Modeling Dry Friction in Multibody Mechanical Systems and Its Applications to the Underactuated Control of an Overhead Crane-Payload System

Document Type : Research Paper

Authors
1 Department of Industrial Engineering, University of Salerno, Via Giovanni Paolo II, 132, 84084 Fisciano, Italy
2 Spin-Off MEID4 s.r.l., University of Salerno, Via Giovanni Paolo II, 132, 84084 Fisciano, Italy
Abstract
This paper presents a comprehensive analysis of dry friction modeling for conducting reliable computer simulations in the MATLAB and SIMULINK environments. In particular, this investigation highlights practical issues relevant to studying underactuated multibody systems and implementing control strategies suitable for this family of mechanical systems within the SIMSCAPE MULTIBODY framework. Another significant contribution of this study is the development of a novel nonlinear control algorithm that combines feedforward and feedback approaches. This enables the control of underactuated dynamical systems, which are typically found in mechanical engineering applications, such as an overhead crane-payload system, serving as the case study analyzed in this work. Beginning with the preliminary analysis of a benchmark problem, a demonstrative example, and a control example, this paper presents several scenarios that consider dry friction and the feedforward plus feedback controller both separately and in combination. To further explore the performance of the adopted control strategy for the case study, critical remarks are provided on the performance of the control policy developed in this study under wind gust conditions, which serve as random disturbances for the crane-payload system. The numerical results presented in this paper for all examined dynamical systems emphasize the importance of accurate dry friction modeling in the overall nonlinear motion control of underactuated multibody mechanical systems.
Keywords
Subjects

Publisher’s Note Shahid Chamran University of Ahvaz remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

[1] Ludema, K., Friction, Wear, Lubrication: A Textbook in Tribology, CRC Press, Boca Raton, 1996.
[2] Piedbœuf, J.C., De Carufel, J., Hurteau, R., Friction and stick-slip in robots: simulation and experimentation, Multibody System Dynamics, 2000, 4, 341–354.
[3] Armstrong-Hélouvry, B., Dupont, P., De Wit, C.C., A survey of models, analysis tools and compensation methods for the control of machines with friction, Automatica, 1994, 30(7), 1083–1138.
[4] Oden, J., Martins, J., Models and computational methods for dynamic friction phenomena, Computer methods in applied mechanics and engineering, 1985, 52(1-3), 527–634.
[5] Meirovitch, L., Fundamentals of vibrations/, McGraw-Hill„ Boston, 2001.
[6] Marques, F., Flores, P., Claro, J.P., Lankarani, H.M., Modeling and analysis of friction including rolling effects in multibody dynamics: a review, Multibody System Dynamics, 2019, 45, 223–244.
[7] Marques, F., Flores, P., Pimenta Claro, J., Lankarani, H.M., A survey and comparison of several friction force models for dynamic analysis of multibody mechanical systems, Nonlinear Dynamics, 2016, 86, 1407–1443.
[8] Farhat, N., Mata, V., Page, A., Díaz-Rodríguez, M., Dynamic simulation of a parallel robot: Coulomb friction and stick–slip in robot joints, Robotica, 2010, 28(1), 35–45.
[9] Kermani, M.R., Patel, R.V., Moallem, M., Friction identification and compensation in robotic manipulators, IEEE Transactions on Instrumentation and Measurement, 2007, 56(6), 2346–2353.
[10] Dahl, P.R., Solid friction damping of mechanical vibrations, AIAA journal, 1976, 14(12), 1675–1682.
[11] Pennestri, E., Valentini, P., Vita, L., Multibody dynamics simulation of planar linkages with dahl friction, Multibody System Dynamics, 2007, 17, 321–347.
[12] Pennestrì, E., Rossi, V., Salvini, P., Valentini, P.P., Review and comparison of dry friction force models, Nonlinear dynamics, 2016, 83, 1785–1801.
[13] De Wit, C.C., Olsson, H., Astrom, K.J., Lischinsky, P., A new model for control of systems with friction, IEEE Transactions on automatic control, 1995, 40(3), 419–425.
[14] Craig, J., Introduction to Robotics Mechanics and Control, Global Edition, Pearson International, Harlow, 2021.
