Stability of an Assembly Project

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This page describes the "Stability of an Assembly" Project from the Coursera course "Modern Robotics, Course 5: Robot Manipulation and Wheeled Mobile Robots."

Program Specification

An assembly consisting of two planar rigid bodies in contact with each other and stationary ground.

You will write a program to determine if an assembly of planar rigid bodies, in frictional contact with each other, can remain standing in gravity (where gravity acts in the direction), or if the assembly must collapse. (See Example 12.11 in Chapter 12.3 of the book.) An example assembly is shown in the image at right.

Your program will take as input:

  • a description of the static mass properties of each of the bodies: the location of the center of mass and the total mass (e.g., in kg); and
  • a description of the contacts. Each contact consists of a list of the two bodies involved in the contact (0 means stationary ground, 1 means body 1, 2 means body 2, etc.); the (x,y) location of the contact; the contact normal direction into the first body involved in the contact (this direction could be specified in degrees or radians, for example); and the friction coefficient at the contact. For example, in the image shown, the single contact between body 1 and body 2 could be specified as (1, 2, 60, 60, 3.1416, 0.5). This means the contact is between body 1 and 2, the contact location is at (60,60), the angle of the contact normal pointing into the first body in the list (body 1) is at pi radians, and the friction coefficient is 0.5. One of the contacts between body 0 (ground) and body 2 could be specified as (2, 0, 72, 0, 1.5708, 0.25), indicating that the contact normal into body 2 is at pi/2 radians and the friction coefficient is 0.25.

The output of your program will be binary: it is either possible or impossible for the assembly to remain standing. Your solution method is likely to use linear programming (linprog in MATLAB, LinearProgramming in Mathematica, or scipy.optimize.linprog in Python). If standing is possible and you find a solution to the contact vector (see Example 12.11), then it is recommended that you also output . This represents one set of contact forces that would keep the assembly standing.

Unlike the form and force closure linear programming tests, where the elements of your contact vector had to be greater than or equal to a positive value, in this problem the elements of your contact vector only need to be nonnegative. (You are only trying to find one solution to the assembly equilibrium equations; in the form and force closure tests, you needed to show that the contacts could create arbitrary wrenches.)

Your biggest job in this programming assignment is taking the specifications of the bodies' mass properties and the contacts and turning them into static equilibrium equations as shown in Example 12.11. These equations are solved by linear programming with the constraint that the contact vector elements must all be nonnegative. If you have $$m$$

Testing Your Program

You will test your program on two sets of contacts, one that yields form closure and one that does not. Ideally the two sets of contacts would differ from each other by a change in just one of the contacts (either its location, its contact normal, or both), just as the sets of contacts in the image on this page differ by just one contact. But, I encourage you to come up with your own sets of contacts; don't just use the contacts shown in the image on this page. For each set of contacts, you should draw the feasible twist cone as rotation-center regions, as shown in the image on this page. Your drawings should show the coordinate frame, so the reader can understand the coordinates for the contact points, and they should confirm the answers provided by your program.

What to Submit

You will submit a single .zip file with the following contents:

  1. Your commented code in a directory called "code." Your code should be lightly commented, so it is clear to the reader what the code is doing. No need to go overboard, but keep in mind your reviewer may not be fluent in your programming language. Your code comments must include an example of how to use the code. You can write your program in any language, provided it is clearly structured, reasonably modular, easy to understand for a reader with no experience in your programming language, and with sufficient comments. Your program can span multiple files, or it can be a single file, if appropriate. If your code is in Mathematica, turn in (a) your .nb notebook file and (b) a .pdf printout of your code, so a reviewer can read your code without having to have the Mathematica software.
  2. A directory called "results". In this directory you should have pictures of the two contact cases you used for testing. Each picture should show the coordinate frame so the coordinates of the contacts are clear; each picture should show the contact locations and normal directions; and each picture should show the body's feasible twist cone as a region of rotation centers using the graphical methods from Chapter 12.1. If the body is in form closure, then there is no twist cone. This directory should also have the output logs of your program when given the input for these two cases, showing your program being called with the input corresponding to your pictures and showing the output (form closure or not form closure).
  3. (OPTIONAL) A plain text file called "README.txt" or other explanatory file. This has any other information that may help the reviewer understand your submission.