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A gravitational search algorithm-based control of an underactuated system with experimental verifications

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Abstract

Underactuated systems, such as the double inverted pendulum (DIP) on a cart, are widely used in various applications such as anthropomorphic robot design, assistive robotics, slosh modeling etc. However, the design of a control strategy for such systems is challenging due to their underactuated, nonlinear and unstable nature. This paper addresses the aforementioned challenges by introducing a linear–quadratic-regulator (LQR) controller assisted by the gravitational search algorithm (GSA) to stabilize the pendulums on the cart, while also allowing it to follow the desired movements and overcoming the unmodeled dynamics of the system. The GSA is used to improve the system performances while maintaining stability. The suitability of the proposed method is demonstrated through simulation and experimental studies, including cases with additional weights on the bob and on the cart and external disturbances. The improvements in system performances are shown through a comparative study with the performances of particle swarm optimization (PSO) and quantum PSO-based LQR controllers. The findings corroborate the efficacy of the proposed method in stabilizing the underactuated double inverted pendulum system on a cart in the nominal operating conditions and situations with additional weights and external disturbances.

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References

  • Ahmad S, Tokhi MO (2008) Modelling and control of a wheelchair on two wheels. In: 2008 second Asia international conference on modelling & simulation (AMS), IEEE, pp 579–584

  • Alimoradpour S, Rafie M, Ahmadzadeh B (2022) Providing a genetic algorithm-based method to optimize the fuzzy logic controller for the inverted pendulum. Soft Comput 26(11):5115–5130

    Article  Google Scholar 

  • Bagheri S, Jafarov T, Freidovich L et al (2016) Beneficially combining LQR and PID to control longitudinal dynamics of a smartfly UAV. In: 2016 IEEE 7th annual information technology, electronics and mobile communication conference (IEMCON), IEEE, pp 1–6

  • Balcazar R, Rubio JDJ, Orozco E et al (2022) The regulation of an electric oven and an inverted pendulum. Symmetry 14(4):759

    Article  CAS  ADS  Google Scholar 

  • Bonifacio SR, Patricio OO, Alexander PG (2018) Robust stabilizing control for the electromechanical triple-link inverted pendulum system. IFAC-PapersOnLine 51(13):314–319

    Article  Google Scholar 

  • Bui HL, Pham MN, Nguyen TTH (2022) Swing-up control of an inverted pendulum cart system using the approach of Hedge-algebras theory. Soft Comput 26(10):4613–4627

    Article  Google Scholar 

  • Chatman Y, Gangadharan S, Schlee K et al (2007) Modeling and parameter estimation of spacecraft fuel slosh with diaphragms using pendulum analogs. In: 30th annual American Astronautical Society (AAS) guidance and control conference, AAS 07-004

  • Chen H, Yang Y, Sun J (2019) Improved genetic algorithm based optimal control for a flying inverted pendulum. In: 2019 3rd international conference on electronic information technology and computer engineering (EITCE), IEEE, pp 1428–1432

  • Deb K (2011) Multi-objective optimisation using evolutionary algorithms: an introduction. In: Wang L, Ng A, Deb K (eds) Multi-objective evolutionary optimisation for product design and manufacturing. Springer, London

    Google Scholar 

  • Ezzat D, Hassanien AE, Ella HA (2021) An optimized deep learning architecture for the diagnosis of COVID-19 disease based on gravitational search optimization. Appl Soft Comput 98:106742

    Article  PubMed  Google Scholar 

  • Ghosh A, Ray AK, Jamshidi M (2021) Control of slosh instability: experimental verification with underactuated double inverted pendulum system. In: 2021 World Automation Congress (WAC), IEEE, pp 100–105

  • Habib MK, Ayankoso SA (2021) Stabilization of double inverted pendulum (DIP) on a cart using optimal adaptive sliding mode control (OASMC). In: 2021 IEEE international conference on mechatronics and automation (ICMA), IEEE, pp 993–999

