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Observer Synthesis for the Adhesion Estimation of a Railway Running Gear
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IFAC PapersOnLine 52-15 (2019) 319–324
Observer Synthesis for the Adhesion Observer Synthesis for the Adhesion Observer Synthesis for the Adhesion Estimation of a Railway Gear Observer Synthesis for Running the Adhesion Estimation of a Railway Running Estimation of a Railway Running Gear Gear Estimation of a Railway Running ∗ ∗ Gear Christoph Schwarz ∗ Alexander Keck ∗
Christoph Schwarz ∗∗ Alexander Keck ∗∗ Christoph Schwarz Alexander Keck Christoph Schwarz ∗ Alexander Keck ∗ ∗ ∗ German Aerospace Center (DLR), Institute of System Dynamics and German Aerospace Center (DLR), Institute of System Dynamics and ∗ ∗ German Aerospace Center (DLR), Institute of [christoph.schwarz, System Dynamics and Control, 82234 Wessling, Germany (e-mail: Control, 82234 Wessling, Germany (e-mail: ∗ German Aerospace Center (DLR), Institute of [christoph.schwarz, System Dynamics and alexander.keck]@dlr.de). Control, 82234 Wessling, Germany (e-mail: [christoph.schwarz, alexander.keck]@dlr.de). Control, 82234 Wessling, Germany (e-mail: [christoph.schwarz, alexander.keck]@dlr.de). alexander.keck]@dlr.de). Abstract: The The presented presented work work illustrates illustrates an an investigation investigation on on the the observer observer synthesis synthesis for for aa highly highly Abstract: nonlinear railway running gear system. The specific focus is longitudinal dynamics and the Abstract: The presented work illustrates an investigation on the observer synthesis for a highly nonlinear railway running gear system. Thean specific focus is on on the longitudinal dynamics the Abstract: The presented work illustrates investigation observer synthesis forinterface aand highly nonlinear railway running gear system. The specific focusbetween is on the longitudinal dynamics and the associated adhesion effects, i.e. the friction conditions wheel and rail. This the associated adhesion effects, i.e. the friction conditions between wheel and rail. This interface nonlinear railway running gear system. The specific focusbetween is on the longitudinal dynamics and the strongly influences the brake and the traction performance of trains and, thus, its detailed associated adhesion effects, i.e. the friction conditions wheel and rail. This interface strongly influences the brake traction performance of wheel trains and,rail. thus, its interface detailed associated adhesion effects, i.e.and thethe friction conditions This consideration is of of crucial crucial importance intraction terms ofperformance safetybetween and comfort. comfort. strongly influences the brake and the of trains and and, thus, its detailed consideration is importance in terms of safety and strongly influences the brake and the traction of gear trains and, thus, its detailed The observer observer design is accomplished accomplished forinthe the experimental running consideration is of crucial importance terms ofperformance safety and comfort. The design is for experimental running gear developed developed at at the the German German consideration is of crucial importance in terms of safety and comfort. The observer design is accomplished for the experimental running gear developed at the Aerospace Center (DLR). The article focuses on the the first first steps steps of of the the observer observer synthesis: synthesis:German (i) the the Aerospace on (i) Center (DLR). The article focuses The observer design is accomplished the experimental running gear developed at (iii) the German set-up of the the observer model, (ii) thefor selection ofthe an first appropriate observer method, and Aerospace Center (DLR). The(ii) article focuses onof steps of observer the observer synthesis: (i) the set-up of observer model, the an appropriate selection method, (iii) and the Aerospace Center (DLR). The article focuses onofthe stepsdetermining of observer the observer synthesis: (i) the validation of the observer observer in aa(ii) simulation environment. The aspect of(iii) theand model set-up of the observer model, the selection an first appropriate method, the validation of the in simulation The determining aspect of the model environment. set-up of the observer model, (ii)simulation theofselection of an appropriate observer method, (iii) and the implementation is the consideration the nonlinear wheel-rail contact. To cover this nonlinear validation of the observer in a environment. The determining aspect of the model implementation isobserver the consideration of the nonlinear wheel-rail contact. To aspect cover this nonlinear validation of the in a simulation environment. The determining of the model implementation is the consideration of the nonlinear wheel-rail contact. To cover this nonlinear behavior an an Extended Extended Kalman Kalman Filter Filter (EKF) (EKF) is is used used in in combination combination with with aa parameter parameter estimator estimator behavior implementation is the consideration the nonlinear cover this nonlinear for the the friction friction characteristics inFilter the of wheel-rail interfaces. In the thecontact. end, theTo received results prove behavior an Extended Kalmanin (EKF) is used inwheel-rail combination with areceived parameter estimator for characteristics the wheel-rail interfaces. In end, the results prove behavior an Extended Kalmanestimates Filter (EKF) is used in combination with areceived parameter estimator that the observer accurately the system behavior and provides reliable information for the friction characteristics in the wheel-rail interfaces. In the end, the results prove that thefriction observer accurately estimates the system andend, provides behavior reliableresults information for the characteristics inand therelated wheel-rail interfaces. In the the received prove on the the adhesion characteristics longitudinal dynamics. that the observer accurately estimates the longitudinal system behavior and provides reliable information on adhesion characteristics and related dynamics. that the observer accurately estimates the system behavior and provides reliable information on the adhesion characteristics and related longitudinal dynamics. Based on on this one upcoming steps DLR Based this findings findings one of of the the upcoming steps at at the the DLR is is to to validate validate the the observer observer in in on adhesion characteristics andtest related longitudinal dynamics. thethe real-time environment rig. Regarding use approach in aa Based on thisenvironment findings one of of the the upcoming steps at the the DLRof isthe to presented validate the observer the real-time of the test rig. Regarding the use of the presented approach in in Based on this findings one of the upcoming steps at the DLR is to validate the observer in train some significant improvements of traction control systems are enabled. A first application the real-time environment of the test rig. Regarding the use of the presented approach in a train some significant improvements of traction controlthe systems enabled. A first application the real-time environment of control the test rig. Regarding use ofare presented approach in a train some significant improvements oftotraction control systems arethe enabled. A first application could be an enhanced brake narrow down the variance of brake distances in daily could be ansignificant enhancedimprovements brake control oftotraction narrow control down the variance of brake A distances in daily train some systems are enabled. first application could be anFurthermore, enhanced brake control tousability narrow of down variance of brake distances in daily operation. an advanced advanced the the condition based monitoring of traction traction operation. Furthermore, an usability of the condition based monitoring of could be an enhanced brake control tousability narrow down the variance of brake distances in daily and brake systems is facilitated with the additional information on the friction conditions. operation. Furthermore, an advanced of the condition based monitoring of traction and brake systems is facilitated with the additional information the friction conditions. operation. Furthermore, an advanced usability of the conditionon based monitoring of traction and brake systems is facilitated with the additional information on the friction conditions. © 2019, IFAC (International Federation Automatic Control) Hosting byonElsevier Ltd. Allconditions. rights reserved. and brake systems is facilitated withofthe additional information the friction Keywords: Rail Rail traffic, traffic, nonlinear nonlinear systems, systems, adhesion, adhesion, Extended Extended Kalman Kalman filter. filter. Keywords: Keywords: Rail traffic, nonlinear systems, adhesion, Extended Kalman filter. Keywords: Rail traffic, nonlinear systems, adhesion, Extended Kalman filter. 