angelika peer, naim bajcinca, christian schweiger institute of … · 2010. 10. 23. · angelika...

Proceedings

of the 3rd International Modelica Conference, Linköping, November 3-4, 2003,

Peter Fritzson (editor)

Paper presented at the 3rd International Modelica Conference, November 3-4, 2003, Linköpings Universitet, Linköping, Sweden, organized by The Modelica Association and Institutionen för datavetenskap, Linköpings universitet All papers of this conference can be downloaded from http://www.Modelica.org/Conference2003/papers.shtml Program Committee Peter Fritzson, PELAB, Department of Computer and Information Science,

Linköping University, Sweden (Chairman of the committee). Bernhard Bachmann, Fachhochschule Bielefeld, Bielefeld, Germany. Hilding Elmqvist, Dynasim AB, Sweden. Martin Otter, Institute of Robotics and Mechatronics at DLR Research Center,

Oberpfaffenhofen, Germany. Michael Tiller, Ford Motor Company, Dearborn, USA. Hubertus Tummescheit, UTRC, Hartford, USA, and PELAB, Department of

Computer and Information Science, Linköping University, Sweden. Local Organization: Vadim Engelson (Chairman of local organization), Bodil Mattsson-Kihlström, Peter Fritzson.

Angelika Peer, Naim Bajcinca, Christian Schweiger Institute of Robotics and Mechatronics, DLR: Physical-based Friction Identification of an Electro-Mechanical Actuator with Dymola/Modelica and MOPS pp. 241-248

Physical-based Friction Identificationof an Electro-Mechanical Actuatorwith Dymola/Modelica and MOPS

Angelika Peer Naim Bajcinca Christian Schweiger

German Aerospace Center (DLR)Institute of Robotics and Mechatronics

Oberpfaffenhofen, 82234 Weßling, Germanyhttp://www.robotic.dlr.de/control/

Abstract

An identification procedure consisting of iterative pa-rameter optimization and model validation tasks usingthe optimization tool MOPS and Dymola/Modelicasimulation environment is presented. This methodis used for modelling of a force-feedback electro-mechanical actuator with Harmonic-Drive gear. Amodelling approach for speed and torque dependentgear losses introduced in a prior work is validated.

1 Introduction

Several objectives, such as model-based control, sim-ulation and design of complex systems require accu-rate system models. Especially, mechanical systemsexhibit complex nonlinear phenomena, e.g. stick-slipeffects, whose modelling may play an essential rolein the dynamics of the whole system. Such complexmodelling tasks require tools, which should providea clear hierarchical model structure, efficient equationsolvers and fast component parametrization. These re-quirements are e.g. fulfilled by Dymola/Modelica sim-ulation environment. Modelica is a physical object-oriented modelling language suitable for modellingand simulation of heterogeneous multi-physical sys-tems. It is designed in such a way, that the user canbuild a physical model in a natural way, as he wouldbuild it in real-world. Additionally, due to symbolicalcode preprocessing, Dymola/Modelica enables real-time simulation of complex physical systems, [OE00].While the structure of a model is physically definedby Modelica, yet for modelling completion, its pa-rameters need to be computed or identified via mea-surements. A convenient environment for parameter

identification is the optimization tool MOPS (Multi-Objective Parameter Synthesis), [JBL+02]. Multi-objective optimization is enhanced by providing ro-bust gradient-free direct-search solvers and an intuitiveuser interface. Parameter optimization with MOPSis especially convenient since different measurementdata can be handled simultaneously in the context ofa multi-objective optimization task with respect to dif-ferent criteria types (typically least-squares).

The main aim of this paper is to present an identifica-tion procedure for accurate modelling in the exampleof an electro-mechanical actuator. Therefore an iden-tification feedback-loop consisting of iterative param-eter optimization and model validation tasks. Whilethe latter is performed in a Dymola/Modelica simula-tion environment, the parameter optimization is donein MOPS.

