Mathematical model of stable equilibrium operation of the flight simulator based on the Stewart platform
Abstract
This paper analyses the available mathematical models of flight simulators based on the Stewart platform. It was found that there is no model that describes the conditions for stable dynamic equilibrium operation of the Stewart platform as a function of a number of important motion parameters. In this context, a new physical model is proposed based on classical models of theoretical mechanics using the d’Alembert formalism, the concept of stable equilibrium of a mechanical system. This model mathematically separates the stable equilibrium of the flight simulator motion system from the general uniformly accelerated motion. The systems of equations obtained in the framework of the model connect the physical and geometrical parameters of the Stewart platform and make it possible to determine the reactions in the upper hinges of the platform support, the limit values of the position angles in the space of the base of the support of the Stewart platform, under which the condition of stable equilibrium operation of the Stewart platform is satisfied. The proposed physical model and the analytical relations obtained on its basis are of great practical importance: the operator controlling the operation of the Stewart platform-based flight simulator can control the range of parameters during training so as not to bring the flight simulator out of stable equilibrium.
Keyword : algorithm of stable operation, limiting angles, equilibrium motion, Stewart platform, analytical solution
How to Cite
Lukianov, P. V., & Kabanyachyi, V. V. (2023). Mathematical model of stable equilibrium operation of the flight simulator based on the Stewart platform. Aviation, 27(2), 119–128. https://doi.org/10.3846/aviation.2023.19264
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
Abramov, N. V., & Mukharlamov, R. G. (2011). Modeling of control processes, stability and stabilization of systems with program constraints. Vesnik of Samara University, 17(2), 130–140. https://doi.org/10.18287/2541-7525-2011-17-2-130-140
Ahmad, M., Hussain, Z. L., Shah, S. I. A., & Shams, T. A. (2021). Estimation of stability parameters for wide body aircraft using computational techniques. Applied Sciences, 11(5), 2087. https://doi.org/10.3390/app11052087
Ahmed, U. (2012). 3-DOF longitudial flight simulation modeling and design using MATHLAB/SIMULINK. Ryerson University Digital Commons.
Andrijevskij, B. R., Arseniev, D. G., Zegzhda, S. A., Kazunin, D. V., Kuznetsov, N. V., Leonov, G. A., Tovstik, P. Ye., & Yushkov, M. P. (2017). Dynamics of the Stewart platform. Vestnik of St Petersburg University. Mathematics. Mechanics. Astronomy, 4(62), 489–505. https://doi.org/10.21638/11701/spbu01.2017.311
Barbashin, E. N. (1970). Funktsiji Lyapunova. Nauka.
Bingul, Z., & Karahan, O. (2012). Dynamic modeling and simulation of Stewart platform. In Serial and parallel robot manipulators – kinematics, dynamics, control and optimization. IntechOpen. https://doi.org/10.5772/32470
Chen, S.-H., & Fu, L.-Ch. (2013). Output feedback sliding mode control for a Stewart platform with a nonlinear observer-based forward kinematics solution. IEEE Transactions on Control Systems Technology, 21(1), 176–185. https://doi.org/10.1109/TCST.2011.2171964
Chandrasekaran, K., Theningaledathill, V., & Hebbar, A. (2021). Ground based variable stability flight simulator. Aviation, 25(1), 22–34. https://doi.org/10.3846/aviation.2021.13564
Das, M. T., & Kumpas, I. (2019). Mathematical modeling and simulation of full flight helicopter simulator. UMAGD, 11(1), 135–140.
Dasgupta, B., & Mruthyunjava, T. S. (1996). A constructive predictor-corrector algorithm for the direct position kinematics problem for a general 6-6 Stewart platform. Mechanism and Machine Theory, 31(6), 799–811. https://doi.org/10.1016/0094-114X(95)00106-9
Dymarek, A., Dzitkowski, T., Herbuś, K., Kost, G., & Ociepka, P. (2014). Geometric analysis of motions exercised by the Stewart platform. Advanced Materials Research, 837, 351–356. https://doi.org/10.4028/www.scientific.net/AMR.837.351
Fichter, E. F. (1986). A Stewart platform based manipulator: General theory and practical construction. The International Journal of Robotics Research, 5(2), 157–182. https://doi.org/10.1177/027836498600500216
Gorbatenko, S. A., Makashov, E. M., Polushkin, Yu. F., & Sheptel L. V. E. (1969). Mehanika polijota. Obshije svedenija. Uravnenija dvizhenija. Mashinostrojenie.