[15] Spong, M.W., Underactuated mechanical systems, B. Siciliano, K.P. Valavanis, eds., Control Problems in Robotics and Automation, Springer Berlin Heidelberg, Berlin, Heidelberg, 135–150.
[16] Pappalardo, C.M., Guida, D., On the dynamics and control of underactuated nonholonomic mechanical systems and applications to mobile robots, Archive of Applied Mechanics, 2019, 89, 669–698.
[17] Wu, L., Ceccarelli, M., A numerical simulation for design and operation of an underactuated finger mechanism for larm hand, Mechanics based design of structures and machines, 2009, 37(1), 86–112.
[18] Wang, J.J., Kumbasar, T., Big bang-big crunch optimized hierarchical sliding-mode control of xz inverted pendulum, Simulation Modelling Practice and Theory, 2018, 86, 25–35.
[19] Pappalardo, C.M., Del Giudice, M., Oliva, E.B., Stieven, L., Naddeo, A., Computer-aided design, multibody dynamic modeling, and motion control analysis of a quadcopter system for delivery applications, Machines, 2023, 11(4), 464.
[20] Choudhry, O.A., Wasim, M., Ali, A., Choudhry, M.A., Iqbal, J., Modelling and robust controller design for an underactuated self-balancing robot with uncertain parameter estimation, Plos One, 2023, 18(8), e0285495.
[21] Javadi, M., Harnack, D., Stocco, P., Kumar, S., Vyas, S., Pizzutilo, D., Kirchner, F., Acromonk: A minimalist underactuated brachiating robot, IEEE Robotics and Automation Letters, 2023, 8(6), 3637–3644.
[22] Zhuo, J., Liu, H., Tian, X., Mai, Q., Self-adjusting prescribed performance h1 trajectory tracking control for underactuated autonomous underwater vehicles, IEEE Access, 2024, 12, 70146–70159.
[23] Blajer, W., Kołodziejczyk, K., Modeling of underactuated mechanical systems in partly specified motion, Journal of Theoretical and Applied Mechanics, 2008, 46, 383–394.
[24] Tian, Z., Yu, L., Ouyang, H., Zhang, G., Sway and disturbance rejection control for varying rope tower cranes suffering from friction and unknown payload mass, Nonlinear Dynamics, 2021, 105(4), 3149–3165.
[25] Li, G., Ma, X., Li, Z., Li,Y., Optimal trajectory planning strategy for underactuated overhead crane with pendulum-sloshing dynamics and full-state constraints, Nonlinear Dynamics, 2022, 109(2), 815–835.
[26] Zhang, M., Ma, X., Chai, H., Rong, X., Tian, X., Li, Y., A novel online motion planning method for double-pendulum overhead cranes, Nonlinear Dynamics, 2016, 85, 1079–1090.
[27] Chen, J., Shi, R., Ouyang, H., Data-driven discrete learning sliding mode control for overhead cranes suffering from disturbances, Nonlinear Dynamics, 2025, 113(4), 3357–3372.
[28] Rigatos, G., Siano, P., Abbaszadeh, M., Nonlinear h-infinity control for 4-dof underactuated overhead cranes, Transactions of the Institute of Measurement and Control, 2018, 40(7), 2364–2377.
[29] Pappalardo, C.M., Guida, D., Control of nonlinear vibrations using the adjoint method, Meccanica, 2017, 52, 2503–2526.
[30] Schuderer, M., Rill, G., Schaeffer, T., Schulz, C., Friction modeling from a practical point of view, Multibody System Dynamics, 2024, 1–18.
[31] Flores, P., Ambrósio, J., Lankarani, H.M., Contact-impact events with friction in multibody dynamics: Back to basics, Mechanism and Machine Theory, 2023, 184, 105305.
[32] Mostaghel, N., Davis,T., Representations of coulomb friction for dynamic analysis, Earthquake Engineering & Structural Dynamics, 1997, 26(5), 541–548.
[33] Pozzi, M., Achilli, G.M., Valigi, M.C., Malvezzi, M., Modeling and simulation of robotic grasping in simulink through simscape multibody, Frontiers in Robotics and AI, 2022, 9, 873558.
