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The involuntary force fluctuations associated with physiological (as distinct from pathological) tremor are an unavoidable component of human motor control. While the origins of physiological tremor are known to depend on muscle afferentation, it is possible that the mechanical properties of muscle-tendon systems also affect its generation, amplification and maintenance. In this paper, we investigated the dependence of physiological tremor on muscle length in healthy individuals. We measured physiological tremor during tonic, isometric plantarflexion torque at 30% of maximum at three ankle angles. The amplitude of physiological tremor increased as calf muscles shortened in contrast to the stretch reflex whose amplitude decreases as muscle shortens. We used a published closed-loop simulation model of afferented muscle to explore the mechanisms responsible for this behaviour. We demonstrate that changing muscle lengths does not suffice to explain our experimental findings. Rather, the model consistently required the modulation of  γ-static fusimotor drive to produce increases in physiological tremor with muscle shortening - while successfully replicating the concomitant reduction in stretch reflex amplitude. This need to control γ-static fusimotor drive explicitly as a function of muscle length has important implications. First, it permits the amplitudes of physiological tremor and stretch reflex to be decoupled. Second, it postulates neuromechanical interactions that require length-dependent γ drive modulation to be independent from α drive to the parent muscle. Lastly, it suggests that physiological tremor can be used as a simple, non-invasive measure of the afferent mechanisms underlying healthy motor function, and their disruption in neurological conditions.

This paper presents our new method, Short Segment-Structural Decomposition SubSpace (SS-SDSS), for the estimation of dynamic joint stiffness from short data segments. The main application is for data sets that are only piecewise stationary. Our approach is to: 1) derive a data-driven, mathematical model for dynamic stiffness for short data segments; 2) bin the non-stationary data into a number of short, stationary data segments; and 3) estimate the model parameters from subsets of segments with the same properties. This method extends our previous state-spacework by recognizing that initial conditions have important effects for short data segments; consequently, initial conditions are incorporated into the stiffness model and estimated for each segment. A simulation study that faithfully replicated experimental conditions delineated the range of experimental conditions for which the method can successfully identify stiffness. An experimental study on the ankle of a healthy subject during a torque matching tasks demonstrated the successful estimation of dynamic stiffness in a slow, time-varying experiment. Together, the simulation and experimental studies demonstrate that the SS-SDSS method is a valuable tool to measure stiffness in functionally important tasks.

Dynamic joint stiffness is a dynamic, nonlinear relationship between the position of a joint and the torque acting about it, which can be used to describe the biomechanics of the joint and associated limb(s). This paper models and quantifies changes in ankle dynamic stiffness and its individual elements, intrinsic and reflex stiffness, in healthy human subjects during isometric, time-varying (TV) contractions of the ankle plantarflexor muscles. A subspace, linear parameter varying, parallel-cascade (LPV-PC) algorithm was used to identify the model from measured input position perturbations and output torque data using voluntary torque as the LPV scheduling variable (SV). Monte-Carlo simulations demonstrated that the algorithm is accurate, precise, and robust to colored measurement noise. The algorithm was then used to examine stiffness changes associated with TV isometric contractions. The SV was estimated from the Soleus EMG using a Hammerstein model of EMG-torque dynamics identified from unperturbed trials. The LPV-PC algorithm identified (i) a non-parametric LPV impulse response function (LPV IRF) for intrinsic stiffness and (ii) a LPV-Hammerstein model for reflex stiffness consisting of a LPV static nonlinearity followed by a time-invariant state-space model of reflex dynamics. The results demonstrated that: (a) intrinsic stiffness, in particular ankle elasticity, increased significantly and monotonically with activation level; (b) the gain of the reflex pathway increased from rest to around 10–20% of subject's MVC and then declined; and (c) the reflex dynamics were second order. These findings suggest that in healthy human ankle, reflex stiffness contributes most at low muscle contraction levels, whereas, intrinsic contributions monotonically increase with activation level.

An accurate model for ElectroMyoGram (EMG)-torque dynamics has many uses. One of its applications which has gained high attention among researchers is its use, in estimating the muscle contraction level for the efficient control of prosthesis. In this paper, the dynamic relationship between the surface EMG and torque during isometric contractions at the human ankle was studied using system identification techniques. Subjects voluntarily modulated their ankle torque in dorsiflexion direction, by activating their tibialis anterior muscle, while tracking a pseudo-random binary sequence in a torque matching task. The effects of contraction bandwidth, described by torque spectrum, on EMG-torque dynamics were evaluated by varying the visual command switching time. Nonparametric impulse response functions (IRF) were estimated between the processed surface EMG and torque. It was demonstrated that: 1) at low contraction bandwidths, the identified IRFs had unphysiological anticipatory (i.e., non-causal) components, whose amplitude decreased as the contraction bandwidth increased. We hypothesized that this non-causal behavior arose, because the EMG input contained a component due to feedback from the output torque, i.e., it was recorded from within a closed-loop. Vision was not the feedback source since the non-causal behavior persisted when visual feedback was removed. Repeating the identification using a nonparametric closed-loop identification algorithm yielded causal IRFs at all bandwidths, supporting this hypothesis. 2) EMG-torque dynamics became faster and the bandwidth of system increased as contraction modulation rate increased. Thus, accurate prediction of torque from EMG signals must take into account the contraction bandwidth sensitivity of this system.

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In the restructuring power industry, establishing an equitable and effective reactive power market with consideration of the voltage security problem is very important to provide a reliable restructured power system. The objective of this paper is to review some of the algorithms that have been proposed for pricing reactive power and present a programming method to solve the optimization problem and obtain reactive power marginal price accordingly. The production cost ofreactive power into the objective function of the optimal power flow problem is included. The results from different case studies show the effects of various factors on reactive power marginal price.

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