Mathematical Error Model Research Articles

A Simulation Study of Active Disturbance Rejection Control for Gravure Register System

According to two-color printing register system working principle, a mathematical model of two-color register error in shaftless driving mode is established, and extended to four-color register error model further more. Aiming to this model, an active disturbance rejection control (ADRC) controller and control system for gravure register are designed, then the simulation system is build also. The simulation results show that the designed register control system is able to realize a high precision control for the multi-color register system, this provides a viable option for gravure printing register system which meets to the requirements of high precision, high speed and low reject rate.

Circular Array of Magnetic Sensors for Current Measurement: Analysis for Error Caused by Position of Conductor.

This paper analyzes the measurement error, caused by the position of the current-carrying conductor, of a circular array of magnetic sensors for current measurement. The circular array of magnetic sensors is an effective approach for AC or DC non-contact measurement, as it is low-cost, light-weight, has a large linear range, wide bandwidth, and low noise. Especially, it has been claimed that such structure has excellent reduction ability for errors caused by the position of the current-carrying conductor, crosstalk current interference, shape of the conduction cross-section, and the Earth’s magnetic field. However, the positions of the current-carrying conductor—including un-centeredness and un-perpendicularity—have not been analyzed in detail until now. In this paper, for the purpose of having minimum measurement error, a theoretical analysis has been proposed based on vector inner and exterior product. In the presented mathematical model of relative error, the un-center offset distance, the un-perpendicular angle, the radius of the circle, and the number of magnetic sensors are expressed in one equation. The comparison of the relative error caused by the position of the current-carrying conductor between four and eight sensors is conducted. Tunnel magnetoresistance (TMR) sensors are used in the experimental prototype to verify the mathematical model. The analysis results can be the reference to design the details of the circular array of magnetic sensors for current measurement in practical situations.

Mathematical model of radio engineering ship system errors in the autolanding problems studies

The paper describes the results of statistical processing of flight path errors recorded on board aircraft during aircraft carrier landing approaches using the S-band radar system and the microwave radio-beacon landing system. The analysis and statistical processing data averaging is carried out. A mathematical model of a random (noise) component of the measurement error of radio engineering landing systems used on aircraft carriers is built.The study introduces a mathematical model based on actual data of full-scale tests, which allows us to carry out a wide range of studies to determine the accuracy of various types aircraft landing approaches, including the presence of ship rocking and spatial evolution of the aircraft. The results of mathematical simulation are presented.

Thermal error robust modeling method for CNC machine tools based on a split unbiased estimation algorithm

In the thermal error compensation technology of CNC machine tools, the core issue is to establish a mathematical model of thermal error with high prediction accuracy and strong robustness. The prerequisite for the mathematical model is choosing optimum temperature sensitive points for modeling. Currently, there are many methods that prioritize reducing the collinearity between temperature sensitive points first before considering the influence weights of temperature sensitive points on thermal error. In this paper, we determined that these methods are unable to ensure that all the temperature sensitive points have high influence weights on thermal error through experimental analysis of the thermal error of Leaderway-V450 CNC machine tools in an idle state. This causes volatility in the temperature sensitive points and decreases the prediction accuracy and robustness of the model. In this regard, we present a new thermal error modeling method called the “Gray relation − Split Unbiased Estimation thermal error robust modeling method”, or the “GR-SUE method”. In the “GR-SUE method”, several temperature sensitive points with the highest influence weights on thermal error are selected directly using the gray relation algorithm. However, the gray relation algorithm causes a significant collinearity problem between temperature sensitive points. Therefore, we improve the multiple linear regression algorithm and propose a split unbiased estimation modeling algorithm to inhibit the influence of collinearity on the prediction accuracy and robustness of the model. Finally, the “GR-SUE method” is verified using numerous thermal error experiments of Leaderway-V450 CNC machine tools at an idle state under different rotation speeds and ambient temperatures. The experimental results show that the “GR-SUE method” can significantly reduce the volatility of temperature sensitive points and improve the prediction accuracy and robustness of the model.

