Homogeneous Transformation Method Research Articles

Shape Sensing of Cantilever Column Using Hybrid Frenet-Serret Homogeneous Transformation Matrix Method.

The Frenet-Serret (FS) framework stands as a pivotal tool in shape sensing for various infrastructures. However, this tool suffers from accumulative errors, particularly at inflection points where the normal vector undergoes sign changes. To minimize the error, the traditional FS framework is modified by incorporating the homogeneous matrix transformation (HMT) method for segments containing inflection points. Additionally, inclination information is also used to calculate the unit tangent vector and the unit norm vector at the start point of each segment. This novel approach, termed the FS-HMT method, aims to enhance accuracy. To validate the effectiveness of the proposed method, a simulation of a cantilever column was conducted using finite element software ANSYS 19.2. The numerical results demonstrate the capability of the proposed method to accurately predict curves with inflection points, yielding a maximum error of 1.1%. Subsequently, experimental verification was performed using a 1 m long spring steel sheet, showcasing an error of 4.9%, which is notably lower than that of the traditional FS framework. Our proposed modified FS framework exhibits improved accuracy, especially in scenarios involving inflection points. These findings underscore its potential as a valuable tool for enhanced shape sensing in practical applications.

Workspace Solution and Space Trajectory Planning of 5-DOF Manipulator

In this paper, a 5-DOF manipulator is designed to carry the pipeline robot. The joint coordinate system and connecting rod parameters of the manipulator are confirmed by D-H method, and the kinematics model of the manipulator is established. The position and direction of the end of the manipulator are deduced by using the homogeneous transformation method. Monte Carlo method is used to solve the workspace of manipulator, and MATLAB is used to draw the workspace cloud map. Set the motion trajectory of manipulator, using joint space trajectory planning, through MATLAB simulation of motion form, get the change of joint. It provides a foundation for trajectory planning, dynamics analysis, motion control and parameter optimization of manipulator.

Multi-Position Calibration Method of Gyroscopes in Inertial Measurement Unit Based on Centrifuge with Position Platform

In order to accurately calibrate the static error model coefficients of three gyroscopes simultaneously, especially the higher-order coefficients, in an inertial measurement unit, an optimal 16-position method is designed. Analysis is conducted using a precision centrifuge with a counter-rotating platform, and coordinate systems are established on centrifuge. Thus, centrifuge error sources, which may affect the calibration accuracy of a gyroscope’s drift coefficients, are analyzed, corresponding to the coordinate systems established. Precise expressions for the input angular rate and the input specific forces on each axis of the three gyroscopes are derived using a homogeneous transformation method. The influence of centrifuge error on the drift coefficients is analyzed, and the calibration accuracy can be improved by compensating for the influence of centrifuge error. The 16-position calibration method is validated through simulation, which establishes that centrifuge errors such as Δθy2t, Δθy2(−Ωt), Δθx2t, Δθx2(−Ωt), and ΔΩ mainly influences the bias MF and first-order drift coefficients MI and MS, but has little impact on higher-order coefficients. The calibration process is simple and short time-consuming, and the calibration accuracy is satisfied to requirement. After compensating for the main centrifuge errors, the 16-position method will further increase the accuracy of an inertial navigation system.

Deep Semantic Parsing of Freehand Sketches With Homogeneous Transformation, Soft-Weighted Loss, and Staged Learning

In this paper, we propose a novel deep framework for part-level semantic parsing of freehand sketches, which makes three main contributions that are experimentally shown to have substantial practical merit. First, we propose a homogeneous transformation method to address the problem of domain adaptation. For the task of sketch parsing, there is no available data of labeled freehand sketches that can be directly used for model training. An alternative solution is to learn from datasets of real image parsing, while the domain adaptation is an inevitable problem. Unlike existing methods that utilize the edge maps of real images to approximate freehand sketches, the proposed homogeneous transformation method transforms the data from domains of real images and freehand sketches into a homogeneous space to minimize the semantic gap. Second, we design a soft-weighted loss function as guidance for the training process, which gives attention to both the ambiguous label boundary and class imbalance. Third, we present a staged learning strategy to improve the parsing performance of the trained model, which takes advantage of the shared information and specific characteristic from different sketch categories. Extensive experimental results demonstrate the effectiveness of the above three methods. Specifically, to evaluate the generalization ability of our homogeneous transformation method, additional experiments for the task of sketch-based image retrieval are conducted on the QMUL FG-SBIR dataset. Finally, by integrating the proposed three methods into a unified framework of deep semantic sketch parsing (DeepSSP), we achieve the state-of-the-art on the public SketchParse dataset.

