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The purpose of this paper is to propose a novel mathematical model to present the three-dimensional tolerance of a discrete surface and to carry out an approach to analyze the tolerance of an assembly with a discrete surface structure. A discrete surface is a special structure of a large surface base with several discrete elements mounted on it, one, which is widely used in complex electromechanical products.

The geometric features of discrete surfaces are separated and characterized by small displacement torsors according to the spatial relationship of discrete elements. The torsor cluster model is established to characterize the integral feature variation of a discrete surface by integrating the torsor model. The influence and accumulation of the assembly tolerance of a discrete surface are determined by statistical tolerance analysis based on the unified Jacobian-Torsor method.

The effectiveness and superiority of the proposed model in comprehensive tolerance characterization of discrete surfaces are successfully demonstrated by a case study of a phased array antenna. The tolerance is evidently and intuitively computed and expressed based on the torsor cluster model.

The tolerance analysis method proposed requires much time and high computing performance for the calculation of the statistical simulation.

The torsor cluster model achieves the three-dimensional tolerance representation of the discrete surface. The tolerance analysis method based on this model predicts the accumulation of the tolerance of components before their physical assembly.

This paper proposes the torsor cluster as a novel mathematical model to interpret the tolerance of a discrete surface.

The purpose of this paper is to present an experimental approach to measure and quantify the three‐dimensional geometrical manufacturing errors on a mass production of parts.

A methodology is developed to model and analyse the combined effect of these errors on a machined feature. Deviation of a machined feature due to the combined errors is expressed in terms of the small displacement torsor (SDT) parameters. Given a tolerance on the machined feature, constraints are specified for that feature to establish a relationship between the tolerance zone of the feature and the torsor parameters. These constraints provide boundaries within which the machined feature must lie. This is used for tolerance analysis of the machined feature. An experimental approach is proposed to measure and quantify the three‐dimensional manufacturing variations as torsors. The results are used to verify the analytical model.

Results show that it is possible to quantify manufacturing dispersions. The paper proposes a measuring method which can be done during the production. In the context of process planning, these experimental data allow us to perform realistic geometrical simulation of manufacturing. The results of this method are torsor components dispersions. Analysis and synthesis of the geometrical simulation of manufacturing are viable with reliable numerical data in order to predict the defects.

To perform realistic geometrical simulation of manufacturing, an experimental approach to measure and quantify the three‐dimensional geometrical manufacturing errors is proposed which is based on the SDT concept.

The purpose of this paper is to carry out an assembly tolerance analysis by means of a combined Jacobian model and skin model shape. The former is based on small displacements modeling of points using 6 × 6 transformation matrices of open kinematic chains in robotics. The latter easily models toleranced features with all kinds of geometric deviations.

This paper presents the procedure of performing tolerance analysis by means of the Jacobian model and skin model shape for assemblies. The point cloud-based discrete representative is able to model the actual toleranced surfaces instead of the ideal or associated ones in an assembly, which brings the simulation tools closer to reality.

The proposed method has the advantage of skin model shape which is suitable for geometric tolerances management along the product life cycle and contact analysis of kinematic small variations, as well as, with the Jacobian, enabling transformation of locally expressed parts deviations to globally expressed functional requirements. The result of the case study shows the accuracy of the method.

The proposed approach has not been developed fully; other functional features such as the pyramid are still ongoing challenges.

It is an effective method for supporting design, manufacturing and inspection by providing a quantitative analysis of the effects of multi-tolerances on the final functional key characteristics and for predicting the quality level.

The paper is original in taking advantages of both Jacobian model and skin model shape to consider all geometric tolerances in assembly.

This paper aims to investigate how form error of journal affects oil film characteristics, which are composed of several parameters including the maximum film pressure, film moment, frictional coefficient and carrying-load capacity.

A new generalized equation based on the small displacement torsor theory is derived, as well as its capability of representing types of form error on the journal, using four specified parameters in a three-dimensional (3D) state. Based on the new generalized equation of form errors, the Reynolds equation is represented and solved numerically using the Swift–Stieber boundary condition.

