Journals / Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi / 2026 / Cilt: 28 Sayı: 2

A review on swing-up methods in inverted pendulums

Ters sarkaçta yükseltme yöntemleri incelemesi

Erman Köybaşı(BALIKESİR ÜNİVERSİTESİ, MÜHENDİSLİK FAKÜLTESİ)0000-0001-7647-0023
Davut Akdaş(BALIKESIR UNIVERSITY, FACULTY OF ENGINEERING, DEPARTMENT OF ELECTRICAL AND ELECTRONIC ENGINEERING)0000-0002-2492-5046
Sabri Bıçakçı(BALIKESIR UNIVERSITY, FACULTY OF ENGINEERING, DEPARTMENT OF ELECTRICAL AND ELECTRONIC ENGINEERING)0000-0002-2334-8515
Pages
484–507
Publication Date
2026-07-31
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AI Summary & Key Takeaways

AI Insight
⚡ TL;DR(Too Long; Didn't Read)

This review provides a broad perspective, ranging from traditional control approaches to AI-driven learning systems, offering a current and comprehensive roadmap for researchers.

📌 3 Key Takeaways

🎯
Objective & Hypothesis:Inverted pendulum systems are considered a fundamental criterion in comparing and validating control methods due to their nonlinear, unstable, and underactuated nature.
📊
Key Findings & Data:In this study, swing-up methods that raise the inverted pendulum from a hanging position to a vertical position are comprehensively examined in light of the literature.
💡
Conclusion & Impact:This review provides a broad perspective, ranging from traditional control approaches to AI-driven learning systems, offering a current and comprehensive roadmap for researchers.
🔬Design:Review & Systematic Survey
📍Scope:Comprehensive Sample
⚙️Variables:Doğrusal ters sarkaç · döner ters sarkaç · sarkaç yükseltme

Abstract

Inverted pendulum systems are considered a fundamental criterion in comparing and validating control methods due to their nonlinear, unstable, and underactuated nature. In this study, swing-up methods that raise the inverted pendulum from a hanging position to a vertical position are comprehensively examined in light of the literature. Linear and rotary inverted pendulum systems are presented along with their mathematical models, and inverted pendulum swing-up methods found in the literature are presented along with the studies conducted,, the methods used to keep the inverted pendulum in equilibrium are listed, and inverted pendulum swing-up methods found in the literature are presented along with the studies conducted. The methods used are evaluated starting from their first proposed studies and including their current applications. In this context, the new roles of classical energy-based methods in soft robotics applications are discussed; the Hedge Algebra approach, which optimizes the computational load in fuzzy logic-based systems, is examined. In addition, the success of Deep Reinforcement Learning algorithms, which offer model-independent control, especially in overcoming "Sim-to-Real" challenges, is discussed through current publications. This review provides a broad perspective, ranging from traditional control approaches to AI-driven learning systems, offering a current and comprehensive roadmap for researchers.

Özet

Ters sarkaç sistemleri, doğrusal olmayan, kararsız ve eksik tahrikli doğası nedeniyle kontrol yöntemlerinin kıyaslanması ve doğrulanmasında temel bir kıstas olarak kabul edilmektedir. Bu çalışmada, ters sarkacın aşağı sarkık konumdan dikey konuma getirilmesini sağlayan yükseltme (swing-up) yöntemleri literatür ışığında kapsamlı bir şekilde incelenmiştir. Lineer ve dönen ters sarkaç sistemleri matematik modelleriyle birlikte sunulmuş, literatürde yer alan ters sarkaç yükseltme yöntemleri yapılan çalışmalarla birlikte verilmiş, ters sarkacın dengede tutulması için kullanılan yöntemler listelenmiş ve literatürde yer alan ters sarkaç yükseltme yöntemleri yapılan çalışmalarla birlikte verilmiştir. Kullanılan yöntemler ilk ortaya konduğu çalışmalardan başlayıp güncel uygulamalarıyla da değerlendirilmiştir. Bu bağlamda, klasik enerji tabanlı yöntemlerin yumuşak robotik (soft robotics) uygulamalarındaki yeni rolleri ele alınmış; bulanık mantık tabanlı sistemlerde hesaplama yükünü optimize eden Hedge Cebiri (Hedge Algebras) yaklaşımı incelenmiştir. Ayrıca, modelden bağımsız kontrol imkânı sunan Derin Pekiştirmeli Öğrenme (Deep Reinforcement Learning) algoritmalarının, özellikle "Simülasyondan Gerçeğe Transfer" (Sim-to-Real) zorluklarını aşmadaki başarısı güncel yayınlar üzerinden tartışılmıştır. Bu derleme çalışması, geleneksel kontrol yaklaşımlarından yapay zekâ odaklı öğrenen sistemlere kadar geniş bir perspektif sunarak, araştırmacılar için güncel ve bütünsel bir yol haritası niteliği taşımaktadır.

