Coordinate Transformation Matrices Research Articles

Meshing characteristics analysis of planetary roller screw mechanism with misalignment

In this study, a meshing model for planetary roller screw mechanism (PRSM) with misalignment is established, considering misalignment angle and misalignment azimuth angle. By incorporating these misalignment variates into space coordinate transformation matrices, the meshing equations of screw-roller and nut-roller interfaces are derived to explore the meshing characteristics of PRSM with misalignment. Furtherly, the maximum misalignment angle of PRSM is obtained by the bisection method to research the limit position in misalignment. The effectiveness of the misalignment meshing model is verified by comparing with a previously published model. In addition, a typical PRSM is taken as an example to investigate the effect of misalignment angle and misalignment azimuth angle on meshing positions and axial clearances of PRSM. The results demonstrate that the meshing characteristics of PRSM with misalignment have an enormous difference from that of PRSM in an ideal state.

Robust Fault Estimation and Tolerant Control for Uncertain Takagi–Sugeno Fuzzy Systems

This research addresses the intricate challenges of fault estimation (FE) and fault-tolerant control (FTC) within a specific subset of T-S fuzzy systems. These systems are characterized by their localized nonlinear models, the presence of unknown inputs, actuator imperfections, and disruptive output disturbances, making them fertile ground for exploration in this study. The contributions of this paper can be summarized as follows: (1) First, we employ coordinate transformation matrices to convert the T-S fuzzy model. This transformation separates the unknown inputs and disturbances at the output. Subsequently, we equip the modified system with a T-S fuzzy adaptive sliding mode observer (ASMO) that serves the dual purpose of fortifying resilience against disruptions and adeptly deducing an extensive spectrum of fluctuating actuator failure signals. (2) In the next step, the insights gained from FE are harnessed to craft a dynamic fuzzy output feedback fault-tolerant controller (DOFFTC). This controller aims to mitigate the effects of actuator errors to maintain the stability of the closed-loop system. The article creates the necessary conditions for the presence of the required ASMO and DOFFTC using H-infinity filtering methods. To address the optimization issue posed by these criteria, we utilize linear matrix inequalities (LMIs) and calculate the required gains for implementation using convex optimization techniques. (3) The study concludes by illustrating the applicability of the proposed techniques with an example employing an inverted pendulum. This paper presents a comprehensive approach to overcoming the challenges of FE and FTC within T-S fuzzy systems. It leverages precise mathematical formulations and optimization strategies to achieve resilient and dependable control, even when confronted with intricate system dynamics.

Simplified transformation matrices of journal bearings in vertical application

Rotodynamic simulation of complex or/and large systems, for instance hydropower machines, may consist of models with many degrees of freedom and require multidisciplinary computations such as fluid-thermal-structure interactions or rotor-stator interactions due to electromagnetic forces. Simulating such systems is often computationally heavy and impractical, especially in the case of optimization or parametric study, where many iterations are required. This has, therefore, created a need for simplified dynamic models to improve computational efficiency without significantly affecting the accuracy of the simulation result. The purpose of this paper is to present simplified coordinate transformation matrices for journal bearings in vertical rotors, which require less computational effort. Matrix multiplications, which appear during coordinate transformation, were eliminated, and the bearing stiffness and damping matrices in the fixed reference frame were represented by local coefficients instead. The dynamic response of a vertical rotor with eight-shoe Tilting pad journal bearings was simulated using the proposed model for two operational conditions, i.e., when the rotor was spinning at constant and variable speeds. The results from the proposed model were compared to those from the original model and validated through experiments. The conclusion was that the presented simulation model is time efficient and can effectively be used in rotordynamic simulations and analyses.

A unified vibration modeling of open cylindrical shell-rectangular plate coupling structures based on the dynamic stiffness method

This paper presents a unified vibration modeling for free and forced vibration analysis of coupled open cylindrical shell-plate structures (COSPS). In the model, the coupled structure is first split into several open cylindrical shells and rectangular plates, and then based on Kirchoff's thin plate theory and Flügge's thin shell theory, the dynamic stiffness (DS) matrix of each substructure is separately established by applying the generalized superposition method and the projection method. Subsequently, according to the continuity and equilibrium conditions of the coupled boundary, the coordinate transformation matrix of each substructure is derived. After obtaining fundamental DS matrices and their coordinate transformation matrices, global DS matrices of various COSPS are assembled using a strategy similar to the finite element method (FEM) without repeating the theoretical derivation. To verify the convergence and reliability of the current formulation, free vibration and forced vibration analysis of three types of coupled structures are carried out, and the results are compared with those from published works and FEM solutions. In addition, an experimental model of a coupled structure is established and an experimental test is performed. The comparison results show that the proposed model is reliable and effective, and its modeling process is more direct and convenient. This work not only greatly expands the application scope of the DSM but also provides a new idea for the vibration analysis of COSPS.

