Synthetic Deviation Correction Method for Tracking Satellite of the SOTM Antenna on High Maneuverability Carriers

Abstract

Using Satcom-On-The-Move (SOTM) antenna on moving carriers to track communication satellites is a rapidly developing technology. The normal running of the SOTM system requires its antenna beam to track the target communication satellite accurately at all times. However, due to the errors of the measurement system, tracking deviation will inevitably occur, especially when the moving carrier is in ahigh maneuvering state, which may cause communication failures. In this paper, we propose a synthetic deviation correction algorithm; when the carrier is in the high maneuvering state, the measurement error is converted into the deviation of the azimuth as well as the pitch of the antenna that needs to be corrected to correct the pointing of the SOTM antenna. Finally, the proposed algorithm is verified by experiments. The experimental results show that the proposed algorithm has a good isolation effect on the high maneuverability of the carrier, which means that the pointing to the communication satellite is more accurate and achieves better communication quality under the high maneuvering state. The effectiveness of the algorithm is illustrated.

1. Introduction

Satellite communication uses artificial Earth satellites as relays so that two or more ground satellite stations can transmit radio wave signals to each other [1,2,3,4]. Compared with base-station-based cellular communication, the coverage of satellite communication is broad, and at the same time, it does not need infrastructure such as a communication base station. For satellite communication, the networking is flexible, and the application range is wide [5]. In early satellite communications, due to the technical condition, antennas 10 m or more in diameter were often required as the ground satellite receiving equipment [6]. In order to realize satellite communication, it is necessary to manually set up and debug the ground satellite receiving equipment, i.e., the antenna, and the antenna is fixed on the ground. The whole process requires a lot of manpower, material resources, and time costs [7]. This satellite communication is generally called the “Satellite Communication Antenna in Static (SCAS)” system [8].

With the improvement of the power of satellite signal receiving–transmitting equipment, as well as the development of signal receiving and demodulating technology of the ground station, the size and the weight of the antenna have decreased significantly. As a consequence, it is easy to align with the satellite, making it convenient to use the SCAS system. However, the azimuth and pitch of the SCAS antenna cannot be adjusted in real time. Consequently, the SCAS system cannot be applied to moving carriers such as aircraft, cars, ships, etc., due to the position and attitude of the carrier constantly changing during the movement. In order to overcome the disadvantage of the SCAS system, a system that realizes satellite communication while the carrier is in motion has appeared in the field of satellite communication, which is usually called the “Satcom-On-The-Move (SOTM)” system [9,10].

In the SOTM system, the antenna for receiving and transmitting satellite signals is fixed on the moving carrier. During the movement of the carrier, the antenna aligns with the satellite in real time by controlling the direction of the antenna so as to establish a stable satellite signal transmission link [11,12,13,14].

Whether the carrier is in a moving or static state, the SOTM system finds the location of the satellite. In addition, it uses the data from the measuring devices, generally including the Global Navigation Satellite System (GNSS), electronic compass, accelerometer, and gyroscope, to calculate the real-time attitude and location of the carrier. Then, the SOTM system adjusts the azimuth and pitch angle of the antenna, making the antenna point to the satellite in real time [15,16].

With the high-speed development of satellite communication technology, the SOTM system will play a very important role. The SOTM system makes moving vehicles, ships, and aircraft realize real-time broadband communication. In order to make the SOTM system work normally, its antenna beam is required to track the target communication satellite accurately at any time. However, no matter what measurement and control technology is adopted, the pointing deviation will inevitably occur. When the moving carrier is in the high maneuvering state, the deviation will become greater. This greater deviation may cause communication obstacles or interrupt the communication system [17]. From the test data of SATPRO M&C Tech Co., Ltd. (SATPRO, a SOTM system manufacturer in Xi’an, China), when the angular velocity of the carrier is greater than 30°/s or the angular acceleration is greater than 40°/s², there may be communication obstacles or communication interruption. There are many reasons for the SOTM antenna pointing deviation, but the azimuth and pitch angle deviation of the antenna beam is an important causation. Our job is to minimize these two deviations in various complex environments.

The application scenarios of the SOTM system are increasing [18,19], and its working conditions are becoming more complex; meanwhile, the working conditions are always unknown and dynamic [20]. Thus, there are still some challenging problems to be solved. As the carrier is in motion, its position and attitude may change dramatically during its movement [21]. For the conventional SOTM servo control system, under this condition, the communication quality may be reduced, and even the communication link may be interrupted, and the experiments of Section 4 verify this conclusion. At present, the research on high maneuverability mainly focuses on carrier attitude estimation and carrier position estimation [22,23,24,25]. The research on the SOTM antenna tracking satellite of high-maneuverability carriers is in the initial stage, and there are few related studies.

Communication satellites, in terms of orbital height, mainly include Geostationary Earth Orbit (GEO) satellites and Low Earth Orbit (LEO) satellites. AGEO satellite is located at a height of 35,800 km above the Earth’s surface, and an LEO satellite is located at a height of less than 5000 km above the Earth’s surface. The orbital period of a GEO satellite is the same as the rotation period of the Earth. Meanwhile, the orbital period of an LEO satellite (less than 128 min) is much less than the rotation period of the Earth [9]. The motion of the LEO satellite is complex, which makes it difficult to align with the satellite [26]. The location of an LEO satellite with respect to the Earth’s surface changes rapidly, the duration of an LEO satellite covering a specific location is short [27], and the SOTM needs to switch the target satellite continuously, which causes communication interruption [28]. At present, the research of the SOTM system with LEO satellite mainly focuses on the antenna and includes the material, structure, etc. of the antenna [29,30,31,32]. The data rate of the SOTM system with the GEO satellite is lower [33], but the GEO satellite provides continuous and stable communication [34], which is important in lots of cases; the research in this paper is based on the GEO satellite.

Aiming at the SOTM system of a mechanical tracking satellite, this paper adopts a sequential least square algorithm to solve the deviation correction problem of a tracking target satellite by the SOTM antenna on a highly mobile carrier.