Sequential Calibration Method of Nonlinear Errors of PIGA on Counter-Rotating Platform Centrifuge

This article proposes a new method for calibrating nonlinear errors of pendulous integrating gyro accelerometer (PIGA) on the precision centrifuge with a counter-rotating platform, which can provide both constant and harmonic input accelerations. First, by analyzing the working principle and error sources of the centrifuge, corresponding coordinate systems are established. With the homogeneous transformation method, the precise input accelerations and angular velocities of PIGA in the centrifuge test are derived, and the error calibration models are established. Then, the sequential calibration method is designed by combining the symmetrical position test with a synchronous counter-rotating test. This sequential method has the advantage of installing PIGA on the centrifuge only once. Not only the nonlinear error coefficients are identified but also the centrifuge errors, the misalignments of PIGA with the centrifuge, and the installation errors of PIGA on the centrifuge are effectively suppressed simultaneously. Experiments are presented and discussed to illustrate the effectiveness of the proposed method.

Effect of detector installation error on the measurement accuracy of multi-degree-of-freedom geometric errors of a linear axis

A linear axis is the key component of several mechanical machines, especially of computer numerical control machine tools. The simultaneous measurement of the geometric motion errors of a linear axis is indispensable to compensate the machine errors, which improves the accuracy of machine tools. The principle of laser collimation is widely used for measuring the multi-degree-of-freedom (MDOF) geometric motion errors of a linear axis. However, installation errors of the detectors affect the measurement accuracy. In this study, we have proposed, for the first time, a measurement model for investigating the influence of all detector installation errors on the accuracy of the simultaneous measurement of MDOF geometric motion errors in a linear axis. The effect of detector installation errors on the horizontal straightness, vertical straightness, pitch, and yaw are analyzed using the homogeneous transformation method, which indicates that angular installation errors of detectors along the X0-axis direction can cause a crosstalk between the two straightness errors and between pitch and yaw errors. In addition, translational installation errors of position-sensitive detectors in the X0-axis direction, which is often called the defocusing amount, has a significant effect on the pitch and yaw measurements. To elucidiate the effect of defocusing, a measurement model was verified by both Zemax simulation and experiments. The experimental results indicate that the maximum deviation of the measured results obtained using the proposed model from that obtained using a Renishaw XL-80 interferometer can be greatly reduced by compensating the measurement errors induced by defocusing.

Kinematics Control and Analysis of Industrial Robot

The robot’s development present situation, basic principle and control system are introduced briefly. Research is mainly focused on the study of the robot’s kinematics and motion control. The structural analysis of a planar articulated robot (SCARA) robot is presented,the coordinate system is established to obtain the position and orientation matrix of the end effector,a method of robot kinematics analysis based on homogeneous transformation method is proposed, and the kinematics solution of the robot is obtained.Establishment of industrial robot’s kinematics equation and formula for positive kinematics by example. Finally,the kinematic analysis of this robot was verified by examples.It provides a basis for structural design and motion control.It has active significance to promote the motion control of industrial robot.

Investigation of sensitivity analysis and compensation parameter optimization of geometric error for five-axis machine tool

To improve the accuracy of five-axis machine tool with a swiveling head, an approach for optimizing compensation values taking into account the sensitivity of position-independent geometric error is proposed in this paper. At first, the synthetic volumetric error model of five-axis machine tool has been established with homogeneous transformation method and multibody system theory, which describes the effect of geometric error components on position and orientation error vector intuitively. Second, the probability distributions of geometric errors in workspace of machine tool are sufficiently taken into account in order to overcome the defects of the analysis at certain locations, and then, global quantitative sensitivity analysis is introduced for determining the effect of each geometric error on precision of machine tool. Next, the optimum values are obtained by multiobjective quality loss and precision robustness trade-offs based on archive-based microgenetic algorithm (AMGA) for geometric error compensation. Finally, the geometric error compensation experiments were carried out, and the results show that the accuracy of measuring trajectories are improved significantly, which reaches 73.7% after error compensation with optimum values based on sensitivity analysis. Hence, the proposed methodology of analysis and compensation are effective for analyzing the effect of geometric errors and improving the precision of machine tool.

A new Multi-position calibration method for gyroscope's drift coefficients on centrifuge

In order to accurately calibrate the gyroscope's drift coefficients, the coordinate systems are established on precision centrifuge with counter-rotating platform, and the corresponding error sources of each coordinate system are analyzed. The precise input angular rate along each axis of gyroscope are derived by homogeneous transformation method and also the nominal input specific forces are given. The precise expression of each drift coefficient is provided combining with gyroscope's error model, input specific forces and angular rates of gyroscope at the 16-position. And the compensation for the identification results can be conducted by the expression. It can be shown that through simulation two coefficients are significantly influenced by centrifuge errors when using the 16-position calibration method, and centrifuge errors have no impacts on other coefficients, which proves that the method can effectively eliminate the impacts of centrifuge errors. Only several centrifuge errors should be considered in actual calibration such as the perpendicularity between the counter rotating platform's revolving axis and the working base, the wobbles of counter rotating platform, and the difference between the angular rate of counter rotating platform and the one of the main axis.

