Coordinate Transformation Theory Research Articles

Approach for predicting and adjusting the pointing accuracy of opto-mechanical systems considering multi-source uncertainty

Pointing accuracy is a critical performance indicator of opto-mechanical systems, directly affecting the systems’ efficiency and application range. This study introduces what we believe to be a novel approach for predicting pointing accuracy and adjusting processes in opto-mechanical systems, considering multi-source uncertainty quantification. First, the relationship between error components and total error is quantified using homogeneous coordinate transformation theory. Second, by applying the Nataf transformation to uncertain variables, a hybrid interval-probabilistic uncertainty quantification model based on generalized polynomial chaos is constructed. Third, by selecting points from the probability distribution domain, a parameterized finite element simulation is conducted to create a pointing accuracy prediction model, obtaining the theoretical limit accuracy for the opto-mechanical system. Finally, considering multi-bolt elastic interactions, an assembly process adjustment model is developed to achieve performance-based assembly process adjustments, and tests are conducted to measure the pointing accuracy of the opto-mechanical system after calibration. Pointing accuracy measurements following calibration showed an improvement from 249” to 117”, an increase of 53.01%, approaching the theoretical limit of 108”. This approach requires only one adjustment to approach optimal accuracy compared to eight adjustments with traditional methods, greatly enhancing assembly efficiency. This study offers a theoretical foundation for predicting and adjusting pointing accuracy in opto-mechanical systems.

Arbitrary target frequency cloaking for flexural waves using deep learning

Differing from electromagnetic and acoustic waves, the governing equation for elastic waves in flexural plates is not form invariant, hindering straightforward cloak design based on coordinate transformation theory. In this work, we propose a novel idea instead of the approximately equivalent transformation method, and employ scattering cancellation techniques to design a multi-layer cylindrical structure for achieving flexural wave cloaking at desired target frequencies. Moreover, we use deep learning to effectively address the time consuming issue dealing with fine-tuning design parameters for the desired response. More importantly, we adopt a method based on a tandem neural network to tackle the one-to-many mapping challenge in inverse design. It not only accurately predicts the scattering spectra of multi-layer structures in advance but also efficiently performs inverse design to obtain the required design parameters for arbitrary target frequency cloaking.

Service Performance Optimization and Experimental Study of a New W-W Type Non-circular Planetary Gear Train

To address the vibration problem in traditional W-W-type non-circular planetary gear trains during operation, this study proposes two new combinations of W-W-type non-circular planetary gear trains. The study focuses on analysing the transmission error and improving service performance. By utilizing the conjugate theory and the coordinate transformation theory, an accurate mathematical model for non-circular gears is derived. The transmission error models for the two combination forms of the non-circular planetary gear train are established using the incremental meshing line method. The analysis also examines the influence of eccentricity on the transmission error. Kinematic analysis of the multi-body dynamics model confirms that the new W-W type non-circular planetary gear train, formed by combining non-circular gears and cylindrical gears, exhibits superior transmission performance. Furthermore, a test analysis is conducted on the roller pumping unit test platform and the indicator diagrams of the pumping unit under various working conditions are obtained. The study concludes that the optimal matching mode for the sucker rod counterweight and motor frequency of the reversing device of the pumping unit is 20 kN and 20 Hz, and the dynamic balance of the pumping unit load can be achieved by adjusting the motor frequency.

Simulation Study on Dynamic Characteristics of the Chain Drive System for Mining Scraper Conveyor Driven by the Permanent Magnet Synchronous Motor

