Modeling and Control of Robotic Manipulators: A Fractional Calculus Point of View


This paper deals with the fractional-order modeling, stability analysis and control of robotic manipulators, namely a single flexible link robotic manipulator (SFLRM) and 2DOF Serial Flexible Joint Robotic Manipulator (2DSFJ). The control law is derived using Pole Placement (PP) method. This paper uses Mittag–Leffler function for the analysis of SFLRM in the time domain. The stability analysis of the fractional model is carried in a transformed \({\Omega }\)-Domain, and from the analysis, it is observed that the response of the fractional model of SFLRM robotic manipulator is stable. The main motive behind this analysis is to understand the fractional behavior of robotic manipulators, and it is well known from literature that most of the real-world systems have their own fractional behavior. Furthermore, a real-time SFLRM and 2DSFJ setups are considered to validate the results obtained and it is found that the control law suggested by PP method improves the settling time of the robotic manipulators.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13


  1. 1.

    Sethi, J.K.; Deb, D.; Malakar, M.: Modeling of a wind turbine farm in presence of wake interactions. In: 2011 International Conference On Energy, Automation And Signal (2011)

  2. 2.

    Patel, R.; Deb, D.: Parametrized control-oriented mathematical model and adaptive backstepping control of a single chamber single population microbial fuel cell. J. Power Sources 396, 599–605 (2018)

    Article  Google Scholar 

  3. 3.

    Nath, A.; Deb, D.; Dey, R.: An augmented subcutaneous type 1 diabetic patient modelling and design of adaptive glucose control. J. Process Control 86, 94–105 (2020)

    Article  Google Scholar 

  4. 4.

    Nasser-Eddine, A.; Huard, B.; Gabano, J.D.; Poinot, T.: A two steps method for electrochemical impedance modeling using fractional order system in time and frequency domains. Control Eng. Pract. 86, 96–104 (2019)

    Article  Google Scholar 

  5. 5.

    Qureshi, S.; Yusuf, A.; Shaikh, A.A.; Inc, M.; Baleanu, D.: Fractional modeling of blood ethanol concentration system with real data application. Chaos Interdiscip. J. Nonlinear Sci. 29(1), 013143 (2019)

    MathSciNet  MATH  Article  Google Scholar 

  6. 6.

    Machado, J.T.; Lopes, A.M.: Fractional-order modeling of a diode. Commun. Nonlinear Sci. Numer. Simul. 70, 343–353 (2019)

    MathSciNet  MATH  Article  Google Scholar 

  7. 7.

    Singh, A.P.; Deb, D.; Agarwal, H.: On selection of improved fractional model and control of different systems with experimental validation. Commun. Nonlinear Sci. Numer. Simul. 79, 104902 (2019)

    MathSciNet  MATH  Article  Google Scholar 

  8. 8.

    Monje, C.A.; Chen, Y.; Vinagre, B.M.; Xue, D.; Feliu-Batlle, V.: Fractional-Order Systems and Controls: Fundamentals and Applications. Springer, Berlin (2010)

    Google Scholar 

  9. 9.

    Caponetto, R.: Fractional Order Systems: Modeling and Control Applications. World Scientific, Singapore (2010)

    Google Scholar 

  10. 10.

    Monje, C.A.; Vinagre, B.M.; Feliu, V.; Chen, Y.: Tuning and auto-tuning of fractional order controllers for industry applications. Control Eng. Pract. 16(7), 798–812 (2008)

    Article  Google Scholar 

  11. 11.

    Luo, Y.; Chen, Y.Q.; Wang, C.Y.; Pi, Y.G.: Tuning fractional order proportional integral controllers for fractional order systems. J. Process Control 20(7), 823–831 (2010)

    Article  Google Scholar 

  12. 12.

    Zhang, X.; Chen, Y.: Admissibility and robust stabilization of continuous linear singular fractional order systems with the fractional order \(\alpha \): the \(0< \alpha < 1\) case. ISA Trans. 82, 42–50 (2018)

    Article  Google Scholar 

  13. 13.

    Sakthivel, R.; Ahn, C.K.; Joby, M.: Fault-tolerant resilient control for fuzzy fractional order systems. IEEE Trans. Syst. Man Cybern. Syst. 49(9), 1797–1805 (2018)

    Article  Google Scholar 

  14. 14.

    Lin, C.; Chen, B.; Shi, P.; Yu, J.P.: Necessary and sufficient conditions of observer-based stabilization for a class of fractional-order descriptor systems. Syst. Control Lett. 112, 31–35 (2018)

    MathSciNet  MATH  Article  Google Scholar 

  15. 15.

    Wang, J.; Shao, C.; Chen, Y.Q.: Fractional order sliding mode control via disturbance observer for a class of fractional order systems with mismatched disturbance. Mechatronics 53, 8–19 (2018)

    Article  Google Scholar 

  16. 16.

    Singh, A.P.; Agarwal, H.; Srivastava, P.: Fractional order controller design for inverted pendulum on a cart system (POAC). WSEAS Trans. Syst. Control 10, 172–178 (2015)

    Google Scholar 

  17. 17.

