Math 202           Study Guide for Chapter 7 and Robotic Manipulators      C.S. Davis

7.1

1.

For a given linear transformation and its standard matrix A,

a. verify the eigenvalues and eigenvectors;    P421#1

b. find the eigenvalues and eigenvectors;    P421#13,17

c. find the eigenspace for each eigenvalue;   P#13,17,55

2.

Find the eigenvalues and eigenvectors for a triangular matrix.    P422#49,51

3.

Demonstrate the Cayley-Hamilton theorem for a given matrix and its characteristic equation.   P421#33

7.2

1.

Determine whether a given matrix A (or C) is diagonalizable and, if so, find the matrix P and the diagonal matrix G such that G = P^-1A P (or P^-1CP) diagonal and tell what G is. P432#,5,13,17,19,23,27

7.3

1.

Determine whether a given matrix is symmetric, orthogonal or neither. P444#3,5,13,15,17

2.

Given a matrix A, find an orthogonal matrix P such that P^T A P diagonalizes A. P444#21,23,25

You are responsible for the subject matter in lectures on the introductory mathematics of robotic manipulators. The following topics are the most important.

1.

Form translation and rotation 4x4 matrices for homogeneous (linear) transformations.

2.

Interpret products of homogeneous transformation matrices in three ways:

a. from right to left (w/r base frame).

b. from left to right (w/r current frame).

c. as columns.

3.

Use the appropriate translation and rotation matrices (rot-trans-trans-rot) to form the A matrix for a given prismatic or revolute link in a manipulator.

4.

Construct a transform graph for a manipulator, form the transform equation and solve the transform equation for a given matrix in the equation.