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Cite This Article

Köybaşı, E., Akdaş, D., Bıçakçı, S. (2026). A review on swing-up methods in inverted pendulums. Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi, Cilt: 28 Sayı: 2, 484–507. https://doi.org/10.25092/baunfbed.1946757

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References (74)

  1. N. Muskinja and B. Tovornik, ‘Swinging up and stabilization of a real inverted pendulum’, IEEE Trans. Ind. Electron., vol. 53, no. 2, pp. 631–639, Apr. 2006, doi: 10.1109/TIE.2006.870667.
  2. M. M. de Almeida and G. V. Raffo, ‘Nonlinear Balance Control of an Inverted Pendulum on a Tilt-rotor UAV∗∗The authors would like to thank the Brazilian research agencies CAPES, CNPq and FAPEMIG for their financial contribution for the accomplishment of this work.’, IFAC-PapersOnLine, vol. 48, no. 9, pp. 168–173, 2015, doi: 10.1016/j.ifacol.2015.08.078.
  3. M. F. P. Polo, M. P. Molina, and J. G. Chica, ‘Swing-up and positioning control of an inverted wheeled cart pendulum system with chaotic balancing motions’, International Journal of Non-Linear Mechanics, vol. 47, no. 6, pp. 655–665, Jul. 2012, doi: 10.1016/j.ijnonlinmec.2011.10.009.
  4. K. Turkoglu and B. Graham, ‘Design, Build and Integration of a Low-Cost Self-Erecting Inverted Pendulum Mechanism’, in AIAA Modeling and Simulation Technologies Conference, Grapevine, Texas: American Institute of Aeronautics and Astronautics, Jan. 2017. doi: 10.2514/6.2017-1762.
  5. Ponce, ‘A REVIEW OF INTELLIGENT CONTROL SYSTEMS APPLIED TO THE INVERTED-PENDULUM PROBLEM’, American Journal of Engineering and Applied Sciences, vol. 7, no. 2, pp. 194–240, Feb. 2014, doi: 10.3844/ajeassp.2014.194.240.
  6. S. Mori, H. Nishihara, and K. Furuta, ‘Control of unstable mechanical system Control of pendulum†’, International Journal of Control, vol. 23, no. 5, pp. 673–692, May 1976, doi: 10.1080/00207177608922192.
  7. K. Furuta, M. Yamakita, and S. Kobayashi, ‘Swing up control of inverted pendulum’, in Proceedings IECON ’91: 1991 International Conference on Industrial Electronics, Control and Instrumentation, Kobe, Japan: IEEE, 1991, pp. 2193–2198. doi: 10.1109/IECON.1991.239008.
  8. Chin Wang Tao, Jinshiuh Taur, J. H. Chang, and Shun-Feng Su, ‘Adaptive Fuzzy Switched Swing-Up and Sliding Control for the Double-Pendulum-and-Cart System’, IEEE Trans. Syst., Man, Cybern. B, vol. 40, no. 1, pp. 241–252, Feb. 2010, doi: 10.1109/TSMCB.2009.2025964.
  9. M. Yamakita, M. Iwashiro, Y. Sugahara, and K. Furuta, ‘Robust Swing Up Control of Double Pendulum’, Proceedings of 1995 American Control Conference - ACC’95, vol. 1, pp. 290–295, 1995, doi: 10.1109/ACC.1995.529255.
  10. T. Glück, A. Eder, and A. Kugi, ‘Swing-up control of a triple pendulum on a cart with experimental validation’, Automatica, vol. 49, no. 3, pp. 801–808, Mar. 2013, doi: 10.1016/j.automatica.2012.12.006.