1. INTRODUCTION INTRODUCTION driver assistance assistance systems systems like like wheel wheel slide slide protection protection and and 1. driver 1. INTRODUCTION driver assistance systems likerisk wheel slide protection and anti-skid systems reduce the of blocking and skidding anti-skid systems systems reduce the risk of blocking and skidding driver likerisk wheel slidethe protection and The improvement improvement1. of ofINTRODUCTION safety and and comfort comfort is is often often the the anti-skid systems reduce wheels.assistance Nevertheless, to further further improve functionality the of blocking skidding The safety wheels. Nevertheless, to improve the and functionality anti-skid systems reduce the risk of blocking and skidding The improvement of safety and comfort is often the basic objective objective in in the the research research on on and and development development of of wheels. further functionality of these these Nevertheless, systems and and to totoenable enable forimprove examplethe condition based basic systems for example condition based The improvement of safety and often the wheels. toenable further improve functionality basic objective in the research oncomfort and development of of mechatronic vehicle concepts. In addition, theisimportance importance of these Nevertheless, systems and todetailed for examplethe condition based monitoring concepts, information on the existing mechatronic vehicle concepts. In addition, the concepts, information on the existing basic objective in the research onconsumes and development of monitoring of these systems and is todetailed enable for example condition based mechatronic vehicle concepts. In addition, the aimportance of aa sustainable sustainable performance that minimum monitoring concepts, detailed information on the existing adhesion conditions necessary. Due to economic economic reasons of performance that consumes aimportance minimum adhesion conditions is necessary. Due to reasons mechatronic vehicle concepts. In addition, the monitoring concepts, detailed information ondaily the existing of aresources sustainable performance that consumes a minimum adhesion was and is still growing. The multifaceted conditions is necessary. Due to economic reasons and the rough environmental influences in railway of aresources was performance and is still that growing. The multifaceted and the rough environmental influences in daily reasons railway sustainable consumes a minimum adhesion conditions is necessary. Duebe to directly economic of resources was of and still growing. TheCenter multifaceted railway activities theisGerman German Aerospace (DLR), and the rough influences in dailyprovided railway operation, this environmental information cannot railway activities of the Aerospace Center (DLR), operation, this information cannot be directly provided of resources was of and isGerman still growing. TheCenter multifaceted and the rough environmental influences inobserver dailyprovided railway railway activities the which are combined in the the long-term research project Next operation, this information cannot be directly Aerospace (DLR), by a sensor. Therefore, a control theoretic might which are combined in long-term research project Next a sensor.this Therefore, a control theoretic observer might railway activities of the German Aerospace Center (DLR), operation, information cannot be directly provided which are combined in the long-term research project Next by Generation Train (NGT), try to solve the partially existing by a sensor. Therefore, a control theoretic observer might be implemented, to estimate the unmeasurable adhesion Generation Train (NGT), try to solve the partially existing be implemented, to estimate the unmeasurable adhesion which combined in the long-term project Next be by aimplemented, sensor.and Therefore, a control observer might Generation Trainthese (NGT), try to solveresearch the partially existing conflictare between aspects. to estimate thetheoretic unmeasurable adhesion conditions related effects like creep forces. conflict between aspects. and related effects like creep forces. Generation Trainthese (NGT), try to solve the partially existing conditions be implemented, to estimate the unmeasurable adhesion conflict between these aspects. conditions and related effects like creep forces. One particular particular focus ofaspects. the railway railway research research at at the the DLR DLR is is conditions In the the automotive automotive field, the like estimation of the the friction friction conflict betweenfocus theseof and related effects creep forces. One the field, the estimation of One particular focus ofand thetechnologies railway research at thethe DLR is In on concepts, methods to exploit great In the automotive field, the estimation of the friction conditions between tire and road is an intensely investion concepts, methods technologies to exploit great between tire is an intensely One particular focus ofand the railway research at thethe DLR is conditions In theproblem, automotive field,and theroad of the investifriction on concepts, methods and technologies to exploit the great potential of mechatronic mechatronic running gears with independently conditions between tire and road an intensely investigated see Rajamani Rajamani etestimation al. is (2012). However, these potential of running gears with independently gated problem, see et al. (2012). However, these on concepts, methods andrunning technologies to exploit the great conditions between tire and road is an intensely investipotential of mechatronic gears with independently rotating wheels identified in Kurzeck et al. (2014). One gated problem, see Rajamani et al. (2012). However, these results cannot be directly adapted, since the adhesion rotating wheels identified in Kurzeck et al. (2014). One cannot see be Rajamani directly adapted, sinceHowever, the adhesion potential of mechatronic gears with gated etsteel al. (2012). these rotating wheels identified in Kurzeck et al.independently (2014). One results of the the most most crucial andrunning determining aspects of railway railway results be directly adapted, since and the steel adhesion effectsproblem, ofcannot the contact contact between wheels rails of crucial and determining aspects of effects of the between steel wheels and rails rotating wheels identified in Kurzeck et al. (2014). One results cannot be directly adapted, since the steel adhesion of the most crucial and determining aspects of railway dynamics is the wheel-rail interface, since it affects the effects of the contact between steel wheels and steel rails significantly differ from the effects between tire and road. dynamics is the wheel-rail interface, since it affects the significantly differ from the effects between tire and road. of the most crucial and the determining aspects of railway effects of the contact between steel wheels and rails dynamics is lateral, the wheel-rail interface, since it affects vertical, the and longitudinal dynamics allthe at significantly differ from the effects between tire steel and road. Regarding the railway sector lot of of the ongoing ongoing research vertical, the lateral, and the longitudinal dynamics all at the railway sector aa lot the research dynamics is lateral, the wheel-rail since itofaffects significantly differ from the effects between tire and road. vertical, the and theinterface, longitudinal dynamics allthe at Regarding once. Concerning the longitudinal dynamics a railway Regarding the railway sector a lot of the ongoing research activities focus focus on on aa phenomenological phenomenological characterization characterization of of once. Concerning theand longitudinal dynamics of a railway vertical, thewheel-rail lateral, the longitudinal at activities Regarding the interface, railway sector of the ongoinginresearch once. the longitudinal a railway vehicleConcerning the contact withdynamics a size sizedynamics ofofonly only aallfew few activities focus on a phenomenological characterization of the wheel-rail wheel-rail whicha is islotoften often executed a labolabovehicle the wheel-rail contact with a of a the interface, which executed in a once. the dynamics aofrailway activities focusinterface, on a with phenomenological characterization of vehicle the wheel-rail contact withtraction a size forces ofofonly a few the squareConcerning centimeters has longitudinal to transmit transmit about wheel-rail which is often set executed in a laboratory environment an extended extended of measurement measurement square centimeters has to traction of about ratory environment withwhich an set of vehicle the of wheel-rail with a higher size forces offorces only a few the wheel-rail interface, is often executed in a labosquare centimeters has contact tounit transmit traction forces of for about 8kN in case a multiple and even loratory environment with an extended set of measurement equipment, see see for for example example Six Six et et al. al. (2015) (2015) and and Chollet Chollet 8kN in centimeters case of a multiple unit and even higher forces lo- equipment, square has to(2014). transmit traction forces of for about ratory environment with an extended of measurement 8kN in casesee of aSchindler multiple unit andIneven forces for lo- equipment, comotives, thishigher context, the maxisee for example Six et al. set (2015) and Chollet (2017). The findings in these publications undoubtedly comotives, see Schindler (2014). In this context, the maxiThesee findings in these undoubtedly 8kN casesee of aSchindler multiple unit andIndepends even forces for lo- (2017). equipment, for example Six publications et al. (2015) and Chollet comotives, thishigher context, the maximumintransmittable transmittable force (2014). strongly on the the adhesion (2017). findings these publications undoubtedly provide aaThe great value ininunderstanding understanding the complex complex wheelmum force strongly depends on adhesion provide great value in the wheelcomotives, see Schindler (2014). In this context, the maxi(2017). The findings in these publications undoubtedly mum transmittable force strongly depends on the adhesion conditions, which are in turn influenced by environmental provide a great value in understanding the complex wheelrail contact. However, they cannot be directly used for conditions, which are in turn influenced byon environmental contact. However, cannot bethe directly used for mum transmittable force strongly depends the adhesion rail provide a great value in they understanding complex wheelconditions, in turn influenced environmental effects, e.g. which water are or leaves on the rail. by Well established rail contact. However, they cannot be directly used for effects, e.g. water or leaves on the rail. Well established conditions, in turnon influenced environmental effects, e.g. which water are or leaves the rail. by Well established rail contact. However, they cannot be directly used for effects, e.g. water or leaves on the rail. Well established 2405-8963 © 2019, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved.
Copyright © 2019 IFAC 844 Copyright 2019 responsibility IFAC 844Control. Peer review©under of International Federation of Automatic Copyright © 2019 IFAC 844 10.1016/j.ifacol.2019.11.694 Copyright © 2019 IFAC 844
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guidance concept, see Heckmann et al. (2016). At the same time this specialty demands for a reliable longitudinal control of the single wheels, since even slight deviations of the brake and traction torques between left and right wheel disturb the performance of the track guidance control. The last hardware part are the four independently rotating wheels with their own in-wheel motors.
