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The work presents and characterizes a frequency digitization circuit used in a multi-parameter, multi-axis inertial measurement unit wholly based on frequency modulated (FM) sensors. With a low quantization noise of $35\mu \mathbf{HZ}\mathrm{ms}$ at 100-Hz output data rate, and a linear full-scale of 1 kHz, the proposed frequency to digital converter (FDC) achieves 149 dB of dynamic range, referred to a 50 Hz sensing bandwidth. The circuit operation is successfully demonstrated on both Lissajous FM gyroscopes and time-switched FM accelerometers, on both in-plane and out-of-plane sensing axes.

High speed optical coherence tomography (OCT) systems with A-scan rates greater than 100 kHz allow for 4D visualizations in applications such as intraoperative OCT. However, traditional triangle or sawtooth waveforms used to drive galvanometer scanners often have frequency content that exceeds the bandwidth of the scanners, leading to distorted scans. Sinusoidal waveforms used to drive resonant scanners also lead to distorted scans due to the nonlinear scan velocity. Additionally, with raster scan patterns, the scanner needs time to stop and reverse direction in between B-scans, leading to significant acquisition dead time. Continuous scan patterns such as constant frequency spiral scanning or Lissajous scanning no longer have acquisition dead times, but suffer from non-uniform sampling across the imaging plane. We previously introduced constant linear velocity (CLV) spiral scanning as a novel scan pattern to maximize the data acquisition efficiency of high speed OCT systems. While this continuous scan pattern has no acquisition dead time and produces more uniform sampling compared to raster scanning, it required significant processing time. We introduce a processing pipeline implemented using CUDA in C++, which drastically reduces the amount of processing time needed, allowing real time visualization of 4D OCT data. To demonstrate its potential utility, we used CLV scanning with a 100 kHz swept-source OCT system to image retinas of enucleated porcine eyes undergoing mock ophthalmic surgery movements. Additionally, we rendered these volumes in virtual reality (VR) in real time, allowing for interactive manipulation and sectioning.

We propose a new concept of single-shot deflectometry for real-time measurement of three-dimensional surface profile using a single composite pattern. To retrieve an accurate phase from one-frame composite pattern, we adapt the Fourier Transform (FT) method and the spatial carrier-frequency phase-shifting (SCPS) technique to our proposed deflectometry. Based on Lissajous figure and ellipse fitting method, we also correct the phase extraction error in SCPS technique by reducing the effect of background and modulation variations. The proposed technique is verified by comparing our measurement results with phase-shifting deflectometry, and the maximum difference between two measurement results is less than 30 nm rms. We also test the robustness to vibration and the measurement capability for dynamic object.

This work aimed to characterize a DBD plasma equipment through optical and electrical measurements, seeking to obtain a greater knowledge of the plasma production process and how it behaves through the adopted parameters, such as frequency and voltage applied between electrodes, at a fixed distance of 1.7 mm. In order to measure them, three different characterization techniques were applied. The first method was the Lissajous figures, a technique quite effective for a complete electrical characterization of DBD equipment. The second technique used was the Optical Emission Spectroscopy, a tool used for the diagnosis of plasma, being it possible to identify the excited species produced in filamentary and diffuse discharge in the plasma. And finally, the triple Langmuir probe technique was used to obtain the electron temperature and electron density. Based on this study, it was possible to identify the equipment efficiency in different regimes. The electron temperature measurement for both systems analyzed were 27.96 eV and 20.69 eV to the filamentary and diffuse regimes, respectively. The density of electrons number to these regimes were 1.09 × 1021 m−3 and 1.56 × 1021 m−3.

The Kustaanheimo-Stiefel transformation of the Kepler problem with a time-dependent perturbation converts it into a perturbed isotropic oscillator of 4-and-a-half degrees of freedom with additional constraint known as bilinear invariant. Appropriate action-angle variables for the constrained oscillator are required to apply canonical perturbation techniques in the perturbed problem. The Lissajous-Kustaanheimo-Stiefel (LKS) transformation is proposed, leading to the action-angle set which is free from singularities of the LCF variables earlier proposed by Zhao. One of the actions is the bilinear invariant, which allows the reduction back to the 3-and-a-half degrees of freedom. The transformation avoids any reference to the notion of the orbital plane, which allowed to obtain the angles properly defined not only for most of the circular or equatorial orbits, but also for the degenerate, rectilinear ellipses. The Lidov-Kozai problem is analyzed in terms of the LKS variables, which allow a direct study of stability for all equilibria except the circular equatorial and the polar radial orbits., Accepted to Celestial Mechanics and Dynamical Astronomy

International audience; The harmonic response of ion-exchange membrane systems under potentiostatic control in the limiting current regime is studied. The transport of an electroneutral binary electrolyte through a cation-exchange membrane and the two diffusion boundary layers adjacent to the membrane is described by means of a Nernst-Planck analysis. The full system is perturbed with a sine voltage of a single frequency superimposed on a direct voltage, and it is obtained the decomposition in a Fourier series of the resulting electric current using the network simulation method. The effect of the different parameters characterizing the perturbing voltage (amplitude, frequency and dc component) on the second harmonic of the resulting current, and their influence on the first-harmonic impedance in a highly biased ion-exchange membrane system, are analyzed and discussed. The behavior of the resistance associated to the first harmonic impedance at the limit of zero frequency has been particularly analyzed and an analytical expression to roughly estimate its dependence on the voltage amplitude has been derived. Auxiliary methods for linearity assessment such as Kramers-Kronig transforms, Lissajous plots and waveforms are also discussed.

