Kinematics modeling analysis of the geostationary satellite monitoring antenna system

Nội dung text: Kinematics modeling analysis of the geostationary satellite monitoring antenna system

1. Science & Technology Development Journal – Engineering and Technology, 4(1):704-712 Open Access Full Text Article Research Article Kinematics modeling analysis of the geostationary satellite monitoring antenna system Quoc-Hoang Pham, Xuan-Hung Le, Manh-Tung Do, Tai-Hoai Thanh Nguyen, Hong-Phong Nguyen, Tien-Trung Vuong, Van-Tuan Pham, Xuan-Bien Duong* ABSTRACT The trend of scientific development in the future cannot fail to mention the great influence ofthe space field, but in the immediate future, the observational satellite systems are related tocommuni- Use your smartphone to scan this cation technology. In fact, in some countries with strong development of communication technol- QR code and download this article ogy 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 manufac- ture of geostationary satellite monitoring antenna systems. The problem of proactive grasping of technology, step by step autonomy in manufacturing technology of telecommunications equip- ment 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 suchas geostationary satellite monitoring antennas. This paper presents the kinematics modeling analysis of the mechanical system of the geosta- tionary 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 kine- matics 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 suc- Advanced Technology Center, Le Quy cessfully applied in practice, especially for dynamics and control system analysis of geostationary Don Technical University, 236 Hoang satellite antenna systems. Quoc Viet, Bac Tu Liem, Hanoi, Key words: geostationary satellite, antenna system, modeling, kinematics Vietnam Correspondence Xuan-Bien Duong, Advanced Technology supply from abroad, ensure national security, and on- Center, Le Quy Don Technical University, INTRODUCTION ward mastering the design and manufacturing tech- 236 Hoang Quoc Viet, Bac Tu Liem, Communication via satellite is the greatly important Hanoi, Vietnam research result from the combination of the two fields nology. Email: duongxuanbien@lqdtu.edu.vn of communication and space science1. The biggest Mathematical modelling and kinematics analysis are History advantage of this invention is its range of information the important steps in design and development of the • Received: 19-09-2020 antenna system for communications and monitoring • transmission and low cost. An indispensable part of Accepted: 05/03/2021 of geostationary satellites. The motion characteristics • Published: 15/03/2021 this satellite communication technology is the satellite antenna receiver and broadcasting system. Up to the and workspace of the antenna system are determined DOI : 10.32508/stdjet.v4i1.770 present time, space technology is still a specific sci- in details by solving the kinematics problem. A few ence that requires a very high level of scientific devel- works related to solving the kinematics and dynam- opment. Therefore, in the world, only a few developed ics problems of geostationary satellite antenna sys- countries are able to develop this field such as Russia, tems were documented 8–12. The basic construction Copyright USA, China, Japan 1–7. of a satellite antenna was described by Bindi 8. Basi- © VNU-HCM Press. This is an open- In Vietnam, autonomy in designing and manufactur- cally, the geostationary satellite antenna system con- access article distributed under the terms of the Creative Commons ing geostationary antenna systems based on the re- sists of a number of basic components which are base Attribution 4.0 International license. sources of domestic equipment is a necessary step to cluster, antenna shafts, and satellite pan cluster. The minimize the cost of applying satellite communica- kinematics and dynamics model was also proposed tion technology, reducing the depends on the level of 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. – En- gineering and Technology; 4(1):704-712. 704
