Nonlinear Muscle Model Research Articles

Dynamics of Cardiovascular Muscle Using a Non-Linear Symmetric Oscillator

In this paper, a complete non-linear symmetric oscillator model using the Hamiltonian approach has been developed and used to describe the cardiovascular conduction process’s dynamics, as the signal generated from the cardiovascular muscle is non-deterministic and random. Electrocardiogram (ECG) signal is a significant factor in the cardiovascular system as most of the medical diagnoses can be well understood by observing the ECG signal’s amplitude. A non-linear cardiovascular muscle model has been proposed in this study, where a modified vanderPol symmetric oscillator-based equation is used. Gone are the days whena non-linear system had been designed using the describing function technique. It is better to design a non-linear model using the Hamiltonian dynamical equation for its high accuracy and flexibility. Varying a non-linear spring constant using this type of approach is more comfortable than the traditional describing function technique. Not only that but different initial conditions can also be taken for experimental purposes. It never affects the overall modeling. The Hamiltonian approach provides the energy of an asymmetric oscillatory system of that cardiovascular conduction system. A non-linear symmetric oscillator was initially depicted by the non-linear mass-spring (two degrees of freedom) model. The motion of an uncertain non-linear cardiovascular system has been solved considering second-order approximation, which also demonstrates the possibility of introducing spatial dimensions. Finally, the model’s natural frequency expression has also been simulated and is composed of the previously published result.

A Non-Linear Control Method to Compensate for Muscle Fatigue during Neuromuscular Electrical Stimulation

Neuromuscular electrical stimulation (NMES) is a promising technique to artificially activate muscles as a means to potentially restore the capability to perform functional tasks in persons with neurological disorders. A pervasive problem with NMES is that overstimulation of the muscle (among other factors) leads to rapid muscle fatigue, which limits the use of clinical and commercial NMES systems. The objective of this article is to develop an NMES controller that incorporates the effects of muscle fatigue during NMES-induced non-isometric contraction of the human quadriceps femoris muscle. Our previous work that used the RISE class of non-linear controllers cannot accommodate fatigue and muscle activation dynamics. A totally new control design approach and associated stability proof is required to derive a new class of NMES control design that accounts for muscle fatigue dynamics and a first-order activation dynamics, in addition to the second-order musculoskeletal dynamics. Motivated from a control method for robotic systems in a strict-feedback form, a backstepping based-non-linear NMES controller was designed to accommodate for the additional muscle activation dynamics. Further, experimentally identified estimates of the fatigue and activation dynamics were incorporated in the control design. The developed controller uses a neural network-based estimate of the musculoskeletal dynamics and error due to fatigue estimation. A globally uniformly ultimately bounded stability is proven the new controller that accounts for an uncertain non-linear muscle model and bounded non-linear disturbances (e.g., spasticity and changing load dynamics). The developed controller was validated through experiments on the left and right legs of 3 able-bodied subjects and was compared with a proportional-derivative (PD) controller and a PD augmented with a neural network. The statistical analysis showed improved control performance compared with the PD controller.

Bilateral Control of Functional Electrical Stimulation and Robotics-based Telerehabilitation.

Currently a telerehabilitation system includes a therapist and a patient where the therapist interacts with the patient, typically via a verbal and visual communication, for assessment and supervision of rehabilitation interventions. This mechanism often fails to provide physical assistance, which is a modus operandi during physical therapy or occupational therapy. Incorporating an actuation modality such as functional electrical stimulation (FES) or a robot at the patient's end that can be controlled by a therapist remotely, to provide therapy or to assess and measure rehabilitation outcomes can significantly transform current telerehabilitation technology. In this paper, a position-synchronization controller is derived for FES-based telerehabilitation to provide physical assistance that can be controlled remotely. The newly derived controller synchronizes an FES-driven human limb with a remote physical therapist's robotic manipulator despite constant bilateral communication delays. The control design overcomes a major stability analysis challenge: the unknown and unstructured nonlinearities in the FES-driven musculoskeletal dynamics. To address this challenge, the nonlinear muscle model was estimated through two neural networks functions that approximated unstructured nonlinearities and an adaptive control law for structured nonlinearities with online update laws. A Lyapunov-based stability analysis was used to prove the globally uniformly ultimately bounded tracking performance. The performance of the state synchronization controller was validated through experiments on an able-bodied subject. Specifically, we demonstrated bilateral control of FES-elicited leg extension and a human operated robotic manipulator. The controller was shown to effectively synchronize the system despite unknown and different delays in the forward and backward channels.