The remainder of this paper is organized as follows: In Section 2, the servo tracking scheme of the SOTM system is introduced; in Section 3, a hybrid deviation correction algorithm is designed for tracking satellites with the SOTM antenna when the carrier is in a high maneuvering state. The main ideas of the algorithm are as follows: (1) In the non-maneuvering and weak maneuvering stages, the original mechanical tracking method is still used; (2) in the high maneuvering stage, the synthetic deviation correction algorithm for mechanical tracking proposed by us is adopted to minimize the tracking deviation. Section 4 verifies the proposed algorithm through experiments. Section 5 provides the conclusions of the full paper.

2. Servo Control of the SOTM Antenna

2.1. Overview of the Development of SOTM

As early as the 1990s, with the development of communication satellite technology, the satellite communication technology of moving carriers received attention. For example, the National Aeronautics and Space Administration (NASA) Glenn Research Center of the United States put the advanced communications technology satellite (ACTS) into Earth orbit from John F. Kennedy Space Center (KSC) in Florida as early as September 1993 and then implemented an ACTS Mobile Terminal (AMT) project of NASA. Its mission is to implement the first satellite test application project immediately after the satellite is put into operation and to demonstrate the communication of trucks moving on the ground communicate through satellites. They developed the K- and Ka-band moving vehicle antenna systems for satellite tracking. The AMT antenna system is composed of a small mechanically controlled reflective antenna and an antenna controller. The two channels of the reflective antenna share an aperture and operate under an antenna cover with a diameter of 23 cm and a height of 10 cm. This antenna tracks satellites as the vehicle moves [35].

The AMT project initially began to operate in the Los Angeles area, and it communicates through ACTS. There is an ACTS master control ground station at the NASA Lewis Research Center in Cleveland, Ohio. AMT uses the spot beam of Louisiana/San Diego for communication. ACTS converts the uplink Ka-band (30 GHz) signal from Cleveland into a K-band (20 GHz) signal and transmits this signal to AMT in Southern California. The Cleveland ground station is required to continuously transmit a pilot signal in the Ka band to check whether a date signal is sent to the AMT. AMT requires the pilot signal to track the satellite with its own antenna. Once the pilot signal is used by all mobile terminals for satellite tracking in the multi-user ACTS mode, ACTS converts the Ka-band uplink signal of AMT into a K-band signal and transmits the signal to the ground station in Cleveland.

In 1996, the television broadcasting service via the Direct Broadcasting Satellite (DBS) was already in operation in Japan, and there were many residential users. In addition, many people want to watch DBS broadcasting even in the car because DBS broadcasting has advantages over terrestrial TV broadcasting, such as high quality, time synchronization, wide coverage, etc. At this time, several dynamic antenna systems for DBS have been developed. These systems have high reception performance, but there is a problem in the fact that these systems are still too high to be installed on the roof of the car. Therefore, great efforts have been made to reduce the height of the antenna system, and a small and short antenna system has been installed on the vehicle [36]. The antenna system needs a simple structure. The conventional antenna system has a mechanical steering unit to track the satellite at pitch angles. It is not necessary to remove the antenna system of such a mechanical unit in order to reduce the height. The horizontal installation of a plane antenna with a large beam inclination angle and a wide beam width is the most suitable way to omit the pitch angle tracking unit. In this case, the system beam inclination angle that needs to be used in most areas of Japan is about 50° from the top, and the pitch angle beam width is greater than 12°. In addition, the satellite tracking function in the azimuth direction is indispensable to an automatic antenna system.

Various satellite tracking methods that combine mechanical tracking with electronic beam control have appeared in various literatures [37,38,39,40], which is the mainstream of the development of communication in motion technology. Among them, some use the single-pulse method, and some use the conical scanning method to track satellite signals. The tracking function is realized by using the gyro data and the received satellite signal strength. Others apply offset beam tracking technology to phased-array antenna systems. This offset beam rotates around the main beam and adds another additional phase to the phase shifter aligned with the main beam. However, the offset beam technique requires a complex and expensive additional alignment phase shifter and a time-consuming alignment process. In addition, when the moving carrier has a strong yaw angle disturbance, the effectiveness of the above tracking method is questionable [14]. Therefore, we need to further study the satellite tracking methods in a strong disturbance environment.

2.2. Principle of the SOTM Servo Control

The antenna needs to be aligned to the target satellite to form the completed communication link during the running of the satellite communication system. As for the SOTM system, the antenna is installed on the moving carrier. The position and attitude will change during the movement of the carrier. The SOTM system calculates the direction of the antenna in real time according to the measured carrier position, attitude information, and the position information of the target satellite. The servo control system adjusts the azimuth angle and pitch angle of the antenna according to the calculated antenna pointing to align the antenna with the satellite.

The overview of the SOTM system is shown in Figure 1. The ground station is fixed on the ground, and it is connected to the command center, Internet Service Provider (ISP), etc., through cables. Meanwhile, the ground station establishes wireless communication links with communication satellites. The SOTM antenna is fixed on the moving carrier. During the movement, the SOTM antenna points to the communication satellite to establish wireless communication links with the communication satellite. At the same time, the SOTM establishes communication links with terminal devices (e.g., cameras, personal computers, cell phones, etc.) through network devices, such as modems or routers. In this way, the complete communication links between the terminal devices and the command center, ISP, etc., are established so that the terminal devices can transmit images, voice, etc., with the command center or access the Internet through the ISP.