The chain drive system represents a critical subsystem within the scraper conveyor. This paper proposes a joint simulation model for the drive system of the scraper conveyor, driven by the permanent magnet synchronous motor, in order to conduct a comprehensive analysis of the dynamic characteristics of the chain drive system during the operational process. Firstly, the dynamic simulation model for the mining scraper conveyor’s chain drive system was established in ADAMS, taking into account its structural characteristics. Then, the mathematical model of the permanent magnet synchronous motor was established using the coordinate transformation theory, and the speed controller based on vector control was designed by using the theory related to sliding mode control. The coupling relationship between the chain drive system of the scraper conveyor and the permanent magnet synchronous motor drive system was investigated. Finally, a joint simulation model of the mechanical system and motor control system was created using ADAMS (View 2019) and MATLAB/Simulink (2020a). The dynamic characteristics of the chain drive system were analyzed, and the three typical working conditions of no load, half load, and rated load were considered. The results show that the contact force between the flat ring and the sprocket undergoes an initial increase, followed by a decrease, and finally another increase. As the load increases from no load to full load, there is a marked increase in the contact force between the loaded side chainrings. Due to the polygon effect, both the speed curve of the permanent magnet drive motor and the contact force curve between the ring chains exhibit periodic fluctuations. The research in this paper provides an idea for the coupling analysis of the scraper conveyor electromechanical system.

Anti-Disturbance Speed Control of Permanent Magnet Synchronous Motor Based on Fractional Order Sliding Mode Load Observer

Based on the fractional order sliding mode load observer, this paper presents an anti-disturbance speed control method of permanent magnet semi-direct drive transmission system for overhead manned equipment. According to the Lagrange equation, the dynamic model of semi-direct drive transmission system is established. Furthermore, the mathematical model of the permanent magnet synchronous motor (PMSM) is established based on the coordinate transformation theory. Subsequently, the fractional order sliding mode observer (FOSMO), which combines nonsingular terminal sliding mode control and fractional order theory is designed to observe the load disturbance changes in the permanent magnet semi-direct drive transmission system. Then, the concept of “active damping” is used to design the speed loop PI controller, and combined with the designed FOSMO to construct the composite speed controller. The designed composite speed controller is further improved by the anti-saturation design and parameter optimization. The results demonstrate that the designed FOSMO can precisely estimate the load disturbance on the PMSM output shaft. Additionally, the designed composite speed controller can realize stable speed control in the case of complex load disturbance, which meets the anti-disturbance speed control requirements of permanent magnet semi-direct drive transmission system, and the robustness of the control system is further improved by adding the anti-saturation link.

Investigation of loaded contact characteristics of planetary roller screw mechanism based on influence coefficient method and machine learning

The performance of the planetary roller screw mechanism (PRSM) is directly determined by the mechanical behavior of the loaded contact. The traditional method based on a finite element modeling to describe the loaded contact characteristics (LCCs) is computationally expensive, while the use of Hertzian contact theory and the thread connection model results in a loss of computational accuracy. As a solution, this study aims to develop a semi-analytic modeling technique that can predict the LCCs of PRSM accurately and efficiently. To achieve this, the study first establishes a mathematical model using the spatial coordinate transformation theory to accurately describe the thread surface. The potential contact area is discretely divided using the differential geometry principle. Then, the linear deformation and nonlinear deformation are matched using node interpolation technique to obtain the accurate compliance of the potential contact area. Finally, the contact forces for all activated contact points are derived by a nonlinear iterative algorithm based on the influence coefficient method. The performance of the proposed model is verified by numerical examples. Further, the sensitivity weights of the geometric parameters to the LCCs are identified utilizing machine learning, and a preferred zone is also constructed to optimize these parameters. This study provides a valuable tool for evaluating and improving the LCCs of PRSM in engineering practice.

Equivalent Transformation Between Axial- and Radial-Flux Permanent-Magnet Machines

In this paper, an equivalent transformation approach for permanent-magnet (PM) electrical machines using coordinate transformation theory is presented. With the proposed method, an axial-flux PM (AFPM) machine can be transformed to an equivalent radial-flux PM (RFPM) machine or vice versa. The main purpose of proposing such a transformation method is to simplify the design of AFPM for specific applications. Besides, according to the working space requirement, the shape of PM machines can be changed while maintaining magnetic performances. The proposed transformation theory is based on the basic magnetic theory and the theoretical derivation is presented in detail. The resulted transformation equations for RFPM and AFPM machines with surface-mounted PMs are friendly to engineers. Moreover, an AFPM machine is used to validate the proposed theory, both FEM and experimental results have good agreement with the results of the proposed method. With the equivalent transformation theory, engineers could design, simulate and evaluate the AFPM machines with an exist well-developed machines.

