1. Introduction
The attitude control for unmanned systems [
1] and an aerospace system are very important, for the precision of the attitude control has great influence on the accuracy and reliability of the system [
2]. Single gimbal control moment gyroscope (SGCMG) is a critical system for the attitude control of a space system that can offer significant accuracy and efficiency control torque for the attitude adjustment and stability of the spacecraft. Its application in spacecraft attitude control has always been a research hot spot [
3,
4,
5]. The output of the rotor used in SGCMG is a constant angular momentum, and the control accuracy mainly depends on the accuracy of the gimbal servo control system. Therefore, the accuracy of the output torque has a significant impact on the performance of attitude control. Generally, the influence of electromagnetic signal on the sensor accuracy of aircraft and spacecraft systems are very small [
6]. In recent years, in order to improve the accuracy of servo control systems, many scholars have focused on servo control algorithms development including robust iterative learning control via adaptive sliding mode control [
7], a variable structure controller and an adaptive feed forward controller [
8] and a cascade extended state observer [
9].
The circular grating is an angular position sensor of the SGCMG control system for measuring the angular position of the gimbal, and the angular velocity is further obtained by the differential processing of sensor signal. Generally, the circular grating with high-resolution and high-precision are used as a system sensor of the satellite to achieve high system accuracy. However, the errors caused by machining and installation [
10,
11,
12] will have significant impact on the measurement accuracy of the circular grating. Improving the measurement accuracy of the circular grating is still an active research subject in various industrial fields. Dateng Zheng et al. [
13] developed the 6 circular grating eccentricity errors model of an articulated arm coordinate measuring machine (AACMM). They also perform calibration and error compensation to improve the measurement accuracy. Ming Chu et al. [
14] proposed a method for circular grating eccentric testing and error compensation for robot joint using double reading heads. Guanbin Gao et al. [
15] studied the mounting eccentric error of the circular grating angle sensors and proposed a compensation method to compensate the error of the joints of an articulated arm coordinate measuring machine. For self-errors of the angle measuring sensor, such as sub-divisional error, Jiawei Yu et al. [
16] established mathematical models for different types of sub-division errors of photoelectric angle encoders. They also proposed an algorithm compensation method based on the established models to improve the tracking precision of a telescope control system. Xianjun Wang [
17] analyzed the cause of angle measuring error of grating for large telescopes and compensated the angle measuring error by resonant equation. Liandong Yu [
18] proposed the harmonic method to compensate for the circular grating angle measurement error of the portable articulated coordinate measurement instrument machines caused by ambient temperature change. Li Xuan et al. [
19] presented a spider-web-patterned scale grating to realize the eccentricity self-detection of the optical rotary encoder by a dual-head scanning unit.
However, the above compensated methods were developed for the ground system and the additional mass and power needed by these methods were not considered. The SGCMG system studied in this paper was used in the civilian micro-agile satellites. The mass of the whole satellite is less than 40 kg. Therefore, the requirements of the SGCMG system used in civilian micro-agile satellites are low mass, low power consumption, low cost and high precision. In general, the methods proposed by previous studies for compensating the eccentricity error of circular grating can be summarized as the hardware [
20] and algorithm [
21] compensation method. The hardware compensation method used multiple reading heads, which were symmetrical, mounted about the center of the circular grating to eliminate the eccentricity error [
22,
23,
24]. The algorithm compensation method developed the compensation model, and the parameters of the model were obtained by the calibration experiments [
25,
26]. The method of using multi reading heads to eliminate eccentricity errors is easy to implement with high precision. However, for there are four SGCMG used in the satellite studied in this paper, and the compensation method should ensure accuracy requirements with relatively low power and mass. Moreover, few studies have focused on the effect of measurement accuracy on the performance of the servo control system, which is especially important for the SGCMG system development.
In general, the former research of scholars can be summarized as:
- (a)
Most researchers have focused on the circular grating’s accuracy of ground systems such as the articulated arm coordinate measuring machine (AACMM) and telescope control system.
- (b)
Many scholars studied the compensation methods and self-calibration [
27,
28] of circular grating by multi reading heads. However, few research studies concerning the algorithm compensation can be found in recent years, and the comparison of the algorithm and hardware method has not been the focus of research on compensation of the circular grating.
- (c)
A search of the literature revealed few studies that improved the performance of the servo control system that considered the accuracy of the circular grating sensors.