We report a high resolution 3D imaging using variable structured illumination via Lissajous scanning MEMS mirror module. The Lissajous scanning MEMS mirror was operated at pseudo-resonant frequencies within the bandwidth. The structured illumination pattern density was controlled by selecting the greatest common divisor (GCD) of two scanning frequencies. Scanning frequency selection with high GCD provides a coarse illumination pattern. The constant pattern was obtained by modulating a laser beam at the least common multiple (LCM) of the two selected scanning frequencies. A low-Q inner mirror provides wide frequency tuning range as well as enables variable structured illumination. 3D stereoscopic image has been successfully obtained by using a variable structured illumination and stereoscopic camera. Structured illumination effectively enhances 3D imaging resolution. This structured illumination module can provide a new breakthrough for 3D MEMS imaging applications.

In the context of Human Spaceflight exploration mission scenario, with the Lunar Orbital Platform- Gateway (LOP-G) orbiting about Earth-Moon Lagrangian Point (EML), Rendezvous and Docking (RVD) operational activities are mandatory and critical for the deployment and utilization of the LOP-G (station assembly, crew rotations, cargo delivery, lunar sample return). There is extensive experience with RVD in the two-body problem: in Low Earth Orbit (LEO) to various space stations, or around quasi-circular Low Lunar Orbits (LLO), the latter by Apollo by means of manual RVD. However, the RVD problem in non-Keplerian environments has rarely been addressed and no RVD has been performed to this date in the vicinity of Lagrangian points (LP) where Keplerian dynamics are no longer applicable. Dynamics in such regions are more complex, but multi-body dynamics also come with strong advantages that need to be further researched by the work proposed here. The aim of this paper is to present methods and results of investigations conducted to first set up strategies for far and close rendezvous between a target (the LOP-G, for example) and a chaser (cargo, crew vehicle, ascent and descent vehicle, station modules, etc.) depending on target and chaser orbit. Semi-analytical tools have been developed to compute and model families of orbits about the Lagrangian points in the Circular Restricted Three Body Problem (CR3BP) like NRHO, DRO, Lyapunov, Halo and Lissajous orbits. As far as close rendezvous is concerned, implementation of different linear and non-linear models used to describe cis-lunar relative motion will be discussed and compared, in particular for NRHO and DRO.

Accepted: 2018-09-29, 資料番号: SA1180338000

The rigid body swing is an important problem for steel catenary risers (SCRs). In addition to many other important issues, the transverse flow direction response is studied in this paper. By extending the load terms of the large deflection slender beam equation, the load of suspension point in the z direction, Morison and rigid body swing are superimposed on the beam equation. On the basis of the above work, a Cable3d subroutine is written to complete the task. Then the structural response is simulated and verified by the Lissajous phenomenon and spectral phase analysis. On the basis of verification, the response is analyzed from an angle of three-dimensional space and the influence coefficient is adopted to evaluate the effect of rigid body swing. The importance of loads is determined by spectral analysis. Phase curve and the change of vibration direction are analyzed by higher orders of frequency. The results show the verification of Lissajous and spectral phase analysis are feasible. The analysis of the spatial response shows the vibration direction of the 140th node is in the same direction as the rigid body swing vector, so the interaction is relatively of more intensity and the influence coefficient is relatively larger. This influence interval of rigid body swing displacement statistical analysis is &minus, 0.02 to 0.02 and the effect is weak. The spectrum analysis indicates there is no resonance between the main load and the bending vibration, and the analysis also shows the main influence load of the transverse flow response in this paper is the top load in the z direction. According to phase analysis, the load has a high order effect on the spectral phase curve of the structure. This paper has drawn a conclusion that rigid body swing has limited effect on transverse flow response, however, it has a relatively strong impact on the middle region of the riser, so it plays an influential role on the safety of the riser to some extent. The key point for this paper is to provide qualitative standards for the verification of rigid body swing through Lissajous graphs, which are central factors to promote the development of rigid body swing. It is hoped that the above research can provide some reasonable suggestions for the transverse flow response simulation of the steel catenary riser.

After more than ten years of researches by different groups, this work summarizes the various frequency-modulated techniques for microelectromechanical system (MEMS) based gyroscopes. It then focuses on the low-power integrated electronics for Lissajous FM angular rate sensors and on the conditions to guarantee high offset stability over temperature. Recent results in terms of scale-factor drift and offset drift over temperature on the first complete Lissajous yaw FM gyroscope with full digital output are reported.

In eddy current testing, the trajectory of the impedance data due to a defect is presented as a Lissajous curve (LC) in the complex plane. This paper proposes a novel analytical model for describing a LC. Further, a new feature extraction method is implemented which automatically computes four geometric features (amplitude, width, angle and symmetry) from Lissajous figures. In addition, six machine learning-based classifiers are used for automatic defect identification based on these features. High detection rates are achieved for both the simulated and experimental data, which demonstrates the flexibility of the analytical model and the validity of the methodology.