The automatic tracking satellite of the SOTM antenna is realized by the SOTM servo control system; its structural block diagram is shown in Figure 2. The core controller is the core of the whole servo control system. The tasks such as data processing and control law calculation are all executed by the core controller. It is also the center of data collection, processing, and forwarding, among other parts of the servo control system. The power supply part is responsible for supplying power to the servo control system. The attitude solution processor comprehensively processes the data of the accelerometer, gyroscope, electronic compass, and GNSS to obtain the attitude information of the carrier, and GNSS provides the location information of the carrier. The core controller calculates the pointing information of the antenna through the location and attitude information of the carrier and the location information of the target satellite. After the calculation of the control law, the control commands about the azimuth and pitch of the antenna are obtained and sent to the drivers of the azimuth system and the pitch system, respectively. The drivers drive the azimuth motor and the pitch motor, respectively, to change the azimuth angle and pitch angle of the antenna so as to realize the coarse pointing of the SOTM antenna to the communication satellite. In addition, the core controller also corrects the azimuth angle and pitch angle of the antenna according to the conical scanning signal of the beacon receiver to achieve accurate pointing to the communication satellite.

The scheme mentioned above is designed when the attitude of the moving carrier changes slowly. In the design process, the high maneuverability of the carrier was not considered too much. When the carrier is in a high maneuvering state, the conical scanning scheme above may fail, resulting in the poor effect of the SOTM antenna in accurately tracking the satellite, and poor communication quality, even causing the SOTM antenna to lose the satellite and failure to establish a communication link with the satellite. When the antenna loses the satellite, the SOTM depends on the attitude and location calculation information of the carrier, the searching satellite speed of the SOTM antenna is limited by the accuracy of the accelerometer, gyroscope, electronic compass, and GNSS, as well as the calculation capability of the attitude calculation processor. For high-maneuvering carriers, this impact will increase with the improvement of maneuverability, which may cause the SOTM antenna to be in the state of satellite searching for a long time so that it is unable to establish a stable communication link with the satellite.

In the next section, we will propose a synthetic-deviation correction method aiming at the difficulties of the SOTM antenna tracking the satellite when the carrier is in a high maneuvering state. The proposed scheme can overcome the impact of high maneuvering, enabling the antenna to track satellites in harsh environments effectively and achieve continuous communication.

3. Synthetic Deviation Correction Algorithm

For the high-maneuvering carrier, there are some problems in the satellite tracking method that only adopts mature mechanical tracking control. The problem is when the carrier is in the high maneuvering state; that is, when the angular velocity and angular acceleration change greatly, it is difficult to track the satellite signal by using the conical scanning method. Here, we propose a synthetic deviation correction method for tracking satellites with the SOTM antenna on the high-maneuvering carrier. The idea is: (1) In the non-maneuvering and weak-maneuvering phases, the satellite tracking method of mechanical tracking is still used because the practice has proved its effectiveness; (2) In the high maneuvering stage, the synthetic deviation correction algorithm for mechanical tracking proposed in this section is adopted to minimize the tracking deviation as much as possible and transition the satellite tracking mode from the strong maneuvering stage to the non-high maneuvering stage smoothly.

3.1. Coordinate System

The carrier SOTM system uses GNSS time service, and its accuracy is to the nanosecond. Next, we will introduce the coordinate systems used in this section, as well as the conversion relationship between the coordinate systems.

(1)

Carrier Cartesian coordinate system and spherical coordinate system.

The Cartesian coordinate system of the carrier is shown in Figure 3. Its origin is at the center of gravity of the carrier, i.e., P in the figure. The $x_r$ axis is fixedly connected to the carrier in the plane of symmetry of the carrier, in the zero position it coincides with the platform axis and points to the front of the carrier movement. The $z_r$ axis is in the plane of symmetry and perpendicular to the $x_r$ axis, in the zero position it points above the carrier directly. The $y_r$ axis, as well as the $x_r$ and $z_r$ axes, form a right-handed system.

The antenna can be rotated in azimuth and pitch to track the communication satellite $S$. The tracking value of the communication satellite in the spherical coordinate system is $\left(r,\alpha ,\beta \right)$, where $r$ is the radial distance, $\alpha$ and $\beta$ is the azimuth angle and pitch angle, respectively.

If the tracking value in the carrier spherical coordinate system is converted to the carrier Cartesian coordinate system, the position vector of the communication satellite in the carrier Cartesian coordinate system is,

$${\mathit{x}}_r=\left[\begin{array}{c}{x}_r\\ y_r\\ z_r\end{array}\right]=\left[\begin{array}{c}r\cos \beta \cos \alpha \\ r\cos \beta \sin \alpha \\ r\sin \beta \end{array}\right]$$

(1)

(2)

Conversion from carrier Cartesian coordinate system to East-North-Up (ENU) coordinate system.

The ENU coordinate system means that the coordinate origin coincides with the center of gravity of the carrier, i.e., point P in Figure 3. The $x_l$ axis points to the due east of the geography, the $y_l$ axis points to the due north of the geography, and the $z_l$ axis deviates from the earth’s center and points to the sky. The $x_l$, $y_l$ and $z_l$ axes are perpendicular to each other, forming a right-handed system.

The attitude of the carrier during movement is expressed as the yaw angle ($\psi$), pitch angle ($\theta$), and roll angle ($\varphi$). The three Euler angles representing the attitude of the carrier have the following conversion relations,

$${\mathit{x}}_l=R_z^T\left(\psi \right)R_y^T\left(\theta \right)R_x^T\left(\varphi \right){\mathit{x}}_r=R_{\psi \theta \varphi }^T{\mathit{x}}_r$$

(2)

where,

$$\begin{array}{c}{R}_z^T\left(\psi \right)=\left[\begin{array}{ccc}\cos \psi & \sin \psi & 0\\ -\sin \psi & \cos \psi & 0\\ 0& 0& 1\end{array}\right]\\ R_y^T\left(\theta \right)=\left[\begin{array}{ccc}\cos \theta & 0& \sin \theta \\ 0& 1& 0\\ -\sin \theta & 0& \cos \theta \end{array}\right]\\ R_x^T\left(\varphi \right)=\left[\begin{array}{ccc}1& 0& 0\\ 0& \cos \varphi & \sin \varphi \\ 0& -\sin \varphi & \cos \varphi \end{array}\right]\end{array}$$

(3)

Thus, the transformation relationship between the ENU coordinate system and the carrier Cartesian coordinate system is,

$$R_{\psi \theta \varphi }^T=R_z^T\left(\psi \right)R_y^T\left(\theta \right)R_x^T\left(\varphi \right)$$

(4)

(3)

Conversion from geodetic coordinate system to ECEF coordinate system.

