Integrating GD&T into dimensional variation models for multistage machining processes

In design of a multistage machining process, process tolerance allocation at each stage and design of process layouts, in particular, the fixture layouts, should be optimized considering dimensional tolerance stackup or variation propagation. When multiple error sources contribute to the tolerance stackup, the dimensionality of the design space could be large and design solution may not be unique. One strategy is to prioritize the allocation of tolerances to different error sources at each stage through proper selection of cost functions. Considering the fact that the cost function selection can be very subjective and the knowledge regarding cost structures is very limited, we propose a hierarchical process design method using error equivalence concept to aggregate multiple error sources together. The main idea is to allocate tolerances to the aggregated error sources at each stage and then further distribute the tolerance to individual error sources through cost analysis. The advantage is two-fold: (1) limiting the impact of cost function selection within individual stages to avoid overhaul of process design caused by subtle change in cost functions, and (2) enabling the optimization of fixture layouts to reduce the overall tolerance stackup due to multiple error sources. To reduce computational load in optimizing process layouts, a computer experiment model is developed to explore a large number of process design alternatives. The robustness of the optimal tolerance is evaluated through sensitivity analysis, which provides guidance for process design. We illustrate the error-equivalence-based process design method by a multistage machining process. The results have shown that the proposed method could significantly reduce the design space and increase the design efficiency.

To describe the configuration of a multi-body system, Cartesian coordinate systems are attached to all bodies comprising the system. Their connections through joints and force elements are efficiently expressed by using 4×4 matrices of the homogeneous transformation, presented by Denavit and Hartenberg in 1955. However, at this time, there is no systematic method to compute velocities and angular velocities using the matrices of such homogeneous transformations. In this paper, homogeneous transformation matrices are identified as a subset of a Lie group, called the special Euclidean group denoted by SE(3). This observation enables the usage of the Lie group theory in multibody kinematics. The effective use of the theory is built upon a platform of a moving frame method as presented in this paper. In this method, for each body-attached Cartesian coordinate system, the coordinate vector basis is written explicitly following Élie Cartan. This moving frame notation enables us to use the Lie algebra of SE(3), denoted by se(3), to compute velocities and angular velocities by minimizing the complexities of the Lie group theory. For kinetics, a variational method is established in se(3) by deriving a relationship between a virtual angular velocities and the corresponding virtual rotational displacements. This constrained variation of virtual angular velocities allows the derivation of the d’Alembert principle of virtual work from Hamilton’s principle for multibody systems. Utilizing this variational tool, we present a systematic computation of equations of motion from Hamilton’s principle. Finally, we reduce the spatial dynamics to planar dynamics and list the simplifications achieved in the two-dimensional problems using SE(2). Then, for a two-degree-of-freedom manipulator the analytical equations of motion are obtained to demonstrate the power of the moving frame method.

Satisfying the quality requirement of products in multistage machining processes (MMPs) is significant and challenging nowadays. In spite of the success of the stream of variation (SoV) theory in variation propagation modelling for MMPs, the absence of elastic deformation variations could be an important factor that limits its application in variable stiffness structure (VSS) workpieces. To this end, a generic variation propagation framework in MMPs for VSS workpieces is established, incorporating the induction and propagation of elastic deformation variations. Region division strategy is adopted according to the characteristics of VSS workpieces, and the analytic solutions of elastic deformation in different regions are solved by contact theory and elastic theory. The effectiveness and accuracy of the proposed model are verified by a six-stage machining process on a four-cylinder engine block, and the proposed model is compared with the conventional SoV model resulting in a significant improvement on quality prediction.

This article aims to compare three different approaches already proposed by the authors regarding the three-dimensional error stack-up analysis in machining. Although the developed error stack-up analysis method comprises of different manufacturing processes, this article is limited to machining operations. This article is built on the previous research of the authors namely the model of manufactured part (MMP) which models the different geometrical deviation 3D impacts on the geometrical and dimensional quality of part produced (error stack-up) in a multi-stage (MS) machining process. In this study, the MMP will be used for developing numerical solution techniques for the aim of tolerance analysis in a MS machining process. The developed solution techniques are classified in two main categories: worst case analysis (WCA) (worst part produced) and stochastic analysis (STA). This article provides an overview of the MMP model and the different solution techniques. The different strategies to simulate the deviation parameters of the model are then discussed. For each of the three proposed solution techniques, its advantages and disadvantages are explored in detail. The performance of the solution techniques is then compared from different points of view (i.e. rapidity, convergence to the global minimum and analysed case … ) through a 3D example, and finally some perspectives are presented.