Harib, K., & Srinivasan, K. (2003). Kinematic and dynamic analysis of Stewart platform-based machine tool structures. Robotica, 21(5), 541–554. https://doi.org/10.1017/S0263574703005046
He, Z., Feng, X., Zhu, Y., Yu, Zh., Li, Zh., Zhang, Y., Wang, Y., Wang, P., & Zhao, L. (2022). Progress of Stewart vibration platform in aerospace micro–vibration control (Review). Aerospace, 9(6), 1–20. https://doi.org/10.3390/aerospace9060324
Heintzman, R. J. (1996). Determination of force cueing requirements for tactical combat flight training devices (Tech. Rep. No. ASC-TR-97-5001). Training Systems Product Group.
Huang, H., He, W., Wang, J., Zhang, L., & Fu, Q. (2022). An all servo-driven bird-like flapping-wing aerial robot capable of autonomous flight. IEEE/ASME Transactions on Mechatronics, 27(6), 1–11. https://doi.org/10.1109/TMECH.2022.3175377
International Civil Аviation Оrganization. (2015). Manual of criteria for the qualification of flight simulation training devices (Doc. 9625 AN/938). ICAO.
Lapiska, C., Ross, L., & Smart, D. (1993). Flight simulation: An overview. Aerospace America, 31(8), 14–33.
Leonov, G. A., Zegzhda, S. A., Zuev, S. M., Ershov, B. A., Kazunin, D. V., Kostygova, D. M., Kuznetsov N. V., Tovstik, P. E., Liu, K., Fitzgerald, J. M., & Lewis, F. L. (1993). Kinematic analysis of a Stewart platform manipulator. IEEE Transactions of Industrial Electronics, 40(2), 282–293. https://doi.org/10.1109/41.222651
Lopes, A. M. (2009). Dynamic modeling of a Stewart platform using the generalized momentum approach. Communications in Nonlinear Science and Numerical Simulation, 14(8), 3389–3401. https://doi.org/10.1016/j.cnsns.2009.01.001
Malkin, I. G. (1966). Teorija ustojchivosti dvizhenija. Nauka.
Markou, A. A., Elmas, S., Filz, G. H. (2021). Revisiting Stewart–Gough platform applications: A kinematic pavilion. Engineering Structures, 249, 113304. https://doi.org/10.1016/j.engstruct.2021.113304
Nanua, P., Waldroon, K. J., & Murthy, V. (1989). Direct kinematic solution of Stewart platform. IEEE Transactions on Robotics and Automation, 6(4), 438–444. https://doi.org/10.1109/ROBOT.1989.100025
Nguyen, C. C., Zhou, Z.-L., Antrazi, S. S., & Campbell, C. E. (1991). Efficient computation of forward kinematics and Jacobian matrix of a Stewart platform-based manipulator. In IEEE Proceedings of the Southaston’91. IEEE. https://doi.org/10.1109/SECON.1991.147884
Ono, T., Eto, R., Yamakawa, J., & Murakami, H. (2022). Analysis and control of a Stewart platform as base motion compensators – Part I: Kinematics using moving frames. Nonlinear Dynamics, 107, 51–76. https://doi.org/10.1007/s11071-021-06767-8
Pausder, H. J., Bouwer, G., von Grunhagen, W., & Holland, R. (1992). Helicopter in-flight simulator ATTHeS: A multi-purpose testbed and its utilization. American Institute of Aeronautics and Astronautics. https://doi.org/10.2514/6.1992-4173
PPO “ERA” (1983). Imitator peregrusok KTS Il-62M3. Eskizno – tekhnichesky project. Shifr 415.40.577. (in Russian).
Petrescu, R. V., Aversa, R., Apicella, A., Kozaitis, S. P., Abu-Lebdeh, T., & Petrescu, F. I. T. (2018). Inverse kinematics of a Stewart platform. Journal of Mechatronics and Robotics, 2(1), 45–59. https://doi.org/10.3844/jmrsp.2018.45.59
Sapunov, Y. A., & Proshin, I. A. (2011). Modelirovanije privoda dinamisheskoho stenda aviacionnogo trenazhera. Aviatseonno-kosmicheskoe mashinostroenie. Izvestija Samarskoho nauchnoho centra Rosijskoi akademiji nauk, 13, 1(2), 337–340.
