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Trilateration: How distance measurements help to work out location

Industry Articles October 13, 2020

Imagine you and some friends are standing in a large square field. Each friend has a way of making a different noise (starter pistol, cymbal, shouting, etc.). You synchronise watches, and one friend goes and stands in each corner of the field. Starting at mid-day, they agree to start making a noise once every 10 seconds.

Now suppose when they start to make noise that two of the sounds arrive simultaneously, shortly followed by the other two. You then move a little in the direction of delayed sounds. Ten seconds later another burst of noise arrives, but this time the delay for the second pair is shorter. If you keep moving you will get to a point where all four noises arrive simultaneously. As you know each person is standing in the corner of the field, you can also work out that you must be standing exactly at the centre.

In terms of GPS navigation, the satellites may not be nicely located in the corners of the sky, but because of the ephemeris data, we can work out their precise location relative to a known co-ordinate system when the signal was sent, and using the C/A code, we can also work out how far we are from that known location. What happens next is a process called trilateration.

Trilateration is a bit like triangulation. With triangulation, you identify a specific point by saying it is at angle of ‘a‘ from point 1 and angle of ‘b‘ from point 2. Lines drawn at those specified angles from each point will cross, and the point at which they cross is the location of our new point.

Triangulation projects lines of unknown length along known angles to find a point.

As long as there is more than one reference point we can identify the location of a new point. So, if we know point x is located at angle of 450 from point 1, and at an -450 from known point 2, the point at which those projected lines intersect must be the location of point x.

Trilateration works in a similar way but uses distance rather than angles to find a point. The other major difference is that with trilateration, you need a minimum of three reference points rather than two to narrow the search down to one location.


One reference point

Trilateration uses lines of known length but unknown angle (circles in other words) to find a point. With one reference point, Point x could be anywhere on it’s perimeter.


Two reference points

When we have two reference points, we know that Point x must be at either of their two intersections, but without a third reference point it is not possible to know which one.


Three reference points

As long as there are more than two reference points we can identify the location of a new point. So, if we know point x is 1 metre from point 1, 1.5 meters from point 2 and 0.75 meters from point 3, the point at which those circles intersect must be the location of point x.

GPS uses the same technique but has to approach things slightly differently. The circles we’ve been looking at are two-dimensional. In real life, things are far more three-dimensional, and that means our intersecting circles become intersecting spheres.

When two spheres intersect, you don’t end up with two points of intersection, you end up with a ring of intersection (imagine two bubbles joined together). If a third bubble joins in, it creates two points where all three intersection rings meet.

Two spheres intersect

When two spheres intersect, the intersection creates a circle (outlined in red).

Three spheres intersect

When three spheres intersect, there are only two points common to all three spheres (one each side).

At this point, you might recall hearing that GPS only needs three satellites in order to generate a position measurement. This statement is both right and wrong. It’s clear from the image above that in order to use trilateration to arrive at a single point, a fourth sphere is required (to show us which point is the correct one). However, as GPS uses an Earth centred, Earth fixed coordinate system, one of the points can be ruled out immediately as it would be well outside Earth’s atmosphere. The other point must, therefore, be the correct one.

So GPS can generate a position measurement using only three satellites, but in order to do that, its internal clock must be accurate otherwise it can’t calculate the distance properly. And correcting the internal clock requires four satellites! That’s why the statement was both right and wrong. As long as the system has used four satellites to correct its internal clock, it can drop back to only three satellites and still make a position estimate—it just can’t do it forever.

To fully answer the question, What is GNSS?‘ we could now discuss GPS position measurements that are based only on the C/A code – ie. Standard Positioning Service (SPS).

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