1.Introduction

Anthropogenic CO2emissions into the atmosphere have been increased considerably by civil aviation given the rapid growth in the air traffic market in recent years.1The aircraft industry has faced economic and environmental issues.2Recently,more electric aircraft3(MEA)and all electric aircraft4,5have received significant interest for developing safer,lower-cost,and greener technologies for next-generation air transportation.6,7With the constant investment in aviation,power-by wire8(PbW)technology eliminates heavy and bulky hydraulic pipes and the pipe vibration issue.Thus,conventional centralized hydraulic,pneumatic,and mechanical networks will be nearly exclusively replaced by an electric power network,which can provide significant advantages in ease of power management,integration,and maintenance,as well as reductions of environmental pollution and fuel burn.9As shown in Fig.1,PbW is introduced step by step into flight controls,landing gears,and engines as the key contribution to MEA,where signal and power are transmitted by electric wires.

Nomenclature

Fig.1 Mapping of power distribution in MEA.8

However,the maturity of PbW technology is lagging.In fact,the real challenge in implementing PbW in MEA is to develop compact,reliable,and electrically powered actuators with the same function to replace conventional hydraulic servo actuators(HSAs).10,11To date,only two categories of PbW actuators,namely,electro-hydrostatic actuators(EHAs)12and electromechanical actuators(EMAs),13have been developed.These actuators may exhibit architectural changes,e.g.,an electric backup hydraulic actuator14(EBHA)and an electric backup mechanical actuator15(EBMA).PbW actuators have already entered into service in the latest commercial airplane programs.16In Airbus A380/A400M/A350,an EHA serves as a backup actuator for primary and secondary flight controls.In Boeing B787,an EMA is partly placed on the front line for secondary flight controls and landing gear braking.17–19

Fig.1 Mapping of power distribution in MEA.8

Compared with an EHA,an EMA completely eliminates the use of hydraulic circuits,thereby increasing its economic,competitive,and environmental advantages.Significant improvements in the performance and maturity of electric motors(EMs),as well as their power drive electronics(PDE)and control,make EMAs more and more attractive.20,21However,a mature EMA for extensive safety critical applications,22particularly for primary flight control,still lacks a historical database and requires considerable effort.23In response to these needs,the aerospace industry is searching for an innovation(either incremental or disruptive)in EMA actuation systems.In particular,linking the motor rotor directly to the load is impossible when mass constraints are severe.The mass of an EM mainly results from its capability to develop torque.Electromagnetic torque is produced by a combination of current and magnetic fields that require heavy materials such as copper,magnets,and iron.The load generally requires a high driving force(typically 20–300 kN)at a relatively low speed(typically 20–500 mm/s);thus,the best option for the actuator mass is to use low-torque/high-speed motors(typically 10,000 r/min for 10 kW)and to introduce mechanical power transformation(MPT)between the EM and the flight control surface(load).24Thus,the safety critical EMA actuation system used in the front line(normal or active mode)has several potential issues that must be considered as a whole.These issues include thermal balance under natural heat dissipation,mechanical balance with reaction forces(housing or load),huge mass reflected by the motor rotor at the load level,tolerance or resistance to jamming,25response to failure by wear or backlash fault,26,27back-drivability,cushioning and damping,mass reduction,and cost saving.

In such a situation,a virtual model for MEA systems and their components should be designed and developed.Several earlier works have mainly focused on the fundamental and theoretical analyses of an EMA and its components,such as dynamic testing,28load distribution modeling,29kinematics and transmission efficiency analysis of nut-screw,30nonlinear elastic compliance analysis,31and failure model development for an EM.32Recently,however,more practical and experimental studies have been conducted.Fu et al.33developed a comprehensive model for the contact position and clearance analysis of standard planetary roller screws.Ma et al.34published their investigations on the thermo-mechanical analysis of an EMA.They considered load distribution as a Frictional heat model that was developed using a 3D finite element method to study the structure parameters,operation conditions,and cooling performances of coolants.

The aforementioned studies analyzed the behavior of an EMA based on the component level.However,designers and manufacturers must meticulously consider the parasitic effects resulting from the imperfections of technology from a system level perspective.These effects include resistance,inductance,friction,backlash,compliance,and temperature.The aforementioned considerations are particularly true for assessing design step changes,cross-links between advanced technology attachment,and reduction of time to market,35which are highly cross-linked Engineering requirements with the following purposes:

(P1)Multi-scale:3D to 1D design(model simplification/linearization)and 1D to 3D design(for sizing activities);

(a)A flight control computer(FCC)ensures that the surface position corresponds to flight law commands.For monitoring purposes,the management of fault monitoring from the actuator is completed by the FCC.

(P3)Multi-physics:electrical(signal and power),electronics,electromagnetic,mechanical(torque and force),solid mechanics,thermal(heat transfer),and fluid(lubrication);

(P4)Multi-system engineering activities:requirements,architecting(functional,advanced,and behavior),sizing(power and con figuration),integration verification and validation,troubleshooting,and training;

阿拉善家畜的体内没有报道存在球虫，西北农林科技大学的教授调查时发现山羊有球虫，且数量很多，用地克珠利威囊效果很好，但是球虫对羊的生长发育影响尚不明确[2]。

(P5)Multi-topic:dynamics,36thermal/mechanical balance,peak/mean electrical power,electrical power network stability and pollution,37energy/power consumption,redundancy38and faults to failure39,40/health monitoring,41nonlinearity analysis42and parameter uncertainty identification,43vibration performance,44and force fighting45/motion synchronization.46

Thus,this deficiency should be compensated by resorting to a model-based system engineering(MBSE)47,48approach(Fig.2 of Ref.49).The requirement,function,logical solution element,and physical element can be easily considered through a standardized model exchange or an open architectural cosimulation process.Although commercial simulation software is widely available,methodologies for model architecting are evidently lacking.Such a deficiency will eventually affect continuity among engineering activities,knowledge capitalization,and the multi-physical nature of actuation systems.Meanwhile,engineers typically fail to apply best practices to take full advantage of this new technology when they address crosslinks in terms of model development,sharing,integration,and capitalization.Accordingly,this study illustrates best practices by using Bond graph formalism50,51as the modeling language to develop EMA models that can be extensively used in MEA.

