Human performance in virtual stabilization of a fractional-order system with reaction delay.

Virtual balancing tasks facilitate the study of human motion control: human reaction to the change of artificially introduced parameters can be studied in a computer environment. In this article, the dynamics of human stick balancing are generalized using fractional-order derivatives. Reaction delay sets a strong limitation on the length of the shortest stick that human subjects can balance. Human processing of visual input also exhibits a memory effect, which can be modelled by fractional-order derivatives. Therefore, we hypothesize a delayed fractional-order PD control of the unstable fractional-order process. The resulting equation of motion is investigated in a dimensionless framework, and stabilizability limits are determined as a function of the dynamics's order. These theoretical limits are then compared with the results of a systematic series of virtual balancing tests performed by 18 subjects. The comparison shows that the theoretical stabilizability limits for controllers with fixed fractional order correspond to the measured data points. The best fit is obtained if the fractional order of the underlying control law is 0.475.

Virtual stick balancing: skill development in Newtonian and Aristotelian dynamics.

Human reaction delay significantly limits manual control of unstable systems. It is more difficult to balance a short stick on a fingertip than a long one, because a shorter stick falls faster and therefore requires faster reactions. In this study, a virtual stick balancing environment was developed where the reaction delay can be artificially modulated and the law of motion can be changed between second-order (Newtonian) and first-order (Aristotelian) dynamics. Twenty-four subjects were separated into two groups and asked to perform virtual stick balancing programmed according to either Newtonian or Aristotelian dynamics. The shortest stick length (critical length, Lc) was determined for different added delays in six sessions of balancing trials performed on different days. The observed relation between Lc and the overall reaction delay τ reflected the feature of the underlying mathematical models: (i) for the Newtonian dynamics Lc is proportional to τ2; (ii) for the Aristotelian dynamics Lc is proportional to τ. Deviation of the measured Lc(τ) function from the theoretical one was larger for the Newtonian dynamics for all sessions, which suggests that, at least in virtually controlled tasks, it is more difficult to adopt second-order dynamics than first-order dynamics.

Virtual stick balancing: sensorimotor uncertainties related to angular displacement and velocity.

Sensory uncertainties and imperfections in motor control play important roles in neural control and Bayesian approaches to neural encoding. However, it is difficult to estimate these uncertainties experimentally. Here, we show that magnitude of the uncertainties during the generation of motor control force can be measured for a virtual stick balancing task by varying the feedback delay, τ. It is shown that the shortest stick length that human subjects are able to balance is proportional to τ  2. The proportionality constant can be related to a combined effect of the sensory uncertainties and the error in the realization of the control force, based on a delayed proportional-derivative (PD) feedback model of the balancing task. The neural reaction delay of the human subjects was measured by standard reaction time tests and by visual blank-out tests. Experimental observations provide an estimate for the upper boundary of the average sensorimotor uncertainty associated either with angular position or with angular velocity. Comparison of balancing trials with 27 human subjects to the delayed PD model suggests that the average uncertainty in the control force associated purely with the angular position is at most 14% while that associated purely with the angular velocity is at most 40%. In the general case when both uncertainties are present, the calculations suggest that the allowed uncertainty in angular velocity will always be greater than that in angular position.