Proposed Error Model Research Articles

Unified framework for geometric error compensation and shape-adaptive scanning in five-axis metrology systems

In response to the escalating demand for precise shape metrology of complex optical surfaces, this study unveils a unified geometric error compensation and trajectory planning framework tailored for high-accuracy five-axis scanning metrology systems, which remains a notably underexplored field compared to error compensation in machine tools. Founded on a unified geometric model, the proposed framework seamlessly integrates a versatile shape-adaptive trajectory planning strategy, a thorough global error sensitivity analysis approach, and an exhaustive geometric error compensation scheme. Leveraging inverse kinematics, an innovative shape-adaptive scanning trajectory generation strategy is mathematically formulated, thereby facilitating adaptable measurement trajectory generation for diverse surface geometries. Employing forward kinematics, an exhaustive geometric error model is established to extensively address the 53 distinct geometric errors in the metrology system. This proposed error model fundamentally augments conventional geometric error models in machine tool by managing not only the geometric errors from the motion system, but also those from the probe and workpiece. To streamline the error compensation procedure, a novel global error sensitivity analysis approach is introduced, identifying both system-oriented and process-oriented sensitive geometric errors for targeted compensation. Experimental validation using a standard ball, which achieved an exceptional 89.35% reduction in the root mean square of the measurement errors, further confirms the feasibility and effectiveness of the proposed framework. By offering an universal trajectory planning, sensitivity analysis and error compensation trinity for five-axis scanning metrology systems, this study sets the stage for precision advancements and design optimization across diverse configurations of metrology systems.

Improvement of photogrammetric joint roughness coefficient value by integrating automatic shooting parameter selection and composite error model

In order to improve the accuracy of the photogrammetric joint roughness coefficient (JRC) value, the present study proposed a novel method combining an autonomous shooting parameter selection algorithm with a composite error model. Firstly, according to the depth map-based photogrammetric theory, the estimation of JRC from a three-dimensional (3D) digital surface model of rock discontinuities was presented. Secondly, an automatic shooting parameter selection algorithm was novelly proposed to establish the 3D model dataset of rock discontinuities with varying shooting parameters and target sizes. Meanwhile, the photogrammetric tests were performed with custom-built equipment capable of adjusting baseline lengths, and a total of 36 sets of JRC data was gathered via a combination of laboratory and field tests. Then, by combining the theory of point cloud coordinate computation error with the equation of JRC calculation, a composite error model controlled by the shooting parameters was proposed. This newly proposed model was validated via the 3D model dataset, demonstrating the capability to correct initially obtained JRC values solely based on shooting parameters. Furthermore, the implementation of this correction can significantly reduce errors in JRC values obtained via photographic measurement. Subsequently, our proposed error model was integrated into the shooting parameter selection algorithm, thus improving the rationality and convenience of selecting suitable shooting parameter combinations when dealing with target rock masses with different sizes. Moreover, the optimal combination of three shooting parameters was offered. JRC values resulting from various combinations of shooting parameters were verified by comparing them with 3D laser scan data. Finally, the application scope and limitations of the newly proposed approach were further addressed.

Thermal error modeling of servo axis based on optimized LSSVM with gray wolf optimizer algorithm

Thermal error of a CNC machine tool has a serious effect on its machining accuracy. In order to reduce the thermal error, establishing an accurate and robust thermal error model is necessary. A new data-driven thermal error model is proposed based on gray wolf optimizer (GWO) and least squares support vector machine (LSSVM). While, three temperature-sensitivity points (TSPs) were picked by grouping search method. With optimizing the hyperparameters γ and σ2 by GWO algorithm, LSSVM is applied to thermal error modeling, which has advantages in dealing with small samples and nonlinear data. The experiments were conducted, and the results showed that the error prediction model constructed by GWO-LSSVM achieved an accuracy of 94.39 % meeting the requirements for error compensation. Then, the stability of the proposed error model is verified. Finally, the LSSVM model is compared with MLR and BP models commonly used in error modeling, and the advantages and applicable occasions are analyzed through experiments.