The results show that the form errors of journal have significant influence on all oil film characteristics. However, the film moment remains nearly unchanged as film characteristics, especially eccentricity ratio, become large. All film characteristics investigated vary periodically as the form error. More importantly, it is found that the film pressure distribution transforms to an asymmetric shape along the axial direction of the bearing, no longer a symmetric shape in the case of two-dimensional (2D) form errors. It is necessary to substitute the 3D form error model, which takes the variations of the film characteristics in axial direction into account, for the 2D model in the designing stage of journal bearings.

First, the effect of the form error of the journal on the performance of hydrodynamic journal bearings is studied in the view of the film characteristics systematically. Secondly, the new generalized equation of form error, derived by SDT theory, is capable of representing any types of form error on the journal, not only representing one type of form error merely.

The conventional statistical method of three-dimensional tolerance analysis requires numerous pseudo-random numbers and consumes enormous computations to increase the calculation accuracy, such as the Monte Carlo simulation. The purpose of this paper is to propose a novel method to overcome the problems.

With the combination of the quasi-Monte Carlo method and the unified Jacobian-torsor model, this paper proposes a three-dimensional tolerance analysis method based on edge sampling. By setting reasonable evaluation criteria, the sequence numbers representing relatively smaller deviations are excluded and the remaining numbers are selected and kept which represent deviations approximate to and still comply with the tolerance requirements.

The case study illustrates the effectiveness and superiority of the proposed method in that it can reduce the sample size, diminish the computations, predict wider tolerance ranges and improve the accuracy of three-dimensional tolerance of precision assembly simultaneously.

The proposed method may be applied only when the dimensional and geometric tolerances are interpreted in the three-dimensional tolerance representation model.

The proposed tolerance analysis method can evaluate the impact of manufacturing errors on the product structure quantitatively and provide a theoretical basis for structural design, process planning and manufacture inspection.

The paper is original in proposing edge sampling as a sampling strategy to generating deviation numbers in tolerance analysis.

The purpose of this study is to establish an assembly accuracy analysis model of deployable arms based on Jacobian–Torsor theory to improve the assembly accuracy. Spacecraft deployable arm is one of the core components of spacecraft. Reducing the errors in assembly process is the main method to improve the assembly accuracy of spacecraft deployable arms.

First, the influence of composite connecting rod, root joint and arm joint on assembly accuracy in the tandem assembly process is analyzed to propose the assembly accuracy analysis model. Second, a non-tandem assembly process of “two joints fixed-composite rod installed-flange gasket compensated” is proposed and analyzed to improve the assembly accuracy of deployable arms. Finally, the feasibility of non-tandem assembly process strategy is verified by assembly experiment.

The experiential results show that the assembly errors are reduced compared with the tandem assembly process. The errors on axes x, y and z directions decreased from 14.1009 mm, 14.2424 mm and 0.8414 mm to 0.922 mm, 0.671 mm and 0.2393 mm, respectively. The errors round axes x and y directions also decreased from 0.0050° and 0.0053° to 0.00292° and 0.00251°, respectively.

This paper presents an assembly accuracy analysis model of deployable arms and applies the model to calculate assembly errors in tandem assembly process. In addition, a non-tandem assembly process is proposed based on the model. The experimental results show that the non-tandem assembly process can improve the assembly accuracy of deployable arms.

This paper aims to clarify the predicting and compensating method of aeroplane assembly. It proposes modeling the process of assembly. The paper aims to solve the precision assembly of aeroplane, which includes predicting the assembly variation and compensating the assembly errors.

The paper opted for an exploratory study using the state space theory and small displacement torsor theory. The assembly variation propagation model is established. The experiment data are obtained by a real small aeroplane assembly process.

The paper provides the predicting and compensating method for aeroplane assembly accuracy.

This paper fulfils an identified need to study how the assembly variation propagates in the assembly process.

This paper aims to comprehensively achieve the requirements of high assembly precision and low cost, a precision-cost model of assembly based on three-dimensional (3D) tolerance is established in this paper.

