Saeed B. Niku - Introduction to Robotics: Analysis, Control, Applications, 3rd Edition

Throughout this book, we use matrices to represent coordinates, frames, objects, and motions. In this appendix, certain characteristics of matrices that we need in our calculations are reviewed. You must already have an understanding of matrix algebra to understand the use of matrices. Therefore, only a simple review is presented here.

A matrix is a collection of m rows and n columns of values, represented in a bracket. The dimensions of the matrix are , and each element of the matrix is referred to as Aij. A matrix whose number of rows and columns are the same is called a square matrix.

The transpose of a matrix Aji is another matrix where elements of each row and column are replaced, as shown:

(A.1)

Matrices can be multiplied by multiplying all the elements of each row by each column and replacing the summation in the corresponding row/column location, as follows:

(A.2)

As you see, the result of an () matrix multiplied by an () matrix is an () matrix. Therefore, the number of columns of the first matrix must be equal to the number of rows of the second matrix. Also, remember that unlike regular algebra, the order of multiplication of matrices may not be changed. In other words, . This can easily be demonstrated by the fact that if A is a () matrix and B is a () matrix, then will yield a (22) matrix, whereas BA will result in a (3 3) matrix, which obviously is different. However, if more than two matrices are to be multiplied, although their order cannot be changed, the result is independent of which pairs of matrices are multiplied first. As a result, the following is true:

(A.3)

(A.4)

(A.5)

A diagonal matrix is a matrix where all, except the diagonal, elements of the matrix are zero. If all diagonal elements are 1, the matrix is a unit matrix, which effectively acts as a 1. Premultiplying or postmultiplying any matrix by a unit matrix results in the same matrix.

Matrix addition can be accomplished by adding each element of one matrix by the corresponding element of the other matrices. Unlike matrix multiplication, addition of matrices is commutative; the order of addition is not important. Obviously, the dimensions of all matrices must be exactly the same for addition. Therefore:

(A.6)

(A.7)

A vector is a onedimensional matrix, either a (1 m) or an (n 1) matrix.

The determinant of a matrix can be calculated as follows:

1. Pick one row or column.
2. Multiply each element in the chosen row or column by the determinant of the matrix that remains after the corresponding row and column of the element are dropped from the matrix, each one with an alternating plus or minus sign.

Calculate the determinant of the following matrix:

First, choose a row or column. In this example, we will pick the first row. The determinant of the matrix is:

This is an important operation in matrix representation of robots. We will use matrix inversions for both inverse kinematics and for differential motions. In this section, two generalpurpose inversion techniques for square matrices are mentioned.

The inverse of a matrix is another matrix such that if the matrix is multiplied by the inverse, the result is a unit matrix. In general, a matrix either has a left inverse or a right inverse. If (where I is a unit matrix), is called a right inverse. If , then is called a left inverse. Generally, the left and right inverse matrices are not the same. However, a square matrix will have the same left and right inverse, such that . In this case,