[34] Mohapatra, S., Srivastava, R., Khera, R., Implementation of a two wheel self-balanced robot using matlab simscape multibody, 2019 Second International Conference on Advanced Computational and Communication Paradigms (ICACCP), IEEE, 1–3.
[35] Bayen, A.M., Siauw, T., An Introduction to MATLAB® Programming and Numerical Methods for Engineers, 2014.
[36] Otto, S., Denier, J., An introduction to programming and numerical methods in MATLAB, 2005.
[37] Berger, E., Friction modeling for dynamic system simulation, Appl. Mech. Rev., 2002, 55(6), 535–577.
[38] Armstrong, B.S., Chen, Q., The z-properties chart, IEEE Control Systems Magazine, 2008, 28(5), 79–89.
[39] Padthe, A.K., Drincic, B., Oh, J., Rizos, D.D., Fassois, S.D., Bernstein, D.S., Duhem modeling of friction-induced hysteresis, IEEE Control Systems Magazine, 2008, 28(5), 90–107.
[40] Cirelli, M., Autiero, M., Belfiore, N.P., Paoli, G., Pennestrì, E., Valentini, P.P., Review and comparison of empirical friction coefficient formulation for multibody dynamics of lubricated slotted joints, Multibody System Dynamics, 2024, 1–22.
[41] Verulkar, A., Sandu, A., Sandu, C., Dopico, D., On effects of continuous event approximations on sensitivities and optimization of multibody systems with friction, EasyChair Preprint no. 13476, Madison, EasyChair, 2024.
[42] Oprea, R.A., Cruceanu, C., Spiroiu, M.A., Alternative friction models for braking train dynamics, Vehicle System Dynamics, 2013, 51(3), 460–480.
[43] The MathWorks Inc., Rotational Friction, https://www.mathworks.com/help/simscape/ref/rotationalfriction.html, 2022.
[44] The MathWorks Inc., Translational Friction, https://www.mathworks.com/help/simscape/ref/translationalfriction.html, 2022.
[45] Ngoc, T.L., Nguyen, T.L., Quasi-physical modeling of robot irb 120 using simscape multibody for dynamicand control simulation, Turkish Journal of Electrical Engineering and Computer Sciences, 2020, 28(4), 1949–1964.
[46] Burkus, E., Awrejcewicz, J., Odry, P., A validation procedure to identify joint friction, reductor self-locking and gear backlash parameters, Archive of Applied Mechanics, 2020, 90(7), 1625–1641.
[47] Boschetti, G., Sinico, T., Designing digital twins of robots using simscape multibody, Robotics, 2024, 13(4), 62.
[48] Tang, Y., Huang, Y., Lindbeck, E., Lizza, S., VanZwieten, J., Tom, N., Yao, W., Wec fault modelling and condition monitoring: A graph-theoretic approach, IET Electric Power Applications, 2020, 14(5), 781–788.
[49] Bettega, J., Richiedei, D., Tamellin, I., Trevisani, A., Model inversion for precise path and trajectory tracking in an underactuated, non-minimum phase, spatial overhead crane, Journal of Vibration Engineering & Technologies, 2023, 11(8), 3841–3857.
[50] Bettega, J., Richiedei, D., Tamellin, I., Trajectory planning through model inversion of an underactuated spatial gantry crane moving in structured cluttered environments, Actuators, vol. 13, MDPI, 176.
[51] Fan, B., Zhang, Y., Sun, L., Wang, L., Liao, Z., Passivity and underactuated modeling-based load energy coupling control for three-dimensional overhead crane systems, Mechatronics, 2023, 96, 103062.
[52] Wang, S., Jin, W., Recursive terminal sliding mode control for the 3d overhead crane systems with motion planning, Mechatronics, 2024, 104, 103267.
[53] Yang, J.H., Yang, K.S., Adaptive coupling control for overhead crane systems, Mechatronics, 2007, 17(2-3), 143–152.
[54] Cibicik, A., Egeland, O., Dynamic modelling and force analysis of a knuckle boom crane using screw theory, Mechanism and Machine Theory, 2019, 133, 179–194.
[55] Kolar, B., Schlacher, K., Flatness based control of a gantry crane, IFAC Proceedings Volumes, 2013, 46(23), 487–492.
