Adaptive Frequency Oscillators Research Articles

Adaptive oscillators support Bayesian prediction in temporal processing.

Humans excel at predictively synchronizing their behavior with external rhythms, as in dance or music performance. The neural processes underlying rhythmic inferences are debated: whether predictive perception relies on high-level generative models or whether it can readily be implemented locally by hard-coded intrinsic oscillators synchronizing to rhythmic input remains unclear and different underlying computational mechanisms have been proposed. Here we explore human perception for tone sequences with some temporal regularity at varying rates, but with considerable variability. Next, using a dynamical systems perspective, we successfully model the participants behavior using an adaptive frequency oscillator which adjusts its spontaneous frequency based on the rate of stimuli. This model better reflects human behavior than a canonical nonlinear oscillator and a predictive ramping model-both widely used for temporal estimation and prediction-and demonstrate that the classical distinction between absolute and relative computational mechanisms can be unified under this framework. In addition, we show that neural oscillators may constitute hard-coded physiological priors-in a Bayesian sense-that reduce temporal uncertainty and facilitate the predictive processing of noisy rhythms. Together, the results show that adaptive oscillators provide an elegant and biologically plausible means to subserve rhythmic inference, reconciling previously incompatible frameworks for temporal inferential processes.

Ankle Foot Orthosis for Patients with Drop Foot Using Shape-Memory-Alloy Actuators

AbstractGait disorders can lower the quality of life of patients. Drop foot, a causative factor of deviated gait patterns, renders patients unable to lift their forefoot towards the body. Hence, a light and compact ankle–foot orthosis (AFO), which is the most common treatment for drop foot, must be designed, especially for patients with impaired lower limb muscles as oxygen consumption increases by 30% per 1.96 N load on their foot. Furthermore, the limited range of ankle angles in the first 10% of the gait cycle (GC) is a major drawback for patients with drop foot compared to healthy individuals. This limited range of ankle angles can be improved by gaining support from an AFO composed of shape memory alloy (SMA) actuators (SMA-AFO). Therefore, in this study, an SMA was used to fabricate a soft actuator to reduce the weight of the AFO. An adaptive frequency oscillator (AO) was implemented in real time for continuous gait phase detection. Walk tests were performed on a treadmill with the SMA-AFO attached to the participants (N = 3). The experimental results showed that the participants could lift their forefoot in the dorsiflexion direction with an ankle angle of 8.75° in the first 10% of the GC. Furthermore, the current required to operate the SMA actuator can be supplied to only 45.3% of the GC, reducing the power consumption. Therefore, the proposed SMA-AFO can be used in patients with drop foot.

Real-Time Gait Event Detection with Adaptive Frequency Oscillators From a Single Head-Mounted IMU

Accurate real-time gait event detection is the basis for the development of new gait rehabilitation techniques, especially when utilizing robotics or virtual reality (VR). The recent emergence of affordable wearable technologies, especially inertial measurement units (IMUs), has brought forth various new methods and algorithms for gait analysis. In this paper, we highlight some advantages of using adaptive frequency oscillators (AFOs) over traditional gait event detection algorithms, implemented a real-time AFO-based algorithm that estimates the gait phase from a single head-mounted IMU, and validated our method on a group of healthy subjects. Gait event detection was accurate at two different walking speeds. The method was reliable for symmetric, but not asymmetric gait patterns. Our method could prove especially useful in VR applications since a head-mounted IMU is already an integral part of commercial VR products.

Adaptive Symmetry Reference Trajectory Generation in Shared Autonomy for Active Knee Orthosis

Gait symmetry training plays an essential role in the rehabilitation of hemiplegic patients. Robotics-based gait training has been widely accepted by patients and clinicians. Reference trajectory generation for the affected side using the motion data of the unaffected side is an important way to achieve this. However, online generation of gait reference trajectory requires the algorithm to provide correct gait phase delay and could reduce the impact of measurement noise from sensors and input uncertainty from users. Based on an active knee orthosis (AKO) prototype, this work presents an adaptive symmetric gait trajectory generation framework for the gait rehabilitation of hemiplegic patients. Using the adaptive nonlinear frequency oscillators (ANFO) and movement primitives, we implement online gait pattern encoding and adaptive phase delay according to the real-time user input. A shared autonomy (SA) module with online input validation and arbitration has been designed to prevent undesired movements from being transmitted to the actuator on the affected side. The experimental results demonstrate the feasibility of the framework. Overall, this work suggests that the proposed method has the potential to perform gait symmetry rehabilitation in an unstructured environment and provide a kinematic reference for torque-assistance AKO.

