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.

Pole balancing on the fingertip: model-motivated machine learning forecasting of falls.

Introduction: There is increasing interest in developing mathematical and computational models to forecast adverse events in physiological systems. Examples include falls, the onset of fatal cardiac arrhythmias, and adverse surgical outcomes. However, the dynamics of physiological systems are known to be exceedingly complex and perhaps even chaotic. Since no model can be perfect, it becomes important to understand how forecasting can be improved, especially when training data is limited. An adverse event that can be readily studied in the laboratory is the occurrence of stick falls when humans attempt to balance a stick on their fingertips. Over the last 20 years, this task has been extensively investigated experimentally, and presently detailed mathematical models are available. Methods: Here we use a long short-term memory (LTSM) deep learning network to forecast stick falls. We train this model to forecast stick falls in three ways: 1) using only data generated by the mathematical model (synthetic data), 2) using only stick balancing recordings of stick falls measured using high-speed motion capture measurements (human data), and 3) using transfer learning which combines a model trained using synthetic data plus a small amount of human balancing data. Results: We observe that the LTSM model is much more successful in forecasting a fall using synthetic data than it is in forecasting falls for models trained with limited available human data. However, with transfer learning, i.e., the LTSM model pre-trained with synthetic data and re-trained with a small amount of real human balancing data, the ability to forecast impending falls in human data is vastly improved. Indeed, it becomes possible to correctly forecast 60%-70% of real human stick falls up to 2.35 s in advance. Conclusion: These observations support the use of model-generated data and transfer learning techniques to improve the ability of computational models to forecast adverse physiological events.

Controlling stick balancing on a linear track: Delayed state feedback or delay-compensating predictor feedback?

A planar stick balancing task was investigated using stabilometry parameters (SP); a concept initially developed to assess the stability of human postural sway. Two subject groups were investigated: 6 subjects (MD) with many days of balancing a 90 cm stick on a linear track and 25 subjects (OD) with only one day of balancing experience. The underlying mechanical model is a pendulum-cart system. Two control force models were investigated by means of numerical simulations: (1) delayed state feedback (DSF); and (2) delay-compensating predictor feedback (PF). Both models require an internal model and are subject to certainty thresholds with delayed switching. Measured and simulated time histories were compared quantitatively using a cost function in terms of some essential SPs for all subjects. Minimization of the cost function showed that the control strategy of both OD and MD subjects can better be described by DSF. The control mechanism for the MD subjects was superior in two aspects: (1) they devoted less energy to controlling the cart's position; and (2) their perception threshold for the stick's angular velocity was found to be smaller. Findings support the concept that when sufficient sensory information is readily available, a delay-compensating PF strategy is not necessary.

Predictor feedback models for stick balancing with delay mismatch and sensory dead zones.

Human stick balancing is investigated in terms of reaction time delay and sensory dead zones for position and velocity perception using a special combination of delayed state feedback and mismatched predictor feedback as a control model. The corresponding mathematical model is a delay-differential equation with event-driven switching in the control action. Due to the sensory dead zones, initial conditions of the actual state cannot always be provided for an internal-model-based prediction, which indicates that (1) perfect prediction is not possible and (2) the delay in the switching condition cannot be compensated. The imperfection of the predictor is described by the delay mismatch, which is treated as a lumped parameter that creates a transition between perfect predictor feedback (zero delay mismatch) and delayed state feedback (mismatch equal to switching delay). The maximum admissible switching delay (critical delay) is determined numerically based on a practical stabilizability concept. This critical delay is compared to a realistic reference value of 230 ms in order to assess the possible regions of the threshold values for position and velocity perception. The ratio of the angular position and angular velocity for 44 successful balancing trials by 8 human subjects was used to validate the numerical results. Comparison of actual human stick balancing data and numerical simulations based on the mismatched predictor feedback model provided a plausible range of parameters: position detection threshold 1°, velocity detection threshold between 4.24 and 9.35°/s, and delay mismatch around 100-150 ms.

Critical delay as a measure for the difficulty of frontal plane balancing on rolling balance board.