Geometric error modeling and compensation for high precision composite optical measurement systems

Composite optical measurement systems are widely used in the field of precision measurement due to their combination of inspection with high accuracy, speed, wide range, real-time, and other advantages. Whereas errors are prevalent in measurements, in order to improve detection accuracy, the systems must be compensated for geometric errors in three-dimensional space. Aiming at the complex situation of multi-probes and multi-zooms in the composite optical measurement system, the current error modelling methods are difficult to be directly applied, so this paper establishes a unified three-dimensional volumetric error model based on the theory of multi-body system and combined with the principle of geometric optics, performs the error verification through the direct measurement method, and finally realises the compensation of geometric error in the continuous space of the whole measurement range. Eventually, the accuracy of the proposed error model and the effectiveness of the error compensation method were verified by a laser interferometer and standard objects to be measured, and the integrated geometric error of the system was decreased by 76.55%, which effectively improved the accuracy of the system. The error modelling and compensation method proposed in this paper provides a new idea for the error compensation of the zoom measurement system, and at the same time, it is universal for the measurement systems of different structures and motion forms, which can be widely used in the field of precision measurement.

Attitude Determination System with Rotation of Gyroscopes

A method of auto-compensation for the influence of constant biases in the output signals of gyroscopes in the system for determining the orientation of an object, based on the rotation of the gyroscopes relative to the object, is considered. The analysis of the obtained models of errors in determining the orientation and the simulation performed show the high efficiency of this method for improving the accuracy of readings. The proposed error model makes it quite easy to evaluate the efficiency of auto-compensation. It is shown that the problem may be reduced to integrating the direction cosine matrix that determine the position of the sensitive elements in the reference coordinate system.

Increasing the Accuracy of Orientation Determination Based on Auto-compensation of Errors

An increasing the accuracy of orientation determination of strapdown system based on the spatial rotation of gyroscopes relative to an object is considered. The subject of analysis is a mathematical model of the measurement errors of orientation angles, taking into account the constant biases of gyroscopes. The proposed error model makes it quite easy to evaluate the efficiency of auto-compensation. It is shown that the problem may be reduced to integrating the direction cosine matrix that determine the position of the sensitive elements in the reference coordinate system. The results of theoretical analysis and simulation confirm the high efficiency of this method for increasing the accuracy of measuring kinematic parameters.

Should data ever be thrown away? Pooling interval-censored data sets with different precision

Data quality is an important consideration in many engineering applications and projects. Data collection procedures do not always involve careful utilization of the most precise instruments and strictest protocols. As a consequence, data are invariably affected by imprecision and sometimes sharply varying levels of quality of the data. Different mathematical representations of imprecision have been suggested, including a classical approach to censored data which is considered optimal when the proposed error model is correct, and a weaker approach called interval statistics based on partial identification that makes fewer assumptions. Maximizing the quality of statistical results is often crucial to the success of many engineering projects, and a natural question that arises is whether data of differing qualities should be pooled together or we should include only precise measurements and disregard imprecise data. Some worry that combining precise and imprecise measurements can depreciate the overall quality of the pooled data. Some fear that excluding data of lesser precision can increase their overall uncertainty about results because lower sample size implies more sampling uncertainty. This paper explores these concerns and describes simulation results that show when it is advisable to combine fairly precise data with rather imprecise data by comparing analyses using different mathematical representations of imprecision. Pooling data sets is preferred when the low-quality data set does not exceed a certain level of uncertainty. However, so long as the data are random, it may be legitimate to reject the low-quality data if its reduction of sampling uncertainty does not counterbalance the effect of its imprecision on the overall uncertainty.