Chaos in a Pendulum Adaptive Frequency Oscillator Circuit Experiment

Adaptive oscillators can learn and encode information in dynamic, plastic states. The pendulum has recently been proposed as the base oscillator of an adaptive system. In a mechanical setup, the horizontally forced pendulum adaptive frequency oscillator seeks a resonance condition by modifying the length of the pendulum's rod. This system stores the external forcing frequency when the external amplitude is small, while it can store the resonance frequency, which is affected by the nonlinearity of the pendulum, when the external amplitude is large. Furthermore, for some frequency ranges, the pendulum adaptive frequency oscillator can exhibit chaotic motion when the amplitudes are large. This adaptive oscillator could be used as a smart vibratory energy harvester device, but this chaotic region could degrade its performance by using supplementary energy to modify the rod length. The pendulum adaptive frequency oscillator’s equations of motions are discussed, and a field-programmable analog array is used as an experimental realization of this system as an electronic circuit. Bifurcation diagrams are shown for both the numerical simulations and experiments, while period-3 motion is shown for the numerical simulations. As little work has been done on the stability of adaptive oscillators, the authors believe that this work is the first demonstration of chaos in an adaptive oscillator.

A Control Framework for Adaptation of Training Task and Robotic Assistance for Promoting Motor Learning With an Upper Limb Rehabilitation Robot

Robot-assisted rehabilitation has been a promising solution to improve motor learning of neurologically impaired patients. State-of-the-art control strategies are typically limited to the ignorance of heterogeneous motor capabilities of poststroke patients and therefore intervene suboptimally. In this article, we propose a control framework for robot-assisted motor learning, emphasizing the detection of human intention, generation of reference trajectories, and modification of robotic assistance. A real-time trajectory generation algorithm is presented to extract the high-level features in active arm movements using an adaptive frequency oscillator (AFO) and then integrate the movement rhythm with the minimum-jerk principle to generate an optimal reference trajectory, which synchronizes with the motion intention in the patient as well as the motion pattern in healthy humans. In addition, a subject-adaptive assistance modification algorithm is presented to model the patient’s residual motor capabilities employing spatially dependent radial basis function (RBF) networks and then combining the RBF-based feedforward controller with the impedance feedback controller to provide only necessary assistance while simultaneously regulating the maximum-tolerated error during trajectory tracking tasks. We conduct simulation and experimental studies based on an upper limb rehabilitation robot to evaluate the overall performance of the motor-learning framework. A series of results showed that the difficulty level of reference trajectories was modulated to meet the requirements of subjects’ intended motion, furthermore, the robotic assistance was compliantly optimized in response to the changing performance of subjects’ motor abilities, highlighting the potential of adopting our framework into clinical application to promote patient-led motor learning.

A unilateral robotic knee exoskeleton to assess the role of natural gait assistance in hemiparetic patients

BackgroundHemiparetic gait is characterized by strong asymmetries that can severely affect the quality of life of stroke survivors. This type of asymmetry is due to motor deficits in the paretic leg and the resulting compensations in the nonparetic limb. In this study, we aimed to evaluate the effect of actively promoting gait symmetry in hemiparetic patients by assessing the behavior of both paretic and nonparetic lower limbs. This paper introduces the design and validation of the REFLEX prototype, a unilateral active knee–ankle–foot orthosis designed and developed to naturally assist the paretic limbs of hemiparetic patients during gait.MethodsREFLEX uses an adaptive frequency oscillator to estimate the continuous gait phase of the nonparetic limb. Based on this estimation, the device synchronically assists the paretic leg following two different control strategies: (1) replicating the movement of the nonparetic leg or (2) inducing a healthy gait pattern for the paretic leg. Technical validation of the system was implemented on three healthy subjects, while the effect of the generated assistance was assessed in three stroke patients. The effects of this assistance were evaluated in terms of interlimb symmetry with respect to spatiotemporal gait parameters such as step length or time, as well as the similarity between the joint’s motion in both legs.ResultsPreliminary results proved the feasibility of the REFLEX prototype to assist gait by reinforcing symmetry. They also pointed out that the assistance of the paretic leg resulted in a decrease in the compensatory strategies developed by the nonparetic limb to achieve a functional gait. Notably, better results were attained when the assistance was provided according to a standard healthy pattern, which initially might suppose a lower symmetry but enabled a healthier evolution of the motion of the nonparetic limb.ConclusionsThis work presents the preliminary validation of the REFLEX prototype, a unilateral knee exoskeleton for gait assistance in hemiparetic patients. The experimental results indicate that assisting the paretic leg of a hemiparetic patient based on the movement of their nonparetic leg is a valuable strategy for reducing the compensatory mechanisms developed by the nonparetic limb.