  • Hazem ZB, Fotuhi MJ, Bingül Z (2020) Development of a fuzzy-LQR and fuzzy-LQG stability control for a double link rotary inverted pendulum. J Frankl Inst 357(15):10529–10556

    Article  MathSciNet  Google Scholar 

  • Hofer M, Muehlebach M, D’Andrea R (2023) The one-wheel cubli: a 3D inverted pendulum that can balance with a single reaction wheel. Mechatronics 91:102965

    Article  Google Scholar 

  • Jacknoon A, Abido MA (2017) Ant colony based LQR and PID tuned parameters for controlling inverted pendulum. In: 2017 international conference on communication, control, computing and electronics engineering (ICCCCEE), IEEE, pp 1–8

  • Jeong S, Takahashi T (2008) Wheeled inverted pendulum type assistant robot: design concept and mobile control. Intell Serv Robot 1(4):313–320

    Article  Google Scholar 

  • Karmouche DCJ, Chavarette FR, deAbreu GLCM, Gonçalves AC (2023) Population evaluation of the adapted particle swarm optimization algorithm applied for control in view of unknown parameter changes in the system. J Braz Soc Mech Sci Eng 45(2):1–18

    Article  Google Scholar 

  • Katebi J, Shoaei-parchin M, Shariati M et al (2020) Developed comparative analysis of metaheuristic optimization algorithms for optimal active control of structures. Eng Comput 36:1539–1558

    Article  Google Scholar 

  • Keo L, Yoshino K, Kawaguchi M et al (2011) Experimental results for stabilizing of a bicycle with a flywheel balancer. In: 2011 IEEE international conference on robotics and automation, IEEE, pp 6150–6155

  • Khalil I, Doyle JC, Glover K (1996) Robust and optimal control. Prentice Hall, Hoboken

    Google Scholar 

  • Kumar S (2021) Controlling an inverted pendulum with policy gradient methods—a tutorial. arXiv preprint arXiv:2105.07998

  • Lakmesari SH, Safipour Z, Mahmoodabadi MJ, Ibrahim Y, Mobayen S (2022) Optimal fuzzy proportional–integral–derivative control for a class of fourth-order nonlinear systems using imperialist competitive algorithms. Complexity 2022

  • Lu J, Yang Y, Jin X (2021) Quadrotor inverted pendulum control based on improved particle swarm optimization. In: 2021 China Automation Congress (CAC), IEEE, pp 6280–6285

  • Magdy M, El Marhomy A, Attia MA (2019) Modeling of inverted pendulum system with gravitational search algorithm optimized controller. Ain Shams Eng J 10(1):129–149

    Article  Google Scholar 

  • Maraslidis GS, Kottas TL, Tsipouras MG et al (2021) A fuzzy logic controller for double inverted pendulum on a cart. In: 2021 6th South-East Europe Design automation, computer engineering, computer networks and social media conference (SEEDA-CECNSM), IEEE, pp 1–8

  • Mohan V, Rani A, Singh V (2017) Robust adaptive fuzzy controller applied to double inverted pendulum. J Intell Fuzzy Syst 32(5):3669–3687

    Article  Google Scholar 

  • Momeni E, Yarivand A, Dowlatshahi MB et al (2021) An efficient optimal neural network based on gravitational search algorithm in predicting the deformation of geogrid-reinforced soil structures. Transp Geotech 26:100446

    Article  Google Scholar 

  • Music Z, Molinari F, Gallenmüller S et al (2019) Design of a networked controller for a two-wheeled inverted pendulum robot. IFAC-PapersOnLine 52(20):169–174

    Article  Google Scholar 

  • Olfati-Saber R (2001) Nonlinear control of underactuated mechanical systems with application to robotics and aerospace vehicles. PhD thesis, Massachusetts Institute of Technology

  • Paredes VC, Hereid A (2022) Resolved motion control for 3D underactuated bipedal walking using linear inverted pendulum dynamics and neural adaptation. In: 2022 IEEE/RSJ international conference on intelligent robots and systems (IROS), IEEE, pp 6761–6767

