# A Precise Modeling Method for PMSM Servo System of Spacecraft
## 1 Introduction  
The permanent magnet synchronous motor (PMSM) servo system is the control and execution mechanism of spacecraft, and its precise modeling is the foundation of control method research, which is of great practical significance for improving the performance of the spacecraft. The modeling methods for spacecraft servo systems mainly include mechanism modeling and experimental modeling. Early modeling methods decomposed the servo system into multiple independent subsystems and submodules, established mechanism equations of the system using existing theories, and optimized the performance through repeated parameter iterations [1]. This modeling method fails to fully reflect the complex characteristics of cross coupling in servo systems, and the iteration time is relatively long, making the research process time-consuming and laborious. With the development of computer technology, the combination of simulation and experimentation in experimental modeling can achieve precise digital simulations of complex servo systems, effectively improving research efficiency. Due to the extremely fast flight speed, complex flight environment and its own unique structure, the PMSM servo system of spacecraft has the complex control characteristics of fast time-varying, strong nonlinear, coupling and uncertainty. As a result, the traditional rigid body hypothesis is no longer valid, and the influence of nonlinear factors such as elasticity, backlash and friction under comprehensive environment such as temperature change, wind tunnel and vibration should be fully considered [2; 3; 4]. There have been relevant studies on the precise modeling of nonlinear factors in PMSM servo systems. [PERSON]. [5] uses the nonlinear differential equation of the flight control actuators to represent its servo system model, thereby improving the performance of the steering gear stiffness. But the fine structure inside the servo system is not reflected. In [6], a function interpolation method based on Chebyshev approximation theory to approximate the kinematic model is proposed, for optimizing the positioning error of the servo system of a flexible joint robotic arm. Compared with the pseudo stiffness modeling method, this method improves the positioning accuracy by an order of magnitude. The singular perturbation method is used in [7] to decouple the dynamic equations of flexible robotic arms, obtaining two subsystems: rigid robot motion and flexible linkage vibration. On this basis, a disturbance observer is designed to suppress the elastic resonance of the flexible robotic arm. An n-dimensional discrete model of a flexible robotic arm servo system is established in [8]. The article studies the full state feedback neural network controller of the servo system and achieves trajectory tracking and vibration suppression. Ref. [9] presents an eigenvalue method to analyze limit cycles in an electric dual-mass drive system with backlash nonlinearity, demonstrating its efficacy across diverse control structures and nonlinear control plants. Backlash is a common problem in the transmission servo system. To address the issues of vibration suppression and torque limitation in servo systems with backlash, a novel torque compensator is proposed in [10] to approximate the system to an inertial rigid system. This method reduces the mechanical vibration of the system under different inertia and torque conditions, ensuring control accuracy and arbitrary torque limitation. The output tracking problem of gear transmission servo systems with backlash nonlinearity is studied in [11]. In order to overcome the nonlinear and non-differentiable characteristics of the backlash, a detailed static softening process is proposed and a backstepping control algorithm is designed to avoid limit cycles caused by backlash nonlinearity. In [12], the prediction and suppression of limit cycle oscillations in a two-mass system with nonlinear backlash is studied.