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The result of simulations of the first-passage time distribution for a discretized equation x ͑ t + 1 ͒ = x ͑ t ͒ + dt ͑ x ͑ t − ͒ + ͒ where is a Gaussian white noise with variance 2 . We have set the threshold at X = 5.0. The
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The inverted pendulum is frequently used as a starting point for discussions of how human balance is maintained during standing and locomotion. Here we examine three experimental paradigms of time-delayed balance control: (1) mechanical inverted time-delayed pendulum, (2) stick balancing at the fingertip, and (3) human postural sway during quiet st...
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... of a first-passage time problem for an unstable fixed point left- hand side of Eq. 15 with reinjection into the interval − − wherever the threshold is crossed. Current interest has focused on the possibility that the left-hand side of Eq. 15 also contains a time delay. This gives rise to a unstable delayed random walk. 32,35,38 As is shown in Fig. 3, the interplay between noise and delay can transiently stabilize the unstable fixed point, i.e., prolong the first-passage time. These effects are interesting in light of measurements of the reaction time and response time when posture is perturbed. 46 In this study it was observed that the neural time delay, i.e., the time interval ...
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... Similar equations are often encountered in modeling noisy feedback systems [1]. Examples range from feedback-controlled Brownian particles [2][3][4][5] and robots [6][7][8] over various biological [9][10][11][12][13] and physical [14][15][16] systems to finance [17]. SDDEs also successfully describe instabilities in car traffic [18] and recently, they have been applied to model other manybody nonequilibrium phenomena with time-delayed interactions such as flocking or swarming [19][20][21][22][23][24][25][26][27]. ...
Stochastic processes with time delay are invaluable for modeling in science and engineering when finite signal transmission and processing speeds can not be neglected. However, they can seldom be treated with sufficient precision analytically if the corresponding stochastic delay differential equations (SDDEs) are nonlinear. This work presents a numerical algorithm for calculating the probability densities of processes described by nonlinear SDDEs. The algorithm is based on Markovian embedding and solves the problem by basic matrix operations. We validate it for a broad class of parameters using exactly solvable linear SDDEs and a cubic SDDE. Besides, we show how to apply the algorithm to calculate transition rates and first passage times for a Brownian particle diffusing in a time-delayed cusp potential.
... Although numerous models exist for human posture (24)(25)(26)(27)(28)(29)(30)(31), many view the human body as a linear system-like an inverted pendulum-with attributes such as time delay, feedback, and a proportional derivative controller. However, Khanian et. ...
Motor Imagery (MI) is gaining traction in both rehabilitation and sports settings, but its immediate influence on human postural control is not yet clearly understood. The focus of this study is to examine the effects of MI on the dynamics of the Center of Pressure (COP), a crucial metric for evaluating postural stability. In the experiment, thirty healthy young adults participated in four different scenarios: normal standing with both open and closed eyes, and kinesthetic motor imagery focused on mediolateral (ML) and anteroposterior (AP) sway movements. A mathematical model was developed to characterize the nonlinear dynamics of the COP and to assess the impact of MI on these dynamics. Our results show a statistically significant increase (p-value<0.05) in variables such as COP path length and Long-Range Correlation (LRC) during MI compared to the closed-eye and normal standing conditions. These observations align well with psycho-neuromuscular theory, which suggests that imagining a specific movement activates neural pathways, consequently affecting postural control. This study presents compelling evidence that motor imagery not only has a quantifiable impact on COP dynamics but also that changes in the Center of Pressure (COP) are directionally consistent with the imagined movements. This finding holds significant implications for the field of rehabilitation science, suggesting that motor imagery could be strategically utilized to induce targeted postural adjustments. Nonetheless, additional research is required to fully understand the complex mechanisms that underlie this relationship and to corroborate these results across a more diverse set of populations.
... Consequently, the nervous system of all animals must learn to control movement based on outdated sensory information and compensate for the influence delays have on self-motion. The delays accompanying the control of different motor effectors, however, are not constant; instead, they vary due to the length and velocity of neural transmission 1,2 and the complexity of the neural networks involved [1][2][3][4][5][6] . Furthermore, delays can lengthen throughout the lifespan due to alterations in nerve conduction, muscle force generation, and neural processing associated with growth 3 , aging 7 , and disease 8,9 . ...
