Kinematics modeling analysis of the geostationary satellite monitoring antenna system

The trend of scientific development in the future cannot fail to mention the great influence of the space field, but in the immediate future, the observational satellite systems are related to communication technology. In fact, in some countries with strong development of communication technology and space technology, the mechanical system of geostationary satellite monitoring antennas has certainly been thoroughly resolved. However, because of a specific technology, the sharing and transferring of design and manufacturing technology to developing countries is a great challenge. It is almost difficult to find published works related to mechanical design calculation and manufacture of geostationary satellite monitoring antenna systems. The problem of proactive grasping of technology, step by step autonomy in manufacturing technology of telecommunications equipment related to space technology has always been the goal of developing countries like Vietnam to limit technology dependence, minimizing technology transfer costs, ensuring national security. The first step in these problems is the autonomous construction of terrestrial transceivers such as geostationary satellite monitoring antennas. This paper presents the kinematics modeling analysis of the mechanical system of the geostationary satellite monitoring antenna. Each component of the antenna system is assumed a rigid body. The mathematical model is built based on multi-bodies kinematics and dynamics theory. The DENAVIT-HARTENBERG (D-H) homogeneous matrix method was used to construct the kinematics equations. The forward kinematics problem is analyzed to determine the position, velocity, acceleration, and workspace of the antenna system with given system motion limits. The inverse kinematics problem is mentioned to determine the kinematics behaviors of the antenna system with a given motion path in the workspace. The numerical simulation results kinematics were successfully applied in practice, especially for dynamics and control system analysis of geostationary satellite antenna systems.


INTRODUCTION
Communication via satellite is the greatly important research result from the combination of the two fields of communication and space science 1 . The biggest advantage of this invention is its range of information transmission and low cost. An indispensable part of this satellite communication technology is the satellite antenna receiver and broadcasting system. Up to the present time, space technology is still a specific science that requires a very high level of scientific development. Therefore, in the world, only a few developed countries are able to develop this field such as Russia, USA, China, Japan 1-7 . In Vietnam, autonomy in designing and manufacturing geostationary antenna systems based on the resources of domestic equipment is a necessary step to minimize the cost of applying satellite communication technology, reducing the depends on the level of supply from abroad, ensure national security, and onward mastering the design and manufacturing technology. Mathematical modelling and kinematics analysis are the important steps in design and development of the antenna system for communications and monitoring of geostationary satellites. The motion characteristics and workspace of the antenna system are determined in details by solving the kinematics problem. A few works related to solving the kinematics and dynamics problems of geostationary satellite antenna systems were documented [8][9][10][11][12] . The basic construction of a satellite antenna was described by Bindi 8 . Basically, the geostationary satellite antenna system consists of a number of basic components which are base cluster, antenna shafts, and satellite pan cluster. The kinematics and dynamics model was also proposed and analyzed under the influence of heat and joint Cite this article : Pham Q, Le X, Do M, Nguyen T T, Nguyen H, Vuong T, Pham V, Duong X. Kinematics modeling analysis of the geostationary satellite monitoring antenna system. Sci. Tech. Dev. J. -Engineering and Technology; 4(1):705-713. error. A system of dynamic equations is built based on the finite element method and Lagrange II system of equations. The mathematical model describing the look angle of the geostationary satellite antenna mentioned by Ogundele 9 is based on the use of two control station models. The adjustment model for the satellite antenna viewing angle was proposed Ogundele 10 . The working principles and classifications of the satellite surveillance antenna can be found in the report of Lida 11 . The geostationary satellite antenna system, called Horn Antenna, was designed and simulated by Shankar 12 ; it was aimed to operate in the high-frequency regions. Researches related to the detailed design and manufacture of geostationary satellite antenna systems have not been well-documented, due to the security issues and design copyrights. This paper presents the kinematics modeling analysis of geostationary satellite antenna systems. The system of kinematics equations of the antenna system is developed, for determination of the workspace through the limited values of joints. The inverse kinematics problem is analyzed to determine the kinematics behaviors of the antenna system with a given motion path in the workspace. The simulation results are obtained based on the numerical calculation methods. The results of this study have important meaning for the dynamic analyzing and the controller designing of the antenna system. The rest of the paper is presented as follows. Firstly, the materials and methods are presented. The mathematical model and the kinematics equations that show the relationship between the joint variables and the pan cluster center of mass position of the antenna system in the workspace are mentioned. Next, the position, velocity, acceleration, and workspace of the antenna system are calculated. Then, some numerical simulation results of the inverse kinematics in order to determine the values of the joint variables to ensure the motion system according to a given path are described. The conclusion is the last part.  Figure 2. Accordingly, the mechanical system consists of two main parts which are the directional cluster and the satellite pan cluster (Figure 1). The movement of the rotating cluster is done by the rotating joint q 1 which is driven by motor 1. The satellite pan cluster height is performed by rotating joint q 2 . This joint is driven by motor 2 through the translational joint A and rotational joint B. However, these joints are only responsible for driving the joint q 2 , so it can be not considered in the kinematics problem. The center of the rotating cluster is G 1 , the satellite pan assembly center is G 2 . The kinematics model of the mechanical system can be constructed as shown in Figure 2. Select a fixed coordinate system (OXY Z) 0 attached to the ground. The (OXY Z) i , (i=1..6) coordinate systems are respectively mounted at the positions shown in Figure 5. The above fixed and local coordinate systems are attached for the purpose of accurately determining the position of any point on the system. In particular, taking point G 2 is the end-effector point representing the satellite pan cluster and it is necessary to determine the position of this point according to the fixed coordinate system.

