Stiffness Model Reduction for Manipulators with

S. K. Mikhel ; A. S. Klimchik

Russian journal of nonlinear dynamics, Tome 17 (2021) no. 3, pp. 347-360.

Résumé

The accuracy of the robot positioning during material processing can be improved if the deformation under the load is taken into account. A manipulator stiffness model can be obtained using various approaches which differ in the degree of detail and computational complexity. Regardless of the model, its practical application requires knowledge of the stiffness parameters of the robot components, which implies solving the identification problem. In this work, we consider a reduced stiffness model, which assumes that the manipulator links are rigid, while the joints are compliant and include both elasticities in the joints themselves and the elastic properties of the links. This simplification reduces the accuracy of the model, but allows us to identify the stiffness parameters, which makes it suitable for practical application. In combination with a double encoders measurement system, this model allows for real-time compensation of compliance errors, that is, the deviation of the real end-effector position from the calculated one due to the deformation of the robot under load. The paper proposes an algebraic approach to determining the parameters of the reduced model in a general form. It also demonstrates several steps that can be done to simplify computations. First, it introduces the backward semianalytical Jacobian computation technique, which allows reducing the number of operations for the manipulator with virtual joints. Second, it provides an algorithm for the calculation of the required intermediate matrices without explicit Jacobian calculation and using more compact expressions. To compare the proposed techniques with the experimental approach, the robot deformation under load is simulated and the tool displacement is estimated. It is shown that both approaches are equivalent in terms of accuracy. While the experimental method is easier to implement, the algebraic approach allows analyzing the contribution of each link in a reduced model of elasticity. Besides, the obtained estimate of the parameters does not depend on the accuracy of measurements and configurations used in the identification process.

Détail

Keywords: robot stiffness, Jacobian evaluation
Mots-clés : identification.

Bibliographie

[1] Alici, G. and Shirinzadeh, B., “Enhanced Stiffness Modeling, Identification and Characterization for Robot Manipulators”, IEEE Trans. on Robotics, 21:4 (2005), 554–564 | DOI

[2] de Luca, A. and Mattone, R., “Sensorless Robot Collision Detection and Hybrid Force/Motion Control”, Proc. of the IEEE Internat. Conf. on Robotics and Automation (Barcelona, Spain, 2005), 999–1004

[3] Deblaise, D., Hernot, X., and Maurine, P., “A Systematic Analytical Method for PKM Stiffness Matrix Calculation”, Proc. of the IEEE Internat. Conf. on Robotics and Automation (Orlando, Fla., USA, 2006), 4213–4219

[4] Han, Z., Yuan, J., and Gao, L., “External Force Estimation Method for Robotic Manipulator Based on Double Encoders of Joints”, ROBIO 2018: IEEE Internat. Conf. on Robotics and Biomimetics (Kuala Lumpur, Malaysia, 2018), 1852–1857

[5] Hasegawa, A., Fujimoto, H., and Takahashi, T., “Robot Joint Angle Control Based on Self Resonance Cancellation Using Double Encoders”, IEEE Internat. Conf. on Advanced Intelligent Mechatronics (Munich, Germany, 2017), 460–465

[6] Izumi, T. and Matsuo, T., Robot System and Robot Control Apparatus, Patent US No. 8 849 455 (30 Sep 2014)

[7] Klimchik, A., Furet, B., Caro, S., and Pashkevich, A., “Identification of the Manipulator Stiffness Model Parameters in Industrial Environment”, Mech. Mach. Theory, 90 (2015), 1–22 | DOI

[8] Klimchik, A. and Pashkevich, A., “Robotic Manipulators with Double Encoders: Accuracy Improvement Based on Advanced Stiffness Modeling and Intelligent Control”, IFAC-PapersOnLine, 51:11 (2018), 740–745 | DOI

[9] Klimchik, A., Pashkevich, A., and Chablat, D., “Fundamentals of Manipulator Stiffness Modeling Using Matrix Structural Analysis”, Mech. Mach. Theory, 133 (2019), 365–394 | DOI

[10] Lynch, K. and Park, F., Modern Robotics: Mechanics, Planning, and Control, Cambridge Univ. Press, Cambridge, 2017, 544 pp.

[11] Mamedov, S., Popov, D., Mikhel, S., and Klimchik, A., “Compliance Error Compensation based on Reduced Model for Industrial Robots”, Proc. of the 15th Internat. Conf. on Informatics in Control, Automation and Robotics (ICINCO'2018), v. 2, 180–191

[12] Mamedov, S. and Mikhel, S., “Practical Aspects of Model-Based Collision Detection”, Front. Robot. AI, 7 (2020), 571574, 15 pp. | DOI

[13] Mikhel, S., Popov, D., Mamedov, S., and Klimchik, A., “Advancement of Robots with Double Encoders for Industrial and Collaborative Applications”, Proc. of the 23rd Conf. of FRUCT Association (Bologna, Italy, 2018), 246–252

[14] Mikhel, S. and Klimchik, A., “Algebraic Approach to the Stiffness Model Reduction for Manipulators with Double Encoders”, Internat. Conf. “Nonlinearity, Information and Robotics” (Innopolis, Russia, 2020), 1–6

[15] Pashkevich, A., Chablat, D., and Wenger, P., “Stiffness Analysis of Overconstrained Parallel Manipulators”, Mech. Mach. Theory, 44:5 (2009), 966–982 | DOI | Zbl

[16] Pashkevich, A., Klimchik, A., and Chablat, D., “Nonlinear Effect in Stiffness Modeling of Robotic Manipulators”, World Acad. Sci. Eng. Technol.. 58 (2009), 168–173

[17] Piras, G., Cleghorn, W., and Mills, J., “Dynamic Finite-Element Analysis of a Planar High-Speed, High-Precision Parallel Manipulator with Flexible Links”, Mech. Mach. Theory, 40:7 (2005), 849–862 | DOI | Zbl

[18] Popov, D. I. and Klimchik, A. S., “Stiffness Modeling for Anthropomorphic Robots”, Comput. Res. Model., 11:4 (2019), 631–651 | DOI

[19] Tsai, J., Wong, E., Tao, J., McGee, H. D., and Akeel, H., Secondary Position Feedback Control of a Robot, Patent US No. 8 473 103, 2013

[20] Sakata, K., Asaumi, H., Hirachi, K., Saiki, K., and Fujimoto, H., “Self Resonance Cancellation Techniques for a Two-Mass System and Its Application to a Large-Scale Stage”, IEEJ J. Ind. Appl., 3:6 (2014), 455–462

[21] Siciliano, B., Sciavicco, L., Villani, L., and Oriolo, G., Robotics: Modelling, Planning and Control, Springer, London, 2010, xxiv, 632 pp.

[22] } {\tt https://github.com/mikhel1984/algebraic_approach