In this paper,we discussed the nonlinear evolution of modulation instability and the steady-state process of induced modulation instability in sine-oscillatory response nonlocal nonlinear media. With plane wave plus perturbation as initial conditions, we simulated the long-term evolution of modulation instability in the nonlocal nonlinear Schrodinger equation with sine-oscillatory response numerically. For the input of modulated wave, the approximate analytical solution of the stable solution of the equation is obtained under the assumption that only the fundamental wave and the first harmonic wave are present. For the input of modulated wave with arbitrary harmonic waves, we obtained the exact numerical solution of the stable solution of the induced modulation instability. MI signifies the exponential growth of a weak perturbation of the amplitude of the wave as it propagates. The gain leads to amplification of sidebands, which breaks up the otherwise uniform wave front and generates fine localized structures. MI means the energy exchange between waves at different frequencies in essence, especially referring to the energy exchange between plane wave and perturbation at high-order harmonics6. The energy of plane wave is transformed into that of perturbation at high-order harmonics. Therefore, perturbation is amplified to lead to the generation of localized structures on plane wave during transmission. Theories have validated that perturbation can only trigger MI in a specific frequency range4. Reported was the first experimental observation of the modulational instability using single-mode fibers5. In nonlinear Kerr media, bright solitons can survive within the frequency range in which MI is generated; while dark soliton exist in the frequency range in which MI does not occur8. On the other hand, MI of a nonlinear system leads to the growth of amplitude of modulated wave when the modulated frequency of the input signals of the modulated wave within the frequency range in which MI is generated. The process is called induced modulation instability10. Hasegawa first proposed a theoretical scheme of generating ultra-high-rate pulse trains by utilizing induced modulation instability in fibers3, which has been experimentally verified by Tai et al.10.Modulation instability (MI) is considered as an important problem in nonlinear physics. A series of studies on MI have been carried out since 1960s, such as fluids11. Beeckman mainly discussed the nonlinear process of induced modulation instability in Kerr media with Gaussian response function16 and exponent response function19 and found that the evolution of optical fields exhibited quasiperiodicity in the process.A majority of researches on MI focus on discussing the linear process of MI during short-term transmission, that is processing the condition under small perturbation by using linear approximation method while ignoring the perturbation of high-order harmonics. However, the increment of perturbation is not much lower than the amplitude of plane wave after optical field evolves for a long term, so linear approximation method is not applicable. It is necessary to come up with new methods to discuss the nonlinear process when the increment of perturbation is not small after perturbation evolving for a long term. Beeckman first explored the nonlinear process of induced modulation instability in nonlocal nonlinear Kerr media during long-term evolution by using analytical and numerical methods20. Recently, researches have shown that it is feasible to attain a sine-oscillatory response function when the boundary of the system satisfies specific conditions in a bounded system21. Linear analysis has been carried out on MI in nonlocal nonlinear Kerr media with sine-oscillatory response6. In this paper, we discuss the long-term (nonlinear) evolution of the optical field under MI in nonlocal nonlinear Schrodinger equation with sine-oscillatory response through numerical method, and the characteristics of the evolution was also discussed. The stable solutions  under induced modulation instability of nonlocal nonlinear Schrodinger equation with sine-oscillatory response was also found and the relationship between the beamwidth in the stable solution and two degrees of freedom (DOF) in the system was surveyed.Sine-oscillatory response function is also a common one apart from Gaussian and exponent response functions of nonlocal nonlinear Kerr media, which was first put forward by Nikolov during the discussion of quadratic solitonz-axis in 1 + 1 dimensional media with sine-oscillatory response satisfies nonlocal nonlinear Schrödinger equation23u is the dimensionless slowly-varying complex amplitude of the optical field, x and z are, respectively, the dimensionless transverse coordinate and the dimensionless evolution coordinate, and the sine-oscillatory response function is expressed as follows21wm being the characteristic length of the nonlinear response. Mathematically, the z coordinate stand for time in equation above25, but it is, physically, the longitudinal space (beam-propagation direction) coordinate for the problem of the optical-beam-propagation28.The evolution of the linearly polarized electric field transmitted along 6. However, linear approximation method is not suitable owing to MI has been nonlinear when the optical field undergoes long-term evolution and the growing increment of perturbation cannot be ignored. Therefore, the nonlinear process of MI during long-term evolution would be discussed by employing numerical method. By taking infinite plane wave with perturbationkx represent plane wave, perturbation and frequency of perturbation, respectively.Previous research has been conducted on MI in nonlocal nonlinear Kerr media with sine-oscillatory response and mainly discusses the linear process of MImentclass2pt{minimI0 denotes the intensity of plane wave)6. The amplitude of the perturbation will grow in the exponential form egz in the initial evolution stage when the frequency of the perturbation is within the range, where g refers to the linear gain coefficient of perturbation. At different frequencies, According to the previous research result of linear analysis on MI, it can be seen that the MI in the system described by Eq.  appears g is infinite under Moreover, from Eq.  that a pMI appears in the range of king Eq.  as initimentclass2pt{minimA comparison was made between the analytical result in the evolutionary process of the optical field treated through linear approximation method and numerically simulated result without linear processing, as shown in Fig. 6. It can be attained through numerical calculation that perturbations did not rise in the initial stage of short-term evolution of optical field when the frequency of perturbations was beyond the above range 11), we cOn condition of introducing normalizing condition mentclasspt{minimaMoreover, the intensity of plane wave in the optical field isThe intensity of first harmonic isBy substituting Eqs.  and 15)15) into When only considering plane wave and first harmonic in the evolutionary process of the optical field, the approximate analytical solution of stable solution can be obtained asSimulation was carried out by taking the result of approximate analytical solution obtained using Eq.  at \documentclass2pt{minimt of Eq. ; by contt of Eq. . It indi29. By taking −4 appeared. In this way, the numerical solution of stable solution can be obtained asThe above approximate analytical solution was obtained in the case of only considering plane wave and first harmonic, and the approximation was suitable when the modulation frequency approached to the cutoff frequency. However, it was necessary to consider more number of harmonics during calculation when modulation frequency significantly departed from the cutoff frequency. Whereas, it was impossible to calculate the analytical solution when considering more enough harmonics, so the calculation was performed through numerical method. Based on normalizing conditionThe numerical solutions at all modulation frequencies were attained by utilizing the aforementioned numerical method. The amplitudes of each harmonic and propagation constant in the numerical solutions are shown in Table mentclass2pt{minimIt can be seen from Table Figure According to Eq. , it can Relation  [that isP6. It meant that At first, the change laws of 23 that Eq. ; the exact numerical solution of stable solution was obtained when inputting the modulated wave with innumerable harmonics. By further discussing the stable solution, it can be found that fewer harmonics needed to be taken into account when calculating a stable solution if the modulation frequency of the stable solution was more approximated to the cutoff frequency in the range in which MI appeared. Additionally, the relationship of the beamwidth in the stable solution with two DOFs (power and nonlinear characteristic length) in the system were attained. On same conditions, the beamwidth reduced with increasing power while rose with the growth of nonlinear characteristic length.4 to simulate the propagation of the optical beams. For the inputs Eqs. (We used the split-step Fourier methoduts Eqs.  and 20)4 to simumentclass2pt{minim29 to obtain the exact numerical solution of stable solution, i.e. the solution of Eq. (−4.We used the Newton iteration method


