The uncertainty of radius estimation in least-squares sphere-fitting,with an introduction to a new summation based method

This paper considers the sensitivity of three sphere-fitting algorithms to real-world measurement errors. It pays particular attention to nominally spherical surfaces, such as those typically measured by tactile and optical profilometers, addressing the limitations of sensor gauge range and angular tolerance. A recently proposed linear circle-fitting algorithm is extended to a sphere-fitting algorithm and its performance compared to two long standing sphere-fitting algorithms; namely linear and non-linear least-squares. Sources of measurement error in optical profilometers are discussed, and user defined scan parameters are optimised based on the results of a designed experiment. The performance of all three sphere-fitting algorithms are tested on a sphere superimposed with varying degrees of surface irregularities in a Monte Carlo simulation; this study shows that both linear routines display a negative skewness in their radius error distribution. Finally, a method of predicting radius uncertainty is offered that considers the surface residual that remains after sphere-fitting and relates this to the radius uncertainty of the chosen algorithm.     

Practical approach to the estimation of the overall mean caliper diameter of a population of spheres and its application to data where small profiles are missed

In the morphometry laboratory a practical, accurate, and computationally simple procedure is needed when the estimation of the number of spherical particles per unit volume (N_v) is pursued. In addition, this procedure should be able to deal with the very real problem of profiles too small to be counted. Two computationally simple methods, the size class analysis method and the mean profile diameter method, were examined in detail. Computer-generated random profiles from various size-distributions of spheres were analysed by both methods. The percents error to be expected with various sphere distributions with a relatively large number of missed small profiles were determined for both methods. The accuracy of the size class analysis method was poor when a significant number of small profiles were missed. In this situation the accuracy of this method could be greatly improved by applying a simple modification to the procedure. A size class was identified which was larger than the largest missed profile but smaller than the diameter of the smallest sphere in the population. All contributions of this size class and all smaller size classes were deleted from the computation of N_v. The mean profile diameter method was found to be difficult to apply to distributions containing missed small profiles. Missed small profiles were handled more predictably by the modified size class analysis method.     

Near-Edge X-ray Absorption Fine Structure Imaging of Spherical and Flat Counterfaces of Ultrananocrystalline Diamond Tribological Contacts: A Correlation of Surface Chemistry and Friction

A recently installed synchrotron radiation near-edge X-ray absorption fine structure (NEXAFS) full field imaging electron spectrometer was used to spatially resolve the chemical changes of both counterfaces from an ultra- nanocrystalline diamond (UNCD) tribological contact. A silicon flat and Si₃N₄ sphere were both coated with UNCD, and employed to form two wear tracks on the flat in a linear reciprocating tribometer. The first wear track was produced using a new, unconditioned sphere whose surface was thus conditioned during this first experiment. This led to faster run-in and lower friction when producing a second wear track using the conditioned sphere. The large depth of field of the magnetically guided NEXAFS imaging detector enabled rapid, large area spectromicroscopic imaging of both the spherical and flat surfaces. Laterally resolved NEXAFS data from the tribological contact area revealed that both substrates had an as-grown surface layer that contained a higher fraction of sp ²-bonded carbon and oxygen which was mechanically removed. Unlike the flat, the film on the sphere showed evidence of having graphitic character, both before and after sliding. These results show that the graphitic character of the sphere is not solely responsible for low friction and short run-in. Rather, conditioning the sphere, likely by removing asperities and passivating dangling bonds, leads to lower friction with less chemical modification of the substrate in subsequent tests. The new NEXAFS imaging spectroscopy detector enabled a more complete understanding of the tribological phenomena by imaging, for the first time, the surface chemistry of the spherical counterface which had been in continual contact during wear track formation.     

Characterization of instrument drift using a spherical artifact

Drift is a common and inevitable error source in measurements. Currently there are two main approaches to address instrument drift in image or area-based measurements, drift calibration with target tracking and active feedback correction. We propose an alternative approach to drift calibration for profilometers, particularly high speed instruments such as confocal microscopes or scanning white light interferometers. The method is based on sequential measurements of a spherical artifact whose diameter is larger than the field of view. A best fit sphere algorithm is used to determine the movement of the spherical artifact's center over time. This reduces drift measurement uncertainty because it uses height data over the full field of view, compared to target tracking strategies that involve tracking small features. Simulation results show that under practical conditions, e.g. with typical noise levels and typical drift rates, this method is quite effective and can yield measurements with low uncertainty. The measurement is demonstrated on a commercial confocal microscope to determine drift rate magnitude and direction.     