In the two-frame sensor calibration problem, the objective is to find rigid-body homogeneous transformation matrices $X,Y$ that best fit a set of equalities of the form $A_i X = Y B_i$ , $i=1, \ldots, N$ , where the $\{(A_i, B_i)\}$ are pairs of homogeneous transformations obtained from sensor measurements. The measurements are often subject to varying levels of noise and the resulting optimization can have numerous local minima that exhibit high sensitivity in the choice of optimization parameters. As a first contribution, we present a fast and numerically robust local optimization algorithm for the two-frame sensor calibration objective function. Using coordinate-invariant differential geometric methods that take into account the matrix Lie group structure of the rigid-body transformations, our local descent method makes use of analytic gradients and Hessians, and a strictly descending fast step-size estimate to achieve significant performance improvements. As a second contribution, we present a two-phase stochastic geometric optimization algorithm for finding a stochastic global minimizer based on our earlier local optimizer. Numerical studies demonstrate the considerably enhanced robustness and efficiency of our algorithm over existing unit quaternion-based methods.

In this dissertation, new algorithms for formation control of multi agent systems (MAS) based on continuum mechanics principles will be suggested. For this purpose, agents of the MAS are considered as particles in a continuum, evolving in R^n, whose desired configuration is required to satisfy an admissible deformation function. Considered is a specific class of mappings that are called homogenous where the Jacobian of the mapping is only a function of time and is not spatially varying. The primary objectives of this dissertation are to develop the necessary theory and its validation on a mobile-agent based swarm test bed that includes two primary tasks: 1) homogenous transformation of MAS and 2) deployment of a random distribution of agents on a desired configuration. Developed will be a framework based on homogenous transformations for the evolution of an MAS in an n-dimensional space (n=1,2, and 3), under1) no inter-agent communication (predefined motion plan), 2) local inter-agent communication, and 3) intelligent perception by agents. In this dissertation, different communication protocols for MAS evolution that are based on certain special features of a homogenous transformation will be developed. It is also aimed to deal with the robustness of tracking of a desired motion by an MAS evolving in R^n. Furthermore, the effect of communication delays in an MAS evolving under consensus algorithms or homogenous maps is investigated. In this regard, the maximum allowable communication delay for MAS evolution is formulated on the basis of eigen-analysis.

The machining quality of a workpiece is determined by many machining process elements, such as the machine tool, fixturing, and process planning, etc. It is especially observed that there is a propagation chain of quality attributes in the multistage machining processes (MMPs). Owing to the propagation chain of quality attributes, the machining quality of the workpiece at a single stage is determined not only by the machining elements of the current stage, but also by the quality attributes of its previous stages. In order to describe the propagation chain, a form-feature-based quality control model for MMPs is proposed. Based on this, a linear error propagation model is established by the use of a rigid-body kinematics method, which can explicitly describe the mapping relationship between machining errors of quality attributes and machining process elements. In order to have some comparison with the error propagation model, the above linear regression model has been built to enable correlated quality attributes to be determined with this error propagation model. A workpiece has been used to show that the error propagation model can effectively describe the propagation chain of the quality attributes.

In a multistage machining process, due to the variation accumulation and interaction between different machining stages, variation analysis and diagnosis for quality-related problems become complicated. In this paper, a fault analysis and prediction method is proposed for connecting engineering design with the activities of process variation sources modeling, dimensional variation analysis, and statistical estimate of variations source. Furthermore, a hypothesis testing method is introduced to determine whether each of potential process faults exists in terms of the statistical significance. At last, an experimental study is provided to illustrate the validity and the significance of the proposed methodology.

The complexity of a multi-stage machining process makes it very difficult to assess process plans, from a geometrical point of view. Today, repeated trial runs or very simple simulations are the methods applied to assess a process plan before production release. Due to the cost and time involved in trial runs and the inaccuracy of current simulations, an alternative means should be investigated. This could involve developing accurate 3D simulation methods. This paper proposes a detailed mathematical formulation of tolerance analysis based on searching for the worst case. The performance of two current optimization methods for worst case identification (genetic algorithm and sequential quadratic programming) will be explored through two 3D examples. This paper builds on the model of manufactured part (the MMP) previously proposed by the authors for modeling the different geometrical deviation impacts on the part produced (error stack-up) in a multi-stage machining process. The model cumulates the impacts of various sources of manufacturing errors, hence enabling tolerance analysis. Previously, the same authors also presented a generic formulation for tolerance analysis based on searching for the worst case using the MMP.

Visions is a collection of APL functions designed to produce two-or three-dimensional graphics and animation. Homogeneous coordinates and transformation matrices have been adopted to permit data, transformations and viewing parameters to be of arbitrary rank and dimension. Drawing is accomplished through the implementation of a shape-independent dyadic draw function which accepts data and transformations as arguments. Clipping, a window-to-viewport transformation, and perspective are incorporated and may be modified under user control. Shape-independent transformation functions are also defined.

