CPSC 452 - Exercise Assignments and Hints
The exercises assigned are from J.J. Craig's text,
"Intro. to Robotics," ed. 2.
Chapter 2 Assignment Due Friday, Jan. 31, 1997
Exercises 2.1, 2.3, 2.6, 2.21, 2.22, 2.33
Hints
Exercise 2.1 - Think of the rotation matrix as operating
on the vector, P, to create first P', then P''. Write the equations relating
P to P' and P' to P''. Then substitute.
Exercise 2.3 - Draw three frames linked by arcs representing
the rotation t-forms. To get to the intermediate {B} (after the first
rotation), you use Rz(theta). Then do same for final {B}. The
product of the R's moving from the original frame, {A}, to {B} is
AB(R) (R super A sub B). Now you need to t-form B(P) to A(P).
You either need AB(R) or its transpose - you decide. Incidentally,
this problem should look a lot like the derivation of the rotation
matrix for ZYX Euler angles.
Exercise 2.6 - There are many ways to do this problem.
Here are 3 suggestions on how to get started:
- Create a frame which aligns it's x-axis on k, i.e. write the
x-, y-, and z-axes in terms of the elements of k. These axes form
a rotation matrix AK(R) taking you from {A} to {K}. Rotate from {K}
using Rx(theta) to a new frame {K'}. The last step is to realize
that the frame you are trying to get to ({A} rotated about k by theta)
must have the same relative t-form to {K'} that the original {A} had
to {K}. Thus using t-forms relating a moving frame starting with {A},
we get Rk(theta) = AK(R) Rx(theta) KA(R). Grind it out and simplify
with trigonometry to get the deisred result.
- Similar idea to the first approach, but AK(R) is constructed by
two rotations, namely: rotation about z until k lies in x-z plane;
rotation about y until k coincides with z; then Rz(theta); inverse
of second rotation; and finally inverse of first rotation. The
sines and cosines of the two angles of rotation needed for the first
two rotations can be found as functions of the elements of k.
Expand and simplify to get the desired result.
- Use geometry to determine the direction cosines of each of the
axes of {A} after rotation about {K} by theta. Rodrigue's Formula
makes this problem trivial.
Exercise 2.21 - Your rotation matrix should be a
function of theta. If you end up with the identity matrix, you
should reread the question.
Exercise 2.22 - Make sure you use 2 different axes
of rotation.
Exercise 2.33 - You should be able to do this
by inspection with maybe a cross product to find the direction cosines
of one axis.