In this paper, we focus on determining the safe operational domain of a coupled actuator–valve configuration. The so-called “smart valves” system has increasingly been used in critical applications and missions including municipal piping networks, oil and gas fields, petrochemical plants, and more importantly, the U.S. Navy ships. A comprehensive dynamic analysis is hence needed to be carried out for capturing dangerous behaviors observed repeatedly in practice. Using some powerful tools of nonlinear dynamic analysis including Lyapunov exponents and Poincaré map, a comprehensive stability map is provided in order to determine the safe operational domain of the network in addition to characterizing the responses obtained. Coupled chaotic and hyperchaotic dynamics of two coupled solenoid-actuated butterfly valves are captured by running the network for some critical values through interconnected flow loads affected by the coupled actuators' variables. The significant effect of an unstable configuration of the valve–actuator on another set is thoroughly investigated to discuss the expected stability issues of a remote set due to others and vice versa.

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The ability to identify physiologic fatigue and related changes in kinematics can provide an important tool for diagnosing fatigue-related injuries. This study examined an exhaustive cycling task to demonstrate how changes in movement kinematics and variability reflect underlying changes in local muscle states. Motion kinematics data were used to construct fatigue features. Their multivariate analysis, based on smooth orthogonal decomposition, was used to reconstruct physiological fatigue. Two different features composed of (1) standard statistical metrics (SSM), which were a collection of standard long-time measures, and (2) phase space warping (PSW)-based metrics, which characterized short-time variations in the phase space trajectories, were considered. Movement kinematics and surface electromyography (EMG) signals were measured from the lower extremities of seven highly trained cyclists as they cycled to voluntary exhaustion on a stationary bicycle. Mean and median frequencies from the EMG time series were computed to measure the local fatigue dynamics of individual muscles independent of the SSM- and PSW-based features, which were extracted solely from the kinematics data. A nonlinear analysis of kinematic features was shown to be essential for capturing full multidimensional fatigue dynamics. A four-dimensional fatigue manifold identified using a nonlinear PSW-based analysis of kinematics data was shown to adequately predict all EMG-based individual muscle fatigue trends. While SSM-based analyses showed similar dominant global fatigue trends, they failed to capture individual muscle activities in a low-dimensional manifold. Therefore, the nonlinear PSW-based analysis of strictly kinematic time series data directly predicted all of the local muscle fatigue trends in a low-dimensional systemic fatigue trajectory. These results provide the first direct quantitative link between changes in muscle fatigue dynamics and resulting changes in movement kinematics.

Tracking or predicting physiological fatigue is important for developing more robust training protocols and better energy supplements and/or reducing muscle injuries. Current methodologies are usually impractical and/or invasive and may not be realizable outside of laboratory settings. It was recently demonstrated that smooth orthogonal decomposition (SOD) of phase space warping (PSW) features of motion kinematics can identify fatigue in individual muscle groups. We hypothesize that a nonlinear extension of SOD will identify more optimal fatigue coordinates and provide a lower-dimensional reconstruction of local fatigue dynamics than the linear SOD. Both linear and nonlinear SODs were applied to PSW features estimated from measured kinematics to reconstruct muscle fatigue dynamics in subjects performing a sawing motion. Ten healthy young right-handed subjects pushed a weighted handle back and forth until voluntary exhaustion. Three sets of joint kinematic angles were measured from the right upper extremity in addition to surface electromyography (EMG) recordings. The SOD coordinates of kinematic PSW features were compared against independently measured fatigue markers (i.e., mean and median EMG spectrum frequencies of individual muscle groups). This comparison was based on a least-squares linear fit of a fixed number of the dominant SOD coordinates to the appropriate local fatigue markers. Between subject variability showed that at most four to five nonlinear SOD coordinates were needed to reconstruct fatigue in local muscle groups, while on average 15 coordinates were needed for the linear SOD. Thus, the nonlinear coordinates provided a one-order-of-magnitude improvement over the linear ones.