Stability of neutral delay differential equations with applications in a model of human balancing

In this paper the exponential stability of linear neutral second order differential equations is studied. In contrast with many other works, coeffcients and delays in our equations can be variable. The neutral term makes this object essentially more complicated for the study. A new method for the study of stability of neutral equation based on an idea of the Azbelev W-transform has been proposed. An application to stabilization in a model of human balancing has been described. New stability tests in explicit form are proposed.

Estimation of Reaction Time During Human Balancing on Rolling Balance Board Based on Mechanical Models

Human balancing on rolling balance board in the sagittal plane is analyzed such that the geometry of the balance board can be adjusted: the radius R of the wheels and the elevation h between the top of the wheels and the board can be changed. These two parameters have a significant influence on the stability of standing on the board as shown by preliminary experiments. The human body was modeled by a single inverted pendulum, while the balance board was considered by the geometry of the mechanical model. Based on literature, it was assumed that the central nervous system (CNS) controls by signals proportional to the angle and angular velocity of the human body and the balance board and is able to tune the feedback gains with 40% accuracy during the balancing process. To take the reaction time into consideration, operation of the CNS was modeled as a delayed proportional-derivative feedback. The critical time delay for the stabilization process is defined such that if the delay is larger than the critical one then no control gains could stabilize the system. Four balance board configurations were chosen with different wheel radius and the corresponding critical time delays were computed based on the mechanical model. Eight young healthy individuals participated in the experiments. Their task was to perform 60 s long balancing trials on each balance board. The reaction time of the participants was estimated by comparing the numerical results obtained for the critical time delay and their successful and unsuccessful balancing trials. The reaction times were found to be in the range of 0.10–0.15 s which are in good agreement with the literature.

Introduction of a Complex Reaction Time Tester Instrument

The reaction time, which is also referred as reflex delay in the literature, is an important factor in human balancing, since reaction time highly affects the ability of self stabilization. Increased reaction time delay may cause dangerous fall-over accidents related to elderly people. Reaction time depends on age, health, everyday activities, the general and actual physical and mental state of the individual and the environmental conditions.The reaction time is considered as a parameter in many of the mathematical models of the neural processes in human balancing. It is beneficial in many cases to estimate the reaction time based on experimental data.The present paper introduces the prototype of a complex reaction time tester instrument. The novelty of the instrument is that the reaction time can be measured in various combinations of sensory organs and reaction movements. The reaction time is defined as the time duration in between the initial time instant of the stimulus of the sensory organs (input signal) and the onset of the response that is typically indicated by a button or a pedal. Another novelty is that the instrument is free of any uncertain time delay, which is not the case for several instruments available.Usually, human simple reaction time is considered to be roughly about 200 ms. The shortest (aural) reaction time for skilled athletes is 85ms. In our measurements the shortest reaction time was 97 ms, and the mean about 190 ms in simple reaction cases. So our collected experimental data are in agreement with the literature.

Periodic Control in a Stick Balancing Problem

Understanding the human balancing is a fundamental question. Investigation of simple tasks can help in this challenging problem. In order to describe the nature of the underlying control mechanism, first of all, the balancing force has to be determined. As a second step one can identify the behaviour of the controller. There are two main problems in the model of the whole control process of balancing, time-delay is unknown and the exact mathematical definition of the control goal is also not known. The explanation for this latter issue the classical inverted pendulum model has 2DoF but only one control forces exists, thus it can be handled as a typical underactuated mechanical system. In under-actuated systems the task of inverse dynamics is not well defined. Some degrees-of-freedom cannot directly be controlled, and the corresponding generalized coordinates depend on the system dynamics only. In this study we model the control mechanism as a time periodically (i.e. clock-driven) switched controller. We investigate the stability properties of the closed-loop system. We show a periodically switched controlled which can be a possible model of human balancing.