Cavities are typical features in aeronautical structural parts and molds. For high-speed milling of multi-cavity parts, a reasonable processing sequence planning can significantly affect the machining accuracy and efficiency. This paper proposes an improved continuous peripheral milling method for multi-cavity based on ant colony optimization algorithm (ACO). Firstly, by analyzing the mathematical model of cavity corner milling process, the geometric center of the corner is selected as the initial tool feed position. Subsequently, the tool path is globally optimized through ant colony dissemination and pheromone perception for path solution of multi-cavity milling. With the advantages of ant colony parallel search and pheromone positive feedback, the searching efficiency of the global shortest processing path is effectively improved. Finally, the milling programming of an aeronautical structural part is taken as a sample to verify the effectiveness of the proposed methodology. Compared with zigzag milling and genetic algorithm (GA)-based peripheral milling modes in the computer aided manufacturing (CAM) software, the results show that the ACO-based methodology can shorten the milling time of a sample part by more than 13%. Cavity shapes in aircraft structure parts and molds are often complex with highly diverse irregularities . They arUsually, we use peripheral milling method to process a single cavity ,8,9. HowAiming at improving the cutting mode of a single cavity, An et al.  used a BTo avoid cracks in the corners of cavities, secondary corner-milling is required . In CNC In order to compensate the machining error of the part surface contour, Yue  proposedHowever, corner-milling is a part of cavities machining. Most researchers used the corner-milling as an independent study and rarely combined corner-milling with whole cavities tool path generation. With the development of artificial intelligence technology, researchers have tried various artificial intelligence algorithms to optimize the milling tool path of multi-cavity parts, such as genetic algorithms (GA) ,8, ant cIn order to solve the optimization problem of surface roughness during part processing, Li et al.  used a nThe above researchers used various methods to achieve the tool path optimization for multi-cavities. However, there are two main factors that affect the efficiency and accuracy of the calculation results. Firstly, the calculation of the spatial distance between different cavities lacks an effective method for identifying the feature position, which is mainly accomplished by manually setting the position coordinate parameters. Secondly, the interference of blend features makes the acquisition of processing boundaries inaccurate, resulting in inaccurate calculation results.To improve milling accuracy and efficiency, in previous work, we proposed a blend feature simplification (BFS) method to accurately obtain the machining boundaries and tool feed position ,26. FirsIn this method, the tool feed position for a single cavity is selected randomly. In order to avoid repeatedly selecting corner tool feed positions in the same cavity, it is necessary to select a tool feed position for each cavity based on BFS. Thus, the optimization of cavities machining sequences is limited. When milling a cavity using the peripheral milling method, there are different options for the tool feed position. In theory, each corner and geometric center both can be selected. If a certain feed point is selected artificially, it is easy to cause a local optimal solution.Hence, this paper tries to use an ACO algorithm to make up for the shortcomings of previous work. In ACO, the pheromone released during the ant travel is used to find the shortest path. Ants with shorter paths release more pheromone. As time progresses, the concentration of pheromone accumulated on the shorter paths gradually increases, and the number of ants that choose the path increases. Finally, the entire ant colony will focus on the best path under the action of positive feedback. At the same time, it corresponds to the optimized solution of the problem. The remainder of this paper is organized as follows. In Generally, the cavity can be processed by ball milling cutters and end milling cutters, as shown in When the tool reaches the milling plane to perform allowance cutting, the cutter feeds above the corner. There are three common milling methods. (1) Zigzag milling. As shown in When processing corners of the cavity, cutting thickness is uneven due to the sudden change in contact area between cutting tool and the part.Therefore, in milling the corners of multi-cavity part, over-cutting or under-cutting often occurs . In ordeT1 to T2 is a stable linear motion during the milling process, and a constant radial depth ea is maintained at this stage. The coordinates of the tool feed position at the corner can be obtained from the tool contour and the part contour. As shown in nP, and the coordinates of the tool feed position nO can be obtained by Formulaes (1) and (2).nR is the corner radius after milling.The tool path from T2 to T3 is the corner machining stage. The tool path is an arc, and the radial cutting depth changes from ascending to descending. We can obtain nO by Formulaes (3) and (4).O1 is the arc center point of the corner contour before machining. The tool path after T3 becomes a stable linear motion again.From nO can be calculated by Formulae (1) and (2). In theory, for the milling of a single cavity, the machining time for different corner infeeds is equal. However, considering the global perspective of multi-cavity part machining, choosing different corner infeeds will affect the empty travel distance from one cavity to the next, thus affecting the part processing time.The tool feed position For island-type cavities, how to choose the optimal combination of tool feed positions becomes a traveling salesman problem (TSP), as shown in For the problem of multi-cavity tool path planning, the influence of cavity machining sequence is significant. Therefore, we use the ACO algorithm to calculate the initial cavities machining sequence in the global scope. When the tool path is globally optimized, a unique feed position needs to be determined for each cavity. Therefore, it is necessary to add the remaining tool feed positions of the same cavity to the tabu list, which avoids obtaining repeated tool feed positions when machining the same cavity.Based on the above analysis, we designed the algorithm shown in Nc = 0 and set the maximum number of iterations Ncmax. Place m ants on n feed positions.Step 1. Initialize all parameters. Let the number of loops m ants select the next feed position. The remaining feed positions of the same cavity are added to the tabu list, and each ant completes the travel of the respective cavity.Step 2. According to the transition probability Nc = Nc + 1 and continue the iteration.Step 3. Store the route distance. Let Step 4. Update the pheromone and empty the tabu list.Nc reaches the maximum number of iterations. If the maximum number of iterations is not reached, return to Step 1 for the next iteration, otherwise, the algorithm ends.Step 5. Judge if kth ant’s transition probability; j is the feed position of unvisited cavity; k is the position of tool feed that each ant can choose after one transfer. The range of allowedk is In Step 2, the probability kth ant at the In order to meet the needs of all ants passing through all cavities, it is necessary to create a data structure called tabu list to store the tool feed positions that the ants have passed. Since there may be more than one feed position for each cavity, the ants should exclude the remaining feed positions of the same cavity before proceeding to the next path selection. In order to avoid the residual information flooding the heuristic information caused by large amounts of residual pheromone, after each ant finishes a cavity or completes the traversal of all cavities , the residual information is updated. The amount of pheromone on each path can be modified askth ant in this cycle.In general, one step . TherefoIn order to verify the effectiveness of the proposed method, the CNC tool path planning of an aircraft structural part shown in  was usedIn this case, the tool feed positions of cavities are obtained by Formulaes (1) and (2). We make use of the application programming interface of the Siemens Unigraphics (UG) software and carry on the secondary development to UG, which realizes the algorithm, as shown in P.Combined with the 3D model as shown in α is , the expected heuristic factor β has a value range of , the pheromone intensity Q value can be selected between 1 and 20, the residual coefficient of the pheromone ρ has little effect on time, when ρ = 0.5, the optimal solution performance is better [P is obtained from the tool feed position matrix D. The control parameters and algorithm codes are shown in The number of ants is determined by the number of cavities. Combined with s better . AccordiNc) is set to 50. As shown in We record the moving length of each ant and select the shortest ones. Then, the pheromone is updated and the pheromone information on the path (considering the pheromone accumulation and volatilization for each movement) is stored to the pheromone matrix. In the end, we empty the tabu list for the next iteration. When the number of iterations reaches the set value, the iteration is stopped and the resulting optimal path is returned. For the purpose of the validity of the algorithm, we implemented a verification experiment in Matlab. The results are shown in In order to verify the advantages of the method proposed in this paper, we performed a comparative simulation experiment in the UG software. The ball end milling cutter we used has a diameter of 10 mm and a length of 75 mm. The main process parameters are given in In The Siemens SINUMERIK 808D CNC system is adopted in the machining experiment. The G codes of tool path shown in As shown in In The machining results in Based on the above experiments, we obtain the following results:This paper proposed and validated a novel NC tool path planning method for high-speed milling of multi-cavity part based on ACO algorithm. Some of the main contributions of this research are listed below.For the milling of inner cavity corners by peripheral milling method, we established a mathematical model of the corner milling process. Based on this, we combine with ACO algorithm to select the optimal tool feed position for each cavity.The ACO algorithm is used to optimize the tool path of multi-cavity part. Compared with the other two commonly used tool path planning methods, the method proposed in this paper can shorten the machining time by more than 13%.In the near future, several issues are worth being further researched: (1) extension of our research to other machining features, e.g., optimization of process routing in machining a group of holes, and (2) exploration the automatic identification technology of blend features to support intelligent NC machining process planning.


