# Gps

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## Content

Satellite frequencies

will seriously degrade the GPS signal for many consumer uses.[79][80] Aviation Week magazine reports that the latest testing (June 2011) confirms "significant jamming" of GPS by LightSquared's system.

Introduction

How GPS Works
Here's how GPS works in six logical steps:

1. The basis of GPS is "triangulation" from satellites. 2. To "triangulate," a GPS receiver measures distance using the travel time of radio signals. 3. To measure travel time, GPS needs very accurate timing which it achieves with some tricks.

4. Along with distance, you need to know exactly where the satellites are in space. High orbits and careful monitoring are the secret. 5. You must correct for any delays the signal experiences as it travels through the atmosphere. 6. Finally (for us), you can now obtain the precise time from the GPS satellites. We'll explain each of these points in subsequent sections. Improbable as it may seem, the whole idea behind GPS is to use satellites in space as reference points for locations here on earth. That's right, by very, very accurately measuring our distance from three satellites we can "triangulate" our position anywhere on earth. Forget for a moment how our receiver measures this distance. We'll get to that later. First consider how distance measurements from three satellites can pinpoint you in space.

Triangulation
    Position is calculated from distance measurements (ranges) to satellites. Mathematically we need four satellite ranges to determine exact position. Three ranges are enough if we reject ridiculous answers or use other tricks. Another range is required for technical reasons to be discussed later.

The Geometry: Suppose we measure our distance from a satellite and find it to be 10,000 miles. Knowing that we're 10,000 miles from a particular satellite narrows down all the possible locations we could be in the whole universe to the surface of a sphere that is centered on this satellite and has a radius of 10,000 miles Next, say we measure our distance to a second satellite and find out that it's 11,000 miles away. That tells us that we're not only on the first sphere but we're also on a sphere that's 11,000 miles from the second satellite. Or in other words, we're somewhere on the circle where these two spheres intersect.

If we then make a measurement from a third satellite and find that we're 12,000 miles from that one, that narrows our position down even further, to the two points where the 12,000 mile sphere cuts through the circle that's the intersection of the first two spheres. So by ranging from three satellites we can narrow our position to just two points in space. This arrangement of satellites is also called a "constellation". To decide which one is our true location we could make a fourth measurement. But usually one of the two points is a ridiculous answer (either too far from Earth or moving at an impossible velocity) and can be rejected without a measurement.

A fourth measurement does come in very handy for another reason however, but we'll tell you about that later. Next we'll see how the system measures distances to satellites.

Measuring Distance
1. Distance to a satellite is determined by measuring how long a radio signal takes to reach us from that satellite. 2. To make the measurement we assume that both the satellite and our receiver are generating the same pseudo-random codes at exactly the same time. 3. By comparing how late the satellite's pseudo-random code appears compared to our receiver's code, we determine how long it took to reach us. 4. Multiply that travel time by the speed of light and you've got distance. Discussion We saw in the last section that a position is calculated from distance measurements to at least three satellites. But how can you measure the distance to something that's floating around in space? We do it by timing how long it takes for a signal sent from the satellite to arrive at our receiver.

The Math In a sense, the whole thing boils down to those "velocity times travel time" math problems we did in high school. Remember the old: "If a car goes 60 miles per hour for two hours, how far does it travel?"
Velocity (60 mph) x Time (2 hours) = Distance (120 miles)

In the case of GPS we're measuring a radio signal so the velocity is going to be the speed of light or roughly 186,000 miles per second. The problem is measuring the travel time

The timing problem is tricky. First, the times are going to be awfully short. If a satellite were right overhead the travel time would be something like 0.06 seconds. So we're going to need some really precise clocks. We'll talk about those soon. But assuming we have precise clocks, how do we measure travel time? To explain it let's use a goofy analogy: Suppose there was a way to get both the satellite and the receiver to start playing "Stairway to Heaven" at precisely 12 noon. If sound could reach us from space then standing at the receiver we'd hear two versions of the 'Stairway to Heaven', one from our

Timing - Achieving Perfect Timing
1. Accurate timing is the key to measuring distance to satellites. 2. Satellites are accurate because they have atomic clocks on board.

But for the triangulation to work we not only need to know distance, we also need to know exactly where the satellites are. In the next section we'll see how we accomplish that.