The paper presents the first frequency-modulated gyroscope (FMG) operated under Lissajous excitation without motion control along the two axes. This operation method is compared to operation under velocity control and under displacement control to show that no significant differences in performance are achieved for the tested axisymmetric structure. At the same time, this operation method can boost the performance in terms of low-power consumption and low-noise by exploiting single-transistor-based oscillators in the system. A 1.6 Hz/Hz ratiometric scale-factor, a sub-0.3% linearity error, a scale-factor stability within 100 ppm/K and a noise of 6.3 mdps/√Hz (the smallest ever reported on FMGs operated in Lissajous mode) are demonstrated.

Scanning MEMS (micro-electro-mechanical system) mirrors are attractive given their potential use in a diverse array of laser scanning display and imaging applications. Here we report on an electrostatic MEMS mirror for high definition and high frame rate (HDHF) Lissajous scanning. The MEMS mirror comprised a low Q-factor inner mirror and frame mirror, which provided two-dimensional scanning at two similar resonant scanning frequencies with high mechanical stability. The low Q inner mirror enabled a broad frequency selection range. The high definition and high frame rate (HDHF) Lissajous scanning of the MEMS mirror was achieved by selecting a set of scanning frequencies near its resonance with a high greatest common divisor (GCD) and a high total lobe number. The MEMS mirror had resonant scanning frequencies at 5402 Hz and 6702 Hz in x and y directions, respectively. The selected pseudo-resonant frequencies of 5450 Hz and 6700 Hz for HDHF scanning provided 50 frames per second with 94% fill factor in 256 &times, 256 pixels. This Lissajous MEMS mirror could be utilized for assorted HDHF laser scanning imaging and display applications.

This paper proposes trajectory planning strategies for online reconfiguration of a multi-agent formation on a Lissajous curve. In our earlier work, a multi-agent formation with constant parametric speed was proposed in order to address multiple objectives such as repeated collision-free surveillance and guaranteed sensor coverage of the area with ability for rogue target detection and trapping. This work addresses the issue of formation reconfiguration within this context. In particular, smooth parametric trajectories are designed for the purpose using calculus of variations. These trajectories have been employed in conjunction with a simple local cooperation scheme so as to achieve collision-free reconfiguration between different Lissajous curves. A detailed theoretical analysis of the proposed scheme is provided. These surveillance and reconfiguration strategies have also been validated through simulations in MATLAB\reg for agents performing parametric motion along the curves, and by Software-In-The-Loop simulation for quadrotors. In addition, they are validated experimentally with a team of quadrotors flying in a motion capture environment.

Purpose To demonstrate, for the first time, the feasibility of obtaining low-latency 3D rigid-body motion information from spherical Lissajous navigators acquired at extremely small k-space radii, which has significant advantages compared with previous techniques. Theory and methods A spherical navigator concept is proposed in which the surface of a k-space sphere is sampled on a 3D Lissajous curve at a radius of 0.1/cm. The navigator only uses a single excitation and is acquired in less than 5 ms. Rotation estimations were calculated with an algorithm from computer vision that exploits a rotation theorem of the spherical harmonics transform and has minimal computational cost. The effectiveness of the concept was investigated with phantom and in vivo measurements on a commercial 3T MRI scanner. Results Scanner-induced in vivo motion was measured with maximum absolute errors of 0.58° and 0.33 mm for rotations and translations, respectively. In the case of real, in vivo motion, the proposed method showed good agreement with motion information from FSL image registrations (mean/maximum deviations of 0.37°/1.24° and 0.44 mm/1.35 mm). In addition, phantom measurements indicated precisions of 0.014° and 0.013 mm. The computations for complete motion information took, on average, 24 ms on an ordinary laptop. Conclusions This work demonstrates a proof of concept for obtaining accurate motion information from small-radius spherical navigators. The method has the potential to overcome several previously reported problems and could help increase the utility of navigator-based motion correction both in research and in the clinic.

An attempt was made to study the dynamics close to the collinear libration point $$ L_{2} $$ of the radial solar sail circular-restricted three-body problem (RSCRTBP) in the Sun–Jupiter System, where the third massless body is a solar sail. We analyse the qausi-periodic (Lissajous solutions) orbits about the libration point $$ L_{2} $$ . The Lindstedt–Poincare approximation for the qausi-periodic orbits was used for numerical simulations. We utilized linear quadratic regulator (LQR) to stabilize the full nonlinear model, and linear state-feedback controller was designed to stabilize the trajectory.

A micro-scanner made of stainless-steel is fabricated via laser cutting technology for 3D Lissajous confocal imaging. The multi-gimbaled structure of the device provides two orthogonal torsional modes and three different out-of-plane modes. Torsional modes can be used to achieve 2D scan and all of the out-of-plane modes can be used in changing the focus of the micro-scanner to achieve a 3D scanning pattern. One of the out-of-plane modes along with two orthogonal torsional modes can be employed for scanning a large depth-stack in sparse fashion while another out-of-plane mode can satisfy a much higher scan fill-rate with less field of view (FOV). Simulations of the micro-scanner are obtained using finite element method (FEM) software and compared with the characterization data gathered from Laser Doppler Vibrometer (LDV). Using various out-of-plane modes, the constructed fill patterns are simulated on MATLAB and fill rates compared.