    Singh, A.; Agrawal, H.: A fractional model predictive control design for 2-d gantry crane system. J. Eng. Sci. Technol. 13(7), 2224–2235 (2018)

    Google Scholar 

  18. 18.

    Mujumdar, A.; Tamhane, B.; Kurode, S.: Fractional order modeling and control of a flexible manipulator using sliding modes. In: 2014 American Control Conference, pp. 2011–2016. IEEE (2014)

  19. 19.

    Kexue, L.; Jigen, P.: Laplace transform and fractional differential equations. Appl. Math. Lett. 24(12), 2019–2023 (2011)

    MathSciNet  MATH  Article  Google Scholar 

  20. 20.

    Lin, S.D.; Lu, C.H.: Laplace transform for solving some families of fractional differential equations and its applications. Adv. Differ. Equ. 2013(1), 137 (2013)

    MathSciNet  MATH  Article  Google Scholar 

  21. 21.

    Sabatier, J.; Farges, C.; Trigeassou, J.C.: Fractional systems state space description: some wrong ideas and proposed solutions. J. Vib. Control 20(7), 1076–1084 (2014)

    MathSciNet  Article  Google Scholar 

  22. 22.

    Li, C.P.; Zhang, F.R.: A survey on the stability of fractional differential equations. Eur. Phys. J. Spec. Top. 193(1), 27–47 (2011)

    Article  Google Scholar 

  23. 23.

    Li, Y.; Chen, Y.; Podlubny, I.: Mittag-Leffler stability of fractional order nonlinear dynamic systems. Automatica 45(8), 1965–1969 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  24. 24.

    Tavazoei, M.S.; Haeri, M.: A note on the stability of fractional order systems. Math. Comput. Simul. 79(5), 1566–1576 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  25. 25.

    Bandyopadhyay, B.; Kamal, S.: Solution, stability and realization of fractional order differential equation. In: Stabilization and Control of Fractional Order Systems: A Sliding Mode Approach, pp. 55–90. Springer, Cham (2015)

  26. 26.

    Singh, A.P.; Kazi, F.S.; Singh, N.M.; Srivastava, P.: PI\(^\alpha \)D\(^\beta \) controller design for underactuated mechanical systems. In: 2012 12th International Conference on Control Automation Robotics and Vision (ICARCV), pp. 1654–1658. IEEE (2012)

  27. 27.

    Dabiri, A.; Poursina, M.; Butcher, E.A.: Integration of divide-and-conquer algorithm with fractional order controllers for the efficient dynamic modeling and control of multibody systems. In: 2018 Annual American Control Conference (ACC), pp. 4201–4206. IEEE (2018)

  28. 28.

    Copot, C.; Muresan, C.I.; Markowski, K.A.: Advances in fractional order controller design and applications. J. Appl. Nonlinear Dyn. 8(1), 1–3 (2019)

    MathSciNet  Google Scholar 

  29. 29.

    Pandey, S.; Dwivedi, P.; Junghare, A.S.: A newborn hybrid anti-windup scheme for fractional order proportional integral controller. Arab. J. Sci. Eng. 43(6), 3049–3063 (2018)

    Article  Google Scholar 

  30. 30.

    Dabiri, A.; Butcher, E.A.: Optimal observer-based feedback control for linear fractional-order systems with periodic coefficients. J. Vib. Control 25(7), 1379–1392 (2019)

    MathSciNet  Article  Google Scholar 

  31. 31.

    Dabiri, A.; Butcher, E.A.; Poursina, M.; Nazari, M.: Optimal periodic-gain fractional delayed state feedback control for linear fractional periodic time-delayed systems. IEEE Trans. Autom. Control 63(4), 989–1002 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  32. 32.

    Gong, Y.; Wen, G.; Peng, Z.; Huang, T.; Chen, Y.: Observer-based time-varying formation control of fractional-order multi-agent systems with general linear dynamics. IEEE Trans. Circuits Syst. II Express Briefs 67, 82–86 (2019)

    Article  Google Scholar 

  33. 33.

    Cortez, A.J.G.; Mendez-Barrios, C.F.; González-Galván, E.J.; MejíaRodríguez, G.; Félix, L.: Geometrical design of fractional PD controllers for linear time-invariant fractional-order systems with time delay. Proc. Inst. Mech. Eng. Part I J. Syst. Control Eng. 233(7), 815–829 (2019)

    Google Scholar 

  34. 34.

    Boubellouta, A.; Boulkroune, A.: Intelligent fractional-order control-based projective synchronization for chaotic optical systems. Soft. Comput. 23(14), 5367–5384 (2019)

    MATH  Article  Google Scholar 

  35. 35.

    Munoz-Hernandez, G.A.; Mino-Aguilar, G.; Guerrero-Castellanos, J.F.; Peralta-Sanchez, E.: Fractional order PI-based control applied to the traction system of an electric vehicle (EV). Appl. Sci. 10(1), 364 (2020)

    Article  Google Scholar 

  36. 36.

    Birs, I.; Muresan, C.; Nascu, I.; Ionescu, C.: A survey of recent advances in fractional order control for time delay systems. IEEE Access 7, 30951–30965 (2019)

    Article  Google Scholar 

  37. 37.