Satellite Tracking
    To use the satellites as references for range measurements we need to know exactly where they are. GPS satellites are so high up their orbits are very predictable. Minor variations in their orbits are measured by the Department of Defense. The error information is sent to the satellites, to be transmitted along with the timing signals.

Discussion Thus far we've been assuming that we know where the GPS satellites are so we can use them as reference points. But how do we know exactly where they are? After all they're floating around 11,000 miles up in space. That 11,000 mile altitude is actually a benefit in this case, because something that high is well clear of the atmosphere. And that means it will orbit according to very simple mathematics. The Air Force has injected each GPS satellite into a very precise orbit, according to the GPS master plan. On the ground all GPS receivers have an almanac programmed into their computers that tells them where in the sky each satellite is, moment by moment. The basic orbits are quite exact but just to make things perfect the GPS satellites are constantly monitored by the Department of Defense. They use very precise radar to check each satellite's exact altitude, position and speed. The errors they're checking for are called "ephemeris errors" because they affect the satellite's orbit or "ephemeris." These errors are caused by gravitational pulls from the moon and sun and by the pressure of solar radiation on the satellites. The errors are usually very slight but if you want great accuracy they must be taken into account. Once the Department of Defense has measured a satellite's exact position, they relay that information back up to the satellite itself. The satellite then includes this new corrected position information in the timing signals it's broadcasting.

So a GPS signal is more than just pseudo-random code for timing purposes. It also contains a navigation message with ephemeris information as well. With perfect timing and the satellite's exact position you'd think we'd be ready to make perfect position calculations. But there's trouble afoot. You can't manage what you don't measure - use GPS fleet tracking

Handling Errors
1. The earth's ionosphere and atmosphere cause delays in the GPS signal that translate into position errors. See a summary of error sources. 2. Some errors can be factored out using mathematics and modeling. 3. The configuration of the satellites in the sky can magnify other errors. 4. Differential GPS can eliminate almost all error. Discussion Up to now we've been treating the calculations that go into GPS very abstractly, as if the whole thing were happening in a vacuum. But in the real world there are lots of things that can happen to a GPS signal that will make its life less than mathematically perfect. To get the most out of the system, a good GPS receiver needs to take a wide variety of possible errors into account. Here's what they've got to deal with. First, one of the basic assumptions we've been using is not exactly true. We've been saying that you calculate distance to a satellite by multiplying a signal's travel time by the speed of light. But the speed of light is only constant in a vacuum. As a GPS signal passes through the charged particles of the ionosphere and then through the water vapor in the troposphere it gets slowed down a bit, and this creates the same kind of error as bad clocks. There are a couple of ways to minimize this kind of error. For one thing we can predict what a typical delay might be on a typical day. This is called modeling and it helps but, of course, atmospheric conditions are rarely exactly typical. Another way to get a handle on these atmosphere-induced errors is to compare the relative speeds of two different signals. This "dual frequency" measurement is very sophisticated and is only possible with advanced receivers. Trouble for the GPS signal doesn't end when it gets down to the ground. The signal may bounce off various local obstructions before it gets to our receiver. This is called multi-path error and is similar to the ghosting you might see on a TV. Good receivers use sophisticated signal rejection techniques to minimize this problem. Trouble for the GPS signal doesn't end when it gets down to the ground. The signal may bounce off various local obstructions before it gets to our receiver.

This is called multi-path error and is similar to the ghosting you might see on a TV. Good receivers use sophisticated signal rejection techniques to minimize this problem. Satellite Errors Even though the satellites are very sophisticated they do account for some tiny errors in the system. The atomic clocks they use are very, very precise but they're not perfect. Minute discrepancies can occur, and these translate into travel time measurement errors. And even though the satellites positions are constantly monitored, they can't be watched every second. So slight position or "ephemeris" errors can sneak in between monitoring times. Basic geometry itself can magnify these other errors with a principle called "Geometric Dilution of Precision" or GDOP. It sounds complicated but the principle is quite simple. There are usually more satellites available than a receiver needs to fix a position, so the receiver picks a few and ignores the rest. If it picks satellites that are close together in the sky the intersecting circles that define a position will cross at very shallow angles. That increases the gray area or error margin around a position. If it picks satellites that are widely separated the circles intersect at almost right angles and that minimizes the error region. Good receivers determine which satellites will give the lowest GDOP.