The assembly precision is related to the tolerance of parts and the deformation of matching surfaces under load. In this paper, the small displacement torsor (SDT) theory is first utilized to analyze the manufacturing tolerances of parts and the assembly deformation deviation of matching surface. In the meanwhile, the extracting method of SDT parameters is proposed and the assembly precision calculation model based on the 3D tolerance is established. Second, an integrated optimization model based on the machining cost, assembly cost (mapping the deviation domain to the SDT domain) and quality loss cost is built. Finally, the practicability of the precision-cost model is verified by optimizing the horizontal machining center.

The assembly deviation has a great influence on cost fluctuation. By setting the optimization objective to maximize the assembly precision, the optimal total cost is CNY 72.77, decreasing by 16.83 per cent from the initial value, which meets economical requirements. Meanwhile, the upper bound of each processing tolerance is close to the maximum value of 0.01 mm, indicating that the load deformation can be offset by appropriately increasing the upper bound of the tolerance, but it is necessary to strictly restrict the manufacturing tolerances of lower parts in a reasonable range.

In this paper, a 3D deviation precision-cost model of assembly is established, which can describe the assembly precision more accurately and achieve a lower cost compared with the assembly precision model based on rigid parts.

This paper aims to develop a mathematical method to analyze the assembly variation of the non-rigid assembly, considering the manufacturing variations and the deformation variations of the non-rigid parts during the assembly process.

First, this paper proposes a deformation gradient model, which represents the deformation variations during the assembly process by considering the forces and the self-weight of the non-rigid parts. Second, the developed deformation gradient models from the assembly process are integrated into the homogenous transformation matrix to model the deformation variations and manufacturing variations of the deformed non-rigid part. Finally, a mathematical model to analyze the assembly variation propagation is developed to predict the dimensional and geometrical variations due to the manufacturing variations and the deformation variations during the assembly process.

Through the case study with a crosshead non-rigid assembly, the results indicate that during the assembly process, the individual deformation values of the non-rigid parts are small. However, the cumulative deformation variations of all the non-rigid parts and the manufacturing variations present a target value (w) of −0.2837 mm as compared to a target value of −0.3995 mm when the assembly is assumed to be rigid. The difference in the target values indicates that the influence of the non-rigid part deformation variations during the assembly process on the mechanical assembly accuracy cannot be ignored.

In this paper, a deformation gradient model is proposed to obtain the deformation variations of non-rigid parts during the assembly process. The small deformation variation, which is often modeled using a finite-element method in the existing works, is modeled using the proposed deformation gradient model and integrated into the nominal dimensions. Using the deformation gradient models, the non-rigid part deformation variations can be computed and the accumulated deformation variation can be easily obtained. The assembly variation propagation model is developed to predict the accuracy of the non-rigid assembly by integrating the deformation gradient models into the homogeneous transformation matrix.

The traditional precision design only takes the influence of geometric tolerance of the parts and does not involve the load deformation in the assembly process. This paper aims to analyze the influence mechanism of flexible parts deformation on the geometric precision, and then to ensure the reliability and stability of the mechanical system.

Firstly, this paper adopts the N-GPS to analyze the influence mechanism of flexible parts deformation on the geometric precision and constructs a coupling 3D tolerance mathematical model of the geometric tolerance and the load deformation deviation based on the SDT theory, homogeneous coordinate transformation theory and surface authentication idea. Secondly, the least square method is used to fit the deformation surface of the mating surface under load so as to complete the conversion from the non-ideal element to the ideal element.

This paper takes the horizontal machining center as a case to obtain the deformation information of the mating surface under the self-weight load. The results show that the deformation deviation of the parts has the trend of transmission and accumulation under the load. The terminal deformation cumulative amount of the system is up to –0.0249 mm, which indicated that the influence of parts deformation on the mechanical system precision cannot be ignored.

This paper establishes a comprehensive 3D tolerance mathematical model, which comprehensively considers the effect of the dimensional tolerance, geometric tolerance and load deformation deviation. By this way, the assembly precision of mechanical system can be accurately predicted.