The rest of this article is organized as follows.Section 2 presents system-level virtual prototyping methodology for EMAs.Power flows and multi-domain disciplines are presented,and current engineering activities and engineering requirements are addressed.In Section 3,replaceable component models(PDE,EM,and MPT)are developed with progressive levels of representativeness.The imperfections of technologies are progressively introduced as parasitic effects in advanced and behavioral models by starting with a functional model that considers only combined physical effects to perform a function.Section 4 illustrates the implementation processes of the proposed models in a causal simulation environment by providing an example of an aileron actuator system.Then,numerical analysis is performed in Section 5 to highlight the interest of various engineering activities.Finally,conclusions summarize the main achievements of this study and offer perspectives for further work.

2.EHA system description

This study deals with a typical direct-drive linear EMA,where the rotating nut of MPT is directly integrated as a part of the EM rotor.The elimination of the intermediate gear box provides the aforementioned EMA with a compact design.This‘in-line” EMA is more attractive than a geared EMA for aerospace applications because of weight reduction and easy geometric integration into the airframe.The schematic of this EMA is presented in Fig.3.This EMA can be developed for flight control and landing gear actuation applications.When applied to flight control,the load is regarded as the surface,which is driven according to pilot/autopilot commands and suffers from aerodynamic disturbances.

Electric power is supplied by the electric power network of the aircraft.Power delivered to the driven load is measured by the PDE according to the command signal from the actuator controller.Electric power is transformed into mechanical power via a rotary EM,and then,mechanical power is transferred to the control surface via a nut-screw MPT.The control of such an EMA is typically based on three main electronic devices as follows.52

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Magnetic hysteresis occurs within ferromagnetic materials.The hysteresis effect happens between the remanence flux density B and the coercivity H(typical B/H curves),as shown in Fig.9.The area of hysteresis domains represents the work performed(per unit volume of material),i.e.,

(P2)Multi-level:continuity among models for airframe,system,equipment,and component suppliers;

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(b)The EMA actuator controller receives the EMA position order from the FCC and sensor information from the EMA;it then processes an associated control loop.In general,EMA control involves a rod position(outer loop),a speed loop(middle loop),and a motor current/torque inner loop,to fulfill performance and stability requirements.Actuator control electronics(ACE)internally generates the pulse width modulation(PWM)command for the PDE as an image of the torque reference.

Fig.2 System-level modeling process and purpose.49

Fig.3 Schematic structure of an EMA actuation system.

(c)The PDE is integrated near the EM into the EMA.The modulated electric power forms the aircraft network to an EM.

Virtual prototyping must be considered only as a through point supporting engineering,not as a final objective.A model is neither universal nor unique.It is developed for a given purpose with an appropriate complexity level.A model is a living object that evolves against need and experience feedback to the extent that it is capitalized.Ideally,a model should be dissociated from its numerical implementation.In many cases,however,this condition cannot be achieved for technological devices with behaviors that exhibit strong nonlinearities and discontinuities,which seriously impact model implementation.

2.1.Modeling procedure

The following questions were raised based on over three decades of experience in system-level virtual prototyping for industrial projects:

绿色建筑工程项目在建造阶段会受到各种不确定因素的干扰，这些因素的产生会造成造价成本的起伏波动，直接和项目最终收益情况关联，对绿色建造工程造成成本预估产生波动的因素主要包括：

The virtual prototyping of an EMA system requires multidisciplinary approaches for preliminary power sizing and performance evaluation,including an estimation of the mass and considerations of the geometric envelope and thermal behavior.Heat in an EMA is generated by different types of power loss(e.g.,conduction,switching,copper,iron,and friction),which are over dissipated toward the local environment.Unlike an HSA system where fluid power also exhibits a cooling effect,an EMA system is a thermally closed-circuit system.The aforementioned physical effects are always multi-domain cross-linked.The models must be balanced(energetically and mechanically)to assess coupled effects.Consequently,the use of Bond graph formalism is particularly interesting for model architecting.The disciplines involved in an EMA study in cross-linked multi domains are shown in Fig.4.

(2)What will be the effects of the function under study?A product is generally designed by combining physical effects to achieve a desired function.

3.1.2.Bond graph models of the PDE and the EM

(4)How are functional and major parasitic effects modeled?Knowledge models(from science)or representation models(from actual or virtual tests)are combined to model functional and dominant parasitic effects.

(5)How is a model implemented in a(generally imposed)simulation tool?The model must be adapted to the solver of the simulation environment.Whenever possible,an option to reuse models/submodels from software libraries should be provided because these models/submodels are generally well-documented,proven,and numerically robust.

2.2.Multidisciplinary effects

(1)What form of engineering must be supported by the simulation of a model?In this manner,the modeling activity will address an appropriate system with a proper complexity level.

Fig.4 Multidisciplinary domain coupling in an EMA.

2.3.Performance requirements

To increase the efficiency of model-based development,the virtual prototyping of an EMA should satisfy the following requirements53:

(R1)Preliminary:The model shall enable the dynamic/static performance of EMA control design,even for component sizing;

(R2)Generality:The model shall consist of generic submodels that can be reused for other modeling purposes;

(R3)Realism:The model shall consider the key physical effects,which significantly affect the performance,at a system-level perspective;

(R4)Balance:The model shall be energetically balanced.Efficiency and output power can be calculated by introducing power loss effects;

(R5)Replaceability:The model shall have standard interfaces to ensure its replaceability in multi-level modeling;

(R6)Faults:The model shall be considered as the response to faults;

(R7)Causality:The model shall be developed to admit various causalities that are consistent with causal and acausal simulation environments.