Precision-Driven Multi-Target Path Planning and Fine Position Error Estimation on a Dual-Movement-Mode Mobile Robot Using a Three-Parameter Error Model

The multi-target path planning problem is a universal problem to mobile robots and mobile manipulators. The two movement modes of forward movement and rotation are universally implemented in integrated, commercially accessible mobile platforms used in logistics robots, construction robots, etc. Localization error in multi-target path tracking is one of the crucial measures in mobile robot applications. In this article, a precision-driven multi-target path planning is first proposed. According to the path's odometry error evaluation function, the precision-optimized path can be discovered. Then, a three-parameter odometry error model is proposed based on the dual movement mode. The error model describes localization errors in terms of the theoretical motion command values issued to the mobile robot, the forward moving distances, and the rotation angles. It appears that the three error parameters follow the normal distribution. The error model is finally validated using a mobile robot prototype. The error parameters can be identified by analyzing the actual moving trajectory of arbitrary movements. The experimental localization error is compared to the simulated localization error in order to validate the proposed error model and the precision-driven path planning method. The OptiTrack motion capture device was used to capture the prototype mobile robot's pose and position data.

Optimal Design of a Kinematically Redundant Planar Parallel Mechanism Based on Error Sensitivity and Workspace

Abstract In this paper, the dimensional optimization of a (2PRR)-R + 2RRR (P and R represent the prismatic and revolute joint, respectively, and the underline indicates that the joint is the actuator) kinematically redundant parallel mechanism is performed by taking the integrated error sensitivity index and workspace as the objectives. Based on the matrix method, a generalized method for error modeling of the planar three-degree-of-freedom (3-DOF) parallel mechanism is proposed. The process of the generalized error modeling method is explained, and the error model of the planar (2PRR)-R + 2RRR kinematically redundant parallel mechanism is established. Based on the proposed error model, the error sensitivity indices of different dimension types are calculated. In order to reduce the error sensitivity and expand the workspace, the elitist non-dominated sorting genetic algorithm (NSGA-II) is used for multi-objective optimization of the mechanism. The comparative analysis between the optimized and the non-optimized mechanism is carried out from three aspects: error sensitivity, distribution of low-error sensitivity area, and area of the workspace. The results show that the optimization algorithm not only expands the workspace of the mechanism but also effectively reduces the error sensitivity in the workspace.

Sensitivity Analysis and Accuracy Synthesis of a Deployable Antenna Considering Clearances and Link Tolerances

Abstract Antenna panels of a satellite are usually deployed and supported by a deployable mechanism. The accuracy of the deployable mechanism is important because it directly affects the performance of the satellite. This article focuses on the accuracy analysis of a deployable antenna considering the coupled effects of joint clearances and link tolerances, and based on which, two accuracy synthesis methods are further discussed. A novel geometric error modeling method is proposed to estimate the error space of the outer panel of the antenna. The advantages of the proposed method are that it can handle the overconstrained constraints efficiently and decouple the positioning error and orientation error. Moreover, the error space is visualized using a sampling method. Based on the proposed error modeling method, the maximum orientation and positioning errors are computed, and following which, sensitives of the antenna accuracy to tolerances considering various sizes of clearances are analyzed. Finally, two optimization methods for accuracy synthesis are discussed, and instructive conclusions are drawn.

Precision Design of Transmission Mechanism of Toggle Press Based on Error Modeling

The position accuracy of the bottom dead centre (BDC) of a slider is a crucial performance index to measure the quality of a multi-link mechanical press. At present, the research on improving the position accuracy of the BDC mainly focuses on error compensation, and there is a lack of research on the tolerance design of the transmission mechanism according to the accuracy requirements of the BDC. In this paper, the motion output accuracy of the press is characterized by the position error of the BDC of the slider and the inclined angle error of the slider. Based on the loop increment method, the error transmission model of the transmission mechanism of a multi-link mechanical press was established. The key error factors that affect the output accuracy of the press were determined by analysis of error sensitivity. On this basis, a precision synthesis method considering manufacturing cost and BDC precision reliability was proposed. Finally, a Monte Carlo simulation and experiments on a toggle mechanical press were conducted to verify the model. The results show that the proposed error model and optimization method are feasible and effective.