2. Science & Technology Development Journal – Engineering and Technology, 4(1):704-712 error. A system of dynamic equations is built based it can be not considered in the kinematics problem. on the finite element method and Lagrange II system The center of the rotating cluster is G1, the satellite of equations. The mathematical model describing the pan assembly center is G2. The kinematics model of look angle of the geostationary satellite antenna men- the mechanical system can be constructed as shown tioned by Ogundele 9 is based on the use of two con- in Figure 2. trol station models. The adjustment model for the Select a fixed coordinate system (OXYZ)0 attached to satellite antenna viewing angle was proposed Ogun- the ground. The (OXYZ)i, (i=1 6) coordinate sys- dele 10. The working principles and classifications of tems are respectively mounted at the positions shown the satellite surveillance antenna can be found in the in Figure 5. The above fixed and local coordinate sys- report of Lida 11. The geostationary satellite antenna tems are attached for the purpose of accurately deter- system, called Horn Antenna, was designed and sim- mining the position of any point on the system. In 12 ulated by Shankar ; it was aimed to operate in the particular, taking point G2 is the end-effector point high-frequency regions. Researches related to the de- representing the satellite pan cluster and it is neces- tailed design and manufacture of geostationary satel- sary to determine the position of this point according lite antenna systems have not been well-documented, to the fixed coordinate system. 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 de- veloped, for determination of the workspace through the limited values of joints. The inverse kinematics problem is analyzed to determine the kinematics be- haviors of the antenna system with a given motion path in the workspace. The simulation results are ob- tained 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 mathe- matical model and the kinematics equations that show the relationship between the joint variables and the Figure 1: Preliminary mechanical system pan cluster center of mass position of the antenna sys- tem in the workspace are mentioned. Next, the posi- tion, velocity, acceleration, and workspace of the an- tenna system are calculated. Then, some numerical simulation results of the inverse kinematics in order to determine the values of the joint variables to en- sure the motion system according to a given path are described. The conclusion is the last part. MATERIALS AND METHODS Consider the preliminary designed geostationary satellite monitoring antenna model as shown in Fig- ure 1 and Figure 2. Accordingly, the mechanical sys- tem consists of two main parts which are the direc- tional cluster and the satellite pan cluster (Figure 1). The movement of the rotating cluster is done by the rotating joint q1 which is driven by motor 1. The satel- Figure 2: Kinematics diagram lite pan cluster height is performed by rotating joint q2. This joint is driven by motor 2 through the trans- lational joint A and rotational joint B. However, these Using the homogeneous matrix rotations and transla- 13,14 joints are only responsible for driving the joint q2, so tions according to the D-H method , the D-H pa- 705