A natural way of a parameter identification task is tosplit it in subtasks by discriminating between differ-ent physical conditions, which primarily excite a cer-tain parameter subset. This paper uses this strategyfor separate identification of linear stiffness, dampingand inertia, as well as, non-linear bearing- and mesh-friction parameters. Thereby, a modelling formalismfor gear friction as proposed in [PSO02] has beenused. The latter work introduces a tabular descrip-tion of friction (loss table), which includes speed- andtorque-dependent gear losses terms, i.e. bearing- andmesh-friction parameters for braking and driving gearconditions. While carrying out of physical conditionsneeded to measure the loss table sets great demandson technical equipment, in this paper it is shown thatidentification is an effective alternative.

The paper is organized as follows. In the next sec-tion the electromechanical force-feedback actuator is

A. Peer, N. Bajcinca, C. Schweiger Friction Identification of an Electro-Mechanical Actuator…

The Modelica Association Modelica 2003, November 3-4, 2003

introduced. Section 3 describes the identification loopwith MOPS and Dymola/Modelica. Section 4 pro-vides the identification of a linear actuator model,including the Dymola/Modelica actuator scheme andlinear parameter identification with MOPS. Section 5recalls the modelling approach of gear losses as pro-posed in [PSO02], which has been further used to ex-tend the linear model by inclusion of nonlinear gearlosses. Finally, concluding remarks and future relatedwork complete the paper.

2 Actuator physical description

This chapter provides the physical description of anelectro-mechanical actuator, see Fig. 1 and Fig. 2,which has been used in a steer-by-wire control struc-ture for force-feedback. In order to avoid rotatingwiring a strain-gauge torque sensor is placed betweenthe motor housing and a fixed console, as shown in thefigure. Thus, in addition to the torque at the outputshaft, a dynamical component resulting from housingrotation is measured, as well.

Figure 1: The force-feedback actuator

stator circular spline

sensor

output shaft

housing

wave generator flexsplineconsole

bearing

rotor

Figure 2: Force-feedback actuator components

Besides the torque sensor, the main component in theforce feedback actuator is a Harmonic Drive serieshollow-shaft gear. In Fig. 3 its main components,Wave Generator, Flexspline and Circular Spline areshown. The teeth on the nonrigid Flexspline and therigid Circular Spline are in continuous engagement.Since the Flexspline has two teeth fewer than the Cir-cular Spline, one revolution of the input causes relativemotion between the Flexspline and the Circular Splineequal to two teeth. With the Circular Spline rotation-ally fixed, the Flexspline rotates in the opposite direc-tion to the input at a reduction ratio equal to one-halfthe number of teeth on the Flexspline. Typical charac-teristics of a Harmonic-Drive gear are high positioningaccuracy, virtually no backlash, periodic torque ripplesand a high gear ratio. One of the main topics of thispaper is modelling of friction losses of this gear usingModelica.

Wave Generator Flexspline Circular Spline

Figure 3: Harmonic Drive gear components

3 Parameter identification with Dy-mola/Modelica and MOPS

Modelica is an object-oriented language for modellingof large, complex and heterogeneous multi-physicalsystems involving mechanical, electrical and hydraulicsubsystems. The engineer can build its model in afraction-by-fraction manner, as he would build it inreal-world, that is link components like motors, pumpsand valves using their physical interfaces. Such a sim-ulation framework is very convenient to use in an iden-tification feedback-loop consisting of parameter op-timization and simulation tasks, as shown in Fig. 4.Thereby, one can perform parameter identification ofspecific components or/and of specific physical con-ditions independently and integrate them easily in thenext identification setup. The insight into physical sys-tem is important for decoupling of different physicalconditions which primarily excite a known set of pa-rameters. Note that using Modelica for physical sim-



ulation may be indispensable for complex systems,since a signal-flow simulation model may be essen-tially influenced by additional physical fractions.