A new orientation relationship observed between MC and ferrite in Ti Mo microalloyed steel produced by ultra-fast cooling

A new crystallographic orientation relationship (OR) between MC (M, Ti, and Mo) and ferrite has been observed in microalloyed steel produced by new-generation thermo-mechanical processing (traditional TMCP involving ultra-fast cooling). The selected area electron diffraction (SAED) patterns present strong evidence for the new OR, that is (111)MC//(110)ferrite and 1¯10MC//1¯10ferrite, which hasn't been reported previously in MC/ferrite system. While the precipitates formed in the specimens with traditional TMCP obey the well-known Baker-Nutting (BN) and Nishiyama-Wassermann (NW) ORs. The coordinate transformation matrices for BN, NW and the new OR have also been calculated, which further confirms the uniqueness of the new OR. The interfacial energy for the carbides and ferrite that obey the new OR is higher compared with that of BN and NW ORs, which can be attributed to the energy accumulated and maintained in austenite deformation and ultra-fast cooling stage, and then leads to un-equilibrium precipitates

Generalized coordinate smoothed particle hydrodynamics with an overset method in total Lagrangian formulation

AbstractThis study proposes a generalized coordinate smoothed particle hydrodynamics (GCSPH) method coupled with an overset method using the total Lagrangian formulation for solving large deformation and crack propagation problems. In GCSPH, the physical space is decomposed into multiple domains, each of which is mapped to a generalized space to avoid coordinate singularities as well as to flexibly change the spatial resolution. The SPH particles are then non‐uniformly distributed in the physical space (e.g., typically in a boundary‐conforming manner) and defined uniformly in each generalized space, similar to the standard SPH. The SPH particles in the generalized and physical spaces are numerically related by coordinate transformation matrices. The use of non‐uniform particle distributions decreases the total number of particles, thus significantly reducing the simulation cost. Three numerical cases: three‐dimensional brittle crack propagation, Taylor impact, and plugging failure, are presented to validate the proposed method. The numerical results are sufficiently accurate compared with the standard total Lagrangian SPH. Furthermore, these results also show the benefits of GCSPH, which not only effectively reduces the computational cost but also eliminates the effects of particle arrangement. These advantages allow GCSPH to perform high‐resolution three‐dimensional problems, which are otherwise costly to perform with the standard SPH.

Research on Fault-Tolerant Field-Oriented Control of a Five-Phase Permanent Magnet Motor Based on Disturbance Adaption

To ensure the high-quality output performance of the five-phase fault-tolerant permanent magnet synchronous motor (FTPMSM) drive system under normal and open-circuit faults and achieve the minimal reconfiguration of the FTPMSM control drive system under various open-circuit faults, in this paper, a fault-tolerant field-oriented control (FOC) strategy based on disturbance adaption is proposed. The speed-loop and current-loop steady-healthy controllers are designed to effectively suppress the torque ripples caused by open-circuit faults and improve the robustness of the drive system to load disturbance and motor parameter variation under fault operation. Moreover, the additional zero-sequence current controller can be omitted. In addition, the modified reduced-order coordinate transformation matrices are proposed to weaken the influence of oscillating neutral. Finally, the fault-tolerant FOC system of the FTPMSM is established, and its experiment is conducted. The experimental results verify the feasibility and effectiveness of the proposed control strategy.

Development a model of robot movement with five degrees of freedom for a warehouse