... Typically, the more contextual factors that overlap between tasks (i.e., goal, movement mechanics, sensory cues, and motor effectors), the more likely learned control policies will generalize [27][28][29][30] . Standing balance involves controlling whole-body motion in both anteroposterior and mediolateral space, with each direction of balance possessing distinct biomechanics 16,17,31,32 , muscle effectors, activation patterns [33][34][35] , and sensorimotor delays 5,17,20,36,37 . Therefore, these differing sensorimotor factors may limit the ability to generalize learning across orthogonal space (H 0 , see Fig. 1), as observed when standing participants adapt their balance to externally imposed perturbations in different directions 38 . ...
... We further assessed whether learning to balance with the 350 ms delay resulted in any aftereffects in normal balance by comparing baseline trials (i.e., 4 ms delay) from pre-and post-learning sessions (Figs. 4,5). In the postlearning baseline trials, angular velocity variability was similar to prelearning trials and participants always remained within the virtual limits. ...
Humans receive sensory information from the past, requiring the brain to overcome delays to perform daily motor skills such as standing upright. Because delays vary throughout the body and change over a lifetime, it would be advantageous to generalize learned control policies of balancing with delays across contexts. However, not all forms of learning generalize. Here, we use a robotic simulator to impose delays into human balance. When delays are imposed in one direction of standing, participants are initially unstable but relearn to balance by reducing the variability of their motor actions and transfer balance improvements to untrained directions. Upon returning to normal standing, aftereffects from learning are observed as small oscillations in control, yet they do not destabilize balance. Remarkably, when participants train to balance with delays using their hand, learning transfers to standing with the legs. Our findings establish that humans use experience to broadly update their neural control to balance with delays.
... Falls in the elderly are a common health issue worldwide and consequently understanding the mechanisms of how humans maintain balance whilst standing is an area of much research [42][43][44][45]. The inverted pendulum has also been used to model and understand this process [44][45][46][47][48][49][50][51][52]. More recent work has also included the use of experiments with robotic manipulanda to investigate how humans can balance items with their hands [53][54][55][56]. ...
... The inverted pendulum has been valuable in understanding human balance while standing [44][45][46][47][48][49][50][51][52]. Future studies within the framework of our pendulum system could further explore and model human behavior in such tasks. ...
We present an inverted pendulum design using readily available V-slot rail components and 3D printing to construct custom parts. To enable the examination of different pendulum characteristics, we constructed three pendulum poles of different lengths. We implemented a brake mechanism to modify sliding friction resistance and built a paddle that can be attached to the ends of the pendulum poles. A testing rig was also developed to consistently apply disturbances by tapping the pendulum pole, characterizing balancing performance. We perform a comprehensive analysis of the behavior and control of the pendulum. This begins by considering its dynamics, including the nonlinear differential equation that describes the system, its linearization, and its representation in the s-domain. The primary focus of this work is the development of two distinct control modes for the pendulum: a velocity control mode, designed to balance the pendulum while the cart is in motion, and a position control mode, aimed at maintaining the pendulum cart at a specific location. For this, we derived two different state space models: one for implementing the velocity control mode and another for the position control mode. In the position control mode, integral action applied to the cart position ensures that the inverted pendulum remains balanced and maintains its desired position on the rail. For both models, linear observer-based state feedback controllers were implemented. The control laws are designed as linear quadratic regulators (LQR), and the systems are simulated in MATLAB. To actuate the physical pendulum system, a stepper motor was used, and its controller was assembled in a DIN rail panel to simplify the integration of all necessary components. We examined how the optimized performance, achieved with the medium-length pendulum pole, translates to poles of other lengths. Our findings reveal distinct behavioral differences between the control modes.
