1. Introduction
The single gimbal control moment gyro (SGCMG) is the primary attitude control actuator of large spacecraft because of good dynamic performance, high control accuracy and large output torque [
1,
2]. As shown in
Figure 1, SGCMG is mainly composed of the high-speed rotor system and a gimbal servo system. The output torque of the SGCMG is
, where
H is the angular momentum of the high-speed rotor, and
is the angular speed of the gimbal servo system [
3,
4]. When the high-speed rotor runs stably, the value of
H remains constant, and the output torque accuracy of the SGCMG is dependent on the precision of the
[
5,
6].
To meet the weight and volume requirements of the gimbal servo system, a harmonic drive is adopted to amplify the output torque, which has the advantages of light weight, small size, high efficiency, large transmission ratio and simple structure [
7,
8]. As shown in
Figure 1, the harmonic drive is a special flexible transmission mechanism, which is composed of a wave generator, a flexspline and a circular spline [
9]. The motion conversion of the harmonic drive is achieved by the elastic deformation of the flex spline [
10]. However, this special transmission structure leads to nonlinear characteristics, such as nonlinear friction, kinematic error and hysteresis [
11]. Among these characteristics, the kinematic error is the major factor that causes the unexpected speed fluctuation of the gimbal servo system [
12,
13]. However, for the complex disturbances, the traditional control method is difficult to achieve satisfactory control results [
14,
15,
16]. To achieve high-precision output speed of the gimbal servo system, it is crucial to suppress the influence of kinematic error on the speed.
In recent decades, many researchers have studied the kinematic error of the harmonic drive. Kennedy et al. proposed that the kinematic error is mainly composed of the pure kinematic error caused by installing and manufacturing assembly and the torsion angle induced by the deformation of the flexspline [
17]. Jia et al. illustrated that the tooth error is the major factor causing the pure kinematic error through the theoretical analysis [
18]. To optimize the design of the conjugate contour teeth, the pure kinematic is analyzed in [
19], and the effects of the load torque on the kinematic characteristics of the meshed teeth pairs is analyzed in [
20]. Moreover, the function of the kinematic error with respect to angular position of the motor rotor was presented in [
18]. Zhu et al. adopted the adaptive joint torque control method to improve the tracking accuracy of the angular position [
21]. However, the algorithm requires the installation of torque sensors, and it is difficult to install additional torque sensors in the universal joint servo system due to the stringent design requirements. In [
22], the kinematic error is converted into a state of the system by means of the Lagrange equation, and then it is observed by ESO. Liu et al. applied the double speed loops control method to suppress the speed fluctuation of the low speed servo system caused by the harmonic drive [
23]. In [
24], Tonshoff et al. proposed a nonlinear proportional-derivative (PD) control algorithm for closed-loop compensation of kinematic error, and the stability of the designed controller was proved by a Lyapunov function. Based on this method, the load angle position error could be close to zero.
Aiming at the periodicity of kinematic error, the iterative learning control [
25] and repetitive control (RC) [
26] have been applied to suppress the speed fluctuation. To suppress the vibration caused by kinematic error, a disturbance observer and a robust speed controller based on coprime decomposition were designed in [
27]. Ma et al. adopted the method of adaptive joint torque to actively and adaptively track, but this method has high requirements for the detection device, and there may be hysteresis [
28]. Li et al. applied the PD-type RC to suppress motion errors and proved the stability of the system through Lyapunov function [
29]. These methods can generate infinity gain at the fundamental and multiplier frequency of the system to suppress or track periodic signals [
30,
31]. For linear continuous time invariant systems, a low-pass filter can be added after the RC to achieve high-precision tracking performance and nice suppression performance of the disturbance [
32]. However, Djouda et al. pointed out that when the harmonic frequency gradually increases, the low-pass filter will make the amplitude attenuation and phase lag of the system appear, which will affect the control accuracy of RC to suppress disturbance [
33]. Ivan Godler et al. applied the composite control method to reduce the disturbance in a uniform velocity system, which combined the acceleration feedback and RC [
12]. However, this method needs to be iterated again when the reference changes. Therefore, it is not suitable for a variable speed system, such as the gimbal servo system. To increase the bandwidth of the system and improve the performance of disturbance rejection, a phase compensator was designed in [
34]. Nevertheless, the traditional plug-in RC cannot achieve better performance when the reference is time-varying. To solve this problem, some researchers applied the adaptive control into RC, and the internal model of RC could be adjusted with the change of disturbance period [
35,
36].