2.4.Model architecting

(3)Advanced Model

3.Multi-level modelling

Model development can be structured using an incremental approach according to requirements.54Model realism can be incremented in a top-down design,and subsystem models must be compatible with bottom-up virtual integration tests.EMA component models can be replaceable and balanced.Complexity is progressively increased regardless of whether each physical effect is considered.

3.1.PDE and EM modeling

The PDE and the EM are the key electric devices in an EMA.The PDE modulates the power transferred between the electric supply bus and the motor through the action of the motor’s winding voltage according to the switching signals sent to the power transistors.The EM performs electromechanical power conversion.The input voltage and current are functionally proportional to the torque and angular velocity,respectively,at the motor shaft.A high-performance permanent magnet synchronous motor(PMSM)is currently applied to an EMA to reduce the complexity of EMA control design.

Fig.6 presents a typical structure of the bridge circuit of the PDE for driving the motor.An equivalent DC motor driven by a chopper that implements the H-bridge can be modeled.For detailed modeling,the PMSM driven by the inverter can be introduced to provide a realistic performance.In both models,the control signal is adapted to generate the duty cycle signal.

Fig.5 Matrix of a model architecture considering engineering requirements and effects.

Fig.6 Co-package structure of a PDE and an EM.

3.1.1.Modeling levels

An incremental modeling approach is used to progressively analyze the physical effects of the PDE and the EM.For model architecting,different modeling levels are suggested as a function of engineering requirements.Four generic levels of model package(the PDE and the EM)can be defined according to the different complexities of physical effects(Fig.7).The levels are organized from the simplest to the most complex,with the associated impacts on accuracy and simulation time cost.The reference torque signal is C*,the current loop is presented,and the current is detected by Bond graph element Df.

(3)What imperfections may significantly alter the addressed performance?The performance of a product is assessed by considering parasitic effects that can never be completely avoided(e.g.,friction,inertia,and winding resistance).From an industrial point of view,the best model is never the most complex.Consequently,only parasitic effects which significantly affect the performance must be considered.

The physical principles acting on the PDE and the EM are multi-domain.These principles include electrical,magnetic,mechanical,and thermal.Technological imperfections(e.g.,power losses and rotor inertia)significantly influence the performance.In most chopper or inverter bridge circuits of the PDE,the basic commutation cell involves a solid-state switch,e.g.,an insulated gate bipolar transistor(IGBT),and a diode that provides freewheeling through an anti-parallel structure.Recent developments in wide band gap semiconductors and highly efficient motors will reduce losses.However,although power losses are minimal,they still significantly affect PDE sizing in the thermal aspect.Temperature affects the dynamic performance,service life,and reliability of the PDE and the EM.It also causes a snowball effect.

Fig.7 Complexity modeling levels of a PDE and an EM.

(1)Perfect Model

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As shown in Fig.8(a),the PDE functionally operates as a perfect and modulated power transformer(MTF)between the DC supply and the EM,which is driven by the current controller to generate a modulating effect of factor(α)as follows:

The EM enables perfect power transformation between the electrical and rotational mechanical domains.The motor shaft torque(Cm)is equal to the motor electromagnetic torque(Cem).A power gyrator(GY)with a parameter of the motor torque constant(Km)is as follows:

(2)Basic Model

As shown in Fig.8(b),a simplified consideration of the power loss in the PDE,typically the efficiency parameter(η),is available in the datasheet of the product supplier.A resistance effect Rddcan be used to express the physical effect that causes voltage to drop(Udd)at the motor supply as

The main source of power losses in the electric domain of the EM is copper loss(also called Joule loss),which can be modeled by the R element in a Bond graph.Power loss can be defined as a voltage drop(Uco)due to the winding resistance(Rw)to current.The winding inductance(Lw)produces an electrical storage(Uis)effect as follows:

When the electric domain effect is considered,voltage supply at the motor level is given as follows:

Rotor inertia and friction are important mechanical effects that affect the propagation of the electromagnetic torque to the motor shaft output.Torque balance is affected by the parasitic rotor inertial torque and friction torque(Cfm).This condition is expressed through the dynamics of the motor shaft as follows:

In accordance with previous performance requirements,the system-level virtual prototyping of an EMA shall follow the system engineering process,and best practices and recommendations must be considered.To facilitate numerical implementation and avoid hard nonlinearities and discontinuous effects,parasitic effects that insignificantly affect the performance are neglected in some cases.As shown in Fig.5,considering physical effects to select the appropriate level of model complexity for model architecting is significant.The virtual prototyping of realism to physical effects has become a practice to work toward multiplicity of engineering requirements,such as natural dynamic analysis,thermal balance,power loss,closed-loop controlled performance,component sizing,weight/cost reduction,and response to faults.However,these forms of engineering require different accuracy levels.Model complexity(i.e.,the number of parameters/variables)affects the rapidity and robustness of simulations.

Fig.8 Multi-level Bond graph models of a PDE combined with an EM.

As shown in Fig.8(c),additional details and realistic physical effects are considered in the PDE and the EM based on Bond graph formalism.Firstly,power losses in the PDE can be divided into three types:on-state conduction,off-state blocking(leakage),and turn-on/turn-off switching losses.In practice,leakage currents generated in off-state power are minor and can be neglected.

Conduction losses occur during the on-state mode when an IGBT or a diode starts conducting and generates a voltage drop Ucd,which is a nonlinear function of the operating current(related to the load current)and the rate current of the IGBT or diode.In the PDE of an EMA,IGBTs or diodes always operate in the low-current range,which is a fairly ‘linear”region of their transfer characteristic,and thus a linear approximation can be used to express the ‘on-state” resistance Ron.Then Ucdby conduction losses can be expressed by zero-current forward threshold voltage Uthplus the voltage drop of Ronas

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A switching loss typically contributes to a significant proportion of the total power loss.It happens during transient‘‘turn-on”/‘turn-off” because an IGBT or a diode cannot switch instantaneously.During switching,a phase lag occurs between current and voltage.The power losses of switching cannot be neglected when the switching frequency is high(typically 10 kHz for aerospace EMAs).In practice,switching loss is regarded as current leak that is directly proportional to the switching frequency(fsw)as

Iron loss in the EM is another important physical effect.The variation of flux density in the magnetic circuit of a motor causes iron loss.The magnetic field in the motor rotor rotates with the rotor at the same velocity.Then,minimal magnetic flux variation generates nearly no iron loss in the rotor.Thus,iron loss mainly occurs in the motor stator and can be typically divided as eddy current loss and hysteresis loss.