The existence of in-phase and anti-phase postural responses to sinusoidal perturbations to the base of support is well known. In this study, we investigate if such coordinated postural responses exist at 'near-sway' perturbations where the perturbation amplitudes are kept within the range of normal sway lengths in healthy adults (n=10). The postural responses are analyzed via bursts of anterior-posterior (AP) 2.5 mm horizontal sinusoidal oscillations of the base of support at sequentially varying frequencies (0.25, 0.375, 0.5, 0.625, 0.75, 1 and 1.25 Hz). The parametric plots of the perturbation signal (platform position) and the response profiles (AP Center of Pressure [APCoP]) show the emergence of elliptical Lissajous patterns as the perturbation frequency is increased from 0.25 Hz to 1.25 Hz. The presence of such characteristic pattern shows the 'lock-in' behavior of APCoP with perturbation signal. These elliptical patterns become more apparent at the center frequencies (0.375 to 0.75 Hz). At the higher frequencies (1 and 1.25 Hz), the Lissajous patterns do exist but are dominated by low-frequency drift. The area and orientation of Lissajous patterns and the phase shifts between perturbation and APCoP show a strong nonlinear decreasing trend with increasing perturbation frequency for both, young (n=5) as well as mature (n=5) adults within the study group. This may suggest that such characteristic, frequency-locked, phased shifted response of healthy posture control could be a fundamental property of a healthy posture control's response to 'near-sway' sinusoidal translations in AP direction.

The quantum variance of an observable is a fundamental quantity in quantum mechanics, and the variance provides additional information other than the average itself. By examining the relation between the particle-current variance $(\delta J)^2$ and the average current $J$ in both closed and open interacting fermionic systems, we show the emergence of a multi-valued Lissajous curve between $\delta J$ and $J$ due to interactions. As a closed system we considered the persistent current in a benzene-like lattice enclosing an effective magnetic flux and solved it by exact diagonalization. For the open system, the steady-state current flowing through a few lattice sites coupled to two particle reservoirs was investigated using a Lindblad equation. In both cases, interactions open a loop and change the topology of the corresponding $\delta J$-$J$ Lissajous curve, showing that this effect is model-independent. We finally discuss how the predicted phenomena can be observed in ultracold atoms, thus offering an alternative way of probing the dynamics of many-body systems., Comment: Main text: 5 pages, 3 figures. Supplemental information: 5 pages, 9 figures

In order to improve the accuracy and expand the range of the phase-shifting algorithm application, a precise three-step phase retrieval algorithm suitable for the interferograms with less than one fringes numbers is proposed. The proposed method improves the accuracy, on the one hand because normalization and subtraction are introduced to reduce background intensity fluctuation, on the other hand because Lissajous ellipse fitting (LEF) method is introduced to reconstruct phase maps from the normalized and Gram-Schmidt (GS) orthonormalized maps. Because the approximation operations for cumulative sum aren’t used in the GS method, the proposed algorithm is suitable for not only interferograms that the fringes numbers are more than one but also interferograms that fringes numbers are less than one. Moreover, the proposed method can be applied to simple and complex patterns. Thus, the proposed algorithm has a wider range of applications. The proposed algorithm’s reliability and performance are confirmed by simulations and experiments.

This work aims to report an algorithm for performing fringe analysis with two phase-stepped fringe patterns. By performing a particular Least Squares Fitting (LSF) it is intended to overcome the restrictions to fringe patterns of having constant background or being high-pass filtered when using Lissajous ellipse fitting. With the new method, the phase step and the fringes background are represented by single coefficients that depend on image position, avoiding the utilization of additional arithmetic operations for its calculation, as done with previous methods, increasing its accuracy and range of applicability. Experimental and simulations are presented to elucidate our proposal.

With the development of measurement technology in the direction of precision, high speed, automation, integration, intelligence, economy, non-contact and multifunction, it has simple structure, small size, low energy consumption and strong adaptability. And other advantages have been widely used. However, the sensor of the capacitive encoder works in the state of low voltage and low current, and is extremely susceptible to the influence of the external electric field and magnetic field, causing problems such as input signal distortion and system failure. With the increasing requirements of the manufacturing industry for stability and accuracy, it is of great significance to analyze the capacitive coding error and improve the accuracy. This article analyzes the error generated by the capacitive encoder, enumerates the influence of DC deviation, amplitude deviation, phase deviation, and high-order harmonics on the interpolation error, and compensates the error signal for the aspects that cause larger errors. The sine and cosine signals are close to the ideal state.

An innovative and completely new technique so called discrete Lissajous functions and recton functions for signal analysis and measurement is explained. The contribution of the study is that it is a new digital method of obtaining information on frequency, change of frequency, phase and phase shift as well as auto-correlation, cross-correlation and energy functions of signals in digital form on-line that may be difficult to be provided in analog form and available for measurement and control applications at each sampling instance directly. The discrete Lissajous figures and recton functions can also be sketched in digital image form for further analysis of signals and systems. They are based on an algorithm utilizing discrete convolutions of discrete time signals. They can be depicted in 3D form. They give more information than classical analog Lissajous figures obtained on an oscilloscope screen. They can be used for several applications from chaotic systems to biomedical applications requiring to finding correlations, energies as well as frequency and phase information of the signals and controlling such systems. In this study, the application of them to systems steady-state, mainly frequency response analysis is explained after giving basic definitions.