3. Science & Technology Development Journal – Engineering and Technology, 4(1):704-712 Table 1: D-H parameters Parameters θi di ai αi O0O1 0 d0 0 0 O1O2 q1 d1 0 0 O2O3 0 0 a2 0 O3O4 0 d3 0 π/2 O4O5 q2 0 a4 −π/2 O5O6 0 d5 0 0 14 rameters tables can be described as shown in Table 1. system (OXYZ)0 through matrix transformation Use the local homogeneous matrix as can be determined as follows Hi =  Di = H1H2 Hi, i = 1 6 (4) cosθ −sinθ cosα cosθ sinα a cosθ [ ]  i i i i i i i θ θ α θ α θ A3×3 P3×1 sin i cos i sin i cos i sin i ai sin i  In(1) which, matrix D = represents the   i 0 1  0 sinαi cosθi di  0 0 0 1 position (P3×1) and direction (A3×3) of local coordi- nate systems relative to the fixed coordinate system. Matrix Hi gives us information about the position Specifically, the positions and directions of the coor- of the (OXYZ)i coordinate system compared to the dinate systems are as follows (OXYZ)i−1 coordinate system. Accordingly, the lo-   1 0 0 0 cal D-H matrices are presented as follows       0 1 0 0  1 0 0 0 D1 =  ;   0 0 1 d0   0 1 0 0  0 0 0 1 H1 =  ;   (5) 0 0 1 d0 cosq −sinq 0 a cosq  1 1 2 1 0 0 0 1  sinq cosq 0 a sinq  D =  1 1 2 1  cosq1 −sinq1 0 0 2     0 0 1 d0 + d1   sinq1 cosq1 0 0 0 0 0 1 H2 =  ; (2)  0 0 1 d1 And,  0 0 0 1   1 0 0 α2 cosq −sinq 0 0    1 1  0 1 0 0   q q    sin 1 cos 1 0 0  H3 =   D3 = ; 0 0 1 0  0 0 1 d0 + d1 0 0 0 1 0 0 0 1   (6) cosq 0 sinq a cosq And,  1 1 2 1     q − q a q  sin 1 0 cos 1 2 sin 1  1 0 0 0 D4 =  0 0 1 d0 + d1 + d3  −  0 0 1 0  0 0 0 1 H4 =  ; 0 0 0 d3 Following Figure 3 0 0 0 1  cosq 0 −sinq a cosq The position of G2 point is determined according to  2 2 4 2  q q a q  the fixed coordinate system as follows sin 2 0 cos 2 4 sin 2  H5 =  ; (3) 0 0 1 0 xG2 = cosq1 (a2 + a4 cosq2 − d5 sinq2)  0 0 0 1 yG2 = sinq1 (a2 + a4 sinq2 − d5 sinq2) (9) 1 0 0 0 z = d + d + d + a sinq + d cosq   G2 0 1 3 4 2 5 2 0 1 0 0  H =   Operate an inspection at a number of special loca- 6 0 0 1 d  5 tions. 0 0 0 1 Position 1: q1 = q2 = 0. This is the position where the From local D-H matrices, the position of the (OXYZ)i direction cluster is in a stationary position. The cen- coordinate system compared to the fixed coordinate terline passes through the basin of the pan parallel to 706
4. Science & Technology Development Journal – Engineering and Technology, 4(1):704-712 Figure 3: Equation (7) and (8) the axis (OX)0 and parallel to the ground. We deduce: xG2 = a2 +a4; yG2 = 0; zG2 = d0 +d1 +d3 +d5. This position is completely consistent with the kinematics model. π Position 2: q1 = 0,q2 = 2 . This is the position of the satellite pan cluster with the centerline of the pan in the direction of vertical (OZ)0. Now: xG2 = a2 − d5; yG2 = 0; zG2 = d0 + d1 + d3 + a4. 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 4: The possible workspace G2 Apply with expected geometric parameters of the an- tenna to design and manufacture as follows d0 = 3.62(m), d1 = 0.87(m), a2 = 0.39(m), d3 = 0.2(m), a4 = 1.15(m), d5 = 0.21(m). Limited joints: 0 ≤ q1 ≤ 2π, 0 ≤ q2 ≤ π/2. The possible workspace of point G2 is shown in Figure 5. The forward kinematics problem is described with in- put is the law of variables of joints q1,q2 and the out- put is the motion law of point G2 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 dia- gram in Figure 6. The given law of variables joints is q1 = πt/5,q2 = π/4. The coordinate zG2 is constant, the trajectory of G2 point is a circle parallel to plane (OXY)0. Figure 6: The law of variable joints 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 The inverse kinematics problem to Figure 10. The position of point G2 is shown in Figure 7. The maximum value along the (OZ)0 axis The inverse kinematics problem is described with is 5.68(m), the (OX)0 and (OY)0 reaches 1(m) . Fig- input as the desired path of the end-effector point ure 8 and Figure 9 show the velocity and acceleration G2(xG2,yG2,zG2) in the workspace and the output of point G2, respectively. The point path of the G2 as the law of the joints variables that satisfy the re- point in the (OXY)0 plane is shown in Figure 10. quired input trajectory. The inverse kinematics prob- 707