  11. I. Fantoni, R. Lozano, and M. W. Spong, ‘Energy based control of the Pendubot’, IEEE Trans. Automat. Contr., vol. 45, no. 4, pp. 725–729, Apr. 2000, doi: 10.1109/9.847110.
  12. Y. Wang and X. Xin, ‘Energy-based Swing-up Control of the Soft Inverted Pendulum with Affine Curvature’, Nonlinear Dyn, vol. 113, no. 16, pp. 21233–21252, Aug. 2025, doi: 10.1007/s11071-025-11276-z.
  13. J.-J. Wang, ‘Simulation studies of inverted pendulum based on PID controllers’, Simulation Modelling Practice and Theory, vol. 19, no. 1, pp. 440–449, Jan. 2011, doi: 10.1016/j.simpat.2010.08.003.
  14. M. El-Bardini and A. M. El-Nagar, ‘Interval type-2 fuzzy PID controller for uncertain nonlinear inverted pendulum system’, ISA Transactions, vol. 53, no. 3, pp. 732–743, May 2014, doi: 10.1016/j.isatra.2014.02.007.
  15. C. Iwendi, M. A. Alqarni, J. H. Anajemba, A. S. Alfakeeh, Z. Zhang, and A. K. Bashir, ‘Robust Navigational Control of a Two-Wheeled Self-Balancing Robot in a Sensed Environment’, IEEE Access, vol. 7, pp. 82337–82348, 2019, doi: 10.1109/ACCESS.2019.2923916.
  16. D. Qian, J. Yi, and D. Zhao, ‘Hierarchical sliding mode control for a class of SIMO under-actuated systems’, Control and Cybernetics, p. 18, 2008.
  17. H. Fukushima, K. Muro, and F. Matsuno, ‘Sliding-Mode Control for Transformation to an Inverted Pendulum Mode of a Mobile Robot With Wheel-Arms’, IEEE Transactions on Industrial Electronics, vol. 62, no. 7, pp. 4257–4266, Jul. 2015, doi: 10.1109/TIE.2014.2384475.
  18. D. I. Barbosa, J. S. Castillo, and L. F. Combita, ‘Rotary inverted pendulum with real time control’, in 2011 IEEE IX Latin American Robotics Symposium and IEEE Colombian Conference on Automatic Control, Oct. 2011, pp. 1–6. doi: 10.1109/LARC.2011.6086851.
  19. W. Li, H. Ding, and K. Cheng, ‘An investigation on the design and performance assessment of double-PID and LQR controllers for the inverted pendulum’, in Proceedings of 2012 UKACC International Conference on Control, Cardiff, United Kingdom: IEEE, Sep. 2012, pp. 190–196. doi: 10.1109/CONTROL.2012.6334628.
  20. S. Kizir, Z. Bingul, and C. Oysu, ‘Fuzzy control of a real time inverted pendulum system’, Journal of Intelligent & Fuzzy Systems, vol. 21, no. 1, 2, pp. 121–133, Jan. 2010, doi: 10.3233/IFS-2010-0441.
  21. S.-E. Oltean, ‘Swing-up and Stabilization of the Rotational Inverted Pendulum Using PD and Fuzzy-PD Controllers’, Procedia Technology, vol. 12, pp. 57–64, Ocak 2014, doi: 10.1016/j.protcy.2013.12.456.
  22. X. Yang and X. Zheng, ‘Swing-Up and Stabilization Control Design for an Underactuated Rotary Inverted Pendulum System: Theory and Experiments’, IEEE Transactions on Industrial Electronics, vol. 65, no. 9, pp. 7229–7238, Sep. 2018, doi: 10.1109/TIE.2018.2793214.
  23. S.-J. Huang and C.-L. Huang, ‘Control of an Inverted Pendulum Using Grey Prediction Model’, IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, vol. 36, no. 2, p. 7, 2000.
