|
I have a question on the GPS positioning algorithm. In all books I've read for 3D positioning we need four satellites, and I don't understand why. We need to calculate three variables: x, y, z. We know when satellite send the signal to earth and when we receive it we can measure the time the signal travel to earth by checking the shift in PRN generator. For what purpose do we need four satellite? |
||||
|
|
|
Just a graphic to add to M'vy's answer. From Geocommons:
Update As R.K. points out, this is not a form of triangulation. Even when GPS is leveraging more than 4 satellites, it is still doing trilateration, as opposed to multilateration, which GPS does not use.
|
|||||
|
|
The major reasons why you need a fourth satellite is for timing corrections. If you know the exact position and speed of the satellites, trilateration will give you indeed 2 points, but one will usually be impossible or with an impossible speed. But a gps receiver uses the time it takes to receive a sattelite signal to determine the distance to that satellite. Even minor errors in the time of your gps receiver will cause huge errors and therefore a large uncertainty band when you have only three satellites. |
|||
|
|
|
You need four satellites because each data from one satellite put you in a sphere around the satellite. By computing the intersections you can narrow the possibilities to a single point. Two satellites intersection places you on a circle. (all points possible) Three satellites intersection places you on two possible points. The last satellite give you the exact location. You can avoid using four satellite if you already know the altitude, for example when you drive, you can use the ground level as the last intersection. But you can't possibly do this in a plane, since you are not bound to the ground. |
|||||
|
|
The fourth satellite is there just to increase accuracy to a point where it would be useable. Although, with 3D Trilateration this is not necessary to calculate a location. GPS although requires this because of the accuracy issue. Resources: |
||||
|
|
|
You actually need to determine four coordinates from the satellites, x, y, z, and t, the time. You cannot use the clock inside the device, because it is much too inaccurate. It is generated by a quartz crystal, while for the desired precision of a few meters you'd need an atomic clock, like those used in the satellites. |
|||
|
>>3 satellites would be enoughGlobal positioning system(s) assume an 'earth centered, earth fixed, x-y-z 3D cartesian coordinate system'. Any location in this 3D space requires no more than 3 components to be completely identified. So, even though 3 spheres we obtain by 3 distance measurements intersect at two different points, one of those points is rendered useless by [ earth centered + earth fixed ] characteristic of the coordinate system GPS assumes; we are interested in locations below the earth's atmosphere. 3 satellites could be used to determine 3 position dimensions with a 'perfect' receiver clock (with an expensive atomic/optical clock). !YES!, you !could have gotten! a 3D position fix with 3 satellites IFF the GPS receiver you are using was equipped with an atomic clock. (The ELIMINATION of the second point, on the bottom left figure of the illustration above, is done "intuitively" as it corresponds to someplace in DEEP SPACE. BECAUSE, there is a reason why GPS satellites are at their specific constellation (~their setup in the sky): !more than! 24 GPS satellites, on 6 orbital planes that are ~20,000 kms above you, and 4 satellites on each plane, 60 degrees between these planes, and 55 degrees inclination with respect to equatorial plane, GIVES YOU 5-8 satellites that you can "connect to" from (almost) any place on earth, and 3 SATELLITES TO GIVE A 3D POSITIONAL FIX ON EARTH. If we are talking about locating things "inside AND outside" of earth, WELL THEN YES, you need at least 1 more satellite to eliminate one of two possible intersection points in the last step. This wasn't the question, was it? In practice, placing expensive clocks in GPS receivers is rarely possible/feasible and 3 space vehicles (SVs, i.e. satellites) can be, instead, used to compute a 2D, horizontal fix (in latitude and longitude) when a certain height (e.g. z-dimension) measurement is ASSUMED; so you get rid of 1 dimensional measurement out of 4 that were originally required. Assumed height can be either the sea level or the altitude of a (normally) altimeter equipped aircraft. It is the height dimension that is chosen to be discarded, because it is the (relatively) least important one among others. Among the 4 required dimensional mesaurements (x,y,z,time), time always needs to be resolved BECAUSE satellite signals (electromagnetic waves) travel at the speed of light and reach receiver in ~0.07 atomic seconds; and thus, a slight inaccuracy in the GPS receiver's relatively cheap internal clock would give a "very wrong" locational fix because of the extra distance the signal is assumed to travel at the extreme speed of light. And, well, the other two dimensions will place the GPS receiver on some (longitude,latitude) pair on the surface of the planet. More than 4 satellites provide better accuracy by introducing additional 'time difference pairs'. 4 dimensional requirements remain, yet number of independent equations increase and exceed 4. This will result in an over determined system of equations with multiple solutions. Over determined systems are !approximated! with numeric methods, e.g. least squares. In this case, the least squares method will give the position (of GPS receiver) that best fits all of the time measurements (with extra dimensions) by minimizing the sum of the squares of errors. (1)
Global Positioning System Overview, Peter H. Dana, Department of Geography, University of Texas at Austin, 1994. (2)
Position Determination with GPS,Dr. Anja Koehne, Michael Wößner,Öko-Institut (Institute for Applied Ecology),Freiburg im Breisgau, Germany (3)
An Underdetermined Linear System for GPS, Dan Kalman (4)
For the colorful illustrations
>> INaccuracy" Four sphere surfaces typically do NOT intersect. Because of this we can say with confidence that when we solve the navigation equations to find an intersection, this solution gives us the position of the receiver along with accurate time thereby eliminating the need for a very large, expensive, and power hungry clock. "
It says "typically" BECAUSE the measurements are INaccurate; otherwise they would intersect at exactly one point. From 4 satellites, you get 4 INaccurate distance measurements. INaccuracy in all these 4 measurements are SAME (=in the same amount) BECAUSE satellites use atomic clocks which keeps them perfectly syncronised among themselves (and accurate with respect to GPS time scale), additionally, the INaccurate clock in the measurements remain the same too, because we are talking about one particular GPS receiver. Since accurate and INaccurate clocks, and thus the INaccuracy, is constant in our measurements, there can be only one correction value that reduces the volume of intersection of 4 spheres to a single point of intersection. That value represents the time INaccuracy. (5) The UTC clock is currently (2012-11-14) 16 seconds behind the GPS clock. (6) How a GPS Receiver Locks, Thomas A. Clark, NASA's Goddard Space Flight Center (7) How Accurate is a Radio Controlled Clock?, Michael A Lombardi, NIST-Time and Frequency Division, Maryland |
|||||
|