Scholten, P. A., van Passen, M. M., Chu, Q. P., & Mulder, M. (2020). Variable stability in-flight simulation system based on existing hardware. JGCD, 43(12), 1–14. https://doi.org/10.2514/1.G005066
Sheng, L., Wan-long, L., Yan-chun, D., & Liang, F. (2006, June 25–28). Forward kinematics of the Stewart platform using hybrid immune genetic algorithm. In Proceedings of the 2006 IEEE International Conference on Mechatronics and Automation (pp. 2330–2335). Luoyang, China. https://doi.org/10.1109/ICMA.2006.257695
Silva, D., Garrido, J., & Riveiro, E. (2022). Stewart platform motion control automation with industrial resources to perform cycloidal and oceanic wave trajectories. Machines, 10(8), 711. https://doi.org/10.3390/machines10080711
Simulator. (n.d.). Kompleksny trenazher ekipazhu litaka L-39. http://simulator.ua/ru/simulators.html
Stewart, D. (1965). A platform with six degrees of freedom. Aircraft Engineering, 38(4), 30–35. https://doi.org/10.1108/eb034141
Šegota, S. B., Andelič, N., Lorencin, I., Saga, M., & Car, Z. (2020). Path planning optimization of six-degree-of-freedom robotic manipulators using evolutionary algorithms. International Journal of Advanced Robotic Systems, (March-April 2020), 1–16. https://doi.org/10.1177/1729881420908076
Tovstik, T. P., & Yushkov, M. P. (2014). Dynamics and control of the Stewart platform. Doklady Akademii Nauk, 458(1), 36–41.
Velasco, J., Calvo, I., Barambones, O., Venegas, P., & Napole, C. (2020). Experimental validation of a sliding mode control for a Stewart platform used in aerospace inspection applications. Mathematics, 8(11), 2051. https://doi.org/10.3390/math8112051
Wang, Zh., He, J., Shang, H., & Gu, H. (2009). Forward kinematics analysis of a six-DOF Stewart platform using PCA and NM algorithm. Industrial Robot, 36(5), 448–460. https://doi.org/10.1108/01439910910980178
Yang, F., Tan, X., Wang, Z., Lu, Zh., & He, T. (2022). A geometric approach for real-time forward kinematics of the general Stewart platform. Sensors, 22(13), 4829. https://doi.org/10.3390/s22134829
Yee, Ch. s., & Lim, K.-b. (1997). Forward kinematics solution of Stewart platform using neural networks. Neurocomputing, 16, 333–349. https://doi.org/10.1016/S0925-2312(97)00048-9
Zhuang, H., Yan, J., & Masory, O. (1998). Calibration of Stewart platforms and other parallel manipulators by minimizing inverse kinematic residuals. Journal of Robotic Systems, 15(7), 395–405. https://doi.org/10.1002/(SICI)1097-4563(199807)15:7<395::AID-ROB2>3.0.CO;2-H
Zhiyong, T., Ma, H., Pei, Zh., Liu, L., & Zhang, J. (2016, July 27–29). A new numerical method for Stewart platform forward kinematics. In The Proceedings of the 35th Chinese Control Conference (CCC) (pp. 6311–6316). Chengdu, China. https://doi.org/10.1109/ChiCC.2016.7554348
Ahmad, M., Hussain, Z. L., Shah, S. I. A., & Shams, T. A. (2021). Estimation of stability parameters for wide body aircraft using computational techniques. Applied Sciences, 11(5), 2087. https://doi.org/10.3390/app11052087
Ahmed, U. (2012). 3-DOF longitudial flight simulation modeling and design using MATHLAB/SIMULINK. Ryerson University Digital Commons.
Andrijevskij, B. R., Arseniev, D. G., Zegzhda, S. A., Kazunin, D. V., Kuznetsov, N. V., Leonov, G. A., Tovstik, P. Ye., & Yushkov, M. P. (2017). Dynamics of the Stewart platform. Vestnik of St Petersburg University. Mathematics. Mechanics. Astronomy, 4(62), 489–505. https://doi.org/10.21638/11701/spbu01.2017.311
Barbashin, E. N. (1970). Funktsiji Lyapunova. Nauka.
Bingul, Z., & Karahan, O. (2012). Dynamic modeling and simulation of Stewart platform. In Serial and parallel robot manipulators – kinematics, dynamics, control and optimization. IntechOpen. https://doi.org/10.5772/32470
Chen, S.-H., & Fu, L.-Ch. (2013). Output feedback sliding mode control for a Stewart platform with a nonlinear observer-based forward kinematics solution. IEEE Transactions on Control Systems Technology, 21(1), 176–185. https://doi.org/10.1109/TCST.2011.2171964
Chandrasekaran, K., Theningaledathill, V., & Hebbar, A. (2021). Ground based variable stability flight simulator. Aviation, 25(1), 22–34. https://doi.org/10.3846/aviation.2021.13564
Das, M. T., & Kumpas, I. (2019). Mathematical modeling and simulation of full flight helicopter simulator. UMAGD, 11(1), 135–140.