  24. S. Bicakci, ‘On the Implementation of Fuzzy VMC for an Under Actuated System’, IEEE Access, vol. 7, pp. 163578–163588, 2019, doi: 10.1109/ACCESS.2019.2952294.
  25. A. A. Saifizul, Z. Zainon, N. A. Abu Osman, C. A. Azlan, and U. F. S. U. Ibrahim, ‘Intelligent Control for Self-erecting Inverted Pendulum Via Adaptive Neuro-fuzzy Inference System’, American J. of Applied Sciences, vol. 3, no. 4, pp. 1795–1802, Apr. 2006, doi: 10.3844/ajassp.2006.1795.1802.
  26. V. Mohan and N. Singh, ‘Performance comparison of LQR and ANFIS controller for stabilizing double inverted pendulum system’, in 2013 IEEE International Conference on Signal Processing, Computing and Control (ISPCC), Solan, India: IEEE, Sep. 2013, pp. 1–6. doi: 10.1109/ISPCC.2013.6663452.
  27. M. Wiklund, A. Kristenson, and K. J. Aström, ‘A New Strategy for Swinging Up an Inverted Pendulum’, IFAC Proceedings Volumes, vol. 26, no. 2, pp. 757–760, Jul. 1993, doi: 10.1016/S1474-6670(17)48372-8.
  28. K. J. Åström and K. Furuta, ‘Swinging up a pendulum by energy control’, Automatica, vol. 36, no. 2, pp. 287–295, Feb. 2000, doi: 10.1016/S0005-1098(99)00140-5.
  29. Y.-H. Chang, C.-W. Chang, C.-H. Yang, and C. W. Tao, ‘Swing Up and Balance Control of Planetary Train Type Pendulum with Fuzzy Logic and Energy Compensation’, International Journal of Fuzzy Systems, vol. 9, no. 2, p. 9, 2007.
  30. X. Xin, S. Tanaka, J. She, and T. Yamasaki, ‘New analytical results of energy-based swing-up control for the Pendubot’, International Journal of Non-Linear Mechanics, vol. 52, pp. 110–118, Jun. 2013, doi: 10.1016/j.ijnonlinmec.2013.02.003.
  31. C. W. Tao, J. S. Taur, C. M. Wang, and U. S. Chen, ‘Fuzzy hierarchical swing-up and sliding position controller for the inverted pendulum–cart system’, Fuzzy Sets and Systems, vol. 159, no. 20, pp. 2763–2784, Ekim 2008, doi: 10.1016/j.fss.2008.02.005.
  32. E. Susanto, A. Surya Wibowo, and E. Ghiffary Rachman, ‘Fuzzy Swing Up Control and Optimal State Feedback Stabilization for Self-Erecting Inverted Pendulum’, IEEE Access, vol. 8, pp. 6496–6504, 2020, doi: 10.1109/ACCESS.2019.2963399.
  33. A. Jain, A. Sharma, V. Jately, B. Azzopardi, and S. Choudhury, ‘Real-Time Swing-Up Control of Non-Linear Inverted Pendulum Using Lyapunov Based Optimized Fuzzy Logic Control’, IEEE Access, vol. 9, pp. 50715–50726, 2021, doi: 10.1109/ACCESS.2021.3058645.
  34. V. Sukontanakarn and M. Parnichkun, ‘Real-Time Optimal Control for Rotary Inverted Pendulum’, p. 11, 2009.
  35. Mun-Soo Park and Dongkyoung Chwa, ‘Swing-Up and Stabilization Control of Inverted-Pendulum Systems via Coupled Sliding-Mode Control Method’, IEEE Trans. Ind. Electron., vol. 56, no. 9, pp. 3541–3555, Sep. 2009, doi: 10.1109/TIE.2009.2012452.
  36. H.-L. Bui, M.-N. Pham, and T.-T.-H. Nguyen, ‘Swing-up control of an inverted pendulum cart system using the approach of Hedge-algebras theory’, Soft Comput, vol. 26, no. 10, pp. 4613–4627, May 2022, doi: 10.1007/s00500-022-06968-2.