Wear analysis and simulation of small module gear based on Archard model

With the development of micro machining technology, small module gears are widely used in the field of high precision technology. In view of the failure problems such as tooth fracture and fatigue pitting caused by dynamic wear in the process of small module gear transmission, this paper takes a micro planetary gear reducer as the research object and establishes the analysis framework of small module gear wear calculation model based on Archard wear model and Hertz contact theory. Considering the influence of installation error and position tolerance, tooth contact analysis (TCA) model of small module gear is established based on the theory of small displacement torsor (SDT) and space coordinate transformation, and the impact of these errors on the dynamic contact trajectory, contact stress, and tooth surface wear is analyzed based on the numerical simulation. The numerical simulation results show that with the increase of the installation error, the contact trajectory will deviate significantly, which will finally result in increased contact stress and wear. Moreover, with the decrease of tolerance level, the service life of the device increases under the same reliability level. The above analysis of tooth surface wear can provide references for the design and installation of a small module gear transmission.

Design of a polyurethane underwater sound absorber based on the BP neural network and genetic algorithm

A coordinate transformation theory for the acoustic cloak design based on pentamode material was proposed by Norris in 2008. This theory avoids the problem of infinite density at the inner boundary of the inertial cloak, but the structural design of the pentamode material was difficult to be achieved. In order to solve this manufacturing problem, a design scheme of a low scattering cloak with only a few layers of polyurethane materials was proposed in this study. First, the material parameter distribution of the acoustic cloak was obtained based on the coordinate transformation theory. On this basis, the material parameter distribution in the two-dimensional annular acoustic cloak was derived by using linear mapping. Through simulation comparison between the uncovered acoustic cloak and covered acoustic cloak, the equivalent scattering characteristics of obstacles covered with the acoustic cloak were significantly reduced, which shows that anisotropic concealed materials have obvious advantages in acoustic control. When the thickness was given and the frequency was fixed, the sound pressure field of the cloak was simulated under the condition of uniform stratification. With the increase in the number of layers, the stealth effect was better, and the complexity of the preparation of metamaterials was increased. In order to obtain the cloak structure with a simple structure and good stealth performance, the BP neural network genetic algorithm and sound field finite element modeling were used. The minimum scattering area was taken as the optimization goal, and we tried to find the design scheme of a low scattering cloak with only a few layers of polyurethane materials. The optimization results show that when the target frequency is 2 KHz and the number of cloak layers is five, the optimized scattering area is 73% less than that of the uniformly layered structure. This shows that anisotropic stealth materials have obvious advantages in sound control. This study can guide the cell configuration design of polyurethane materials in different positions and greatly simplify the complexity of the manufacturing process of stealth materials.

Microstructure Design and Performance Optimization of Constant Modulus Pentamode Materials Acoustic Cloak

Underwater acoustic stealth has great scientific research value. According to acoustic coordinate transformation theory, the acoustic stealth cloak based on pentamode materials can realize underwater broadband acoustic stealth. However, due to the correlation between the density and modulus of pentamode materials and the changes in the parameters of each layer of the acoustic stealth cloak, a large amount of structural optimization work is required for the pentamode material to meet the specific parameter requirements, which significantly increases the difficulty of the pentamode acoustic stealth cloak design. To simplify the design process, inspired by the calculation of equivalent modulus by representation volume element, this article proposed a pentamode material configuration with independent variation of density and modulus and designed a 1 m radius acoustic stealth cloak with a specific structure of pentamode materials matching the coordinate transformation equation of constant modal mapping. After simulation calculation and optimization design, in the range of a/λ from 0 to 1, the average total scattering cross-section of the cavity with a radius of 0.5 m covered by the acoustic stealth cloak is 0.858; the average total scattering cross-section of the cavity is 19.718 after removing the pentamode material acoustic stealth cloak. The results of the study are expected to provide some method references for simplifying the design process of the pentamode material acoustic stealth cloak and the microstructure design of the pentamode materials.