The principles underlying smooth and effortless human walking while maintaining stability as well as the ability to quickly respond to unexpected perturbations result from a plethora of well-balanced parameters, most of them yet to be determined. In this paper, we investigate criteria that may be useful for benchmarking stability properties of human walking. We perform dynamic reconstructions of human walking motions of unimpaired subjects and subjects walking with transfemoral prostheses from motion capture recordings using optimal control. We aim at revealing subject-specific strategies in applying dynamics in order to maintain steady gait considering irregularities such as deviating gait patterns or asymmetric body segment properties. We identify foot placement with respect to the Instantaneous Capture Point as the strategy globally applied by the subjects to obtain steady gait and propose the Residual Orbital Energy as a measure allowing for benchmarking human-like gait toward confident vs. cautious gait. Human and human-like walking motions form an important and challenging class of motions with respect to dynamics and control. The development of efficient measures for benchmarking bipedal locomotion is an important topic for many fields of research, ranging from human motion studies in biomechanics or medical fields to the development and control of humanoid robots, exoskeletons, prostheses etc. Among others, benchmarking allows to define walking standards, measure progress of human walking during therapy, define training goals or to compare different robot platforms, alternative prostheses models or tunings for a patient, different exoskeleton technologies or control algorithms, etc. Good benchmarking measures allow to shift from a purely qualitative comparison of motions to a more useful quantitative one that precisely assesses the quantities in a motion that are considered relevant for quality, thus giving helpful insights for improvement.There are many different types of benchmarking measures. In the KoroiBot project, which studied human locomotion with the intention to improve walking qualities of humanoid robots, we have distinguished three different groups of key performance indicators (KPIs) to benchmark locomotion , i.e., the projection of the Center of Mass on the ground plane, within the borders of the Base of Support (BoS), i.e., the convex hull spanned by the contact points of the system with the ground, can be used as a very simple requirement for static stability of bipedal systems , introduced by Vukobratovic and Branislav (Center of Pressure (CoP). Maintaining the ZMP within the borders of a subset of the BoS has been used by various projects to control the walking motion of a humanoid robot , introduced by Pratt et al. (Instantaneous Capture Point (ICaP).This work focuses on describing the human foot placement strategy in terms of the velocity-based t et al. , Koolen t et al. , and Hoft et al. , since it et al. , Koolen t et al. , and Krat et al.  and desct et al. . Furthert et al. ,b, 2014.Angular Momentum applied by the human walkers about the center of mass. It has been observed by Herr and Popovic  an unimpaired female subject, (B) an unimpaired male subject as well as (C) a male subject walking unilaterally with a transfemoral prosthesis on the right side. Some characteristics of the subjects are included in Table The recordings include a full stride beginning with the Clinic for Orthopedics and Trauma Surgery at Heidelberg University HospitalHeidelberg University Orthopedic Hospital equipped with a Vicon Motion Systems Ltd.The kinematic part of the walking motion of the subjects has been recorded using marker-based motion capturing. The recordings have been gathered in the The subjects' gait has been recorded at self-selected walking speeds. The gait appearance of the unimpaired subjects can be considered healthy, symmetric and without any physical limitations. The impaired subject has been individually fitted with a prosthetic knee which also includes a customized socket and appropriately selected prosthetic components. This subject has been provided on the right side with the Össur hf.  Rheo propelvis, left/right thigh, left/right shank, left/right foot, mid/upper trunk, left/right upper arm, left/right lower arm, left/right hand, and head, respectively.The human body is modeled as a 13-segment multibody model with 34 degrees of freedom (DoFs). The rigid bodies represent the body segments right/left hip, right/left ankle, Lumbo-Sacral joint, right/left shoulder, and Cervicale as well as the 1-DoF joints right/left elbow and right/left knee, respectively. The absolute translation and orientation of the entire system with respect to the global frame in Euclidean space is defined by the six DoFs for the absolute translation and orientation of the pelvis segment.The rigid bodies are connected by the 3-DoF joints Xiphoid joint and the right/left wrists. Hence, zero DoFs are assumed between the middle and upper trunk as well as between the lower arms and the hands, respectively, and the model can be reduced to the model used in this study. The dynamic model parameters for all subjects were obtained using the regression equations provided by de Leva  the contact Jacobian G(q) has full rank and (3) can be uniquely solved.For non-redundant constraints v− before the collision to the generalized velocity v+ after the collision. The transition is determined byWhenever the model gains contact with the ground, the perfectly rigid foot model causes discontinuous transitions from the generalized velocity Λ denotes the contact impulse.where the first line determines the change of the system's momentum caused by the collision and optimal control problem (OCP) in a least-squares (LSQ) sense. In order to minimize the dimension of the optimal control problem the motion is reasonably fitted along the generalized coordinates of the model instead of the Cartesian coordinates.In this paper, unimpaired and prosthetic human walking motions are reconstructed by fitting the motions of subject-specific dynamic models to motion capture data by formulating and solving a large-scale multi-phase The reference motions for the optimal control problems are created by converting the measured motion capture data from marker trajectories in Cartesian space into trajectories in joint angle space. This can be performed by approximating the motion of a subject-specific multibody model such that the distance between virtual markers defined on the model and the appropriate measured marker positions are minimized for each time frame considered. The fit is performed in a least-squares sense and considers the entire kinematic chain of the multibody model j and the joint angles W = diag(wl), wl > 0, l = 1…nact and the factor γu. The ODEs Equation (9) describe the model dynamics in each phase where the right hand sides x(t) as well as the controls u(t) are covered by the path constraints Equation (11). The path constraints for the generalized coordinates x(t) are chosen to reflect the ranges for the typical walking motions. Additional constraints that, e.g., ensure physical feasibility such as unilateral ground contacts as well as switching conditions for phases are contained in the interior point constraints Equations  which distinguish the several gait phases from each other.The first term of the objective function Equation (8) minimizes the sum of squared differences between the model joint angles direct multiple-shooting method and a piecewise linear control discretization. The multiple-shooting state parameterization transforms the original boundary value problem into a set of initial value problems with corresponding continuity and boundary conditions. The multiple-shooting method is implemented in the software package MUSCOD-II  by the motions of the upper and lower body for their contribution to the full body angular momentum in order to reveal individual strategies to compensate for asymmetric dynamic properties of the body or unbalanced habitual gait patterns in steady walking.Capturability concept introduced by Pratt et al. (Instantaneous Capture Point (ICaP)An intuitive approach to quantify and control stability in human walking is motivated as a response to an unexpected loss of balance, e.g., when a sudden perturbation occurs. Since walking can be considered a perpetual falling motion followed by a well-timed and well-placed step, we analyze the walking motion based on the t et al. , Hof et t et al.  and Koolt et al.  and the Center of Pressure rcop, the velocity ṙcom of the pendulum's mass and its angular eigenfrequencyl. In addition, we study the Orbital EnergyElip of the Linear Inverted Pendulum Model (LIPM) underlying the walking system which we normalize by dividing the quantities related to positions and lengths by each subject's leg length (hip to ankle) such thatwith the current position of the Residual Orbital EnergyWe analyze the normalized orbital energy The results for the angular momentum applied by the subjects in the upper body, lower body and full body, respectively, in the frontal, sagittal, and horizontal planes are shown in Figure Subject A has balanced momenta in the frontal and horizontal plane during the left swing phase Figure . HoweverThe angular momentum of Subject B in the frontal plane is regular and symmetric mostly established by a balanced pendulum motion of the upper body Figure . The angSubject C shows well-balanced angular momenta in the frontal and horizontal plane in the time instances right before the heel strike events of both feet Figure . AlthougThe angular momenta in the sagittal plane have typical profiles for all subjects. The whole body momentum is created by forward swinging motions of the arms and legs and is partially dissipated at heel strike.The normalized residual orbital energy Considering the Residual Orbital Energy Using the Residual Orbital Energy In this work, we have investigated benchmarking criteria that help to quantitatively assess the stability of walking motions. As a first test, we have applied them to the walking motions of two unimpaired subjects and one subject walking with a transfemoral prosthesis which have been reconstructed from motion capture recordings using multibody dynamics and optimal control methods. The reconstructed walking motions have been analyzed for their dynamics and findings are gathered on how unbalanced habitual gait patterns can lead to irregular walking motions. On the other hand, the analysis provides insights into the individual strategies applied by the subject walking with the prosthesis to compensate for his asymmetric dynamic properties of the lower limbs and achieve a perfectly balanced walking motion.For all subjects, the ICaP is shown to be consistently approached by the swing foot even if the walking paths deviate in lateral direction. The subjects choose their step locations such that the ICaP is located in anterior-medial direction of the foot and maintain a Residual Orbital Energy at heel strike >0 in order to facilitate the forward propulsion and lateral oscillation characteristic for human walking. The Residual Orbital Energy combines the subject's distance maintained between the CoP and the ICaP at each step with the gait velocity which, in human walking, is controlled to a great part by the rate of change of angular momentum.In our proposed method we simultaneously interpret foot placement with respect to the ICaP as well as the Residual Orbital Energy. By deconstructing the walking motion into these parts we are able to reveal hidden phenomena in gait which superficially appears regular and symmetric. Regarding these criteria, the walking motion of both unimpaired subjects turn out to be irregular while the impaired subject's gait is well-controlled and well-balanced. We suspect that the unimpaired subjects' awareness of their physical capabilities provides them with enough confidence to allow for less conscious gait. On the other hand, we suspect that the impaired subject is aware of his limited ability to control his prosthetic leg and, therefore, follows a more cautious and conscious approach to walking.The results are encouraging, but the criteria obviously remain to be further tested and validated on large sets of data. A thorough evaluation of these criteria based on existing whole-body walking data , e.g., from data bases, KoroiBot Motion Capture DatabaseThis study was carried out in accordance with the recommendations of the ethics guidelines 158/2006 issued by the ethics committee of the University of Heidelberg with written informed consent from all subjects. All subjects gave written informed consent in accordance with the Declaration of Helsinki. The protocol was approved by the ethics committee of the University of Heidelberg.K-LH developed the hypotheses, developed and executed the procedure for data processing and analysis, and wrote the manuscript. SW designed the experiment, collected data and advised in data analysis, KM advised in data processing and analysis and contributed to writing the manuscript.The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.