... Various switching models for intermittent control have been considered and reported in the literature (Morasso et al, 2020;Asai et al, 2009;Nomura et al, 2013). While some of the proposed models in the literature are better representations of the switching strategy employed by the CNS, we choose simple switching strategies based on ankle angle (Kowalczyk et al, 2012;Milton et al, 2009) as they are more suitable for analysis of the effect of changing thresholds and control gains. In the control feedback (active torque), we consider one consolidated time delay δ representing the accumulation of various time delays in the sensorimotor feedback system (taken to be 0.1 s (Chagdes et al, 2013)). ...
Purpose: The central nervous system (CNS) employs an intermittent control strategy in humanquiet stance and other balancing activities as stated in the literature. We investigate the less exploredaspect of how the intermittent control strategy changes as humans practice balancing on an unstableunrestrained platform.
Method: We perform human subject experiments with a rocker board, obtain the Center of Pres-sure (COP), ankle and rocker board angle, and EMG data for the Medial Gastrocnemius (MG) and Tibialis Anterior (TA) muscles for the right leg (active leg). We analyze the trends in various postur-ography, kinematic, and muscle-firing parameters while taking reference from simulation analysis todraw insights about the nature of changes in the intermittent control strategy with practice.
Result: Multi-modal or skewed histograms of COP, power law in power spectrum, and 2-slope diffu-sion stabilogram are observed in all data. Total kernel density of the peaks in the COP histogramsincrease with practice trials and increases with decrease in intermittent threshold during simulation.The low-frequency slope sL of the ankle angle power spectrum decreases with practice and decreaseswith decrease in intermittency threshold in simulations. Average Energy AE and Simple SquareIntegral SSI decrease with practice. Wavelet spectrum of MG and TA muscle EMG show intermit-tent bursts in the signals.
Conclusion: The intermittent control strategy is retained throughout the balanc-ing practice on the rocker board. The trends in parameters with practice are bestexplained by a decrease in intermittency threshold in the control strategy with practice.
... If τ → 0, the trivial solution is asymptotically stable for p > 1 and d > 0. Conversely, for τ > 0, the trivial equilibrium becomes unstable for too large p and d values. Several studies [60,61] investigated the stability of this and similar models. In particular, it was proven that, in most cases, the system loses stability through a subcritical Andronov-Hopf bifurcation [62,63], as illustrated in Fig. 12 for d = 2 and τ = 0.5. ...
This paper proposes a novel method for predicting the presence of saddle-node bifurcations in dynamical systems. The method exploits the effect that saddle-node bifurcations have on transient dynamics in the surrounding phase space and parameter space, and does not require any information about the steady-state solutions associated with the bifurcation. Specifically, trajectories of a system obtained for parameters close to the saddle-node bifurcation present local minima of the logarithmic decrement trend in the vicinity of the bifurcation. By tracking the logarithmic decrement for these trajectories, the saddle-node bifurcation can be accurately predicted. The method does not strictly require any mathematical model of the system, but only a few time series, making it directly implementable for gray- and black-box models and experimental apparatus. The proposed algorithm is tested on various systems of different natures, including a single-degree-of-freedom system with nonlinear damping, the mass-on-moving-belt, a time-delayed inverted pendulum, and a pitch-and-plunge wing profile. Benefits, limitations, and future perspectives of the method are also discussed. The proposed method has potential applications in various fields, such as engineering, physics, and biology, where the identification of saddle-node bifurcations is crucial for understanding and controlling complex systems.
... In this case, disturbances on the robot can be controlled through the whole teleoperation pipeline and human commands. By speculating on the simplified robot model, i.e., LIPM, studies shows that if the time delay of the system is higher than a critical time delay, the robot destabilizes [125]. The critical time delay τ c is relevant to the robot CoM height, i.e., τ c = β z 0 g from (2). ...
Teleoperation of humanoid robots enables the integration of the cognitive skills and domain expertise of humans with the physical capabilities of humanoid robots. The operational versatility of humanoid robots makes them the ideal platform for a wide range of applications when teleoperating in a remote environment. However, the complexity of humanoid robots imposes challenges for teleoperation, particularly in unstructured dynamic environments with limited communication. Many advancements have been achieved in the last decades in this area, but a comprehensive overview is still missing. This survey article gives an extensive overview of humanoid robot teleoperation, presenting the general architecture of a teleoperation system and analyzing the different components. We also discuss different aspects of the topic, including technological and methodological advances, as well as potential applications.
