I've been following this paper (notably the section on Fang's Method) in an attempt to achieve a solution to the problem of trilateration using the TDOA technique.
I'm hoping that someone experienced in Fang / TDOA can lend me a helping hand. For some reason, my implementation is returning incorrect roots to the final quadratic. Here's the code I've written so far:
#include <stdio.h>
#include <math.h>
struct Point {
double x;
double y;
};
inline double sqr(double n) {
return n * n;
}
// r1 and r2 are the TDOA of the sound impulse to p1 and p2, respectively
void fang(double r1, double r2) {
// transmitter coords
Point tx = {0.7, -0.1};
// receiver coordinates
Point p0 = {0, 0};
Point p1 = {1.7320508075688772, 0};
Point p2 = {0.8660254037844388, 1.5};
// linear coefficients
double g = ((r2 * (p1.x/r1)) - p2.x) / p2.y;
double h = (sqr(p2.x) + sqr(p2.y) - sqr(r2) + r2 * r1 * sqr(1 - (p1.x / r1))) / (2 * p2.y);
// quadratic coefficents
double d = -(1 - sqr(p1.x / r1) + sqr(g));
double e = p1.x * (1 - sqr(p1.x / r1)) - (2 * g * h);
double f = (sqr(r1) / 4) * sqr(1 - sqr(p1.x / r1)) - sqr(h);
double result_x = (-e - sqrt(sqr(e) - (4 * d * f))) / (2 * d);
}
int main() {
// these values have been calculated a-priori, from the known transmitter coords
double r1 = 0.32977743096231715;
double r2 = 0.90148404145971694;
fang(r1, r2);
}
Ultimately I'd expect the x_result
to be equal to the transmitter's x coordinate (tx.x == 0.7
), but frustratingly the result is ā0.237
.
An outline of my exact problem (and it's solution, where the two hyperbolas intersect) can be viewed geometrically in the below graph:
Any help would be hugely appreciated!
The paper gives the following calculation for h
with the Fang method:
Your code incorrectly squares the entire (1 - (p1.x / r1))
expression, instead of just the (p1.x / r1)
part. Moreover, you use the wrong values (p2
and p1
) instead of the correct ones (p3
and p2
). To fix, simply change h
to:
double h = (sqr(p3.x) + sqr(p3.y) - sqr(r3) + r3 * r2 * (1 - sqr(p2.x / r2))) / (2 * p3.y);