Hitherto acoustic cloaking devices, which conceal objects externally, depend on objects' characteristics. Despite previous works, we design cloaking devices placed adjacent to an arbitrary object and make it invisible without the need to make it enclosed. Applying sequential linear coordinate transformations leads to a non-closed acoustic cloak with homogeneous materials, creating an open invisible region. Firstly, we propose to design a non-closed carpet cloak to conceal objects on a reflecting plane. Numerical simulations verify the cloaking effect, which is completely independent of the geometry and material properties of the hidden object. Moreover, we extend this idea to achieve a directional acoustic cloak with homogeneous materials that can render arbitrary objects in free space invisible to incident radiation. To demonstrate the feasibility of the realization, a non-resonant meta-atom is utilized which dramatically facilitated the physical realization of our design. Due to the simple acoustic constitutive parameters of the presented structures, this work paves the way toward realization of non-closed acoustic devices, which could find applications in airborne sound manipulation and underwater demands. Not long after, Cummer and Schurig presented a two-dimensional (2D) acoustic cloak by illustrating the analogy between 2D time harmonic Maxwell equations and acoustic wave equations2. Later, a similar coordinate transformation method was further extended to design of 3D acoustic cloaks by Chen and Chan3. The same relations through the acoustic scattering theory were also derived in4 by Cummer et al. Due to the potential applications of cloaking in a variety of both acoustics and electromagnetics scenarios, numerous schemes have been devoted to theoretical development and fabrication of cloaking devices10. The major difficulty in ideal cloak implementation is the requirement of extreme material properties, which is resulted from the mapping of a point or line to the cloaked region in the cylindrical or spherical cloak, respectively. To remove this obstacle, the concept of carpet cloak was proposed11. Carpet cloak or ground plane cloak is a device that restores the signature of the target as if the incident wave reflects from a mirror plane. The carpet cloak is designed by employing a coordinate transformation from a flat sheet to the cloaked region that creates a bump with the mirror plane, resolving the need to extreme materials’ parameters. Later, inspired by the carpet cloak concept, unidirectional free space cloak was proposed to represent the cloaking effect for a specified direction of propagation12. The strategy to unidirectional cloak design is based on the property of the mirror plane to be invisible when probed by a plane wave propagating parallel to it.Invisibility cloak is one of the most attractive research topics due to its exotic properties in deflecting the waves around objects. A powerful approach to achieve invisibility devices is the coordinate transformation method firstly utilized by Pendry et al. for the design of ideal electromagnetic (EM) cloak12 with the remarkable advantage of minimizing the anisotropy of obtained materials15. However, the inhomogeneous structure of quasi-conformal cloaks leads to a difficult fabrication process and neglecting the weak anisotropy causes a lateral shift in the reflected wave16. Another disadvantage is the size of this type of cloaks, which is bulky compared to that of the target. To overcome these challenges, linear transformation based carpet cloak20 and unidirectional cloak23 were proposed. Cloaks based on linear coordinate transformation have homogeneous constitutive parameters with finite anisotropy, which obviates the need for space dependent materials. So far, in addition to huge amount of theoretical investigations to advance of cloaks28, several studies experimentally demonstrated acoustic carpet cloaks via homogeneous fluid-like materials. For example, in airborne acoustics, perforated plastic plates have been applied for 2D29 and 3D30 ground plane cloaks and for underwater acoustics, steal strips31 and brass plates32 have been proposed. In addition, acoustic unidirectional cloaks were implemented via composites of metals and porous materials for a multi-layered host medium33 and meta-fluid structured by slab-shape units for air host34.The first proposal of carpet cloak and unidirectional cloak was presented through quasi-conformal mapping35 presented "external" cloaking and illusion devices for the EM framework. Because of relevant acoustic demands, not long after the first proposal, the idea was extended to acoustics by Zhu et al.36. The key enabling feature of designing such external or “non-closed” invisibility devices is the complementary media concept, which is designed to conceal a predefined object40. However, complementary media based cloaks depend on the shape and material properties of the hidden object42. Therefore, any changes in description of object or any movements disturb the cloaking effect. Consequently, the dependence of the cloaking device on target and inevitable inhomogeneity of obtained materials are the remaining challenges that make the implementation of conventional external cloaks practically nonrealistic.Although considerable progress has been made in invisibility cloaks, all conventional cloaking devices are “interior” cloaks and thus prevent the target from interacting with outer world, which is a restriction for the use of these applicable devices. In order to obviate this drawback, Lai et al.Differ from conventional design, in this paper; we design non-closed acoustic cloaks (NCACs), in which their materials are feasibly independent of the target geometry and its constitutive materials. The proposed technique is based on applying sequential linear transformations to create a non-closed invisible region positioned on a reflecting plane. Therefore, any standing or moving object in the hidden region will be acoustically invisible without being blinded. Due to the nature of linear coordinate transformation, the obtained materials of NCAC are homogeneous which results in easy to fabricate non-closed carpet cloaks. Moreover, the idea of non-closed devices is further extended to design a free space unidirectional acoustic cloak that can conceal any arbitrary objects. The full wave numerical simulations using COMSOL finite element solver verify the expected behavior of NCACs. Finally, to give a more realistic point of view, the required constitutive materials are realized with the aid of non-resonant acoustic meta-atoms.18 has a hidden region underneath a sound hard boundary bump where any arbitrary object located becomes invisible. The main purpose is to design an acoustic cloak, which, in addition to attempting to avoid enclosing objects, its constitutive parameters being independent of the material, geometry and position of objects. Figure To start with a conception of idea, it is well understood that the conventional carpet cloak18 will be proposed. Without loss of generality, the problem is discussed in 2D framework while it could be extended to 3D case. As demonstrated in Fig. 35 and is actually an SHB mimicking structure, without any imposed sound hard boundary condition. Considering the geometrical symmetry of the scheme, the problem is discussed only in the left half space of the reference and real spaces and a same method is applied to the right side. Analytically, a linear coordinate transformation that maps At the first step, the conventional carpet cloak3, our design strategy is valid for any desired operating frequency and the chosen frequency, is completely arbitrary. By taking the left side of problem's geometry as the reference, at the first and second steps of the design, the acoustic constitutive materials for conventional cloaks and the SSHB region are In order to verify the effect of carpet cloak with SSHBs, numerical simulations are carried out using the COMSOL Multiphysics finite element solver. All simulations are performed by adopting the acoustic pressure field in the frequency of 3.34 kHz, and the host medium is chosen as air with 43. Thereupon, the compressing transformations map Finally, we can open a window in the carpet cloak with SSHBs to make its hidden region non-closed. Subsequently, at the third step, the carpet cloak with SSHBs and its surrounding fluid in the reference space ,y2 Fig. a, are co35 that convey the route of wave in the real space to the compressed space by employing a linear transformation. The linear coordinate mapping folds the black dashed lines representing the reference space to black solid lines in the real space. The significant result of this mapping is the invariance of waves’ path from the reference space to the real space  created on the structure are extended, which leads to increasing the value of negative constitutive parameters. The presented approach surmounts resorting to spatially-varying constitutive parameters and object-dependent performance of the cloaking devices. The homogeneous material parameters of the proposed devices significantly facilitate the realization of acoustic external cloaking devices. To give a more realistic point of view, the required materials for cloaking devices are realized with the aid of non-resonant acoustic meta-atoms. Due to these benefits, the proposed structures could find applications in varied scenarios such as making submarines invisible from sonar with non-blinded or fenestrate structures. As a proof-of-concept demonstration, the proposed NCAC design is carried out at a selected frequency. It should be noted that the strategy itself is not dependent on the choice of the frequency and the design can be updated for any desired operating frequency. The impact of loss is minimized due to the non-resonance meta-atom. As the meta-atoms are dispersive, the bandwidth of the device can be limited for a practical design. To increase the bandwidth, achromatic metamaterials with inverse design techniques may be used to decrease the frequency dependence of the effective properties. In addition, based on the effective medium theory, the dimension of meta-atom must be much smaller than the operating wavelength. To apply the concept to higher frequencies, the size of meta-atoms should be decreased which is feasible due to the advanced technology of fabrication. The presented method can also be applied to other acoustic devices, such as acoustic cavities, waveguides, illusion devices, etc., to make them non-closed to outer world that could be useful in future acoustic demands.Supplementary Information


