A modified homotopy optimization for parameter identification in dynamic systems with backlash discontinuity


Model-based control considers system dynamics to solve challenging control problems; recently, the amount of activity in developing model-based controllers is growing, specifically in rehabilitation robotics. The performance of this controller depends on how accurate the system dynamics has been modeled. Dynamic parameter identification (DPI) of the systems is required for optimal performance of the model-based controller. Current DPI methods are more suitable for systems with continuous dynamics. If any type of discontinuity (e.g., backlash) is present in the system, the DPI may have numerical problems for convergence. In this work, we propose a modified homotopy optimization to identify parameters of a system with mechanical discontinuity (i.e., backlash). The performance of the proposed method was first evaluated through a computer simulation on a system with sandwiched backlash. Results of the DPI showed that the proposed homotopy optimization can identify the discontinuous system parameters with a good accuracy. It was found that ignoring the backlash in the system dynamics imposes large errors in the system DPI. After verifying the proposed method using computer simulations, the DPI was implemented to identify the parameters of a rehabilitation robot with actuator backlash. The proposed method provided a better estimate of the system parameters compared to the no-backlash DPI of the experimental robot. Despite the noise in velocity and acceleration due to the numerical differentiation of the sampled angle measurements, the forward dynamics results are quite accurate for all of the tested configurations with the discontinuous backlash model.

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  1. 1.

    Brown, P., McPhee, J.: A continuous velocity-based friction model for dynamics and control with physically meaningful parameters. J. Comput. Nonlinear Dyn. 11(5), 054502 (2016). https://doi.org/10.1115/1.4033658

    Article  Google Scholar 

  2. 2.

    Chen, S., Billings, S.A., Grant, P.M.: Non-linear system identification using neural networks. Int. J. Control 51(6), 1191–1214 (1990). https://doi.org/10.1080/00207179008934126

    Article  MATH  Google Scholar 

  3. 3.

    Ding, M., Hirasawa, K., Kurita, Y., Takemura, H., Takamatsu, J., Mizoguchi, H., Ogasawara, T.: Pinpointed muscle force control in consideration of human motion and external force. In: 2010 IEEE International Conference on Robotics and Biomimetics (ROBIO), pp. 739–744. IEEE (2010). https://doi.org/10.1109/ROBIO.2010.5723418

  4. 4.

    Enikov, E., Stepan, G.: Microchaotic motion of digitally controlled machines. J. Vib. Control 4(4), 427–443 (1998). https://doi.org/10.1177/107754639800400405

    MathSciNet  Article  MATH  Google Scholar 

  5. 5.

    Ghannadi, B., Mehrabi, N., Sharif Razavian, R., McPhee, J., Razavian, R.S., McPhee, J.: Nonlinear model predictive control of an upper extremity rehabilitation robot using a two-dimensional human-robot interaction model. In: IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), pp. 502–507. IEEE, Vancouver, British Columbia, Canada (2017). https://doi.org/10.1109/IROS.2017.8202200

  6. 6.

    Jiang, Z.H., Ishida, T., Sunawada, M.: Neural network aided dynamic parameter identification of robot manipulators. In: 2006 IEEE International Conference on Systems, Man and Cybernetics, pp. 3298–3303. IEEE (2006). https://doi.org/10.1109/ICSMC.2006.384627

  7. 7.

    Ljung, L.: System Identification: Theory for User, 2nd edn. PTR Prentice Hall Information and System Sciences Series (1987). https://doi.org/10.1016/0005-1098(89)90019-8

  8. 8.

    Maciejasz, P., Eschweiler, J., Gerlach-Hahn, K., Jansen-Troy, A., Leonhardt, S.: A survey on robotic devices for upper limb rehabilitation. Journal of NeuroEngineering and Rehabilitation 11(3) (2014). https://doi.org/10.1186/1743-0003-11-3

  9. 9.

    Nordin, M., Gutman, P.O.: Controlling mechanical systems with backlash–a survey. Automatica 38(10), 1633–1649 (2002). https://doi.org/10.1016/S0005-1098(02)00047-X

    MathSciNet  Article  MATH  Google Scholar 

  10. 10.

    Nyarko, E.K., Scitovski, R.: Solving the parameter identification problem of mathematical models using genetic algorithms. Appl. Math. Comput. 153(3), 651–658 (2004). https://doi.org/10.1016/S0096-3003(03)00661-1

    MathSciNet  Article  MATH  Google Scholar 

  11. 11.

    Proietti, T., Crocher, V., Roby-Brami, A., Jarrasse, N.: Upper-limb robotic exoskeletons for neurorehabilitation: a review on control strategies. IEEE Rev. Biomed. Eng. 9, 4–14 (2016). https://doi.org/10.1109/RBME.2016.2552201

    Article  Google Scholar 

  12. 12.