5. Science & Technology Development Journal – Engineering and Technology, 4(1):704-712 Figure 5: Forward kinematics calculation diagram Figure 7: Position of G2 point Figure 9: Acceleration of G2 point Figure 8: Velocity of G2 point Figure 10: The path of G2 on the (OXY)0 708
6. Science & Technology Development Journal – Engineering and Technology, 4(1):704-712 lem can be solved using either the analytical method The derivative of the two-sided derivative (12) respect or the numerical method. On the one hand, the an- to time alytic method allows us to use the kinematics equa- . . x(t) = J(q)q(t) + J (q)q(t) (16) tion transforms (9) to find q1,q2. However, the solv- ing process will encounter transcendent trigonomet- From (16) ric functions, so there are many satisfying results. The . . problem of choosing the right answer to the system J (q)q(t) = x(t) − J(q)q(t) (17) configuration is not a simple problem and takes a lot Put (15) into (17) of time. Sometimes this approach is not feasible for . . complex systems. On the other hand, the numeri- J (q)q(t) = x(t) − J(q)J+(q)x(t) (18) cal method uses modern algorithms to solve prob- lems according to the approximation method. The Then the joints acceleration vector can be written as . . outstanding advantage of this method is the feasibil- q(t) = J+ (q)x(t) − J+ (q)J (q)J+ (q)x(t) (19) ity and response to the requirements of the system configuration, which can solve complex problems that The velocity and acceleration vectors can be calculated the analytical method cannot meet. The limitations of from Eq. (15) and Eq. (19) if know q(t) at the time of . the numerical method are errors. However, with the investigation and x(t),x(t),x(t). strong development of computer science and mathe- Consider the desired motion of point G2 as follows matics, this problem is almost completely solved com- π xG2 = cos t (m); pared to the requirement of the problem. The algo- π4 rithm of adjusting the generalized vector 14–17 is ap- y = sin t (m); (20) G2 4 plied to solve this problem. zG2 = 5.68 (m) To facilitate the presentation of the generalized prob- lem, some vectors are defined as follows Operate the inverse kinematics problem in MATLAB [ ] T software, the results of this problem are described in x(t) = xG2 yG2 zG2 ; Figure 11 to Figure 16. The values of variables joints [ ] (10) T are shown in Figure 11. The joints errors are presented q(t) = q1 q2 qn in Figure 12. The velocity and acceleration of joints Relationship between coordinates of G2 point in the also described in Figure 13 and Figure 14, respec- workspace and joints coordinates in the joint space tively. The position of end-effector point G2 can be re- can be described as follows calculated through the forward kinematics equations with received q(t) is shown in Figure 15 with small x = f (q) (11) position error which is presented in Figure 16. The The derivative of the two-sided derivative (11) respect 3D model of antenna system (Figure 17) can be simu- to time lated in the MATLAB software by using the results of . ∂ f . . the inverse kinematics problem. x = q = J (q)q (12) ∂q where,   ∂ f1 ∂ f1 ∂ f1    ∂q ∂q ∂q  ∂ f  1 2 n  J (q) = ∂ =   (13) q  ∂ f ∂ f ∂ f  3 3 3 ∂q1 ∂q2 ∂qn 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 [ ]− J+ (q) = JT (q) J (q)JT (q) 1 (14) Then from the expression (14), the joints velocity is Figure 11: The positions of joints determined as . . q(t) = J+ (q)x(t) (15) 709
7. Science & Technology Development Journal – Engineering and Technology, 4(1):704-712 Figure 15: The position of G2 point Figure 12: The joints errors Figure 13: The joints velocities Figure 16: The position error of G2 Figure 17: The motion trajectory of point G2 in Figure 14: The joints accelerations MATLAB 710
