## 1 Introduction
[13] illustrate the relationship between the position and orientation deviations of the part and the surface deviations of the part after assembly is completed. A deviation model for multi-part assembly is developed from the perspective of manufacturing errors, using the amount of deviation at the contact points as the controlling factor. Based on the finite element method, [PERSON] et al. [14] propose an assembly method that considers the contact effects between parts and clamping elements, i.e., the enhanced dimensionality reduction method, for a highly non-linear part assembly system. In [15], the authors use a two-dimensional finite element model under linear elasticity to model the stress intensity factor in the static crack domain, analyze two typical fracture model problems and then use standard and multistage Monte Carlo methods combined with selective refiners to analyze several error tolerations, which significantly reduces the calculation cost. It is worth noting that, with the development of the field of robotics engineering, many of the basic theories applied to robot dynamics systems have also been widely applied to the field of assembly deviation [16]. In [17], by introducing the concept of virtual joints, the Jacobian matrix was introduced into the tolerance transfer model for the first time, enriching the research gap in the field of multi-level assembly tolerance and the deviation control of parts. In [18], a novel method for the deviation analysis of three-dimensional parts is presented, using the Jacobian matrix model to describe the deviation of a partially parallel structure, and the deviation of a partially parallel structure consisting of a cylindrical contact pair and a planar contact pair is modelled through the combination of the parameters of the rotary quantities. In [19], the relative positioning method is combined with a quadratic planning technique to unify the calculation of series and parallel chains between parts in a complex assembly process. The use of discrete geometry based on point cloud technology instead of the traditional ideal surface model allows for the calculation of the cumulative assembly deviations in the Jacobi model in a simple way. Among the many studies on multi-level assembly deviation theory, the stream of variation (SoV) theory is a very promising basic theory for deviation analysis and control. [PERSON] and [PERSON] [20] argued that product deviations in a production system would accumulate gradually and were the first to propose the theory of assembly deviation flow to investigate the mechanism of part assembly deviation propagation in a multi-level production system in terms of both the prediction and diagnosis of deviations. So far, a variety of deviation flow transmission and prediction theories have emerged, including the state space method, which is a classical approach in modern control theory. Using a \"3-2-1\" positioning scheme, [PERSON] and [PERSON] [21] first applied the state space method to the field of deviation transmission modelling, and the deviation transmission model established by the two scholars included key concepts, such as tool positioning error, part accumulation error and repositioning error. They also defined an observation equation to reveal the relationship between the observation vector and the state vector in a multi-level machining and manufacturing system. In [22], a deviation transfer and control model is proposed for multi-level linear dynamic assembly systems and different types of assembly joints by means of a state transformation model approach. In short, the introduction of the state space method presents a new way of solving the deviation transfer problem, allowing researchers to establish simple observation equations to predict and diagnose the transfer and accumulation of deviations in assembling complex assembly systems and to predict and react correctly to the accumulation of deviations in real time for each assembly step of the part.