Example 10

Goal

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

Situation

In this case, consider the situation where an unknown station 3 is visible from two known stations 1 and 2 and where the angles \theta_{23} and \theta_{13} are measured, as shown below.

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. Where it simplifies things however, you may take the results obtained in earlier examples without re-deriving them.

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

Guidance

Because you’re bringing together several models here, this is another situation where I’d offer the same guidance I did at the bottom of the section on Linearizing the model for a simple trilateration.