Строгие методы преобразования систем координат

Determination of coordinate systems’ parameters transformation has always been and remains the core of surveyors’ practical and theoretical works. However, in most cases, the issue is interpreted under the assumption that the angular rotation of the coordinate axes and the similarity parameter are very small, which enables a significant simplifying of the algorithm. In this paper the strict theory of determining the 7 and 9 parameters of coordinate systems transformation within the framework of nonlinear method of least squares with limitations is considered. Strict algorithm is described and numerical experiments are performed. The connection of the coordinate systems transformation theory with the general one of mathematical statistics, known under the name of Procrust analysis, is specified. However, in general, artificial neural network theory is recommended for conversion of one coordinate system to another. This generally eliminates the need for a prior determination of the transformation’s appropriate parameters, inevitably associated with hypotheses for the transformation model, but quickly and accurately solves the main problem.

Dual-function thermoelastic cloak based on coordinate transformation theory

• Transformation theory is extended to thermoelastic framework for dual-function cloak design. • Two transformation schemes with asymmetric or symmetric elastic tensors are given. • Invisibility of different cloaks is validated by FEM and observed in both temperature and displacement fields. In past years, various cloaks have attracted the attention of many scientists but they usually remain confined to a single function. Cloaks combining multiple functions, however, are desirable. In this paper, we design thermoelastic cloaks with dual functionality by the coordinate transformation technique. The transformation of the thermoelastic wave equations for cloaking and explicit expressions for the required material parameters are established. Symmetrization of the elastic tensor is applied using Norris’ gauge matrix to emphasize the feasibility of the designs. Two different operating conditions, transient elastic wave propagation and steady heat transfer, are adopted in numerical calculations for the designed cloaks. It is shown that, on one hand, from the perspective of observer, whether the cloak is symmetrized or not, its external response is identical and invisibility is reliable. On the other hand, it is destined that the symmetrization of the cloak would be accompanied by the increase of internal displacement, which may need to be paid attention to in the actual design. In addition, the temperature effects of both cloaks are consistent. The work in this paper may pave a way toward the realization of thermoelastic cloaks, thus broadening the research scope for metamaterials.

A novel hierarchical iterative hypothesis strategy for intrinsic parameters calibration of laser structured-light weld vision sensor

In automatic welding, multi-line laser structured-light vision sensors provide more adequate information to compared with traditional single-line structured light and are gradually being widely designed. However, when calibrating the sensor, a global calibration approach should be adopted in order that the laser intrinsic parameters follow the inherent constraints. In this paper, a Lagrange solution is shown and its unsuitability for general cases is qualitatively analyzed. We then propose a novel hierarchical iterative hypothesis (HIH) strategy to deal with issues with the following conditions. First, the configuration and pose (including rotation and translation) of a 3D model need to be solved. Second, the model can be obtained by scaling a known model in several dimensions. Third, part of sampling points of the model can be acquired. The proposed hypothesis is introduced based on the coordinate transformation theory. Hierarchical iteration, including global and local iteration, is elaborated mathematically, which utilizes singular value decomposition (SVD) and gradient descent technology. The process of applying hierarchical iterative hypothesis strategy HIH is illustrated, and a reconfirmation for transformation is introduced. The steps in the intrinsic parameters calibration for weld vision sensor are described. A calibration experiment is witnessed that the ambiguity on the intersection issue is reduced to compared with ordinary least square method for structured light calibration. Several quality measures were introduced that allow us to judge the performance of our method. Two experiments for verifying calibration results reveal that the proposed strategy earns a great calibration performance that can meet weld seam tracking needs.

Three-Dimensional Omnidirectional Acoustic Illusion

Acoustic waves are ubiquitous in everyday human experience, therefore, precise control over the deformation of acoustic waves is always extremely desirable, which can be used, for example, to transform or hide objects from incident waves. Acoustic illusion devices are generally implemented by transformation acoustics, which can deceive ears or sonar systems. Challenges remain, for example, the complex and extreme material parameters prescribed by coordinate transformation theory make the implementations particularly difficult, even with the help of acoustic metamaterials. Here, the method based on Fabry-Perot resonances offers a feasible solution for achieving three-dimensional (3D) omnidirectional passive acoustic illusion. We theoretically demonstrate 3D acoustic illusion via Mie theory, a reduced version is further designed numerically and implemented experimentally. Our work opens possibilities for the implementation of modern acoustic illusion devices in the future, such as camouflage for antisonar detection.

