Least Squares Trilateralization – 2D

Trilateralization is basically a distance-distance intersection using two or more points and corresponding measurements from them to an unknown point.

Working with just two points can yield 0, 1 or 2 solutions.

Case 1: No Solutions

• Given
  
  • Two points with fixed X,Y ( Points A and B)
  • Two measurements from fixed Points (A &B ) to the Unknown Point (P)
• Results
  • If the sum of the measurements from A to P and B to P is less than the distance A to B then there cannot be an intersection

Case 2: One Solution

• Given
  
  • Two points with fixed X,Y ( Points A and B)
  • Two measurements from fixed Points (A &B ) to the Unknown Point (P)
• Results
  • The sum of the measurements from point A to P and B to P, equals the distance (A to B), thus point P must lie on the line AB

Case 3: Two Solutions

• Given
  • Two points with fixed X,Y ( Points A and B)
  • Two measurements from fixed Points (A &B ) to the Unknown Point (P)
• Results
  • If the sum of the measurements from A to P and B to P, are greater than the distance (A to B) then there exists two possible solutions. Circle-Circle Intersection

The least squares part of this occurs when you have 3 or more fixed points, and a measurement from each to the unknown point ( P ). Like shown here:

Assume Points A, B, C and D are fixed. Point P is unknown.

Chain Rule _ Dist Equation – Derivative

A different approach this time. Instead of showing each iteration in expanded form, I am going to introduce the matrices in the form of MathCAD functions. We can them solve them in a MathCAD Program Block.

Above is the MathCAD Program Block. Where we define the program, like a function, where it takes the matrix ‘pt’, the column vector ‘P’, and the double ‘tol’ and an integer ‘n’. However, MathCAD does not have strict variable declaration but the way I access the variables will error out if they aren’t passed in the method described.

SolveIt yields us the solution for point P and the variance from the measurements.

*EDIT – I found and corrected error in the J Matrix above. If you are revisiting the site be sure to refresh the page. Sorry.*

Dave