Fig. 1. Running gear on the DLR test rig. the real-time applications the presented article addresses. Other publications on the wheel-rail interface investigate the lateral dynamics, single wheelsets, or one specific type of friction conditions (wet or dry), like Strano and Terzo (2018), Hussain et al. (2013), and Chen et al. (2011). In this work an observer approach for the scaled experimental running gear at DLR will be described, which is general but still simple due to the low order of the underlying observer model. Therefore, some aspects of the hardware configuration and the sensor equipment of the test rig are highlighted in the following section along with the observer model. The simulation and optimization environments of the observer are presented in detail in Section 3. To validate the observer and demonstrate its robust, reliable, and accurate performance it is applied to simulation data in Section 4. The last section draws a conclusion, underlines the contribution to the field of mechatronic railway concepts, and depicts the upcoming tasks to be tackled.
The longitudinally relevant measurement equipment at the test rig includes converters for the wheel rotations ωij with i ∈ [l, t] (leading, trailing) and j ∈ [r, l] (right, left) as well as for the roller rotation ωR . These angular velocities and the translational equivalent vR , respectively, are usually measured on real trains and, therefore, are used as observer inputs in the following sections. In addition, force-torque sensors are integrated in the wheel modules that measure forces and torques in all three directions. The information of the force-torque sensors on the longitudinal and the vertical forces FF T S,x,ij and FF T S,z,ij are used to validate the developed observer at the test rig. Finally, some limitations of the test rig are denoted. First of all, the influence of the longitudinal acceleration v˙ on the wheel load deviation h (1) ∆Fz,i = ±M v˙ , 2l with the mass of the running gear M , the height of the center of gravity h, and the wheel base l cannot be investigated at the test rig, since there is no actual longitudinal motion. Furthermore, in contrast to the NGT concept the experimental running gear is neither equipped with friction brakes nor with a secondary suspension. Thus, the influence of the vertical dynamics on the longitudinal motion is out of scope of the presented observer design.
2. RUNNING GEAR SYSTEM ON THE ROLLER RIG
2.2 Observer Model Synthesis
In the first subsection a short overview on the running gear and roller rig hardware is shown and some peculiarities are highlighted. A detailed discussion of the corresponding simulation environment can be seen in Schwarz et al. (2015). The second subsection presents the analytic formulation of the observer model, which includes a nonlinear friction characteristic for the wheel-rail contact.
The focus of the presented investigation on the longitudinal dynamics leads to a nonlinear system formulation with a linear output equation x) + B u x˙ = f (x y = C x. (2)
2.1 Hardware of the Running Gear on the Roller Rig The test facility is mechanically divided into the experimental running gear and the red rollers, see Fig. 1. The rotation of the two rollers is coupled via a tooth belt, so that there is only one roller velocity vR = ωR · rR , with the roller angular velocity ωR and the roller radius rR = 0.18 m. The velocity vR represents the longitudinal speed of a real train on a track. To block this longitudinal motion at the roller rig the connection between the rig and the running gear frame is realized as a lemniscate guidance. The frame can be charged with additional weights to test different loading conditions and in consequence different wheel-rail characteristics. At the front and the rear end of the frame there is an axle bridge mounted via a leaf spring guidance. These guidances also serve as vertical springs in case of no additional loads. The axle bridges on which the wheels are mounted have got a nearly undamped yaw degree of freedom, what is essential for the lateral track 845
T The state vector x = [ωlr , ωll , ωtr , ωtl , ωR ] considers the four angular wheel velocities and the roller velocity. However, the angles of the wheels and the roller are neglected, since they do not affect the dynamics at all. The system order is intentionally kept at a minimum to minimize the required computational effort and simplify the transfer to control units commonly used in railway applications. The motor torques τij are defined as inputs u so that the input matrix is 1 0 0 0 JW 1 0 0 0 JW , 1 (3) B = 0 0 0 J W 1 0 0 0 JW 0 0 0 0
with the moment of inertia JW = 0.0244 kg·m2 of a wheel with respect to its lateral axis. According to the measurement equipment described in the foregone section
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Table 1. Parameters of the friction model µ0 [−] µ∞ [−] s BP ol m kS [−] kA [−] c11 [−] a · b mm2 rW [m] G [GP a]
0.3 0.11 0.2 0.5 0.9 4.346 0.1 0.1 80.8
max. friction coefficient for w = 0 min. friction coefficient for w = ∞ reduction factor of the friction coeff. reduction factor in the slip area red. factor in the adhesion area Kalker coefficient in long. direction product of contact ellipse semi-axes wheel radius shear modulus
the output matrix C is the identity matrix I ∈ R5×5 . Concerning the primary goal of identifying the friction parameters this definition of C is reasonable even though it might seem unusual in the context of observers. The friction interface between wheel and rail as a crucial aspect regarding the longitudinal dynamics is implemented following the theory postulated in Polach (2000). There the dynamics of the friction coefficients µij are µij = µ0 (1 − AP ol ) e−|BP ol wij | + AP ol , (4)
, the other parameters described in with AP ol = µµ∞ 0 Table 1, and the slip velocity wij = vR + ωij rW .
(5)
In the roller rig context the velocities of the wheels and the roller are added in Equation (5), since they have got an opposite rotational direction. Using Equation (4) the resulting friction force in longitudinal direction is kA · ij 2 · Fz,ij · µij Fx,ij = 2 + arctan (kS · ij ) π 1 + (kA · ij ) (6) with ij =
A G · π · a · b · kS +k · c11 2 sx,ij . 4Fz,ij · µ
(7)
The remaining parameters can be seen in Table 1 and the longitudinal slip is wij (8) sx,ij = vR with wij as described in Equation (5). Combining the above equations in the moment equilibria of the wheels and the roller results in the following system representation 1 (rW · Fx,ij + τij ) JW f = r (9) , R Fx,ij JR i j
with the moment of inertia of the rollers JW = 2.42 kg·m2 .
This stringent formulation of a running gear with only five states neglects the disturbance of coupler forces, which occur in current train sets due to unbalanced traction or braking forces of the single wagons. Anyhow, the presented approach especially addresses the facilitation of advanced traction systems, so it seems acceptable to disregard the coupler effects. In the end, the observer model is implemented in the object-oriented and equationbased language Modelica. This offers essential advantages, when the system might be extended to take electronic, pneumatic and thermal aspects into account. 846
Fig. 2. Scheme of the combined observer structure. 3. OBSERVER SYNTHESIS FOR THE RUNNING GEAR SYSTEM The focus of this work is not only on the dynamics estimation but also on the identification of the adhesion conditions. Therefore, an observer is synthesized that consists of a state observer and a parameter estimator (PE) that determines the friction coefficients µij . The structure of the combined observer is presented in Fig. 2. The nonlinearities illustrated in Equations (4) and (6) have to be considered in the choice of the applied observer approach. A well-established method for observing nonlinear systems is the Extended Kalman Filter (EKF). The EKF specifically takes the noise properties of the system x) + B u + n x x˙ = f (x y = C x + ny (10) n into account. The additive Gaussian noise terms x and n y are incorporated into the observer algorithm by the diagonal covariance matrices Q ∈ R5×5 and Z ∈ R5×5 . The set-up of the EKF with its noise consideration is done in anticipation of the later use on the test rig. For a detailed discussion of the general Kalman Filter approach the interested reader is referred to Simon (2006). In this work the implementation of the EKF bases upon the algorithm presented in Brembeck et al. (2014). To keep the observer structure as simple as possible the parameter estimator is defined as a linear, time-invariant feedback µ = P · ∆yy ∆µ ˆ lr ωlr − ω pl,W 0 0 0 pl,R ω −ω ˆ ll 0 pl,R ll 0 pl,W 0 ˆ tr , (11) = · ωtr − ω 0 0 pt,W 0 pt,R ω −ω ˆ tl tl 0 0 0 pt,W pt,R ωR − ω ˆR
µ = [∆µlr , ..., ∆µtl ]T , with the friction correction vector ∆µ the output estimation error ∆yy = y − yˆ, and the feedback matrix P . Due to the chosen structure of P the deviation in the angular velocity of a wheel does only affect the friction correction of that specific wheel. In contrast, the µ via pi,R . To keep roller speed error influences the entire ∆µ the optimization problem, which is described at the end of this section, manageable the feedback parameters for the right and left wheel-rail contact are the same. Existing methods, like Wenzel et al. (2006), combine two Kalman Filters, what offers great benefits in case of slowly time varying parameters. However, in the discussed context of adhesion estimation the variation of the friction coefficient is often abrupt, e.g. during the entering or exit of a tunnel,
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oscillating behavior is induced by the initialization error of 10% in vinitial .