The object of research is a mathematical model describing the movement of a robot with five degrees of freedom for a warehouse. The work was aimed at analyzing the kinematic structure of the manipulator, on the basis of which the base and local coordinate systems were selected, as well as further formalized recording of the kinematic equations in matrix form. It is noted that one of the most problematic places is that the algorithms for controlling the robot most often contain local rules for the interaction of robots between themselves and the external environment, and emergent behavior is manifested as a result of the application of these rules, which does not have a formal description. Therefore, it is important to modernize the models describing the motion of a robot with five degrees of freedom for a warehouse. Using the matrix method, the sequence of constructing coordinate systems is described and its mathematical description is given, which will make it possible to eliminate this drawback in the future. The computer implementation of the developed algorithms was carried out using methods for processing matrix data structures. The principle of constructing a kinematic model of a robot is presented, with the help of which the main coordinate transformation matrices are obtained for robot with five degrees of freedom, and the possibility of taking into account the size of the gaps in the joints is shown. The resulting model is obtained, which is proposed for use in building control algorithms for a robot with an automatic gap selection, as well as in robot calibration. This is due to the fact that the proposed model has a number of features, in particular, the basic coordinate system and the coordinate system of each link of robot with five degrees of freedom are taken into account. This makes it possible to obtain the values of the indicators for the projection of the robot position vector in the initial state, in the rotation of the fourth link at a well-defined angle and in the case of a vertically straightened manipulator. Compared to similar known studies, this provides advantages such as minimizing errors in position, speed and motion accuracy. The object of research is a mathematical model describing the movement of a robot with five degrees of freedom for a warehouse. The work was aimed at analyzing the kinematic structure of the manipulator, on the basis of which the base and local coordinate systems were selected, as well as further formalized recording of the kinematic equations in matrix form. It is noted that one of the most problematic places is that the algorithms for controlling the robot most often contain local rules for the interaction of robots between themselves and the external environment, and emergent behavior is manifested as a result of the application of these rules, which does not have a formal description. Therefore, it is important to modernize the models describing the motion of a robot with five degrees of freedom for a warehouse. Using the matrix method, the sequence of constructing coordinate systems is described and its mathematical description is given, which will make it possible to eliminate this drawback in the future. The computer implementation of the developed algorithms was carried out using methods for processing matrix data structures. The principle of constructing a kinematic model of a robot is presented, with the help of which the main coordinate transformation matrices are obtained for robot with five degrees of freedom, and the possibility of taking into account the size of the gaps in the joints is shown. The resulting model is obtained, which is proposed for use in building control algorithms for a robot with an automatic gap selection, as well as in robot calibration. This is due to the fact that the proposed model has a number of features, in particular, the basic coordinate system and the coordinate system of each link of robot with five degrees of freedom are taken into account. This makes it possible to obtain the values of the indicators for the projection of the robot position vector in the initial state, in the rotation of the fourth link at a well-defined angle and in the case of a vertically straightened manipulator. Compared to similar known studies, this provides advantages such as minimizing errors in position, speed and motion accuracy.

Generalized set-theoretic interval observer using element-wise nonnegativity transformation

This paper proposes a generalized set-theoretic interval observer (GSIO) for robust state estimation of discrete linear time-invariant (LTI) systems with bounded uncertainties. Including the previous set-theoretic interval observer (SIO) as a special case, the proposed GSIO only requires to transform the system matrix of the first subsystem to be nonnegative. Based on this idea, a nonsmooth H∞ optimization method is used to design a nonnegative subsystem matrix for the first subsystem based on an element-wise nonnegativity transformation. The advantage of this method is that it can design a nonnegative subsystem matrix by transforming each of its elements to be nonnegative, which does not impact any structural constraints on matrix elements. This implies that more extra design degrees of freedom could be obtained from the coordinate transformation and observer gain matrices, which can optimize state estimation performance of GSIO (e.g., suppress the effects of disturbances and noises). In this way, we could not only achieve an expected balance between robust state estimation conservatism and computational complexity but also better robust state estimation performance. Besides, under the framework of the proposed GSIO, this paper further proves its existence, gives its stability condition and compares it with the set-valued observer (SVO) and the interval observer (IO) in state estimation conservatism, which together forms a complete theoretic system for the proposed GSIO. At the end of this paper, numerical examples are used to illustrate the effectiveness of the GSIO.

Analysis of Tan-Sun Coordinate Transformation System for Three-Phase Unbalanced Power System

This paper mainly analyzes the characteristics of Tan-Sun coordinate transformation system, which is an extended version of Clarke coordinate transformation system suitable to be applied in three-phase unbalanced power system. The main purpose of Tan-Sun coordinate transformation system is to convert three-phase unbalanced alternating-current quantities into two orthogonal alternating-current quantities with the same amplitude, and these two quantities can be further converted into two direct-current quantities without second harmonic components through Park transformation. Tan-Sun coordinate transformation system can realize the direct-current decoupling of three-phase unbalanced currents, and the strict decoupling of the model of three-phase unbalanced power system, so the control methods developed for three-phase balanced power system can be migrated into three-phase unbalanced power system perfectly. In this paper, Tan-Sun and inverse Tan-Sun coordinate transformation matrices are illustrated at first. The essence of Tan-Sun coordinate transformation system is then presented, which can be divided into two coordinate transformation processes, and the corresponding coordinate transformation matrices are provided by detailed analysis. The relationship between Tan-Sun and Clarke coordinate transformation systems is also discussed. Finally, the simulation and experimental results are provided to validate the effectiveness of the adaptive Tan-Sun phase-locked loop method and the superiority of Tan-Sun coordinate transformation system.