The relation between balancing performance on rolling balance board and reaction time is investigated. Ten young healthy adults performed balancing trials on a rolling balance board with different wheel radius R and stance widths d in the frontal plane. A 2- and a 3-degree-of-freedom models subject to delayed state feedback with a single lumped reaction delay were created in order to describe hip and hip-lumbar strategies. The critical delays of the underlying models, for which state feedback stabilization is still possible, were determined. This critical delay is a good measure of the difficulty of the balancing task: the smaller the critical delay, the more difficult the balancing task. Subjects' response time to visual stimuli correlates with the critical delay of the models, therefore, it can be used as an indirect indicator for balancing skill. Hip-lumbar strategy was found to be superior to pure hip strategy in the sense that it allows larger reaction delay. Overall, rolling balance board is a simple and convenient tool to assess human's balancing skill. The difficulty of the task can be tuned by employing different wheel radius with different stance widths while it can be measured by a single number: the critical delay of the underlying model. Improvement in balancing skill therefore can easily be monitored during balance therapy and physical rehabilitation.

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.

Response to perturbation during quiet standing resembles delayed state feedback optimized for performance and robustness.

Postural sway is a result of a complex action-reaction feedback mechanism generated by the interplay between the environment, the sensory perception, the neural system and the musculation. Postural oscillations are complex, possibly even chaotic. Therefore fitting deterministic models on measured time signals is ambiguous. Here we analyse the response to large enough perturbations during quiet standing such that the resulting responses can clearly be distinguished from the local postural sway. Measurements show that typical responses very closely resemble those of a critically damped oscillator. The recovery dynamics are modelled by an inverted pendulum subject to delayed state feedback and is described in the space of the control parameters. We hypothesize that the control gains are tuned such that (H1) the response is at the border of oscillatory and nonoscillatory motion similarly to the critically damped oscillator; (H2) the response is the fastest possible; (H3) the response is a result of a combined optimization of fast response and robustness to sensory perturbations. Parameter fitting shows that H1 and H3 are accepted while H2 is rejected. Thus, the responses of human postural balance to "large" perturbations matches a delayed feedback mechanism that is optimized for a combination of performance and robustness.

The effects of sensory quantization and control torque saturation on human balance control.

The effect of reaction delay, temporal sampling, sensory quantization, and control torque saturation is investigated numerically for a single-degree-of-freedom model of postural sway with respect to stability, stabilizability, and control effort. It is known that reaction delay has a destabilizing effect on the balancing process: the later one reacts to a perturbation, the larger the possibility of falling. If the delay is larger than a critical value, then stabilization is not even possible. In contrast, numerical analysis showed that quantization and control torque saturation have a stabilizing effect: the region of stabilizing control gains is greater than that of the linear model. Control torque saturation allows the application of larger control gains without overcontrol while sensory quantization plays a role of a kind of filter when sensory noise is present. These beneficial effects are reflected in the energy demand of the control process. On the other hand, neither control torque saturation nor sensory quantization improves stabilizability properties. In particular, the critical delay cannot be increased by adding saturation and/or sensory quantization.

Rolling balance board of adjustable geometry as a tool to assess balancing skill and to estimate reaction time delay.

The relation between balancing performance and reaction time is investigated for human subjects balancing on rolling balance board of adjustable physical parameters: adjustable rolling radius R and adjustable board elevation h. A well-defined measure of balancing performance is whether a subject can or cannot balance on balance board with a given geometry (R, h). The balancing ability is linked to the stabilizability of the underlying two-degree-of-freedom mechanical model subject to a delayed proportional-derivative feedback control. Although different sensory perceptions involve different reaction times at different hierarchical feedback loops, their effect is modelled as a single lumped reaction time delay. Stabilizability is investigated in terms of the time delay in the mechanical model: if the delay is larger than a critical value (critical delay), then no stabilizing feedback control exists. Series of balancing trials by 15 human subjects show that it is more difficult to balance on balance board configuration associated with smaller critical delay, than on balance boards associated with larger critical delay. Experiments verify the feature of the mechanical model that a change in the rolling radius R results in larger change in the difficulty of the task than the same change in the board elevation h does. The rolling balance board characterized by the two well-defined parameters R and h can therefore be a useful device to assess human balancing skill and to estimate the corresponding lumped reaction time delay.

Establishing metrics and control laws for the learning process: ball and beam balancing.