Timekeeping - Bringing precise time to the world
Although GPS is well-known for navigation, tracking, and mapping, it's also used to disseminate precise time, time intervals, and frequency. Time is a powerful commodity, and exact time is more powerful still. Knowing that a group of timed events is perfectly synchronized is often very important. GPS makes the job of synchronizing our clocks easy and reliable. There are three fundamental ways we use time. As a universal marker, time tells us when things happened or when they will. As a way to synchronize people, events, even other types of signals, time helps keep the world on schedule. And as a way to tell how long things last, time provides and accurate, unambiguous sense of duration. GPS satellites carry highly accurate atomic clocks.

And in order for the system to work, our GPS receivers here on the ground synchronize themselves to these clocks. That means that every GPS receiver is, in essence, an atomic accuracy clock. Astronomers, power companies, computer networks, communications systems, banks, and radio and television stations can benefit from this precise timing. One investment banking firm uses GPS to guarantee their transactions are recorded simultaneously at all offices around the world. And a major Pacific Northwest utility company makes sure their power is distributed at just the right time along their thousands of miles of transmission lines. Beagle Software offers the Star Sync product which sets any Windows computer to the GPS clock and can serve as a time source for computers on a network. We hope you've enjoyed our brief discussion about the technology behind GPS.

How GPS Works

Global Positioning System (GPS) Technology and Cars

GPS can aid in the recovery of a stolen car. Integrated with the automobile security system, GPS can notify the car owner by phone or e-mail when the car alarm is triggered, and indicate the location of the car (10). An owner can contact the GPS unit from outside the car using a cell phone or via the Internet, and can start the car engine remotely and unlock the door, solving the problem of frozen locks. GPS also monitors the car condition, and issues warnings when the battery is low or when it is time for an oil change (11). The Ford automobile company has developed a new telematic system through GPS on the Ford Focus, that will alert emergency services when an airbag deploys. The emergency services can locate the vehicle quickly and provide assistance to an injured driver or passengers (12). In addition to its applications in private vehicles, GPS is being used by commercial shipping companies to speed the delivery of cargo. It allows companies to track their fleets, record the movement of their trucks, and control route planning (13). Car rental companies can equip their cars with GPS technology to find stolen rental cars and charge customers for dangerous conduct as speeding (14). Commercial fishermen can use GPS receivers to keep track of the best spots where they have caught fish. GPS technology is also a valuable aid to targeting locations for military personnel, firefighters, and construction workers (3). The development of GPS technology opens the door to the design of an unmanned vehicle. Equipped with laser and infrared scanners to "see" its environment, such a vehicle could be used for tasks that are considered too time consuming or hazardous for human drivers. Students in the electrical and electronic engineering department of the Nanyang Technological University in Singapore are working on this technology (15). Though GPS technology is gaining popularity in cars, some problems still exist. The biggest problem is the blockage of signal transmission by obstacles such as mountains, high buildings, tunnels, urban canyons, or thick-branched trees. Multipath signals generated by reflections from nearby surfaces or fences can also interfere with the GPS data. Some methods have been introduced to improve these disadvantages. Dead Reckoning (DR) is a way to keep tracking vehicles during periods of blocked transmission. DR combines directional and distance information from a heading sensor in the car and from the car's odometer, and calculates the current position of the vehicle by computing the course steered and speed over the ground from a last known position. DR is used to improve accuracy when GPS transmission is available, in addition to being a backup when transmission is blocked. However, its accuracy tends to drop if the car travels for an extended period without receiving GPS signals (16). GPS technology is advancing very quickly. New models are being developed every year to correct problems in the older versions, focusing on improved accuracy, better reception, and even more user-friendly features.