Spherical EBSD

Spheres, or more usually spherical surfaces, are important in electron backscatter diffraction. Both electron backscatter patterns (EBSPs) and pole figure texture data are more accurately represented on the spherical surface, S²; and unit quaternions, which are the optimal method for orientation calculations, exist on the surface of the hypersphere, S³. This paper is split into two distinct parts. The first shows a little of the history of the EBSP technique, including the use of spheres to assemble a spherical Kikuchi map (SKM) and as calibration artefacts. The second part relates new developments in the analysis of EBSPs on the surface of a sphere, a new spherical Hough transform and ideas for fully automatic, ab initio analysis of unknown phases using collections of EBSPs assembled as spherical Kikuchi maps.     

Research on transmission principle and kinematics analysis for involute spherical gear

A new kind of transmission-type spherical gear called ‘ring involute spherical gear mechanism’ is introduced. Compared to the famous TRALLFA spherical gear, this new spherical gear has an involute tooth profile, and a ring tooth that is distributed continuously on the surface of the sphere. This allows the gear to overcome two disadvantages of the TRALLFA spherical gear: the drive principle error and the manufacturing difficulty. The new transmission and the formation principle of the tooth’s surface of the new spherical gear mechanism are first introduced, then another mechanism, called disk rack, is introduced, which is derived from the spherical gear mechanism when one of the spherical gear’s tooth number reaches infinity. To make the research more convenient, every part of the new spherical gear mechanism is named. In the following sections, some problems are discussed, such as the assembly form, the construction characteristics, the correct meshing condition, the continuum transmission condition and so on. Furthermore, the paper deduces the surface formula of the conjugate teeth profiles, which proves that the conjugate teeth profiles is also one of the ringed involute spherical gear. In order to analyze the relationship between two coordinate systems, which is attached respectively to the diving spherical gear and the driven spherical gear, the orientation cosine matrix method is utilized. By series rotational transformation, the kinematics model and inverse kinematics model are deduced. Using the method for calculating the transmission ratio of planet gear train, the relationship of the two oscillating angle between output axis and the bracket is established. Based on the research, the kinematics graphic simulation of spherical gear mechanism and disk rack are made respectively. The results prove correctness of the kinematics model. __________ Translated from Chin. J. Mech. Eng., 2005, 41(5) (in Chinese)     

Study of surface roughness effects in elastohydrodynamic lubrication of rolling line contacts using a deterministic model

A sinusoidal surface roughness model is adopted for the analysis of the effects of roughness amplitude and wavelength on pressure profile, film shape, minimum film thickness and coefficient of friction in a steady state EHL line contact. The influence coefficients used for the evaluation of surface displacements are calculated by utilizing a numerical method based on Fast Fourier Transform. Significant reduction is observed in the minimum film thickness due to surface roughness. Such reduction is quantified by roughness correction factor, C_R, and a relationship between C_R and non-dimensional surface roughness amplitude A is derived as: C_R=1−0.7823A^0.8213. This equation may prove to be of interest from designer's viewpoint. The friction coefficient is found to increase appreciably with increasing amplitude and decreasing wavelength of surface roughness.     

材质气孔率和气孔大小与表面粗糙度的数学模拟与特征

为提高光学系统的成像质量,研究了光学系统中镜片表面粗糙度与镜片材质中的颗粒大小、缺陷、气孔率、气孔大小的关系.通过几何方法,建立了气孔率与气孔大小和数量的关系模型,分别给出了单位体积和单位面积中气孔率与气孔大小和数量的关系表达式.最后用数学方法建立了材质中气孔率、气孔大小与表面粗糙度的定量模型,给出了表面粗糙度与材质气孔率和气孔大小的关系表达式,表明表面粗糙度近似与材质中的气孔率成正比,与气孔大小成正比.     