Table 2. Overview on the optimization cases vinitial µ0 [−]
m s
plant model 0.1 0.3
Case 1 0.1 0.3
Case 2 0.09 0.3
Case 3 0.1 0.2
and usually indeterministical. This conceptual difference together with the aforementioned limited computational capacity legitimates the presented parameter estimator design. After the description of the observer concept some remarks on the optimization of the observer parameters are made. To receive a robust and accurate observer a multi-case optimization is implemented using the Modelica Optimization Library, see Pfeiffer (2012). The three optimization cases vary in their initial conditions as can be seen in Table 2. Each case bases upon the acceleration/deceleration scenario depicted in Fig. 3. Since the results of the leading and trailing wheels are almost identical, only the leading wheels are shown in the following. In the upper plot the longitudinal velocities of the roller and the wheels are illustrated, which represent the measurements y . The speed of 5 ms in 1:5 scale is equivalent to approximately 40 km h in real scale. A descent of the friction coefficients from 0.3 to 0.2 in the time interval t ∈ [3s; 4.2s] leads to the reduced acceleration of the roller in this period. In addition, the friction coefficients are reduced in the same way for t ∈ [11s; 12.2s] but due to the lower deceleration the effect on the roller is less severe. The lower plot shows on the one hand the resulting longitudinal friction forces Fx,lj , what clearly highlights the reduced transmittable force in the low adhesion period. On the other hand the input torques u = τ are illustrated. To optimize the observer parameters the diagonal entries of Q and Z as well as the parameters pi,W and pi,R are defined as tuner variables. Consequently, there are 14 tuner variables. The objective function z is defined as the maximum value of the case objectives zk x −x µ −µ zk = |ccx · (x ˆ )| + |ccµ · (µ ˆ )| dt, (12)
with the index k = 1, 2, 3 for the investigated case. The vectors c x and c µ weight the deviations between measured and observed variables. By using the maximum value of zk it is intended to receive an observer, that works well in all scenarios. The resulting parameters are Q = diag(0.1, 0.1, 0.0006, 0.0001, 0.001), Z = diag 0.9, 0.8, 0.08, 0.1, 1e−5 , and 0.6 0 0 0 1.0 0 0.6 0 0 1.0 P = . (13) 0 0 0.8 0 1.5 0 0 0 0.8 1.5 4. RESULTS OF THE OBSERVER TESTS
In this section some significant results of the observer synthesis are depicted and discussed. All in all the state estimation shows a high accuracy (estimation error ≈ 1%) in every case. Anyhow, this is to be expected, since each of the five states is measured. In addition, the results of Case 1 resemble Case 2, so that only the latter will be presented in the following. The only difference to be mentioned occurs in the first few time steps of Case 2 where an 847
Fig. 4 shows the friction coefficient (upper plot) and longitudinal creep force results (lower plot) of Case 2. First of all, the friction results verify the functionality of the designed observer in estimating the adhesion conditions. The descents in the actual friction coefficient are almost ideally reflected by both wheels, what proves the accuracy. The only remarkable deviation occurs at about 5.4s, when the roller speed reaches the wheel speed and the longitudinal creep force goes from its maximum to zero, compare Fig. 3. The same oscillation appears in the estimation error of the longitudinal creep force, as the lower plot illustrates. Nevertheless, in both signals the deviation is diminished after a short period of time and, thus, it is not rated critical. The last noticeable aspect of Case 2 is the increasing creep force estimation error after 16s. This behavior might come from the slip definition in Equation (8), where the division by vR causes numeric problems in case of vR ≈ 0 ms . This problem is not critical as well, since the adhesion conditions are not relevant after the running gear comes to a stop. The outcome of Case 3 is illustrated in Fig. 5. The creep force error is left out, since there are no relevant differences to Case 2. Comparing the friction results the estimation in the periods with the reduced friction coefficient is just as accurate as in Fig. 4. However, the other phases display major differences. Anyhow, the difference in the periods without braking and accelerating can be neglected, since during this operating conditions the adhesion has got no influence on the dynamics at all. Regarding the longitudinally relevant segments t ∈ [10s; 11s] ∪ [12.2s; 16s] the friction coefficients are also lower than the actual value. The explanation for this behavior might lie in the parameter estimator, which changes the friction coefficient only by the minimally required value that leads to ∆yy = 0 . In the given scenario this means that during the braking process the minimum coefficient of friction, which is necessary to transmit the actual creep force, is approximately 0.25. Looking at the creep force results in Fig. 3 supports this thesis, since the braking forces (≈ 16N) have smaller values than the accelerating forces (≈ 23N). To sum up, the robustness of the presented observer concept as well as the accuracy of the state estimation is verified by different test cases. The reliability of the parameter estimator depends on the initialization error of the friction coefficient. However, in critical phases with a low actual µ the estimator gives accurate results in all of the cases. 5. CONCLUSIONS AND OUTLOOK The previous sections describe the design process of an observer for the longitudinal dynamics of a railway running gear. Since the wheel-rail friction interface strongly affects the longitudinal dynamics, it is considered in the observer synthesis. The presented tests show a reliable and accurate performance of the observer for different scenarios. In the end, this work underlines the great potential of mechatronic components in the field of railway systems and especially the benefits of observers in terms of improved longitudinal dynamics. To validate the performance of the observer in a realtime environment, the observer will be implemented at
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Fig. 3. Test scenario for the validation of the adhesion observer.
Fig. 4. Results of the observer in case 2. the roller rig at DLR. Subsequent to the test rig validation the observer will be tested with measurement data from track tests. Another challenge to be tackled is the adaption of the adhesion observer, so that it is capable of handling also other friction effects, for example in brake components. Therefore, existing models of brake friction dynamics, like Ostermeyer (2003), might be integrated in the presented approach without structural changes. However, this extension requires a specific consideration of the time behavior of the two effects. Thus, the variation of the friction conditions in brake components is comparably slow in relation to the partially abrupt changes in the wheel-rail friction coefficient. Finally, the future work has to define 848
strategies and use cases that exploit the additional information on the adhesion. Examples are the implementation of a monitoring strategy for brake components and also the improvement of existing wheel slide protection and anti skid systems.
ACKNOWLEDGEMENTS The authors would like to thank Dr. Jonathan Brembeck for his valuable comments and support in the observer design.