  37. X. Liu, S. Wang, X. Li, and Z. Cui, ‘Balance Controller Design for Inverted Pendulum Considering Detail Reward Function and Two-Phase Learning Protocol’, Symmetry, vol. 16, no. 9, p. 1227, Sep. 2024, doi: 10.3390/sym16091227.
  38. X. Bajrami, F. Kaçiu, E. Shala, and R. Likaj, ‘Real-Time Swing-up of a Linear Inverted Pendulum Using Reinforcement Learning’, mech, vol. 31, no. 2, pp. 123–135, May 2025, doi: 10.5755/j02.mech.39202.
  39. K. Furuta, M. Yamakita, and S. Kobayashi, ‘Swing-up Control of Inverted Pendulum Using Pseudo-State Feedback’, Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering, vol. 206, no. 4, pp. 263–269, Nov. 1992, doi: 10.1243/PIME_PROC_1992_206_341_02.
  40. S. McGilvray, ‘Self-Erecting Inverted Pendulum: Swing up and Stabilization Control’, IEEE Student paper contest, p. 29, 2002.
  41. J. Rubı́, Á. Rubio, and A. Avello, ‘Swing-up control problem for a self-erecting double inverted pendulum’, IEE Proceedings - Control Theory and Applications, vol. 149, no. 2, pp. 169–175, Mar. 2002, doi: 10.1049/ip-cta:20020326.
  42. A. Khashayar, M. A. Nekoui, and H. AhangarAsr, ‘Stabilization of Rotary Inverted Pendulum Using Fuzzy Logic’, IJIIP, vol. 2, no. 4, pp. 23–31, Dec. 2011, doi: 10.4156/ijiip.vol2.issue4.3.
  43. R. Urniezius and E. geguzis, ‘Hybrid Fuzzy Logic and Adaptive LQR Controller for Swing-up, Positioning and Stabilization of Inverted Pendulum’, Electronics and Electrical Engineering, Lithuania, vol. 20, pp. 11–15, Nisan 2014, doi: 10.5755/j01.eee.20.3.6675.
  44. H. C. T. Nguyen and A. W. Shen, ‘Using Hybrid And LQR Method Control Of A Self-Erecting Rotary Inverted Pendulum System Based On PIC 18F443’, vol. 2, p. 6, 2011.
  45. M. Park, Y.-J. Kim, and J.-J. Lee, ‘Swing-up and LQR stabilization of a rotary inverted pendulum’, Artif Life Robotics, vol. 16, no. 1, pp. 94–97, Jun. 2011, doi: 10.1007/s10015-011-0897-9.
  46. M. W. Spong and D. J. Block, ‘The Pendubot: a mechatronic system for control research and education’, in Proceedings of 1995 34th IEEE Conference on Decision and Control, New Orleans, LA, USA: IEEE, 1995, pp. 555–556. doi: 10.1109/CDC.1995.478951.
  47. P. Melba Mary and N. S. Marimuthu, ‘Design of intelligent hybrid controller for swing-up and stabilization of rotary inverted pendulum’, J. Eng. Appl. Sci. (Asian Res. Publ. Netw.), vol. 3, no. 4, p. 11, 2008.
  48. P. M. Mary and N. S. Marimuthu, ‘Minimum Time Swing up and Stabilization of Rotary Inverted Pendulum Using Pulse Step Control’, Iranian journal of fuzzy systems, vol. 6, no. 3, p. 15, 2009.
  49. J. Yi, N. Yubazaki, and K. Hirota, ‘Upswing and stabilization control of inverted pendulum system based on the SIRMs dynamically connected fuzzy inference model’, Fuzzy Sets and Systems, vol. 122, no. 1, pp. 139–152, Aug. 2001, doi: 10.1016/S0165-0114(00)00049-X.
  50. S. V. Radhamohan, ‘FUZZY SWING-UP AND STABILIZATION OF REAL INVERTED PENDULUM USING SINGLE RULEBASE’, p. 9, 2005.