[56] Lobe, A., Ettl, A., Steinböck, A., Kugi, A., Flatness-based nonlinear control of a three-dimensional gantry crane, IFAC-PapersOnLine, 2018, 51(22), 331–336.
[57] Peng, K., Garcia, A., Ferri, A., Modeling and control of crane payload lift-off and lay-down operations, FME Trans., 2016, 44(3), 237–248.
[58] Wang, M., Liu, J., Modeling and vibration suppression for overhead crane in planar space with nonlinear time-varying actuator faults and uncertain control directions, Nonlinear Dynamics, 2024, 1–15.
[59] Khair Al-Solihat, M., Al Saaideh, M., Al-Rawashdeh, Y.M., Al Janaideh, M., On investigating dynamic coupling in floating platform and overhead crane interactions: modeling and control, Nonlinear Dynamics, 2024, 1–17.
[60] Bello, M., Mohamed, Z., Efe, M., Ishak, H., Modelling and dynamic characterisation of a double-pendulum overhead crane carrying a distributedmass payload, Simulation Modelling Practice and Theory, 2024, 134, 102953.
[61] Lopez Rojas, A.D., Mendoza-Trejo, O., Padilla-García, E.A., Ortiz Morales, D., Cruz-Villar, C.A., La Hera, P., Design, rapid manufacturing and modeling of a reduced-scale forwarder crane with closed kinematic chain, Mechanics Based Design of Structures and Machines, 2023, 51(12), 6748–6773.
[62] Alfares, M., Alhazza, K., Comparative analysis on the performance of different types of input-and command-shaping controllers in minimizing payload residual vibration of an overhead crane with an inclined supporting track, International Journal of Mechanical System Dynamics, 2024, 4(1), 22–33.
[63] Lu, B., Fang, Y., Sun, N., Sliding mode control for underactuated overhead cranes suffering from both matched and unmatched disturbances, Mechatronics, 2017, 47, 116–125.
[64] Zhang, S., He, X., Chen, Q., Zhu, Z., Partially saturated coupling-based control for underactuated overhead cranes with experimental verification, Mechatronics, 2019, 63, 102284.
[65] Schatz, J., Caverly, R.J., Payload trajectory tracking of a 5-dof tower crane with a varying-length hoist cable: A passivity-based adaptive control approach, Mechatronics, 2023, 94, 103027.
[66] Ku, N., Ha, S., Roh, M.I., Crane modeling and simulation in offshore structure building industry, International Journal of Computer Theory and Engineering, 2014, 6(3), 278.
[67] Hong, K.S., Shah, U.H., Dynamics and control of industrial cranes, Springer, 2019.
[68] Li, K., Liu, M., Yu, Z., Lan, P., Lu, N., Multibody system dynamic analysis and payload swing control of tower crane, Proceedings of the Institution of Mechanical Engineers, Part K: Journal of Multi-body Dynamics, 2022, 236(3), 407–421.
[69] Chu, Y., Æsøy, V., Ehlers, S., Zhang, H., Integrated multi-domain system modelling and simulation for offshore crane operations, Ship Technology Research, 2015, 62(1), 36–46.
[70] Khorshid, E., Al-Fadhli, A., Optimal command shaping design for a liquid slosh suppression in overhead crane systems, Journal of Dynamic Systems, Measurement, and Control, 2021, 143(2), 021005.
[71] De Simone, M.C., Veneziano, S., Pace, R., Guida, D., Multibody analysis of sloshing effect in a glass cylinder container for visual inspection activities, Applied Sciences, 2024, 14(11), 4522.
[72] Jin, G., Wang, S., Gao,Y., Sun, M., Chen, H., Sun,Y., Dynamic analysis and experimental research on anti-swing control of distributed mass payload for marine cranes, Journal of Marine Science and Engineering, 2025, 13(6), 1112.
[73] Jin, B., Zeng, J., Gao, P., Zhang, H., Ge, S., Dynamic modeling and validation of dual-cable double-pendulum systems for gantry cranes, Machines, 2025, 13(8), 676.
[74] Jaiswal, S., Poursina, M., Historical evolution of heavy machinery and a general role of multibody dynamics, Machines, 2025, 13(8), 741.