Development and Learning Control of a Human Limb With a Rehabilitation Exoskeleton

This paper describes a novel development of a lower limber exoskeleton for physical assistance and rehabilitation. The developed exoskeleton is a motorized leg device having a total of 4 DOF with hip, knee, and ankle actuated in the sagittal plane. The exoskeleton applies forces and learns the impedance parameters of both robot and human. An adaptive control scheme by incorporating learning control approaches into the exoskeleton system is developed to help the leg movement on a desired periodic trajectory and handle periodic uncertainties with known periods. The proposed control approach does not require a muscle model and can be proven to yield asymptotic stability for a nonlinear muscle model and an exoskeleton model in the presence of bounded nonlinear disturbances (e.g., spasticity and fatigue). The performance of the controller is demonstrated through closed-loop experiments on human subjects. The experiments illustrate the ability of the exoskeleton to enable the leg shank to track single and multiple period trajectories with different periods and ranges of motion.

The role of mechanical resonance in the neural control of swimming in fishes

The bodies of many fishes are flexible, elastic structures; if you bend them, they spring back. Therefore, they should have a resonant frequency: a bending frequency at which the output amplitude is maximized for a particular input. Previous groups have hypothesized that swimming at this resonant frequency could maximize efficiency, and that a neural circuit called the central pattern generator might be able to entrain to a mechanical resonance. However, fishes swim in water, which may potentially damp out many resonant effects. Additionally, their bodies are elongated, which means that bending can occur in complicated ways along the length of the body. We review previous studies of the mechanical properties of fish bodies, and then present new data that demonstrate complex bending properties of elongated fish bodies. Resonant peaks in amplitude exist, but there may be many of them depending on the body wavelength. Additionally, they may not correspond to the maximum swimming speed. Next, we describe experiments using a closed-loop preparation of the lamprey, in which a preparation of the spinal cord is linked to a real-time simulation of the muscle and body properties, allowing us to examine resonance entrainment as we vary the simulated resonant frequency. We find that resonance entrainment does occur, but is rare. Gain had a significant, though weak, effect, and a nonlinear muscle model produced resonance entrainment more often than a linear filter. We speculate that resonance may not be a critical effect for efficient swimming in elongate, anguilliform swimmers, though it may be more important for stiffer carangiform and thunniform fishes.

Assessing the Perception of Human-Like Mechanical Impedance for Robotic Systems

As physical interactions between robots and humans become more common, there is a growing need to design robots that are kinesthetically perceived as human-like. One approach to implement human-like mechanical impedance is to physically simulate models of the human neuromuscular system. However, the level of model complexity needed to achieve perception of human-like properties is unknown. The purpose of this study was to develop an objective assessment of a human subject's ability to discriminate kinesthetically between a model-defined impedance and that produced by human muscle. The assessment was based upon signal detection theory, by which the ability to discriminate between two classes of stimuli is analyzed statistically. With this assessment, we tested the hypothesis that a nonlinear muscle model is necessary to obtain perception of human-like muscle mechanical impedance. Fifteen subjects were presented with a mechanical impedance of either a simulated muscle model or electrically stimulated wrist muscle and were asked to decide if they were interacting with a “machine” or “human.” The impedances were randomized for a total of 30 presentations. The results showed that a robot that stimulates either linear viscoelastic properties or a nonlinear Hill muscle model could be distinguished from a human wrist muscle by almost all subjects. However, the subjects' ability to discriminate between the Hill model and human muscle was significantly less, which may have been due to larger overall impedance of the viscoelastic model. These results are important for the design of robots that emulates mechanical impedances that are perceived as human-like.