Sensing Mechanism and Error Analysis of a Capacitive Long-Range Displacement Nanometer Sensor Based on Time Grating

In this paper, the sensing mechanism of a novel capacitive nanometer sensor based on the time grating approach is investigated. A mathematical model of the sensor is established using electric field coupling theory and the area integral method, which indicates that the measured displacement of the object is proportional to the phase shift of the output signal. High frequency time pulses serve as the measurement standard to realize the phase detection, and the displacement is measured by counting the time pulses. To evaluate the performance of the proposed sensor, periodic measurement errors are analyzed in detail. The primary periodic errors are quantified through the derivation of a mathematical error model. Experiments performed with a prototype sensor allow the causes of periodic first, second, and fourth harmonic errors to be traced back to cross interference, installation misalignment, and the effects of the electric field, respectively. After adopting a multilayer structure, adjusting the installation, and increasing the gap width to 0.3 mm, the primary periodic errors are sequentially eliminated. Therefore, the reasonability of the mathematical model and error analysis is validated by the designed experiments. Finally, the experimental results demonstrate that the measurement accuracy attains a value of ±200 nm over a 200 mm measurement range. This paper provides the theoretical guidance for the optimal design of high-performance time grating capacitive sensors.

Investigation on shaping machining method for deep hole keyway based on on-line symmetry detection and compensation

Machining of deep hole keyway often suffers from low processing accuracy, low processing symmetry degree between keyway and the reference plane and is difficult to control. In order to enhance the processing quality, a mathematical model of space symmetry error was established based on space projection, orthogonal combination and linear fitting methods so as to study rotatable and shift-able space objects. The space symmetry error was separated into included angle error and displacement error between the fitting straight line and the benchmark line. Through included angle rotation and location measurement along with quantitative translation, the space symmetry error compensation was realized. On the basis of the proposed methods, a Double combined manual self-centering fixture (DCMSF) was designed to realize the radial self-centering and axial location for effective clamping of deep hole keyway, and the reference plane of built-in straight slot was derived out to make it detectable and controllable. Meanwhile, an Online multi-degree of freedom symmetry detection device (OMFSDD) and corresponding Numerical control (NC) programs were also designed for automatic detection and control of symmetry error. Subsequently, mass production of high symmetric deep hole keyway were realized by means of NC gear shaper servo control of rotating shaft angle compensation and bi-directional cutter back-off amount translation compensation, and combination of NC control of fixed angle processing. Deep hole keyway processing experiments indicate that the proposed method can steadily keep the symmetry within 0.03 mm. Thus, the rationality and accuracy of the proposed method was validated which provides a novel approach to high precision deep hole keyway processing.

Model of bit error rate for laser communication based on superconducting nanowire single photon detector

The high-speed deep space communication is one of the key technologies for deep space exploration. Laser communication system equipped with sensitivity of single photon will improve existing deep space communication speed. However, laser communication at single photon level needs to consider not only the effect of transmission environment, but also the performance of used single photon detector and the photon number distribution. As a new single photon detector, superconducting nanowire single photon detector (SNSPD) outperforms the traditional semiconducting SPDs at near infrared wavelengths, and has high detection efficiency, low dark count rate, low timing jitter, high counting rate, etc. The SNSPD can be used for detecting single photons efficiently, rapidly and accurately. In this paper, we introduce the system detection efficiency and dark count rate of SNSPD based on the photoelectric detecting model without considering the effect of atmospheric turbulence, establish the mathematical model of bit error, and put forward the formula of system bit error rate. What should be emphasized is that the bit error rate is an important parameter for measuring the performance of laser communication system. Error is partly from background thermal radiation and circuit electromagnetic interference; in addition, error appears when photons reach the surface of device without being absorbed to successfully produce resistance area or photons are absorbed but there occurs no response. As a result, the calculation of bit error rate includes the whole process of photoelectric conversion. In order to analyze how to affect the size of system bit error rate, first we simulate two factors of the formula, i.e., light intensity and laser pulse repetition frequency. The results show that the light intensity has the greatest influence on error bit rate. With the light intensity increasing from 0.01 to 1000 photon/pulse, the error bit rate significantly decreases from 10-1 to 10-7 level. The influence of laser pulse repetition frequency is restricted by the light intensity, which declines with the increase of pulse repetition frequency. Then we measure the error bit rate experimentally, which validates the simulation model. However, when increasing light intensity or speed, experimental bit error rate is about 10-4 times higher than simulation result. The reason may be that the insufficiency of actual communication modulation extinction ratio of optical signal to the background noise through optical fiber increases the dark count rate. The above model and experimental results could be the foundation of high-speed deep space laser communication such as moon-earth and Mars-earth based on SNSPD.