MOPS

Dymola

optimization

simulationparameter-adaptation

parameters

simulationresults

measurements

Figure 4: Identification loop

The parameter optimization in the fraction-by-fractionidentification procedure is done using the optimiza-tion tool MOPS (Multi-Objective Parameter Synthe-sis), [JBL+02]. Basically, MOPS provides a multi-objective optimization environment for the design ofsystems with a large amount of parameters and cri-teria, but it may be used equally well for parame-ter estimation in identification problems. The multi-criteria optimization problem in MOPS is handled byreformulating it as a standard Nonlinear ProgrammingProblem (NLP) with equality, inequality and boundconstraints. MOPS uses several available gradient-free direct-search solvers, which are more robust com-pared to though more efficient gradient-based solvers.These include algorithms such as sequel quadratic pro-gramming (SQP), Quasi-Newton, pattern search, sim-plex method and genetic algorithms. To overcomethe problem of local minima to some extent, solversbased on statistical methods or genetic algorithms canbe alternatively used. An identification problem maybe formulated as a multi-objective optimization prob-lem, whereby measured data corresponding to differ-ent physical conditions or/and inputs define a set ofoptimization objectives. Different scalar or/and vectorcriteria may be defined, e.g least-square-error, error-vector, etc.

4 Linear model

4.1 Actuator Modelica Model

Fig. 5 represents a Modelica modelling setup of theelectro-mechanical actuator. Since a Harmonic-Drivegear can be classified as a typical sun-carrier-ring plan-etary gear (Wave Generator corresponding to the sun,Circular Spline to the carrier and Flexspline to thering), a planetary gear component from the Modelica

rotational mechanics library has been used for its mod-elling. The torque balance and angular equations ofHarmonic-Drive are modelled as follows,

τC = (n−1)τW

τF = −nτW

ϕW = (1−n)ϕC +nϕF ,(1)

withτC: torque at the Circular SplineτF : torque at the Flexsplinen: gear ratioϕC: Circular Spline angleϕF : Flexspline angle.

Note that in the linear model the losses of this compo-nent are neglected.

sensor motor gear

cS

dS

JS JR JGn cG

dG

JL

load

Figure 5: Linear physical model of the actuator

The following listing introduces the physical descrip-tion of the setup parameters.n: gear transmission ratioJL: gear output inertiadS: sensor dampingcS: sensor stiffnessdG: gear dampingcG: gear stiffnessJR: rotor inertiaJS: stator (housing) inertia.

4.2 Parameter identification

Two different physical conditions are respectively dis-criminated for parameter identification of the linearand nonlinear actuator model. In the linear model thefriction losses in Harmonic-Drive gear are neglected.In order to match the physical model as close as possi-ble to such a linear one, the non-linear effects excitedon Harmonic-Drive gear are minimized by fixing theoutput shaft.With the output shaft keeping fixed, load inertia,JL in Dymola/Modelica model in Fig. 5 has no dy-namical effect. While several parameters, such asHarmonic-Drive gear ratio (n = 50), theemf motorconstant (Km = 0.7 Nm/rad), torque sensor stiffness(cs = 130000Nm/rad) and Flexspline stiffness (cG =25500Nm/rad) are given by the manufacturer, the rest



of parameters, i.e. sensor damping (dS), motor hous-ing inertia (JS) and Flexspline damping (dG) need tobe identified. Their initial values for optimization areset to reasonable values estimated by some simple ex-periments,

dS = 0.6 Nm s/rad,

JS = 0.003kgm2,

dG = 50Nm s/rad.

Thereby, as input data in Fig. 4 are used measure-ments corresponding to a set of current inputs (step,sinusoidal and PRBS) of different amplitudes and fre-quencies and torque response is measured by the sen-sor. After 18 successive iterations of the identificationfeedback-loop in Fig. 4, the parameter values listed be-low result,

dS = 2.66Nm s/rad,

JS = 0.003039kgm2,

dG = 70.625Nm s/rad.

The respective optimization history is shown in Fig. 6.Further in Fig. 7 several validation results for differentinput and measurement data are collected.

