# Main Page

### Welcome to Robotics !

This Wiki provides support for selected mathematical topics in context of the Robotics lecture. Each topic is presented in a single article which is divided in subarticles. The articles can be read separately or corresponding to the table of contents. For some of the articles there are exercises as self-check. In these cases hints are attached that lead to the exercise. Interactive three-dimensional applets and simulations illustrate the particular contents and help to understand them.

### Important topics

 Vector algebra This article gives a brief explanation of vectors and vector algebra. After a short introduction to vector algebra unit vectors and simple arithmetic operations are presented. Afterwards the dot product and the cross product are briefly explained.

 Matrices This article gives a brief explanation of matrices and basic arithmetic algebra. After a brief introduction multiplication with a scalar and computing the transpose is described. Then the approaches for addition and multiplication of matrices to each other are presented. Conclusively the minors and cofactors and the determinant of a matrix are described.

 Matrix inversion After describing the preconditions for the existence of an inverse and its definition, two procedures to determine the inverse of a matrix, the Gauß-Jordan-Algorithm and the Adjugate Formula, are introduced and clarified by examples.

 Transformations In this article general transformations used in the context of robotics and the underlying mathematics are described. The two basic types of transformation are translation and rotation. To be able to apply all types of transformations by matrix multiplication, homogeneous coordinates are introduced. Based on the two basic transformations, combinations of transformations are possible. Additionaly a special matrix inversion method is presented for inverse transformation.

 Three-Angle Representations Three angles are enough to describe the orientation of an object in three-dimensional space. But there are two different ways to define these angles, the notation of Roll-Pitch-Yaw and of Euler angles.

 Quaternions In this chapter quaternions are introduced, with which rotations can be represented. First, the basic properties of a quaternion are presented. Then pure and unit quaternions followed by the rules for addition and multiplication are explained. Quaternions are usually used to represent rotations but can also be used for the realization of transformations in general.

 Denavit-Hartenberg Convention The Denavit-Hartenberg convention covers methods to assign coordinate frames to the links of serial link manipulators and to describe the spatial relationship between them by the four Denavit-Hartenberg parameters. Based on the A-matrices, the transformation of the end-effector with respect to the manipulators base frame can easily be determined and used for further computations. Typical examples for links are presented to illustrate the usage of the convention.

### MATLAB

 MATLAB is a powerful software for mathematical computations and simulations. Accordingly the articles of this Wiki are accompanied by explanations how to use MATLAB for the particular topics of mathematics and robotics. The first MATLAB article provides some introducing information and an overview of the MATLAB articles . To facilitate a better access to the actually commercial MATLAB software, a web-based MATLAB interface is provided within this Wiki.

## Did you know... ?

... that you can determine the inverse of a matrix with the Gauß-Jordan-Algorithm? more

### Hint

The Robotics Wiki is extended continuously. For that reason some articles are provided with the hint, that they are still under development. Please consider that these articles are possibly incomplete and might contain errors.