The contribution of this paper is to propose an error compensation method for the angle measurement of the SGCMG system used in satellite. Only one reading head was used and the angle measurement errors were compensated based on the calibration experiment to verify the accuracy of the algorithm method was almost the same as the hardware method. Moreover, a SGCMG servo system model for velocity control accuracy simulation was investigated to prove that improving the accuracy of circular grating can improve the accuracy of the servo control system. In the next section the design of a single gimbal control moment gyroscope system is described. In
Section 3, the source of circular grating errors was studied, and the mathematic models of tilt and eccentricity error are also presented.
Section 4 introduces the circular grating calibration experiment. Detailed results of the calibration experiment and accuracy analysis of the proposed method are given in
Section 5, followed by conclusions in
Section 6.
3. Eccentricity Error Modeling
The circular grating error is the difference between the measured angle value of the reading head and the actual angle value. In general, the errors caused by the angle measurement of circular grating consists of two parts: self-errors and installation errors. The self-errors include the graduation accuracy and sub-divisional error [
31], which are the periodic systematic errors that are sourced from the circular grating and the reading head. Generally, these errors can be decreased by using high-precision circular gratings and reading heads [
32]. The installation errors mainly include the tilt and eccentricity error, which are caused by the installation of the circular grating and reading head. The grating tilt is caused by installation tilt and shaft sloshing, as shown in
Figure 2a. The shaft sloshing is caused by the attitude control of the satellite and the micro-vibration of the flywheel rotating. The influence of tilt error on measurement accuracy is very small in comparison with eccentricity error. Therefore, tilt error is usually negligible. The mathematic model of tilt error is presented briefly in
Appendix A.
The eccentricity error is caused by the non-coincidence between the geometric center of the circular grating and the rotational center of the measured shaft [
33,
34]. The value of the eccentricity error changes periodically with the rotation of the shaft. Because the eccentricity error is determined after the installation of the reading head and circular grating, the eccentricity error can be modeled by geometric method.
Figure 2b shows the relationship between the actual rotation angle
β of the circular grating and the measurement angle
α of the reading head. A is the geometric center of the circular grating; O is the rotational center of measurement system; P is the zero angular position; C is the angle measuring position where the reading head is installed. Because the lines OP and AC intersect,
where
γ and
θ are the angle value of ∠OPA and ∠OCA, respectively.
According to the sine theorem, the relationship between side length and the angle of triangle ΔOAP and ΔOAC are given as
where
ε is the value of eccentricity (the distance between the geometric center and rotational center) and
r is the radius of the circular grating;
φ represents the direction of the eccentricity.
Then,
γ and
θ can be expressed as follows:
The grating measurement error of reading head is given as
where
δ is the measurement error of the circular grating.
By substituting Equations (5), (8) and (9) into (10), one can obtain,
If
r is no less than 52 mm and the eccentricity
ε is controlled within 0.01 mm, then
Based on the small-angle approximation theory, the simplification of the Equation (11) is given as
where the sine value of the
δ approaches 0, and the cosine value of
δ approaches 1. Then
is approximated as follows:
The Equation (13) can be reduced to
in which
6. Conclusions
In order to improve control precision of the SGCMG, a general and systematic methodology is presented to compensate the measurement error of circular grating. A calibration experiment was proposed to measure the error of the circular grating. The interactions among the measurement error, compensation accuracy and control accuracy were investigated. Both the algorithm and hardware compensation method were performed, and a comparison and appraisal were made for both methods. In general, therefore, it seems that the proposed method was effective, offering compensation for the measurement error of the circular grating with only one reading head. The results of this study indicate that the error calibration and compensation have achieved accuracy solutions for measuring and predicting the measurement error of circular grating. Based on the results of this study, the following main conclusion can be drawn:
- (1)
Eccentricity error is the main source of measurement error of circular grating.
- (2)
The key step in the proposed method is that the error calibration process includes calibration and fitting of the measurement error to provide accurate fitting compensation models to predict measurement errors.
- (3)
The accuracy of the algorithm method was almost same with the hardware method in this study. We also conducted fitting processing with a two-term Fourier compensation model, and the accuracy of the measurement was not obviously improved.
Generally speaking, we observe that the algorithm compensation method proposed in this paper effectively offers good potential to be applied to the angle measurement of circular grating used in the space system in order to meet the requirements of lower-mass and high accuracy.