A new methodology for determining the stress state of the plastic region in machining with restricted contact tools

In rigid-plastic slip-line theory, once the geometry of the slip-line field is established, the stress state of the plastic region (including the primary and secondary deformation zones) in restricted contact machining is governed by the hydrostatic pressure P_A (at a point on the intersection line of the shear plane and the work surface to be machined) and the frictional shear stress τ on the tool rake face. Based on the recently established universal slip-line model and a detailed study of six representative machining cases, a new methodology for determining the stress state of the plastic region, i.e. maximum value principle, is presented in this paper. According to this principle, the stress state of the plastic region can be determined by giving both P_A and τ their theoretical maximum permissible values. The theoretical maximum permissible values of P_A and τ can be found by satisfying four mechanical and geometrical constraint conditions under which the universal slip-line model applies. A comprehensive assessment factor is introduced in this paper. It is shown that the three machining parameters investigated in this present study, i.e. cutting force ratio, chip thickness ratio, and chip back-flow angle can be simultaneously considered to form a comprehensive criterion to compare predicted and experimental results. The applicable range of the maximum value principle is also discussed.     

基于球形目标的激光位移传感器光束方向标定

搭建了非接触式的三坐标测量系统以便精密测量三维型面。将激光位移传感器通过具有两个回转轴的回转体安装在测量机的Z轴上,从而可根据待测表面的形状来调整传感器的方位。为了使传感器在各个方位上实现测量功能,提出了基于球形目标的光束方向标定方法,并详细阐述了其数学原理。标定时,驱动测量机使传感器分别沿测量机的X,Y和Z轴做等间距步进,根据步长和激光束长度的变化建立方程组求解出激光束所在直线的单位方向向量。最后,多次测量尺寸参数已知的六面体标准块规,检验了该测量系统的重复性。结果显示,该系统的测量不确定度为0.048mm;测量另一直径已知的被测球时,传感器在各个方位上的误差小于0.05mm,表明所提出的标定方法使测量系统达到了逆向工程的使用要求。得到的数据表明,本文所提出的方法有较高的标定精度和较好的重复性,为实现三维型面的快速扫描测量奠定了基础。     

Dynamic measurement of first-order spatial derivatives of deformations by digital shearography

This paper presents a simple spatial phase shift shearography based on the Michelson interferometer. A novel digital shearography set-up with a large angle of view, which is based on a 4f system is demonstrated. In the system, the Michelson interferometer is used as a shearing device to generate a shearing distance by tilting a small angle in one of the two mirrors. In fact spherical wave fronts become plane after going through the first lens in 4f system. Tilting the mirror in the Michelson interferometer also generates spatial carrier frequency. Sinusoid fitting method is applied to evaluate the phase. This system can generate a phase map of shearography by using only a single image. The effects of shearing angle, the choice of algorithm and comparison result are discussed in detail. The theory and the application are presented.     

A surface motor-driven planar motion stage integrated with an XYθZ surface encoder for precision positioning

This paper describes a surface motor-driven XY planar motion stage equipped with a newly developed XYθ_Z surface encoder for sub-micron positioning. The surface motor consists of four linear motors placed on the same surface, two pairs in the XY-axes. The magnetic array and the stator winding of the linear motor are mounted on the platen (the moving element) and the stage base, respectively. The platen can be moved in the X-direction by the X-linear motors, and in the Y-direction by the Y-linear motors. It can also be rotated about the Z-axis if the X- or Y-linear motors generate a moment about the Z-axis. The surface encoder consists of two two-dimensional angle sensors and an angle grid with two-dimensional sinusoidal waves on its surface. The angle grid is mounted on the platen of the stage which is levitated by air-bearings. The angle sensors and the air-bearing pads are fixed on the stage base so that the motion of the platen is not affected by the electronic cables and air hoses. The XY-positions and θ_Z rotation of the platen can be obtained from the angle sensor outputs with resolutions of approximately 20 nm and 0.2′′, respectively. The surface encoder is placed inside the stage so that the stage system is very compact in size. Experimental results indicate that precision positioning can be carried out independently in X, Y and θ_Z with resolutions of 200 nm and 1′′, respectively.     

