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Über dieses Buch

This book presents a three-dimensional model of the complete unicycle–unicyclist system. A unicycle with a unicyclist on it represents a very complex system. It combines Mechanics, Biomechanics and Control Theory into the system, and is impressive in both its simplicity and improbability. Even more amazing is the fact that most unicyclists don’t know that what they’re doing is, according to science, impossible – just like bumblebees theoretically shouldn’t be able to fly.

This book is devoted to the problem of modeling and controlling a 3D dynamical system consisting of a single-wheeled vehicle, namely a unicycle and the cyclist (unicyclist) riding it. The equations of motion are derived with the aid of the rarely used Boltzmann–Hamel Equations in Matrix Form, which are based on quasi-velocities. The Matrix Form allows Hamel coefficients to be automatically generated, and eliminates all the difficulties associated with determining these quantities. The equations of motion are solved by means of Wolfram Mathematica. To more faithfully represent the unicyclist as part of the model, the model is extended according to the main principles of biomechanics. The impact of the pneumatic tire is investigated using the Pacejka Magic Formula model including experimental determination of the stiffness coefficient.

The aim of control is to maintain the unicycle–unicyclist system in an unstable equilibrium around a given angular position. The control system, based on LQ Regulator, is applied in Wolfram Mathematica.

Lastly, experimental validation, 3D motion capture using software OptiTrack – Motive:Body and high-speed cameras are employed to test the model’s legitimacy. The description of the unicycle–unicyclist system dynamical model, simulation results, and experimental validation are all presented in detail.



Chapter 1. Introduction

This chapter provides crucial information that was found to be relevant to the present study. The related information regarding the unicycle, such as design of particular types, unicycle performance during riding and a brief history of the unicycle evolution, is presented. Additionally, technical aspects of the unicycle approach in designing and controlling of walking robots are discussed.
Michał Niełaczny, Barnat Wiesław, Tomasz Kapitaniak

Chapter 2. Model of the Unicycle-Unicyclist System

In this chapter, a complete model of the unicycle-unicyclist system is introduced. The classical mechanics approach based on Euler angles and the Boltzmann–Hamel equation is applied to derive the equations of motion. To identify the geometry of the system, 3-D scans of the unicyclist body were made. In tyre modelling, the Pacejka theory is used. Additionally, the control performed by unicyclist to maintain the vertical position is considered.
Michał Niełaczny, Barnat Wiesław, Tomasz Kapitaniak

Chapter 3. Numerical and Experimental Validation of the Model

Validation is a documented procedure aimed at confirming that all the processes, equipment, materials, procedures, activities and systems actually lead to the planned results. First, the model developed in Chap. 2 was validated in the numerical simulations and next, on the basis of the experimental data obtained during a real unicycle ride. Due to the complexity of the system, before the 3-D validation was used, the 2-D one had been performed in order to verify the direction of investigations and simulations.
Michał Niełaczny, Barnat Wiesław, Tomasz Kapitaniak

Chapter 4. Concluding Remarks

The first author was delighted to investigate his point of interest from a scientific perspective and to convince everyone that passion and science can go hand in hand.
Michał Niełaczny, Barnat Wiesław, Tomasz Kapitaniak


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