Yunxian Chu

On the symmetric location problem

Accurate and efficient localization of symmetric features plays an important role in dimensional inspection of machined parts and machining of partially finished workpieces. We present a geometric theory for efficient and accurate localization of symmetric features. First, we show that the configuration space of a symmetric feature can be naturally identified with the homogeneous space SE/sub (3/)/G/sub o/ of the Euclidean group SE/sub (3/), where G/sub o/ is the symmetry group of the feature. Then, we explore the geometric structure of the homogeneous space and present a simple and unifying algorithm for symmetric localization. Finally, we give simulation results illustrating several unique features of the algorithm: 1) implementational simplicity; 2) robustness with respect to initial conditions; 3) high accuracy in computed results; and 4) computational efficiency.

A geometric theory of form, profile, and orientation tolerances

Abstract This article develops a geometric theory which unifies the formulation and computation of form (straightness, flatness, cylindricity, and circularity), profile and orientation tolerances stipulated in ANSI Y14.5M standard. The theory rests on an important observation that a toleranced feature exhibits a symmetry subgroup G 0 under the action of the Euclidean group, SE (3). Thus, the configuration space of a toleranced (or a symmetric) feature can be identified with the homogeneous space SE (3)/ G 0 of the Euclidean group. Geometric properties of SE (3)/ G 0 , especially its exponential coordinates carried over from that of SE (3), are analyzed. We show that all cases of form, profile and orientation tolerances can be formulated as a minimization or constrained minimization problem on the space SE (3)/ G 0 , with G 0 being the symmetry subgroup of the underlying feature. We develop a simple geometric algorithm, called the Symmetric Minimum Zone (SMZ) algorithm, to unify the computation of form, profile, and orientation tolerances. Finally, we use numerical simulation results comparing the performances of the SMZ algorithm against the best known algorithms in the literature.

Workpiece localization and computer aided setup system

Workpiece localization and similar problems have attracted many researcher in robotics, computer vision and image processing research fields. Workpiece localization algorithms have close relations with workpiece setup in manufacturing process. However when these algorithms apply to a computer aided setup (CAS) system, many problems, like online measurement planning and error compensation etc., should be solved first. In this paper, a CAD model based CAS system will be introduced. This system is based on an open architecture CNC machine. Some problems in implementation are also addressed, such as algorithm divergence. The system has used a new method to avoid the possible divergence of localization algorithms. New error compensation method has been applied in the localization algorithms, instead of directly compensating the measurement data. Experimental results will be given to show the effect of these new methods and the efficiency of the CAS system.

A geometric method for computation of datum reference frames

A datum reference frame (DRF) is a coordinate system used to locate and orient part features. Constructing a DRF from a set of datum features is a complicated process involving: a) specifying a valid combination and the precedence of the datum features which define the DRF; b) developing datum from datum features of the part; and c) determining the position and orientation of the DRF from the datums. We develop a geometric theory for establishing DRFs. The theory is based on the observation that a datum feature such as a plane, a cylinder or a sphere has a symmetry subgroup G/sub 0/ under the action by the group SE(3) of rigid motions in R/sup 3/. Thus, the configuration space of a datum feature can be identified with the homogeneous space SE(3)/G/sub 0/, and the problem of datum development can be posed as a minimization problem in SE(J)/G/sub 0/. We give conditions under which a datum feature qualifies to be a secondary or a tertiary datum. We present a sequential procedure that transforms the primary, secondary and tertiary datum problems as a minimization or a constrained minimization problem in the homogeneous spaces of SE(3). We develop simple algorithms to solve these problems, and give simulation results illustrating efficiency and simplicity of the approach.