Optimal Construction Scheme for Controlling Construction Errors of a Cable-strut Tensile Structure

In a real cable-strut tensile structure, construction errors are inevitable. To explore the optimal construction scheme and control construction errors, this study optimized the construction scheme of a cable-strut tensile structure. First, a mathematical model of the element length error was investigated based on stochastic theory. By combining the balance equation, geometric equation and physical equation, the fundamental relationship between the pre-stress deviation and element length error was derived. Because the pre-stress in the active cable can be controlled exactly and the cable’s pre-stress deviations were zero during construction, the relationship between the pre-stress deviation of a passive cable and the element length error was obtained. Then, the statistical characteristics of the pre-stress deviation were obtained under different construction schemes using statistical theory. Finally, an example was analysed as a case study. The study showed that different elements have different error sensitivities and that different construction schemes have different error effects. Using the method proposed in this paper, the error effect of different construction schemes can be analysed, and the optimal construction scheme, with lower error effects and lower construction costs, can be selected for actual construction projects.

Optimization of the construction scheme of the cable-strut tensile structure based on error sensitivity analysis

Optimization of the construction scheme of the cable-strut tensile structure based on error sensitivity analysis is studied in this paper. First, the element length was extracted as a fundamental variable, and the relationship between element length change and element internal force was established. By setting all pre-stresses in active cables to zero, the equation between the pre-stress deviation in the passive cables and the element length error was obtained to analyze and evaluate the error effects under different construction schemes. Afterwards, based on the probability statistics theory, the mathematical model of element length error is set up. The statistical features of the pre-stress deviation were achieved. Finally, a cable-strut tensile structure model with a diameter of 5.0 m was fabricated. The element length errors are simulated by adjusting the element length, and each member in one symmetrical unit was elongated by 3 mm to explore the error sensitivity of each type of element. The numerical analysis of error sensitivity was also carried out by the FEA model in ANSYS software, where the element length change was simulated by implementing appropriate temperature changes. The theoretical analysis and experimental results both indicated that different elements had different error sensitivities. Likewise, different construction schemes had different construction precisions, and the optimal construction scheme should be chosen for the real construction projects to achieve lower error effects, lower cost and greater convenience.

The Pre-Travel Error Study of Electrical Trigger Probe in On-Machine Measurement

On-machine measurement is a high-efficient measure method in NC machining, and there are pre-travel errors during measurement. What is more, these errors tend to present anisotropy, and affect the measurement accuracy. This paper aimed at measuring the anisotropy of probe’s pre-travel error. First, the standard ball is measured by changing the probe angle with the same machine tool moving direction, and the pre-travel errors with different probe angles under the same machine tool errors are obtained. Then, the machine tool backlash errors are measured and compensated, and the pre-travel errors with different probe angle are obtained. Finally, the mathematical model of pre-travel error is established. The method proposed in this paper can be used to determine probe’s pre-travel error with anisotropy.

Study on the Generalized Holo-Factors Mathematical Model of Dimension-Error and Shape-Error for Sheet Metal in Stamping Based on the Back Propagation (BP) Neural Network