Phase shifting profilometry (PSP) is an effective technique to reconstruct the three-dimensional shape of object. In general, PSP needs three or more fringe patterns with phase-shifting accurately known to extract the phase distribution of objects. Therefore, the scene and the test objects should remain stationary during capturing the fringe patterns. However, the phase shifts may be unknown in an actual PSP measurement system, especially when measuring the moving object, that is, the phase-shift error may be introduced during the obtaining of the phase-shifting fringe patterns of moving object. In the dynamic measurement scenario, the use of fewer fringe patterns can realize the faster measurement speed and suppress the phase shift error introduced by the moving object. In this paper, a two-step PSP algorithm is proposed based on Lissajous ellipse fitting (LEF). The proposed method uses only two fringe patterns to extract the phase distribution of the object and can suppress the phase shift error caused by the moving object. However, in a practical PSP system, the spatiotemporally varying background intensity and modulation also significantly affect the phase accuracy extracted by LEF, and thus three error-suppressing methods are proposed to reduce the phase error caused by the non-uniform background intensity and modulation. In order to verify the effectiveness of the three error-suppressing methods, we analyze and compare their performances of error suppression under different background intensities and modulations. The advantages of three error-suppressing methods can be summarized as follows. 1) The mean and modulation correction technique has greater advantage than the other two when the background intensity and modulation vary with time. 2) When the background intensity and modulation are relevant to pixel position and the number of fringe patterns, the empirical mode decomposition normalization can more effectively suppress the influence of the non-uniform background intensity and modulation. In experiment, a two-step phase-shifting dynamic measurement based on LEF is conducted. Compared with the traditional PSP which needs at least three fringe patterns, the two-step PSP algorithm successfully extracts the phase with only two fringe patterns and suppresses the phase shift error caused by the motion of the object. Compared with Fourier transform profilometry (FTP), the two-step PSP algorithm can obtain very accurate phase distribution and retain many phase details.

In this paper we present a miniaturized pre-calibration based forward-viewing Lissajous scanning fiber probe for endoscopic optical coherence tomography (OCT). The probe is based on an asymmetric fiber cantilever driven by the piezoelectric bender to realize the two-dimensional (2D) Lissajous scanning, which can realize a relatively large scanning range under a low driving voltage. A capillary metal tube is mounted at the end of the main fiber to reduce the resonant frequency of the fiber cantilever. The relationship between the filling rate and the side-lobe number of the Lissajous scanning pattern is studied, and a method of selecting the orthogonal resonant frequency of the Lissajous scanning is proposed. Through the numerical simulation by COMSOL software, the structural parameters of the asymmetric fiber cantilever are determined. The orthogonal resonant frequencies of the asymmetric fiber cantilever are 169 Hz and 122 Hz. The lengths of the main imaging fiber, the auxiliary fiber and the metal capillary tube are 15.94 mm, 4.49 mm and 2 mm, respectively. The probe is fully packaged in a metal tube for endoscopic imaging. The focal spot and the working distance are 25 µm and 5 mm, respectively. The field of view is larger than 1.5 mm × 1.5 mm. The total rigid length and the outer diameter of the probe are 35 mm and 3.5 mm, respectively. The stability and repeatability of the Lissajous scanning trajectory, and the imaging stability with the rotation of the probe are investigated and verified. The probe is incorporated into a 50 kHz swept source OCT system. The axial resolution of the endoscopic OCT is 10.3 μm, and the imaging frame rate is 1 FPS (frames per second). The maximum signal-to-noise ratio of the imaging system is 110 dB. The imaging performance of the probe is validated by the 2D en-face and three-dimensional volumetric OCT imaging of the high scattering sample and the biological tissue. The probe can be used for the endoscopic imaging of the human tooth. From the result we can distinguish the dental enamel, dental essence and the dental calculus. The developed forward-viewing Lissajous scanning fiber probe is expected to be used in dental applications such as early calculus detection.

This paper presents the motion of a harmonically excited dynamical system with three degrees of freedom (3-DOF) in which it consists of a connected rigid body with a damped spring pendulum whose suspension point moves in a Lissajous curve path. Multiple scales method is utilized to obtain the asymptotic solutions of the equations of motion up to third approximation. Some types of resonances and the conditions of solvability for the steady state solutions have been clarified in light of the achieved modulation equations. The temporal representation of the achieved solutions and resonance curves are presented in some plots to show the good effect of the distinct parameters on the dynamical motion of the investigated system. The numerical solutions of the governing system of motion are gained utilizing the Runge-Kutta method from fourth order. The comparison between these solutions and the analytical ones reflects the good accuracy of the analytical solutions and the used perturbation techniques.

For expected applications of fast tool servo (FTS) and vibration machining, a 3-axis positioning device with low interference motions is proposed in this paper. The positioning device was composed of a XY stage and a Z-axis stage, which were actuated by piezoelectric (PZT) actuators combined with specially-designed symmetric flexure hinges. Through fundamental experiments, when the applied voltage was 50 V, the displacements along the X-, Y-, and Z-axes were measured as 6.35 μm, 6.61 μm, and 10.12 μm, respectively, with the corresponding small percentages of interference displacement of 3.80%, 4.02%, and 3.30%. In addition, the resonant frequencies were obtained as 1.06 kHz, 0.65 kHz, and 0.54 kHz. To examine control performances, a real-time control system considering hysteresis effect of PZT actuators was implemented by the field-programmable gate array (FPGA) modules to conduct tracing controls for sinusoidal waveform, 3D Lissajous motion, and 3D spiral motion. The tracing errors along 3-axis actuations were under 30 nm. The performances of a 3-axis positioning device were well demonstrated. Future work is to perform machining examinations on a machine tool.