Eddy current is generated by reversing the magnetic field in iron.This field induces a voltage that produces eddy currents due to the electrical resistance of iron.Magnetic quantities cannot be accessed during measurements;hence,the effect of eddy current is commonly expressed as an equivalent power loss that is reflected in the mechanical domain.The eddy current torque(Ced)can be modeled by the first member of the Steinmetz equation55as a function of the eddy current constant ked,namely,the magnetic flux density(Bm)and the angular velocity(ωm)as follows:

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Another physical effect in the EM is the cogging torque(Ccg),which occurs in the magnetic domain.The variation in the air gap permeance of the stator teeth and slots above the magnets during rotor rotation generates a torque ripple,which is also known as the detent torque.The cogging torque can be described as a pure sign function of a nonlinear spring effect.The AC element in the Bond graph can be used,as shown in Fig.8(c).A system-level representation model for Ccgcan be expressed versus the rotor/stator relative angle θ.This model is parameterized by the number npof motor pole pairs and the cogging factor λ that is applied to the rated torque Cn,which is given by

在某项目调试过程中，紧挨着RPT的设备D3频繁离线，硬件更换之后，故障仍然存在。通过排查MVB线路及连接器，没有发现明显异常，每段线路阻抗均为(1±10%)×120 Ω，符合IEC 61375标准要求。故障线路拓扑示意如图1所示，总线主BA和离线的从设备D3分别位于中继器RPT的两侧。

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Fig.9 Examples of B/H curves with hysteresis loss.56

Modeling the cogging effect may be important for two reasons.Firstly,the cogging torque can be used as a functional effect in certain actuation applications to avoid backdrivability.Secondly,cogging generates a torque ripple during operation,the frequency of which depends on the relative rotor/stator velocity.This condition may excite the natural dynamics of the EMA and its mechanical environment,thereby potentially leading to vibrations and noise emission.

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The power losses of the PDE and the EM generate heat and increase temperature in a closed volume.Electrical resistance is sensitive to the operating temperature,e.g.,the on-state resistance characteristics of IGBTs and diodes in the PDE and winding resistance in the EM.This condition will cause a snowball effect of power losses.However,the effect of temperature on electrical resistance is nonlinear and poorly documented in data sheets.In the absence of accurate data,a linear dependence is assumed for each type of resistance,related to the initial temperature(T0)and the operating temperature(T1)as

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Furthermore,an increase in temperature may decrease the performance of the magnets in the EM,which reduces the motor torque constant(Km).A Bond graph element(De)is used to modulate the generator(MGY)parameter(Km)in the Bond graph model shown in Fig.8(c).Similar to electrical resistance,this element may also be modeled as a linear dependency to temperature as follows:

Motor friction highly depends on temperature.The modeling of this effect at the system level has been addressed in our previous work.57The effect of temperature on friction is poorly documented in supplier data sheets.

(4)Behavioral Model

不同特性的生物保鲜剂对水产品的保鲜机理不尽相同。有些生物保鲜剂含有抗菌活性物质[5]，如茶多酚、壳聚糖、双歧杆菌、鱼精蛋白和海藻酸钠等，能够抑制或杀死水产品中的腐败菌，减缓TVB-N值的上升，保持水产品鲜度；有些生物保鲜剂具有抗氧化作用，防止水产品中不饱和脂肪酸等氧化造成品质劣变[5]，如葡萄糖氧化酶、乳酸链球菌素和竹叶抗氧化物[6]等；还有些生物保鲜剂如茶多酚和植酸等能够抑制酶的活性[5]，防止水产品的变色，保证良好的感官品质。此外，像壳聚糖、谷氨酸胺转氨酶和蜂胶等保鲜剂还可以在水产品表面形成一层保护膜，防止微生物侵染，并减少水分损失，保持水产品的品质[2]。

As the flux density drops below the saturation point,the relationship between current and magnetic flux in ferromagnetic materials ceases to be linear;a given current generates less magnetic flux than expected.The EM torque constant is a function of the magnetic flux(ψ),and the look-up table approach can be used to identify the relation as follows:

In addition,the inductance of motor windings(Lm)is affected by magnetic saturation.Thus,as shown in Fig.8(d),the EM torque constant is modulated by ψ and the operating temperature,and the winding inductance is also modulated by ψ.

Previous models simplified the EM as an equivalent DC model,which is driven by a chopper-type PDE.This model considers a realistic three-phase PMSM motor,which is currently widely implemented in the aerospace industry.A typical structure when a three-phase PMSM is associated with a three-leg(leg A,leg B,leg C)inverter PDE is shown in Fig.6.The three legs of the PDE are driven by a field-oriented control(FOC)58strategy to generate the PWM voltage(V1to V6)applied to the motor phases.The Bond graph models of the inverter PDE and the three-phase PMSM are presented in Fig.8(d).The driven signals are elaborated by direct and inverse Park transformations,which link space vectors associated with the motor phases in a fixed motor frame(three-phase A,B,C)to a rotating frame(direct-quadrature transformation d-q)attached to the relative rotor/stator position(θ).Control signals are transformed into on/off states,which resort to the PWM,to drive the solid-state switches of the legs.

