Balancing with Reflex Delay

Abstract

In sport, unstable equilibria of mechanical systems often have to be stabilized by the human operators, i.e. by the sportsmen. A typical basic example for this is the selfbalancing of the human body. The analysis of the problem of balancing an inverted pendulum proves that the human operator has to apply a quite complicated control strategy if he wants to achieve his goal in the presence of the time delay of his/her reflexes. It is a rule of thumb, that increasing time delay tends to destabilize any dynamical system. To avoid the instability naturally occurring in the mechanical system and also caused by the time delay, the human operator has to choose the control parameters from a narrow region which can be found only after more or less practicing. Above a critical value of the reflex delay, the balancing is impossible. We consider the simplest possible model describing the "man-machine" system when somebody places the end of a stick on his fingertip, and tries to move the lowest point of the stick in a way that its upper position should be stable. The stability analysis of the. mechanical model provides surprisingly simple and clear results in spite of the fact that modelling the time delay makes the corresponding differential equation infinite dimensional in mathematical sense. Conclusions can be derived in the following respects: - The stability chart in the space of the control parameters explains the complicated process the operator should carry out when balancing the stick. - The calculated critical delay of reflexes depends on the mechanical parameters and shows a good agreement with the experimental results. - By means of the above results we also get some in-view into the work of the organ called "labyrinthus" in the inner ear which helps in self balancing of the human body even when our eyes are closed. These conclusions may provide a good basis for developing new tests to check the sportsmen's reflex delays and balancing abilities.