8. Science & Technology Development Journal – Engineering and Technology, 4(1):704-712 The dynamics equation and control system problems DUONG implemented research methods, calculated has been built and solved based on the results of the the program and drafted the manuscript. kinematics modeling analysis above. Figure 18 shows REFERENCES the entire antenna system fabricated and is in the pro- 1. Maral G. Satellite communications systems - Systems, Tech- cess of assembly. The details of this issue will be pre- niques and Technology, Fifth Edition. John Wiley and Sons, sented in the next studies in the near future. Ltd Publication. 2009;. 2. Satcom and Antenna technologies division at www.cpii.com [Access: October 2020];. 3. VincorTM Product Data and Specification Archives: www.catal og.vincor.com [Access: October 2020];. 4. Hubble Space Telescope: www.nasa.gov [Access: October 2020];. 5. Featured SATCOM Products: www.satcom-services.com [Ac- cess: October 2020];. 6. Antenna Systems: www.viasat.com/products/antenna-syste ms [Access: October 2020];. 7. Satellite Communications equipment: www.digisat.org [Ac- cess: October 2020] ;. 8. Bindi Y, Dong L. Dynamics analysis and control of a spacecraft Figure 18: Geostationary satellite surveillance an- mechanism with joint clearance and thermal effect. Precision tenna was fabricated Motion Systems: Modeling, Control, and Applications, Else- vier Inc. 2019;p. 163–215. Available from: 1016/B978-0-12-818601-5.00017-2. 9. Ogundele DA, et al. Mathematical Modelling of Antenna Look Angles of Geostationary Communications Satellite Using Two Models of Control Stations, 3rd International Conference on CONCLUSIONS Advanced Computer Theor and Engineering (ICACTE). 2010;p. In conclusion, this paper presents the kinematics 236–240. 10. Ogundele DA, et al. Model validation and analysis of an- modeling of antenna systems for geostationary satel- tenna look angles of a geostationary satellite, International lite communications and monitoring, with the fo- Conference on Computer Science and Automation Engineer- cus on analyzing the kinematics problems. The ing (CSAE), IEEE. 2012;p. 509–513. Available from: org/10.1109/CSAE.2012.6272824. workspace, position, velocity and acceleration of the 11. Lida T. Satellite Communications Antenna Concepts and En- pan cluster center of mass are calculated by using the gineering, Handbook of Satellite Applications, Springer Sci- ence+Business Media New York. 2015;PMID: 25435485. Avail- limit value of joints and solving the forward kinemat- able from: ics problem. The rules of joints are determined en- 12. Shankar SG, Reddy KV. Design and Simulation of Horn An- suring the given trajectory of pan cluster center of tenna in x-Ku Band for Satellite Communications. Interna- tional Journal of Research in Science & Technology (IJRST). mass in the workspace through analyzing the inverse 2014;1(10). kinematics problem. The numerical simulation re- 13. Spong MW, Hutchinson S, Vidyasagar M. Robot modeling and sults kinematics were successfully applied in dynam- Control, First edition. New York, USA. 2001;. 14. Khang NV. Dynamics of Multi-bodies. Hanoi Science and Tech- ics and control analyzing and in practice. nology Publishing House. 2007;. 15. Khang NV, et al. Inverse kinematic and dynamic analysis CONFLICT OF INTEREST of redundant measuring manipulator BKHN-MCX-04. Vietnam Journal of Mechanics, VAST. 2010;32:15–26. Available from: All of authors have no conflict on interest in publish- ing of the paper 16. My CA, et al. Inverse Kinematics Analysis of Welding Robot IRB 1520ID Using Algorithm for Adjusting the Increments of AUTHOR CONTRIBUTION Generalized Vector. The 5th International Conference on Re- search in Intelligent and Computing in Engineering. Springer Quoc-Hoang PHAM proposed ideas, research Singapore. 2020;Available from: 981-15-7527-3. methods, quality monitoring and correcting the 17. Bien DX, et al. Optimize the Feed Rate and Determine the manuscript. Xuan-Hung LE, Manh-Tung DO, Joints Torque for Industrial Welding Robot TA 1400 Based Tai-Hoai Thanh NGUYEN, Hong-Phong NGUYEN, on Kinematics and Dynamics Modeling, International Journal of Mechanical Engineering and Robotics Research (IJMERR). Van-Tuan PHAM, and Tien-Trung VUONG de- 2020;9(9):1335–1340. Available from: signed and fabricated the system. Xuan-Bien 18178/ijmerr.9.9.1335-1340. 711