Redundant Parallel Beam Multiaxis Force Sensor—Accuracy Space

The principle of the redundant parallel beam multi-axis force sensor is a transformation from displacement domain to force domain. This paper proposed a general force error model of a distributed force measurement system with redundant multi-axis force sensors based on the measurement <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$/$ </tex-math></inline-formula> surveying adjustment method to transform displacement sensor errors to force errors. The optimum accuracy space is defined according to the error model. According to the accuracy space, the torque error increase sharply when the measuring point is far away from the optimum accuracy space, but the force error remains constant. A long-distance arrangement of displacement sensors can increase the size of the optimum accuracy space and a high-precision displacement sensor can increase the measurement accuracy in the whole space. A deformation filter can minimize the affection of the loading platform deformation on the measurement accuracy. A dual-measurement system is recommended to increase both the force accuracy and the torque accuracy when the measuring point is far from the optimum accuracy space of the main multi-axis force sensing system. The error point cloud is used to verify the proposed error model.

Optimum synthesis of four-bar mechanisms considering clearance-induced uncertainty

This paper presents a multi-objective optimization model for the motion synthesis of four-bar mechanisms considering the effects of revolute joint clearances. A geometric method is proposed to compute the maximum clearances-induced positioning and orientation errors. The proposed error modeling method is validated using a simple example, and the results reveal its advantages of high accuracy and fast computation compared with the widely used existing methods. A multi-objective optimization model is developed to design the four-bar mechanism with the best accuracy. The positioning and orientation errors are modeled separately and equally. Moreover, three constraints are included in the optimization to ensure that the synthesized mechanism is a compact crank-rocker or double-crank mechanism free from branch and order defects. A study case of synthesizing a four-bar mechanism passing through 6 motion points is investigated to illustrate the proposed method. Finally, conclusions and a brief discussion are given.

Performance Analysis of Nonuniformly Arranged Array With Position Errors

In this letter, focusing on the diversity of position error generation of distributed array with different interelements’ distances, we presents a number of mathematic models of position error to give further analysis on performance of optimized distributed array. Proposed models define a tolerance analysis tool to research the uncertainties of nonuniform array’s pattern with position errors. A set of representative numerical experiments are reported and discussed to point out the effectiveness and accuracy of proposed error models. Position error tolerance of distributed array is also researched to give specific error limitation under certain demand on performance. Meanwhile, with the study of position error tolerance, it is helpful to make performance analysis of optimized distributed array with position errors more precise.

Optimizing the Energy Efficiency of Unreliable Memories for Quantized Kalman Filtering

This paper presents a quantized Kalman filter implemented using unreliable memories. We consider that both the quantization and the unreliable memories introduce errors in the computations, and we develop an error propagation model that takes into account these two sources of errors. In addition to providing updated Kalman filter equations, the proposed error model accurately predicts the covariance of the estimation error and gives a relation between the performance of the filter and its energy consumption, depending on the noise level in the memories. Then, since memories are responsible for a large part of the energy consumption of embedded systems, optimization methods are introduced to minimize the memory energy consumption under the desired estimation performance of the filter. The first method computes the optimal energy levels allocated to each memory bank individually, and the second one optimizes the energy allocation per groups of memory banks. Simulations show a close match between the theoretical analysis and experimental results. Furthermore, they demonstrate an important reduction in energy consumption of more than 50%.

Error model and simulation for multisource fusion indoor positioning

Seamless positioning services are of a critical concern in building smart cities. In a multisource fusion indoor positioning system, providing the guidance information for the deployment of positioning sources is a key technology, which can optimize the infrastructure resources to provide higher positioning accuracy. The error models of single-source positioning such as the received signal strength (RSS) fingerprint and the pedestrian dead reckoning (PDR) should be extended to meet the requirement of multisource indoor positioning for positioning error estimation. This paper proposes a model that combines the RSS fingerprint and PDR positioning error models for fusion positioning error simulation, which weights the PDR and RSS fingerprint positioning results and calculates the mean square error for the fusion positioning according to their positioning variances. This model is also used to establish an indoor positioning simulation system. To validate the proposed model, an experiment is performed which compared the actual positioning errors using the fusion positioning with the errors of the simulate model. The results show that the actual positioning error curves and the error curve predicted by the model are consistent. As a result, the proposed error model provides a solution for optimizing the deployment of positioning sources.

