Example 7

Goal

Your goal in this example is to extend the earlier examples based on the distance equation to obtain the linearized model for a simple trilateration.

Situation

We first looked at the simple case of measuring the distance between two points in Example 2 and then extended it in Example 6 to derive its linearized form. We did the latter in considerable detail so as to demonstrate good practice – and what I would expect you to submit when coming up with such a solution.

Building on these examples, I want you to consider the simple trilateration shown in the figure below. That is, the case where:

  • the coordinates of three points, 1, 2, and 3, are known in a planar 2D coordinate system
  • we seek to find the coordinates of a fourth point, 4
  • we will do so by measuring the distances d_{14}d_{24}, and d_{34}

Problem

I want you derive the linearized functional model for this example in the form \mathbf{A}\boldsymbol{\delta} - \mathbf{e} + \mathbf{w} = 0.

I want you to be as explicit in doing so as we were in Example 6, e.g. by: sketching the situation; stating what is given; stating what is required; showing your work and which equations you use; and putting boxes around your final answers.

To keep things simple, you may assume we only measure each distance once.

Guidance

In general I’d recommend doing the following for this kind of problem where you’re building a model from models you developed earlier (several distance measurements in this case, each based on the earlier-developed distance model):

a) A summary of the situation including the sketch

b) A quick accounting for what are the values u, n, and r for the whole thing

c) An explicit statement of what are the vectors \mathbf{x}, \mathbf{x}_0, \boldsymbol{\delta}, and \mathbf{l}_{measured} in terms of the variables they represent, and what sizes they are (even though their sizes should be clear by looking at what’s in the matrices)

d) The corresponding nonlinear functional model, \mathbf{F}(\mathbf{x},\mathbf{l}_{true}) = \mathbf{0} as a vector function, which means a set of r equations like this:

    \begin{equation*} \begin{bmatrix} f_1(\mathbf{x},\mathbf{l}_{true}) = 0 \\ f_2(\mathbf{x},\mathbf{l}_{true}) = 0 \\ \vdots \\ f_r(\mathbf{x},\mathbf{l}_{true}) = 0 \end{bmatrix} \end{equation*}

e) An expression for the design matrix, \mathbf{A}, in terms of the partial derivatives needed to derive it for the case in the sketch. Include an indication of what size it is (although, again, this should be clear from looking at it).

f) An expression for the full \mathbf{A} matrix.

g) An expression for the misclosure vector, \mathbf{w}, including an indication of its size too.

h) Expressing the full linearized model in the form: \mathbf{A}\boldsymbol{\delta} - \mathbf{e} + \mathbf{w} =\mathbf{0}

Notes:

  • Keep in mind that you’re not solving for the unknowns here. You’re only developing the linearized functional model.
  • Further, for the \mathbf{A} and \mathbf{w} matrices, you don’t need to calculate the values of each element in the matrices at this point. In fact, you don’t have all the data required for this. Rather, you’re just clearly articulating the calculations that would need to be made – so you can make them later in the lab.
  • Think of these steps as making algorithm notes that you can code.