Stick Balancing Research Articles

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.

The Critical Length is a Good Measure to Distinguish between Stick Balancing in the ML and AP Directions

Seven novice subjects participated in experiments of stick balancing on a linear track in the anterior-posterior (AP) and the medio-lateral (ML) directions. The goal of the experiments was to test how the depth perception in the subjects' AP direction affects balancing performance compared to balancing in the ML direction, where depth perception does not play a role. It is easier to balance longer sticks than shorter ones, therefore balancing performance is measured by the length of the shortest stick that subjects can balance. Subjects were found to be able to balance shorter sticks in the ML direction than in the AP direction: the ratio of the shortest stick lengths in the ML direction relative to the AP direction was in average 0.53. Thus, the additional challenge posed by depth perception in the AP direction is clearly observable. Additionally, repeated trials were carried out for 5 consecutive days to assess the development of balancing skill by using stabilometry analysis. The maximal balance time of the subjects significantly increased with the days of practice.

Machine learning forecasting of extreme fluctuations in a human balancing task

Here we consider the problem of predicting falls using real data from human stick balancing experiments implementing k-nearest neighbors, random forest, support vector machine, and a neural network. Carrying out two different approaches to evaluate the temporal series, it is shown that, in one case, prefall segments can be classified with an F1 score larger than 0.97, using the neural network and the random forest; being able to detect changes in the pattern preceding a fall around 0.83 s before. An additional approach can identify the temporal position of fall precursors in a best-case scenario with an F1 score >0.9, using the neural network, at times around 0.83 s before. In general, we found that the classifiers behave well when dealing with this complex classification task. Best performing methods in both approaches coincide in determining a falling precursor at 0.83 s before the escape — coinciding with previous results. These results may help establishing a reference for future works using other algorithms best suited to predict complex temporal series.

Noise and Financial Stylized Facts: A Stick Balancing Approach.

In this work, we address the beneficial role of noise in two different contexts, the human brain and financial markets. In particular, the similitude between the ability of financial markets to maintain in equilibrium asset prices is compared with the ability of the human nervous system to balance a stick on a fingertip. Numerical simulations of the human stick balancing phenomenon show that after the introduction of a small quantity of noise and a proper calibration of the main control parameters, intermittent changes in the angular velocity of the stick are able to reproduce the most basilar stylized facts involving price returns in financial markets. These results could also shed light on the relevance of the idea of the "planetary nervous system", already introduced elsewhere, in the financial context.

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.

Special issue on PID control in the information age: Theoretical advances and applications

Special issue on PID control in the information age: Theoretical advances and applications

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.

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.

Critical parameters for the robust stabilization of the inverted pendulum with reaction delay: State feedback versus predictor feedback

AbstractThe critical length that limits stabilizability for delayed proportional‐derivative‐acceleration (PDA) feedback and for predictor feedback (PF) is analyzed for the inverted pendulum paradigm. The aim of this work is to improve the understanding of human balancing tasks such as stick balancing on the fingertip, which can be modeled as a pendulum cart system. The relation between the critical length of the balanced stick and the reaction time delay in the presence of sensory uncertainties, which are modeled as static parameter perturbations in the control gains, is investigated rigorously. Robust stabilizability analysis is performed using the real structured stability radius. Performance is assessed by the length of the shortest pendulum (critical length) that can still be balanced for a fixed reaction delay. For both PDA feedback and PF control with delay mismatch, it is observed that the relation between the critical length and the reaction delay remains quadratic in the presence of perturbations on the control gains (of fixed size). Numerical comparison shows that predictor feedback is superior over PDA feedback in terms of critical length: shorter pendulum can be balanced by PF than by PDA feedback for the same reaction delay and for the same static parameter perturbation. Furthermore, it is found that both control concepts are more sensitive to the change in the feedback delay than on the same relative change in the parameter uncertainties. Interpretation to human balancing suggests that it is more challenging for the nervous system to cope with reaction delay than with sensory uncertainties.