MATERIALS AND METHODS
rameters tables can be described as shown in Table 1.
Use the local homogeneous matrix as Matrix H i gives us information about the position of the (OXY Z) i coordinate system compared to the (OXY Z) i−1 coordinate system. Accordingly, the local D-H matrices are presented as follows And, From local D-H matrices, the position of the (OXY Z) i coordinate system compared to the fixed coordinate system (OXY Z) 0 through matrix transformation 14 can be determined as follows represents the position (P 3×1 ) and direction (A 3×3 ) of local coordinate systems relative to the fixed coordinate system. Specifically, the positions and directions of the coordinate systems are as follows And, Following Figure 3 The position of G 2 point is determined according to the fixed coordinate system as follows Operate an inspection at a number of special locations. Position 1: q 1 = q 2 = 0. This is the position where the direction cluster is in a stationary position. The centerline passes through the basin of the pan parallel to  (7) and (8) the axis (OX) 0 and parallel to the ground. We deduce: This position is completely consistent with the kinematics model. Position 2: q 1 = 0, q 2 = π 2 . This is the position of the satellite pan cluster with the centerline of the pan in the direction of vertical (OZ) 0 . Now: This position is also completely suitable for the kinematics model. Thus, the results of the modeling of the system ensure reliability and accuracy.

RESULTS AND DISCUSSION
The forward kinematics problem  Figure 5. The forward kinematics problem is described with input is the law of variables of joints q 1 , q 2 and the output is the motion law of point G 2 or any point on the system in workspace including position, velocity, and acceleration. this problem can be solved by using the MATLAB calculation software according to the diagram in Figure 6. The given law of variables joints is q 1 = πt/5, q 2 = π/4. The coordinate z G2 is constant, the trajectory of G 2 point is a circle parallel to plane (OXY ) 0 . The input of the forward kinematics problem is the law of the joint variables and is shown in Figure 6. The results of this problem analysis are shown in Figure 7 to

The inverse kinematics problem
The inverse kinematics problem is described with input as the desired path of the end-effector point G 2 (x G2 , y G2 , z G2 ) in the workspace and the output as the law of the joints variables that satisfy the required input trajectory. The inverse kinematics prob-     lem can be solved using either the analytical method or the numerical method. On the one hand, the analytic method allows us to use the kinematics equation transforms (9) to find q 1 , q 2 . However, the solving process will encounter transcendent trigonometric functions, so there are many satisfying results. The problem of choosing the right answer to the system configuration is not a simple problem and takes a lot of time. Sometimes this approach is not feasible for complex systems. On the other hand, the numerical method uses modern algorithms to solve problems according to the approximation method. The outstanding advantage of this method is the feasibility and response to the requirements of the system configuration, which can solve complex problems that the analytical method cannot meet. The limitations of the numerical method are errors. However, with the strong development of computer science and mathematics, this problem is almost completely solved compared to the requirement of the problem. The algorithm of adjusting the generalized vector 14-17 is applied to solve this problem. To facilitate the presentation of the generalized problem, some vectors are defined as follows Relationship between coordinates of G 2 point in the workspace and joints coordinates in the joint space can be described as follows The derivative of the two-sided derivative (11) respect to time where, The matrix J(q) of size 3xn is called a Jacobi matrix. For the redundant system, it is common to choose the pseudo-inverse matrix of the rectangular matrix J(q) as Then from the expression (14), the joints velocity is determined as The derivative of the two-sided derivative (12) respect to time ..
Then the joints acceleration vector can be written as .. ..
The velocity and acceleration vectors can be calculated from Eq. (15) and Eq. (19) if know q(t) at the time of investigation and x (t) , .
x (t). Consider the desired motion of point G 2 as follows Operate the inverse kinematics problem in MATLAB software, the results of this problem are described in Figure 11 to Figure 16. The values of variables joints are shown in Figure 11. The joints errors are presented in Figure 12. The velocity and acceleration of joints also described in Figure 13 and Figure 14, respectively. The position of end-effector point G 2 can be recalculated through the forward kinematics equations with received q(t) is shown in Figure 15 with small position error which is presented in Figure 16. The 3D model of antenna system ( Figure 17) can be simulated in the MATLAB software by using the results of the inverse kinematics problem. The dynamics equation and control system problems has been built and solved based on the results of the kinematics modeling analysis above. Figure 18 shows the entire antenna system fabricated and is in the process of assembly. The details of this issue will be presented in the next studies in the near future.

CONCLUSIONS
In conclusion, this paper presents the kinematics modeling of antenna systems for geostationary satellite communications and monitoring, with the focus on analyzing the kinematics problems. The workspace, position, velocity and acceleration of the pan cluster center of mass are calculated by using the limit value of joints and solving the forward kinematics problem. The rules of joints are determined ensuring the given trajectory of pan cluster center of mass in the workspace through analyzing the inverse kinematics problem. The numerical simulation results kinematics were successfully applied in dynamics and control analyzing and in practice.