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

The falling cat is an interesting theme to pursue, in which geometry, mechanics, and control are in action together. The connection theory in differential geometry can provide rigorous definitions of rotation and vibration for many-body systems. This book applies the connection theory to a many-body system to show that vibrational motions can result in rotations without performing rotational motions and then to the cat model consisting of jointed rigid bodies. Three torque inputs are applied as control inputs to perform a half turn and halt the motion upon landing.

Format: Paperback / softback
Length: 182 pages
Publication date: 24 April 2021
Publisher: Springer Verlag, Singapore

The captivating subject of the falling cat brings together geometry, mechanics, and control in a fascinating interplay. It is widely recognized that cats possess an incredible ability to right themselves when thrown into the air in an upside-down position, almost always landing on their feet. However, if cats are not given a non-vanishing angular momentum at the initial instant, they cannot rotate during their motion, limiting their movement in the air to mere vibration. Yet, astonishingly, cats manage to accomplish a half turn without rotation when landing on their feet. To unravel this intriguing mystery, a deep understanding of rotations and vibrations is essential. The connection theory in differential geometry offers rigorous definitions of rotation and vibration for many-body systems, providing a valuable framework for exploring the falling cat phenomenon.

Deformable bodies, such as those of cats, present a unique challenge in mechanical treatment. A feasible approach to understanding the falling cat is to start with many-body systems and gradually progress to rigid bodies, and then 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 demonstrate that vibrational motions can lead to rotations without actual rotational motions. This geometric setting serves as the foundation for developing mechanics of many-body systems and jointed rigid bodies, taking into account the cat's ability to deform its body.

To account for the cat's deformability, three torque inputs are applied as control inputs, ensuring that the angular momentum vanishes. Following the port-controlled Hamiltonian method, a control is designed for the model cat to perform a half turn and halt its motion upon landing. Additionally, the book provides a brief review of control systems, emphasizing their significance.

Weight: 312g
Dimension: 157 x 236 x 15 (mm)
ISBN-13: 9789811606878
Edition number: 1st ed. 2021