Fuzzy control with amplitude/pulse-width modulation of nerve electrical stimulation for muscle force control

The main goal of this study was to study the performance of fuzzy logic controllers combined with simplified hybrid amplitude/pulse-width (AM/PW) modulation to regulate muscle force via nerve electrical stimulation. The recruitment curves with AM/PW and AM modulations were constructed for the calf muscles of rabbits. Integrated with the modulation methods, a proportional-integral-derivative (PID) and three fuzzy logic controllers were designed and applied for the electrical stimulation of tibial nerves to control the ankle torque under isometric conditions. The performance of the two modulation methods combined with the four controllers was compared when the ankle was fixed at three positions for both in vivo experiments and model simulations using a nonlinear muscle model. For the animal experiments, AM/PW modulation performed better than AM modulation alone. The fuzzy PI controller performed marginally better and was resistant to external noises, though it tended to have a larger overshoot. The performance of the controllers had a similar trend in the three different joint positions, and the simulation results with the nonlinear model matched the experimental results well. In conclusion, AM/PW modulation improved controller performance, while the contribution of fuzzy logic was only marginal.

Active muscle response using feedback control of a finite element human arm model

Mathematical human body models (HBMs) are important research tools that are used to study the human response in car crash situations. Development of automotive safety systems requires the implementation of active muscle response in HBM, as novel safety systems also interact with vehicle occupants in the pre-crash phase. In this study, active muscle response was implemented using feedback control of a nonlinear muscle model in the right upper extremity of a finite element (FE) HBM. Hill-type line muscle elements were added, and the active and passive properties were assessed. Volunteer tests with low impact loading resulting in elbow flexion motions were performed. Simulations of posture maintenance in a gravity field and the volunteer tests were successfully conducted. It was concluded that feedback control of a nonlinear musculoskeletal model can be used to obtain posture maintenance and human-like reflexive responses in an FE HBM.

Reaching Movements of a Redundant Musculoskeletal Arm: Acquisition of an Adequate Internal Force by Iterative Learning and Its Evaluation through a Dynamic Damping Ellipsoid

This paper presents a proposal of an iterative learning control method for a musculoskeletal arm to acquire adequate internal force in order to realize human-like natural movements. Additionally, a dynamic damping ellipsoid at the end-point is introduced to evaluate internal forces obtained through the iterative learning scheme. Our previous works presented that a human-like smooth reaching movement using a musculoskeletal redundant arm model can be achieved by introducing a nonlinear muscle model and 'the virtual spring–damper hypothesis'. However, to date, the internal forces have been determined heuristically. As described in this paper, to determine internal forces more systematically, an iterative learning control method is used for acquisition of an adequate dynamic damping ellipsoid according to a given task. It is presented that the learning control method can perform effectively to realize given tasks, even though strong nonlinear characteristics of the muscles exist. After acquiring a given task, the dynamic damping ellipsoid is introduced to evaluate the relation between a damping effect generated by the acquired internal forces and a trajectory of the end-point. Some numerical simulations are performed and the usefulness of the learning control strategy, despite strong nonlinearity, is demonstrated through these results.

Linear Homeomorphic Models for Muscles in the Head–Neck Region

The linear homeomorphic muscle model proposed by Enderle and coworkers for the rectus eye muscle is fitted to reflect the dynamics of muscles in the head-neck complex, specifically in muscles involved in gaze shifts. This parameterization of the model for different muscles in the neck region will serve to drive a 3D dynamic computer model for the movement of the head-neck complex, including bony structures and soft tissues, and aimed to study the neural control of the complex during fast eye and head movements such as saccades and gaze shifts. Parameter values for the different muscles in the neck region were obtained by optimization using simulated annealing. These linear homeomorphic muscle models provide non-linear force-velocity profiles and linear length tension profiles, which are in agreement with results from the more complex Virtual Muscle model, which is based on Zajac's non-linear muscle model.