Footwork is the most fundamental skill in badminton, involving the ability of acceleration or deceleration and changing directions on the court, which is related to accurate shots and better game performance. The footwork performance in-field is commonly assessed using the total finished time, but does not provide any information in each direction. With the higher usage of the smartphones, utilizing their built-in inertial sensors to assess footwork performance in-field might be possible by providing information about body acceleration in each direction. Therefore, the purpose of this study was to evaluate the feasibility of a smartphone-based measurement system on badminton six-point footwork. The body acceleration during the six-point footwork was recorded using a smartphone fixed at the belly button and a self-developed application in thirty badminton players. The mean and maximum of the acceleration resultant for each direction of the footwork were calculated. The participants were classified into either the faster or slower group based on the finished duration of footwork. Badminton players who finished the footwork faster demonstrated a greater mean and maximum acceleration compared to those who finished slower in most directions except for the frontcourt directions. The current study found that using a smartphone’s built-in accelerometer to evaluate badminton footwork is feasible. Badminton is a popular racket sport globally . The perAn efficient evaluation of footwork performance is crucial in order to provide quantitative data on in-field performance. However, to date, methods for the assessment of footwork performance on the badminton court remain preliminary. Typically, only the total time to finish the six-direction footwork is obtained using a stopwatch, without collecting any more detailed information, such as the speed of movement in each direction . Depth aMost relevant studies in laboratory settings have used optical motion analysis systems to investigate the swing motions involved in shots ,9, but oTo overcome these limitations, inertial measurement units (IMUs) are increasingly being utilized because of their smaller size, simple operation, and lack of location limitation . The intFew studies have utilized IMUs for assessing agility performance or confirmed that IMUs can provide reliable and valid values to represent the shuttle run speed and ability to change directions ,17,18. CHowever, some limitations remain when using conventional IMUs on badminton courts, including the power supply for the personal computer and sensor station, as well as the limited range of data transmission via Bluetooth. Although IMUs are considerably more affordable than motion analysis systems, their convenience and availability remain a concern for coaches and players . As the Therefore, the primary purpose of this study was to develop and verify the feasibility of a smartphone-based badminton footwork performance measurement tool. The current findings could benefit both players and coaches during their daily training and follow up by providing them with quantitative information of each direction of the footwork performance, and a specific training program could be designed accordingly.p > 0.05). Players in the fast group had a significantly shorter duration for completing the footwork than the slow group did (p < 0.001).A total of 30 badminton players participated in the study, including 22 men and 8 women . The aveThe application (app) was designed using the Android Studio 4 to acquire data from the accelerometer of the smartphone and to be operated by the users themselves. Because the Android system has a higher market share worldwide than other operating systems, we decided to develop the app under the Android operating system to maximize the future usability . The selThe ASUS Zenfone 6 was used in this study because of the steadiness of the built-in sensor sampling rate compared to other brands such as hTC in our pilot study. The built-in accelerometer was selected because related literature has proven the utility of accelerometers in evaluating sports-related performance ,24. The AR) was calculated using the following formula:The acceleration data obtained from the accelerometer were processed using the MATLAB R2019a  software. Footwork performance was indicated as the mean and maximum of the acceleration resultants in each direction of the footwork task and during the whole task. The acceleration resultant was calculated, since the body was turning three-dimensionally while performing footwork ,26,27. TThis study was approved by the Institutional Review Board of National Cheng Kung University Hospital. Eligible participants were informed of the purpose and experimental procedures of the study and signed a consent form prior to participation. Data collection was conducted on a standard badminton court in the university. The functions of the app were first introduced to each participant. Participants were required to be able to operate the app independently. Subsequently, the smartphone was attached to the belly button (the standardized location) with a waistband . The foot-tests were used to compare the demographic differences between the fast and slow groups, as well as to determine whether the smartphone system could discriminate the differences in footwork performance between the two groups. In addition, the effect size (Cohen’s d) and 95% confidence interval of each variable of interest were reported. All the data were analyzed using SPSS . The level of significance was set a priori at α = 0.05.The normality assumption for parametric statistical analysis was tested using the Shapiro–Wilk test. Data are presented as means and standard deviations. Independent The results showed that utilizing the smartphone’s built-in accelerometer could differentiate the performance between the faster and slower groups based on the mean and maximum acceleration result. The means of the acceleration result for the entire footwork task and in each direction are presented in The maximums of the acceleration result for the entire footwork task and in each direction are presented in The current study aimed to assess the feasibility of a straightforward footwork evaluation system utilizing a smartphone’s built-in sensor. To extend the usage to the general public, this system was specially designed for ease of use and with an embedded calculation function. Efficient performance of six-point footwork on the court is associated with competition results and is vital for badminton players. However, to date, no precise and convenient assessment method is available for assessing footwork on the court other than the total completion time of the task. In this study, we developed a smartphone-based assessment system that employs the built-in accelerometer and a self-developed app to quantify the on-court footwork performance in each direction. The current results revealed that the smartphone’s built-in accelerometer and the developed app could be used to differentiate the performance between the players who finished the six-point footwork faster and slower according to the parameters of the average and maximum acceleration achieved.Researchers have employed in-field inertial sensors for measuring movements or performance in a wide range of sports, such as baseball, ice skating, running, and tennis. Spatiotemporal parameters, center of mass kinematics, segmental orientation and joint kinematics have been analyzed . AccelerTo the best of our knowledge, no study has utilized acceleration to quantify the performance of badminton footwork. Only a few studies have used accelerometers to examine speed or agility during sports. In 2015, McGinnis et al. recruited 32 participants from the local university population to analyze the agility performance by using an accelerometer. The inertial sensor was fixed to the participants’ sacrum with a belt before a slalom run. The course was composed of seven cones spaced 5 m apart to define five changes of direction. The displacement, velocity, and acceleration of the participants were measured. The results indicated that inertial sensors could provide quantified biomechanical metrics for speed and agility performance in agility assessment tasks . In 2017In this study, the players who finished the footwork faster demonstrated a larger mean and maximum acceleration result than did the players who finished slower in most directions, except for the right and left frontcourt. The results support that badminton players may have different performance among each direction of the footwork, and it is necessary to develop a method for assessing the footwork performance in each direction. Players who finished the footwork slower still had a similar acceleration ability to the players who finished faster in the forward directions. The possible explanation is that the right or left frontcourt directions, alongside the forehanded or backhanded swing, are maybe the most common and easily performed movements among all six directions for players. Forward stepping and lunging is a more intuitive direction for individuals to initiate acceleration. A relatively simple footwork strategy in the frontcourt directions may not challenge most players in this study. Therefore, no differences in the frontcourt directions between the two groups were found. The discrepancies between the two groups were observed in the sideward and backward directions, suggesting agility training for those who finished footwork slower could emphasize on these specific directions.The findings of the current study highlight that using the built-in accelerometer could provide more information on footwork performance in each direction than only the completion time using a stopwatch. Badminton players may have different acceleration or deceleration ability depending on the footwork directions. The readily available smartphone and our self-developed app could easily provide the badminton players information of their acceleration or deceleration ability in each direction of footwork. Such a system assists the players and coaches to notice their direction of agility weakness on the court. A scientific-based and individualized training program may thereby be designed and implemented. The self-developed app system could also record the footwork performance of an individual day to day. The long-term changes of footwork performance could be followed which may assist the coaches to examine the effects of different training programs or help to monitor the health or injury conditions of the players. The self-developed app in the current study, which acquires the body acceleration data from the built-in accelerometer of a smartphone, enables a more comprehensive assessment of footwork performance in each direction of the court and may have considerable implications for sports in-fieldSeveral limitations in this study should be acknowledged. Only built-in accelerometer data were obtained; thus, only the linear aspect of performance was evaluated. Performance when changing directions or turning was not assessed in this study. Although our results showed that the acceleration resultant of the body could provide sufficient information on footwork in each direction, adding the angular data would definitely provide even more details of the agile performance. Future research could consider obtaining the gyroscope data from the smartphone to assess the angular performance of footwork.The sequence of the footwork directions was not randomized in this study, and all participants were required to move from the right frontcourt in a clockwise direction to the left frontcourt. The clockwise design was to avoid the additional cognitive demand to the participants while memorizing the order of the directions or reacting to sudden instructions in different directions. The results might be less affected by the variability of the participants. However, a non-random order of footwork directions may induce bias such as fatigue or learning. The randomization of footwork directions is required in future studies. The overall finished duration of the footwork in this study was measured using a stopwatch, which may over- or underestimate the time and thereby influence the results. Future studies may consider using the light barriers to obtain more accurate and reliable time data for the footwork.Finally, the badminton players recruited in this study were mainly from the university campus and were, therefore, relatively young. This lack of diversity in participants might limit the generalizability of the current findings. Future studies could consider recruiting players from diverse sources with various skill levels, including amateur and professional players.The current study confirms that using a smartphone’s built-in accelerometer to evaluate badminton performance is feasible. The self-developed app system can acquire body acceleration data and provide more information regarding each direction of footwork. Badminton players who finished the six-point footwork faster demonstrated larger mean and maximum acceleration results than players who finished slower in most directions of the footwork task, except for the frontcourt directions. This system could be an easy to use alternative assessment system for the badminton court which might further benefit players and coaches when designing their training programs.