There are some complicated interactions among different stages in multistage machining processes (MMPs), so it is important and necessary to explore the inherent mechanism of machining error propagation for improving product quality. With respect to this issue, a machining error propagation network (MEPN) of MMPs is proposed and its complexity is discussed. In order to establish this MEPN, the machining form feature (MFF) of the workpiece, machine tool, fixture, and cutting tool (i.e. machining element, ME) are defined as different network nodes. The associated relationship among different nodes is mapped by graph theory, according to the computer aided process planning (CAPP) information of the workpiece. Based on this, complex networks theory is introduced to analyse the characteristics of the established MEPN, and several measuring indices of error propagation at different nodes are proposed to quantify the interactions among different nodes. The synthesized propagation index (SPI) is then established to identify the key nodes and the SPI variation curve is used to quantify these key nodes' influences on the MEPN. Finally, a box part of a missile launcher is used to illustrate the proposed methods.

The simultaneous measurement and identification of position-independent geometric errors (PIGEs) and position-dependent geometric errors (PDGEs) of dual rotary axes in five-axis machine tools remain a significant challenge in precision machining. To address this issue, this study develops a geometric error measurement system for the dual rotary axes of a five-axis machine tool by integrating a scanning probe, a block, and a calibration sphere. Unlike conventional touch-trigger probes, the scanning probe acquires continuous measurement signals along the surfaces of the block and the calibration sphere, enabling the extraction of richer geometric features and providing sufficient data for the simultaneous analysis of multiple geometric error components. The proposed measurement system is implemented on a five-axis machine tool, Uni5X-400, manufactured by FALCON Machine Tools Co., Ltd., to measure the geometric information of the calibration sphere and the block and thereby identify 20 PIGEs and PDGEs associated with the dual rotary axes. A kinematic model of the measurement system is established using homogeneous coordinate transformation matrices. By applying forward and inverse kinematics, the spatial positions of the calibration sphere and block are derived, forming an explicit relationship between volumetric errors and geometric errors. The resulting error identification equations are solved using the least squares method. Experimental results demonstrate that the proposed approach can efficiently and accurately identify the geometric errors of the dual rotary axes and maintain robust reliability under real-world operating conditions, making it suitable for practical calibration and accuracy improvement of five-axis machine tools.

Chapter 10 - Open kinematic chains

Auto-body closure panel-fitting positions at the final stage of body-in-white (BIW)’s hierarchical assembly. It is such a critical process that contributes greatly to automotive dimensional integrity and quality. Poor fitting results in unsatisfied gap width and flushness that would cause functional problems and make adverse effect on vehicle quality perceived by customers. Although this problem has been studied by several authors in the past, two aspects still require further research. One is the improvement of fit quality measure suitable for complicated contours of closure panel and body opening’s profile. The other is the effect of weld-induced distortion during fitting process. In this paper, a novel computer-aided approach is proposed to address these above two problems. Firstly, panel fitting is mathematically modeled as a relative position optimization between panel and body opening which adopts translation and transposition parameters in homogeneous transformation as independent variables. Hausdorff distance between two continuous closed panel curves is firstly employed as the measure to evaluate the fitting quality of panel-opening profile contour. Secondly, since hinge axis plays important role for panel-opening relative position, its deviation from nominal design intent could not be neglected. After obtaining weld-induced hinge distortion using numerical analysis for these subassembly process, deviation of closure panel profile contour from design stage is calculated by homogeneous transformation during optimization and factors resulting from this physical thermal effects on gap width and flushness are taken into account and compensated. For the reason that the panel-fitting optimal function contains more than three variables, it is not suitable for applying algorithm with gradient descent or direct search. Genetic algorithm is used to search optimal fitting variables in a global and parallel way. Case study of certain car door fitting shows that the proposed method can effectively improve panel-fitting quality and accuracy.

A new method compensating geometric error components of a four-axis horizontal machine tool is presented in this paper. Homogeneous transformation matrices (HTMS) and error conversion are integrated in the compensation strategy. A mathematical model which contains 29 geometric error components is established based on HTMS and the errors in X, Y and Z directions are obtained through calculation. The errors in three directions are compensated by shifting the corresponding axis. But the configuration of this machine tool is X-axis, Z-axis, B-axis, C-axis without Y-axis, so the errors in X, Y and Z directions need to be converted into X, Z and C directions which is different from the traditional machine tools. The errors after conversion are used to compensation directly. This approach is significant for the error modeling and compensation which is an easy and efficient way to improve the precision of the four-axis machine tools.