Understanding how dexterity improves with practice is a fundamental challenge of motor control and neurorehabilitation. Here we investigate a ball and beam implementation of a dexterity puzzle in which subjects stabilize a ball at the mid-point of a beam by manipulating the angular position of the beam. Stabilizability analysis of different biomechanical models for the ball and beam task with time-delayed proportional-derivative feedback identified the angular position of the beam as the manipulated variable. Consequently, we monitored the changes in the dynamics with learning by measuring changes in the control parameters. Two types of stable motion are possible: node type (nonoscillatory) and spiral type (oscillatory). Both types of motion are observed experimentally and correspond to well-defined regions in the parameter space of the control gains. With practice the control gains for each subject move close to or on the portion of the boundary which separates the node-type and spiral-type solutions and which is associated with the rightmost characteristic exponent of smallest real part. These observations suggest that with learning the control gains for ball and beam balancing change in such a way that minimizes overshoot and the settling time. This study provides an example of how mathematical analysis together with careful experimental observations can shed light onto the early stages of skill acquisition. Since the difficulty of this task depends on the length of the beam, ball and beam balancing tasks may be useful for the rehabilitation of children with dyspraxia and those recovering from a stroke.

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.

Acting together, destabilizing influences can stabilize human balance.

The causes of falling in the elderly are multi-factorial. Three factors that influence balance stability are the time delay, a sensory dead zone and the maximum ankle torque that can be generated by muscular contraction. Here, the effects of these contributions are evaluated in the context of a model of an inverted pendulum stabilized by time-delayed proportional-derivative (PD) feedback. The effect of the sensory dead zone is to produce a hybrid type of control in which the PD feedback is switched ON or OFF depending on whether or not the controlled variable is larger or smaller than the detection threshold, Π. It is shown that, as Π increases, the region in the plane of control parameters where the balance time (BT) is greater than 60 s is increased slightly. However, when maximum ankle torque is also limited, there is a dramatic increase in the parameter region associated with BTs greater than 60 s. This increase is due to the effects of a torque limitation on over-control associated with bang-bang type switching controllers. These observations show that acting together influences, which are typically thought to destabilize balance, can actually stabilize balance. This article is part of the theme issue 'Nonlinear dynamics of delay systems'.

Microchaos in human postural balance: Sensory dead zones and sampled time-delayed feedback.

Models for the stabilization of an inverted pendulum figure prominently in studies of human balance control. Surprisingly, fluctuations in measures related to the vertical displacement angle for quietly standing adults with eyes closed exhibit chaos. Here we show that small-amplitude chaotic fluctuations ("microchaos") can be generated by the interplay between three essential components of human neural balance control, namely time-delayed feedback, a sensory dead zone, and frequency-dependent encoding of force. When the sampling frequency of the force encoding is decreased, the sensitivity of the balance control to changes in the initial conditions increases. The sampled, time-delayed nature of the balance control may provide insights into why falls are more common in the very young and the elderly.

Saturation limits the contribution of acceleration feedback to balancing against reaction delay.

A nonlinear model for human balancing subjected to a saturated delayed proportional-derivative-acceleration (PDA) feedback is analysed. Compared to the proportional-derivative (PD) controller, it is confirmed that the PDA controller improves local stability even for large feedback delays. However, it is shown that the saturated PDA controller typically introduces subcritical Hopf bifurcation into the system, which can also lead to falling for large enough perturbations. The subcriticality becomes stronger as the acceleration feedback gain increases or the saturation torque limit decreases. These explain some features of human balancing failure related to the increased reaction delay of inactive elderly people.

Quantization improves stabilization of dynamical systems with delayed feedback.

We show that an unstable scalar dynamical system with time-delayed feedback can be stabilized by quantizing the feedback. The discrete time model corresponds to a previously unrecognized case of the microchaotic map in which the fixed point is both locally and globally repelling. In the continuous-time model, stabilization by quantization is possible when the fixed point in the absence of feedback is an unstable node, and in the presence of feedback, it is an unstable focus (spiral). The results are illustrated with numerical simulation of the unstable Hayes equation. The solutions of the quantized Hayes equation take the form of oscillations in which the amplitude is a function of the size of the quantization step. If the quantization step is sufficiently small, the amplitude of the oscillations can be small enough to practically approximate the dynamics around a stable fixed point.

Control at stability's edge minimizes energetic costs: expert stick balancing.