When the constant power constraint coordinates are used,Park transformation(MTF),the actual three-phase motor current to the d-q coordinate current,and the MTF parameters are modulated by θ,which provides

When Ohm’s law is applied to various phases in the PMSM,stator phases A,B,C tensions are then replaced by the d-q tension to give the following:

The stator cyclic inductance and resistance of the d-q axis can be easily defined in the Bond graph.In a direct drivetype EMA,when a surface mount-type PMSM is integrated,and the relation between the inductance and resistance of the d/q axis is as follows:

Thus,in the Bond graph model shown in Fig.8(d),the modulate gyrator parameter(β)of the motor’s d/q axis can be expressed as

The motor stator current is given as

To reduce copper loss in the motor,the torque angle(δ)is unchanged at 90°,and the direct axis current is zero.This strategy is most commonly used for surface PMSM control.It leaves only the torque or quadrature axis current in place.Thus,this strategy has constant direct axis flux linkages(ψ)to produce a constant motor torque,which is given by

3.2.MPT modeling

MPT is a key component in the virtual prototyping of an EMA.A rotary-to-linear power transformation is achieved using a nut-screw mechanism.This transformation aims to convert the torque and rotational velocity of the EM into force and linear speed to drive the flight control surface,as shown in Fig.10.

Fig.10 Schematic of MPT(roller type nut-screw).38

3.2.1.Topology decomposition

MPT can be developed as an integrated model in which the nut-screw,bearings(cylinder pairs),joints(hinges),shaft,and end stop are considered at their power path.For the behavioral function,MPT can be regarded as having two functional types of motion on the same axis:rotation from the EM rotor and translation to the rod.The motions of the PDE and the EM can be ignored and simply studied.Topology models with the same sectional view of product geometric arrangement are proposed,as shown in Fig.11(left).This figure illustrates the proposed two-degree of freedom (2-DOF)architecture and the corresponding functional power flow through the MPT model based on Bond graph formalism.59As illustrated in Fig.11(right),it is suggested to use the generic component with four ports(① to④)to connect two different bodies.The generic sub-model is able to model the interactions between the four power bonds associated with body 1 and body 2,for rotational and translational motions.In order to ensure energy balance and energy conservation as they represent the energy exchange between subsystems,a fifth thermal port(⑤)is added to introduce the heat generation.In this MPT concept,modeling the thermal balance and functions of the anti-rotation rod and the rotor axial thrust bearing is interesting.In addition,for a simplified study of thermal behavior,the EMA housing and rod are assumed as thermal bodies that receive the heat generated by power losses from the EM and MPT subsystem and exchanges it with the environment.

3.2.2.Bond graph models of MPT

When the candidate locations of the generic effects on MPT are considered,the actual model involves a series of four effects from the motor to the surface:global inertia/mass,perfect nut-screw,friction loss,and compliance effects(which can represent backlash,preload,or pure compliance).In the proposed MPT architecture,friction loss and compliance effects are the most significant and can be modeled at different complexity levels(linear/nonlinear and continuous/discontinuous).Inertia and mass effects on MPT can either be neglected or modeled depending on the orders of the EM rotor and surface rotors.The effects of the bearings,joints,and end stop will not be explicitly modeled,but can be considered as a part of the MPT model.Consequently,the proposed generic MPT model involves four mechanical power ports(rotation and translation of the nut and screw).

Fig.11 Mechanical and energy balance decomposition for MPT.

(1)Perfect model

MPT is firstly expressed as a perfect model without any parasitic effect.Friction,compliance,inertia,and relative motion are ignored.ATF element with causality in the Bond graph is shown in Fig.12(a).In the proposed causality,the nut-screw receives the torque(Cm)from the EM and transfers the amount of force(Fs)as an output to drive the surface.A perfect nut-screw model achieves a pure power transformation ratio(2π/l)as follows:

where l is the screw pitch or lead.

(2)Basic Model

The main interest of a basic model lies in its linearity,which is useful for preliminary control design based on a linear control theory.At this level,three major parasites are introduced:nut-screw inertia(Jns),viscous friction loss(fe),and basic compliance effect including the contact elastic force(Fe)and the damping force(Fdm).The corresponding Bond graph model is shown in Fig.12(b),in which the proposed causalities use the same interfaces of the power and signal ports as those of the previous functional model.

The inertia effect generates an inertial torque as follows:

The friction effect generates a friction force.From the point of view of a control engineer,the simplest explanation is to describe friction loss as being proportional to the operating velocity,which is affected by a viscous coefficient(fvc)as follows:

The compliance effect makes MPT compliant because of the elastic deformation of solids under mechanical stress,particularly at the contact locations.A simple compliance model consists of pure spring(ke)and damping(de)effects.The contact elastic force(Fe)and the damping force(Fdm)are typically considered linear functions of the relative displacement(xe)and the relative velocity(ve),respectively,as follows:

(3)Advanced Model

In advanced modeling,the nonlinear friction model(influences of velocity,load force,and temperature)is firstly upgraded to a modulated resistance(MRS)element in the Bond graph as it dissipates power.The nonlinear compliance model(pure spring,backlash,and preloading effects)is considered a modulated capacitance(MC)element.Then,the mechanical power losses of friction(Pf)and compliance damping(Pd)can be introduced through heat flows and connected to an additional thermal port.Consequently,temperature can be easily used as a time-variable input in the energy loss models.Lastly,the proposed MPT model architecture at this level involves a thermal port that is connected to the EMA thermal model.Heat generation can be regarded as an output to improve the thermal model for the EMA by maintaining energy balance in the entire system.Thus,temperature sensitivities in friction and compliance models(dilatation)can be modeled using a thermal port temperature variable.Fig.12(c)describes the Bond graph modeling of the advanced MPT model.Causalities are kept consistent with those of the two previous model levels.

The mathematical expressions for the nonlinear effects of friction,compliance,and temperature sensitivity were presented in our previous work.57Confidence is improved in the current advanced level of MPT modeling,which is consistent with the energy balance point of view.

Fig.12 Multi-level Bond graph model for MPT.