Triangulation
In trigonometry and geometry, triangulation is the process of determining the location of a point by measuring angles to it from known points at either end of a fixed baseline, rather than measuring distances to the point directly (trilateration). The point can then be fixed as the third point of a triangle with one known side and two known angles. Triangulation can also refer to the accurate surveying of systems of very large triangles, called triangulation networks. This followed from the work of Willebrord Snell in 1615–17, who showed how a point could be located from the angles subtended from three known points, but measured at the new unknown point rather than the previously fixed points, a problem called resectioning. Surveying error is minimized if a mesh of triangles at the largest appropriate scale is established first. Points inside the triangles can all then be accurately located with reference to it. Such triangulation methods were used for accurate large-scale land surveying until the rise of global navigation satellite systems in the 1980s. Applications Optical 3d measuring systems use this principle as well in order to determine the spatial dimensions and the geometry of an item. Basically, the configuration consists of two sensors observing the item. One of the sensors is typically a digital camera device, and the other one can also be a camera or a light projector. The projection centers of the sensors and the considered point on the object’s surface define a (spatial) triangle. Within this triangle, the distance between the sensors is the base b and must be known. By determining the angles between the projection rays of the sensors and the basis, the intersection point, and thus the 3d coordinate, is calculated from the triangular relations.

Distance to a point by measuring two fixed angles

Triangulation may be used to calculate the coordinates and distance from the shore to the ship. The observer at A measures the angle α between the shore and the ship, and the observer at B does likewise for β . With the length l or the coordinates of A and B known, then the law of sines can be applied to find the coordinates of the ship at C and the distance d.

The coordinates and distance to a point can be found by calculating the length of one side of a triangle, given measurements of angles and sides of the triangle formed by that point and two other known reference points. The following formulas apply in flat or Euclidean geometry. They become inaccurate if distances become appreciable compared to the curvature of the Earth, but can be replaced with more complicated results derived using spherical trigonometry.

Calculation

Therefore

Using the trigonometric identities tan α = sin α / cos α and sin(α + β) = sin α cos β + cos α sin β, this is equivalent to:

From this, it is easy to determine the distance of the unknown point from either observation point, its north/south and east/west offsets from the observation point, and finally its full coordinates.

History

Liu Hui (c. 263), How to measure the height of a sea island. Illustration from an edition of 1726

Gemma Frisius's 1533 proposal to use triangulation for mapmaking

Nineteenth-century triangulation network for the triangulation of Rhineland-Hesse

Triangulation today is used for many purposes, including surveying, navigation, metrology, astrometry, binocular vision, model rocketry and gun direction of weapons. The use of triangles to estimate distances goes back to antiquity. In the 6th century BC the Greek philosopher Thales is recorded as using similar triangles to estimate the height of the pyramids by measuring the length of their shadows and that of his own at the same moment, and comparing the ratios to his height (intercept theorem);[1] and to have estimated the distances to ships at sea as seen from a clifftop, by measuring the horizontal distance traversed by the line-ofsight for a known fall, and scaling up to the height of the whole cliff.[2] Such techniques would have been familiar to the ancient Egyptians. Problem 57 of the Rhind papyrus, a thousand years earlier, defines the seqt or seked as the ratio of the run to the rise of a slope, i.e. the reciprocal of gradients as measured today. The slopes and angles were measured using a sighting rod that the Greeks called a dioptra, the forerunner of the Arabic alidade. A detailed contemporary collection of constructions for the determination of lengths from a distance using this instrument is known, the Dioptra of Hero of Alexandria (c. 10–70 AD), which survived in Arabic translation; but the knowledge became lost in Europe. In China, Pei Xiu (224–271) identified "measuring right angles and acute angles" as the fifth of his six principles for accurate map-making, necessary to accurately establish distances;[3] while Liu Hui (c. 263) gives a version of the calculation above, for measuring perpendicular distances to inaccessible places.[4][5] In the field, triangulation methods were apparently not used by the Roman specialist land surveyors, the agromensores; but were introduced into medieval Spain through Arabic treatises on the astrolabe, such as that by Ibn al-Saffar (d. 1035).[6] Abu Rayhan Biruni (d. 1048) also introduced triangulation techniques to measure the size of the Earth and the distances between various places.[7] Simplified Roman techniques then seem to have co-existed with more sophisticated techniques used by professional surveyors. But it was rare for such methods to be translated into Latin (a manual on Geometry, the eleventh century Geomatria incerti auctoris is a rare exception), and such techniques appear to have percolated only slowly into the rest of Europe.[6] Increased awareness and use of such techniques in Spain may be attested by the medieval Jacob's staff, used specifically for measuring angles, which dates from about 1300; and the appearance of accurately surveyed coastlines in the Portolan charts, the earliest of which that survives is dated 1296.

