Nonlinear optimal control of an active transfemoral prosthesis using state dependent Riccati equation approach

Document Type : Research Article

Authors

1 Faculty of Electrical, Biomedical and Mechatronics Engineering, Qazvin Branch, Islamic Azad University, Qazvin, Iran

2 Department of Electrical Engineering, AmirKabir University of Technology

Abstract

Nowadays, scientific and technological advances have created the ability to replace prosthetic legs with amputated limbs, which the design of a suitable controller is still being discussed by researchers. Therefore, according to the importance of the subject, in this paper, a combination of a nonlinear optimal control method based on the state-dependent Riccati equation approach with the integral state control technique is proposed for an active prosthetic leg for transfemoral amputees. The main objective of this paper is to optimize the energy consumption of the robot/prosthesis system and desirable tracking of the vertical displacement in hip and thigh and knee angles. Also, due to the robustness properties of the suggested controller is investigated sensitivity analysis against ±30% parametric uncertainty and compared with robust adaptive impedance control. The performance of the controller is assessed for both point-to-point motion and tracking modes by considering the saturation bounds of control signals. Finally, the simulation results show a decrease in control effort, desirable performance in tracking, and relatively good robustness in the presence of parametric uncertainty and constant disturbance. Numerical results indicate a significant reduction in energy consumption and total cost in this method compared to the robust adaptive impedance control.
 