Dasgupta, B., & Mruthyunjava, T. S. (1996). A constructive predictor-corrector algorithm for the direct position kinematics problem for a general 6-6 Stewart platform. Mechanism and Machine Theory, 31(6), 799–811. https://doi.org/10.1016/0094-114X(95)00106-9
Dymarek, A., Dzitkowski, T., Herbuś, K., Kost, G., & Ociepka, P. (2014). Geometric analysis of motions exercised by the Stewart platform. Advanced Materials Research, 837, 351–356. https://doi.org/10.4028/www.scientific.net/AMR.837.351
Fichter, E. F. (1986). A Stewart platform based manipulator: General theory and practical construction. The International Journal of Robotics Research, 5(2), 157–182. https://doi.org/10.1177/027836498600500216
Gorbatenko, S. A., Makashov, E. M., Polushkin, Yu. F., & Sheptel L. V. E. (1969). Mehanika polijota. Obshije svedenija. Uravnenija dvizhenija. Mashinostrojenie.
Harib, K., & Srinivasan, K. (2003). Kinematic and dynamic analysis of Stewart platform-based machine tool structures. Robotica, 21(5), 541–554. https://doi.org/10.1017/S0263574703005046
He, Z., Feng, X., Zhu, Y., Yu, Zh., Li, Zh., Zhang, Y., Wang, Y., Wang, P., & Zhao, L. (2022). Progress of Stewart vibration platform in aerospace micro–vibration control (Review). Aerospace, 9(6), 1–20. https://doi.org/10.3390/aerospace9060324
Heintzman, R. J. (1996). Determination of force cueing requirements for tactical combat flight training devices (Tech. Rep. No. ASC-TR-97-5001). Training Systems Product Group.
Huang, H., He, W., Wang, J., Zhang, L., & Fu, Q. (2022). An all servo-driven bird-like flapping-wing aerial robot capable of autonomous flight. IEEE/ASME Transactions on Mechatronics, 27(6), 1–11. https://doi.org/10.1109/TMECH.2022.3175377
International Civil Аviation Оrganization. (2015). Manual of criteria for the qualification of flight simulation training devices (Doc. 9625 AN/938). ICAO.
Lapiska, C., Ross, L., & Smart, D. (1993). Flight simulation: An overview. Aerospace America, 31(8), 14–33.
Leonov, G. A., Zegzhda, S. A., Zuev, S. M., Ershov, B. A., Kazunin, D. V., Kostygova, D. M., Kuznetsov N. V., Tovstik, P. E., Liu, K., Fitzgerald, J. M., & Lewis, F. L. (1993). Kinematic analysis of a Stewart platform manipulator. IEEE Transactions of Industrial Electronics, 40(2), 282–293. https://doi.org/10.1109/41.222651
Lopes, A. M. (2009). Dynamic modeling of a Stewart platform using the generalized momentum approach. Communications in Nonlinear Science and Numerical Simulation, 14(8), 3389–3401. https://doi.org/10.1016/j.cnsns.2009.01.001
Malkin, I. G. (1966). Teorija ustojchivosti dvizhenija. Nauka.
Markou, A. A., Elmas, S., Filz, G. H. (2021). Revisiting Stewart–Gough platform applications: A kinematic pavilion. Engineering Structures, 249, 113304. https://doi.org/10.1016/j.engstruct.2021.113304
Nanua, P., Waldroon, K. J., & Murthy, V. (1989). Direct kinematic solution of Stewart platform. IEEE Transactions on Robotics and Automation, 6(4), 438–444. https://doi.org/10.1109/ROBOT.1989.100025
Nguyen, C. C., Zhou, Z.-L., Antrazi, S. S., & Campbell, C. E. (1991). Efficient computation of forward kinematics and Jacobian matrix of a Stewart platform-based manipulator. In IEEE Proceedings of the Southaston’91. IEEE. https://doi.org/10.1109/SECON.1991.147884
Ono, T., Eto, R., Yamakawa, J., & Murakami, H. (2022). Analysis and control of a Stewart platform as base motion compensators – Part I: Kinematics using moving frames. Nonlinear Dynamics, 107, 51–76. https://doi.org/10.1007/s11071-021-06767-8
Pausder, H. J., Bouwer, G., von Grunhagen, W., & Holland, R. (1992). Helicopter in-flight simulator ATTHeS: A multi-purpose testbed and its utilization. American Institute of Aeronautics and Astronautics. https://doi.org/10.2514/6.1992-4173
PPO “ERA” (1983). Imitator peregrusok KTS Il-62M3. Eskizno – tekhnichesky project. Shifr 415.40.577. (in Russian).