    Sharifi, M., Behzadipour, S., Vossoughi, G.: Nonlinear model reference adaptive impedance control for human-robot interactions. Control Eng. Pract. 32, 9–27 (2014). https://doi.org/10.1016/j.conengprac.2014.07.001

    Article  Google Scholar 

  13. 13.

    Soderstrom, T., Stoica, P.G.: System Identification. Prentice-Hall International, Englewood Cliffs (1988)

    Google Scholar 

  14. 14.

    Sreenivasa, M., Ayusawa, K., Nakamura, Y.: Modeling and identification of a realistic spiking neural network and musculoskeletal model of the human arm, and an application to the stretch reflex. IEEE Trans. Neural Syst. Rehabil. Eng. 24(5), 591–602 (2016). https://doi.org/10.1109/TNSRE.2015.2478858

    Article  Google Scholar 

  15. 15.

    Su, C.Y., Stepanenko, Y., Svoboda, J., Leung, T.: Robust adaptive control of a class of nonlinear systems with unknown backlash-like hysteresis. IEEE Trans. Autom. Control 45(12), 2427–2432 (2000). https://doi.org/10.1109/9.895588

    MathSciNet  Article  MATH  Google Scholar 

  16. 16.

    Tao, G., Kokotovic, P.V.: Adaptive Control of Systems with Actuator and Sensor Nonlinearities. Wiley-Interscience, New York (1996)

    Google Scholar 

  17. 17.

    Tao, G., Ma, X., Ling, Y.: Optimal and nonlinear decoupling control of systems with sandwiched backlash. Automatica 37(2), 165–176 (2001). https://doi.org/10.1016/S0005-1098(00)00153-9

    MathSciNet  Article  MATH  Google Scholar 

  18. 18.

    Taware, A., Tao, G.: Control of Sandwich Nonlinear Systems. Lecture Notes in Control and Information Sciences, vol. 288. Springer, Berlin (2006)

    Google Scholar 

  19. 19.

    Thanh, T.D., Kotlarski, J., Heimann, B., Ortmaier, T.: Dynamics identification of kinematically redundant parallel robots using the direct search method. Mech. Mach. Theory 52, 277–295 (2012). https://doi.org/10.1016/j.mechmachtheory.2012.02.002

    Article  Google Scholar 

  20. 20.

    Vyasarayani, C.P., Uchida, T., Carvalho, A., McPhee, J.: Parameter identification in dynamic systems using the homotopy optimization approach. Multibody Syst. Dyn. 26(4), 411–424 (2011). https://doi.org/10.1007/s11044-011-9260-0

    MathSciNet  Article  MATH  Google Scholar 

  21. 21.

    Vyasarayani, C.P., Uchida, T., McPhee, J.: Single-shooting homotopy method for parameter identification in dynamical systems. Phys. Rev. 85(3), 036,201 (2012). https://doi.org/10.1103/PhysRevE.85.036201

    Article  Google Scholar 

  22. 22.

    Wu, J., Wang, J., You, Z.: An overview of dynamic parameter identification of robots. Robot. Comput. Integr. Manuf. 26(5), 414–419 (2010). https://doi.org/10.1016/j.rcim.2010.03.013

    Article  Google Scholar 

  23. 23.

    Ye, M.: Parameter identification of dynamical systems based on improved particle swarm optimization. In: Intelligent Control and Automation, pp. 351–360. Springer Berlin Heidelberg (2006). https://doi.org/10.1007/978-3-540-37256-1

  24. 24.

    Zhou, J., Wen, C.: Nonsmooth nonlinearities. In: Adaptive Backstepping Control of Uncertain Systems, Lecture Notes in Control and Information Sciences, vol. 372, pp. 83–96. Springer Berlin Heidelberg, Berlin, Heidelberg (2008). https://doi.org/10.1007/978-3-540-77807-3

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We thank the anonymous reviewer who provided very helpful suggestions to improve the paper. This work was funded by the Canada Research Chairs (CRC) program and the Natural Sciences and Engineering Research Council of Canada (NSERC). The authors wish to thank Quanser Inc. for providing the upper extremity rehabilitation robot and Toronto Rehabilitation Institute (TRI) for collaborating.

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Correspondence to Borna Ghannadi.



The DPI of the simulation setup is also done by other conventional optimization methods: Bound Optimization by Quadratic Approximation (BOBYQA) and Pattern Search (PS) in Table 4, and Particle Swarm Optimization (PSO) and Genetic Algorithm (GA) in Table 5. The objective functions for these methods are similar to the SQP method.

Table 4 Simulation setup DPI values using BOBYQA and PS optimization methods
Table 5 Simulation setup DPI values using PSO and GA optimization methods

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Ghannadi, B., Sharif Razavian, R. & McPhee, J. A modified homotopy optimization for parameter identification in dynamic systems with backlash discontinuity. Nonlinear Dyn 95, 57–72 (2019). https://doi.org/10.1007/s11071-018-4550-1

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  • Parameter identification
  • System dynamics
  • Sandwiched backlash
  • Homotopy optimization
  • Rehabilitation robot