Precision improvement in sheet metal stamping has been the concern that the stamping researchers have engaged in. In order to improve the forming precision of sheet metal in stamping, this paper devoted to establish the generalized holo-factors mathematical model of dimension-error and shape-error for sheet metal in stamping based on BP neural network. Factors influencing the forming precision of stamping sheet metal were divided, altogether ten factors, and the generalized holo-factors mathematical model of dimension-error and shape-error for sheet metal in stamping was established using the back-propagation algorithm of error based on BP neural network. The undetermined coefficients of the model previously established were soluble according to the simulation data of sheet punching combined with the specific shape based on the BP neural network. With this mathematical model, the forecast data compared with the validate data could be obtained, so as to verify the fine practicability that the previously established mathematical model had, and then, it was shown that the generalized holo-factors mathematical model of size error and shape-error had fine practicality and versatility. Based on the generalized holo-factors mathematical model of error exemplified by the cylindrical parts, a group of process parameters could be selected, in which forming thickness was between 0.713 mm and 1.335 mm, major strain was between 0.085 and 0.519, and minor strain was between −0.596 and 0.319 from the generalized holo-factors mathematical model prediction, at the same time, the forming thickness, the major strain, and the minor strain were in good condition.

Measurement uncertainty evaluation of conicity error inspected on CMM

The cone is widely used in mechanical design for rotation, centering and fixing. Whether the conicity error can be measured and evaluated accurately will directly influence its assembly accuracy and working performance. According to the new generation geometrical product specification(GPS), the error and its measurement uncertainty should be evaluated together. The mathematical model of the minimum zone conicity error is established and an improved immune evolutionary algorithm(IIEA) is proposed to search for the conicity error. In the IIEA, initial antibodies are firstly generated by using quasi-random sequences and two kinds of affinities are calculated. Then, each antibody clone is generated and they are self-adaptively mutated so as to maintain diversity. Similar antibody is suppressed and new random antibody is generated. Because the mathematical model of conicity error is strongly nonlinear and the input quantities are not independent, it is difficult to use Guide to the expression of uncertainty in the measurement(GUM) method to evaluate measurement uncertainty. Adaptive Monte Carlo method(AMCM) is proposed to estimate measurement uncertainty in which the number of Monte Carlo trials is selected adaptively and the quality of the numerical results is directly controlled. The cone parts was machined on lathe CK6140 and measured on Miracle NC 454 Coordinate Measuring Machine(CMM). The experiment results confirm that the proposed method not only can search for the approximate solution of the minimum zone conicity error(MZCE) rapidly and precisely, but also can evaluate measurement uncertainty and give control variables with an expected numerical tolerance. The conicity errors computed by the proposed method are 20%-40% less than those computed by NC454 CMM software and the evaluation accuracy improves significantly.

An error compensation method for a linear array sun sensor with a V-shaped slit

Existing methods of improving measurement accuracy, such as polynomial fitting and increasing pixel numbers, cannot guarantee high precision and good miniaturization specifications of a microsun sensor at the same time. Therefore, a novel integrated and accurate error compensation method is proposed. A mathematical error model is established according to the analysis results of all the contributing factors, and the model parameters are calculated through multi-sets simultaneous calibration. The numerical simulation results prove that the calibration method is unaffected by installation errors introduced by the calibration process, and is capable of separating the sensor’s intrinsic and extrinsic parameters precisely, and obtaining accurate and robust intrinsic parameters. In laboratorial calibration, the calibration data are generated by using a two-axis rotation table and a sun simulator. The experimental results show that owing to the proposed error compensation method, the sun sensor’s measurement accuracy is improved by 30 times throughout its field of view (±60° × ±60°), with a RMS error of 0.1°.

A study on the optimal selection of spur slice cutter parameters and machining parameters

Preliminary study on slicing technology and design theory of slicing cutter shows that there is deviation between the machined tooth surface and the theoretical tooth surface, called theoretical machining error. This deviation is affected by cutter parameters and machining parameters. To improve machining accuracy, a suitable combination of parameters should be ascertained. On the other hand, interference between major flank face of the cutter and machined tooth surface of the workpiece could occur with inappropriate parameters. In addition, the constraint conditions of enough addendum thickness and no top cut also raise a claim to parameter optimization. For this purpose, the mathematical model of theoretical machining error is built. The judgment method of interference is proposed. The mathematical discriminants of enough addendum thickness and no top cut are presented. On this basis, the selectable ranges of parameters are determined by the constraint conditions. Then, orthogonal experiment is carried out to select an optimal combination of parameters by analyzing the theoretical machining error. Machining experiment is performed with the selected parameters. The results prove that the proposed optimal selection method is effective and applicable.

