# Preliminary Mathematical Concepts

• Pål Johan From
• Jan Tommy Gravdahl
• Kristin Ytterstad Pettersen
Chapter
Part of the Advances in Industrial Control book series (AIC)

## Abstract

This chapter presents the main mathematical tools used throughout the book. The necessary mathematical background is presented, which in turn will be used to develop mathematical models that accurately describe the mechanical behavior of our systems, which is the main goal of mathematical modeling. The models are derived so that they are well suited for analysis, simulation, and control of the mechanical system.

In this chapter the most important tools from Lie theory and differential geometry are presented in a brief but concise manner. The theory is presented in such a way that it can serve as an introduction to differential geometry for roboticists not familiar with the topic. We focus on the tools and properties required for modeling mechanical systems with focus on presenting the reader with a well-defined framework well suited to derive a singularity-free formulation of the kinematics and dynamics of complex mechanical systems. Even though the main mathematical tools are presented in this chapter, the readers that are not interested in a deep mathematical analysis of these systems, but rather a more practical approach, can go straight to Chap.  on rigid body modeling.

## Keywords

Reference Frame Rigid Body Multibody System Configuration Space Inertial Frame
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© Springer-Verlag London 2014

## Authors and Affiliations

• Pål Johan From
• 1
• Jan Tommy Gravdahl
• 2
• Kristin Ytterstad Pettersen
• 2
1. 1.Department of Mathematical Sciences and TechnologyNorwegian University of Life SciencesÅsNorway
2. 2.Department of Engineering CyberneticsNorwegian Univ. of Science & TechnologyTrondheimNorway