Gait Pattern Identification and Phase Estimation in Continuous Multilocomotion Mode Based on Inertial Measurement Units

In the field of lower limb exoskeletons, it is essential to accurately estimate the gait phase of humans. Many methods have been proposed to estimate the gait phase, but only a few studies have considered the multi-locomotion mode. This paper proposes a novel inertial measurement unit(IMU)-based method to estimate the gait phase of a pilot in continuous multi-locomotion mode. The method includes gait pattern recognition based on long short-term memory (LSTM), continuous phase estimation based on a dual adaptive frequency oscillator(DAFO), threshold-based toe-off event detection and a rule-based gait phase synchronization module. First, we used the LSTM-based network to identify four gait patterns including standing, level ground walking, upstairs and downstairs. Next, the DAFO was used to obtain the continuous gait phase of the pilot. Then, we detected the gait events in different gait modes. Finally, the continuous gait phase was synchronized according to the gait events. The experimental result shows that the gait pattern classification accuracy using 5 IMUs is 98.58% and the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal {F}_{\!{1}}$ </tex-math></inline-formula> score reaches 0.9875. The proposed DAFO model can maintain good stability when multiple gait modes are frequently switched, significantly improving the problem of slow convergence and the poor robustness of single adaptive frequency oscillator(SAFO) models. Toe-off gait events of 492 steps are all detected and the average error at the detected gait events in different gait modes is 15.34± 40.58 ms.

The pendulum adaptive frequency oscillator

Adaptive oscillators are a type of nonlinear oscillator that are capable of learning and storing information in plastic states. Here, a typical mechanical pendulum is modified to have an adjustable rod length to create a pendulum adaptive frequency oscillator. Since the resonance frequency of the pendulum is a function of the rod length, this allows the pendulum to learn and encode frequency information from an external source. An experimental pendulum adaptive frequency oscillator is designed and constructed, and its performance is compared to numerical simulations. This nonlinear pendulum was approximated as a Duffing oscillator through the method of multiple scales to determine the physical constants of the experiment by using a curve fit. Utilizing the pendulum adaptive frequency oscillator’s dynamics, this system is able to learn a resonance condition and store this information in the rod length. This causes the system to seek resonance, even with considerable nonlinearity. As pendulums can be used to harvest energy, this type of adaptation could be used to further exploit vibratory energy sources.

Complex waveform estimation using adaptive frequency oscillators

Automatic learning and emulation of a complex periodic waveform is demonstrated using a nonlinear dynamical system containing a single adaptive frequency oscillator (AFO) with feedback. The AFO uses a phase-lock loop (PLL) to match the fundamental frequency and phase of a periodic input signal. Nonlinear functions of the AFO state provide an extended basis to represent the waveform shape, and amplitudes for the extended basis are automatically estimated using feedback. Significantly, the approach also works if the AFO state is not sinusoidal but is itself complex shaped. Numerical simulations demonstrate successful estimation of a triangle wave and a complex waveform constructed with multiple harmonics using both Hopf and van der Pol AFOs.

Stochastic effects on a Hopf adaptive frequency oscillator

This paper explores the stochastic dynamics of a Hopf adaptive frequency oscillator when driven by noise. Adaptive oscillators are nonlinear oscillators that store information via plastic states. As noise is ubiquitous in physical systems, it is important to gain an understanding of the stochastic effects on adaptive oscillators. Previously, it has been shown that a simplified analysis of the Fokker–Planck equation results in affecting the plastic frequency state of these oscillators. However, when the full Fokker–Planck equation is considered, new behaviors are observed due to changes in oscillation amplitudes in addition to frequencies. The plastic frequency state of these oscillators may benefit from enhanced learning due to small amplitudes of noise, converge to incorrect values for medium amplitudes of noise, and even collapse to zero in the limit of large amplitudes of noise. Interestingly, not all averaged states collapse equally, which leads a two dimensional limit cycle to collapse into single dimensional oscillations when considering the averaged dynamics. These behaviors are compared analytically through the Fokker–Planck equation, numerically using the Euler–Maruyama simulations, and finally validated experimentally using an analog, electronic circuit. These results show that noise can enhance, mislead, or even reduce the dimensionality of the averaged adaptive Hopf oscillator.