The ability to coordinate the simultaneous movements of our arms is limited by a coalition of constraints. Some of these constraints can be overcome when the task conceptualisation is improved. The present study investigated how the movement preparation of bimanual reaching movements was affected by integrated visual feedback of the responses. Previous research has shown that the preparation of bimanual asymmetric movements takes longer than bimanual symmetric movements. The goal of the present study was to determine whether integrated, Lissajous feedback could eliminate this bimanual asymmetric cost. Fifteen participants made unimanual and bimanual symmetric and asymmetric reaches with separate feedback, where there was a cursor and a target for each hand. Participants also made bimanual symmetric and asymmetric movements with integrated feedback; a single cursor and a single target represented the locations and goals of both arms in this condition. The results showed a bimanual asymmetric cost with separate feedback, and that this cost persisted with integrated feedback. We suggest that integrated feedback improved continuous and discrete bimanual movements in other experiments by facilitating error detection and correction processes. We hypothesise that the bimanual asymmetric cost persisted in the present experiment because the uncertainty associated with choice reaction time prevented the facilitated error processing from improving the preparation of the next trial.

The interaction of the Lissajous singularity and singular Lissajous line carried by non-paraxial polychromatic beams is studied. It has been found that with propagation the singular Lissajous line breaks into more than two Lissajous singularities, pairs creation and annihilation of Lissajous singularities, the handedness reversal and change in the degree of polarization of Lissajous singularities, as well as the change in the shape of Lissajous figures may take place.

The feasibility and limitation of directly using the Lissajous figure and ellipse fitting technology to correct the phase extraction error in generalized data reduction algorithm (GDRA) for phase extraction of randomly phase-shifted interferograms are analyzed and discussed. By combining Lissajous calibration technology, which represents the transformative process of Lissajous ellipse to circle (ETC), with advanced iterative algorithm (AIA) we propose a novel generalized phase shifting algorithm (GPSA), and here it is abbreviated as ETCI method. The phase distribution and phase shifts that extracted from randomly phase shifted interferograms by use of ETCI are more accurate and the whole process is far faster than AIA. Additionally, proposed method is less sensitive to non-uniform background intensity and modulation amplitude. Numerical simulations are conducted to evaluate the performance of ETCI, and some influential factors are elaborated. The experimental results further indicate proposed method is suitable for truly random phase shifted interferograms.

The dynamics of a fault with heterogeneous friction is studied by employing a discrete fault model with two asperities of different strengths. The average values of stress, friction and slip on each asperity are considered and the state of the fault is described by the slip deficits of the asperities as functions of time. The fault has three different slipping modes, corresponding to the asperities slipping one at a time or simultaneously. Any seismic event produced by the fault is a sequence of n slipping modes. According to initial conditions, seismic events can be different sequences of slipping modes, implying different moment rates and seismic moments. Each event can be represented geometrically in the state space by an orbit that is the union of n damped Lissajous curves. We focus our interest on events that are sequences of two or more slipping modes: they show a complex stress interchange between the asperities and a complex temporal pattern of slip rate. The initial stress distribution producing these events is not uniform on the fault. We calculate the stress drop, the moment rate and the frequency spectrum of the events, showing how these quantities depend on initial conditions. These events have the greatest seismic moments that can be produced by fault slip. As an example, we model the moment rate of the 1992 Landers, California, earthquake that can be described as the consecutive failure of two asperities, one of which has a double strength than the other, and evaluate the evolution of stress distribution on the fault during the event.

In this paper, a new method for achieving absolute position information is proposed. As opposed to the common approach of matching magnetoresistive (MR) sensors to specific pole pitch patterns on medium, this paper validates the possibility of mismatching between the two with the same element spacing in MR sensors over pole pitch variation from 1.1 to 1.3 mm in two-row magnetic patterns. The first row is the incremental row, which has the same 1 mm pole pitch as the sensor. The second row serves as the absolute row with the tweaked pole pitch pattern. With different pole pitches in the two rows, an offset between each pole exists before the two different pole pitches reach their common multiple. Through analyzing Lissajous profiles resulted from both rows, it is found that the relation between two rows on each position is unique, and hence can be used as a mean to determine absolute position.

Dough blended with rocket leaves powder was subjected to small and large amplitude oscillatory shears. Small amplitude oscillatory shear data were fitted to a discrete relaxation model of elastic solids and to a critical gel model. The small amplitude relaxation spectrum was thereafter used to calculate the LAOS predictions of various large deformation models. The LAOS theoretical calculations using the Phan-Thien model showed good agreement with the first harmonic stress data, and only qualitative agreement with the third and the fifth harmonic stress values. Lissajous curves showed dissimilarity in shape between the experimental data and Phan-Thien model. The network model of Sim et al. (2003). Did not have the butterfly shape displayed in the Phan-Thien model, but it provided a worse fit to stress harmonics than the Phan-Thien model. An improved damage function was proposed, where time effect on network damage was taken into consideration, and fits to stress harmonics and to Lissajous stress-strain curves were significantly improved.