(4)Behavioral Model

To increase realism,studies on support motion and faults to failure should be considered.A behavioral model includes mechanical faults(e.g.,jamming and free-play)and parasitic motion(e.g.,relative displacement because of bearings and joints),as shown in Fig.12(d).

(i)Mechanical Faults:The simulation of faults that may occur in MPT is of utmost importance when it aims to assess the fault-to-failure mechanism and the response to-failure mechanism of an actuation system that may involve faulty MPT.Two main faults are identified:jamming(no possible motion)and free-play(no force transmitted to the load).

Jamming to failure:Numerous methods are currently proposed to provide EMA tolerance or resistance to jamming,such as in HSAs or EHAs.These methods either apply the redundancy concept,including several transmission paths to set a backup mode,or remove or decrease the number of gear mechanisms in EMAs.The development of direct-drive EMAs using roller screws is currently a hot issue.However,although it is lower in direct-drive EMAs,the risk of jamming remains an important source of potential hazard.The capability to assess the response of a system that involves an EMA(e.g.,flight controls of landing gears)to a jamming fault in the EMA through simulation is important during systems development.Jamming faults should also be simulated to assess the advantages of health monitoring features.

In the proposed model,jamming can be triggered as a signal input to modulate the friction model parameters(MRS)to force stiction.This approach is preferred over introducing a piloted brake into the model for the following reasons:a brake does not exist physically,and increasing friction is consistent with modeling the progressive degradation of mechanical efficiency(e.g.,resulting from lack of lubrication).Consequently,jamming can be modeled by increasing the force in the friction model in response to a jamming signal,which can be calculated as follows:

Free-play to failure:Free-play(or free-run)is another fault caused by mechanical rupture(or considerable increase in backlash).This fault is difficult to control with a high accuracy in a closed loop.The response of a system that involves an EMA to the effect of free-play is particularly important to fulfill safety requirements.Free-play can lead to flutter or shimmy,whereas excessive free-play can cause instability of the internal control loops(e.g.,EMA speed loop).These destabilizing effects are undesirable in aerospace applications.Many activities are focused on estimating and compensating the backlash effect in control strategies.

In the proposed model,free-play can also be triggered using a signal input to change the parameters of MC.Preload can be progressively decreased,whereas backlash(xk0)can be progressively increased,by increasing the compliance parameter xkwhich is consequently altered according to the free-play signal(xfp)input as follows:

(ii)Parasitic Motion:Power transformation does not operate purely between motor shaft rotation and rod translation.Strictly speaking,it is applied to the relative rotational speed between the nut and the screw(or vice versa)and the relative translational velocity between the screw and the nut(or vice versa).This process can be added to the model by considering the rotational and translational motions of the nut-screw support,namely,the anti-translation function of the MPT rotating part and the anti-rotation function of the MPT translating part.Consequently,two power interfaces(mechanical translational and rotational power bonds)are added to the Bond graph model.In this manner,imperfect bearings and joints(e.g.,reproducing their compliance and friction)can be considered,and the reaction torque(Ch)and force(Fh)can be accessed to perform antirotation and anti-translation functions(e.g.,within the EMA housing,airframe,or by the load itself).Therefore,the relative rotational velocity(ωnr)for nut/rotor and the relative linear velocity(vnr)for screw to surface are calculated(represented by a ‘0” junction in the Bond graph model).

Support rotation:The relative rotational velocity between the rotor/nut and support bearings can be modeled as follows:

Support translation:The relative translational velocity between the screw and support bearings is given by

All the proposed features finally result in an MPT model with both energy balance and mechanical balance.This model is also the preliminary virtual prototype of a 2-DOF motion MPT that can connect bearing and joint support interfaces.Moreover,this behavioral model can be extended to prognostics and health monitoring studies of EMAs.

4.Implementation of models

The flight surface is subjected to aerodynamic action to illustrate the proposed component models.The subsequent developments of EMA models are based on a cascading multiloop control architecture.Gpis the function of the position controller,and Gvis the function of the velocity controller.The speed limitation (ωlim)and the torque limitation(Clim)are introduced for safety consideration,as shown in Fig.13.

The driven load(e.g., flight control surface)is modeled by combining its inertial effect(equivalent mass)and the compliance of attachment to the EMA and the external load,e.g.,originating from aerodynamic forces.As shown in Fig.13,the proposed EMA model can be regarded as a separate package with multiple levels.These levels are independent and replaceable(the same input power/signal port from the controller and output power/signal port to the surface).Although causal choices have been made in the previous section,each architecture is adapted to both causal(AMESim)and noncausal(Dymola)simulation environments.The implementation of the different levels of the EMA model is detailed as follows.

4.1.Preliminary control synthesis model

The EMA model should be simple,low-order,and linear when focusing on system preliminary control design or functional simulation engineering activities.This situation indicates that the current loop,PDE,electromagnetic part of the EM,and nonlinear mechanical effect of MPT are neglected,as shown in Figs.9(a)and 12(b).At this simulation level,detailed dynamics,thermal effects,and electrical supply are not considered.The torque reference(C*)generated by the EMA controller is directly applied to the rotor shaft inertia as a perfect EM and as a torque source.By contrast,the parasitic inertia effect is not neglected because it is the main driver of the closed-loop performance.EM inertia and MPT inertia are merged into a single lumped inertia on which pure viscous friction is applied.EM power flows to a perfect MPT(nutscrew)and is transmitted to the load through a compliant airframe structure that is considered using a linear spring-damper model.The simple package model of PDE,EM,and MPT is shown in Fig.14.

Fig.13 Model implementation areas in the EMA actuation system.

Fig.14 Model for preliminary control synthesis.