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Fig. 5. Results of the observer in case 3. REFERENCES Brembeck, J., Pfeiffer, A., Fleps-Dezasse, M., Otter, M., Wernersson, K., and Elmqvist, H. (2014). Nonlinear state estimation with an extended fmi 2.0 co-simulation interface. In Proceedings of the 10th International Modelica Conference, 53–62. Link¨ oping University Electronic Press, Lund, Sweden. Chen, H., Ishida, M., Namura, A., Baek, K.S., Nakahara, T., Leban, B., and Pau, M. (2011). Estimation of wheel/rail adhesion coefficient under wet condition with measured boundary friction coefficient and real contact area. Wear, 271(1-2), 32–39. Chollet, H. (2017). Estimation of the distribution of the wheel-rail friction coefficient in curves during the dynotrain project. In Proceedings of the 25th International Symposium on Dynamics of Vehicles on Roads and Tracks (IAVSD 2017), 791–795. CRC Press, Rockhampton, Australia. Heckmann, A., Schwarz, C., B¨ unte, T., Keck, A., and Brembeck, J. (2016). Control development for the scaled experimental railway running gear of DLR. In Proceedings of the 24th Symposium of the International Association for Vehicle System Dynamics (IAVSD 2015). CRC Press, Graz, Austria. Hussain, I., Mei, T., and Ritchings, R. (2013). Estimation of wheelrail contact conditions and adhesion using the multiple model approach. Vehicle System Dynamics, 51(1), 32–53. Kurzeck, B., Heckmann, A., Kaiser, I., Wesseler, C., and Keck, A. (2014). Potentials of actively controlled independently rotating wheel pairs in high speed traffic - results from the Next Generation Train project. ZEVrail, 138(8), 304–312. Ostermeyer, G. (2003). On the dynamics of the friction coefficient. Wear, 254(9), 852–858. Pfeiffer, A. (2012). Optimization library for interactive multi-criteria optimization tasks. In Proceedings of the 9th International Modelica Conference, 669–680. Link¨ oping University Electronic Press, Munich, Germany. Polach, O. (2000). A fast wheel-rail forces calculation computer code. Vehicle System Dynamics, 33, 728–739. Rajamani, R., Phanomchoeng, G., Piyabongkarn, D., and Lew, J. (2012). Algorithms for real-time estimation of individual wheel tire-road friction coefficients. IEEE/ASME Transactions on Mechatronics, 17(6), 1183–1195. 849
Schindler, C. (ed.) (2014). Handbuch SchienenfahrzeugeEntwicklung, Produktion, Instandhaltung. DVV Media Group GmbH, Hamburg, 1st edition. Schwarz, C., Heckmann, A., and Keck, A. (2015). Different models of a scaled experimental running gear for the DLR RailwayDynamics Library. In Proceedings of the 11th International Modelica Conference, 441– 447. Link¨oping University Electronic Press, Versailles, France. Simon, D. (ed.) (2006). Optimal State Estimation: Kalman, H Infinity, and Nonlinear Approaches. Wiley & Sons, Hoboken, New Jersey, 1st edition. Six, K., Meierhofer, A., M¨ uller, G., and Dietmaier, P. (2015). Physical processes in wheelrail contact and its implications on vehicletrack interaction. Vehicle System Dynamics, 53(5), 635–650. Strano, S. and Terzo, M. (2018). On the real-time estimation of the wheel-rail contact force by means of a new nonlinear estimator design model. Mechanical Systems and Signal Processing, 105, 391–403. Wenzel, T., Burnham, K., Blundell, M., and Williams, R. (2006). Dual extended kalman filter for vehicle state and parameter estimation. Vehicle System Dynamics, 44(2), 153–171.
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Observer Synthesis for the Adhesion Observer Synthesis for the Adhesion Observer Synthesis for the Adhesion Estimation of a Railway Gear Observer Synthesis for Running the Adhesion Estimation of a Railway Running Estimation of a Railway Running Gear Gear Estimation of a Railway Running ∗ ∗ Gear Christoph Schwarz ∗ Alexander Keck ∗
Christoph Schwarz ∗∗ Alexander Keck ∗∗ Christoph Schwarz Alexander Keck Christoph Schwarz ∗ Alexander Keck ∗ ∗ ∗ German Aerospace Center (DLR), Institute of System Dynamics and German Aerospace Center (DLR), Institute of System Dynamics and ∗ ∗ German Aerospace Center (DLR), Institute of [christoph.schwarz, System Dynamics and Control, 82234 Wessling, Germany (e-mail: Control, 82234 Wessling, Germany (e-mail: ∗ German Aerospace Center (DLR), Institute of [christoph.schwarz, System Dynamics and alexander.keck]@dlr.de). Control, 82234 Wessling, Germany (e-mail: [christoph.schwarz, alexander.keck]@dlr.de). Control, 82234 Wessling, Germany (e-mail: [christoph.schwarz, alexander.keck]@dlr.de). alexander.keck]@dlr.de). Abstract: The The presented presented work work illustrates illustrates an an investigation investigation on on the the observer observer synthesis synthesis for for aa highly highly Abstract: nonlinear railway running gear system. The specific focus is longitudinal dynamics and the Abstract: The presented work illustrates an investigation on the observer synthesis for a highly nonlinear railway running gear system. Thean specific focus is on on the longitudinal dynamics the Abstract: The presented work illustrates investigation observer synthesis forinterface aand highly nonlinear railway running gear system. The specific focusbetween is on the longitudinal dynamics and the associated adhesion effects, i.e. the friction conditions wheel and rail. This the associated adhesion effects, i.e. the friction conditions between wheel and rail. This interface nonlinear railway running gear system. The specific focusbetween is on the longitudinal dynamics and the strongly influences the brake and the traction performance of trains and, thus, its detailed associated adhesion effects, i.e. the friction conditions wheel and rail. This interface strongly influences the brake traction performance of wheel trains and,rail. thus, its interface detailed associated adhesion effects, i.e.and thethe friction conditions This consideration is of of crucial crucial importance intraction terms ofperformance safetybetween and comfort. comfort. strongly influences the brake and the of trains and and, thus, its detailed consideration is importance in terms of safety and strongly influences the brake and the traction of gear trains and, thus, its detailed The observer observer design is accomplished accomplished forinthe the experimental running consideration is of crucial importance terms ofperformance safety and comfort. The design is for experimental running gear developed developed at at the the German German consideration is of crucial importance in terms of safety and comfort. The observer design is accomplished for the experimental running gear developed at the Aerospace Center (DLR). The article focuses on the the first first steps steps of of the the observer observer synthesis: synthesis:German (i) the the Aerospace on (i) Center (DLR). The article focuses The observer design is accomplished the experimental running gear developed at (iii) the German set-up of the the observer model, (ii) thefor selection ofthe an first appropriate observer method, and Aerospace Center (DLR). The(ii) article focuses onof steps of observer the observer synthesis: (i) the set-up of observer model, the an appropriate selection method, (iii) and the Aerospace Center (DLR). The article focuses onofthe stepsdetermining of observer the observer synthesis: (i) the validation of the observer observer in aa(ii) simulation environment. The aspect of(iii) theand model set-up of the observer model, the selection an first appropriate method, the validation of the in simulation The determining aspect of the model environment. set-up of the observer model, (ii)simulation theofselection of an appropriate observer method, (iii) and the implementation is the consideration the nonlinear wheel-rail contact. To cover this nonlinear validation of the observer in a environment. The determining aspect of the model implementation isobserver the consideration of the nonlinear wheel-rail contact. To aspect cover this nonlinear validation of the in a simulation environment. The determining of the model implementation is the consideration of the nonlinear wheel-rail contact. To cover this nonlinear behavior an an Extended Extended Kalman Kalman Filter Filter (EKF) (EKF) is is used used in in combination combination with with aa parameter parameter estimator estimator behavior implementation is the consideration the nonlinear cover this nonlinear for the the friction friction characteristics inFilter the of wheel-rail interfaces. In the thecontact. end, theTo received results prove behavior an Extended Kalmanin (EKF) is used inwheel-rail combination with areceived parameter estimator for characteristics the wheel-rail interfaces. In end, the results prove behavior an Extended Kalmanestimates Filter (EKF) is used in combination with areceived parameter estimator that the observer accurately the system behavior and provides reliable information for the friction characteristics in the wheel-rail interfaces. In the end, the results prove that thefriction observer accurately estimates the system andend, provides behavior reliableresults information for the characteristics inand therelated wheel-rail interfaces. In the the received prove on the the adhesion characteristics longitudinal dynamics. that the observer accurately estimates the longitudinal system behavior and provides reliable information on adhesion characteristics and related dynamics. that the observer accurately estimates the system behavior and provides reliable information on the adhesion characteristics and related longitudinal dynamics. Based on on this one upcoming steps DLR Based this findings findings one of of the the upcoming steps at at the the DLR is is to to validate validate the the observer observer in in on adhesion characteristics andtest related longitudinal dynamics. thethe real-time environment rig. Regarding use approach in aa Based on thisenvironment findings one of of the the upcoming steps at the the DLRof isthe to presented validate the observer the real-time of the test rig. Regarding the use of the presented approach in in Based on this findings one of the upcoming steps at the DLR is to validate the observer in train some significant improvements of traction control systems are enabled. A first application the real-time environment of the test rig. Regarding the use of the presented approach in a train some significant improvements of traction controlthe systems enabled. A first application the real-time environment of control the test rig. Regarding use ofare presented approach in a train some significant improvements oftotraction control systems arethe enabled. A first application could be an enhanced brake narrow down the variance of brake distances in daily could be ansignificant enhancedimprovements brake control oftotraction narrow control down the variance of brake A distances in daily train some systems are enabled. first application could be anFurthermore, enhanced brake control tousability narrow of down variance of brake distances in daily operation. an advanced advanced the the condition based monitoring of traction traction operation. Furthermore, an usability of the condition based monitoring of could be an enhanced brake control tousability narrow down the variance of brake distances in daily and brake systems is facilitated with the additional information on the friction conditions. operation. Furthermore, an advanced of the condition based monitoring of traction and brake systems is facilitated with the additional information the friction conditions. operation. Furthermore, an advanced usability of the conditionon based monitoring of traction and brake systems is facilitated with the additional information on the friction conditions. © 2019, IFAC (International Federation Automatic Control) Hosting byonElsevier Ltd. Allconditions. rights reserved. and brake systems is facilitated withofthe additional information the friction Keywords: Rail Rail traffic, traffic, nonlinear nonlinear systems, systems, adhesion, adhesion, Extended Extended Kalman Kalman filter. filter. Keywords: Keywords: Rail traffic, nonlinear systems, adhesion, Extended Kalman filter. Keywords: Rail traffic, nonlinear systems, adhesion, Extended Kalman filter. 