An IMUs-Based Extended Kalman Filter to Estimate Gait Lower Limb Sagittal Kinematics for the Control of Wearable Robotic Devices

Inertial sensors have gained relevance as wearable sensors to acquire the kinematics of human limbs through fusion sensor algorithms and biomechanical models. However, there are some limitations to the use of Inertial Measurement Units in the control of wearable robotic devices: 1) Some approaches use magnetometer readings to estimate the orientation of the sensor, and, as a result, they are prone to errors due to electromagnetic interferences; 2) Biomechanical model-based approaches require complex and time-consuming calibration procedures. In order to address these issues, this paper proposes an Extended Kalman Filter to estimate sagittal lower limb kinematics during gait, based on gyroscopes and accelerometers and without requiring any calibration or sensor alignment process. As magnetometer measurements are not involved, this method is not affected by electromagnetic disturbances. Our approach calculates the knee rotation axis in real-time, and it estimates hip and ankle sagittal axes considering that the movements in that plane occur around parallel axes. We carried out an experimental validation with eight healthy subjects walking on a treadmill at different velocities. We obtained waveform RMS errors about 3.8°, 3.6°, and 4.8° for hip, knee, and ankle in the sagittal plane. We also assessed the performance of this method as a tool for controlling lower-limb robotic exoskeletons by detecting gait events or estimating the phase and frequency of the gait in real-time through an Adaptive Frequency Oscillator. The average RMS delay in the detection of gait events was lower than 60 ms, and the RMSE in the estimation of the gait phase was about 3% of the gait cycle. We conclude that the described method could be used as a controller for wearable robotic devices.

Slow-fast Dynamics of Strongly Coupled Adaptive Frequency Oscillators

Oscillators have two main limitations: their synchronization properties are limited (i.e., they have a finite synchronization region) and they have no memory of past interactions (i.e., they return...

Lower-Limb Exoskeleton With Variable-Structure Series Elastic Actuators: Phase-Synchronized Force Control for Gait Asymmetry Correction

Series elastic actuators (SEAs) can provide accurate force control and backdrivability in physical human–robot interaction. Control of SEA-generated forces or torques makes allowance for the user's own volitional control and allows implementing a wide variety of assistive strategies. A novel force control method for a SEA-driven lower-limb assistive exoskeleton is presented. The device features variable-structure SEAs coupled via Bowden cables. The actuator alternates between two discrete levels of stiffness depending on the amplitude of the commanded force. The algorithm features a switching force-tracking control based on the forward-propagating Riccati equation. A disturbance-rejection component increases the device's transparency in zero assistance mode. The force control was used to implement an assistive strategy that aims to correct the asymmetric gait typical of stroke survivors. Assistive joint torques synchronize with the user's gait by means of an adaptive frequency oscillator, which extracts the continuous phase and frequency of the patient's gait using data from both the paretic and the healthy sides. The control was tested with healthy subjects wearing the exoskeleton while subject to a simulated knee flexion impairment. The control proved effective in restoring spatial and temporal knee flexion symmetry to levels comparable to unobstructed gait.

The Swing Control of Knee Exoskeleton Based on Admittance Model and Nonlinear Oscillator

It is important to control exoskeleton to act synchronously with the user for movement assistance. A knee exoskeleton was constructed for gait rehabilitation training, and a control method based on adaptive frequency oscillator (AFO) was presented for swing motion of knee exoskeleton. Firstly, during the learning mode, the user flexed and extended with the exoskeleton leg together, and the exoskeleton leg was not actuated. The phase of the exoskeleton angle extracted, and the envelope curve of the user’s EMG is generated based on AFO. Then, during power-assisted mode, the assistive torque is generated according to the angle phase on-line and the restructuring muscle torque. At last, the assistive torque as a feedback signal was input to the admittance model for the swing control of the knee exoskeleton. The experiments of swing control showed that knee exoskeleton can follow the user’s motion well.

Design and Experimental Validation of a 3-DOF Underactuated Pendulum-Like Robot

This article presents the design and the experimental validation of a novel 3-degree-of-freedom (DOF) pendulum-like cable-driven robot capable of executing point-to-point motions by leveraging partial feedback linearization control and on-line trajectory planning based on adaptive frequency oscillators (AFOs). Unlike most cable-suspended parallel robots, which rely on at least n actuated cables to control n DOF, the proposed robot is capable of performing 3-DOF point-to-point motions, from a starting pose to a goal one within its dynamic workspace, by means of two actuators only. Feedback linearization allows the dynamics of the variable-length pendulum to be decoupled from the orientation of the end effector, enabling the device to use parametric excitation to control the oscillations of the variable-length pendulum, akin to a playground swing. A pool of AFOs is introduced to enable smooth, lag free on-line estimations of the current phase of the pendulum oscillation to inform the on-line planner and the parametric excitation controller. Experimental results demonstrate feasibility of the proposed design and control approach.

