How to solve 3 TDOA equations with 3 variables x,y,z

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I have a set of equations which looks like that, how to find x,y,z by coding while all other variables are known?

Accepted Answer

Torsten
Torsten on 21 Jan 2022
Code for symbolic solution:
syms x y z
x1 = 0.60; y1 = 0.00; z1 = 0.70;
x2 = 0.20; y2 = 1.30; z2 = 0.70;
x3 = 0.80; y3 = 1.05; z3 = 0.20;
x4 = 0.00; y4 = 0.25; z4 = 0.20;
a = 0.19845; b = 0.08791; c = 0.11492;
eqn1 = a==sqrt((x-x2)^2+(y-y2)^2+(z-z2)^2) - sqrt((x-x1)^2+(y-y1)^2+(z-z1)^2);
eqn2 = b==sqrt((x-x3)^2+(y-y3)^2+(z-z3)^2) - sqrt((x-x1)^2+(y-y1)^2+(z-z1)^2);
eqn3 = c==sqrt((x-x4)^2+(y-y4)^2+(z-z4)^2) - sqrt((x-x1)^2+(y-y1)^2+(z-z1)^2);
S = solve([eqn1,eqn2,eqn3],[x,y,z])
xnum = double(S.x)
ynum = double(S.y)
znum = double(S.z)
Code for fsolve:
X0 = [1,1,1];
x1 = 0.60; y1 = 0.00; z1 = 0.70;
x2 = 0.20; y2 = 1.30; z2 = 0.70;
x3 = 0.80; y3 = 1.05; z3 = 0.20;
x4 = 0.00; y4 = 0.25; z4 = 0.20;
a = 0.19845; b = 0.08791; c = 0.11492;
fun=@(x,y,z)[a-(sqrt((x-x2)^2+(y-y2)^2+(z-z2)^2) - sqrt((x-x1)^2+(y-y1)^2+(z-z1)^2)), ...
b-(sqrt((x-x3)^2+(y-y3)^2+(z-z3)^2) - sqrt((x-x1)^2+(y-y1)^2+(z-z1)^2)), ...
c-(sqrt((x-x4)^2+(y-y4)^2+(z-z4)^2) - sqrt((x-x1)^2+(y-y1)^2+(z-z1)^2))];
X = fsolve(@(x)fun(x(1),x(2),x(3)),X0);
x = X(1)
y = X(2)
z = X(3)

More Answers (1)

Torsten
Torsten on 21 Jan 2022
If there are no specialized methods to solve the TDOA equations (did you take a look into the literature ?), I suggest trying
syms x1 x2 x3 x4 y1 y2 y3 y4 z1 z2 z3 z4 x y z a b c
eqn1 = a==sqrt((x-x2)^2+(y-y2)^2+(z-z2)^2) - sqrt((x-x1)^2+(y-y1)^2+(z-z1)^2);
eqn2 = b==sqrt((x-x3)^2+(y-y3)^2+(z-z3)^2) - sqrt((x-x1)^2+(y-y1)^2+(z-z1)^2);
eqn3 = c==sqrt((x-x4)^2+(y-y4)^2+(z-z4)^2) - sqrt((x-x1)^2+(y-y1)^2+(z-z1)^2);
S = solve([eqn1,eqn2,eqn3],[x,y,z])
If this is not successful, use "fsolve".
  3 Comments
Ebrahim Abdullah Ahmed Albabakri
This is the output of solve
S =
struct with fields:
x: [1×1 sym]
y: [1×1 sym]
z: [1×1 sym]
fsolve shows an error
Error using fsolve (line 180)
FSOLVE requires the following inputs to be of data type double: 'X0'.
and This is the code
syms x1 x2 x3 x4 y1 y2 y3 y4 z1 z2 z3 z4 x y z a b c
x1 = 0.60; y1 = 0.00; z1 = 0.70;
x2 = 0.20; y2 = 1.30; z2 = 0.70;
x3 = 0.80; y3 = 1.05; z3 = 0.20;
x4 = 0.00; y4 = 0.25; z4 = 0.20;
a = 0.19845; b = 0.08791; c = 0.11492;
eqn1 = a==sqrt((x-x2)^2+(y-y2)^2+(z-z2)^2) - sqrt((x-x1)^2+(y-y1)^2+(z-z1)^2);
eqn2 = b==sqrt((x-x3)^2+(y-y3)^2+(z-z3)^2) - sqrt((x-x1)^2+(y-y1)^2+(z-z1)^2);
eqn3 = c==sqrt((x-x4)^2+(y-y4)^2+(z-z4)^2) - sqrt((x-x1)^2+(y-y1)^2+(z-z1)^2);
S = solve([eqn1,eqn2,eqn3],[x,y,z])

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