Petrescu, R. V., Aversa, R., Apicella, A., Kozaitis, S. P., Abu-Lebdeh, T., & Petrescu, F. I. T. (2018). Inverse kinematics of a Stewart platform. Journal of Mechatronics and Robotics, 2(1), 45–59. https://doi.org/10.3844/jmrsp.2018.45.59
Sapunov, Y. A., & Proshin, I. A. (2011). Modelirovanije privoda dinamisheskoho stenda aviacionnogo trenazhera. Aviatseonno-kosmicheskoe mashinostroenie. Izvestija Samarskoho nauchnoho centra Rosijskoi akademiji nauk, 13, 1(2), 337–340.
Scholten, P. A., van Passen, M. M., Chu, Q. P., & Mulder, M. (2020). Variable stability in-flight simulation system based on existing hardware. JGCD, 43(12), 1–14. https://doi.org/10.2514/1.G005066
Sheng, L., Wan-long, L., Yan-chun, D., & Liang, F. (2006, June 25–28). Forward kinematics of the Stewart platform using hybrid immune genetic algorithm. In Proceedings of the 2006 IEEE International Conference on Mechatronics and Automation (pp. 2330–2335). Luoyang, China. https://doi.org/10.1109/ICMA.2006.257695
Silva, D., Garrido, J., & Riveiro, E. (2022). Stewart platform motion control automation with industrial resources to perform cycloidal and oceanic wave trajectories. Machines, 10(8), 711. https://doi.org/10.3390/machines10080711
Simulator. (n.d.). Kompleksny trenazher ekipazhu litaka L-39. http://simulator.ua/ru/simulators.html
Stewart, D. (1965). A platform with six degrees of freedom. Aircraft Engineering, 38(4), 30–35. https://doi.org/10.1108/eb034141
Šegota, S. B., Andelič, N., Lorencin, I., Saga, M., & Car, Z. (2020). Path planning optimization of six-degree-of-freedom robotic manipulators using evolutionary algorithms. International Journal of Advanced Robotic Systems, (March-April 2020), 1–16. https://doi.org/10.1177/1729881420908076
Tovstik, T. P., & Yushkov, M. P. (2014). Dynamics and control of the Stewart platform. Doklady Akademii Nauk, 458(1), 36–41.
Velasco, J., Calvo, I., Barambones, O., Venegas, P., & Napole, C. (2020). Experimental validation of a sliding mode control for a Stewart platform used in aerospace inspection applications. Mathematics, 8(11), 2051. https://doi.org/10.3390/math8112051
Wang, Zh., He, J., Shang, H., & Gu, H. (2009). Forward kinematics analysis of a six-DOF Stewart platform using PCA and NM algorithm. Industrial Robot, 36(5), 448–460. https://doi.org/10.1108/01439910910980178
Yang, F., Tan, X., Wang, Z., Lu, Zh., & He, T. (2022). A geometric approach for real-time forward kinematics of the general Stewart platform. Sensors, 22(13), 4829. https://doi.org/10.3390/s22134829
Yee, Ch. s., & Lim, K.-b. (1997). Forward kinematics solution of Stewart platform using neural networks. Neurocomputing, 16, 333–349. https://doi.org/10.1016/S0925-2312(97)00048-9
Zhuang, H., Yan, J., & Masory, O. (1998). Calibration of Stewart platforms and other parallel manipulators by minimizing inverse kinematic residuals. Journal of Robotic Systems, 15(7), 395–405. https://doi.org/10.1002/(SICI)1097-4563(199807)15:7<395::AID-ROB2>3.0.CO;2-H
Zhiyong, T., Ma, H., Pei, Zh., Liu, L., & Zhang, J. (2016, July 27–29). A new numerical method for Stewart platform forward kinematics. In The Proceedings of the 35th Chinese Control Conference (CCC) (pp. 6311–6316). Chengdu, China. https://doi.org/10.1109/ChiCC.2016.7554348