4.2.Development of the PDE model

A perfect three-leg inverter can be selected and considered as a modulated power source to simplify sizing and control application activities in PDE.This method provides access to power drawn from the electrical power supply and develops the current loop of the three-phase EM.The EM model uses the standard model that introduces the resistance and inductance of the windings.At this level,PWM dynamics and power losses are not considered.Each leg is modeled as a voltage source that is modulated by the associated output of the reverse Park transformation;current feedback is modulated by the direct Park transformation and introduced into the current controller.58The associated model is shown in Fig.15.

High-frequency dynamics,pollution of the DC supply,and high-frequency PWM switching shall be modeled when the focus is on power losses(e.g.,switching loss),and the expense of the simulation load shall be accepted.The proposed model reuses the standard library as much as possible and replaces the former perfect modulated power source.The PWM module is driven by the reverse Park transformation block and generates six-switching orders sent to power transistors.Each transistor(IGBT)model and diode model are also obtained from the standard library.They separately output conduction and switching losses.All losses are summed up to determine the overall PDE power losses that are sent to the thermal behavior port.These detailed changes are shown in Fig.16.

4.3.Development of the EM model

The standard three-phase EM model(PMSM)always considers copper loss and is already integrated into the current commercial simulation software library.However,this model does not include iron loss,cogging torque,magnetic saturation,or hysteresis effect.As shown in Fig.8(d),these effects must be considered when developing an advanced EM model.Thus,a subcomponent EM is designed with the same type of interfaces but with improved accuracy of the motor model(Fig.17),which can replace the standard motor model.This EM implements Eqs.(9)–(11)using standard blocks from the library for signals(functions of one variable)and for mechanics(e.g.,torque sum,friction,and sensors).A specific model is created to provide the speed sum that corresponds to the 0 junction in the Bond graph.

4.4.Development of the MPT model

In accordance with the Bond graph model in Fig.8(c),the basic model shown in Fig.14 of a perfect nut-screw with linear friction is replaced with the advanced model presented in Fig.18 or the behavioral model illustrated in Fig.19.

Fig.15 Implementation of the advanced model that considers the three-phase current loop and controller.

Fig.16 Implementation of the advanced inverter model for power loss and switching dynamics.

In the advanced MPT model(Fig.18),the proposed friction model is implemented by making the friction force dependent on the transmitted force,temperature,and sliding velocity(force and velocity are captured through sensors models).This dependence can be defined either by parametric functions or by using look-up tables.The compliance model is implemented by moving to the ‘signal” world,thereby explicitly separating flow and effort variables through a standard AMESim model.A standard AMESim thermal power bond is then introduced,which collects power losses(friction model and structural damping of the compliance model)and inputs temperature into the friction(impact on the friction factor)and compliance(impact on backlash)models.

The behavioral MPT model(Fig.19)is based on the Bond graph model in Fig.11.Jamming and free-play faults are optionally introduced via external signals that affect friction and compliance parameters,respectively.In addition,the relative rotational and translation positions between two mechanical components are available.Two mechanical ports associated with nut housing for anti-translation and with screw housing for anti-rotation can be explicitly introduced for connection with the bearings and joints for 2-DOF modeling to conduct a detailed study.

5.Numerical investigation and analysis

The previous sections have architected EMA component models using the incremental method.The present section presents the influences of multi-level models on simulated responses.The flight control surface is simply modeled as an equivalent translating mass(Ms)to which air load is applied.In addition,the anchorage of the EMA housing to the wing and the EMA rod to the load connection is assumed to be a single-structure compliance of a linear spring and damper.The parameters of the EMA controllers and driven load are listed in Table 1.

Fig.17 Block diagram of the advanced model that considers the three-phase current loop and controller.

Fig.18 Block diagram of the advanced MPT model.

Fig.19 Implementation of the behavioral MPT model in AMESim.

Table 1 EMA controller and equivalent load parameters.

Parameters Symbols Units Values Proportional gain of position controller 45 Proportional gain of velocity controller Kprad·mm-1/s 0.47 Velocity limitation ωlimrad/s 314 Torque limitation ClimN·m 10 Proportional gain of current controller Kv(N·m·s/rad)KipV/A 67.8 Integral gain of current controller KiiV·s-1/(A·m)17.7 Structural stiffness ksN/m 5 ×107Structural damping BsN·s/m 1×104Equivalent flight surface mass Mskg 600

The parameters of the PDE and the EM are shown in Table 2,which are established based on our previously established knowledge.54The parameters of MPT are provided in Table 3,which can be found either in product datasheets60for inertia effect,from our former study52for friction effect,or assumed from engineering experience for compliance effect.In practice,this information is of prior interest for concurrent engineering and for performance sensitivity studies.

The numerical simulation presented in this section is valid for simulating the position-controlled EMA of a commercialaircraft aileron under the same mission pro file,in which the multi-level PDE,EM,and MPT models are individually assessed in the time domain with a single closed-loop simulation.This simulation study can support various engineering requirements,such as control design,energy consumption or thermal balance,dynamic performance,power supply pollution,fault injection,and mechanical balance.

Table 2 Parameters of the PDE and EM models.

Parameters Symbols Units Value DC supply UsV 565 PWM switching frequency fswkHz 10 Total on-state resistance of IGBTs and diodes in each leg RonΩ0.14 Total forward threshold voltage of IGBTs and diodes in each leg UthV 1.7 IGBTs and diodes switching on energy losses EonJ 0.008 IGBTs and diodes switching offenergy losses EoffJ 0.008 Motor pole pairs np4 Motor rotor inertia included nut-screw Jmkg m2 0.0017 Motor torque constant Km0N m/A 1.64 Motor nominal torque CnN m 10 Motor resistance RmΩ2.26 Motor inductance LmmH 18.2 Motor hysteresis loss Steinmetz constant khy5.8×10-3Motor eddy current Steinmetz constant ked9.3×10-6Motor cogging torque factorλ 0.01

Table 3 Parameters of the MPT model.

