# Introduction

Transformation matrices are used to describe the position and orientation of one coordinate frame with respect to another. The matrix is made of a position vector and a rotation matrix.

# Table of Contents

# Position Vector

A position vector is simply the coordinates of one frame compared to another. In robotics cartesian coordinates are generally used to describe position as follows:

(1)# Rotations

It is not possible to represent a rotation of coordinate systems with a vector, because the comutativity rule will not hold. This is illustrated by the book in Fig. 1 [1]. The same rotations are made for each case, just in a different order.

Figure 1 [1]

## Rotation Matrix

The rotation matrix is used to describe rotations of coordinate systems. The formulation assumes right-handed angle measurements. The rotation matrix describing the projection of coordinate $o_{1}x_{1}y_{1}z_{1}$ onto $o_{0}x_{0}y_{0}z_{0}$ is given by:

(2)Because of its form it has the following special properties:

- $\mid R \mid = 1$

- $R^{T} = R^{-1}$

- Rows and Columns are orthogonal

### Basic Rotation Matrices

Basic rotation matrices describe the rotation around one axis. The three basic rotation matrices are:

(3)### Composition of Basic Rotation Matrices

Any complex rotation can be decomposed into a number of simple one axis rotations which can be combined through multiplication. Post multiplication is used to combine rotation matrices relative to the current frame [2] :

(6)Athough less common, rotations are also possible about fixed axes, in which premultiplication is used.

# Transformation Matrix

The transformation matrix is a way to express the position and pose of a body in one matrix. Once $o_{1}^{0}$ and $R_{1}^{0}$ are know transformation matrix $T_{1}^{0}$ can be written as:

(10)Transformation matrices can be combined through post multiplication like rotation matrices.

*Analytical Dynamics*. McGraw-Hill.

*Robot Modeling and Control*. Hoboken: John Wiley & Sons, Inc.