Let the geographical coordinates of the geodetic coordinate system of the carrier origin, i.e., point P, be $\left({\lambda }_P,{\vartheta }_P,H_P\right)$, as shown in Figure 4, where, ${\lambda }_P$ is the geographic longitude, ${\vartheta }_P$ is the geographic latitude, $H_P$ is the altitude, and its corresponding Earth-Centered, Earth-Fixed (ECEF) coordinate value is,

$${\mathit{x}}_P=\left[\begin{array}{c}{x}_P\\ y_P\\ z_P\end{array}\right]=\left[\begin{array}{c}\left(C_P+H_P\right)\cos {\vartheta }_P\cos {\lambda }_P\\ \left(C_P+H_P\right)\cos {\vartheta }_P\sin {\lambda }_P\\ \left(C_P\left(1-e^2\right)+H_P\right)\sin {\vartheta }_P\end{array}\right]$$

(5)

where,

$$C_P=\frac{E_q}{\sqrt{1-e^2{\sin }^2{\vartheta }_P}}$$

(6)

$$e=\sqrt{\frac{a^2-b^2}{a^2}}$$

(7)

$E_q$ is the equatorial radius of the earth, $e$ is the eccentricity of the earth, $a$ is the long half axis of the earth ellipsoid, and $b$ is the short half axis of the earth ellipsoid.

(4)

Conversion from ENU coordinate system to the ECEF coordinate system.

For the communication satellite $S$, the coordinate value ${\mathit{x}}_l$ of the carrier in the ENU coordinate system has been obtained from Equation (2). Since the longitude $\left({\lambda }_P\right)$ and latitude $\left({\vartheta }_P\right)$ are known, the coordinate system origin is translated first, and then the coordinate value ${\mathit{x}}_l$ in the ENU coordinate system can be converted to the coordinate value ${\mathit{x}}_S$ in the ECEF coordinate system through three sequential coordinate axis rotation changes. That is,

$${\mathit{x}}_S=G_{\lambda \vartheta P}{\mathit{x}}_l+{\mathit{x}}_P$$

(8)

where,

$$G_{\lambda \vartheta P}=\left[\begin{array}{ccc}-\sin {\lambda }_P\cos {\vartheta }_P& -\sin {\vartheta }_P& -\cos {\lambda }_P\cos {\vartheta }_P\\ -\sin {\lambda }_P\sin {\vartheta }_P& \cos {\vartheta }_P& -\cos {\lambda }_P\sin {\vartheta }_P\\ \cos {\vartheta }_P& 0& -\sin {\lambda }_P\end{array}\right]$$

(9)

${\mathit{x}}_S$ is the position of the communication satellite $S$ in the ECEF coordinate system.

(5)

Conversion from ECEF coordinate system to geodetic coordinate system.

Assume that the position vector of the communication satellite $S$ in the ECEF coordinate system is ${\mathit{x}}_S={\left(x_S,y_S,z_S\right)}^T$, which needs to be converted into a geodetic coordinate system, then its geodetic coordinate system coordinate is $\left({\lambda }_P,{\vartheta }_P,H_P\right)$,

$$\{\begin{array}{l}{\lambda }_P=\arctan \frac{y_S}{x_S}\hfill \\ {\vartheta }_S=\arctan \left[\frac{z_S+e_1^{2}b\sin {\phi }_S^3}{\sqrt{x_S^2+y_S^2}-e^{2}a\cos {\phi }_S^3}\right]\hfill \\ H_S=\frac{\sqrt{x_S^2+y_S^2}}{\cos {\vartheta }_S}-C_S\hfill \end{array}$$

(10)

where,

$$\begin{array}{l}{e}_1=\sqrt{\frac{a^2-b^2}{b^2}}\hfill \\ C_S=\frac{E_q}{\sqrt{1-e^2{\sin }^2\left({\vartheta }_S\right)}}\hfill \\ {\phi }_S=\arctan \frac{Z_S·a}{\sqrt{x_S^2+y_S^2}·b}\hfill \end{array}$$

(11)

$a$ is the long half axis of the earth ellipsoid, and $b$ is the short half axis of the earth ellipsoid.

3.2. Deviation Correction Algorithm

First, make the following settings.

(1)

The tracked communication satellite is a geosynchronous satellite. The position of the geosynchronous satellite, which is relative to the Earth, will change slightly, and the ground control system will fine-tune the satellite’s position during its life cycle. The time when the moving carrier is in the high maneuvering state is far less than the satellite life cycle, and the distance between the carrier and the satellite is far greater than the small displacement of the synchronous satellite. Therefore, it can be considered that the coordinates of the communication satellite in the ECEF coordinate system remain unchanged, that is, ${\mathit{x}}_S\left(t\right)={\overline{\mathit{x}}}_S$ is a constant value.

(2)

A navigation and positioning system composed of GNSS and inertial devices is installed on the carrier; that is, at any time $t$, the center of the gravity position ${\mathit{x}}_P\left(t\right)$ of the carrier can be considered to beobtained accurately so that the distance between the carrier and the satellite can be obtained accurately,

$$r\left(t\right)=\sqrt{{\left({\overline{x}}_S-x_P\left(t\right)\right)}^2+{\left({\overline{y}}_S-y_P\left(t\right)\right)}^2+{\left({\overline{z}}_S-z_P\left(t\right)\right)}^2}$$

(12)

(3)

At any time $t$, the navigation system can obtain the measured yaw angle $\psi \left(t\right)$, pitch angle $\psi \left(t\right)$, and roll angle $\varphi \left(t\right)$ of the carrier. The carrier can also obtain the measured azimuth angle $\alpha \left(t\right)$ and pitch angle $\beta \left(t\right)$ of the antenna relative to the Cartesian coordinate system of the carrier, but deviations exist. Here, since the resulting deviation is converted into the antenna azimuth angle deviation $\Delta \alpha \left(t\right)$ and the pitch angle deviation $\Delta \beta \left(t\right)$, it is considered that $R_{\psi \theta \varphi }^T$ can be calculated as no deviation.

