Homework #4 - Skew symmetric matrices, angular velocity,
and the manipulator Jacobian
Due: Oct. 16, 2008.
This first part of this
homework will lead you through some of the more interesting
aspects of SO(3) and so(3).
If you find the results beautiful and intriguing, track down a copy of
the robotics text by Murray, Li and Sastry or the 3D Computer vision
book by Ma et al. to see more applications of the matrix
exponential and its relationship to SO(3) and so(3).
The second part of the homework is less beautiful, but perhaps more
useful. For it, you will compute several manipulator
Jacobians, and investigate singularities.
For this assignment, feel free to use a symbolic math package (e.g., Robotica)
to do the tedious algebra, but be sure to explain your work.
The Assignment
Solve the following problems
4-4
4-7 and 4-8
4-9 This one is not nearly so difficult as it may first seem.
4-10 and 4-11
4-12 This one should not require loads of algebra.
4-16, 4-17, 4-18
4-19 Also, give a geometric explanation for the singularities
of the SCARA arm.
4-20 Also, give a geometric explanation for the singularities
of the cylindrical arm.