Особенности применения матриц преобразования и уравнений связи при анализе несимметричных повреждений

Kron’s tensor-topological method being advanced by many Russian and foreign researchers opens wide opportunities for solving applied problems in complex electric power systems by means of tensor analysis and diakoptics. Its application to the analysis and calculation of asymmetric and complex types of faults occurring in power supply systems has its own peculiarities which are due to the need for transformations of variables (coordinate systems). In addition to the problems of defining transformation matrices and constraint equations, there are problems of combining individual parts of the system represented in different coordinates. Difficulties arise from the need to transform these matrices when constructing equivalent circuits. The paper discusses simplest examples to show some features of how coordinate transformation matrices, coupling matrices and constraint equations can be jointly applied, as well as how they can be differentiated when modeling and analyzing non-symmetric faults in power supply systems.

Digitization modeling and CNC machining for cone-generated double-enveloping worm drive

Digitization modeling and CNC machining method of cone-generated double-enveloping worm drive are proposed in this study. The modeling coordinate frame is developed, and coordinate transformation matrices are also built for different coordinate systems. Next, the common normal vectors and relative velocity vectors of the conical cutter and worm and worm and worm wheel are derived by using differential geometry, kinematic analysis, and coordinate transformation method; then, the mathematical models of the worm drive are established applying spatial meshing principle. In addition, the solving algorithms for the complex multivariate nonlinear constrained mathematical models of the worm drive are developed according to the worm drive’s surface-generating characteristic; then, the meshing point cloud that represents the tooth profile of the worm drive are calculated based on the developed algorithms; finally, the digital CAD models of the worm drive could be constructed on the basis of the point cloud in the modeling software system. The CNC machining code is generated based on the accurate digital CAD models, and the surfaces of the worm drive could be machined in the CNC machining center. The modeling and machining experiment results demonstrate the validity and accuracy of the proposed modeling and CNC machining method for cone-generated double-enveloping worm drive.

Static analysis of thin-walled space frame structures with arbitrary closed cross-sections using transfer matrix method

This paper develops a new transfer matrix method for the static analysis of thin-walled space frame structures with arbitrary closed single-cell sections, that is, thin-walled beam transfer matrix method (TBTMM), which expands the advantages of the general transfer matrix method (TMM). In this method, the bimoment B and the rate of twist of the beam φ(x) are taken into account based on the Vlasov's beam theory and Kollbrunner-Hajdin assumption, and the bending and twisting can be calculated separately combined with the characteristic of Maillart's shear center. Using this method, the transfer matrix of single beam is obtained through Laplace transform and inverse Laplace transform at first, and then gets the coupling equations of the whole structure with respect to coordinate transformation matrices and joint coupling matrices. Some numerical examples are given and compared with the general TMM and the finite element method (B-FEM and S-FEM) results to validate the developed method.

Modeling of forming technological errors in processing by gear shaping machine

In the paper, the mathematical model for evaluation the accuracy of the cut gear depending on the initial error of the technological system, is presented. The model is based on the coordinate transformation matrices, variations matrices and considers the deviations of the base and the real surfaces. The technique allows evaluating the possibilities of the technological system to achieve the specified accuracy.

Mathematical Model for Assessing the Accuracy of Processed Gears on Gear Shaping Machines

This paper examines the impact of the geometric and kinematic errors of gear shaping machines for the accuracy of processed wheels. There is a problem to establish a correlation between the errors of the technological system with the gear accuracy parameters defined by standards. In this article, the equation of the machined surface is proposed based on the coordinate transformation matrices and equations of involute surface of the tool in the form of a matrix equation. On the basis of this equation the machine output precision is given in the form of matrix equation. The impact of errors of the technological system elements is displayed by a matrix of variations, components of which are the projection of errors on the axis. Output errors are applied on the machined surface via the projection on the line of action. Next, the obtained values are translated into a standardized deviation of the gear accuracy. This enables forecasting the possibility of obtaining the products of certain accuracy on this machine, as well as to control the specific parameters of the technological system to ensure the accuracy and stability of the process.

