I have been making some experiments to get a precise position using three beacons.
Results of trilateration
Unluckily, the results were very disappointing in terms of quality. There were mainly two issues:
- In non-controlled environments, where you can find metals, and other objects that affect the signal, the received signal strength of the beacons changes so often that it seems impossible to get error range below 5 meters.
- Depending on the way that the user is handling the receiver device, the readings can change a lot as well. If the user puts his/her hand over the bluetooth antenna, then the algorithm will have low signals as input, and thus the beacons will supposed to be very far from the device. See this image to see the precise location of the Bluetooth antenna.
Possible solutions
After talking with an Apple engineer who actively discouraged me to go down this way, the option I feel more inclined to use right now is brute force. Try to set up a beacon every X meters (X being the maximum error tolerated in the system) so we can track on this beacons grid the position of a given device by calculating which beacon on the grid is the closest to the device and assuming that the device is on the same position.
Trilateration algorithm
However, for the sake of completeness, I share below the core function of the trilateration algorithm. It’s based on the paragraph 3 (“Three distances known”) of this article.
- (CGPoint)getCoordinateWithBeaconA:(CGPoint)a beaconB:(CGPoint)b beaconC:(CGPoint)c distanceA:(CGFloat)dA distanceB:(CGFloat)dB distanceC:(CGFloat)dC {
CGFloat W, Z, x, y, y2;
W = dA*dA - dB*dB - a.x*a.x - a.y*a.y + b.x*b.x + b.y*b.y;
Z = dB*dB - dC*dC - b.x*b.x - b.y*b.y + c.x*c.x + c.y*c.y;
x = (W*(c.y-b.y) - Z*(b.y-a.y)) / (2 * ((b.x-a.x)*(c.y-b.y) - (c.x-b.x)*(b.y-a.y)));
y = (W - 2*x*(b.x-a.x)) / (2*(b.y-a.y));
//y2 is a second measure of y to mitigate errors
y2 = (Z - 2*x*(c.x-b.x)) / (2*(c.y-b.y));
y = (y + y2) / 2;
return CGPointMake(x, y);
}