Starting from the basic definition, the explicit expression for the handedness of Lissajous singularities in polychromatic vector optical fields is derived, which is illustrated and used to study the handedness of Lissajous singularities in free space and propagation through an astigmatic lens. It is found that the handedness is not only related to the Stokes parameters S 3i ( i = 1, 2, ……, n ) of constituting components of the polychromatic vector electric field , but also to their frequencies and the amplitude factors.

In this work, we perform an investigation on the multiple harmonics suppression for optical encoders based on generalized grating imaging. We firstly analyse the formation of the harmonic distortion of the encoder signals and evaluate the interpolation errors caused by the higher harmonic signals. The result shows that the harmonic distortion of the encoder signals depends mainly on the higher harmonic components in the interference fringe. In addition, it shows that the higher harmonic signals influence the Lissajous figure of the encoder signals significantly and introduce a relatively large interpolation error. Then, conditions for eliminating the higher harmonic signals are studied, and optical filtering methods using specially designed index grating are proposed. We explain the filtering principle in detail and present three forms of index grating for multiple harmonics suppression. In particular, we show the patterns of the index grating for eliminating the dominant third and fifth harmonics...

This research study analysed the rheological properties of fresh food foams based on egg white protein, and with the addition of apple pectins and xanthan gum. The rheological analysis was carried out using the large amplitude oscillatory shear (LAOS) technique. From the obtained results two types of Lissajous figures were constructed, describing the elastic and viscous properties. The obtained figures were subjected to geometrical decomposition, which resulted in the determination of the stress values characteristic of nonlinear purely elastic and purely viscous properties. The Fast Chebyshev Transform allowed for the calculation of the Chebyshev coefficients, and further detailed analysis of rheological behaviour of foams as a function of strain amplitude. The determination of the values of elastic and viscous Chebyshev coefficients allowed for the interpretation, with high resolution, of nonlinear rheological properties of the obtained foams.

Interfacial rheological measurements often show in their nonlinear Lissajous-plots rhombus or saddle-like shapes indicating complex local deformation behaviour. A strong interacting silica particle and an almost not interacting clay particle were studied in respect to their interfacial rheological properties. Large amplitude oscillation shear measurements were performed with a bicone geometry and combined with optical measurements, from which particle tracks were calculated. A correlation was found between the appearance of shear localisation and Lissajous-plot shapes. Silica particles showed shear localisation at the bicone edge and rhombic plateaus in the Lissajous-plot, while the shear localisation for the clay particles was observed at the cup's wall as saddle-like shaped Lissajous-plots.

Dielectric Barrier Discharge (DBD) is a promising technique for producing plasma in atmospheric pressure and is gaining emphasis in several areas, especially in the biomedical and textile industries, among other. This is due to the fact that this plasma does not reach high temperatures, enabling its use on thermally sensitive materials. However, there is still the need for the development of research relating to the field of physical mechanisms of non-thermal plasma. In this work, we studied the behavior of plasma in DBD equipment, varying distance between electrodes. For this study, two distinct characterization techniques were used. The first, the Lissajous Figures method, is effective and precise in the electrical characterization of DBD equipment. The second technique used, Optical Emission Spectroscopy (OES) is an effective tool in the diagnosis of plasma. This tool enables the identification of excited species in plasma. With the data produced, it was possible to identify an appropriate distance for producing plasma with a higher intensity and homogeneity of active species.

Dispersed viscoelastic microsphere systems have preliminarily been applied in oil fields with gratifying successes. In this paper, a type of viscoelastic microsphere system with an average particle size of 100.6 μm was synthesized by employing the inverse suspension polymerization. The synthesized viscoelastic microspheres could swell 25.42 times due to the three-dimension structure. The rheological properties of dispersed viscoelastic microsphere system were researched by dynamic strain amplitude scanning measurements and the energy dissipation behaviors were investigated through mathematical method. The results showed that the dispersed system exhibited “shear-thickening” behavior when the strain was greater than the critical strain at different fixed angular frequency. The area enclosed by LISSAJOUS curves or normalized LISSAJOUS curves increased with the increasing of the strain at fixed angular frequency. The relationship between the dissipated energy ( E d ) and strain amplitude ( γ max ) is: E d = K ( γ max ) α . And the relationship between the dissipated energy exponent ( α ) and the fixed angular frequency( ω ) is: α ∝ ω 0.131±0.009 . Therefore, as the strain increases, the clusters grow up gradually and block the flow of the system, which results in “shear-thickening” and the demand of additional energy.

A location method based on the Lissajous figure is proposed for fiber-optic distributed disturbance sensor based on Mach–Zehnder interferometer. The disturbance is located by the fiber-optic distributed disturbance sensor due to the time delay between the two signals from clockwise and counter-clockwise Mach–Zehnder interferometers. In the Lissajous figure of the two signals, a fitting ellipse with constant semi-major and azimuths is induced to evaluate the time delay. The relationship between the time delay and semi-minor of the ellipse is obtained by linear fitting. Then, the time delay can be calculated by the semi-minor of the fitting ellipse. In the experiment with 2 km long fiber and 4 MHz sampling frequency, the maximal location errors with Lissajous figure based method and current cross correlation function based method, without phase generation carrier technique, are 75 m and 1.25 km, respectively. It seems to be that our proposed Lissajous figure based method is a useful for improving fiber-optic distributed disturbance sensor.

