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.
Author(s): Toshihiro Iwai
Series: Lecture Notes in Mathematics 2289
Edition: 1
Publisher: Springer Nature Singapore
Year: 2021
Language: English
Pages: 182
City: Singapore
Tags: Geometric Mechanics, Control Theory, Cat
Preface
Contents
1 Geometry of Many-Body Systems
1.1 Planar Many-Body Systems
1.2 Rotation and Vibration of Planar Many-Body Systems
1.3 Vibrations Induce Rotations in Two Dimensions
1.4 Planar Three-Body Systems
1.5 The Rotation Group SO(3)
1.6 Spatial Many-Body Systems
1.7 Rotation and Vibration for Spatial Many-Body Systems
1.8 Local Description of Spatial Many-Body Systems
1.8.1 Local Product Structure
1.8.2 Local Description in the Space Frame
1.8.3 Local Description in the Rotated Frame
1.9 Spatial Three-Body Systems
1.10 Non-separability of Vibration from Rotation
2 Mechanics of Many-Body Systems
2.1 Equations of Motion for a Free Rigid Body
2.2 Variational Principle for a Free Rigid Body
2.3 Lagrangian Mechanics of Many-Body Systems
2.4 Hamel's Approach
2.5 Hamiltonian Mechanics of Many-Body Systems
3 Mechanical Control Systems
3.1 Electron Motion in an Electromagnetic Field
3.2 The Inverted Pendulum on a Cart
3.3 Port-Hamiltonian Systems
3.4 Remarks on Optimal Hamiltonians
4 The Falling Cat
4.1 Modeling of the Falling Cat
4.2 Geometric Setting for Rigid Body Systems
4.3 Geometric Setting for Two Jointed Cylinders
4.3.1 The Configuration Space
4.3.2 Geometric Quantities
4.3.3 Summary and a Remark on the Geometric Setting
4.4 A Lagrangian Model of the Falling Cat
4.5 A Port-Controlled Hamiltonian System
4.6 Execution of Somersaults
4.7 Remarks on Control Problems
5 Appendices
5.1 Newton's Law of Gravitation, Revisited
5.2 Principal Fiber Bundles
5.3 Spatial N-Body Systems with N≥4
5.4 The Orthogonal Group O(n)
5.5 Many-Body Systems in n Dimensions
5.6 Holonomy for Many-Body Systems
5.7 Rigid Bodies in n Dimensions
5.8 Kaluza–Klein Formalism
5.9 Symplectic Approach to Hamilton's Equations
5.10 Remarks on Related Topics
5.10.1 Quantum Many-Body Systems
5.10.2 Geometric Phases and Further Reading
5.10.3 Open Dynamical Systems and Developments
Bibliography
Index
