Several people used the Lagrangian method to derive the dynamics but used the wrong calculation for the moment of inertia. When calculating the kinetic energy of a body, use the following equation:
This is just the sum of the translational and rotational kinetic energy. The moment of inertia I should be calculated about the center of mass. Where people erred was in calculating the moment of inertia about the inboard joint. To see why this is wrong, consider a single-link pendulum consisting of a thin rod of uniform density with mass M and length L. See this hyperphysics page. The moment of inertia about the end is
So that the total energy is
The reason you can't do this with multi-joint arms is that since the rest of the arm is moving, you are not instantaneously rotating about the joint. We must separately compute the energy due to linear and rotational motion. The position of the center of mass is
Its velocity is
So the total energy of the system is
Which simplifies to
Which matches the previous calculation. This offers some insight into the the parallel axis theorem
See the file ke1.m for a method to do this automatically in Matlab. Very little need be done by hand, and one thing I want you to learn from this class is how to use software to make mistakes impossible. That brings us to the second mistake people made, which was in deriving equations manually. Don't do this! Dr. Atkeson's pages and the material posted on these pages should have enough to teach you how to use Mathematica or Matlab to avoid making mistakes in deriving your dynamics.
I was disappointed in the effort most people put into verifying their dynamics.
A few people attempted the extra credit, but not all of them gave a satisfactory answer. The truth is that the finger constraint in the assignment oversimplifies the biology. You can bend the main joint in your finger without bending the most distal joint much, but the converse is not true. This assignment has an enlightening picture.
Remarks on grading: Almost all assignments made a mistake on the dynamics or at least did an unconvincing job of verify them. Some people whose work was interesting got penalized for sloppy or incomplete writeups. I want to see thorough writeups that clearly document the decisions made and the details of the approaches taken. Don't be shy to include lots of plots, but say something about each of them. I graded using letters for this assignment.
- A
- 3
- A-
- 3
- B+
- 1
- B
- 3