Strategy to Decrease the Angle Measurement Error Introduced by the Use of Circular Grating in Dynamic Torque Calibration.

A circular grating angle encoder is a key component in the dynamic torque calibration system. To improve the accuracy of an angle measurement, in this paper, the source of the angle measurement error of the circular grating is analyzed; an eccentricity error model and an inclination error model are proposed, respectively; further, these two models are combined to establish a total error model. Through the simulation study with the models, the conditions, in which the eccentricity error or inclination error can be ignored, are discussed. The calibration and compensation methods of the angle measurement error are given, and a progressive error compensation function which integrates the first harmonic fitting and the second harmonic fitting is obtained. An experiment is performed to verify the proposed calibration and compensation methods. The peak-to-peak value of the compensated angle measurement error of the single reading head can be reduced by about 93.76%, which approximates to the error of the mean value of the double reading heads. The experimental results show that the error calibration and compensation method based on the proposed error model can effectively compensate the angle measurement error of the circular grating with a single reading head, and obtain a high-precision measurement angle.

Geometric analysis of measurement errors in a surface metrology class with closed-loop probes

This paper presents a novel measurement error model for closed-loop surface metrology. In these surface metrology systems, the motion of the probe tip is strictly constrained on a parallel surface that is separated from the measured sample surface of constant displacement by closed-loop controllers. Examples of such metrology systems include constant signal mode scanning probe microscopes and coordinate measuring machines. The combined effect of volumetric errors and surface tracking controllers on the final measurement error was analyzed using an error model developed based on multi-body system theory and surface geometry. A comprehensive overview of the errors of a typical large-scale scanning tunneling microscopy system is provided as an example. Simulations were performed to illustrate the influence of the systematic errors on the final accuracy. Experiments were also conducted on the measurement of a typical micro-structured surface to both verify the proposed error model and establish a possible compensation scheme.

Error modeling and singularity analysis of planar parallel mechanisms using optimized expression of joint clearances and link deformation

A fast error modeling method is proposed to analyze the pose deviation of 3-RPR planar parallel mechanisms with multiple revolute joint clearances. The pose error arisen from clearances and limb deformation is modeled from the point of inverse kinematics and estimated numerically. Manipulator poses are represented by elements of the Lie group SE(2) and the discontinuous deformation of limbs due to clearances are modeled by the Heaviside function, which is then approximated with the hyperbolic tangent function. After establishing the error model, an efficient error compensation method is introduced. The proposed error modeling and accuracy analysis method are validated by case study of a 3-RPR mechanism. The results show that the hyperbolic tangent function runs faster than the step function and it provides better convergence. Finally, the singularity of the 3-RPR mechanism with load is analyzed and the results reveal the effects of loads and clearances on mechanism singularities.

A High-Accuracy Method for Calibration of Nonorthogonal Angles in Dual-Axis Rotational Inertial Navigation System

The dual-axis rotational inertial navigation system (RINS) can compensate the errors of inertial sensors by rotating the inertial measurement unit (IMU), thereby improving the accuracy of the inertial navigation system. However, in practical applications, nonorthogonal angles exist between the rotating axes, resulting in a decrease in navigation accuracy. In this paper, the error model of dual-axis RINS is established, which includes all nonorthogonal angles. Based on the nonorthogonal angle error model, the transformation matrix between the frames is derived, and the necessity of nonorthogonal angle calibration is discussed. The observability of the system is analyzed, and the degree of observability obtained indicates that the proposed nonorthogonal angles are observable. Then, a new nonorthogonal angle calibration method is proposed, which uses the dual-axis rotation mechanism. The proposed calibration method can calibrate all nonorthogonal angles with high accuracy, and the minimum estimation error is less than 10%. Simulations and experiments prove the effectiveness of the proposed error model and the calibration method, and demonstrate that the navigation accuracy can be significantly improved compared with the previous method.