## ECE 470: Introduction to Robotics Homework #2

### More Properties of SO(3)

These problems will explore in a bit more depth some properties of rotation matrices
• Solve problem 2-19. This problem investigates the meaning of the Eigenvalues and Eigenvectors of a rotation matrix.
• Solve problem 2-25. This problem deals with the number of ways to parameterize a rotation using three angles.
• Solve problem 2-43. This problem investigates the commutativity of composition of rotations.

### Composition of Rotations

• Solve problems 2-10, 2-11, 2-12, and 2-13. The problems will give you practice with the composition of transformations.
• Solve problems 2-14 and 2-15.

### Axis/Angle Parameterization of SO(3)

• Solve problems 2-21 and 2-23.

### Complex Number Parameterization of SO(2)

• Solve problems 2-26 and 2-27. This problem shows how complex numbers can be used to represent orientation in the plane.
• Solve problems 2-28, 2-29, 2-30. These problems show how complex numbers can be generalized to quaternions, and how quaternions can be used to represent rotations.
• Solve problems 2-31, 2-32 and 2-33. These problems introduce basic algebraic manipulations of quaternions.
• Solve problems 2-34 and 2-35. These problems show how quaternions can be used to perform coordinate transformations.