装配机器人位姿误差建模与补偿研究

针对装配机器人在运行过程中因发生速度、加速度的突变以及冲击、振动等不良现象而致使其定位精度低的问题，提出了一种误差补偿方法。首先，运用D-H法构建各关节的坐标变换矩阵，推导机器人的数学模型；其次，釆用传递矩阵法构建基于D-H参数的机器人误差模型及表达式；最后，釆用摄动补偿法对误差模型进行补偿，建立误差测量平台进行实测，用最小二乘算法计算辨识出机器人结构参数误差，并在此基础上进行误差补偿实验，实验表明所提出的误差补偿方法能够有效地提高装配机器人末端的定位精度，对装配机器人的进一步研究提供参考。 An error compensation method is proposed for the problem of low positioning accuracy that the assembly robot has due to the sudden change of velocity and acceleration and the bad phenomena such as shock and vibration. Firstly, DH is used to construct the coordinate transformation matrices of each joint, and the mathematical model of the robot is deduced. Secondly, the transfer matrix method is used to construct the robot error model and expression based on DH parameters. Finally, the error model is calculated by perturbation compensation method. The experimental results show that the proposed error compensation method can effectively improve the positioning accuracy of the end of the assembly robot, and it is a good tool for the assembly of the robot. The error compensation method is used to calculate the position error of the robot. Further study provides a reference.

Performance evaluation of laser tracker kinematic models and parameter identification

A new kinematic model for a laser tracker is presented in this paper. This model obtains the kinematic parameters by the coordinate transformation matrices between successive reference systems based on the Denavit-Hartenberg method. The ASME B89.4.19 Standard provides some ranging tests, length measurement system tests and two-face system tests that can be performed to analyze the performance of the laser tracker. However, these tests take a lot of time and require specialized equipment. Another problem is that the end user cannot apply the manufacturer’s model because he cannot measure physical errors. The kinematic model developed has been compared with a geometric model based on modelling physical errors. To do this, the laser tracker kinematic model has been presented and validated using a mesh with synthetic reflector coordinates and known error parameters. The laser tracker has then been calibrated, in an easy and fast way, with experimental data using the measurements obtained by a coordinate measuring machine as nominal values. The calibration has been performed with both the kinematic model presented in this work and the geometric model based on physical errors. A comparison of both calibrations has been made, analyzing the performance of both models. Finally, a sensitivity analysis of the length measurement system tests is presented to recommend the more suitable positions to perform the calibration procedure.

A Comprehensive Model of a Tracker System Based on Motion Equations of a Two Degree-of-Freedom Gimbal System

In this paper a comprehensive model of a tracker system including derive subsystem is presented. Since the main part of the under consideration system consists a two Degree-Of-Freedom (DOF) gimbal subsystem, at first, the model of a two-axis gimbal is considered. For this purpose, after introducing the coordinate systems and different transformation matrices, the related equations of pitch and yaw axes by considering the friction torques, mass unbalances and cable restraint torques are derived. Moreover, all disturbance terms and the methods for their reduction or elimination are investigated. Then, the derive system of the gimbal including a DC motor with a gear is modeled and governing overall equations of the whole system is obtained. In order to have a model with practical considerations, the model of rate gyro which is used for measuring the angular velocities is also included. Finally, the simulation results of the under consideration case study by using the derived comprehensive dynamical equations of this paper are provided. Analysis of these results are used to synthesis the behavior of the practical system and choosing the suitable control structure in order to achieve stabilization and tracking objectives.

Jointly Optimal High-Order Error Feedback and Realization for Roundoff Noise Minimization in 1-D and 2-D State-Space Digital Filters

Joint optimization of high-order error feedback and state-space realization for minimizing roundoff noise at filter output subject to l2-scaling constraints is investigated for one- and two-dimensional state-space digital filters. Linear algebraic techniques that convert the problems at hand into an unconstrained optimization problem are explored, and an efficient quasi-Newton algorithm is then applied to solve the unconstrained optimization problem iteratively. In this connection, closed-form formulas are derived for fast and accurate gradient evaluation. Finally, case studies are presented to demonstrate that the high-order error feedback does offer much improved performance and that the proposed joint optimization is superior relative to a sequentially optimized system where the state-space coordinate transformation and high-order error feedback matrices are optimized separately.

Four-Dimensional Wind Speed Model for Adequacy Assessment of Power Systems With Wind Farms

With high wind-power penetration, power-system operators require accurate wind-power output from wind farms for system operation. A more accurate four-dimension wind speed model is proposed in this paper to simulate the effects of terrain and time delay of wind speed. Two coordinate transformation matrices are proposed to easily incorporate wind direction in wind-speed simulation. The effects of random failures and wake of wind turbine generators on wind-speed distribution in a wind farm are also considered in the proposed model. A sequential Monte Carlo simulation technique is developed to evaluate adequacy of power systems with wind farms using the proposed model. The modified IEEE RTS is analyzed to illustrate the models and technique. The simulation results show that those factors have significant impact on power output of a wind farm and system reliability.