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Hamilton-Jacobi inequality robust neural network control of a mobile wheeled robot. (English) Zbl 1440.70007

Summary: The work presented here presents a new approach to determine a robust neural network follow-up motion control of a mobile wheeled robot. The solution is a result of a Hamilton-Jacobi inequality, enabling synthesis of control of a non-linear object in terms of input to output stability. By applying Lyapunov’s theory of stability, it was demonstrated that all signals are limited, while the determined control provides a relatively high accuracy of actuated motion. The weights of the neural network were updated in real time and online. The produced simulation solutions confirmed the efficiency of the approach contemplated here.

MSC:

70E60 Robot dynamics and control of rigid bodies
93D05 Lyapunov and other classical stabilities (Lagrange, Poisson, \(L^p, l^p\), etc.) in control theory
93C85 Automated systems (robots, etc.) in control theory
70B15 Kinematics of mechanisms and robots
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[1] [1] Beeler, SC, Tran, HT, Banks, HT. Feedback control methodologies for nonlinear systems. J Optimiz Theor Appl 2000; 107: 1-33. · Zbl 0971.49023
[2] [2] Hendzel, Z. An adaptive critic neural network for motion control of a wheeled mobile robot. Nonlin Dyn. December2007; 50(4): 849-55. · Zbl 1170.70316
[3] [3] Hendzel, Z, Trojnacki, M. Neural network identifier of a four-wheeled mobile robot subject to wheel slip. J Automat Mob Robot Intell Syst 2014; 8: 24-30.
[4] [4] Hendzel, Z, Trojnacki, M. Neural network control of a four-wheeled mobile robot subject to wheel slip. In: Mechatronics: ideas for industrial applications (ed Kacprzyk, J ). Chapter 19. Series: Advances in Intelligent Systems and Computing. The Netherlands: Printforce, 2015, 187-201.
[5] [5] Fierro, R, Lewis, FL. Control of a nonholonomic mobile robot using neural networks. IEEE Trans Neural Networks 1998; 9: 589-600.
[6] [6] Lewis, FL, Jagannathan, S, Yesildirek, A. Neural network control of robot manipulators and nonlinear systems. London: Taylor & Francis, 1999.
[7] [7] Liu, J, Wang, X. Advanced sliding mode control for mechanical systems. Beijing: Tsinghua University Press, Springer, 2012. · Zbl 1253.93002
[8] [8] Giergiel, J, Zylski, W. Description of motion of a mobile robot by Maggie’s equations. J Theor App Mech 2005; 43: 511-521.
[9] [9] Yildirim, Ş . Vibration control of suspension systems using a proposed neural network. J Sound Vib 2004; 277: 1059-1069.
[10] [10] Giorgio, I., Culla, A, Del Vescovo, D. Multimode vibration control using several piezoelectric transducers shunted with a multiterminal network. Arch Appl Mech 2009; 79: 859-879. · Zbl 1176.74128
[11] [11] Giorgio, I., Galantucci, L, Della Corte, A. Piezo-electromechanical Smart Materials with distributed arrays of Piezoelectric Transducers: current and upcoming applications. Int J Appl Electromagn Mech 2015; 47: 1051-1084.
[12] [12] Pepe, G, Carcaterra, A, Giorgio, I. Variational Feedback Control for a nonlinear beam under an earthquake excitation. Math Mech Solids 2016; 21: 1234-1246. · Zbl 1370.74092
[13] [13] Rosi, G, Pouget, J, dell’Isola, F. Control of sound radiation and transmission by a piezoelectric plate with an optimized resistive electrode. Eur J Mech A Solids 2010; 29: 859-870. · Zbl 1481.74564
[14] [14] Shen, H, Qiu, JH, Ji, HL. A low-power circuit for piezoelectric vibration control by synchronized switching on voltage sources. Sensors Actuators A Phys 2010; 161: 245-255.