9. Tạp chí Phát triển Khoa học và Công nghệ – Kĩ thuật và Công nghệ, 4(1):704-712 Open Access Full Text Article Bài nghiên cứu Phân tích mô hình động học hệ thống Anten giám sát vệ tinh địa tĩnh Phạm Quốc Hoàng, Lê Xuân Hùng, Đỗ Mạnh Tùng, Nguyễn Tài Hoài Thanh, Nguyễn Hồng Phong, Vương Tiến Trung, Phạm Văn Tuân, Dương Xuân Biên* TÓM TẮT Xu hướng phát triển khoa học trong tương lai không thể không kể đến sức ảnh hưởng to lớn của lĩnh vực không gian vũ trụ, trước mắt là các hệ thống vệ tinh liên quan đến công nghệ viễn thông. Use your smartphone to scan this Trên thực tế, ở một số nước có nền công nghệ thông tin liên lạc và công nghệ vũ trụ phát triển QR code and download this article mạnh mẽ thì vấn đề thiết kế, chế tạo hệ thống cơ khí anten giám sát vệ tinh địa tĩnh chắc chắn đã được giải quyết triệt để. Tuy nhiên, vì là một công nghệ đặc thù nên việc chia sẻ và chuyển giao công nghệ thiết kế và chế tạo cho các nước đang phát triển không hề dễ dàng. Hầu như rất khó tìm thấy các công trình đã công bố liên quan đến tính toán thiết kế cơ khí và chế tạo hệ thống anten giám sát vệ tinh địa tĩnh. Vấn đề chủ động nắm bắt công nghệ, từng bước tự chủ công nghệ chế tạo thiết bị viễn thông liên quan đến công nghệ vũ trụ luôn là mục tiêu của các nước đang phát triển như Việt Nam quan tâm và hướng tới nhằm hạn chế sự phụ thuộc công nghệ, giảm thiểu chi phí trang bị, chuyển giao công nghệ, đảm bảo bí mật quốc gia. Bước đầu tiên trong những vấn đề này là việc chủ động tự nghiên cứu thiết kế và chế tạo các thiết bị thu phát trên mặt đất như anten giám sát vệ tinh địa tĩnh. Bài báo này trình bày việc phân tích mô hình động học cho hệ thống anten giám sát vệ tinh địa tĩnh. Mỗi thành phần của hệ thống anten được giả thiết là vật rắn tuyệt đối. Mô hình toán học được xây dựng dựa trên lý thuyết động học và động lực học hệ nhiều vật. Phương pháp ma trận chuyển đổi thuần nhất DENAVIT-HARTENBERG (D-H) được sử dụng để xây dựng các phương trình động học. Bài toán động học thuận được phân tích để xác định vị trí, vận tốc, gia tốc và không gian làm việc của hệ thống anten với các giới hạn chuyển động góc của hệ thống đã cho. Bài toán động học ngược được đề cập để xác định các ứng xử động học của hệ thống anten với quỹ đạo chuyển động cho trước trong không gian làm việc. Kết quả tính toán và mô phỏng số động học đã được ứng dụng thành công trong thực tế, đặc biệt là ứng dụng để phân tích bài toán động lực học và bài toán điều khiển cho hệ thống anten vệ tinh địa tĩnh. Trung tâm Công nghệ, Đại học Kỹ thuật Từ khoá: Vệ tinh địa tĩnh, hệ thống anten, mô hình hóa, động học Lê Quý Đôn, 236 Hoàng Quốc Việt, Bắc Từ Liêm, Hà Nội, Việt Nam Liên hệ Dương Xuân Biên, Trung tâm Công nghệ, Đại học Kỹ thuật Lê Quý Đôn, 236 Hoàng Quốc Việt, Bắc Từ Liêm, Hà Nội, Việt Nam Email: duongxuanbien@lqdtu.edu.vn Lịch sử • Ngày nhận: 19-09-2020 • Ngày chấp nhận: 05/03/2021 • Ngày đăng: 15/03/2021 DOI : 10.32508/stdjet.v4i1.770 Bản quyền © ĐHQG Tp.HCM. Đây là bài báo công bố mở được phát hành theo các điều khoản của the Creative Commons Attribution 4.0 International license. Trích dẫn bài báo này: Hoàng P Q, Hùng L X, Tùng D M, Thanh N T H, Phong N H, Trung V T, Tuân P V, Biên D X. Phân tích mô hình động học hệ thống Anten giám sát vệ tinh địa tĩnh. Sci. Tech. Dev. J. - Eng. Tech.; 4(1):704-712. 712