Analytical Modeling of the Closed-Loop Operation and Quench Behavior for Superconducting Electrodynamic Suspension Train

Superconducting electrodynamic suspension (EDS) as one of the most promising ultrahigh-speed transportations has been widely followed with great interest in the past decades, and service safety under high-speed operation is always the focus of academic attention. Thereinto, an evaluation method of quench behavior can effectively reduce and even avoid the risky accident in advance for the EDS train, possessing in a more realistic sense. In this article, we established an analytical model for the superconducting EDS with cross-connection, fully considering the arbitrary attitudes of the bogie and closed-loop operation of onboard superconducting magnets (SCMs), on the basis of spatial coordinate transformation and dynamic circuit theory. Meanwhile, the data of the finite-element model and experiment, regarded as references, validate the availability of the proposed analytical model. Afterward, we systematically analyzed and revealed the end effect caused by the SCMs and characteristics of 3-D electromagnetic interactions from the perspective of normal operation and quench conditions, respectively. The results show that the quench behaviors of SCMs have a significant influence on the operating characteristics of trains and are dependent upon the installation position of the disabled SCMs, providing an invaluable tool for the studies of the train dynamics under quench conditions.

Analysis and optimization of vehicle handling stability with considering front and rear suspension hard point coordinates

For the front double wishbone suspension, starting from the space analytic geometry, a method of describing the movement law of the its is proposed and kinematic analysis is performed, and then the expressions of front wheel alignment parameters are derived by the vector method. Based on the theory of space coordinate transformation, a method of analyzing the kinematics of the five-link suspension is proposed, and the toe angle and camber angle of the rear wheels are derived. The virtual kingpin of the rear suspension is approximated by the instant screw axis method to obtain the expressions of the kingpin alignment parameters. The sensitivity analysis of the impact of the hard point coordinates on the suspension performance indicators is obtained, and the hard point coordinates that are most sensitive to the front and rear suspension performance indicators are obtained. And this is used as the analysis object to explore the influence of the hard point coordinate changes on the wheel alignment parameters. The frequency response index of the vehicle is derived from the vehicle dynamics model, and the change law of the frequency response index of the vehicle in the low and high frequencies is obtained based on swept-sine input simulation. Finally, the sensitivity analysis of the vehicle handling stability is carried out on the hard point coordinates, and the more sensitive hard point coordinates are selected as the optimization variables. The vehicle frequency characteristic index is used as the optimization target, and the NSGA-II algorithm is used for multi-objective optimization. After optimization, the result shows that the handing stability of the vehicle in the low frequency range has been improved.

Influences of the workpiece pose errors on gear skiving accuracy

This paper investigates the influence of the workpiece pose errors on gear skiving accuracy. First, based on the theory of homogeneous coordinate transformation, the mapping relationship between the tooth flank of the workpiece and the skiving tool is established according to the principle of gear skiving. Second, the workpiece pose errors are defined, and the relationship between each workpiece pose error and cutting depth is analyzed. According to the kinematic chain of the machine tool, a method for measuring the workpiece pose errors based on the actual inverse kinematic model is proposed. Third, the tooth deviations caused by individual workpiece pose errors and the coupling workpiece pose errors are studied. Then, the sensitive errors are obtained by numerical simulation. Finally, the numerical analysis is verified by gear skiving experiments, and the proposed method can be used to identify the possible workpiece pose errors. Furthermore, the particle swarm optimization algorithm is introduced to guide the workpiece setting pose errors in the gear skiving process.