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.20

5

10

15

20

25

time / s

torq

ue /

Nm

measured data

final optimizationresultintermedial optimizationresults

Figure 6: Optimization step response history of linearmodel

5 Nonlinear model

5.1 Gear losses

The nonlinear model of Harmonic-Drive Gear used inthis paper is based on the friction modelling approachproposed in [PSO02]. Therefore, the gear modelimplemented in componentLossy Planetary of

the Modelica Power Train library includes a torque-dependent (due to mesh friction in the gear teeth con-tact) and speed-dependent friction (due to bearing fric-tion). Similar to the standard Modelica friction model,the three modesforward sliding, stuckandbackwardsliding are available. The friction torque∆τ for thesliding modes is given by Table 1, wherebyτW denotesthe driving torque,τbf the bearing friction andηmf themesh friction coefficient.

ωW τW ∆τ => 0 ≥ 0 (1−ηmf1)τW + |τbf1| (= ∆τmax1≥ 0)> 0 < 0 (1−1/ηmf2)τW + |τbf2| (= ∆τmax2≥ 0)< 0 ≥ 0 (1−1/ηmf2)τW−|τbf2| (= ∆τmin1≤ 0)< 0 < 0 (1−ηmf1)τW−|τbf1| (= ∆τmin2≤ 0).

Table 1:∆τ = ∆τ(ωW,τW) in sliding mode

It can be shown, that the linear torque equations in (1)are extended by the friction component,∆τ as follows,

τF = −n(τW−∆τ)τC = (n−1)τW−n∆τ. (2)

The typical relationship betweenτW and ∆τ is illus-trated in Fig. 8 for both the sliding and the stuck modeand in combination in Fig. 9.

∆τ

τW

∆τmax1, ωW > 0

∆τmax2, ωW > 0

∆τmin2, ωW < 0 ∆τmin1, ωW < 0

ωW > 0

∆τ(ωW=0)

∆τ

ωW

τW

τW

Figure 8: friction torque in sliding and stuck mode

The parameters to be provided are the stationary gearration and tablelossTable to define the gear losses,see Table 2.Wheneverηmf1, ηmf1, τbf1 or τbf2 are needed, they aredetermined by interpolation inlossTable . The inter-face of this Modelica model is therefore defined as

parameter Real i = 1;parameter Real lossTable[:,5]

= [0, 1, 1, 0, 0];

using the unit gear ratio and no losses as a default.

5.2 Parameter Identification

This section deals with identification of thelossTable in Table 1. For mesh friction, it is



time / s

Sine 750mA, 10Hz

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0 0.05 0.1 0.15 0.2�-5

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Figure 7: Validation results linear model

|ωW| ηmf1 ηmf2 |τbf1| |τbf2|...

......

......

Table 2: Format of tablelossTable

natural to assume no-loss (ideal gear) conditions asinitial values. Besides, it is relatively difficult to setan experimental setup for its measurement, sinceadditional drives have to be installed on the outputshaft for covering the whole set of conditions asdescribed in Table 2. While measurement of bearingfriction is not essentially simpler, it may be roughlyassumed that,

τb f1 ≈ τb f2.

However, assuming ideal conditions as initial onesmay cause difficulties in optimization of bearing-friction, since the solvers are required to change theinitial structure by including additional damping intothe model. Fortunately, using the above assumptioninitial values are relatively easily estimated in a setupwith free rotation of the output shaft (no external load)at different constant velocities. Given that torque sen-sor sits between the input and output bearing friction,it can see just the output bearing friction. Thus, assum-ing that the torque generated on the motor shaft bal-ances the net (both input and output) bearing friction

speed input torque

fric

tion

torq

ue

1

-1-1

-1

1

1

Figure 9: Friction model

(output load effects due to the Flexspline inertia are ne-glected), motor current can be used for its estimation.Fig. 10 presents the estimation results correspondingto rotation in both directions. From this curve theτb f1,i.e. τb f2 are read as initial values for the optimization.Note that the above figure indicates clearly the appear-ance of the Stribeck effect when switching from stuckto sliding mode.