Calibration for the sensitivity of multi-beam angle sensor using cylindrical plano-convex lens

A highly sensitive and compact multi-beam angle sensor (MBAS), which utilizes the principle of operation of an autocollimator, was developed to detect the differential of the local slope components (angle difference) of a point on the mirror surface and using Fourier series, we can obtain the profile data from the angle difference. In order to investigate the application of the MBAS for high precision aspheric surface measurements, two types of calibration methods using plane mirror and cylindrical plano-convex lens has been proposed to measure the sensitivity of the MBAS. The calibration data analysis results using plane mirror agree well with the measurement results of the cylindrical plano-convex lens data. Comparison of the two methods confirms that the second method (using cylindrical plano-convex lens) is more adapted for measurement with ultra high level of uncertainty. Further, the second method is simple, corresponding to a direct calculate in the sensitive parameters aiming to minimize the cost.     

A method for the selection of algorithms for form characterization of nominally spherical surfaces

This paper presents a method for the selection of algorithms for the form characterization of nominally spherical surfaces. The form characterization is applied to discrete data obtained from a three-dimensional (3-D) high-precision measurement system. In this paper, five sphere-fitting algorithms are outlined and compared. The fitting algorithms under investigation are: linear and nonlinear least-squares sphere fit, minimum zone sphere fit, four-point sphere fit, and sphere fit by error curve analysis. A method for selection of the best-fit algorithm, based on a spectral analysis of the surface irregularity is suggested and applied to the surface measurement of worn electrical switching contacts.     

球面拼接的相对曲率半径测量

拼接镜面加工制造的最大难题是匹配各子镜,即使各子镜的曲率半径完全匹配,使其能够等效成一个单一的主镜,对于球面拼接镜,就是要求各子镜的曲率半径相同。本文提出了一种使用Shack-Hartmann传感器和高精度球径仪对球面子镜的相对曲率半径进行测量的新方法,并建立了一个实验系统。该方法先使用共焦调整方法使各子镜共焦,然后再用S-H传感器测量子镜的轴向离焦量,轴向调整压电陶瓷促动器,使由传感器测得的离焦量接近于0,最后对子镜进行一次共焦调整之后,就可以使用高精度球径仪测量出各子镜之间的相对曲率半径差。实验中,拼接镜由三块对边长300mm的正六边形子镜组成,子镜面形为球面,设计曲率半径为2000mm。分析及实验表明,该方法测得的球面拼接子镜的相对曲率半径差精度约为 ,方法适用于大型球面拼接镜面望远镜各子镜相对曲率半径的检测。     

旋转导向钻具导向节轴承圆周间隙角研究*

导向节轴承是旋转导向钻井工具可控弯接头中传递扭矩和钻压的关键部件,导向节轴承中钢球与沟道之间的间隙将影响可控弯接头中导向轴偏置角的准确性和稳定性。根据导向节轴承的结构组成及钢球与沟道的结构特点,建立钢球与沟道截面的几何模型和沟道曲面的曲面方程及数学模型;根据钢球与沟道截面的几何模型,利用MATLAB软件计算导向节轴承中壳内钢球和套内钢球与沟道之间的圆周间隙角的大小。结果表明:导向架内、外球面沟道面为椭圆曲面,无论导向轴朝哪个方向偏置都不影响钢球的圆周间隙角大小;导向轴偏置角为0时,钢球总的圆周间隙角恒定不变且最大,偏置角不为0时,圆周间隙角随转角的增大呈周期性变化并呈现出波动性;套内钢球与沟道之间的圆周间隙角的波动性大于壳内钢球的圆周间隙角的波动性,圆周间隙角的波动性是导致导向轴转动过程中产生振动和噪声的重要原因。     

A novel CNC grinding method for the rake face of a taper ball-end mill with a CBN spherical grinding wheel

A novel method using a CBN spherical grinding wheel to grind the rake face of a taper ball-end mill and the configuration of corresponding CNC tool grinder are presented. This method utilizes the self-adaptation characteristics of a sphere to decrease the number of simultaneous cooperative axes of a CNC tool grinder and to smooth the rake face on the transition area between the taper and the ball-end of the mill. In order to obtain an accurate normal rake angle, which is one of the key factors affecting tool cutting performance, a moving coordinate system based on the required normal rake angle and the cutting edge was established. Then, by the proposed moving coordinate system, an algorithm to determine the position and orientation of a spherical grinding wheel, the basis of CNC code generation, is proposed and the relevant formulations are deduced. The 3D simulation of rake face grinding for a taper ball-end mill with constant helical angle indicates that the number of simultaneous cooperative axes of the CNC tool grinder is decreased from five to four and the smooth transition of the rake face is realized by the proposed method herein.     