  • Pei J, Rothhaar PM (2015) Demonstration of launch vehicle slosh instability on a pole-cart platform. In: AIAA atmospheric flight mechanics conference, p 2712

  • Quanser (2017) Laboratory guide: linear double inverted pendulum experiment. https://www.made-for-science.com/de/quanser/?df=made-for-science-quanser-linear-double-inverted-pendulum-coursewarestud-matlab.pdf. Online; Accessed 27 Mar 2023

  • Rashedi E, Nezamabadi-Pour H, Saryazdi S (2009) GSA: a gravitational search algorithm. Inf Sci 179(13):2232–2248

    Article  Google Scholar 

  • Rodríguez-Molina A, Mezura-Montes E, Villarreal-Cervantes MG et al (2020) Multi-objective meta-heuristic optimization in intelligent control: a survey on the controller tuning problem. Appl Soft Comput 93:106342

    Article  Google Scholar 

  • Saleem O, Abbas F, Iqbal J (2023) Complex fractional-order LQIR for inverted-pendulum-type robotic mechanisms: design and experimental validation. Mathematics 11(4):913

    Article  Google Scholar 

  • Shah MHM, Mehedi IM, Al-Saggaf UM et al (2021) Balancing a rotary double inverted pendulum using two-loops adaptive controller based on dynamic inversion technique. In: 2020 8th international conference on intelligent and advanced systems (ICIAS), IEEE, pp 1–6

  • Shehab M, Khader AT, Laouchedi M et al (2019) Hybridizing cuckoo search algorithm with bat algorithm for global numerical optimization. J Supercomput 75:2395–2422

    Article  Google Scholar 

  • Shehab M, Abualigah L, Al-Hamad H et al (2020) Moth-flame optimization algorithm: variants and applications. Neural Comput Appl 32:9859–9884

    Article  Google Scholar 

  • Shehab M, Mashal I, Momani Z et al (2022) Harris hawks optimization algorithm: variants and applications. Arch Comput Methods Eng 29(7):5579–5603

    Article  Google Scholar 

  • Shi X, Gao J, Lu Y, Tian D, Liu Y (2021) Biped walking based on stiffness optimization and hierarchical quadratic programming. Sensors 21(5):1696

    Article  PubMed  PubMed Central  ADS  Google Scholar 

  • Shima TJ, Bashir HA (2021) Performance assessment of particle swarm optimisation and differential evolution algorithms on integral sliding mode control of cart-inverted pendulum. Int J Nonlinear Dyn Control 2(1):1–18

    Article  Google Scholar 

  • Shivalingegowda C, Jayasree P (2021) Hybrid gravitational search algorithm based model for optimizing coverage and connectivity in wireless sensor networks. J Ambient Intell Humaniz Comput 12(2):2835–2848

    Article  Google Scholar 

  • Singh O, Ghosh A, Ray AK (2022) Two, four, and five parameters estimation based modelling of Si cell, mono-crystalline and poly-crystalline PV modules. Silicon 14:12191–12202

    Article  CAS  Google Scholar 

  • Singhal NK, Swarup A (2019) Performance improvement of inverted pendulum using optimization algorithms. In: 2019 3rd international conference on electronics, communication and aerospace technology (ICECA), IEEE, pp 316–322

  • Stanovov V, Akhmedova S, Semenkin E (2019) Automatic design of fuzzy controller for rotary inverted pendulum with success-history adaptive genetic algorithm. In: 2019 international conference on information technologies (InfoTech), IEEE, pp 1–4

  • Susanto E, Wibowo AS, Rachman EG (2020) Fuzzy swing up control and optimal state feedback stabilization for self-erecting inverted pendulum. IEEE Access 8:6496–6504

    Article  Google Scholar 

  • Thakar PS, Bandyopadhyay B, Gandhi P (2016) Sliding mode control for underactuated slosh–container system using collocated and non-collocated linearization. In: 2016 Indian control conference (ICC), IEEE, pp 157–164