  51. F. Kiliç, S. Bi̇Cakci, and H. Güneş, ‘Ters sarkacın uyarlamalı kazançlı bağımsız yüzeyli kayan kip yöntemi ile denetimi’, Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi, vol. 21, no. 2, pp. 610–622, Jun. 2019, doi: 10.25092/baunfbed.632774.
  52. K. Graichen, M. Treuer, and M. Zeitz, ‘Swing-up of the double pendulum on a cart by feedforward and feedback control with experimental validation’, Automatica, vol. 43, no. 1, pp. 63–71, Jan. 2007, doi: 10.1016/j.automatica.2006.07.023.
  53. S. Biçakçi, F. Kiliç, and H. Güneş, ‘Eksik tahrikli bir sistemin karınca kolonisi tabanlı optimal PD kontrolü: Deneysel bir uygulama’, Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi, vol. 25, no. 2, pp. 710–730, Jul. 2023, doi: 10.25092/baunfbed.1163250.
  54. L.-H. Chang and A.-C. Lee, ‘A hybrid controller design for bi-axial inverted pendulum system’, Int. J. Robust Nonlinear Control, vol. 19, no. 5, pp. 512–531, Mar. 2009, doi: 10.1002/rnc.1328.
  55. D. Chatterjee, A. Patra, and H. K. Joglekar, ‘Swing-up and stabilization of a cart–pendulum system under restricted cart track length’, Systems & Control Letters, vol. 47, no. 4, pp. 355–364, Nov. 2002, doi: 10.1016/S0167-6911(02)00229-3.
  56. K. Sakurama, S. Hara, and K. Nakano, ‘Swing-up and stabilization control of a cart– pendulum system via energy control and controlled Lagrangian methods’, Elect. Eng. Jpn., vol. 160, no. 4, pp. 24–31, Sep. 2007, doi: 10.1002/eej.20534.
  57. Xin Xin and M. Kaneda, ‘Analysis of the energy-based control for swinging up two pendulums’, IEEE Trans. Automat. Contr., vol. 50, no. 5, pp. 679–684, May 2005, doi: 10.1109/TAC.2005.846598.
  58. T. Albahkali, R. Mukherjee, and T. Das, ‘Swing-Up Control of the Pendubot: An Impulse–Momentum Approach’, IEEE Trans. Robot., vol. 25, no. 4, pp. 975–982, Aug. 2009, doi: 10.1109/TRO.2009.2022427.
  59. F. Camastra et al., ‘A fuzzy decision system for genetically modified plant environmental risk assessment using Mamdani inference’, Expert Systems with Applications, vol. 42, no. 3, pp. 1710–1716, Feb. 2015, doi: 10.1016/j.eswa.2014.09.041.
  60. G. J. Klir and B. Yuan, Fuzzy sets and fuzzy logic: theory and applications. Upper Saddle River, N.J: Prentice Hall PTR, 1995.
  61. J.-H. Yang, S.-Y. Shim, J.-H. Seo, and Y.-S. Lee, ‘Swing-up control for an inverted pendulum with restricted cart rail length’, Int. J. Control Autom. Syst., vol. 7, no. 4, pp. 674–680, Aug. 2009, doi: 10.1007/s12555-009-0419-x.
  62. A. K. Pal and T. Nestorovic, ‘Swing Up and Balance of an Inverted Pendulum Using Reinforced Learning Approach Coupled With a Proportional-Integral-Derivative Controller’, in 2022 International Conference on Electrical, Computer, Communications and Mechatronics Engineering (ICECCME), Maldives, Maldives: IEEE, Nov. 2022, pp. 1–6. doi: 10.1109/ICECCME55909.2022.9988506.