Stick balancing on the fingertip is a complex voluntary motor task that requires the stabilization of an unstable system. For seated expert stick balancers, the time delay is 0.23 s, the shortest stick that can be balanced for 240 s is 0.32 m and there is a [Formula: see text]° dead zone for the estimation of the vertical displacement angle in the saggital plane. These observations motivate a switching-type, pendulum-cart model for balance control which uses an internal model to compensate for the time delay by predicting the sensory consequences of the stick's movements. Numerical simulations using the semi-discretization method suggest that the feedback gains are tuned near the edge of stability. For these choices of the feedback gains, the cost function which takes into account the position of the fingertip and the corrective forces is minimized. Thus, expert stick balancers optimize control with a combination of quick manoeuvrability and minimum energy expenditures.

Extension of Stability Radius to Neuromechanical Systems With Structured Real Perturbations.

The ability of humans to maintain balance about an unstable position in a continuously changing environment attests to the robustness of their balance control mechanisms to perturbations. A mathematical tool to analyze robust stabilization of unstable equilibria is the stability radius. Based on the pseudo-spectra, the stability radius gives a measure to the maximum change of the system parameters without resulting in a loss of stability. Here, we compare stability radii for a model for human frontal plane balance controlled by a delayed proportional-derivative feedback to two types of perturbations: unstructured complex and weighted structured real. It is shown that: 1) narrow stance widths are more robust to parameter variation; 2) stability is maintained for larger structured real perturbations than for unstructured complex perturbations; and 3) the most robust derivative gain to weighted structured real perturbations is located near the stability boundary. It is argued that stability radii can effectively be used to compare different control concepts associated with human motor control.

Sensory uncertainty and stick balancing at the fingertip.

The effects of sensory input uncertainty, [Formula: see text], on the stability of time-delayed human motor control are investigated by calculating the minimum stick length, [Formula: see text], that can be stabilized in the inverted position for a given time delay, [Formula: see text]. Five control strategies often discussed in the context of human motor control are examined: three time-invariant controllers [proportional-derivative, proportional-derivative-acceleration (PDA), model predictive (MP) controllers] and two time-varying controllers [act-and-wait (AAW) and intermittent predictive controllers]. The uncertainties of the sensory input are modeled as a multiplicative term in the system output. Estimates based on the variability of neural spike trains and neural population responses suggest that [Formula: see text]-13 %. It is found that for this range of uncertainty, a tapped delay-line type of MP controller is the most robust controller. In particular, this controller can stabilize inverted sticks of the length balanced by expert stick balancers (0.25-0.5 m when [Formula: see text] s). However, a PDA controller becomes more effective when [Formula: see text]. A comparison between [Formula: see text] for human stick balancing at the fingertip and balancing on the rubberized surface of a table tennis racket suggest that friction likely plays a role in balance control. Measurements of [Formula: see text], and a variability of the fluctuations in the vertical displacement angle, an estimate of [Formula: see text], may make it possible to study the changes in control strategy as motor skill develops.

Acceleration feedback improves balancing against reflex delay.

A model for human postural balance is considered in which the time-delayed feedback depends on position, velocity and acceleration (proportional-derivative-acceleration (PDA) feedback). It is shown that a PDA controller is equivalent to a predictive controller, in which the prediction is based on the most recent information of the state, but the control input is not involved into the prediction. A PDA controller is superior to the corresponding proportional-derivative controller in the sense that the PDA controller can stabilize systems with approximately 40 per cent larger feedback delays. The addition of a sensory dead zone to account for the finite thresholds for detection by sensory receptors results in highly intermittent, complex oscillations that are a typical feature of human postural sway.

Delayed feedback of sampled higher derivatives.

Systems where the present rate of change of the state depends on the past values of the higher rates of change of the state are described by so-called advanced functional differential equations (AFDEs). In an AFDE, the highest derivative of the state-space coordinate appears with delayed argument only. The corresponding linearized equations are always unstable with infinitely many unstable poles, and are rarely related to practical applications due to their inherently implicit nature. In this paper, one of the simplest AFDEs, a linear scalar first-order system, is considered with the delayed feedback of the second derivative of the state in the presence of sampling in the feedback loop (i.e. in the case of digital control). It is shown that sampling of the feedback may stabilize the originally infinitely unstable system for certain parameter combinations. The result explains the stable behaviour of certain dynamical systems with feedback delay in the highest derivative.