Department of Precision and Microsystems Engineering Model based friction compensation for an electromechanical Stewart platform actuator Name: M. W. van der Kooij Report no: EM 11.034 Coach: ir. E.J.H. de Vries Professor: Prof. dr. D.J. Rixen Specialization: Automotive Type of report: Master Thesis Date: Delft, November 1, 2011 Model based friction compensation for an electromechanical actuator of a Stewart platform Maarten Willem van der Kooij s i s e h T e c n e i c S f o r e t s a M Model based friction compensation for an electromechanical actuator of a Stewart platform Master of Science Thesis For the degree of Master of Science in Mechanical Engineering at Delft University of Technology Maarten Willem van der Kooij November 1, 2011 Faculty of Mechanical, Maritime and Materials Engineering (3mE) Delft University of · Technology The work in this thesis was supported by Moog B.V. located in Nieuw Vennep. Their coop- eration is hereby gratefully acknowledged. Copyright © All rights reserved. Abstract MoogintheNetherlandsproduces, amongotherproducts, motionplatformsthatcanbeused for flight and drive simulation. Generally, the motion platforms are driven by electromechan- ical actuators. The performance of these actuators is limited by friction, which is a force that opposes the direction of relative velocity. Naturally, this is most noticeable when the direction of relative velocity changes, as it implies a sudden change in the friction force. This sudden change in friction force makes it difficult to control a zero velocity crossing smoothly. When the motion platform is used for flight or drive simulation this nonsmooth zero velocity crossing can be felt by the user, making the simulation less realistic. Of course, one can pre- dict a zero velocity crossing and apply some extra force on the system during a zero velocity crossing. Currently a simple algorithm is used to do exactly this, but unfortunately things are not that easy. This algorithm turned out to be insufficient in predicting the extra force for all zero velocity crossing situations the actuator is subject to. Therefore the purpose of this thesis is to identify a more elaborate friction model and use this model for feedforward friction compensation to ensure a smoother zero velocity crossing. The first part of this thesis gives an overview of the friction modeling literature of the past decades. Two models are of special interest: the LuGre model as proposed by Canudas de Wit et al. in 1995 and the DNLRX model proposed by Rizos and Fassios in 2005. In the second part of this thesis the two models of interest are identified on a 1 degree of freedom test set-up, consisting of one motion platform actuator connected to a test-rig. Since it is not possible to directly measure the friction with sufficient accuracy a system identification is performed. All terms in the equation of motion of the system are identified, except for the friction. The friction signal is then measured as being the residuals of the equation of motion and the two models of interest are identified on this data. It is shown that the DNLRX model is able to properly predict the friction during a zero velocity crossing whereas the LuGre model gives better predictions outside of the zero velocity regime. The system model including either one of the models is able to predict the motor current based on the actuator position, velocity and acceleration with an average error of less than 0.09 ampere. The third part of this work focusses on the implementation of the friction models. Both the models are used for feedforward friction compensation out of the range they were identified Master of Science Thesis ii on: at varying actuator positions with a variety of accelerations during the zero velocity crossing. Afterinitialmeasurementsthetwomodelsareslightlyadjusted. TheLuGremodelis adjustedtoreducetheovercompensationitgives,theDNLRXmodelisadjustedtoreducethe number of parameters. When using the LuGre model for feedforward friction compensation the commanded velocity tracking error is reduced by 69 %. When using the DNLRX model, the adjusted DNLRX model or the adjusted LuGre model the commanded velocity tracking error reduction is around 80 %. Master of Science Thesis Table of Contents Preface xi 1 Introduction 1 1-1 Moog. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1-2 Problem Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1-3 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1-4 Outline of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 I Theory 7 2 Friction Phenomena, Models and Compensation Methods 9 2-1 Experimentally Observed Phenomena . . . . . . . . . . . . . . . . . . . . . . . . 9 2-2 Friction Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2-3 Friction Compensation Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3 Friction Model Identification Methods 35 3-1 LuGre Parameter Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3-2 DNLRX Parameter Identification . . . . . . . . . . . . . . . . . . . . . . . . . . 40 II Experimental Identification 43 4 Experimental Set-up 45 4-1 Hardware and Working Principle of the Set-up . . . . . . . . . . . . . . . . . . . 45 4-2 Acquiring a Set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 4-3 Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 4-4 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 Master of Science Thesis iv Table of Contents 5 Identification of System Parameters 63 5-1 Decoupled Identification of System Parameters . . . . . . . . . . . . . . . . . . 64 5-2 Coupled Identification of System Parameters and Sliding Friction . . . . . . . . . 67 5-3 Absolute Values for the System Parameters . . . . . . . . . . . . . . . . . . . . 76 5-4 Validation of Identified Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 5-5 Conclusions on the System Identification and Method . . . . . . . . . . . . . . . 83 6 Identification of Friction Models 87 6-1 Motor Cogging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 6-2 LuGre Parameter Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 6-3 DNLRX Parameter Identification . . . . . . . . . . . . . . . . . . . . . . . . . . 95 6-4 Comparison of the LuGre, DNLRX and the Static Models . . . . . . . . . . . . . 99 III Model implementation and experimental validation 101 7 Friction Compensation 103 7-1 Implementation of the Models and Adjustments to the Set-up . . . . . . . . . . 103 7-2 Initial Comparison of the Compensation Methods . . . . . . . . . . . . . . . . . 106 7-3 Adjusting the Friction Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 7-4 An Extensive Comparison of the Compensation Methods . . . . . . . . . . . . . 117 8 Conclusions and Recommendations 123 8-1 Relating back to the Objectives of the Thesis . . . . . . . . . . . . . . . . . . . 123 8-2 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 Appendices 126 A Data of the experimental set-up 129 A-1 Testrig and actuator data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 A-2 Datasheet of the Phase motor . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 A-3 Measurements done on the torque constant . . . . . . . . . . . . . . . . . . . . 138 B Calculations 139 B-1 Bristle stiffness calculation according to Lischinsky . . . . . . . . . . . . . . . . 139 B-2 System identification for pre selected fixed inertias . . . . . . . . . . . . . . . . . 141 B-3 Gradient, Hessian and covariance matrix at the system identification optimum . . 142 B-4 Calculating the covariance matrix from the Hessian . . . . . . . . . . . . . . . . 143 B-5 Equations of motion of the 1-DOF testrig . . . . . . . . . . . . . . . . . . . . . 146 C Measurements 149 C-1 System Identification Validation Measurements . . . . . . . . . . . . . . . . . . 149 C-2 Dynamic Parameter Identification Measurement . . . . . . . . . . . . . . . . . . 152 Master of Science Thesis