  • Thiagarajan K, Manapakkam Anandan M, Stateczny A et al (2021) Satellite image classification using a hierarchical ensemble learning and correlation coefficient-based gravitational search algorithm. Remote Sens 13(21):4351

    Article  ADS  Google Scholar 

  • Wang JJ, Liu GY (2019) Hierarchical sliding-mode control of spatial inverted pendulum with heterogeneous comprehensive learning particle swarm optimization. Inf Sci 495:14–36

    Article  MathSciNet  Google Scholar 

  • Waszak M, Łangowski R (2020) An automatic self-tuning control system design for an inverted pendulum. IEEE Access 8:26726–26738

    Article  Google Scholar 

  • Wolpert DH, Macready WG (1997) No free lunch theorems for optimization. IEEE Trans Evol Comput 1(1):67–82

    Article  Google Scholar 

  • Wright S, Nocedal J (1999) Numerical optimization. Springer, Berlin

    Google Scholar 

  • Xu B, Lyu Y, Gadsden SA (2018) Estimation and control of a double-inverted pendulum. In: The Canadian Society for mechanical engineering international congress 2018, CSME-SCGM, pp 1–5

  • Xu H, Jiang S, Zhang A (2021) Path planning for unmanned aerial vehicle using a mix-strategy-based gravitational search algorithm. IEEE Access 9:57033–57045

    Article  Google Scholar 

  • Yang Y, Zhang HH, Voyles RM (2023) Optimal fractional-order proportional–integral–derivative control enabling full actuation of decomposed rotary inverted pendulum system. Trans Inst Meas Control. 45(10):1986–1998. https://doi.org/10.1177/01423312221146606

  • Yaren T, Kizir S (2021) Power-based modelling and control: experimental results on a cart-pole double inverted pendulum. Turk J Electr Eng Comput Sci 29(3):1736–1750

    Article  Google Scholar 

  • Younes Z, Alhamrouni I, Mekhilef S et al (2021) A memory-based gravitational search algorithm for solving economic dispatch problem in micro-grid. Ain Shams Eng J 12(2):1985–1994

    Article  Google Scholar 

  • Zhu S, Liu S, Feng S, Huang M, Sun B (2022) An optimization method for the inverted pendulum problem based on deep reinforcement learning. J Phys Conf Ser 2296(1):012008

    Article  Google Scholar 

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  1. Electrical and Electronics Engineering, National Institute of Technology Sikkim, Ravangla, Sikkim, 737139, India

    Arabinda Ghosh & Anjan Kumar Ray

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  1. Arabinda Ghosh
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Appendix A

Appendix A

System parameters of the considered system

\(m_\textrm{p1}=0.072\) kg, \(m_\textrm{p2}=0.127\) kg, \(L_\textrm{p1}=0.2095\) m, \(L_\textrm{p2}=0.3365\) m, \(l_\textrm{p1}=0.1143\) m, \(l_\textrm{p2}=0.1778\) m, \(m_\textrm{c}=0.70313\) kg, \(m_\textrm{h}=0.17\) kg, \(\beta _\textrm{p1}=0.0024\) N m s/rad, \(\beta _\textrm{p2}=0.0024\) N m s/rad, \(\beta _\textrm{eq}=4.3\) N m s/rad, \(K_g=3.71\), \(K_t=0.00768\) N m/A, \(K_\textrm{m}=0.00767\) V s/rad, \(R_\textrm{m}=2.6\) \(\Omega \), \(r_\textrm{mp}=0.00635\) m, \(\eta _g\)=1, \(\eta _\textrm{m}=1\), \(g=9.8\) m/s\(^2\).

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Ghosh, A., Ray, A.K. A gravitational search algorithm-based control of an underactuated system with experimental verifications. Soft Comput 28, 3353–3369 (2024). https://doi.org/10.1007/s00500-023-08606-x

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