[75] Cao, M., Xu, M., Gao, Y., Wang, T., Deng, A., Liu, Z., Advanced control for shipboard cranes with asymmetric output constraints, Journal of Marine Science & Engineering, 2025, 13(1).
[76] Siciliano, B., Sciavicco, L., Villani, L., Oriolo, G., Robotics: Modelling, planning and control, vol. 0, 2009.
[77] Lynch, K.M., Park, F.C., Modern robotics, Cambridge University Press, Cambridge, England, 2017.
[78] Jazar, R.N., Theory of applied robotics, Springer, New York, NY, 2nd ed., 2010.
[79] Seifried, R., Dynamics of underactuated multibody systems: Modeling, control and optimal design, vol. 205, 2014.
[80] Lozano, R., Fantoni, I., Non-linear control for underactuated mechanical systems, Communications and Control Engineering, Springer, London, England, 2001.
[81] The MathWorks Inc., Spatial Contact Force, https://www.mathworks.com/help/sm/ref/spatialcontactforce.html, 2022.
[82] Eldirdiry, O., Zaier, R., Al-Yahmedi, A., Bahadur, I., Alnajjar, F., Modeling of a biped robot for investigating foot drop using matlab/simulink, Simulation Modelling Practice and Theory, 2020, 98, 101972.
[83] The MathWorks Inc., Translational Multibody Interface, https://www.mathworks.com/help/simscape/ref/translationalmultibodyinterface.html, 2022.
[84] The MathWorks Inc., Rotational Multibody Interface, https://www.mathworks.com/help/simscape/ref/rotationalmultibodyinterface.html, 2022.
[85] Pappalardo, C.M., Magaldi, T., Masucci, L., La Regina, R., Naddeo, A., Virtual prototyping, multibody dynamics, and control design of an adaptive lift table for material handling, Journal of Applied and Computational Mechanics, 2024, 10(4), 659-693.
[86] Pappalardo, C.M., Guida, D., A comparative study of the principal methods for the analytical formulation and the numerical solution of the equations of motion of rigid multibody systems, Archive of Applied Mechanics, 2018, 88, 2153–2177.
[87] Slotine, J.J.E., Li, W., Applied nonlinear control, Prentice hall Englewood Cliffs, NJ, 1991.
[88] Betts, J.T., Practical Methods for Optimal Control and Estimation Using Nonlinear Programming, Second Edition, Society for Industrial and Applied Mathematics, Philadelphia, 2010.
[89] Kirk, D.E., Optimal control theory: an introduction, Courier Corporation, Mineola, New York, 2004.
[90] Lewis, F.L., Vrabie, D., Syrmos, V.L., Optimal control, John Wiley & Sons, Hoboken, New Jersey, 2012.
[91] Wang, Y.P., Liao, W.H., Lee, C.L., A state-space approach for dynamic analysis of sliding structures, Engineering Structures, 2001, 23(7), 790–801.
[92] Shabana, A.A., Dynamics of multibody systems, Cambridge University Press, Cambridge, 2020.
[93] Shabana, A.A., Theory of vibration: an introduction, Springer, Cham, Switzerland, 2018.
[94] The MathWorks Inc., Spring and Damper Force, https://www.mathworks.com/help/sm/ref/springanddamperforce.html, 2022.
[95] Rabinowicz, E., Stick and slip, Scientific American, 1956, 194(5), 109–119.
[96] McMillan, A., A non-linear friction model for self-excited vibrations, Journal of sound and vibration, 1997, 205(3), 323–335.
[97] Rill, G., Schaeffer, T., Schuderer, M., Lugre or not lugre, Multibody System Dynamics, 2024, 60(2), 191–218.
[98] Marques, F., Flores, P., Lankarani, H.M., On the frictional contacts in multibody system dynamics, Multibody Dynamics: Computational Methods and Applications, 2016, 67–91.
[99] Siqueira, T.M., Coda, H.B., Improved friction model applied to plane sliding connections by a large deformation fem formulation, Latin American Journal of Solids and Structures, 2023, 20, e472.
[100] Shabana, A.A., Computational dynamics, John Wiley & Sons, The Atrium, Southern Gate, Chichester, West Sussex, United Kingdom, 2009.