Xinmin Lai

A comprehensive study of three dimensional tolerance analysis methods

Abstract Three dimensional (3D) tolerance analysis is an innovative method which represents and transfers tolerance in 3D space. The advantage of 3D method is taking both dimensional and geometric tolerances into consideration, compared with traditional 1/2D tolerance methods considering dimensional tolerances only. This paper reviews four major methods of 3D tolerance analysis and compares them based on the literature published over the last three decades or so. The methods studied are Tolerance-Map (T-Map), matrix model, unified Jacobian–Torsor model and direct linearization method (DLM). Each of them has its advantages and disadvantages. The T-Map method can model all of tolerances and their interaction while the mathematic theory and operation may be challenging for users. The matrix model based on the homogeneous matrix which is classical and concise has been the foundation of some successful computer aided tolerancing software (CATs), but the solution of constraint relations composed of inequalities is complicated. The unified Jacobian–Torsor model combines the advantages of the torsor model which is suitable for tolerance representation and the Jacobian matrix which is suitable for tolerance propagation. It is computationally efficient, but the constraint relations between components of torsor need to be considered to improve its accuracy and validity. The DLM is based on the first order Taylor’s series expansion of vector-loop-based assembly models which use vectors to represent either component dimensions or assembly dimensions. Geometric tolerances are operated as dimensional tolerances in DLM, which is not fully consistent with tolerancing standards. The results of four models with respect to an example are also listed to make a comparison. Finally, a perspective overview of the future research about 3D tolerance analysis is presented.

Assembly sequence planning of automobile body components based on liaison graph

Purpose – This paper presents an algorithm for the generation of mechanical assembly sequences.Design/methodology/approach – The algorithm employs an adjacency matrix, and uses three different mathematical patterns of subassemblies to generate automatically all geometrical feasible assembly sequences.Findings – This algorithm cannot only generate automatically all geometrical feasible assembly sequences but also reduce the number of sequences.Originality/value – Assembly modeling is more completed than that of previous research; the method is able to automatically generate all possible assembly sequences and be implemented easily with program; and this paper makes use of exact mathematical equations describing every subgroup of the three‐subassembly patterns.

A framework for an automotive body assembly process design system

This paper describes a framework for an automotive body assembly process design system. It is a computer-aided intelligent system that can automatically generate the optimal joint types and assembly sequences for the best dimensional quality. The backbone of this system is the Case-Based Reasoning (CBR) methodology, which works by searching through a case library created from previous designs whose identifying features resemble the current case. Algorithms for initial solution generation, dimension chain generation, joint design selections, assembly sequence generation, and tolerance analysis and optimization are developed. Based on the framework, a software tool called Body Build Advisor, or BBA, is developed. The software allows process designers to analyze candidate assembly schemes and achieve the best assembly process design prior to having detailed knowledge of geometry of the parts, and thus is ideal for architectural process design. In addition, the system has the advantage of an open structure that can be easily modified and adapted to accommodate existing assemblies and to suggest areas for improvement.

A small displacement torsor model for 3D tolerance analysis of conical structures

The small displacement torsor model is a classic three-dimensional tolerance analysis method. It uses three translational vectors and three rotational vectors to represent tolerance information in three-dimensional Euclidean space. However, the target features of this model mainly focused on planes and cylinders in previous studies. Little attention is invested to conical features and their joints which are used widely and more complex than the planar and cylindrical features. The objective of this article is to present a three-dimensional mathematical method of tolerance representation about conical surfaces and their joints based on the small displacement torsor model, and propose a mathematical model of variations and constraint relations of components of the small displacement torsor for conical surfaces caused by geometric tolerances limited by its tolerance zone. In addition, a simple example involving conical structures is used to demonstrate three-dimensional conical tolerance propagation. Both deterministic and statistical results are obtained by this model.

Design of an electromagnetic prismatic joint with variable stiffness

This paper aims to propose an electromagnetic prismatic joint with variable stiffness. The joint can absorb the sudden shocks and improve the natural dynamics of robotics. The ability of regulating the output stiffness can also be used for force control in industrial applications.,Unlike some existing designs of variable stiffness joints (VSJs) in which the stiffness regulation is implemented using the stiffness adjustment motor and mechanisms, the main structure of the electromagnetic VSJ is a permanent magnet (PM) arranged inside coaxial cylinder coils. The adjustment of input current can cause the change of magnetic force between the PM and the cylinder coils, and thus leads to the variation of output stiffness.,According to the theoretical model, the output stiffness of the electromagnetic VSJ is linearly proportional to the input current. The experiments further indicate that the current-controlled stiffness can make the stiffness variation response of this VSJ more rapid for practical applications. Due to the large damping introduced by the copper-based self-lubrication bearings, the VSJ shows good properties in motion positioning and trajectory tracking.,In summary, the electromagnetic VSJ is compact in size and light in weight. It is possible to realize the online adaptability to work conditions with dynamic load by using this VSJ.

Compliance Analysis and Synthesis of a Serial Manipulator with Redundant DOFs Used in Tokamak

Serial manipulator with redundant DOF joints performs the maintenance and inspection tasks in Tokamak, which needs good stiffness behavior in vertical direction and appropriate compliance in horizontal planar. According to this demand, this paper analyses the Cartesian compliance of carrying manipulator (6R) and the whole serial manipulator (10R), respectively. Then an equivalent full DOF serial mechanism which performs the same compliance is proposed. Based on the equivalent robotic arm, the desired compliance behavior is realized by adding several passive springs without changing the parameters of the original ten joints.

Tolerance analysis of mechanical assemblies based on the product of exponentials formula

Many existing tolerancing models deal with the tolerance analysis of mechanical assemblies using linear stack-up function, which will cause truncation errors for nonlinear assembly problem. This article proposes a new method for describing the variation propagation based on the product of exponentials formula. Both the length deviations and the angle deviations are uniformly represented by twist coordinates. Accordingly, the nonlinear stack-up functions relating the assembly functional requirements to the manufacturing variations can be derived, and no truncation errors generated. A three-dimensional assembly case is studied exploiting the proposed method and the direct linearization method, respectively. It appears that the product of exponentials formula can bring consistent tolerance analysis results with the exact geometric solutions, whereas there are some small errors in the analysis results of direct linearization method. The precision of tolerance estimation via the product of exponentials formula is improved by 0.52% at the most compared with that by the direct linearization method. Due to the elimination of local datum reference frames, the product of exponentials formula can also lead to a more concise description of tolerance accumulation than the direct linearization method.