(4)

Since the sampling rate of the carrier to the communication satellite is not less than 50 Hz, it is considered that the azimuth angle deviation $\Delta \alpha \left(t\right)$ and pitch angle deviation $\Delta \beta \left(t\right)$ are constant values, i.e., $\Delta \alpha \left(t\right)=\Delta \alpha$, $\Delta \beta \left(t\right)=\Delta \beta$. The purpose of the deviation correction algorithm in this section is to estimate the size of these two deviations to correct the tracking process.

Figure 5 is a schematic diagram of the process of carrier tracking communication satellites. Since the system uses digital electronic devices, we need to obtain a discrete-time model, where $\Delta t$ represents the sampling time interval; each tracking datum has a time mark, $k$ represents the time sequence, and the tracking time sequence is shown in Figure 6.

The tracking process of the system is described as follows:

(1)

At each step, $k$, the mechanical tracking system of the carrier completes the coarse tracking to the target satellite and provides the corresponding coarse measurement value of azimuth angle $\widehat{\alpha }\left(k\right)$ and pitch angle rough measurement $\widehat{\beta }\left(k\right)$. The converted azimuth deviation is $\Delta \alpha \left(k\right)=\Delta \alpha$, the pitch angle deviation is $\Delta \beta \left(k\right)=\Delta \beta$. Then, the position vector of the target satellite in the carrier Cartesian coordinate system is:

$${\widehat{\mathit{x}}}_r\left(k\right)+\Delta {\widehat{\mathit{x}}}_r\left(k\right)=\left[\begin{array}{c}r\left(k\right)\cos \left(\widehat{\beta }\left(k\right)+\Delta \beta \right)\cos \left(\widehat{\alpha }\left(k\right)+\Delta \alpha \right)\\ r\left(k\right)\cos \left(\widehat{\beta }\left(k\right)+\Delta \beta \right)\sin \left(\widehat{\alpha }\left(k\right)+\Delta \alpha \right)\\ r\left(k\right)\sin \left(\widehat{\beta }\left(k\right)+\Delta \beta \right)\end{array}\right]$$

(13)

Since the azimuth angle deviation and pitch angle deviation in the sampling period can be considered as constant values, and this deviation is small, then,

$$\{\begin{array}{l}\cos \left(\widehat{\beta }\left(k\right)+\Delta \beta \right)\approx \cos \widehat{\beta }\left(k\right)-\sin \widehat{\beta }\left(k\right)\Delta \beta \hfill \\ \cos \left(\widehat{\alpha }\left(k\right)+\Delta \alpha \right)\approx \cos \widehat{\alpha }\left(k\right)-\sin \widehat{\alpha }\left(k\right)\Delta \alpha \hfill \\ \sin \left(\widehat{\beta }\left(k\right)+\Delta \beta \right)\approx \sin \widehat{\beta }\left(k\right)+\cos \widehat{\beta }\left(k\right)\Delta \beta \hfill \\ \sin \left(\widehat{\alpha }\left(k\right)+\Delta \alpha \right)\approx \sin \widehat{\alpha }\left(k\right)+\cos \widehat{\alpha }\left(k\right)\Delta \alpha \hfill \end{array}$$

(14)

Through calculation, we can find that,

$$\{\begin{array}{c}r(k)\cos (\widehat{\beta }(k)+\Delta \beta )\cos (\widehat{\alpha }(k)+\Delta \alpha )\approx r(k)\cos \widehat{\beta }(k)\cos \widehat{\alpha }(k)-r(k)(\sin \widehat{\beta }(k)\cos \widehat{\alpha }(k)\Delta \beta +\cos \widehat{\beta }(k)\sin \widehat{\alpha }(k)\Delta \alpha )\\ r(k)\cos (\widehat{\beta }(k)+\Delta \beta )\sin (\widehat{\alpha }(k)+\Delta \alpha )\approx r(k)\cos \widehat{\beta }(k)\sin \widehat{\alpha }(k)-r(k)(\sin \widehat{\beta }(k)\sin \widehat{\alpha }(k)\Delta \beta -\cos \widehat{\beta }(k)\cos \widehat{\alpha }(k)\Delta \alpha )\\ r(k)\sin (\widehat{\beta }(k)+\Delta \beta )\approx r(k)\sin \widehat{\beta }(k)+r(k)\cos \widehat{\beta }(k)\Delta \beta \end{array}$$

(15)

Therefore, the tracking deviation of the satellite at time $k$ can be approximately expressed as follows:

$$\Delta {\mathit{x}}_r\left(k\right)\approx r\left(k\right)\left[\begin{array}{c}-\sin \widehat{\beta }\left(k\right)\cos \widehat{\alpha }\left(k\right)\Delta \beta -\cos \widehat{\beta }\left(k\right)\sin \widehat{\alpha }\left(k\right)\Delta \alpha \\ -\sin \widehat{\beta }\left(k\right)\sin \widehat{\alpha }\left(k\right)\Delta \beta +\cos \widehat{\beta }\left(k\right)\cos \widehat{\alpha }\left(k\right)\Delta \alpha \\ \cos \widehat{\beta }\left(k\right)\Delta \beta \end{array}\right]=\widehat{\mathsf{\Lambda }}\left(k\right)\Delta \mathit{e}$$

(16)

where,

$$\begin{array}{l}\widehat{\mathsf{\Lambda }}\left(k\right)=\left[\begin{array}{cc}-\sin \widehat{\beta }\left(k\right)\cos \widehat{\alpha }\left(k\right)& -\cos \widehat{\beta }\left(k\right)\sin \widehat{\alpha }\left(k\right)\\ -\sin \widehat{\beta }\left(k\right)\sin \widehat{\alpha }\left(k\right)& \cos \widehat{\beta }\left(k\right)\cos \widehat{\alpha }\left(k\right)\\ \cos \widehat{\beta }\left(k\right)& 0\end{array}\right]\hfill \\ \Delta \mathit{e}=\left[\begin{array}{c}\Delta \beta \\ \Delta \alpha \end{array}\right]\hfill \end{array}$$