Biphasic soft materials are challenging to model by nature. Ongoing efforts are targeting their effective modeling and simulation. This work uses experimental atomic force nanoindentation of thick hydrogels to identify the indentation forces are a function of the indentation depth. Later on, the atomic force microscopy results are used in a GENERIC general equation for non-equilibrium reversible–irreversible coupling (GENERIC) formalism to identify the best model conserving basic thermodynamic laws. The data-driven GENERIC analysis identifies the material behavior with high fidelity for both data fitting and prediction. Thanks to the recent progress in simulation technology and computing power, the mechanical behavior of biological tissues is nowadays one of the most active research topics. However, many biological tissues are biphasic by nature, which renders their effective modeling and simulation a challenging issue, even with the impressive progress achieved recently . Human cDifferent efforts are undertaken to model soft materials. In , the autAll the aforementioned studies use experimental results to tune previously defined models, either using finite element simulation or analytical derivations. However, the derived models will always lack the exact truth due to the missing knowledge or information, and the errors involved in experimental measurements. Moreover, models derived to reproduce a given test situation may work pretty well in that specific test. However, their predictive ability in different loading cases, boundary conditions, or other basic variations in the problem, may not be as accurate as desired. This fact has lead to the development of data-driven models to overcome the need to predefine a “model”, as well as hybrid-twins models aiming to correct established models with experimental results . Data-drDespite being popular nowadays, data-driven models and hybrid ones sometimes can violate the basic established principles of physics, like mass and energy conservation, entropy production, etc., especially when the data sample size is not large enough and/or contains multiple outliers. In fact, no physical constraint is currently integrated in machine learning techniques to alleviate the possibility of violating the basic principles of thermodynamics. Recent efforts aim to create a physically informed neural network for supervised learning, through integrating the physical differential equation error in the neural network optimization ,17. An aRecently, the GENERIC formalism is used to derive data-driven modeling, while preserving basic thermodynamics laws . Such anThis work consists of a first attempt to model soft materials, hydrogels, for instance, using the GENERIC formalism and experimental data. This work aims to formulate a new constitutive model using experimental data and the conservation of thermodynamic quantities. The experimental setup consists of a unidirectional nanoindentation test with atomic force microscopy, using a spherical rigid indenter. The experimental device measures the displacement and the spring back force at the indenter. The indented specimen is a highly porous hydrogel medium, which is a material of high interest currently . In factThe work starts with a review of the experimental indentation process at hand, giving the reaction forces in a thick hydrogel poroviscoelastic material, as a function of the process input parameters: the indentation depth and velocity of the indenter. Later on, a modeling of the process is performed and then the GENERIC formalism used in this work is detailed along with the identified model. Finally, the numerical results are illustrated and discussed.In this section, we review the experimental setup used to identify the mechanical behavior of a thick hydrogel. The experiment uses atomic force microscopy nanoindentation (AFM nanoindentation) to identify the mechanical response of a hydrogel. AFM is usually modeled as a cantilever beam supporting a semi-spherical indenter of radius F at the base of the spring, which corresponds to the force in the spring. Therefore one may easily write:The specimen indentation depth is z and F at every time step R the indentation radius defined as:The experiment is repeated for five different indentation rates, Moreover, we note that the initial hydrogel specimen height Under the framework of non-equilibrium thermodynamics, the GENERIC formalism establishes a completely general equation of the dynamics of a system under reversible and irreversible conditions, as dictated by the evolution of energy and entropy, respectively ,23,27. M1.t. The choice of [L is the so-called Poisson matrix and will be responsible for the reversible (Hamiltonian) part of the evolution of the system.E t, represents the energy of the system, as a function of its particular state at time M represents the friction matrix, responsible for the irreversible part of the evolution of the system.S represents the entropy of the system for the particular choice of variables The GENERIC structure of the evolution equations for an arbitrary problem isEquation  is supplz in In what follows, if there is no risk of confusion, and for the sake of readability, we will omit the subscript By choosing The objective is to model the biphasic hydrogels using GENERIC formalisms, thus to obtain the GENERIC Equation  for the Since data Equation  is firstThe set of variables chosen for each As can be seen in , differeEquation  and zexpThe just-introduced procedure must not be seen as a model fitting procedure. Indeed, the number of values to determine is much higher than the usual number of parameters in a suitable model. What we seek with this procedure is a method of machine learning the GENERIC expression of the problem. The result is the numerical value of these GENERIC building blocks and not the particular values of any model parameter. Those are considered constant for each experiment, although a piece-wise linear variation along time is equally possible, in general.Once the minimization problem has been solved, and given an initial For practitioners used to employ experimental results for constitutive law fitting purposes, the procedure outlined above could seem intricate and unclear. Particularly, regression of the terms in Equation  could soNote, however, that some terms in Equation  should sFrom a practical point of view, the development of finite element time integration schemes deriving directly from GENERIC expressions is something already developed by different authors, particularly I. Romero and coworkers. Equation  could beThe experimental measurements are taken every w as a function of the number of measurements taken from the beginning of the experiment, which is proportional to the time t since measurements are taken every 15 ms. We can clearly see an excellent comparison of the derived GENERIC model with the experimental indentation depth, with an average relative error of w using explicit Euler’s derivation. Using the generic formalism, we can also show the indenter velocity The relative errors for different GENERIC formalism modeling, for the three quantities of interests, are illustrated in w for both materials in All experiments have low relative errors in general. We can clearly see negligible relative error for the indentation depth N is the number of measurements in the experiment using a given indentation rate.In this section we use the aforementioned GENERIC regression to predict the experimental results for an indentation velocity Using the linear interpolation approach, we predict the GENERIC formalism matrices. Eventually, using a more accurate interpolation scheme is an option for interested users (Kriging for example). The prediction results are illustrated in This work consists of a first attempt to explore the possibilities of using the GENERIC formalism with data-driven identification, to model the indentation of thick hydrogels. The results show a good match with the experimental data. The use of three different matrices to identify in the model gives an advantage and flexibility of identifying a model with 27 parameters. The 27 parameters  are eventually enough to reproduce the best behavior of the material. Indeed, not all the identified parameters are independent. For instance, the symmetry of the approach , or a coDifferent matrices chniques . The resThe choice of using the GENERIC formalism appears to be a suitable approach for identifying models with challenging mechanical behavior but without prior exact knowledge of the constitutive equations. It also shows good predictive abilities for unfitted experimental results, as illustrated in In this work, we investigate the possibility of using the GENERIC formalism for modeling non-trivial material behavior. For instance, the work focuses on modeling thick hydrogels using only the conservation of thermodynamic laws. The results show a good correlation with the experimental ones for different experiments. The predictive ability of the model is illustrated in


The reliability of aero engine has a direct impact on the flight safety of the whole plane. With the continuous improvement of performance requirements of aero engines, the related fatigue and reliability problems also appear. For the fatigue failure characteristics of the typical component (compressor disk) in an aero engine, the fatigue reliability of its multi-site damage structure in service is analyzed by using probability cumulative damage criterion in this paper. The probability distribution definitions of life, damage and damage threshold are discussed and the relationship among them is also introduced by the new proposed criterion. Meanwhile, a method to determine the probability distribution of cumulative damage threshold and probability life prediction is carried out, based on which a hierarchical index system of statistical analysis and reliability modeling principle on the system level is further constructed for compressor disk. At the end of the paper, a certain cruise of fighter plane is analyzed to verify the validity of the new model. Emphasizing the difference between the compressor disk and traditional component, the new reliability analysis model developed in this study is basically reasonable for most of the load histories for the compressor disk, other than the traditional one, especially for the changeable and complex cruise missions. The reliability of an aero engine has direct effects on the safety of the plane. With the continuous improvement of performance requirements for aero engines, the engine rotors with lighter structure mass, larger workload and higher endurance temperature are required, for which the related fatigue and reliability issues are subsequently exposed . Among vP-S-N) curve [In recent years, multi-site damage has caused great concern. Moreover, relevant studies on multi-crack structures have been conducted on different