Nonlinear Neuromuscular Electrical Stimulation Tracking Control of a Human Limb

A high-level objective of neuromuscular electrical stimulation (NMES) is to enable a person to achieve some functional task. Towards this goal, the objective of the current effort is to develop a NMES controller to produce a knee position trajectory that will enable a human shank to track any continuous desired trajectory (or constant setpoint). A nonlinear control method is developed to control the human quadriceps femoris muscle undergoing nonisometric contractions. The developed controller does not require a muscle model and can be proven to yield asymptotic stability for a nonlinear muscle model in the presence of bounded nonlinear disturbances (e.g., spasticity, delays, fatigue). The performance of the controller is demonstrated through a series of closed-loop experiments on human subjects. The experiments illustrate the ability of the controller to enable the leg shank to track single and multiple period trajectories with different periods and ranges of motion, and also track desired step changes with changing loads.

The Use of a Nonlinear Muscle Model in Explaining the Relationship Between Duration, Amplitude, and Peak Velocity of Human Rapid Movements

Rapid human movements exhibit a quasilinear relationship between their amplitude and maximum velocity and a log-like relationship between their amplitude and duration. The authors demonstrate that those well-observed relations can be obtained with a simple nonlinear muscle model and a pulse-step control scheme. That result encourages the use of nonlinear musculoskeletal models with simple control schemes for modeling human ballistic movements.

The control of shoulder muscles during goal directed movements, an inverse dynamic analysis

Fast goal directed arm movements in the sagittal plane were analyzed with a three-dimensional shoulder model with 95 muscle elements. Dynamics of the muscle elements were described by a third-order nonlinear muscle model. Muscle forces and activation were estimated using the method of inverse muscular dynamics, an optimization scheme which uses only very limited computational power. Most model results were similar to the EMG but some differences between model results and EMG were found in muscles where the EMG activity was subject dependent. For the movement studied, the thoracoscapular muscles were shown to deliver about 40% of the energy required for the acceleration of the arm during anteflexion and about 22% during retroflexion. Activity of thoracoscapular muscles was also required to ensure contact between the thorax and the scapula which is important for the mechanical stability of the shoulder. The rotator cuff muscles were found to deliver about 19% of the energy required for the acceleration of the arm during anteflexion and about 8% during retroflexion.

The 3‐D‐tongue FEM model revised

As a first application of a newly developed object‐oriented finite element code that is specialized for the modeling of biological tissues we have been reimplementing the geometry of the tongue model by S. Kirtani et al. (‘‘A computational model of the tongue,’’ Ann. Bul. Res. Inst. of Logopedics and Phoniatrics, Univ. Tokyo, Vol. 10, pp. 243–251). The new implementation uses large strain and dynamic modeling of the tissue, includes the effects of inertia and observes precisely a weak formulation of the incompressibility constraint of the tissue. The muscle activation input for the current model is made similar to the original computation. However, a different, more realistic nonlinear muscle model is used, in which the active stress depends on the instantaneous strain and strain rate in the direction of the muscle fibers. The implementation methods and computational results will be presented. [Work supported by the Whitaker Foundation and by ATR‐International.]

Time optimality in the control of human movements.

In a simulation study the control of maximally fast goal directed movements has been analyzed. For a simple linear model it is shown that the presence of a third input block reduces the movement duration. The time optimal size of the third block depends on the ratio of a neuromuscular time constant (first-order lag) and movement time. As a second step a non-linear muscle model was simulated. By an optimization of input parameters it was found that the time optimal input, as expected, switches between maximal agonist and maximal antagonist activation. As for the linear model, a third phase was required for an optimal movement. It was found that the third phase serves to compensate the slowly decaying antagonist force. Also an input similar to experimentally found activation patterns was simulated. This input contains a silent period between the first two bursts and the second and the third burst have submaximal amplitudes. This input led to a near time optimal movement with a duration 9% larger than the minimal duration but with largely reduced muscle forces. This suggests that a criterion is minimized which also takes into account the effort spent. Including gravity in the model indicates optimality of a silent period between the third phase and a final agonist activity to resist gravity. When assuming different dynamics for agonist and antagonist, the optimal switch times for agonist and antagonist no longer coincide, also after the three block pattern some extra activity is required to obtain a cancellation of the slowly decaying force in agonist and antagonist.