(17)

(2)

At step $k$, the yaw of the carrier is $\psi$, and the pitch and the roll of the carrier is $\theta$ and $\varphi$, respectively. The tracking position of the communication satellite in the carrier ENU coordinate system is:

$${\widehat{\mathit{x}}}_l\left(k\right)+\Delta {\mathit{x}}_l\left(k\right)=R_{\psi \theta \varphi }^T\left({\widehat{\mathit{x}}}_r\left(k\right)+\Delta {\widehat{\mathit{x}}}_r\left(k\right)\right)=R_{\psi \theta \varphi }^T{\widehat{\mathit{x}}}_r\left(k\right)+R_{\psi \theta \varphi }^T\widehat{\mathsf{\Lambda }}\left(k\right)\Delta \mathit{e}$$

(18)

Then, the tracking deviation vector is:

$$\Delta {\mathit{x}}_l\left(k\right)=R_{\psi \theta \varphi }^T\widehat{\mathsf{\Lambda }}\left(k\right)\Delta \mathit{e}$$

(19)

According to Equation (8), the position of the communication satellite in the ECEF coordinate system is:

$${\widehat{\mathit{x}}}_S\left(k\right)+\Delta {\mathit{x}}_S\left(k\right)=G_{\lambda \vartheta P}\left({\widehat{\mathit{x}}}_l\left(k\right)+\Delta {\mathit{x}}_l\left(k\right)\right)+{\mathit{x}}_P\left(k\right)$$

(20)

then,

$$\Delta {\mathit{x}}_S\left(k\right)=G_{\lambda \vartheta P}\Delta {\mathit{x}}_l\left(k\right)=G_{\lambda \vartheta P}{R}_{\psi \theta \varphi }^T\widehat{\mathsf{\Lambda }}\left(k\right)\Delta \mathit{e}=\widehat{\mathsf{\Sigma }}\left(k\right)\Delta \mathit{e}$$

(21)

where,

$$\begin{array}{l}{G}_{\lambda \vartheta P}=\left[\begin{array}{ccc}-\sin {\lambda }_P(k)\cos {\vartheta }_P(k)& -\sin {\vartheta }_P(k)& -\cos {\lambda }_P(k)\cos {\vartheta }_P(k)\\ -\sin {\lambda }_P(k)\sin {\vartheta }_P(k)& \cos {\vartheta }_P(k)& -\cos {\lambda }_P(k)\sin {\vartheta }_P(k)\\ \cos {\vartheta }_P(k)& 0& -\sin {\lambda }_P(k)\end{array}\right]\\ \widehat{\mathsf{\Sigma }}(k)=G_{\lambda \vartheta P}{R}_{\psi \theta \varphi }^T\widehat{\mathsf{\Lambda }}(k)\end{array}$$

(22)

(3)

Deviation estimation of carrier tracking satellite.

At every step $k$, according to Equation (20), the communication satellite position calculated by the carrier according to the measurement information is as follows:

$${\widehat{\mathit{x}}}_S\left(k\right)=G_{\lambda \vartheta P}{\widehat{\mathit{x}}}_l\left(k\right)+{\mathit{x}}_P\left(k\right)=G_{\lambda \vartheta P}{R}_{\psi \theta \varphi }^T{\widehat{\mathit{x}}}_r\left(k\right)+{\mathit{x}}_P\left(k\right)$$

(23)

Since ${\mathit{x}}_S\left(t\right)={\overline{\mathit{x}}}_S$ is a known constant, the tracking deviation value of the communication satellite in the ECEF coordinate system can always be calculated as follows:

$$\Delta {\mathit{x}}_S\left(k\right)={\overline{\mathit{x}}}_S-{\widehat{\mathit{x}}}_S\left(k\right)$$

(24)

There are two measurable sequences

$$\{\begin{array}{l}\begin{array}{cccc}\Delta {\mathit{x}}_S\left(k-n\right)& \Delta {\mathit{x}}_S\left(k-n+1\right)& \cdots & \Delta {\mathit{x}}_S\left(k\right)\end{array}\hfill \\ \begin{array}{cccc}\widehat{\mathsf{\Sigma }}\left(k-n\right)& \widehat{\mathsf{\Sigma }}\left(k-n+1\right)& \cdots & \widehat{\mathsf{\Sigma }}\left(k\right)\end{array}\hfill \end{array}$$

(25)

Let,

$$\begin{array}{c}\Delta X_S\left(k\right)=\left[\begin{array}{c}\Delta {\mathit{x}}_S\left(k-n\right)\\ \Delta {\mathit{x}}_S\left(k-n+1\right)\\ ⋮\\ \Delta {\mathit{x}}_S\left(k\right)\end{array}\right]\\ \widehat{Y}\left(k\right)=\left[\begin{array}{c}\widehat{\mathsf{\Sigma }}\left(k-n\right)\\ \widehat{\mathsf{\Sigma }}\left(k-n+1\right)\\ ⋮\\ \widehat{\mathsf{\Sigma }}\left(k\right)\end{array}\right]\end{array}$$

(26)

then,

$$\Delta X_S\left(k\right)=\widehat{Y}\left(k\right)\Delta \mathit{e}$$

(27)

We use the least square matrix method to estimate the deviation vector

$$\Delta \widehat{\mathit{e}}\left(k\right)={\left[{\widehat{Y}}^T\left(k\right)\widehat{Y}\left(k\right)\right]}^{-1}{\widehat{Y}}^T\left(k\right)\Delta X_S\left(k\right)$$

(28)

