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Geometry, Mechanics, and Control in Action for the Falling Cat

  • Book
  • © 2021

Overview

Authors:
  1. Toshihiro Iwai

    1. Kyoto University, Kyoto, Japan

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  • Gives undergraduate and graduate students alike a vivid idea of connection theory in action for many-body systems
  • Shows that geometric mechanics for many-body systems works well in dealing with the falling cat problem
  • Adapts geometric mechanics for the falling cat problem in the formulation of the port-controlled Hamiltonian system

Part of the book series: Lecture Notes in Mathematics (LNM, volume 2289)

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Table of contents (5 chapters)

  1. Front Matter

    Pages i-x
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  2. Geometry of Many-Body Systems

    • Toshihiro Iwai
    Pages 1-67

  3. Mechanics of Many-Body Systems

    • Toshihiro Iwai
    Pages 69-94

  4. Mechanical Control Systems

    • Toshihiro Iwai
    Pages 95-109

  5. The Falling Cat

    • Toshihiro Iwai
    Pages 111-136

  6. Appendices

    • Toshihiro Iwai
    Pages 137-176

  7. Back Matter

    Pages 177-184
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Keywords

About this book

The falling cat is an interesting theme to pursue, in which geometry, mechanics, and control are in action together. As is well known, cats can almost always land on their feet when tossed into the air in an upside-down attitude. If cats are not given a non-vanishing angular momentum at an initial instant, they cannot rotate during their motion, and the motion they can make in the air is vibration only. However, cats accomplish a half turn without rotation when landing on their feet.  In order to solve this apparent mystery, one needs to thoroughly understand rotations and vibrations.  The connection theory in differential geometry can provide rigorous definitions of rotation and vibration for many-body systems. Deformable bodies of cats are not easy to treat mechanically. A feasible way to approach the question of the falling cat is to start with many-body systems and then proceed to rigid bodies and, further, to jointed rigid bodies, which can approximate the body of a cat.In this book, the connection theory is applied first to a many-body system to show that vibrational motions of the many-body system can result in rotations without performing rotational motions and then to the cat model consisting of jointed rigid bodies. On the basis of this geometric setting, mechanics of many-body systems and of jointed rigid bodies must be set up. In order to take into account the fact that cats can deform their bodies, three torque inputs which may give a twist to the cat model are applied as control inputs under the condition of the vanishing angular momentum. Then, a control is designed according to the port-controlled Hamiltonian method for the model cat to perform a half turn and to halt the motion upon landing. The book also gives a brief review of control systems through simple examples to explain the role of control inputs.

Reviews

“This is a unique book, probably first of its kind … . The concepts of rotation and vibration is thoroughly examined and explained. … The appendices contain advanced material concerning many-body systems and related topics together with Newton’s law of gravitation. This is an interesting book!” (Girish Kumar Ramaiah, zbMATH 1472.70001, 2021)

Authors and Affiliations

  • Kyoto University, Kyoto, Japan

    Toshihiro Iwai

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