would be miles off. We'd probably notice something was not right when we started running into rocks, but nothing in the calculations would tell us.

Now this is where our trigonometry trick can help: Let us add another

measurement to the calculation. In our two dimensional example, that means a third


satellite. Let us say in reality (if we had perfect clocks) satellite C is eight seconds from our true position. Now let us add our one-second offset to the drawing and see what happens. The dotted lines show the "pseudo-ranges" caused by our fast clock. The phrase "pseudo-range" is used in GPS circles to describe ranges that contain errors (usually timing errors). Notice that while A's and B's fast times still intersect at XX, C's fast time is nowhere near that point. So there's no point (position) that can really be five seconds from A, seven seconds from B and nine seconds from C. There's no physical way those measurements can intersect.

The little computers in our GPS receivers are programmed such that when they get a series of measurements that cannot intersect at a single point, they realize something is wrong. And they assume the cause is that their internal clock is off—that it has some offset. So the computer starts subtracting (or adding) time, the same amount of time from all the measurements. It keeps trimming off time from all the measurements until it hits on an answer that lets all the ranges go through one point. In essence, it "discovers" that by subtracting one second from all three measurements it can make the circles intersect at a point. And from this it assumes that its clock is one second fast. In another words GPS receiver is self calibrate the measurements and quickly compute the clock offset.

But the idea is the same, by adding one more measurement we can cancel out any

consistent clock error our receivers might have. In three dimensions, if we make four measurements it cancels out any error. And that's a very significant number to remember because it means that you cannot get a truly accurate position until you have four satellites above the horizon around you.

Adapted from: 1. GPS A Guide to the Next Utility. By Trimble. 2.Understanding the GPS. An introduction to the Global Positioning System. What it is and how it works. Gregory T. French. Geo Research, Inc. Bethesda, MD.

R. Khosla Fall Semester 2015 1

Principles of GPS

Principles of GPS…

• How the Global Positioning System works is, conceptually, really very simple

• The GPS system is based on a distance measuring (satellite ranging) system

• That means that we find ‘our position’ on earth by measuring our distance from a group of satellites in space

• Lets start with a simple example of triangulation:

• Assume that Sam's boat travels at a speed of 6 miles per hour. Sam has determined that his favorite fishing spot in the Lake is:

10 minutes from dock A

7.5 minutes from dock B, and

15 minutes from dock C



• The first step is to calculate how far the fishing spot is from each of the docks

• Speed of 6 miles/ hr = 0.1miles/minute

• DISTANCE = SPEED X TIMEDistance to dock A:

• 0.1miles/minute X 10 minutes = 1 mile

Distance to dock B:• 0.1miles/minute X 7.5 minutes = 0.75 miles

Distance to dock C:• 0.1miles/minute X 15 minutes = 1.5 miles


http://www.blindfishingboat.com/?p=274


Dock A

Dock B

Dock C

Where is Sam’s fishing spot? 1 mile from dock A ¾ mile from dock B 1 ½ mile from dock C


Sam’s fishing spot

Somewhere on this line

Somewhere in these spots

Dock B

Dock C

Dock A


1. Determine the time required to reach the fishing spot from each of 3 boating docks on the lake

2. Assume that the speed of the boat is constant at 6 miles per hour

3. Calculate the distance in miles, from each of the three docks

From this example, we can define the steps required to locate the fishing spot as follows:


4. Locate each of the boating docks already marked on the map

5. Determine the fishing spot location by drawing circles on the map around the three docks, which were equivalent to calculated distances

These are essentially the same steps a GPSreceiver uses to determine its position

In the example above, the ranging method wasused to determine the location of the fishing spotin 2-dimensions based on distances from3- reference points, by intersecting 3-circles at asingle point

Likewise, GPS provides positions information inthree dimensions by using signals fromfour (or more reference points) satellites


• Lets have a look how GPS works

• Lets assume we are lost on the planet earth and we are trying to locate ourselves using GPS

• If we know that we are a certain distance from satellite A, say 11,000 miles, that really narrows it down where in the whole universe we can be

• It tells us we must be somewhere on an imaginary sphere that is centered on the satellite and that has a radius of 11,000 miles




Principles of GPS…Single range known20 000 km radiusSecond range known22 000 km radiusThird range known21 000 km radiusFourth range known20 500 km radius

• By ranging from three satellites we can narrow down where we are to just two points in space

• But how do we decide which one of those two points is our true location?

