## 1 Introduction  
The core components of rotating machinery, such as the high-pressure compressor of an aero-engine, are typically assembled by several single-stage rotors [1, 2, 3]. The machining errors of a single-stage rotor will propagate and accumulate during the assembly, which makes the actual assembly location of the rotor deviate from its ideal design location and the center of mass of the rotor deviate from the central inertia principal axis of the assembly, resulting in unbalance [4, 5]. The unbalance caused by improper assembly is the main reason for vibration faults of rotating machinery. The traditional way to eliminate unbalance is to measure the unbalance of the rotor with a dynamic balancing machine and then grind the rotor on a pre-set balanced surface [6, 7]. This often requires repeated test runs and repairs to ensure the rotor meets the vibration requirements. This method not only costs time but also changes the initial unbalance of the single-stage rotor with the balanced surfaces. If the multi-stage rotor is reassembled, the unbalance of the single-stage rotor with balanced surfaces should be measured again. Therefore, there is an urgent need for an assembly optimization method that can not only predict the cumulative error of a multi-stage rotor after assembly but also achieve optimal matching of the assembly orientations of rotors at different stages to improve the assembly efficiency and one-time assembly qualified rate of multi-stage rotors. The assembly optimization method of multi-stage rotors has been derived in many studies seeking to minimize the assembly cumulative errors. [PERSON] et al. [8] proposed an assembly error propagation model for 2-D rotational parts. For the first time, they came up with a strategy of straight-build assembly. To be specific, the line of centers of rotors at diverse stages can be as straight as possible by altering the assembly orientations of these rotors. In [9], five assembly optimization methods were evaluated to reduce the error build-up in a straight-build assembly of 2D rectangular components. [PERSON] et al. [10] proposed another strategy of a parallelism-build assembly based on the above 2-D error propagation model. In line with such a strategy, the angular error between the actual and nominal mounting surfaces was minimized for the last-stage rotor. Such a 2-D error propagation model was further modified into a 3-D model by [PERSON] et al. [11]. Then, they assumed that the key dimensions of rotors at various stages obeyed a normal distribution within a specified tolerance zone, and the eccentricity error of assembled rotors were predicted by tolerance analysis. [PERSON] et al. [12] derived a progressive eccentricity error optimization model, and the corresponding physical significance still conformed to the concept of a straight-build assembly. [PERSON] et al. [13] also probed into how the number of the circumferential assembly orientations affects the cumulative error of assembly. By constructing a probability density function of cumulative eccentric error of the last-stage rotor, [PERSON] et al. [14] compared the global optimization results obtained by probability and the Monte Carlo method. [PERSON] et al. [15] proposed a solution to partial parallel chains for error analysis in aero-engine assembly. [PERSON] et al. [16] proposed a neural network model for predicting the concentricity and perpendicularity of a multi-stage rotor. Depending on the location and orientation tolerance of rotor parts, an assembly error propagation model of a multi-stage rotor was constructed by [PERSON] et al. [17] to achieve the optimal assembly orientations of rotors at different stages. [PERSON] et al. [18] also derived the same error propagation model according to the coordinate propagation principle, which can provide guidance for the maintenance of aero-engines. In all the above studies, the optimization objectives and known conditions all focused on the geometric attributes of rotors.