Talk:Pseudorange

message: Remarks.
Pseudoranges are relative distances, the ranges (distances) to known position where all ranges with the same error. Also called relative distances. Distances can be expressed in the measure of distance (m) or in the timedelay time (s). distance = c*time. (The assumption here is that the speed of light is always c. Errors caused by the layers around the earth are ignored).
Triangulation, is calculating with triangles base on the knowledge of angles (and at least one side) of the triangle. In surveing this was a standard technique, whole area's could be measured and mapped using this system.
Trilateration, is calculating with triangles base on the knowledge where the length of the sides is known. When used for positioning some people assume (or require) that Trilateration is based on exactly three known point and a fourth unknown point.
Pseudorange calculation is Not based on Trilateration, because the length of the sides of the triangles is Not known. In Pseudorange calculation in 3D at least 4 positions are needed with the relative distance to each, to determine a point. In GPS these four positions can be supplied by 4 satellites or by 3 satellites and a known hight. (The know hight is not realy a pseudo range, but the distance to the middle of the earth without the relative error, the other signals/distances have the same error).
Implementation.
A fairly simple and effective method to implement the pseudorange calculation works as follows.
:1.Use an estimated position as a starting point, this can be the last known point, but for example for GPS the middle of the earth can be choosen.
:2.Measure the relative distance to the known locations (With GPS these are the satellites). This measuring is often done by measuring the time a signal travels. All measured distances should have the same error.
:3.Calculate the distance for all the known locations to the estimated postition.
:4.For each of the satellites calculate the difference between the estimated position (from 3) and the actual ranges (from 2).
:5.Now find the (relative) x,y,z coordinate which would give such an error. (Explained further on).
:6.Now correct the estimated position with this information.
:7.After a second repeat the process with new position information. (After a few seconds you should have a fix where the mathematical error is almost zero even if you are moving fast, offcourse other errors remain).
:5. Finding the x,y,z which would give the error.
We are seeking for an x,y,z and the error, this is four parameters. We also have four signals from four known location. This means we can make four equations. So we have four equitions and four unknowns. Four equations with four unknowns are easy to solve if the equations are lineair. But equations with distances (spheres) or with distance differences (hyperboloids) or not lineair. But the pheres which we are working with are huge, at least 3 times the size of the earth, as long as we look locally the earth is pretty plane. So for our equitions we are going to assume the the signals arrive as a plane. Doing this we have four equitions with four unknown. This can easely be solved. This results in an how much our estimate was wrong in x,y,z and the relative distance. But this result is not exact, because of the assumption of a plane. If we start of with an near enough place to the point which is the actual point, one iteration is enough. Otherwise a few iterations can be used. 
Pseudorange calculation implemented with lineair equations has the following advantages.
:1. It is simple to implement.
:2. The result is very close to the actual point.
:3. The calculation is relative simple and fast.
:4. Using a fairly close approcimation only one iteration, gives an accurate result.
:5. For GPS use the centre of the earth can be used as a first estimate. So no actual location is needed.
Are there alternatives to do the same calculations?
Yes, the calculation can be done with intersecting spheres and estimated timing errors.
Yes, the calculation can be done with the intersecting of the hyperboloids formed by the time differences from known location.
Yes, there are some other methods which can be used. Four example an analogue mechanical way would be building a model of four known positions. Make ropes with the relative length of the distances all with the same extra length. Connect the ropes at one point, each rope goes from this connecting point towards the know location, and from there to a central point where a weight is pulling the rope tight. (The length of each rope has to be adjusted by the distance from the known location to the central point.) If all ropes are tight the position where the ropes meet should be the actual location. it could be that you have to pull the rope to get it in the relative position. (This could be done by a weigth, but then the location should be below the know locations.) A pulley system then has to make sure that the other side is also using a weight. (this method is completely rediculous, because it is not practical and does not give detailled information. But it would work in principle.).
But, none of these methods can deliver a result as fast and as accurate as the shown implementation can.
WORK IN PROGRESS.
Crazy Software Productions 13:32, 18 September 2007 (UTC)
thread_title:  Pseudoranges some remarks and suggested alterations.