• We do that by making a fourth measurement from another satellite



• There are other ways of deciding our true location:

• We can make an intelligent assumption, because one of the two points is a ridiculous answer. Reason: the incorrect point may not be close to the earth

• Or if you're sure of your altitude, like mariners are (they know they're at sea level), you can eliminate one of the satellite measurements

• However, the computers in the GPS receivers have various mathematical techniques for distinguishing the correct point from the incorrect one


1. Determine the time required for satellite signal to reach the GPS unit/receiver

2. Assume that the speed of light is constant at 186,000 miles per second

3. Receiver calculates the distance from each of the four or more satellites

4. Locate each of the satellites on specified orbits

5. Make corrections about the position and read out the coordinates

In simple terms, the steps required in finding position on earth is as follows:


GPS is a Distance/Ranging system

Operates on the Principle of Trilateration

Satellites transmit unique Radio waves

Receivers passively receive SV signal

Receivers measure time for signal to reach it

Distance computed by D = V x TΔ

V = C = 300,000 Km/Sec (186,000Mi/Sec)

Summary: How does GPS work?


Two-way Ranging

Sound Navigation and Ranging

One-way Ranging

Sound Navigation and Ranging

Source: http://standeyo.com/NEWS/07_Earth_Changes/070705.apoc.weather.html

Source: http://www.drawshop.com/imgdetail.php?auid=19045

Speed of sound =1mi/5sec ~ 720mi/hr

Accurate positioning requires very precise measurement

Only takes 6/100 sec. for SV signal to reach ground

At 300,000 Km/s

1/1,000,000 sec error => 300 M Pos. error

Satellites have very precise Atomic Clocks

Receivers have only ‘Inexpensive’ Quartz clocks

What if clock is off by just tiny fraction of a second

Why 4 satellites?


5 seconds

6 seconds

Assumption : Receiver clock and SV clock in perfect sync

= True position

True time/range

5 seconds

6 seconds

Receiver clock “fast” by 1 second

= True position = Incorrect “perceived” position

Receiver time 1 second fast

With additional SV, there is no place where all three radii intersect

Receiver sees problem and slows time until they do

= True position = Incorrect “perceived” position

5 seconds

7 seconds

6 seconds

Addition of a SV Time/Range

Ll1


• So the computer starts subtracting (or adding) time, the same amount of time from all the measurements


• The little computers in our GPS receivers are programmed such that when they get a series of measurements that cannot intersect at a single point, they realize something is wrong

• They assume the cause is that their internal clock is off—that it has some offset

Receiver

Satellite

Time difference

Pseudo Random Noise

Receiver recognizes a satellite by SVN (Satellite Vehicle Number)Or PRN (Psuedo Random Noise/Code) coded on each satellite signal

A short repeating PRN code sample


The Code Is The Key

• Receiver “Knows” each SV’s Unique Code

• Receiver generates replica of each code internally

• Receiver then measures the “Lag-Time” of SV code

Satellite

ReceiverLag-time

No correlation with a different PRN code

Partial correlation of identical receiver and satellite PRN codes


Full correlation (code-phase lock) of receiver and satellite PRN codes

http://www.sourcival.ch/referral; http://dailycaller.com/2010/05/21/soccer-referee-stabbed-by-heckler/

D = V x T∆V = Speed of sound = 352 yard/sec

0.11 seconds

.

.

0.17 seconds

Frequency

Wave length

Amplitude

Basic Signal Structure


Source http://www.nrao.edu/index.php/learn/radioastronomy/radiocommunication

Frequency modulation

Source http://www.nrao.edu/index.php/learn/radioastronomy/radiocommunication

Phase modulation

Two GPS Services

Use of only L1 Band

Use of both L1 and L2 Band

SPS: Standard Positioning Services

PPS: Precise Positioning Services


Each Satellite Transmits On Two Frequencies

L1 Carrier 1575.42 MHz and L2 Carrier 1227.60 MHz

Superimposed on these carriers are Pseudo-Random, Binary, Bi-Phase Modulation Codes called PRN (Pseudo-Random Noise) Codes

unique to each satellite

Coarse Acquisition Code (C/A-Code): Standard Positioning

Precise, or Protected Code (P-Code) Precise Positioning

Basic GPS Signal Structure

Documents

Lecture 04 Principles of GPS - Home-Precision …...Read Chapter 2. (Pages 9-15) of the TEXT book: Precision farming A guide for Agriculturists. By John Deere. PRINCIPLE OF GLOBAL