1. INTRODUCTION INTRODUCTION driver assistance assistance systems systems like like wheel wheel slide slide protection protection and and 1. driver 1. INTRODUCTION driver assistance systems likerisk wheel slide protection and anti-skid systems reduce the of blocking and skidding anti-skid systems systems reduce the risk of blocking and skidding driver likerisk wheel slidethe protection and The improvement improvement1. of ofINTRODUCTION safety and and comfort comfort is is often often the the anti-skid systems reduce wheels.assistance Nevertheless, to further further improve functionality the of blocking skidding The safety wheels. Nevertheless, to improve the and functionality anti-skid systems reduce the risk of blocking and skidding The improvement of safety and comfort is often the basic objective objective in in the the research research on on and and development development of of wheels. further functionality of these these Nevertheless, systems and and to totoenable enable forimprove examplethe condition based basic systems for example condition based The improvement of safety and often the wheels. toenable further improve functionality basic objective in the research oncomfort and development of of mechatronic vehicle concepts. In addition, theisimportance importance of these Nevertheless, systems and todetailed for examplethe condition based monitoring concepts, information on the existing mechatronic vehicle concepts. In addition, the concepts, information on the existing basic objective in the research onconsumes and development of monitoring of these systems and is todetailed enable for example condition based mechatronic vehicle concepts. In addition, the aimportance of aa sustainable sustainable performance that minimum monitoring concepts, detailed information on the existing adhesion conditions necessary. Due to economic economic reasons of performance that consumes aimportance minimum adhesion conditions is necessary. Due to reasons mechatronic vehicle concepts. In addition, the monitoring concepts, detailed information ondaily the existing of aresources sustainable performance that consumes a minimum adhesion was and is still growing. The multifaceted conditions is necessary. Due to economic reasons and the rough environmental influences in railway of aresources was performance and is still that growing. The multifaceted and the rough environmental influences in daily reasons railway sustainable consumes a minimum adhesion conditions is necessary. Duebe to directly economic of resources was of and still growing. TheCenter multifaceted railway activities theisGerman German Aerospace (DLR), and the rough influences in dailyprovided railway operation, this environmental information cannot railway activities of the Aerospace Center (DLR), operation, this information cannot be directly provided of resources was of and isGerman still growing. TheCenter multifaceted and the rough environmental influences inobserver dailyprovided railway railway activities the which are combined in the the long-term research project Next operation, this information cannot be directly Aerospace (DLR), by a sensor. Therefore, a control theoretic might which are combined in long-term research project Next a sensor.this Therefore, a control theoretic observer might railway activities of the German Aerospace Center (DLR), operation, information cannot be directly provided which are combined in the long-term research project Next by Generation Train (NGT), try to solve the partially existing by a sensor. Therefore, a control theoretic observer might be implemented, to estimate the unmeasurable adhesion Generation Train (NGT), try to solve the partially existing be implemented, to estimate the unmeasurable adhesion which combined in the long-term project Next be by aimplemented, sensor.and Therefore, a control observer might Generation Trainthese (NGT), try to solveresearch the partially existing conflictare between aspects. to estimate thetheoretic unmeasurable adhesion conditions related effects like creep forces. conflict between aspects. and related effects like creep forces. Generation Trainthese (NGT), try to solve the partially existing conditions be implemented, to estimate the unmeasurable adhesion conflict between these aspects. conditions and related effects like creep forces. One particular particular focus ofaspects. the railway railway research research at at the the DLR DLR is is conditions In the the automotive automotive field, the like estimation of the the friction friction conflict betweenfocus theseof and related effects creep forces. One the field, the estimation of One particular focus ofand thetechnologies railway research at thethe DLR is In on concepts, methods to exploit great In the automotive field, the estimation of the friction conditions between tire and road is an intensely investion concepts, methods technologies to exploit great between tire is an intensely One particular focus ofand the railway research at thethe DLR is conditions In theproblem, automotive field,and theroad of the investifriction on concepts, methods and technologies to exploit the great potential of mechatronic mechatronic running gears with independently conditions between tire and road an intensely investigated see Rajamani Rajamani etestimation al. is (2012). However, these potential of running gears with independently gated problem, see et al. (2012). However, these on concepts, methods andrunning technologies to exploit the great conditions between tire and road is an intensely investipotential of mechatronic gears with independently rotating wheels identified in Kurzeck et al. (2014). One gated problem, see Rajamani et al. (2012). However, these results cannot be directly adapted, since the adhesion rotating wheels identified in Kurzeck et al. (2014). One cannot see be Rajamani directly adapted, sinceHowever, the adhesion potential of mechatronic gears with gated etsteel al. (2012). these rotating wheels identified in Kurzeck et al.independently (2014). One results of the the most most crucial andrunning determining aspects of railway railway results be directly adapted, since and the steel adhesion effectsproblem, ofcannot the contact contact between wheels rails of crucial and determining aspects of effects of the between steel wheels and rails rotating wheels identified in Kurzeck et al. (2014). One results cannot be directly adapted, since the steel adhesion of the most crucial and determining aspects of railway dynamics is the wheel-rail interface, since it affects the effects of the contact between steel wheels and steel rails significantly differ from the effects between tire and road. dynamics is the wheel-rail interface, since it affects the significantly differ from the effects between tire and road. of the most crucial and the determining aspects of railway effects of the contact between steel wheels and rails dynamics is lateral, the wheel-rail interface, since it affects vertical, the and longitudinal dynamics allthe at significantly differ from the effects between tire steel and road. Regarding the railway sector lot of of the ongoing ongoing research vertical, the lateral, and the longitudinal dynamics all at the railway sector aa lot the research dynamics is lateral, the wheel-rail since itofaffects significantly differ from the effects between tire and road. vertical, the and theinterface, longitudinal dynamics allthe at Regarding once. Concerning the longitudinal dynamics a railway Regarding the railway sector a lot of the ongoing research activities focus focus on on aa phenomenological phenomenological characterization characterization of of once. Concerning theand longitudinal dynamics of a railway vertical, thewheel-rail lateral, the longitudinal at activities Regarding the interface, railway sector of the ongoinginresearch once. the longitudinal a railway vehicleConcerning the contact withdynamics a size sizedynamics ofofonly only aallfew few activities focus on a phenomenological characterization of the wheel-rail wheel-rail whicha is islotoften often executed a labolabovehicle the wheel-rail contact with a of a the interface, which executed in a once. the dynamics aofrailway activities focusinterface, on a with phenomenological characterization of vehicle the wheel-rail contact withtraction a size forces ofofonly a few the squareConcerning centimeters has longitudinal to transmit transmit about wheel-rail which is often set executed in a laboratory environment an extended extended of measurement measurement square centimeters has to traction of about ratory environment withwhich an set of vehicle the of wheel-rail with a higher size forces offorces only a few the wheel-rail interface, is often executed in a labosquare centimeters has contact tounit transmit traction forces of for about 8kN in case a multiple and even loratory environment with an extended set of measurement equipment, see see for for example example Six Six et et al. al. (2015) (2015) and and Chollet Chollet 8kN in centimeters case of a multiple unit and even higher forces lo- equipment, square has to(2014). transmit traction forces of for about ratory environment with an extended of measurement 8kN in casesee of aSchindler multiple unit andIneven forces for lo- equipment, comotives, thishigher context, the maxisee for example Six et al. set (2015) and Chollet (2017). The findings in these publications undoubtedly comotives, see Schindler (2014). In this context, the maxiThesee findings in these undoubtedly 8kN casesee of aSchindler multiple unit andIndepends even forces for lo- (2017). equipment, for example Six publications et al. (2015) and Chollet comotives, thishigher context, the maximumintransmittable transmittable force (2014). strongly on the the adhesion (2017). findings these publications undoubtedly provide aaThe great value ininunderstanding understanding the complex complex wheelmum force strongly depends on adhesion provide great value in the wheelcomotives, see Schindler (2014). In this context, the maxi(2017). The findings in these publications undoubtedly mum transmittable force strongly depends on the adhesion conditions, which are in turn influenced by environmental provide a great value in understanding the complex wheelrail contact. However, they cannot be directly used for conditions, which are in turn influenced byon environmental contact. However, cannot bethe directly used for mum transmittable force strongly depends the adhesion rail provide a great value in they understanding complex wheelconditions, in turn influenced environmental effects, e.g. which water are or leaves on the rail. by Well established rail contact. However, they cannot be directly used for effects, e.g. water or leaves on the rail. Well established conditions, in turnon influenced environmental effects, e.g. which water are or leaves the rail. by Well established rail contact. However, they cannot be directly used for effects, e.g. water or leaves on the rail. Well established 2405-8963 © 2019, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved.