Extinctions in time-delayed population maps, fallings, and extreme forcing

It is known that random population maps with time delay undergo a noise-mediated transition that produces the loss of its structural stability. Such a transition is optimized for a specific value of the noise correlation time of an Ornstein–Uhlenbeck forcing, as a consequence of the coupling of the involved deterministic and stochastic time scales. Here, it is shown that the deterministic time scale is related to the dynamics of the system close to a stability boundary. The escaping process depicts a survival distribution function similar to the one observed in human stick balancing, a task known to involve a truncated Levy forcing. Here, it is shown that such extreme distribution favours the stabilisation of an inverted pendulum, when compared with a normal forcing, and the system parameters are close to a stability boundary. This outcome suggests that an unstable dynamics may temporarily avoid an extinguishing transition if the extreme forcing is able to tune the system parameters at specific time delay values (of physiological significance) close to the stability boundary. These results remark the relevance of feedbacks close to the stability edge on survival and extinction.

State-space intermittent feedback stabilization of a dual balancing task

Balancing the body in upright standing and balancing a stick on the fingertip are two examples of unstable tasks that, in spite of strong motor and sensory differences, appear to share a similar motor control paradigm, namely a state-space intermittent feedback stabilization mechanism. In this study subjects were required to perform the two tasks simultaneously, with the purpose of highlighting both the coordination between the two skills and the underlying interaction between the corresponding controllers. The experimental results reveal, in particular, that upright standing (the less critical task) is modified in an adaptive way, in order to facilitate the more critical task (stick balancing), but keeping the overall spatio-temporal signature well known in regular upright standing. We were then faced with the following question: to which extent the physical/biomechanical interaction between the two independent intermittent controllers is capable to explain the dual task coordination patterns, without the need to introduce an additional, supervisory layer/module? By comparing the experimental data with the output of a simulation study we support the former hypothesis, suggesting that it is made possible by the intrinsic robustness of both state-space intermittent feedback stabilization mechanisms.

Comparison of Pixel-based Position Input and Direct Acceleration Input for Virtual Stick Balancing Tests

A virtual stick balancing environment is developed using a computer mouse as input device. The development process is presented both on the hardware and software level. Two possible concepts are suggested to obtain the acceleration of the input device: discrete differentiation of the cursor position measured in pixels on the screen and by direct measurements via an Inertial Measurement Unit (IMU). The comparison of the inputs is carried out with test measurements using a crank mechanism. The measured signals are compared to the prescribed motion of the mechanism and it is shown that the IMU-based input signal fits better to the prescribed motion than the pixel-based input signal. The pixel-based input can also be applied after additional filtering, but this presents an extra computational delay in the feedback loop.

Experimental estimation of tactile reaction delay during stick balancing using cepstral analysis

The control mechanism of the human nervous system during stick balancing is analyzed using signal processing techniques. It is assumed that during stick balancing the inclination angle, the angular velocity and the angular acceleration of the stick are all perceived and used for feedback. Taking into account the human reaction time delay, the balancing process is modelled by a delayed proportional-derivative-acceleration feedback. The corresponding governing equation is a neutral delay-differential equation. Two different delays are used in the model: one for visual perception associated with position and velocity feedback and one for tactile perception associated with acceleration feedback. The feature of neutral delay-differential equations, namely, that the delay of the neutral part can be detected in the cepstrum of the output signal motivates the study that acceleration feedback can be identified using cepstral analysis. Time histories of balancing sticks of different weights (made of wood, aluminum and steel) and different colors (red, gray and yellow) by human subjects are analyzed in order to stimulate mechanoreceptors at the fingertip and visual perception at different extent. Visual delays were measured by standard reaction time tests. The tactile delays were estimated by cepstral analysis of the recorded motion and were found to have similar tendency in their average and their standard deviation as the visual delays.

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.

Preliminary results on a dual balancing task.

Balancing a stick on the fingertip while standing upright is a dual balancing task that requires the integration of two independent control systems. This is a completely novel experimental paradigm for attempting to understand how the brain deals with equilibrium in a general way. Preliminary results are presented and interpreted in the framework of an intermittent control policy extended to include a dual balancing task. In particular the study reveals that the upright standing (the less critical task) is modified in an adaptive way, in order to facilitate the more critical task, i.e. stick balancing.