The speed fluctuation caused by the kinematic error is position period, and its frequency is twice the frequency of a motor angular position. Considering the periodic characteristics of speed fluctuation, a novel composite control method combining the RC and the acceleration feedback is proposed in this paper. The RC is designed in the position domain based on the domain conversion [
37]. By this method, even if the reference speed changes, the frequency of the speed fluctuation is still fixed, and the RC does not need to reconstruct the internal model. Therefore, the algorithm can converge very quickly. The acceleration feedback can increase the damping of the system and further reduce the speed fluctuation. For convenience of reading, the performance and features of the relevant typical methods and the proposed method are summarized in
Table 1.
The rest of this article is organized as follows.
Section 2 discusses the dynamic modeling of the gimbal servo system with harmonic drive. The compound control method is presented in
Section 3, which combines acceleration feedback with position domain RC. The effectiveness and superiority of this method are verified by the simulation and experimental results in
Section 4. In
Section 5, some important conclusions are summarized.
2. Analysis of The Gimbal Servo System with Harmonic Drive
The structure of the gimbal servo system of SGCMG is shown in
Figure 2. It is mainly composed of the motor-side encoder, bearings, permanent magnet synchronous motor (PMSM), harmonic drive, high-speed rotor system and load-side resolver.
According to [
38], the PMSM can be simplified as a DC motor, and the equivalent electrical circuit is shown in
Figure 3.
On the basis of
Figure 3, the balance equation of the voltage can be expressed as:
where
is the control voltage of the PMSM,
is the output torque of the PMSM,
R is the stator phase resistance,
is the armature current,
L is the phase inductance,
is the back electromotive force coefficient, and
is the angular position of the PMSM. The output torque
of the PMSM can be represented as:
where
is the torque coefficient.
According to [
38], the structure diagram of the gimbal servo system with harmonic drive is shown in
Figure 4, and the dynamic model can be established as follows:
where
is the output torque of the harmonic drive,
is the torsional stiffness,
represents the torsional angle,
is the angular position of the load,
and
stand for the speed of the motor side and load side, respectively,
and
are the rotational inertia of the motor and load side, respectively,
and
represent the damping coefficient of the motor side and load side, respectively,
is other interference torque, and
N is the ideal gear ratio of the harmonic drive.
To obtain the high-precision output speed of the gimbal servo system, double closed-loop feedback control methodology is applied in the gimbal servo system of SGCMG [
38]. The inner is a current loop; the outer is a speed loop, and their loop controllers are all designed as a traditional PI controller which is described as
and
, respectively. The parameters of the
and
are shown in
Table 2, which are calculated by the pole assignment method and fine-tuned by analyzing the Bode diagram.
Combining (
1) and (
3), the basic control block diagram of the gimbal servo system is shown in the
Figure 5, where
,
and
.
From
Figure 5, the load speed error caused by the kinematic error can be written as:
where
According to [
39], the model of the kinematic error can be described as:
where
(
i = 1, 2, 3, …) is the magnitude which is related to the speed of the motor and the load torque.
From (
5), it can be seen that the kinematic error is related to the frequency of the motor’s angular position, which has a fixed frequency in the position domain. The frequency is defined as:
where
f is the frequency with period
T in the time domain, and
is the frequency in the position domain.
The FFT analysis of the different speed conditions in the position domain are shown in
Figure 6. It can be seen from
Figure 6 that the speed fluctuation at different speeds has obvious constant frequency characteristics in the position domain. The main components are first-harmonic
, second-harmonic
and third-harmonic
, and the other order harmonic content is relatively small. Therefore, we designed the position RC to suppress the three main order harmonics of the speed fluctuation.