陀螺仪漂移系数的多位置离心机标定方法及误差分析

为了准确标定陀螺仪漂移系数，本文在带反转平台的精密离心机上建立坐标系，并在考虑离心机误差源的情况下分析了坐标系间的位置关系。用齐次变换法推导出了陀螺仪各轴上精确的角速度输入并给出了比力输入标称值，再结合陀螺仪静态误差模型推导出含有陀螺仪漂移系数的各次谐波幅值表达式，通过离心机提供两种向心加速度获得陀螺仪各采样时刻的输出，应用谐波分析法可以求得各次谐波幅值大小，从而利用最小二乘估计对各漂移系数进行辨识。试验仿真表明采用带反转平台的离心机对陀螺仪漂移系数标定时可以有效的规避离心机误差，离心机误差只对于DI、DS和DOS的标定有影响，且影响非常大，须在考虑离心机误差的情况下才能够实现精确标定；对于其他系数标定没有影响。实际标定时需要精确测量反转平台回转轴与盘面的垂直度误差，反转平台回转轴倾角回转误差以及反转平台角速率与主轴角速率的大小之差等以满足标定精度要求。 In order to calibrate the drift coefficients of gyro precisely, in the paper, the coordinate systems on precision centrifuge with counter-rotating platform were established, and the positional relationships were analyzed in consideration of centrifuge errors. The accurate angular velocity along each axis of gyro was derived by using homogeneous transformation method and the nominal value of input specific force was given, and each harmonic amplitude expression containing drift coefficients of gyro was derived combining the static error model of gyro. The output at each sampling time of gyro was acquired by providing two centripetal accelerations on centrifuge, and each harmonic amplitude’s value was calculated by using harmonic analysis, thus each drift coefficient could be identified by least squire evaluation. The simulation results show that the method using the counter-rotating platform can avoid the centrifuge errors’ influences and the errors mainly impact on the DI, DS and Dos significantly which should be precisely calibrated in consideration of centrifuge errors; and the errors have no impacts on other coefficients. In real calibration, the perpendicularity between rotating axis of counter-rotating platform and the disk surface, the wobbles of rotating axis of counter-rotating platform and the difference between the angular rate of counter-rotating platform and the main axis should be measured precisely in order to meet the requirement.

Mechanical Design and Kinematics Analysis of a Hydraulically Actuated Manipulator

This paper gives a mechanism design of a six DOF hydraulically actuated manipulator firstly. Then its DH frames and link parameters are given. Next, its forward kinematic equations are derived according to homogeneous transformation method. Fourthly, the analytical solutions of its inverse kinematics are solved by given the position and posture of the end-effector simultaneously. The posture of the end-effector is given with three z-y-z Euler angles for they have obvious geometry meanings and are easy to be measured. In addition, the correctness of the inverse kinematic equations is verified in Simulink by comparing many sets of randomly produced joint variables in workspace and their corresponding inverse solutions.

On-machine volumetric-error measurement and compensation methods for micro machine tools

The CNC micro machine tool is an essential of the micro machining process for manufacturing miniature high-tech products. A software-based error compensation method based on measured volumetric errors that can pre-compensate cutting path can further enhance a machine’s accuracy without increasing its manufacturing cost. Considering the limited workspace of a micro machine tool and the instrument cost, an on-machine and vision-based measurement method integrating machine vision method with use of 3 COMs and homogeneous transformation method was proposed in this study. Sensitivity analysis was conducted to analyze the influence of the CMOS pixel error to the measurement accuracy. To improve the machining accuracy, the measured errors were directly compensated to the machining trajectory with use of an error compensation method that consists of an element-free error interpolation scheme and a recursive software error compensation scheme. Experiments on a toggle-type micro machine tool were conducted to verify the effectiveness of the proposed methods. The results have shown that the feasibility and effectiveness of the proposed methods. The tracking errors of the machine was further improved from (x = 7.9 μm, y = 4.1 μm) to (x = 5 μm, y = 3.7 μm).