Control Force Recalculation for Balancing Problems

Understanding the mechanism of human balancing is a scientifically challenging task. In order to describe the nature of the underlying control mechanism, the control force has to be determined experimentally. A main feature of balancing tasks is that the open-loop system is unstable. Therefore, reconstruction of the trajectories using the measured control force is difficult, since measurement inaccuracies, noise and numerical errors increase exponentially with time. In order to overcome this problem, a new approach is proposed in this paper. In the presented technique, first the solution of the linearized system is used. As a second step, an optimization problem is solved which is based on a variational principle. A main advantage of the method is that there is no need for the numerical differentiation of the measured data for the calculation of the control forces, which is the main source of the numerical errors. The method is demonstrated in case of a human stick balancing.

Stabilization of a Cart Inverted Pendulum: Improving the Intermittent Feedback Strategy to Match the Limits of Human Performance.

Stabilization of the CIP (Cart Inverted Pendulum) is an analogy to stick balancing on a finger and is an example of unstable tasks that humans face in everyday life. The difficulty of the task grows exponentially with the decrease of the length of the stick and a stick length of 32 cm is considered as a human limit even for well-trained subjects. Moreover, there is a cybernetic limit related to the delay of the multimodal sensory feedback (about 230 ms) that supports a feedback stabilization strategy. We previously demonstrated that an intermittent-feedback control paradigm, originally developed for modeling the stabilization of upright standing, can be applied with success also to the CIP system, but with values of the critical parameters far from the limiting ones (stick length 50 cm and feedback delay 100 ms). The intermittent control paradigm is based on the alternation of on-phases, driven by a proportional/derivative delayed feedback controller, and off-phases, where the feedback is switched off and the motion evolves according to the intrinsic dynamics of the CIP. In its standard formulation, the switching mechanism consists of a simple threshold operator: the feedback control is switched off if the current (delayed) state vector is closer to the stable than to the unstable manifold of the off-phase and is switched on in the opposite case. Although this simple formulation is effective for explaining upright standing as well as CIP balancing, it fails in the most challenging configuration of the CIP. In this work we propose a modification of the standard intermittent control policy that focuses on the explicit selection of switching times and is based on the phase reset of the estimated state vector at each switching time and on the simulation of an approximated internal model of CIP dynamics. We demonstrate, by simulating the modified intermittent control policy, that it can match the limits of human performance, while operating near the edge of instability.

Parametric study of virtual stick balancing based on a delayed PD model

Virtual stick balancing by human subjects is analyzed with special attention to the reaction time delay and the sensory dead zones. A pendulum-cart system displayed on a computer screen is controlled via an optical computer mouse. Realization of the balancing task involves additional computational delay in the feedback loop. The measurement results are compared to numerical simulations based on a model with delayed proportional-derivative (PD) feedback. Balancing skills are measured using the critical length, i.e., the length of the shortest stick that subjects can balance. The critical length obtained by the virtual balancing tests is larger by a factor of 2 ~ 3.5 than the critical length for mathematical model with delayed PD control. Time domain simulations for a model with dead zones both for the position and for the velocity show a limit cycle type of oscillation rather than a realistic noisy (or chaotic) motion.

Stick Balancing with Feedback Delay, Sensory Dead Zone, Acceleration and Jerk Limitation

A simplified model of stick balancing on the fingertip subjected to predictor feedback is investigated, which accounts for three important modeling issues: (1) feedback delay; (2) the sensory dead zone; and (3) limitation of the control force corresponding to the maximum acceleration and the maximum jerk of human hand movement. Eight different cases (± sensory dead zone, ± acceleration limitation, ± jerk limitation) are compared by estimating the maximum balance time out of five time-domain simulations with different initial conditions. It is shown that the region of linear stability in the plane of control parameters is reduced by the presence of dead zone, not affected by limitations on hand acceleration, but is increased by limitations on the jerk.

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.