## 1 Introduction  
As the functions of spacecrafts have become increasingly numerous and powerful, their energy demand has increased sharply. Sun-tracking solar arrays are adopted by more and more spacecrafts as energy sources to meet the high power demand. A sun-tracking solar array is mainly composed of a solar array driving assembly (SADA) and a solar array. The SADA acts as an actuator to drive the rotation of the solar array and to keep it oriented toward the sun in order to efficiently generate electricity to meet the energy requirements of the spacecraft. Although a sun-tracking solar array is an effective solution to the increasing energy demand of spacecrafts, the fluctuations of speed and the driving torque of the SADA will cause the oscillatory motion of the flexible solar array. The fluctuant driving torque of the SADA and the oscillatory motion of the solar array will act on the main body of the spacecraft and cause micro-vibrations. Spacecraft micro-vibration is low-amplitude and wide-band, and it will significantly reduce the spacecraft pointing accuracy [1]. There are many potential causes of disturbance on the spacecraft, among which the flywheel [2; 3] and the control moment gyros [4; 5] with middle and high frequency disturbance frequencies are considered to be the largest sources. Scholars have conducted a lot of relevant research and had many achievements with regard to the flywheel and control moment gyros [6; 7; 8]. On the other hand, the sun-tracking solar array is also a source of disturbance, and the disturbance it causes is mainly concentrated on the low-frequency band, which has the characteristics of a wide-frequency band and lasts for a long period of time [9]. This affects the performance of the spacecraft payloads and even the attitude stability of the satellite in some cases [10; 11]. The disturbance torque model of the sun-tracking solar array is the key to predicting the influence of the disturbance on the spacecraft and to developing the disturbance suppression methods. The SADA is the disturbance source of the sun-tracking solar array, which is generally composed of a stepping motor and a harmonic reducer [12]. Stepping motors are the core component of the SADA. Many scholars have carried out theoretical and simulation research on the vibration of stepping motors. [PERSON] proposed the electromagnetic vibration model of the stepping motor [13]. [PERSON] revealed the electromagnetic vibration mechanism of the stepping motor and proposed a method of reducing the amplitude of electromagnetic vibration [14]. [PERSON] defines the expression of the electromagnetic vibration frequency of the stepping motor [15]. [PERSON] inferred that the electromagnetic field between the stator and rotor of the stepping motor could be equivalent to a viscous damping system of an electromagnetic spring and named the stiffness of the electromagnetic spring as electromagnetic stiffness [16]. Although many achievements have been made in the study of the vibration of the stepping motor, there is little research on the vibration characteristics of the SADA. [PERSON] proposed the dynamic model of SADA based on the dynamic model of the stepping motor established by [PERSON], and adopted the adaptive current compensation method to improve the performance [17]. [PERSON] proposed the disturbance torque model of a SADA-driven rigid load for the first time and measured the disturbance torque emitted by the SADA driving four rigid loads to verify the model [9]. However, the model ignored the friction torque of the harmonic reducer and the conductive slip ring and the rigid-flexible coupling between the SADA and the solar array. [PERSON] pointed out that the effects of the rigid-flexible coupling could lead to the instability of the controlled system [18]. Therefore, it is necessary to propose a comprehensive disturbance torque model for SADA in driving the flexible solar array.