A comprehensive study of three dimensional tolerance analysis methods

• Introduce four major 3D tolerance analysis models briefly.
• Make a comprehensive comparison and discussion between them.
• Expound the connotation of 3D tolerance analysis.
• Present a perspective overview of the future research about 3D tolerance analysis.

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.