... If τ → 0, the trivial solution is asymptotically stable for p > 1 and d > 0. Conversely, for τ > 0, the trivial equilibrium becomes unstable for too large p and d values. Several studies [43,44] investigated the stability of this and similar models. In particular, it was proven that, in most cases, the system loses stability through a subcritical Andronov-Hopf bifurcation [45,46], as illustrated in Figure 12 for d = 2. ...
This paper proposes a novel method for predicting the presence of saddle-node bifurcations in dynamical systems. The method exploits the effect that saddle-node bifurcations have on transient dynamics in the surrounding phase space and parameter space, and does not require any information about the steady-state solutions associated with the bifurcation. Specifically, trajectories of a system obtained for parameters close to the saddle-node bifurcation present local minima of the logarithmic decrement trend in the vicinity of the bifurcation. By tracking the logarithmic decrement for these trajectories, the saddle-node bifurcation can be accurately predicted. The method does not strictly require any mathematical model of the system, but only a few time series, making it directly implementable for gray- and black-box models and experimental apparatus. The proposed algorithm is tested on various systems of different natures, including a single-degree-of-freedom system with nonlinear damping, the mass-on-moving-belt, a time-delayed inverted pendulum, and a pitch-and-plunge wing profile. Benefits, limitations, and future perspectives of the method are also discussed. The proposed method has potential applications in various fields, such as engineering, physics, and biology, where the identification of saddle-node bifurcations is crucial for understanding and controlling complex systems.
... After a disturbance, physiological processes take about 130 ms before ankles begin producing reactive balance-correcting joint moments (4,5). This long latency limits a person's ability to regain postural equilibrium (6,7) and is primarily due to the time required for neural conduction and processing (8). To compensate for these morphological limitations, researchers have begun developing wearable exoskeletons to improve human balance during everyday life (9)(10)(11)(12)(13)(14). ...
Maintaining balance throughout daily activities is challenging because of the unstable nature of the human body. For instance, a person's delayed reaction times limit their ability to restore balance after disturbances. Wearable exoskeletons have the potential to enhance user balance after a disturbance by reacting faster than physiologically possible. However, "artificially fast" balance-correcting exoskeleton torque may interfere with the user's ensuing physiological responses, consequently hindering the overall reactive balance response. Here, we show that exoskeletons need to react faster than physiological responses to improve standing balance after postural perturbations. Delivering ankle exoskeleton torque before the onset of physiological reactive joint moments improved standing balance by 9%, whereas delaying torque onset to coincide with that of physiological reactive ankle moments did not. In addition, artificially fast exoskeleton torque disrupted the ankle mechanics that generate initial local sensory feedback, but the initial reactive soleus muscle activity was only reduced by 18% versus baseline. More variance of the initial reactive soleus muscle activity was accounted for using delayed and scaled whole-body mechanics [specifically center of mass (CoM) velocity] versus local ankle-or soleus fascicle-mechanics, supporting the notion that reactive muscle activity is commanded to achieve task-level goals, such as maintaining balance. Together, to elicit symbiotic human-exoskeleton balance control, device torque may need to be informed by mechanical estimates of global sensory feedback, such as CoM kinematics, that precede physiological responses.
... In the various fields including mathematics, biology, physics, engineering, economics, and so on, there have been interests in investigating the effect of delays in the system. [1,2,3,4,5,6,7,8,9,10,11,12,13,14]). Typically, delays introduce oscillations and complex behaviors to otherwise simple and well-behaved systems. ...
We present here a connection between the solutions of the transcendental trigonometric equation and the Lambert W function. This connection emerged through an analysis of resonant conditions with a recently proposed simple delay differential equation that shows transient oscillatory behaviors. We investigate and present the connection both analytically and numerically.