Design and Analysis of Novel 3-RPC Flexible Precision Platform

Based on 3-RPC parallel mechanism, a novel model of flexible precision platform is proposed, which is driven by piezoelectric ceramic actuator. And moving pair, cylinder, connecting rod motion pair of structure are optimized based on the initial configuration. Then using the FEA simulations, static analysis of the same size of the initial configuration and the optimization is analysed. Using Homogeneous transformation method, the forward and inverse kinematics of 3-RPC is gained. Obtained the relationship of flexible precision platform driving by piezoelectric ceramic actuator between the input of move pair and the micro displacement output of the upper platform, and loading test by finite element software, testing the solution results of the forward and inverse kinematics. Based on design principle, the error of principle of the precision platform is analyzed and calculated the displacement error on function and non-function direction, while comparing the output displacement and the size of the error. And the correctness of the analysis results is validated by FEA simulations. The results of the study show that the flexible precision platform has several advantages in the aspects of kinematics decouple, structure simple, control easily and a good application prospect.

THREE-DIMENSIONAL POLYHEDRAL INVISIBLE CLOAK CONSISTING OF HOMOGENEOUS MATERIALS

Invisible cloak with its amazing functions has been turned into reality due to the advent of transformation optics during the past few years. However, the inhomogeneity and singularity of electromagnetic parameters in cloak are still the main bottlenecks for practical realization. In this paper, we propose a scheme of three-dimensional polyhedral invisible cloak to overcome these shortcomings by using a linear homogeneous transformation method. The constitutive parameters of the polyhedral cloak are homogeneous and anisotropic, which are relatively easy for realization. Numerical simulations demonstrate that good invisibility performance can be achieved for any polarization wave. Our work provides a novel approach to simplify three-dimensional cloak in practice.

Broadband polygonal invisibility cloak for visible light

Invisibility cloaks have recently become a topic of considerable interest thanks to the theoretical works of transformation optics and conformal mapping. The design of the cloak involves extreme values of material properties and spatially dependent parameter tensors, which are very difficult to implement. The realization of an isolated invisibility cloak in the visible light, which is an important step towards achieving a fully movable invisibility cloak, has remained elusive. Here, we report the design and experimental demonstration of an isolated polygonal cloak for visible light. The cloak is made of several elements, whose electromagnetic parameters are designed by a linear homogeneous transformation method. Theoretical analysis shows the proposed cloak can be rendered invisible to the rays incident from all the directions. Using natural anisotropic materials, a simplified hexagonal cloak which works for six incident directions is fabricated for experimental demonstration. The performance is validated in a broadband visible spectrum.

Modelling of the 3D machining geometric defects accounting for workpiece vibratory behaviour

In this paper, we present a three-dimensional manufacturing tolerancement model. Several researchers have interested to modelling the machining geometric defects. The most researchers are limited to kinematic and static study. Only some works are evoked the dynamic effects, especially the influence of the chatter phenomenon on the roughness of the machined surface. In this context, the paper presents a contribution for modelling and quantification of the machining geometric defects where the machining dynamic effects are considered. A developed method is established based on Homogeneous Transformation Method in subject to determine the kinematical deviations caused by part locating and relocating. The dynamic displacements due to clamping and machining forces are defined using Finite Element Method. The numerical results are then compared to published experimental results. (Received in July 2010, accepted in February 2011. This paper was with the authors 2 months for 1 revision.)

Modeling and Analysis of Multi-station Dimensional Deviation Stream Based on Linear State Space Model

In order to solve the problem of implicit expression of system matrix in the state space model which describes the transfer of product dimensional deviation,and existence of nonlinear factors in traditional multi-station manufacturing systems,part model consists of surface orientation vector,position orientation vector,size vector and its variation model are given on the basis of definitions of part,fixture and machine coordinate systems;by using homogeneous transformation method and matrix theory,mutual transformation between three coordinate systems and linear state space model with explicit system matrix are derived.The relationship built between dimensional deviation and its corresponding deviation source enhances the usability of traditional state space model,and the effectiveness of the model is verified by a case study.

Computational analysis of robot kinematics, dynamics, and control using the algebra of rotations

The basic quantities required for computations in robot kinematics, dynamics, and control are defined. By modifying the rotational operator in terms of the Euler parameters and appropriately choosing the zero reference position, the method of the algebra of rotations becomes more efficient than the homogeneous transformation method for computing these basic quantities. As a result, the computational efficiency is improved by about 50|% for robot kinematics, and by about 20|% for robot dynamics and control, if the method of the algebra of rotations is used instead of the homogeneous transformation method.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

A statistical analysis of positional errors of a multiaxis machine tool

This paper addresses one of the most fundamental problems in precision engineering: machine tool repeatability. The paper presents a synthesis and transformation method for the prediction of accuracy and repeatability of any kinematic machines based on the error information in each axis, which can be measured by using a measurement device such as laser interferometer. It also introduces a procedure to calculate the probability of positioning errors, with a specified limit. based on the homogeneous transformation method in conjunction with a statistical synthesis technique for mean and variance composition. An example shows the actual calculation of probability curve with respect to different tolerance values.