In this first part of this lab, you’re asked to work with pen and paper to derive a number of functional models that are key to geomatics networks, and then to linearize them. This part of the lab prepares you for Part 2, but it has also been designed to help you succeed on your related quiz.

Even though I want to see you do this work from scratch, I would recommend checking out your textbook(s), our lecture notes, and the topics linked below. No point in reinventing the wheel when the goal is to get intimate with it and how it’s used.

Step 1: Review and redo the distance model

In class we derived and linearized the functional model for the case where the distance is measured between two unknown points. I want you to start this lab by reviewing the notes you took in class and by reviewing this online version of the same. Then I want you to write out the full solution as the first part in your submission for Part 1 of this lab.

In general in Part 1 of this lab, I want you to be as explicit about showing your work as I was in that example, 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. Think of what you’re doing as creating a reference manual for yourself.

Step 2: Derive and linearize the basic trilateration model

Guidance and directions for this are found here. Here you’ll practice extending the model from Step 1 for the case where the distance is measured from three known points to an unknown point – a simple trilateration.

Step 3: Derive and linearize the azimuth model

Guidance and directions for this are found here. You’ll develop a model for the case where an azimuth is to be measured from one point to another, again assuming the more general case where the coordinates of both points are unknown.

Step 4: Derive and linearize the angle model

Guidance and directions for this are found here. You’ll model the situation where an angle is measured between two points from a third point, as the difference of two observed azimuths from Step 3.

Step 5: Derive and linearize the basic intersection model

Guidance and directions for this are found here. You’ll model the case where two angles (Step 4) are measured from the line connecting two known points, to a third point with unknown coordinates – what we refer to as a simple intersection model.

Step 6: Go deeper on our width of the room example

In question 5 of the self-assessment problems found here I set out the challenge to explore the use of least squares to estimate the width of the room. I’d like you to answer the questions posed there and submit them as part of this lab. Some of you may have done this in Lab 1. If that’s the case then please just summarize your results.

Step 7: Develop some coding guidance

Look back at the work you’ve done so far and develop some coding guidance for your future self. Under the heading “Guidance I have for myself on coding this” in your report, I want you to answer questions such as:

  • What should be put into functions (vs. the main program) if I’m going to develop code that spits out these linearized observation equations in a way I can use them?
  • In doing so, what functions should I create?
  • What input and output should those functions have if the goal (for now) is to augment \mathbf{A} and \mathbf{w} with each added measurement?
  • Other considerations for coding?

Once it’s released, you may look ahead to Part 2 of this lab to help catalyze your thinking here.

Step 8: Submit your work

When you’ve completely finished this part you need to submit it.

To do this, submit it according to the directions provided. In short, I would expect your report to contain the following:

  • Introductory comments and preamble, as required by the report template
  • Comments referring to the appendix which can contain your work.

Step 9: Move to Part 2 of this lab when you’re ready

Once you’ve finished all of this you can proceed to Part 2 in the next lesson using the buttons below.