[15] [15] Alessandroni, S, Andreaus, U, dell’Isola, F. Piezo-electromechanical (PEM) Kirchhoff-Love plates. Eur J Mech A Solids 2004; 23: 689-702. · Zbl 1065.74562
[16] [16] Huber, N, Tsakmakis, Ch. A neural network tool for identifying the material parameters of a finite deformation viscoplasticity model with static recovery. Comput Meth Appl Mech Eng 2001; 191: 353-384. · Zbl 0991.74019
[17] [17] Turco, E. Tools for the numerical solution of inverse problems in structural mechanics: review and research perspectives. Eur J Environ Civ Eng 2017; 21: 509-554.
[18] [18] Placidi, L, Andreaus, U, Della Corte, A. Gedanken experiments for the determination of two-dimensional linear second gradient elasticity coefficients. Z Angew Math Phys 2015; 66: 3699-3725. · Zbl 1386.74018
[19] [19] Andreaus, U, dell’Isola, F, Giorgio, I. Numerical simulations of classical problems in two-dimensional (non) linear second gradient elasticity. Int J Eng Sci 2016; 108: 34-50. · Zbl 1423.74089
[20] [20] Placidi, L, Andreaus, U, Giorgio, I. Identification of two-dimensional pantographic structure via a linear D4 orthotropic second gradient elastic model. J Eng Math 2017; 103: 1-21. · Zbl 1390.74018
[21] [21] Reiher, JC, Giorgio, I, Bertram, A. Finite-element analysis of polyhedra under point and line forces in second-strain gradient elasticity. J Eng Mech 2017; 143: 04016112-1-13.
[22] [22] Giorgio, I, Andreaus, U, dell’Isola, F. Viscous second gradient porous materials for bones reconstructed with bio-resorbable grafts. Extreme Mech Lett 2017; 13: 141-147.
[23] [23] Giorgio, I, Andreaus, U, Scerrato, D. A visco-poroelastic model of functional adaptation in bones reconstructed with bio-resorbable materials. Biomech Model Mechanobiol 2016; 15: 1325-1343.
[24] [24] Giorgio, I., Scerrato, D. Multi-scale concrete model with rate dependent internal friction. Eur J Environ Civ Eng 2017; 21: 821-839.
[25] [25] Berke, L, Patnaik, SN, Murthy, PL. Application of artificial neural networks to the design optimization of aerospace structural components. NASA: Scientific and Technical Information Program, 13, 1993.
[26] [26] Saxena, S, Pathak, KK. Application of artificial neural networks for fully stressed design of Pratt and Howe Truss. Int J of New Technologies in Science and Engineering 2015; 2: 94-103
[27] [27] Greco, L, Cuomo, M. On the force density method for slack cable nets. Int J Solids Struct 2012; 49: 1526-1540.
[28] [28] dell’Isola, F, Lekszycki, T, Pawlikowski, M. Designing a light fabric metamaterial being highly macroscopically tough under directional extension: first experimental evidence. Zeitschrift fur angewandte Mathematik und Physik ZAMP 2015; 66: 3473-3498. · Zbl 1395.74002
[29] [29] Carcaterra, A, dell’Isola, F, Esposito, R. Macroscopic description of microscopically strongly inhomogenous systems: a mathematical basis for the synthesis of higher gradients metamaterials. Arch Ration Mech Anal 2015; 218: 1239-1262. · Zbl 1352.37193
[30] [30] Spagnuolo, M, Barcz, K, Pfaff, A. Qualitative pivot damage analysis in aluminum printed pantographic sheets: numerics and experiments. Mech Res Commun 2017; 83: 47-52.
[31] [31] dell’Isola, F, Steigmann, D, Della Corte, A. Synthesis of fibrous complex structures: designing microstructure to deliver targeted macroscale response. Appl Mech Rev 2016; 67: 060804.
[32] [32] dell’Isola, F, Della Corte, A, Giorgio, I. Pantographic 2D sheets: discussion of some numerical investigations and potential applications. Int J Non Lin Mech 2016; 80: 200-208.