On the estimation of particle number

Two methods are proposed for estimating the number of separated particles within a solid structure per unit volume of structure, N_v. Apart from being arranged with independence of any size parameter, no special assumptions upon the size, shape and orientation of the particles are made. The first method is based on the identity N_V = (N_A)_u ? μ_u^?1, where (N_A)_u is the mean number of particle sections per unit area of a plane probe T_u which is uniform random within the structure and perpendicular to a given direction u, whereas μ_u is the mean particle caliper length along u. The second method uses N_V = A_A?V^?1, where A_A is the mean areal fraction of the particles per unit area of section, whereas v is the mean particle volume. The estimation of (N_A)_u, μ_u, and v requires the examination of parallel serial sections above and below T_u. Particle model reconstructions are not needed, however. Previous approaches to the problem are discussed.     

Nanometer profile measurement of large aspheric optical surface by scanning deflectometry with rotatable devices: Uncertainty propagation analysis and experiments

High-accuracy mirrors and lenses with large dimensions are widely used in huge telescopes and other industrial fields. Interferometers are widely used to measure near flat surfaces and spherical optical surfaces because of their high accuracy and high efficiency. Scanning deflectometry is also used for measuring optical near flat surfaces with sub-nanometer uncertainty. However, for measuring an aspheric surface with a large departure from a perfect spherical surface, both of these methods are difficult to use. The key problem for scanning deflectometry is that high-accuracy autocollimators usually have a limited measuring range less than 1000″, so it cannot be used for measuring surfaces having a large slope. We have proposed a new method for measuring large aspheric surfaces with large slopes based on a scanning deflectometry method in which rotatable devices are used to enlarge the measuring range of the autocollimator. We also proposed a method to connect the angle data which is cut by the rotation of the rotatable devices. An analysis of uncertainty propagation in our proposed method was done. The result showed that when measuring a large aspheric surface with a diameter over 300 mm and a slope of 10 arc-deg, the uncertainty was less than 10 nm. For the verification of our proposed method, experimental devices were set up. A spherical optical mirror with a diameter of 35 mm and curvature radius of 5000 mm was measured. The measuring range of the autocollimator was successfully enlarged by our proposed method. Experimental results showed that the average standard deviation of 10 times measurement was about 20 nm.     

A study of the cutting modes in the grooving of tungsten carbide

In this paper, the cutting modes for grooving a tungsten carbide work material are investigated and presented. The grooving tests were carried out on an inclined workpiece surface using a solid CBN tool on a CNC lathe. The experimental results indicated that there was a transition from a ductile mode cutting to a brittle mode cutting in the grooving of tungsten carbide workpiece material as the depth of cut was increased from zero to a critical value. Ductile mode cutting is identified by the machined workpiece surface texture and the material removal ratio f _ab -ratio of the average of the volume of material removed to the volume of the machined groove. Scanning electron microscopy (SEM) observations on the machined workpiece surfaces indicated that there are three cutting modes in the grooving of tungsten carbide as the depth of cut increased: a ductile mode, a semi-brittle mode and a brittle mode. The ductile cutting mode depends on the stress in the cutting region, i.e., whether or not the shear stress in the chip formation region is greater than the critical shear stress for the chip formation ( _slip > _c ), and whether or not the fracture toughness of the work material is larger than the stress intensity factor (K ₁<K _c ). When ( _slip < _c ) and (K₁>K _c ), crack propagation dominates, the chip formation and the cutting mode are brittle.Nomenclature A ₁ , A ₂ A cross-section areas of the ridge - A _V A cross-section area of the groove - A _W The value of A _V subtracted by A ₁+A ₂ - F _X The horizontal force - F _Z The vertical force - K _C The fracture toughness - K _I The stress intensity factor - f _ab The work material removal ratio - f _n The normal cutting force - f _t The tangential cutting force - The inclined angle - _c The critical shear stress for dislocation - _slip The shear stress in chip formation zone