Research of a High Precision Time Synchronization Method Based on GPS Calibration

With the higher requirement for clock accuracy in power system, the accuracy and stability of wild area punctuality in GPS cannot meet the demands of the high–precision measuring instruments in power system. Firstly, this paper establishes the mathematical model of crystal oscillator clock error using polynomial fitting method. Then, it estimates the parameters of polynomial model with generalized least squares method. Thirdly, it updates the error-model of crystal oscillator clock dynamically for modifying its predicted error. The result shows, this method can not only predict accurate time error, but also filter out random errors well in GPS.

Fast field calibration of MIMU based on the Powell algorithm.

The calibration of micro inertial measurement units is important in ensuring the precision of navigation systems, which are equipped with microelectromechanical system sensors that suffer from various errors. However, traditional calibration methods cannot meet the demand for fast field calibration. This paper presents a fast field calibration method based on the Powell algorithm. As the key points of this calibration, the norm of the accelerometer measurement vector is equal to the gravity magnitude, and the norm of the gyro measurement vector is equal to the rotational velocity inputs. To resolve the error parameters by judging the convergence of the nonlinear equations, the Powell algorithm is applied by establishing a mathematical error model of the novel calibration. All parameters can then be obtained in this manner. A comparison of the proposed method with the traditional calibration method through navigation tests shows the classic performance of the proposed calibration method. The proposed calibration method also saves more time compared with the traditional calibration method.

Research on Cam Contour Error of Grinding Machining

In order to obtain higher cam quality, the research on the cam contour error is studied. The cam is machined in the way of X-C biaxial linkage motions. The linear motor drives the grinding wheel mechanism to get the motion of X axis, and the motion of C axis is the rotating of cam driven by the torque motor. Because of the servo-system-lag of the two axes, the cam contour error is formed in the X-C biaxial linkage motions. Moreover, the following position error of X axis and C axis is not same as the cam contour error. The relationship between axis following position error and cam contour error is studied. The mathematical model of cam contour error is constructed, the relationship between the cam contour error and the following position error are obtained. At last, the conclusion which the cam contour error can be controlled is made, although the following position error exists at the same time. To design the contour error controller for higher quality cam is based on the above conclusion.

Biofuels temperature measurement error in termoanemometrical flowmeters

The paper considers the measurement of biofuel temperature error, which is based on using a thermoelectric converter in thermoanemometric flowmeter. Quantitatively, these measurement errors were expressed by a mathematical model. Therefore, for more efficient use of thermal converters, necessary calculations of TAF errors were carried out, and the analysis of the feasibility of this TAF was conducted. As a result of the studies, measurement accuracy and performance, which ensures a continuous real-time measurement process were determined. A new mathematical model of thermoanemometric flowmeter errors was developed, and analytical expressions for calculations of its basic errors were obtained. Therefore, to improve the flowmeter accuracy, more sophisticated algorithmic procedures (approximation of biofuel temperature measurement results based on the least squares method and applying artificial neural networks) were used.This ensures measurement accuracy of volumetric motor fuel flow rate of 1.0...1.5 % at the heater temperature measurement accuracy of 1 %. Using a set of thermal methods and creating new flowmeter structures allow to make biofuel flow rate measurements fully automated with high accuracy and performance.

Error Modeling Analysis of Face-Gear NC Grinding Machine

Based on the multi-body system theories and used the homogeneous coordinate transform. Systematically analyzed the geometric error and thermal error of face gear NC grinding machine. Established the face gear NC grinding machine of grinding mathematical model of comprehensive error according with the mechanism grinding errors,To provide a theoretical basis of the error modeling analysis and the error compensation on NC grinding Machine Tool of face-gear.

Influence Analysis of Machine Tool’s Turntable Angle Errors to the Roundness of Machined Hole

Machining accuracy has always been a key problem to machine tool. Machine errors are caused by a variety of error factors among which the geometric error is a key element. An accuracy model of the 3-axis CNC machine tool is established in this paper with multi-body system (MBS).And based on the accuracy model, the mathematical model of roundness error also built up. At last, the influence of CNC machine tools turntable angle error to roundness error of machined hole is analyzed.