Note that at this time, the deviation estimation $\Delta \widehat{\mathit{e}}\left(k\right)={\left[\begin{array}{cc}\Delta \widehat{\beta }\left(k\right)& \Delta \widehat{\alpha }\left(k\right)\end{array}\right]}^T$ of the pointing and tracking angle of the communication satellite at time $k$ has been obtained, and the tracking angle can be adjusted by using this deviation estimation. For the purpose of sequential correction, for the next time $k+1$, let

$$\begin{array}{c}\Delta X_S\left(k+1\right)=\left[\begin{array}{c}\Delta {\mathit{x}}_S\left(k-n+1\right)\\ \Delta {\mathit{x}}_S\left(k-n+2\right)\\ ⋮\\ \Delta {\mathit{x}}_S\left(k+1\right)\end{array}\right]\\ \widehat{Y}\left(k+1\right)=\left[\begin{array}{c}\widehat{\mathsf{\Sigma }}\left(k-n+1\right)\\ \widehat{\mathsf{\Sigma }}\left(k-n+2\right)\\ ⋮\\ \widehat{\mathsf{\Sigma }}\left(k+1\right)\end{array}\right]\end{array}$$

(29)

We can also find that,

$$\Delta \widehat{\mathit{e}}\left(k+1\right)={[{\widehat{Y}}^T\left(k+1\right)\widehat{Y}\left(k+1\right)]}^{-1}{\widehat{Y}}^T\left(k+1\right)\Delta X_S\left(k+1\right)$$

(30)

In order to track the target satellite accurately, the algorithm can run continuously until the carrier is in a non-high maneuvering state. Note that the estimated radian value must be converted into an angle value when returning to the application.

4. Experiment

In this section, the comparison experiment method is used to verify the effectiveness of the scheme proposed in this paper. In this experiment, a 60 cm aperture SOTM antenna is used, and the experiment site is shown in Figure 7.

Location: SATPRO M&C Tech Co., Ltd., Weiyang District, Xi’an, China

Time: 2 August 2022

Weather: Sunny, 37 °C

Communication Satellite: YATAI 6D Geosynchronous Satellite

Yaw range of the swing platform: $-15°$–$15°$

Pitch range of the swing platform: $-15°$–$15°$

Roll range of the swing platform: $-15°$–$15°$

The angles of yaw, pitch, and roll run in a sinusoidal manner with a period of 3 s, the maximum angular velocity is 31.4159°/s, and the maximum angular acceleration is 65.7974°/s². From the test data of SATPRO, the swing platform may be in a high maneuvering state during the experiment.

An external laptop is connected to the SOTM antenna through a network cable to verify whether it can communicate with the satellite normally.

This experiment is carried out by using the conventional servo control scheme and the scheme of the synthetic deviation correction algorithm proposed in this paper.

During the experiment, when the swing platform is in the normal maneuvering state, the dynamic communication antenna is in the steady-state and accurate tracking state. At this time, the communication of the SOTM system is normal, as shown in Figure 8. The communication between the SOTM user terminal, i.e., personal computer and China Telecom gateway, is normal. Then, the swing platform is put into the high maneuvering state (the motion cycle of each axis is 3 s, which is the maximum maneuvering state of the swing platform used in the experiment), and the duration is 300 s.

As shown in Figure 9 and Figure 10, the azimuth angle and pitch angle deviation of the SOTM antenna are calculated during the running of the SOTM system. The high maneuvering state lasts from the time of 50 s to 350 s. In the normal state stage, other than the high maneuvering state, the azimuth angle and pitch angle deviation converted by the SOTM antenna are all 0.

Figure 11 shows the azimuth angle of the SOTM antenna measured during the running of the SOTM system using the conventional method. It can be seen that when the carrier is in the state of high maneuvering at the time of 50 s, the measured azimuth angle of the SOTM antenna fluctuates greatly until the high maneuvering state of the carrier ends at the time of 350 s. From the moment of 350 s, the azimuth angle of the SOTM antenna gradually returns to the normal state.

Figure 12 shows the azimuth angle of the SOTM antenna measured during the running of the SOTM system using the synthetic deviation correction algorithm. It can be seen from the figure that from the time when the carrier is in the high maneuvering state at the time of 50 s to the end of the high maneuvering state at the time of 350 s, the deviation and fluctuation of the azimuth angle of the SOTM antenna are small, and the azimuth angle of the SOTM antenna maintains a relatively stable state during the whole movement of the carrier.

According to Figure 11 and Figure 12, when the carrier is in a high maneuvering state, the scheme using the synthetic deviation correction algorithm obtains a more stable SOTM antenna azimuth than the conventional scheme. It can be concluded that the synthetic deviation correction algorithm proposed in this paper played a positive role in suppressing the high maneuverability of the carrier.

Figure 13 shows the pitch angle of the SOTM antenna measured during the running of the SOTM system using the conventional method. It can be seen that when the carrier is in a high maneuvering state at the time of 50 s, the measured pitch angle of the SOTM antenna fluctuates greatly until the high maneuvering state of the carrier ends at the time of 350 s, which is the same as the azimuth angle of the SOTM antenna above. From the time of 350 s, the pitch angle of the SOTM antenna gradually returns to the normal state.

Figure 14 shows the pitch angle of the SOTM antenna measured during the running of the SOTM system using the synthetic deviation correction algorithm. It can be seen from the figure that from the time when the carrier is in the high maneuvering state at the time of 50 s to the end of the high maneuvering state at the time of 350 s, the deviation and fluctuation of the pitch angle of the SOTM antenna are small, and the pitch angle of the SOTM antenna maintains a relatively stable state during the whole movement of the carrier.

According to Figure 12 and Figure 14, when the carrier is in a high maneuvering state, the scheme using the synthetic deviation correction algorithm obtains a more stable SOTM antenna pitch than the conventional scheme. It can be concluded that the synthetic deviation correction algorithm proposed in this paper played a positive role in suppressing the high maneuverability of the carrier, and the tracking effect of the SOTM antenna to the communication satellite is better.