  63. M. Riedmiller, ‘Neural Reinforcement Learning to Swing-up and Balance a Real Pole’, in 2005 IEEE International Conference on Systems, Man and Cybernetics, Waikoloa, HI, USA: IEEE, 2005, pp. 3191–3196. doi: 10.1109/ICSMC.2005.1571637.
  64. J. Baek, C. Lee, Y. S. Lee, S. Jeon, and S. Han, ‘Reinforcement learning to achieve real-time control of triple inverted pendulum’, Engineering Applications of Artificial Intelligence, vol. 128, p. 107518, Feb. 2024, doi: 10.1016/j.engappai.2023.107518.
  65. Y. Oh, T. Lee, S. Ryoo, K. C. Koh, S. Han, and Y. S. Lee, ‘Reinforcement Learning to Achieve Real-time Control of a Quadruple Inverted Pendulum’, Int. J. Control Autom. Syst., vol. 23, no. 9, pp. 2797–2806, Sep. 2025, doi: 10.1007/s12555-025-0235-y.
  66. S. Israilov et al., ‘Reinforcement learning approach to control an inverted pendulum: A general framework for educational purposes’, PLoS ONE, vol. 18, no. 2, p. e0280071, Feb. 2023, doi: 10.1371/journal.pone.0280071.
  67. Z. Ben Hazem, ‘Study of Q-learning and deep Q-network learning control for a rotary inverted pendulum system’, Discov Appl Sci, vol. 6, no. 2, p. 49, Feb. 2024, doi: 10.1007/s42452-024-05690-y.
  68. Y. Dai, K. Lee, and S. Lee, ‘A real-time HIL control system on rotary inverted pendulum hardware platform based on double deep Q-network’, Measurement and Control, vol. 54, no. 3–4, pp. 417–428, Mar. 2021, doi: 10.1177/00202940211000380.
  69. R. S. Bhourji, S. Mozaffari, and S. Alirezaee, ‘Reinforcement Learning DDPG–PPO Agent-Based Control System for Rotary Inverted Pendulum’, Arab J Sci Eng, vol. 49, no. 2, pp. 1683–1696, Feb. 2024, doi: 10.1007/s13369-023-07934-2.
  70. R. Hernandez, R. Garcia-Hernandez, and F. Jurado, ‘Modeling, Simulation, and Control of a Rotary Inverted Pendulum: A Reinforcement Learning-Based Control Approach’, Modelling, vol. 5, no. 4, pp. 1824–1852, Nov. 2024, doi: 10.3390/modelling5040095.
  71. P. Böhm, P. Pounds, and A. C. Chapman, ‘Low-cost Real-world Implementation of the Swing-up Pendulum for Deep Reinforcement Learning Experiments’, 2025, arXiv. doi: 10.48550/ARXIV.2503.11065.
  72. X. Xu, ‘Reliable Transfer of Reinforcement Learning Policies to Swing Up and Stabilize an Inverted Pendulum’, Ph.D., North Carolina State University, United States -- North Carolina, 2025. [Online]. Available: https://www.proquest.com/dissertations-theses/reliable-transfer-reinforcement-learning-policies/docview/3309846628/se-2?accountid=15410
  73. J. Awrejcewicz, G. Wasilewski, G. Kudra, and S. A. Reshmin, ‘An experiment with swinging up a double pendulum using feedback control’, J. Comput. Syst. Sci. Int., vol. 51, no. 2, pp. 176–182, Apr. 2012, doi: 10.1134/S1064230712020037.
  74. N. Adhikary and C. Mahanta, ‘Integral backstepping sliding mode control for underactuated systems: Swing-up and stabilization of the Cart–Pendulum System’, ISA Transactions, vol. 52, no. 6, pp. 870–880, Nov. 2013, doi: 10.1016/j.isatra.2013.07.012.