An admittance controller based on assistive torque estimation for a rehabilitation leg exoskeleton

A rehabilitation exoskeleton leg was constructed for gait training, and a method for exoskeleton leg swing control based on admittance model was studied. The man–machine interaction torque, the inertia compensator torque, and the exoskeleton’s assistive torque act as the inputs to the admittance control system for realizing the effective assistance to the wearer. The exoskeleton’s assistive torque is generated by estimating net torque exerted by the muscle based on adaptive frequency oscillator. A variable swing frequency experiment for simulating physical rehabilitation exercise was implemented. The results show that the inertia compensator adjusting the compensation varies with the swing frequency. The control system provides effective assistance to the wearer. The coordinated control experiment of hip and knee joints was also implemented. The results show that the interaction forces are all controlled in a reasonable and small range, and there is a good coordination between the hip joint and knee joint during swing motion.

Robot adaptation to human physical fatigue in human–robot co-manipulation

In this paper, we propose a novel method for human---robot collaboration, where the robot physical behaviour is adapted online to the human motor fatigue. The robot starts as a follower and imitates the human. As the collaborative task is performed under the human lead, the robot gradually learns the parameters and trajectories related to the task execution. In the meantime, the robot monitors the human fatigue during the task production. When a predefined level of fatigue is indicated, the robot uses the learnt skill to take over physically demanding aspects of the task and lets the human recover some of the strength. The human remains present to perform aspects of collaborative task that the robot cannot fully take over and maintains the overall supervision. The robot adaptation system is based on the Dynamical Movement Primitives, Locally Weighted Regression and Adaptive Frequency Oscillators. The estimation of the human motor fatigue is carried out using a proposed online model, which is based on the human muscle activity measured by the electromyography. We demonstrate the proposed approach with experiments on real-world co-manipulation tasks: material sawing and surface polishing.

Adaptive Natural Oscillator to exploit natural dynamics for energy efficiency

We present a novel adaptive oscillator, called Adaptive Natural Oscillator (ANO), to exploit the natural dynamics of a given robotic system. This tool is built upon the Adaptive Frequency Oscillator (AFO), and it can be used as a pattern generator in robotic applications such as locomotion systems. In contrast to AFO, that adapts to the frequency of an external signal, ANO adapts the frequency of reference trajectory to the natural dynamics of the given system. In this work, we prove that, in linear systems, ANO converges to the system’s natural frequency. Furthermore, we show that this tool exploits the natural dynamics for energy efficiency through minimization of actuator effort. This property makes ANO an appealing tool for energy consumption reduction in cyclic tasks; especially in legged systems. We also extend the proposed adaptation mechanism to high dimensional and general cases; such as n-DOF manipulators. In addition, by investigating a hopper leg in simulation, we show the efficacy of ANO in face of dynamical discontinuities; such as those inherent in legged locomotion. Furthermore, we apply ANO to a simulated compliant robotic manipulator performing a periodic task where the energy consumption is drastically reduced. Finally, the experimental results on a 1-DOF compliant joint show that our adaptive oscillator, despite all practical uncertainties and deviations from theoretical models, exploits the natural dynamics and reduces the energy consumption.

Demonstration learning of robotic skills using repeated suggestions learning algorithm

Abstract In this paper a new model of nonlinear dynamical system based on adaptive frequency oscillators for learning rhythmic signals is implemented by demonstration. This model uses coupled Hopf oscillators to encode and learn any periodic input signal. Learning process is completely implemented in the dynamics of adaptive oscillators. One of the issues in learning in such systems is constant number of oscillators in the feedback loop. In other words, the number of adaptive frequency oscillators is one of the design factors. In this contribution, it is shown that using enough number of oscillators can help the learning process. In this paper, we address this challenge and try to solve it in order to learn the rhythmic movements with greater accuracy, lower error and avoid missing fundamental frequency. To reach this aim, a method for generating drumming patterns is proposed which is able to generate rhythmic and periodic trajectories for a NAO humanoid robot. To do so, a programmable central pattern generator is used which is inspired from animal’s neural systems and these programmable central pattern generators are extended to learn patterns with more accuracy for NAO humanoid robots. Successful experiments of demonstration learning are done using simulation and a NAO Real robot.