This book demonstrates how delay differential equations (DDEs) can be used to compliment the laboratory investigation of human balancing tasks. This approach is made accessible to non-specialists by comparing mathematical predictions and experimental observations. For example, the observation that a longer pole is easier to balance on a fingertip than a shorter one demonstrates the essential role played by a time delay in the balance control mechanism. Another balancing task considered is postural sway during quiet standing.
With the inverted pendulum as the driver and the feedback control depending on state variables or on an internal model, the feedback can be identified by determining a critical pendulum length and/or a critical delay. This approach is used to identify the nature of the feedback for the pole balancing and postural sway examples. Motivated by the question of how the nervous system deals with these feedback control challenges, there is a discussion of ‘’microchaotic’’ fluctuations in balance control and how robust control can be achieved in the face of uncertainties in the estimation of control parameters. The final chapter suggests some topics for future research.
Each chapter includes an abstract and a point-by-point summary of the main concepts that have been established. A particularly useful numerical integration method for the DDEs that arise in balance control is semi-discretization. This method is described and a MATLAB template is provided.
This book will be a useful source for anyone studying balance in humans, other bipedal organisms and humanoid robots. Much of the material has been used by the authors to teach senior undergraduates in computational neuroscience and students in bio-systems, biomedical, mechanical and neural engineering.
Author(s): Tamás Insperger, John Milton
Series: Lecture Notes on Mathematical Modelling in the Life Sciences
Publisher: Springer
Year: 2021
Language: English
Pages: 161
City: Cham
Preface
Acknowledgements
Contents
Chapter 1 Introduction
1.1 Organization
1.2 Resources
Chapter 2 Background
2.1 The Inverted Pendulum
2.2 Time-delayed Feedback Control
2.3 Stability Analysis
2.3.1 PD Feedback
2.3.2 PDA Feedback
2.3.3 Predictor Feedback
2.4 Critical Parameters
2.4.1 PD Feedback
2.4.2 PDA Feedback
2.4.3 Predictor Feedback
2.5 Summary
Chapter 3 Pole Balancing at the Fingertip
3.1 Pendulum-cart Model
3.1.1 The Control Problem: Stabilization Angular Displacement
3.1.2 Equations of Motion
3.1.3 Estimation of m0
3.1.4 Physical Constraints
3.1.5 Measurement of τ
3.2 Feedback Identification
3.3 Skill Acquisition
3.4 Over-control
3.5 Summary
Chapter 4 Sensory Dead Zones: Switching Feedback
4.1 Time Scales for Balance Control
4.1.1 Vertical Displacement Angle
4.1.2 Fingertip Speed
4.2 Sensory Dead Zones
4.2.1 Dead Zones in Pole Balancing
4.2.2 Estimating the Sensory Dead Zone
4.2.3 Dead Zone Benefits
4.3 Model: Pole Balancing on the Fingertip
4.3.1 Feedback Identification
4.3.2 Edge of Stability
4.3.3 Why the Pole Falls?
4.4 Intermittent Control
4.5 Summary
Chapter 5 Microchaos in Balance Control
5.1 Semi-discretization
5.2 First-order Models
5.3 Quail Map
5.4 Microchaotic Map
5.4.1 Permanent Microchaos
5.4.2 Transient Microchaos
5.5 Hayes Equation
5.6 Postural Sway: Eurich-Milton Model
5.6.1 Case 1: Continuous Control (R→∞)
5.6.2 Case 2: Digital Control (R = 0)
5.6.3 Case 3: Semi-discretized Control (0 < R < ∞)
5.7 Bifurcations
5.8 Summary
Chapter 6 Postural Sway during Quiet Standing
6.1 Postural Sway
6.2 Inverted Pendulum Models for Postural Sway
6.3 Sensory Dead Zone
6.4 Time Delay
6.5 “Pinned” Inverted Pendulum Model
6.6 Sensory Dead Zones and Torque Saturation
6.7 Chaotic Sway
6.8 Frontal Plane Balance Control: Stance Width
6.9 Summary
Chapter 7 Stability Radii and Uncertainty in Balance Control
7.1 Rectangular Tiling
7.2 D-curve Slicing
7.3 ε-Pseudospectrum
7.4 Comparison of the Three Approaches
7.5 Measuring Uncertainty Radii
7.6 Stability Radii for Frontal Plane Balance as Stance Width Changes
7.7 Summary
Chapter 8 Challenges for the Future
8.1 Derivative Control
8.2 Different Feedback Delays in the Feedback Loop
8.3 Act-and-Wait Control
8.4 Ball and Beam Balancing
8.5 Balancing on Balance Boards
8.6 Skill Acquisition
8.7 Stochastic Perturbations
8.8 Falls
Appendix A Semi-discretization Method
Appendix B Stability Radii: Some Mathematical Aspects
References
Index