This paper presents a novel analysis of nonlinear circuits useful for the computation of the reactive power from only terminal measurements. The method computes the reactive power of unknown circuits based on the physical understanding of Maxwell equations. The proposed method is able to identify instantaneous quantities (circuit parameters, powers, etc.) for two physical equivalent models: series and parallel circuits. The parallel equivalent model shows advantages for multi-subsystems. Several numerical examples are provided for validation of the proposed method and to illustrate step-by-step the calculation details. To be compatible with digital instrumentation, a time discrete formulation is used for all calculations. All examples are simulated in MATLAB and the EMTP (Electro-Magnetic Transients Program).

This paper contributes to the theory behind the analysis of strain-controlled large amplitude oscillatory shear (LAOStrain) using Fourier transform rheology as well as Lissajous plots and Pipkin diagrams. It applies LAOStrain to both a reduced form of the Cross model (known as the “r Cross model”) and a Maxwell model whose viscous part is defined by the r Cross model (reffered to as the “r Cross - Maxwell model” in this paper). The analytical determination of the loss modulus and higher harmonics of the r Cross model as mathema- tical functions simplifies the discussion thereof. This in turn facilitates the determination of characteristic points which are directly related to material parameters and therefore enable the identification of both the material parameters of the rCross model and their physical significance. A method for identifying material parameters can be presented which do not require minimising residuals of a non-linear optimisation scheme. It was not possible to identify an analytical solution to the non-linear viscoelastic constitutive equations of the r Cross - Maxwell model, which are therefore solved numerically. The characteristic points identified for the r Cross - Maxwell model are based on the analytical solutions arrived at for the r Cross and Maxwell models. This means that it is also possible to both identify the material parameters of the r Cross - Maxwell model and determine their physical significance. The method of material parameter identification developed is explained using the example of a solvent-borne alkyd primer and the characteristic points determined for strain amplitude dependent storage and loss moduli.

Various approaches to the calculation of the components of the dynamic modulus measured at large amplitude oscillatory shear are compared. The method of Fourier analysis using Chebyshev polynomials, the determination of moduli via the differentiation of stress at the points of zero and maximum deformations, and the integration of the Lissajous figures are considered. A comparative analysis of material-nonlinearity measures obtained through various methods is performed. The measures of viscous and elastic nonlinearities, the mechanical-loss tangent of the material, and the power of its response are compared. The investigation is conducted for model samples of polyisobutylene in viscous-flow and high-elastic relaxation states and polyisoprene filled with nanoparticles of silicon oxide.

The polarization errors of large aperture dynamic interferometers based on the polarization phase shifting method are mainly coming from the effects of imperfect polarized elements and birefringence of large elements. Using the Lissajous ellipse fitting algorithm to correct the influence of the polarized device can effectively eliminate single and double frequency print through errors. We develop a wave plate model for analyzing the birefringence effect, and on this basis, we establish the relationship between the calculated phase and the ideal phase distribution. Experiments are carried out on a 600mm aperture Fizeau interferometer and then compared with the result acquired through the wavelength tuning method. The difference between PV is only 0.002λ.

High-resolution fluorescent microscopic imaging techniques are in high demand to observe detailed structures or dynamic mechanisms of biological samples. Structured illumination microscopy (SIM) has grabbed much attention in super-resolution imaging due to simple configuration, high compatibility with common fluorescent molecules, and fast image acquisition. Here, we report Lissajous scanning SIM (LS-SIM) by using a high fill-factor Lissajous scanning micromirror and laser beam modulation. The LS-SIM was realized by a Lissajous scanned structured illumination module, relay optics, and a conventional fluorescent microscope. The micromirror comprises an inner mirror and an outer frame, which are scanned at pseudo-resonance with electrostatic actuation. The biaxial scanning frequencies are selected by the frequency selection rule for high fill-factor (> 80%) Lissajous scanning. Structured illumination (SI) was then realized by modulating the intensity of a laser beam at the least common multiple (LCM) of the scanning frequencies. A compact Lissajous scanned SI module containing a fiber-optic collimator and Lissajous micromirror has been fully packaged and coupled with relay optics and a fiber-based diode pumped solid state (DPSS) laser including acousto-optic-modulator (AOM). Various structured images were obtained by shifting the phase and orientation of the illumination patterns and finally mounted with a conventional fluorescent microscope. The LS-SIM has experimentally demonstrated high-resolution fluorescent microscopic imaging of reference targets and human lung cancer cell PC-9 cells. The LS-SIM exhibits the observable region in spatial frequency space over 2x, the line-edge sharpness over 1.5x, and the peak-to-valley (P-V) ratio over 2x, compared to widefield fluorescent microscopy. This method can provide a new route for advanced high-resolution fluorescent microscopic imaging.

In order to achieve the high accuracy and high speed in phase shifting interferometry, a phase extraction approach by combining advanced principal component analysis and Lissajous ellipse fitting (APCA-LEF) is proposed. It can obtain accurate phase distribution with only two no pre-filtering phase shifted interferograms, and it costs less time simultaneously. It removes the restriction that PCA needs more than three interferograms with well distributed phase shifts to subtract relatively accurate mean. Moreover, adjacent pixels taken part in the APCA process increases the accuracy by suppressing the effect of noise, also, it is suitable for different levels of noises. Last but not least, if the high accuracy is required, the phase shift would be best to far away from 0 rad and π rad. The simulations and experiments verify the correctness and feasibility of APCA-LEF.