Gemma Frisius and triangulation for mapmaking
On land, the Dutch cartographer Gemma Frisius proposed using triangulation to accurately position far-away places for map-making in his 1533 pamphlet Libellus de Locorum describendorum ratione (Booklet concerning a way of describing places), which he bound in as an appendix in a new edition of Peter Apian's best-selling 1524 Cosmographica. This became very influential, and the technique spread across Germany, Austria and the Netherlands. The astronomer Tycho Brahe applied the method in Scandinavia, completing a detailed triangulation in 1579 of the island of Hven, where his observatory was based, with reference to key landmarks on both sides of the Øresund, producing an estate plan of the island in 1584.[8] In England Frisius's method was included in the growing number of books on surveying which appeared from the middle of the century onwards, including William Cunningham's Cosmographical Glasse (1559), Valentine Leigh's Treatise of Measuring All Kinds of Lands (1562), William Bourne's Rules of Navigation (1571), Thomas Digges's Geometrical Practise named Pantometria (1571), and John Norden's Surveyor's Dialogue (1607). It has been suggested that Christopher Saxton may have used rough-and-ready triangulation to place features in his county maps of the 1570s; but others suppose that, having obtained rough bearings to features from key vantage points, he may have estimated the distances to them simply by guesswork.[9]

Willebrord Snell and modern triangulation networks
The modern systematic use of triangulation networks stems from the work of the Dutch mathematician Willebrord Snell, who in 1615 surveyed the distance from Alkmaar to Bergen op Zoom, approximately 70 miles (110 kilometres), using a chain of quadrangles containing 33 triangles in all. The two towns were separated by one degree on the meridian, so from his measurement he was able to calculate a value for the circumference of the earth – a feat celebrated in the title of his book Eratosthenes Batavus (The Dutch Eratosthenes), published in 1617. Snell calculated how the planar formulae could be corrected to allow for the curvature of the earth. He also showed how to resection, or calculate, the position of a point inside a triangle using the angles cast between the vertices at the unknown point. These could be measured much more accurately than bearings of the vertices, which depended on a compass. This established the key idea of surveying a large-scale primary network of control points first, and then locating secondary subsidiary points later, within that primary network. Snell's methods were taken up by Jean Picard who in 1669–70 surveyed one degree of latitude along the Paris Meridian using a chain of thirteen triangles stretching north from Paris to the clocktower of Sourdon, near Amiens. Thanks to improvements in instruments and accuracy, Picard's is rated as the first reasonably accurate measurement of the radius of the earth. Over the next century this work was extended most notably by the Cassini family: between 1683 and 1718 Jean-Dominique Cassini and his son Jacques Cassini surveyed the whole of the Paris meridian from Dunkirk to Perpignan; and between 1733 and 1740 Jacques and his son César Cassini undertook the first triangulation of the whole country, including a re-surveying of the meridian arc, leading to the publication in 1745 of the first map of France constructed on rigorous principles.

Triangulation methods were by now well established for local mapmaking, but it was only towards the end of the 18th century that other countries began to establish detailed triangulation network surveys to map whole countries. The Principal Triangulation of Great Britain was begun by the Ordnance Survey in 1783, though not completed until 1853; and the Great Trigonometric Survey of India, which ultimately named and mapped Mount Everest and the other Himalayan peaks, was begun in 1801. For the Napoleonic French state, the French triangulation was extended by Jean Joseph Tranchot into the German Rhineland from 1801, subsequently completed after 1815 by the Prussian general Karl von Müffling. Meanwhile, the famous mathematician Carl Friedrich Gauss was entrusted from 1821 to 1825 with the triangulation of the kingdom of Hanover, for which he developed the method of least squares to find the best fit solution for problems of large systems of simultaneous equations given more real-world measurements than unknowns. Today, large-scale triangulation networks for positioning have largely been superseded by the Global navigation satellite systems established since the 1980s. But many of the control points for the earlier surveys still survive as valued historical features in the landscape, such as the concrete triangulation pillars set up for retriangulation of Great Britain (1936–1962), or the triangulation points set up for the Struve Geodetic Arc (1816–1855), now scheduled as a UNESCO World Heritage Site.

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