Figure 15 shows the real-time Automatic Gain Control (AGC) level of the SOTM antenna measured during the running of the SOTM system using the conventional method. It can be seen that when the carrier is in the state of high maneuvering state at the time of 50 s, the measured real-time AGC level of the SOTM antenna fluctuates greatly, and the real-time AGC level of the SOTM antenna drops significantly until the end of the high maneuvering state of the carrier at the time of 350 s, indicating that the communication quality of the system has decreased significantly. Where, from 100 s to 120 s, the real-time AGC level of the SOTM antenna drops below 1.9, which indicates that the communication of the SOTM system is interrupted during this time period. From 350 s, the real-time AGC level of the SOTM antenna gradually returns to the normal state.

Figure 16 shows the real-time AGC level of the SOTM antenna measured during the running of the SOTM system using the synthetic deviation correction algorithm proposed in this paper. It can be seen that from the beginning time when the carrier is in a high maneuvering state at the time of 50 s to the end of the high maneuvering state at the time of 350 s, the real-time AGC level of the dynamic communication antenna decreases and fluctuates very little, and the real-time AGC level of the SOTM antenna maintains a relatively stable state during the whole movement of the carrier, that is, the communication quality of the dynamic communication system decreases very little.

According to Figure 15 and Figure 16, when the carrier is in a high maneuvering state, the scheme using the synthetic deviation correction algorithm obtains a more stable real-time AGC level than the conventional scheme; that is, the communication quality obtained by the SOTM system is relatively stable, and the communication quality decreases less. It can be concluded that the synthetic deviation correction algorithm proposed in this paper has a great inhibitory effect on the high maneuverability of the carrier, and the communication quality obtained by the SOTM system is relatively stable.

Figure 17 shows the real-time SNR of the SOTM antenna measured during the running of the SOTM system using the conventional method. It can be seen that when the carrier is in a high maneuvering state at the time of 50 s, the measured real-time SNR of the SOTM antenna fluctuates greatly, and the real-time SNR of the SOTM antenna drops significantly until at the end of the high maneuvering state of the carrier at the time of 350 s. In addition, from 100 s to 120 s, the real-time SNR of the SOTM system is zero for a small period of time. It can be seen from the system setting that the satellite loss of the SOTM antenna occurs within this period of time; that is, the communication between the SOTM system and the communication satellite is interrupted during this period of time, which is consistent with the real-time AGC level of the SOTM antenna shown in Figure 14, this indicates that the communication quality of the system has significantly decreased. From the time of 350 s, the real-time SNR of the mobile communication antenna gradually returns to the normal state.

Figure 18 shows the real-time SNR of the SOTM antenna measured during the running of the SOTM system using the synthetic deviation correction algorithm. It can be seen that from the time when the carrier is in a high maneuvering state at the time of 50 s to the end of the high maneuvering state at the time of 350 s, the real-time SNR of the SOTM antenna decreases and fluctuates very little, and the real-time SNR of the SOTM antenna maintains a relatively stable state during the whole movement of the carrier; that is, the communication quality of the SOTM system decreases very little, and there is no communication interruption occurs.

According to the results shown in Figure 15, Figure 16, Figure 17 and Figure 18, when the carrier is in a high maneuvering state, the scheme using the synthetic deviation correction algorithm obtains a more stable real-time AGC level and real-time SNR than the conventional scheme; that is, the communication quality obtained by the SOTM system is relatively stable, and the communication quality decreases less. However, under the same carrier movement state, the communication quality of the SOTM system using the conventional method has significantly decreased and fluctuated, and even short-time communication interruption has occurred. It can be concluded that the synthetic deviation correction algorithm proposed in this paper played a positive role in suppressing the high maneuverability of the carrier, and the system communication quality obtained by the SOTM system is relatively stable. In addition, this method can reduce the possible satellite loss of the SOTM antenna and the communication interruption of the SOTM system caused by the high maneuverability of the carrier.

From the experiments above, it can be seen that the synthetic deviation correction algorithm proposed in this paper has a strong isolation effect on the high maneuvering state of the carrier compared with the conventional method. Under the same carrier movement, after adopting the synthetic deviation correction algorithm, the pointing of the SOTM antenna is more stable, and the tracking effect for communication satellites is better. On the other hand, for the communication quality, after the synthetic deviation correction algorithm is adopted, although the high maneuvering state of the carrier impacts the communication quality of the SOTM system to a certain extent, the decline is small, and the communication quality of the SOTM system is relatively stable, there are no large fluctuations. In contrast, for the conventional method, when the carrier is in a high maneuvering state, the accuracy of the pointing and tracking of the SOTM antenna to the satellite fluctuates greatly, which has a great impact on the communication quality of the SOTM system. In the whole high maneuvering state of the carrier, the communication quality of the SOTM system decreases significantly, and even short-time satellite loss of the SOTM antenna and communication interruption of the SOTM system occur.

5. Conclusions

Aiming at the difficulties faced by SOTM antenna mechanical tracking under the condition of high maneuverability, we proposed a synthetic deviation-correction algorithm. The algorithm is based on the least square algorithm and is simple to implement. The algorithm does not contain complex mathematical operations and can be implemented by a general microprocessor. In this paper, the designed algorithm is verified by experiments. The experimental results show that the synthetic deviation correction algorithm has a good isolation effect on the high maneuverability of the carrier. After the algorithm is adopted, the pointing and the tracking of the SOTM antenna to the communication satellite are more stable under the same high maneuvering state of the carrier. Meanwhile, the SOTM system obtains better communication quality; it can avoid the loss of satellite of the SOTM antenna and the communication interruption of the SOTM system caused by a conventional method to a certain extent. The synthetic deviation correction method proposed in this paper is based on the low-precision attitude of the carrier in the high maneuvering state to correct the antenna azimuth and pitch angle, and large errors will still be introduced in the algorithm. In future research, a feasible scheme is to first correct the carrier attitude to obtain a higher precision carrier attitude. Based on the higher precision carrier attitude obtained from the correction, a better scheme may be obtained by synchronously adopting the synthetic deviation correction method.