aspects. There are lots of studies reporting on multi-crack structures in different aspects such as residual strength , fatigueN) curve ,21,22,23N) curve ,25,26. TS-N curve, cyclic stress-strain curve, standard deviation and probability density distribution of load history, as well as rain-flow counting method are applied to obtain the cyclic load spectrum block. After that, Petryna et al. [The problem of fatigue failure probability under variable amplitude load history is more complicated, but various studies are limited to specific parts or structures with multi-site damage rarely being taken into account ,28,29,30a et al.  proposeda et al.  introduca et al.  further S–N curve, which can be used for the multi-site damage of aero-engine compressor disk. Based on the median cumulative damage criterion, the probabilistic life prediction method of multi-site damage structure is proposed, and the reliability analysis model is established accordingly.In this paper, a method (named median cumulative damage criterion) is introduced to calculate the probability fatigue cumulative damage and probability distribution of critical damage value based on the median The fatigue life at specific cyclic stress level is a random variable, usually obeying the Weibull distribution or lognormal distribution. The distribution of fatigue life under variable amplitude load will show a more complex form. In order to estimate and predict the probability life of components under variable amplitude load, the probability characteristics of fatigue damage and the related cumulation method should be studied. Firstly, the probability distribution of fatigue damage and the expression of its threshold under variable amplitude load was put forward, as well as the calculation method of damage cumulation corresponding to random critical damage.is, the fatigue life of a structure is a random variable with median value if(N). After in load samples are applied on the structure, the failure probability can be calculated byUnder the specified cyclic stress level iN as a deterministic quantity iN), the stress cycles in can be taken as a random variable, represented by the symbol in, by which the relationship In order to facilitate the damage equivalent transformation under different stress levels and the cumulation calculation of probability damage, the probability equivalent transformation between the deterministic stress cycles and random variables of fatigue life should be carried out. By assuming the random variable iN is a random variable, the damage i/Nin of stress cycles in is also a random variable, expressed as Obviously, since such a transformation can be realized, the probability damage can be calculated based on the median life value and the corresponding random variable of stress cycle. Similarly, according to the equivalence relationship between From the perspective of damage threshold, the failure probability under specified cyclic stress amplitude can be expressed asN is a random variable, according to Equation (3), the cumulative damage threshold can be interpreted as a random variable. According to the conventional and deterministic linear cumulative damage criterion, the damage threshold is independent of stress level. In order to consider the failure probability, the relationship between threshold of damage random variable and stress level needs to be analyzed firstly in probabilistic meaning. Coincidentally, our previous study [Since us study  has showTherefore, the probability distribution of cumulative damage threshold can be determined based on the life distribution at any stress level .The conventional cumulative damage is defined under the condition that the life and load history are both deterministic. Since the life is a random variable, damage caused by the certain load history is also a random variable. To obtain probabilistic life or reliable life, it is necessary to formulate an appropriate probability cumulative damage criterion. As mentioned above, fatigue damage accumulation can be calculated according to the median life value. Then the probability life can be obtained based on the probability distribution of median cumulative damage and cumulative damage threshold, which is the prime target of this section.jn cycles under stress js can be equivalent to the fatigue damage generated by in cycles under stress isFollowing the linear damage accumulation theory, the median life value under different stress levels is converted to calculate the equal damage stress cycles under corresponding stress levels, which means the fatigue damage generated by Based on the assumption that the distribution of damage critical value of the same material is independent of the stress level, the basic property of "same damage leads to the same failure probability" can be further deducedEquation (5) is the basis of new probabilistic damage accumulation criterion defined in this study. Based on this basic relationship, the failure probability under variable amplitude cyclic stress is given byThis just means that the failure probability under variable amplitude stress is the probability that the linear cumulative damage value calculated based on the median life value is greater than the critical damage random variable.m stress distributions, the ith stress distribution is expressed as if(s) with in load samples. On the basis of the linear cumulative damage calculation formula (Miner rule), the cumulative damage  of component calculated by the median life at each stress level is given by0s are the median S-N curve equation and fatigue limit of component material, respectively. For the component with M damaged sites, the cumulative damage at the jth damaged site can be obtained as followsith stress distribution on the jth site and the corresponding number of loads, respectively.For multi-site damage structures, fatigue damage of each damage site is different under the variable amplitude loads, as well as the damage evolution process, caused by the different stress histories at different damage sites. Supposing that the load history of a structure is composed of j isf(cd) and F(cD) are respectively the probability density function and cumulative probability distribution function of damage threshold calculated under the selected stress level.By substituting the results into Equation (6), it can be obtained that the failure probability of damaged site β and η are shape parameter and scale parameter of Weibull distribution, respectively.If the life follows two-parameter Weibull distribution at the base stress level, the probability density function can be expressed asBased on the definition of median damage threshold (j can be expressed asSubstituting Equations (11) and (12) into Equation (9), in the process of loading, the specific expression form of failure probability of the damaged site M damaged sites are expressed as Equations (14) and (15), respectively.According to the probabilistic algorithm of series system, the failure probability and reliability of component with For the probabilistic fatigue life prediction model proposed in this study, a specific example was employed to verify its applicability in this section. Referring to the published literature  and the μs and standard deviation σs) determined by the characteristics of aero engine output power at each cruise phase. The statistical parameters of the spoke hole on the compressor disk  in five phases are listed in −1 unit is employed due to the hours-long cruise mission, which is investigated in our study. Besides, according to the actual cruise characteristics and service conditions of the aero engine, the rotating speed v and working time t of the aero engine rotor at the corresponding phase are also given to describe the load number of the compressor disk at each phase, as listed in Similarly, based on S-N curve distribution of 961 steel is shown in S-N curve of material can also be fitted, as shown in The structural sketch of the compressor disk is shown in Then, it can be transformed into the form in Equation (7)jth spoke hole (damage site) in each cruise phaseiv is the rotating speed in the ith cruise phase, and the value is shown in j-spoke hole (damage site) in a cruise mission (including five cruise phases) is given byit is the cruise time of the ith cruise phase, and the specific value is shown in σ criterion" to make a finite amendment to the upper limit of the integral in Equation (19). The modified result is shown asμsi and σsi are the mean and standard deviation of load distribution in the ith cruise phase, respectively. By substituting the analysis results into Equations (14) and (15), the failure probabilities and reliabilities of single spoke hole j and compressor disk with 8 spoke holes in a cruise mission can be expressed byResulting from the analysis above, the corresponding relationship between the median life of material and stress level (Equation (17)) is substituted into Equation (8) to obtain the cumulative damage on the β = 3.4569, η = 690230 and Based on the previous test data analysis in  and EquaAccording to the calculation results, the fatigue life of the compressor disk is much lower than that of a spoke hole (damaged site) under the given failure probability due to the effect of uncertainty life. Apparently, there is an obvious difference between the lives of structure and one damaged site when the randomness becomes larger and service life becomes longer.Moreover, it can be concluded that the reliability of a compressor disk calculated by an integral solution method remains constant as long as the statistical average parameters of all phases in these cruise missions are the same. However, reliabilities calculated by the subsection solution method for these cruise missions are different as the results are obtained based on the parameters of all independent phases in these cruise missions. Seen from E is defined with the results of the new model reference.0R is the reliability of the new method; 1R is the reliability of the stress-strength interference model; minR is the reliability limit of compressor disk in the design period, minR = 0.99 here. The error index of the stress-strength interference model, compared with the new method, in one cruise mission is shown in In order to fully embody the difference of the subsection solution method proposed in this paper, the analysis result of the stress-strength interference model is also calculated to compare with the result of the new model. In the calculation process of the stress-strength interference model, assume that the degeneration of material strength follows an exponential criterion, which can be obtained from S–N curve is introduced, based on which the probabilistic life prediction method of multi-site damage structure is proposed, and the reliability analysis model is established accordingly.A method to calculate the probability fatigue cumulative damage and probability distribution of critical damage value based on the median Due to the complex working environment applied to aero engine, a subsection solution method is innovatively proposed by dividing a cruise mission into several phases with different loading environments.To analyze the reliability of compressor disk before the incidents occur, a new reliability analysis model is developed in the paper. The major characteristics of this study are as follows:S–N curve and life distribution under a certain stress level are required for material performance data, for which it is convenient for engineering application. At the end of the paper, a certain cruise of fighter plane is analyzed to verify the validity of the new model. Emphasizing the difference between compressor disk and traditional component, the new reliability analysis model developed in this study is basically reasonable to most of the load history for a compressor disk than the stress-strength interference model, especially for the changeable and complex cruise mission. Moreover, further studies also show that the proposed model can provide a more secure result for the compressor disk in long cruise missions.In the using process, only median