Copyright © 2019 IFAC 844 Copyright 2019 responsibility IFAC 844Control. Peer review©under of International Federation of Automatic Copyright © 2019 IFAC 844 10.1016/j.ifacol.2019.11.694 Copyright © 2019 IFAC 844
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guidance concept, see Heckmann et al. (2016). At the same time this specialty demands for a reliable longitudinal control of the single wheels, since even slight deviations of the brake and traction torques between left and right wheel disturb the performance of the track guidance control. The last hardware part are the four independently rotating wheels with their own in-wheel motors.
Fig. 1. Running gear on the DLR test rig. the real-time applications the presented article addresses. Other publications on the wheel-rail interface investigate the lateral dynamics, single wheelsets, or one specific type of friction conditions (wet or dry), like Strano and Terzo (2018), Hussain et al. (2013), and Chen et al. (2011). In this work an observer approach for the scaled experimental running gear at DLR will be described, which is general but still simple due to the low order of the underlying observer model. Therefore, some aspects of the hardware configuration and the sensor equipment of the test rig are highlighted in the following section along with the observer model. The simulation and optimization environments of the observer are presented in detail in Section 3. To validate the observer and demonstrate its robust, reliable, and accurate performance it is applied to simulation data in Section 4. The last section draws a conclusion, underlines the contribution to the field of mechatronic railway concepts, and depicts the upcoming tasks to be tackled.
The longitudinally relevant measurement equipment at the test rig includes converters for the wheel rotations ωij with i ∈ [l, t] (leading, trailing) and j ∈ [r, l] (right, left) as well as for the roller rotation ωR . These angular velocities and the translational equivalent vR , respectively, are usually measured on real trains and, therefore, are used as observer inputs in the following sections. In addition, force-torque sensors are integrated in the wheel modules that measure forces and torques in all three directions. The information of the force-torque sensors on the longitudinal and the vertical forces FF T S,x,ij and FF T S,z,ij are used to validate the developed observer at the test rig. Finally, some limitations of the test rig are denoted. First of all, the influence of the longitudinal acceleration v˙ on the wheel load deviation h (1) ∆Fz,i = ±M v˙ , 2l with the mass of the running gear M , the height of the center of gravity h, and the wheel base l cannot be investigated at the test rig, since there is no actual longitudinal motion. Furthermore, in contrast to the NGT concept the experimental running gear is neither equipped with friction brakes nor with a secondary suspension. Thus, the influence of the vertical dynamics on the longitudinal motion is out of scope of the presented observer design.
2. RUNNING GEAR SYSTEM ON THE ROLLER RIG
2.2 Observer Model Synthesis
In the first subsection a short overview on the running gear and roller rig hardware is shown and some peculiarities are highlighted. A detailed discussion of the corresponding simulation environment can be seen in Schwarz et al. (2015). The second subsection presents the analytic formulation of the observer model, which includes a nonlinear friction characteristic for the wheel-rail contact.
The focus of the presented investigation on the longitudinal dynamics leads to a nonlinear system formulation with a linear output equation x) + B u x˙ = f (x y = C x. (2)
2.1 Hardware of the Running Gear on the Roller Rig The test facility is mechanically divided into the experimental running gear and the red rollers, see Fig. 1. The rotation of the two rollers is coupled via a tooth belt, so that there is only one roller velocity vR = ωR · rR , with the roller angular velocity ωR and the roller radius rR = 0.18 m. The velocity vR represents the longitudinal speed of a real train on a track. To block this longitudinal motion at the roller rig the connection between the rig and the running gear frame is realized as a lemniscate guidance. The frame can be charged with additional weights to test different loading conditions and in consequence different wheel-rail characteristics. At the front and the rear end of the frame there is an axle bridge mounted via a leaf spring guidance. These guidances also serve as vertical springs in case of no additional loads. The axle bridges on which the wheels are mounted have got a nearly undamped yaw degree of freedom, what is essential for the lateral track 845
T The state vector x = [ωlr , ωll , ωtr , ωtl , ωR ] considers the four angular wheel velocities and the roller velocity. However, the angles of the wheels and the roller are neglected, since they do not affect the dynamics at all. The system order is intentionally kept at a minimum to minimize the required computational effort and simplify the transfer to control units commonly used in railway applications. The motor torques τij are defined as inputs u so that the input matrix is 1 0 0 0 JW 1 0 0 0 JW , 1 (3) B = 0 0 0 J W 1 0 0 0 JW 0 0 0 0
with the moment of inertia JW = 0.0244 kg·m2 of a wheel with respect to its lateral axis. According to the measurement equipment described in the foregone section
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321
Table 1. Parameters of the friction model µ0 [−] µ∞ [−] s BP ol m kS [−] kA [−] c11 [−] a · b mm2 rW [m] G [GP a]
0.3 0.11 0.2 0.5 0.9 4.346 0.1 0.1 80.8
max. friction coefficient for w = 0 min. friction coefficient for w = ∞ reduction factor of the friction coeff. reduction factor in the slip area red. factor in the adhesion area Kalker coefficient in long. direction product of contact ellipse semi-axes wheel radius shear modulus
the output matrix C is the identity matrix I ∈ R5×5 . Concerning the primary goal of identifying the friction parameters this definition of C is reasonable even though it might seem unusual in the context of observers. The friction interface between wheel and rail as a crucial aspect regarding the longitudinal dynamics is implemented following the theory postulated in Polach (2000). There the dynamics of the friction coefficients µij are µij = µ0 (1 − AP ol ) e−|BP ol wij | + AP ol , (4)
, the other parameters described in with AP ol = µµ∞ 0 Table 1, and the slip velocity wij = vR + ωij rW .
(5)
In the roller rig context the velocities of the wheels and the roller are added in Equation (5), since they have got an opposite rotational direction. Using Equation (4) the resulting friction force in longitudinal direction is kA · ij 2 · Fz,ij · µij Fx,ij = 2 + arctan (kS · ij ) π 1 + (kA · ij ) (6) with ij =
A G · π · a · b · kS +k · c11 2 sx,ij . 4Fz,ij · µ
(7)
The remaining parameters can be seen in Table 1 and the longitudinal slip is wij (8) sx,ij = vR with wij as described in Equation (5). Combining the above equations in the moment equilibria of the wheels and the roller results in the following system representation 1 (rW · Fx,ij + τij ) JW f = r (9) , R Fx,ij JR i j
with the moment of inertia of the rollers JW = 2.42 kg·m2 .