    Guo, Y.; Ma, B.L.: Global sliding mode with fractional operators and application to control robot manipulators. Int. J. Control 92(7), 1497–1510 (2019)

    MathSciNet  MATH  Article  Google Scholar 

  38. 38.

    Haghighi, A.; Ziaratban, R.: A non-integer sliding mode controller to stabilize fractional-order nonlinear systems. Adv. Differ. Equ. 2020, 1–19 (2020)

    MathSciNet  Article  Google Scholar 

  39. 39.

    Raouf, F.; Maamar, B.; Mohammad, R.: Control of serial link manipulator using a fractional order controller. Int. Rev. Autom. Control 11(1), 1–6 (2018)

    Article  Google Scholar 

  40. 40.

    Ivanescu, M.; Popescu, N.; Popescu, D.; Channa, A.; Poboroniuc, M.: Exoskeleton hand control by fractional order models. Sensors 19(21), 4608 (2019)

    Article  Google Scholar 

  41. 41.

    Sanz, A.; Etxebarria, V.: Composite robust control of a laboratory flexible manipulator. In: Proceedings of the 44th IEEE Conference on Decision and Control, pp. 3614–3619. IEEE (2005)

  42. 42.

    Etxebarria, V.; Sanz, A.; Lizarraga, I.: Control of a lightweight flexible robotic arm using sliding modes. Int. J. Adv. Rob. Syst. 2(2), 11 (2005)

    Article  Google Scholar 

  43. 43.

    Mujumdar, A.A.; Kurode, S.: Second order sliding mode control for single link flexible manipulator. In: International Conference on Machines and Mechanisms (2013)

  44. 44.

    Shitole, C.; Sumathi, P.: Sliding DFT-based vibration mode estimator for single-link flexible manipulator. IEEE/ASME Trans. Mechatron. 20(6), 3249–3256 (2015)

    Article  Google Scholar 

  45. 45.

    Mujumdar, A.; Tamhane, B.; Kurode, S.: Observer-based sliding mode control for a class of noncommensurate fractional-order systems. IEEE/ASME Trans. Mechatron. 20(5), 2504–2512 (2015)

    Article  Google Scholar 

  46. 46.

    Ahmad, M.A.; Mohamed, Z.; Ismail, Z.H.: Hybrid input shaping and PID control of a flexible robot manipulator. J. Inst. Eng. 72(3), 56–62 (2009)

    Google Scholar 

  47. 47.

    Pham, D.T.; Koç, E.; Kalyoncu, M.; Tınkır, M.: Hierarchical PID controller design for a flexible link robot manipulator using the bees algorithm. Methods Genet. Algorithm 25, 32 (2008)

    Google Scholar 

  48. 48.

    Jnifene, A.; Andrews, W.: Experimental study on active vibration control of a single-link flexible manipulator using tools of fuzzy logic and neural networks. IEEE Trans. Instrum. Meas. 54(3), 1200–1208 (2005)

    Article  Google Scholar 

  49. 49.

    Sun, C.; He, W.; Hong, J.: Neural network control of a flexible robotic manipulator using the lumped spring-mass model. IEEE Trans. Syst. Man Cybern. Syst. 47(8), 1863–1874 (2016)

    Article  Google Scholar 

  50. 50.

    Sun, C.; Gao, H.; He, W.; Yu, Y.: Fuzzy neural network control of a flexible robotic manipulator using assumed mode method. IEEE Trans. Neural Netw. Learn. Syst. 99, 1–14 (2018)

    MathSciNet  Google Scholar 

  51. 51.

    Forbes, J.R.; Damaren, C.J.: Single-link flexible manipulator control accommodating passivity violations: theory and experiments. IEEE Trans. Control Syst. Technol. 20(3), 652–662 (2011)

    Article  Google Scholar 

  52. 52.

    Talole, S.E.; Kolhe, J.P.; Phadke, S.B.: Extended-state-observer-based control of flexible-joint system with experimental validation. IEEE Trans. Industr. Electron. 57(4), 1411–1419 (2009)

    Article  Google Scholar 

  53. 53.

    Quanser Inc.: SRV02 Rotary Flexible Link User Manual (2011)

  54. 54.

    Haubold, H.J.; Mathai, A.M.; Saxena, R.K.: Mittag-Leffler functions and their applications. J. Appl. Math. 2011, 298628 (2011).

Download references


This research was funded by the European Regional Development Fund in the Research Centre of Advanced Mechatronic Systems project, grant number \(CZ.02.1.01/0.0/0.0/16\_019/0000867\) within the Operational Programme Research, Development and Education.

Author information



Corresponding author

Correspondence to Abhaya Pal Singh.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Singh, A.P., Deb, D., Agrawal, H. et al. Modeling and Control of Robotic Manipulators: A Fractional Calculus Point of View. Arab J Sci Eng (2021).

Download citation


  • \(\Omega \)-domain
  • Fractional stability
  • Single flexible link robotic manipulator
  • 2DOF serial flexible joint robotic manipulator
  • Robotics
  • Pole placement control