[33] [33] Slotine, J-J.E, Li, W. Applied nonlinear control. Englewood Cliffs, NJ: Prentice Hall, 1991. · Zbl 0753.93036
[34] [34] Abu-Khalaf, M, Lewis, FL. Nonlinear H2/H∞ constrained feedback control. London: Springer, 2006. · Zbl 1116.93028
[35] [35] Van Der Schaft, AJ. L2-gain analysis of nonlinear systems and nonlinear state feedback H∞ control. IEEE Trans Autom Contr 1992; 37: 770-784. · Zbl 0755.93037
[36] [36] Van Der Schft, AJ . L_2-gain and passivity techniques in nonlinear control. London: Springer, 2000. · Zbl 0937.93020
[37] [37] Fariwata, SS, Filev, D, Langari, R. Fuzzy control. Chichester, England: John Wiley Sons, Ltd, 2000. · Zbl 0935.00024
[38] [38] Nash, J. Non-cooperative games. Ann Math 1951; 2: 286-295. · Zbl 0045.08202
[39] [39] Basar, T, Bernard, P. H_∞ optimal control and related minimax design problems. Boston: Birkhäuser, 1995. · Zbl 0835.93001
[40] [40] Lewis, FL, Vrabie, DL, Syroms, VL. Optimal control. 3rd ed.Hoboken, NJ: Wiley & Sons, 2012. · Zbl 1284.49001
[41] [41] Hendzel, Z, Szuster, M. Approximate dynamic programming in the sensor-based navigation of the wheeled mobile robot. Solid State Phenom 2015; 220-221: 60-66.
[42] [42] Hendzel, Z, Burghardt, A, Szuster, M. Reinforcement learning in discrete neural control of the underactuated system. Part I, LNAI 7894. Berlin Heidelberg: Springer ICAISC, 2013, 64-75.
[43] [43] Hendzel, Z, Szuster, M. Adaptive dynamic programming methods in control of wheeled mobile robot. Int J Appl Mech Eng 2012; 17: 837-851. · Zbl 1221.70043
[44] [44] Hendzel, Z, Penar, P. Application of differential games in mechatronic control system. Int J Appl Mech Eng 2016; 21: 867-878.
[45] [45] Krener, AJ. The local solvability of a Hamilton-Jacobi-Bellman PDE around a non hyperbolic critical point. SIAM J Contr Optim 2001; 39: 1461-1484. · Zbl 1001.49026
[46] [46] Aliyu, MDS . Nonlinear H-infinity control, Hamiltonian systems and Hamilton-Jacobi equations. Boca Raton, FL: CRC Press, 2011. · Zbl 1243.93002
[47] [47] Cheng, D, Hu, X, Shen, T. Analysis and design of nonlinear control systems. Heidelberg: Springer Science & Business Media, 2011.
[48] [48] Yang, X, Liu, D, Ma, H. Online approximate solution of HJI equation for unknown constrained-input nonlinear continuous-time systems. Int J 2016; 328: 435-454. · Zbl 1391.49053
[49] [49] Isidori, A, Kang, W. H_∞ control via measurement feedback for general nonlinear systems. IEEE Trans Autom Contr 1995; 40: 466-472. · Zbl 0822.93029
[50] [50] Liu, J. Radial basis function (RBF) neural network control for mechanical systems. Beijing: Tsinghua University Press and Berlin Heidelberg: Springer, 2013. · Zbl 1277.93003
[51] [51] Chen, YH. Equations of motion of mechanical systems under servo constraints: the Maggi approach. Mechatronics 2008; 18: 208-2017.
[52] [52] Dopico, D, Zhu, Y, Sandu, A. Direct and adjoint sensitivity analysis of multibody systems using Maggi’s equations. In: Proceedings of the ASME 2013 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference IDETC/CIE 2013, 4-7 August, Portland, Oregon, USA, 2013.
[53] [53] Jalón, JG, Alfonso Callejo, A, Hidalgo, AF. Efficient solution of Maggi’s equations. J Comput Nonlin Dynam 2011; 7: 021003.
[54] [54] Kurdila, A, Papastavridis, JG, Kamat, MP. Role of Maggi’s equations in computational methods for constrained multibody systems. J Guid Contr Dynam 1990; 13: 113-120. · Zbl 0712.70027
[55] [55] Soltakhanov, AhKh, Yushkov, MP, Zegzhda, SA.
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