Keywords

Main Subjects


[1] SF. Tabatabi Ghomshe, R. Osqueizadehi, SH. Navabi, Trans-Tibial Amputee Gait Correction through Real-Time Visual Feedback, Journal of Sport Biomechanics, 1(3) (2016) 25-32. (in Persian)
 [2] V. Azimi, D. Simon, H. Richter, Stable robust adaptive impedance control of a prosthetic leg, In Dynamic Systems and Control Conference (Vol. 57243, p. V001T09A003), American Society of Mechanical Engineers, (2015).
 [3] SM. Moosavi, Derivative-free Kalman filter-based control of nonlinear systems with application to transfemoral prostheses, (Doctoral dissertation, Cleveland State University), (2017).
[4] H. Richter, D. Simon, WA. Smith, S. Samorezov, Dynamic modeling, parameter estimation and control of a leg prosthesis test robot, Applied Mathematical Modeling, 39(2) (2015) 559-73. 
[5] V. Azimi, D. Simon, H. Richter, SA. Fakoorian, Robust composite adaptive transfemoral prosthesis control with non-scalar boundary layer trajectories, In 2016 American Control Conference (ACC), (2016) 3002-3007. IEEE. 
[6] D. Ebeigbe, D. Simon, H. Richter, Hybrid function approximation based control with application to prosthetic legs, In 2016 Annual IEEE Systems Conference (SysCon), (2016) 1-6. IEEE. 
[7] V. Azimi, S. Abolfazl Fakoorian, T. Tien Nguyen, D. Simon, Robust adaptive impedance control with application to a transfemoral prosthesis and test robot. Journal of Dynamic Systems, Measurement, and control, 140(12) (2018).
[8] V. Azimi, T. Shu, H. Zhao, R. Gehlhar, D. Simon, AD. Ames, Model-based adaptive control of transfemoral prostheses: theory, simulation, and experiments. IEEE Transaction on Systems, Man, and Cybernetics: Systems, (2019).
[9] A. Bavarsad, A. Fakharian, MB. Menhaj, Optimal Sliding Mode Controller for an Active Transfemoral Prosthesis Using State-Dependent Riccati Equation Approach, Arabian Journal for Science and Engineering, (2020) 1-14. https://doi.org/10.1007/s13369-020-04563-x
[10] JJ. Slotine, W. Li, Applied nonlinear control. Englewood Cliffs, NJ: Prentice hall; (1991).
[11] V. Azimi, A. Fakharian, Robust Mixed-Sensitivity Gain-Scheduled H∞ Tracking Control of a Nonlinear Time-Varying IPMSM via a T-S Fuzzy Model, In2012 9th France-Japan & 7th Europe-Asia Congress on Mechatronics (MECATRONICS)/13th Int'l Workshop on Research and Education in Mechatronics (REM), (2012) 345-352. IEEE.
[12] V. Azimi, MB. Menhaj, A. Fakharian, Fuzzy Mixed-Sensitivity Control of Uncertain Nonlinear Induction Motor. Majlesi Journal of Electrical Engineering, 8(2) (2014).
[13] V. Azimi, A. Fakharian, MB. Menhaj, Position and Current Control of an IPMSM by Using Loop-Shaping Methodology: Blending of H Mixed-Sensitivity Problem and T-S Fuzzy Model Scheme. Journal of Dynamic Systems, Measurement, and Control, 135(5) (2013).
[14] A. Fakharian, V. Azimi, Robust Mixed-Sensitivity H Control for a Class of MIMO Uncertain Nonlinear IPM Synchronous Motor via T-S Fuzzy Model. In2012 17th International Conference on Methods & Models in Automation & Robotics (MMAR), (2012) 546-551.
[15] V. Azimi, MB. Menhaj, A. Fakharian, Tool Position Tracking Control of a Nonlinear Uncertain Flexible Robot Manipulator by Using Robust H2/H∞ Controller via T-S Fuzzy Model, Sadhana, 40(2) (2015) 307-333.
  [16] T. Çimen, State-dependent Riccati equation (SDRE) control: A survey, IFAC Proceedings Volumes, 41(2) (2008) 3761-75.
[17] HT. Banks, BM. Lewis, HT. Tran, Nonlinear feedback controllers and compensators: A state-dependent Riccati equation approach, Computational Optimization and Applications, 37(2) (2007) 177-218. 
[18] JR. Cloutier, DT. Stansbery, The capabilities and art of state-dependent Riccati equation-based design, In Proceedings of the 2002 American Control Conference, (IEEE Cat. No. CH37301), Vol. 1 (2002) 86-91.
 [19] T. Çimen, SP. Banks, Nonlinear Optimal Tracking Control with Application to Super-tankers for Autopilot Design. Automatica, 40(11) (2004) 1845-1863.
 [20] A. Fakharian, MT. Hamidi Beheshti, A. Davari, Solving the Hamilton-Jacobi-Bellman equation using Adomian decomposition method, International Journal of Computer Mathematics, 87(12) (2010) 2769-85.
 [21] A. Fakharian, MT. Hamidi Beheshti, Solving Linear and Nonlinear Optimal Problem Using Modified Adomian Decomposition Method. Journal of Computer & Robotics, 1(1) (2010).
[22] J. Jung, SY. Park, SW. Kim, YH. Eun, YK. Chang, Hardware in- the-loop Simulations of Spacecraft Attitude Synchronization using the State-dependent Riccati Equation Technique, Advances in Space Research, 51(3) (2013) 434-449.
 [23] F. Ornelas-Tellez, JJ. Rico, R. Ruiz-Cruz, Optimal tracking for state-dependent coefficient factorized nonlinear systems, Asian Journal of Control, 16(3) (2014) 890-903.
 [24] H. Ghane, MB. Menhaj, Pseudo linear systems: stability analysis and limit cycle emergence. Journal of Control Engineering and Applied Informatics, 16(2) (2014) 78-89.
 [25] M. Innocenti, F. Baralli, F. Salotti, A. Caiti, Manipulator path control using SDRE, InProceeding of the 2000 American Control Conference, ACC (IEEE Cat. No.00CH36334) Vol. 5 (2000) 3348-3352. IEEE.
[26] S. Kiliçaslan, Tracking control of elastic joint parallel robots via state-dependent Riccati equation, Turkish Journal of Electrical Engineering & Computer Sciences, 23 (2) (2015) 522-538.
[27] M. Xin, SN. Balakrishnan, Z. Huang, Robust state dependent Riccati equation based robot manipulator control. InProceeding of the 2001 IEEE International Conference on Control Applications, (2001) 369-374.
[28] MH. Korayem, SR. Nekoo, State-dependent differential Riccati equation to track control of time-varying systems with state and control nonlinearities, ISA Transactions, 57 (2015)117-135.
[29] MH. Korayem, M. Irani, S. RAFINEKOU, Analysis of manipulators using SDRE: A closed loop nonlinear optimal control approach, (2010).
[30] MH. Korayem, SR. Nekoo, Finite-time state-dependent Riccati equation for time-varying nonaffine systems: Rigid and flexible joint manipulator control, ISA Transaction, 54 (2015) 125-144.
 [31] H. Beikzadeh, HD. Taghirad, Stability analysis of the discrete-time difference SDRE state estimator in a noisy environment, In 2009 IEEE International Conference on Control and Automation, (2009) 1751-1756. IEEE.
 [32] M. Habibnejad Korayem, S. Rafee Nako, N. Yousefi Lademakhi, The SDRE controller and estimator design for flexible joint manipulators in presence of noise and disturbance, Modares Mechanical Engineering, 16(8) (2016) 1-12. (in Persian)
[33] SS. Moosapour, G. Alizadeh, S. Khanmohammadi, Three-Dimensional Optimal Robust Guidance Law Design for Missile Using Sliding-Mode Control and SDRE Control. Journal of Control, 6(2) (2012) 55-64.
 [34] AK. Sedigh, Modern Control Systems, (2003). (in Persian)