A haptic interface based on electrical muscle stimulation (EMS) has huge potential in terms of usability and applicability compared with conventional haptic interfaces. This study analyzed the force response characteristics of forearm extensor muscles for EMS-based haptic rendering. We introduced a simplified mathematical model of the force response, which has been developed in the field of rehabilitation, and experimentally validated its feasibility for haptic applications. Two important features of the force response, namely the peak force and response time, with respect to the frequency and amplitude of the electrical stimulation were identified by investigating the experimental force response of the forearm extensor muscles. An exponential function was proposed to estimate the peak force with respect to the frequency and amplitude, and it was verified by comparing with the measured peak force. The response time characteristics were also examined with respect to the frequency and amplitude. A frequency-dependent tendency, i.e., an increase in response time with increasing frequency, was observed, whereas there was no correlation with the amplitude. The analysis of the force response characteristics with the application of the proposed force response model may help enhance the fidelity of EMS-based haptic rendering. Virtual reality (VR) is an immersive sensory experience simulated in a computer-generated virtual world. In VR applications, the virtual world, which refers to the contents in a virtual space, should be felt in an immersive and interactive way  for a usIn previous studies, numerous haptic devices have been developed to provide a high-fidelity haptic feedback in various applications including teleoperations ,8,9,10, The lack of force response models is one of the important issues in EMS-based haptic rendering. The force response model is the mathematical representation of the relation between the electrical stimulation and the resulting force sensation. In EMS-based haptic rendering, the force response model is necessary to determine the appropriate electrical stimulation to deliver the desired force sensation to the user. However, the mechanism involved in muscle contractions induced by an electrical stimulation is complex and has not been sufficiently investigated. Moreover, it is difficult to develop the model that can be generally applied to the public because the sensitivity to electrical stimulations varies person-to-person due to the diverse physical conditions of individuals. The same stimulation may be experienced differently by different people. Most previous studies on EMS-based haptic rendering have used pre-determined stimulus patterns ,16,23,24In the field of rehabilitation, EMS has been utilized to treat impaired muscle contraction caused by neuromuscular injuries. Various force response models ,28,29,30The simplified force response model developed for lower limb muscles in the field of rehabilitation was introduced for EMS-based haptic rendering, and its feasibility for the forearm extensor muscles was experimentally verified. Thus, the model can be utilized for EMS-based haptic rendering given its feasibility for upper limb muscles.The force response characteristics, including the peak force and response time, were identified through the experiment based on a quantitative evaluation using the force response model. The peak force–amplitude relationship was assumed as an exponential function and verified by comparing the estimated peak force with the experimentally measured peak force. The response time characteristics were also identified by performing a quantitative evaluation using the force response model fitted to the experimental data. The presented results are expected to contribute to our understanding of the transient and steady-state features of the muscle contraction force response.This paper introduces a force response model that can represent the transient response of an electrically elicited muscle contraction force, and experimentally validated the feasibility of the model for haptic applications. The force response characteristics of the forearm muscles, including the peak force and response time, with respect to the amplitude and frequency of the electrical stimulation were investigated through the experiment involving ten healthy subjects. The experimental results were analyzed quantitatively using the proposed force response model. The main contributions of the paper are as follows.In EMS-based haptic rendering, a haptic sensation is delivered through muscle contraction elicited by an electrical stimulation. This section presents the force response model of the muscle contraction elicited by the electrical stimulation. Based on the previous force model proposed by Ding et al. , we presThe force response model representing the relationship between the given electrical stimulation and the elicited muscle contraction force can be simply expressed as a function of the electrical stimulation parameters as follows.Waveform  ,47 and l[<20 Hz) ,49 signaAmplitude  represents the biophysical procedure for developing muscle contraction forces with respect to the calcium concentration derived from a linear spring, damper, and motor connected in series [2+-troponin complex [The muscle force response to an external electrical stimulation can be modeled using two first-order differential equations as follows ,51.(2)dectively . Equatio follows .(4)u(t)neration  as well  complex  between The force scale factor he pulse . In prevhe pulse ,53,54,55scomfort  and painscomfort , the avahreshold . In thisThe force response model was experimentally validated for the forearm extensor muscles which can be used for presenting the haptic sensation related to the wrist flexion motion frequently occurring in various VR applications. Two important features of the force response for EMS-based haptic rendering, including the peak force and response time, with respect to the amplitude and frequency of the electrical stimulation were also identified in the experiment.Ten healthy subjects, nine male and one female , participated in the experiment. The experimental protocol was approved by the Konkuk University Institutional Review Board (7001355-201909-HR-337). All the subjects were instructed not to exercise excessively the day before the experiment. In addition, the purpose, procedure, and risks of the experiment were explained in detail.Four EMS parameters : At the beginning of the experiment, the subject’s right arm is fixed on the experimental set up as presented in previous section.STEP 2 (Electrode attachment): After fixing the subject’s right arm, one directed the subject to do wrist extension, and find the point that shows the greatest contraction by visual inspection. Two square electrodes with a side length of 5 cm attached at the designated point with intervals of 6–7 cm to cover four motor points of the wrist extensor muscles as shown in STEP 3 : After determining stimulation amplitudes  and (3) with the estimated parameter values. The fourth-order Runge–Kutta method with a fixed time interval of 0.2 ms was used for the numerical integration. The initial values of the state variables C1r values . To avoifunction ,30.(8)CThe force response characteristics of the forearm extensor muscle were identified using the experimental data. First, the feasibility of the exponential estimation of the peak force was evaluated by comparing the estimated peak force value with the measured value. The root-mean-square-error (RMSE), normalized RMSE (NRMSE) and the coefficient of determination  , the accIdentifying the features of the response time is important to deliver the desired force feedback at the exact timing in real-time haptic applications. In this study, the response time of the force was evaluated at various frequencies and amplitudes using the mathematical model fitted to the experimental data. The slope of the force response decreases as it reaches to the peak force that results as shown in the shape of the force responses. To evaluate the response time, mplitude .The proposed model can be used to estimate the muscle force response according to the frequency and amplitude. However, there remains a limitation that should be resolved in the future research. The force response characteristics can be influenced by the posture, but this study only considered the force response of isometric muscle contraction in a fixed posture. Correlation between the force response and the posture should be investigated to be applicable to more diverse situations in the future research. In addition, the force response characteristics presented in this paper does not represent the force response of actual muscle contraction, but overall force response of wrist extension caused by co-contraction of forearm extensor muscles. For precise control of each muscle contraction, further research is required to measure the level of individual muscle contraction using sensors such as piezoresistive sensors.In this study, we analyzed the force response characteristics of forearm extensor muscles for EMS-based haptic rendering by introducing a simplified force response model with amplitude modulation using an exponential function. The proposed force response model can be utilized to predict not only the transient behavior but also the steady-state characteristics of the force response for determining appropriate EMS parameters that can provide the desired haptic sensation. The features of the PF and RT with respect to the frequency and amplitude were identified using the experimental force response data, and this result can be utilized to implement EMS-based haptic rendering in various haptic applications.