On the Hybrid Localization/Envelopment Problem

The problem of aligning the CAD model of a workpiece such that all points measured on the finished surfaces of the workpiece match closely to corresponding surfaces on the model while all unmachined surfaces lie outside the model is referred to as the hybrid localization/envelopment problem. The hybrid problem has important applications in setting up for machining of partially finished workpieces. This paper gives a formulation of the hybrid localization/envelopment problem, and presents a simple algorithm for computing its solutions. First, we show that when the finished surfaces of a workpiece are inadequate to fully constrain the rigid motions of the workpiece, then the set of free motions remaining must form a subgroup G0 of the Euclidean group SE(3). This allows us to decompose the hybrid problem into a (symmetric) localization problem on the homogeneous space SE(3)/G0 and an envelopment problem on G0. While the symmetric localization problem is solved using the fast symmetric localization (FSL) algorithm developed in one of our earlier papers, the envelopment problem is solved by computing the solutions of a sequence of linear programming (LP) problems. We derive explicitly the LP problems, and apply standard linear programming techniques to solve the LP problems. We present simulation results to demonstrate the effectiveness of our method for the hybrid problem.

A geometric approach of form tolerance formulation and evaluation

We first present a unified geometric formulation of form tolerances using an extended least-squares (E-LSQ) approach. Then, using properties of configuration space of symmetric features an iterative algorithm is developed to solve the E-LSQ problem. Simulation results show that the algorithm not only keeps computational efficiency of the least-squares approach, but also possesses of computational accuracy of min-max algorithms. The outstanding properties of the geometric formulation include: 1) coordinate-free representation; 2) in conformance to the tolerance standard; and 3) easy implementation.

A CAD-based probing and localisation method for arbitrarily fixed workpiece

In this paper an efficient method for automatic probing and localisation of 3D workpiece arbitrarily fixed to the machine table of a five-axis milling machine is presented. First, by formulating localisation problem as minimizing a least-square question an explicit solution is obtained. Based upon this solution and sensitivity analysis, we present some guidelines and develop a CAD-based automatic probing strategy. By means of the knowledge about the approximate location of workpiece computed by 3D localisation algorithm as estimator, together with the CAD model of workpiece, we then generate online a new collision free probing trajectory to probe additional point on a neighbourhood of the previous sampling point. Utilizing the newly measurement point, we improve the Euclidean transformation which in turn is used together With the CAD model, to generate another collision free probing path to probe more points on the surfaces of workpiece. This process continuous until the workpiece is accurately located. Experimental results show that this algorithm overcomes the need of home surface identification process and is thus suitable for real-time implementation in manufacturing or inspection process.

A geometric approach to establishment of datum reference frames

A datum reference frame is a coordinate system used to locate and orient port features. Constructing a datum reference frame from a set of features is a complicated process involving: 1) specifying a valid combination and the precedence of the datum features which define the datum reference frame; 2) developing from datum features of the part; and 3) determining the position and orientation of the datum reference frame from the data. In this paper, we develop a geometric theory for establishing datum reference frames, and present a sequential procedure that transforms the primary, secondary and tertiary datum problems as a minimization or a constrained minimization problem. We develop simple algorithms to solve these problems, and give simulation results illustrating efficiency and simplicity of the approach.

Geometric formulation of orientation tolerances

The rapid proliferation of coordinate measuring machines (CMM) triggered the need for precise and rigorous formulations of each tolerance concept. In this paper, we employ the concept of configuration space of symmetric features to define each type of datum features and orientation tolerances. The establishment of datum features and orientation tolerances are formulated in an linear programming approach. Using properties of Lie groups and Lie algebra, a simple, unified and coordinate free algorithm for datum establishment and orientation tolerance is developed. The results of the algorithm comply to the definitions stipulated in the standard ANSI Y14.5M.

A geometric algorithm for hybrid localization/inspection/machinability problem

We first propose a hybrid localization/inspection/machinability problem. Next, we formulate the hybrid problem using differential geometric theory and the minimax method. Then, we develop a methodology for treating localization, online inspection and machinability of workpieces simultaneously. Using the geometric properties of the hybrid problem, the hybrid problem is decoupled into a (symmetric) localization/inspection problem and a machinability problem. Then both problems are formulated as constrained optimization problems and are solved by a sequence of linear programming problems. Finally, we present simulation results to demonstrate the efficiency of our method for the hybrid problem.