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The following are some recommended applied self-assessment questions for the lesson called Linearizing the parametric functional model. They’re intended for you to work through to test your own understanding of the key concepts we covered there.

Question 1

If you have the following functional model:

l_{true} = f(x) = 3x + 6x^2 + 9x^3 in mm

What is the corresponding linearized form \mathbf{A}\boldsymbol{\delta} - \mathbf{e} + \mathbf{w} =\mathbf{0} if:

x^0 = 1.04 mm

l_{meas} = 20.203625 mm

Because there is only one measurement here, what I’m really asking for is for you to write out A\delta - e + w =0 (which is the same as A(x-x^0) - (l_{meas}-l_{true}) + w =0), i.e. the work here is in obtaining numerical values for A and w. Please show your work.

I get the following: A = – 44.683200 (unitless) and w = 0.470249 mm.

Question 2

Imagine the following situation where we want to estimate the unknown coordinates of point 4:

Imagine that we know the coordinates of the other points to be as follows:

    \begin{equation*} \begin{bmatrix} E_1 \\ N_1 \\ \end{bmatrix} = \begin{bmatrix} -87.413 \\ 56.988 \\ \end{bmatrix} m \end{equation*}

    \begin{equation*} \begin{bmatrix} E_2 \\ N_2 \\ \end{bmatrix} = \begin{bmatrix} 153.156 \\ 159.927 \\ \end{bmatrix} m \end{equation*}

    \begin{equation*} \begin{bmatrix} E_3 \\ N_3 \\ \end{bmatrix} = \begin{bmatrix} 124.599 \\ -29.212 \\ \end{bmatrix} m \end{equation*}

And we know the measured distances to be as follows:

    \begin{equation*} \mathbf{l}_{meas} = \begin{bmatrix} d_{14} \\ d_{24} \\ d_{34} \\ \end{bmatrix} + \mathbf{e} = \begin{bmatrix} 152.416 \\ 146.095 \\ 94.651 \\ \end{bmatrix} m \end{equation*}

a) In this situation, what are n, r, and u?

b) What is the functional model that applies for each row of the vector function \mathbf{F}, i.e. for each measurement? Write this in terms of general points A and B as we did in the example in the topic video.

c) Is this linear or non-linear in \mathbf{x}?

d) What is the full linearized form of the functional model in a)? This should have the form: \mathbf{A}\boldsymbol{\delta} - \mathbf{e} + \mathbf{w} =\mathbf{0} and should be written out as set of vectors and matrices into which you could plug in and compute the numerical values.

e) Write out the same linearized model, but now with the numerical values corresponding to the given situation. Here you can use the following approximate coordinates for point 4:

    \begin{equation*} \mathbf{x}^0 = \begin{bmatrix} E_4^0 \\ N_4^0 \\ \end{bmatrix} = \begin{bmatrix} 64 \\ 43 \\ \end{bmatrix} m \end{equation*}

I get the following linearized form of this functional model:

    \begin{equation*} \begin{bmatrix} -0.9958 & 0.0920 \\ 0.6063 & 0.7952 \\ 0.6428 & -0.7660 \\ \end{bmatrix} \boldsymbol{\delta} - \mathbf{e} + \begin{bmatrix} 0.3582 \\ -0.9448 \\ 0.3811 \\ \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \\ \end{bmatrix} \end{equation*}

which doesn’t need to be but could also be written more explicitly as follows (I just find expanding the terms helpful as a reminder of what they mean):

    \begin{equation*} \begin{bmatrix} -0.9958 & 0.0920 \\ 0.6063 & 0.7952 \\ 0.6428 & -0.7660 \\ \end{bmatrix} \begin{bmatrix} \delta E_4 \\ \delta N_4 \\ \end{bmatrix} - \begin{bmatrix} e_{14} \\ e_{24} \\ e_{34} \\ \end{bmatrix} + \begin{bmatrix} 0.3582 \\ -0.9448 \\ 0.3811 \\ \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \\ \end{bmatrix} \end{equation*}

If you’re one of my students, then you’re expected to answer these on your own and submit them according to the directions provided in class, i.e. you don’t  need to submit them through this website. Don’t forget that our TA and I are both here to help you in the associated lab (and/or tutorial) sessions.

I want you to put your answers to these conceptual questions into a single document. I don’t mind if you hand write it or type it out – do what works best for you.

But I want you to use the same document for all of the self-assessment questions you do before the due date, i.e. you will be asked to hand them all in together. This means you should keep things well organized with clear headings so you (and your TA) can figure out which solutions refer to which questions. Good self-assessment documents use headings and some even provide the links, e.g. the URL for this page. This helps you keep track and go back and forth quickly between the problems and solutions.

Also, keep in mind that you will be asked to submit them through D2L, so you’ll have to scan any handwritten documents.

You can click through to other self-assessments or lessons (if any) using the button below, and return here whenever you wish.