HW3 Spring 2024

Due Thursday 3/21 11:59 PM Eastern.

Instructions

In this homework, we are ready to start solving some nonlinear trajectory optimization problems. We will explore two common methods in questions one and two, followed by an open ended question where you may choose to use whichever technique you prefer to accomplish a set of control objectives.

1. Solve for a cartpole swingup trajectory using NonLinear Programming (NLP) with IPOPT via direct collocation.

2. Solve for an aggressive quadrotor aerobatic manuever using iLQR.

3. Use whichever tool(s) you prefer to generate collision free trajectories for three planar quadrotors.

NOTE: UNIT TESTS ARE NOT FULLY COMPREHENSIVE!! PASSING THEM DOES NOT GUARANTEE FULL CREDIT IF CODE IMPLEMENTATION IS INCORRECT.

Submission Instructions

Checklist

Before you submit your completed homework PDF, remember to complete all the checklist items below:

• Make sure text and code, including special characters, are completely legible.
• Make sure code is completely visible (i.e., no cropped lines). If we can't see it, then we can't grade it... Look at section below for details on remedying this.
• Make sure plots and unit tests have been outputted and rendered properly. Meshcat visuals are exempt.
• DO NOT ALTER UNIT TEST CASES. We check and can tell...
• Assign questions to each respective page of your PDF in Gradescope, including the text of the problem.

Fixing Cropped Code

If your code is cropped in the PDF, then there are multiple remedies for this:

1. Split line around binary operator (e.g., +):

L_grad = FD.gradient(f, x) +
                transpose(FD.jacobian(c, x))*λ

2. Split line around parenthesis:

L_grad = (FD.gradient(f, x)
                + transpose(FD.jacobian(c, x))*λ)

3. Split line after assignment operator (e.g., =):

L_grad = 
    FD.gradient(f, x) + transpose(FD.jacobian(c, x))*λ

4. Combine 1-3 to split into more lines:

L_grad = 
    FD.gradient(f, x) +
    transpose(FD.jacobian(c, x))*λ

5. Assign terms to more variables:

cost_grad = FD.gradient(f, x)
constraint_jac_T = transpose(FD.jacobian(c, x))

L_grad = cost_grad + constraint_jac_T*λ

Export to PDF

Feel free to use any method you'd like to export your Jupyter notebook as a PDF (with all checklist items completed) and submit on gradescope.

We recommend the following method of converting your Jupyter notebook to a PDF because it requires no additional installs (hopefully). It's slightly involved, but it is the most consistent in our experience.

1. Open the Jupyter notebook in your favorite web browser (not VS Code) with IJulia.
2. Go through the submission checklist, and make sure all relevant items are completed.
3. In the top left corner of the Jupyter menu bar, do File -> Save and Export Notebook As -> HTML. It should download an HTML file.
4. Open the downloaded HTML file in your favorite web browser.
5. Open up the browser's print menu and select Save as PDF.
6. Save and combine PDFs.
7. Submit on Gradescope, and make sure to assign the right pages to all questions.

Others

If HTML to PDF does not work, feel free to try other methods: https://mljar.com/blog/jupyter-notebook-pdf/.

Notes

• These questions will have long outputs for each cell, remember you can use cell -> all output -> toggle scrolling to better see all of the output without having to scroll.

• IPOPT can take a long time for certain problems. In Jupyter Notebooks, IPOPT will not output anything until IPOPT is finished. If you find that IPOPT is hanging for a long time, it may be that IPOPT is running and taking a very long time (which likely means something else is wrong). To see the output of IPOPT in real time, you have to call the script from the Julia REPL. To do this, you can run the following in the REPL:

using NBInclude
@nbinclude("Q1.ipynb")