Freeform surface topography model for ultraprecision turning under the influence of various errors

This work establishes an accurate freeform surface topography model considering geometric errors, tracking errors, and cutting forces. Regarding geometric errors, based on the theory of multibody systems and homogeneous coordinate transformation, the transfer relationship between various geometric errors and the tool error in the workpiece coordinate system is described. Using the continuous acceleration Position-Velocity-Time (PVT) interpolation method, the offset discrete points caused by geometric errors are interpolated and rediscretized, and the accurate mathematical model between the tool errors (position and posture) in the workpiece coordinate system and the topography of the freeform surface is established. Based on the model, the influence of geometric errors on the contour errors of freeform surfaces is accurately calculated, and the contribution and similarity of the effects of various geometric errors on the contour errors of freeform surfaces are analyzed. For dynamic-dependent factors, a second-order system with two degrees of freedom on the X or Z axis is established. According to the simplified result of the equivalent system, the contour error of the freeform surface caused by the tracking error and cutting force is analyzed. For the tracking error, under the condition of constant rotation velocity of the spindle, the change of the slope of the freeform surface along the circumferential direction causes the change in the velocity and acceleration of the motion axis; thus, the corresponding area of the surface has a larger contour error. For cutting force, the change of the slope of the freeform surface along the circumferential direction changes the tool working rake angle, the tool working clearance angle, and the shearing angle, resulting in a change in the cutting force. Moreover, the change of cutting speed is accompanied by the change of cutting force. An experiment verifies the effectiveness of the proposed modeling method. In addition, the modeling method provides an effective solution for the calculation of the contribution rate of different error sources to the contour error of the freeform surface.

Kinematic Modeling of Six-Axis Industrial Robot and its Parameter Identification: A Tutorial

In advanced industrial applications, like machining, the absolute positioning accuracy of a six-axis robot is indispensable. To improve the absolute positioning accuracy of an industrial robot, numerical compensation based on positioning error prediction by the Denavit and Hartenberg (D-H) model has been investigated extensively. The main objective of this study is to review the kinematic modeling theory for a six-axis industrial robot. In the form of a tutorial, this paper defines a local coordinate system based on the position and orientation of the rotary axis average lines, as well as the derivation of the kinematic model based on the coordinate transformation theory. Although the present model is equivalent to the classical D-H model, this study shows that a different kinematic model can be derived using a different definition of the local coordinate systems. Subsequently, an algorithm is presented to identify the error sources included in the kinematic model based on a set of measured end-effector positions. The identification of the classical D-H parameters indicates a practical engineering application of the kinematic model for improving a robot’s positioning accuracy. Furthermore, this paper presents an extension of the present model, including the angular positioning deviation of each rotary axis. The angular positioning deviation of each rotary axis is formed as a function of the axis’ command angles and the direction of its rotation to model the effect of the rotary axis backlash. The identification of the angular positioning deviation of each rotary axis and its numerical compensation are presented, along with their experimental demonstration. This paper provides an essential theoretical basis for the error source diagnosis and error compensation of a six-axis robot.

A variable inductor based harmonic filter design for multi-phase renewable energy systems with double closed-loop control

Compared to L- or LC- type filters, LCL-type filters are superior in high-frequency harmonic attenuation ability. But the filtering inductors are designed with fixed values in traditional principles, which aim to limit the harmonic amplitude at the rated power. However, power converters may have to operate under load-shedding conditions due to the intermittent of both renewable energy and power consumers, which will worsen the harmonic issue on account of a skimpy filtering inductance. In this paper, in order to adapt the light-load conditions to a certain degree, an LCL-type filter with a variable converter-side inductor is designed for a multi-phase inverter system. The variable inductor can obtain about four times inductance at low currents compared to the inductance at the rated power. Meanwhile, a systematic double closed-loop control strategy is applied based on the provided extended coordinate transformation theory in multi-phase system, which can be generally applied in any n-phase systems. And in order to meet steady-state performance requirements of the system with the variable inductor, a system calibration method is proposed and introduced to design the controller parameters. Taking a five-phase inverter as an example, feasibility and validity of the proposed scheme have been verified based on the finite-element analysis (FEA), PLECS simulation platform and practical experiments.