For completion of thelossTable the identificationprocedure is repeated for different constant velocities.Each identification step corresponds to a row in thelossTable .



�-300 �-200 �-100 0 100 200 300�-0.2

�-0.15

�-0.1

�-0.05

0

0.05

0.1

0.15

0.2

speed at the input shaft / rad/s

fric

tion

torq

ue /

Nm

Stribeck effect

Figure 10: Bearing friction measurement

5.3 Model validation

For illustration purposes the row corresponding to therotor speed of 10rad/s will be discussed. Based onthe previous discussion the initial values are chosen tobe,

ηm f1 = 1,

ηm f2 = 1

τb f1 = 0.09

τb f2 = 0.09.

After 103 optimization/simulation iterations in Fig. 4these parameters converge to the values,

ηm f1 = 0.923,

ηm f2 = 0.864

τb f1 = 0.058

τb f2 = 0.058.

For model validation the authors have set the setupshown in Fig. 12, whereby a defined torque at the out-put shaft has been applied by an excentric load. Differ-ent load conditions may be realized by varying the loadradius. The Dymola/Modelica actuator model corre-

cS

dS

JS JR + JGn cG

dG

JL

load

Figure 11: Modelica model with excentric load

Figure 12: Excentric load experiment

|ωW| ηmf1 ηmf2 |τbf1| |τbf2|10 0.979 0.945 0.086781 0.08678115 0.9625 0.92125 0.090313 0.08843820 0.854 0.847 0.0565 0.049

Table 3: Format of tablelossTable

sponding to the physical situation in Fig. 11 is aug-mented as shown in the above figure, by making useof the new Multi-body Modelica Library, [OEM03].

0 5 10 15 20 25 30-30

-20

-10

0

10

20

30

time / s

torq

ue /

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measured data

final optimizationresultintermedial optimizationresults

Figure 13: Optimization history of nonlinear model

The identification history of the loop in Fig. 4 for therow 10rad/sassumingτb f1 = τb f2 is shown in Fig. 13.

In a next identification step the assumptionτb f1 =τb f2 is removed. Table 3 shows three rows oflossTable corresponding to the rotor speeds of 10,15 and 20rad/s. Notice thatτb f1 ≈ τb f2.

Finally, Fig. 14 collects the validation results for dif-ferent input current signals.



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Figure 14: Validation results for nonlinear model

6 Conclusions

It is shown that iterative parameter optimization withMOPS and model validation using Dymola/Modelicais a powerful identification environment. This methodis used for modelling of a force-feedback electro-mechanical actuator with Harmonic-Drive gear. Amodelling approach for speed and torque dependentgear losses introduced in a prior work is validated. Fu-ture work might include identification of dynamicalfriction models. The procedure presented in this pa-per may be applied for dynamics identification of othergear technologies.

References[JBL+02] JOOS, Hans-Dieter ; BALS, Johann ; LOOYE, Gert-

jan ; SCHNEPPER, Klaus ; VARGA, Andras: A multi-objective optimisation-based software environment forcontrol systems design. In:IEEE International Sympo-sium on Computer Aided Control System Design Pro-ceedings. Glasgow, September 2002

[OE00] OTTER, Martin ; ELMQVIST, Hilding: Modelica —Language, Libraries, Tools, and Conferences. In:Sim-ulation News Europe(2000), December

[OEM03] OTTER, Martin ; ELMQVIST, Hilding ; MATTSSON,Sven E.: The New Modelica MultiBody Library. In:Proceedings of the 3rd International Modelica Confer-ence. Linkoping, November 2003

[PSO02] PELCHEN, Christoph ; SCHWEIGER, Christian ; OT-TER, Martin: Modeling and Simulating the Efficiencyof Gearboxes and of Planetary Gearboxes. In:Pro-ceedings of the 2nd International Modelica Confer-ence. Oberpfaffenhofen, March 2002, S. 257–266



angelika peer, naim bajcinca, christian schweiger institute of … · 2010. 10. 23. · angelika...

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