This stringent formulation of a running gear with only five states neglects the disturbance of coupler forces, which occur in current train sets due to unbalanced traction or braking forces of the single wagons. Anyhow, the presented approach especially addresses the facilitation of advanced traction systems, so it seems acceptable to disregard the coupler effects. In the end, the observer model is implemented in the object-oriented and equationbased language Modelica. This offers essential advantages, when the system might be extended to take electronic, pneumatic and thermal aspects into account. 846
Fig. 2. Scheme of the combined observer structure. 3. OBSERVER SYNTHESIS FOR THE RUNNING GEAR SYSTEM The focus of this work is not only on the dynamics estimation but also on the identification of the adhesion conditions. Therefore, an observer is synthesized that consists of a state observer and a parameter estimator (PE) that determines the friction coefficients µij . The structure of the combined observer is presented in Fig. 2. The nonlinearities illustrated in Equations (4) and (6) have to be considered in the choice of the applied observer approach. A well-established method for observing nonlinear systems is the Extended Kalman Filter (EKF). The EKF specifically takes the noise properties of the system x) + B u + n x x˙ = f (x y = C x + ny (10) n into account. The additive Gaussian noise terms x and n y are incorporated into the observer algorithm by the diagonal covariance matrices Q ∈ R5×5 and Z ∈ R5×5 . The set-up of the EKF with its noise consideration is done in anticipation of the later use on the test rig. For a detailed discussion of the general Kalman Filter approach the interested reader is referred to Simon (2006). In this work the implementation of the EKF bases upon the algorithm presented in Brembeck et al. (2014). To keep the observer structure as simple as possible the parameter estimator is defined as a linear, time-invariant feedback µ = P · ∆yy ∆µ ˆ lr ωlr − ω pl,W 0 0 0 pl,R ω −ω ˆ ll 0 pl,R ll 0 pl,W 0 ˆ tr , (11) = · ωtr − ω 0 0 pt,W 0 pt,R ω −ω ˆ tl tl 0 0 0 pt,W pt,R ωR − ω ˆR
µ = [∆µlr , ..., ∆µtl ]T , with the friction correction vector ∆µ the output estimation error ∆yy = y − yˆ, and the feedback matrix P . Due to the chosen structure of P the deviation in the angular velocity of a wheel does only affect the friction correction of that specific wheel. In contrast, the µ via pi,R . To keep roller speed error influences the entire ∆µ the optimization problem, which is described at the end of this section, manageable the feedback parameters for the right and left wheel-rail contact are the same. Existing methods, like Wenzel et al. (2006), combine two Kalman Filters, what offers great benefits in case of slowly time varying parameters. However, in the discussed context of adhesion estimation the variation of the friction coefficient is often abrupt, e.g. during the entering or exit of a tunnel,
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oscillating behavior is induced by the initialization error of 10% in vinitial .
Table 2. Overview on the optimization cases vinitial µ0 [−]
m s
plant model 0.1 0.3
Case 1 0.1 0.3
Case 2 0.09 0.3
Case 3 0.1 0.2
and usually indeterministical. This conceptual difference together with the aforementioned limited computational capacity legitimates the presented parameter estimator design. After the description of the observer concept some remarks on the optimization of the observer parameters are made. To receive a robust and accurate observer a multi-case optimization is implemented using the Modelica Optimization Library, see Pfeiffer (2012). The three optimization cases vary in their initial conditions as can be seen in Table 2. Each case bases upon the acceleration/deceleration scenario depicted in Fig. 3. Since the results of the leading and trailing wheels are almost identical, only the leading wheels are shown in the following. In the upper plot the longitudinal velocities of the roller and the wheels are illustrated, which represent the measurements y . The speed of 5 ms in 1:5 scale is equivalent to approximately 40 km h in real scale. A descent of the friction coefficients from 0.3 to 0.2 in the time interval t ∈ [3s; 4.2s] leads to the reduced acceleration of the roller in this period. In addition, the friction coefficients are reduced in the same way for t ∈ [11s; 12.2s] but due to the lower deceleration the effect on the roller is less severe. The lower plot shows on the one hand the resulting longitudinal friction forces Fx,lj , what clearly highlights the reduced transmittable force in the low adhesion period. On the other hand the input torques u = τ are illustrated. To optimize the observer parameters the diagonal entries of Q and Z as well as the parameters pi,W and pi,R are defined as tuner variables. Consequently, there are 14 tuner variables. The objective function z is defined as the maximum value of the case objectives zk x −x µ −µ zk = |ccx · (x ˆ )| + |ccµ · (µ ˆ )| dt, (12)
with the index k = 1, 2, 3 for the investigated case. The vectors c x and c µ weight the deviations between measured and observed variables. By using the maximum value of zk it is intended to receive an observer, that works well in all scenarios. The resulting parameters are Q = diag(0.1, 0.1, 0.0006, 0.0001, 0.001), Z = diag 0.9, 0.8, 0.08, 0.1, 1e−5 , and 0.6 0 0 0 1.0 0 0.6 0 0 1.0 P = . (13) 0 0 0.8 0 1.5 0 0 0 0.8 1.5 4. RESULTS OF THE OBSERVER TESTS
In this section some significant results of the observer synthesis are depicted and discussed. All in all the state estimation shows a high accuracy (estimation error ≈ 1%) in every case. Anyhow, this is to be expected, since each of the five states is measured. In addition, the results of Case 1 resemble Case 2, so that only the latter will be presented in the following. The only difference to be mentioned occurs in the first few time steps of Case 2 where an 847
Fig. 4 shows the friction coefficient (upper plot) and longitudinal creep force results (lower plot) of Case 2. First of all, the friction results verify the functionality of the designed observer in estimating the adhesion conditions. The descents in the actual friction coefficient are almost ideally reflected by both wheels, what proves the accuracy. The only remarkable deviation occurs at about 5.4s, when the roller speed reaches the wheel speed and the longitudinal creep force goes from its maximum to zero, compare Fig. 3. The same oscillation appears in the estimation error of the longitudinal creep force, as the lower plot illustrates. Nevertheless, in both signals the deviation is diminished after a short period of time and, thus, it is not rated critical. The last noticeable aspect of Case 2 is the increasing creep force estimation error after 16s. This behavior might come from the slip definition in Equation (8), where the division by vR causes numeric problems in case of vR ≈ 0 ms . This problem is not critical as well, since the adhesion conditions are not relevant after the running gear comes to a stop. The outcome of Case 3 is illustrated in Fig. 5. The creep force error is left out, since there are no relevant differences to Case 2. Comparing the friction results the estimation in the periods with the reduced friction coefficient is just as accurate as in Fig. 4. However, the other phases display major differences. Anyhow, the difference in the periods without braking and accelerating can be neglected, since during this operating conditions the adhesion has got no influence on the dynamics at all. Regarding the longitudinally relevant segments t ∈ [10s; 11s] ∪ [12.2s; 16s] the friction coefficients are also lower than the actual value. The explanation for this behavior might lie in the parameter estimator, which changes the friction coefficient only by the minimally required value that leads to ∆yy = 0 . In the given scenario this means that during the braking process the minimum coefficient of friction, which is necessary to transmit the actual creep force, is approximately 0.25. Looking at the creep force results in Fig. 3 supports this thesis, since the braking forces (≈ 16N) have smaller values than the accelerating forces (≈ 23N). To sum up, the robustness of the presented observer concept as well as the accuracy of the state estimation is verified by different test cases. The reliability of the parameter estimator depends on the initialization error of the friction coefficient. However, in critical phases with a low actual µ the estimator gives accurate results in all of the cases. 5. CONCLUSIONS AND OUTLOOK The previous sections describe the design process of an observer for the longitudinal dynamics of a railway running gear. Since the wheel-rail friction interface strongly affects the longitudinal dynamics, it is considered in the observer synthesis. The presented tests show a reliable and accurate performance of the observer for different scenarios. In the end, this work underlines the great potential of mechatronic components in the field of railway systems and especially the benefits of observers in terms of improved longitudinal dynamics. To validate the performance of the observer in a realtime environment, the observer will be implemented at
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Fig. 3. Test scenario for the validation of the adhesion observer.
Fig. 4. Results of the observer in case 2. the roller rig at DLR. Subsequent to the test rig validation the observer will be tested with measurement data from track tests. Another challenge to be tackled is the adaption of the adhesion observer, so that it is capable of handling also other friction effects, for example in brake components. Therefore, existing models of brake friction dynamics, like Ostermeyer (2003), might be integrated in the presented approach without structural changes. However, this extension requires a specific consideration of the time behavior of the two effects. Thus, the variation of the friction conditions in brake components is comparably slow in relation to the partially abrupt changes in the wheel-rail friction coefficient. Finally, the future work has to define 848
strategies and use cases that exploit the additional information on the adhesion. Examples are the implementation of a monitoring strategy for brake components and also the improvement of existing wheel slide protection and anti skid systems.
ACKNOWLEDGEMENTS The authors would like to thank Dr. Jonathan Brembeck for his valuable comments and support in the observer design.
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