Parameters Symbols Units Value Nut–screw inertia integrated with rotor Jnkg·m20.0017 Rod mass Mtkg 1 Lead of screw l mm 2.54 Elastic stiffness of nut-screw keN/m 3× 108Damping factor of nut-screw deN·s/m 1× 104

5.1.Interest for control design

The performance requirements are expressed in terms of closed-loop stability,dynamics,and accuracy for both position pursuit and load rejection.A pilot step demand Xcfor the aileron position(10 mm at time t=0.1 s,6.6%full stroke)and an external aerodynamic force FL(10 kN at time t=1 s,65%rated output force)are applied to the following simulations.

Friction is an important parasitic effect that influences the system performance.Fig.20 compares the load position simulated by different MPT models(i.e.,functional,basic,and advanced)that introduce different friction effects(linear or nonlinear).Both simulated responses are stable.Firstly,friction increases the damping of the EMA system and affects dynamic performances when compared with the perfect(no friction)model.Secondly,the perfect model considers in finite stiffness in MPT,whereas the basic and advanced models introduce realistic nut-screw compliance.The effects of compliance clearly reflect the load rejection performance by increasing load oscillations,particularly in the basic and advanced models.

Fig.21 shows the electromagnetic torque of the motor versus time to highlight the differences among the MPT models.The motor torque/current is saturated longer when a more realistic friction model is considered.Significant differences in responses also occur at extremely low velocities.

5.2.Interest of the PWM switching dynamics

Fig.20 Comparison of dynamic performances.

Fig.21 Comparison of torque sources power flow.

PWM shall be modeled by increasing the simulation load when focusing on the high-frequency dynamics and pollution of the DC supply.The responses of the PWM switching and nons witching models are assessed based on the Bond graph models in Figs.15 and 16,respectively.

EM phase voltage and current computed by the switching and non-switching models are compared in Figs.22 and 23,respectively.The non-switching models mean that switching dynamics are not computed and variables are the averages of the actual quantities during the switching period.The harmonics caused by the high-frequency switching dynamics can only be simulated in switching models.

In Fig.22,PMSM phase-A voltages are simulated,and the switching model switches between fixed values at high frequency and presents the shape of the real phase voltage.With the non-switching model,only the average value can be observed,and it does not fit exactly the voltage objective,because semiconductor voltage drops cause a distortion effect.In Fig.23,PMSM phase-A currents are simulated,and the current of the switching model has noise at the switching frequency.The value of the non-switching model current is slightly higher,because the non-switching model does not take into account the phenomenon that the PWM introduces a delay of half of the switching period.

In addition,the d-axis and q-axis currents under the fieldoriented control can be studied.As shown in Fig.24,for the PWM switching model,the q-axis dynamic current is nearly null at time 0.24 s because the surface achieves the peak overshoot of the load displacement and has a whistle stop.

5.3.Interest of power consumption analysis

When analysis is focused on the power losses and energy consumption of MPT at the system-level simulation of an EMA,the basic model fails to reproduce the contribution of friction,whereas the proposed advanced model for MPT enables mechanical designers to assess the impact for comparative analysis.Fig.25 shows the specific mission(Xcand FL)and the load position response(Xs).A trapezoidal position pro file is required at a maximum speed of 125 mm/s.The external force is increased from 0 N at time 0.1 s to 15 kN at time 1 s.

As shown in Fig.26,when the final surface position is reached(no speed but high load),friction loss is null,and high-speed and high-addition loads exhibit the highest power losses.The MPT friction loss represents the highest source(70%)of total energy loss,which highlights the importance of developing an advanced friction model for MPT.This result is consistent with observations made in practice for various research projects.

Fig.22 Comparison of the phase-A voltage in a standard PMSM.

Fig.23 Comparison of the phase-A current in a standard PMSM.

Fig.24 Comparison of the d/q axis current via FOC control.

5.4.Interest for wear/aging and preloading

Fig.25 Mission pro file for power consumption analysis.

The following analysis illustrates the interest of the proposed models for wear/aging by increasing backlash and considering the preloading effect.Firstly,the proposed behavior models dynamically modify the transmission backlash by acting on parameter xk0to simulate the effect of wear versus service.This feature of the model is illustrated in Fig.27,where backlash is changed from null to 60 μm,and then to 0.3 mm.Backlash has a minimal effect when contacts are loaded in a single direction;hence,only the pursuit part of the EMA response is plotted.In the presence of a backlash,a nonlinear limit cycle occurs around the demanded rod position,thereby resulting in an unstable position control.

Secondly,the preloading effect is simulated by the MPT behavioral model.The responses are shown in Fig.28,which introduces the following functions:null preload/backlash(xk0=0),backlash(xk0=0.3 mm),and then a preload of 3 kN(xk0=-0.06 mm).Backlash results in surface chattering and affects position accuracy.Preload eliminates this effect(but increases friction)on the surface and affects surface rapidity.

5.5.Interest for faults to failure

The EMA control methodology should be validated and verified in the presence of faults because of critical safety requirements.Thus,the simulation of faults to failure will facilitate real validation of experimental tests.One issue of faults to failure in an EMA is free-play.The response to a free-play fault can be simulated by setting the backlash with a value higher than twice the nut-screw stroke(i.e.,150 mm).In this manner,the mechanical link between the motor rotor and the load is nonexistent.An example of the simulation result is plotted in Fig.29 using the behavioral MPT model,in which free-play is initiated at time 0.2 s when a speed is fully established.The external load is null in this simulation.

As shown in the Fig.29,rod displacement cannot follow the position demand and continues to move because of the kinetic energy stored in the rotor that is dissipated into heat by frictional effects.The motor speed becomes rapidly saturated because the controller increases the required torque in response to the wrong position.

Jamming fault is another important issue in an EMA in the aspect of security.In the proposed behavioral MPT model,jamming can be simulated by increasing the friction force parameter(Fjm).

Fig.26 Power consumption and loss analysis.

Fig.27 Various backlash responses in the behavioral model.

Fig.28 Wear/aging responses of backlash or preloading.

Fig.30 shows the simulated surface position when jamming is forced by adding a Coulomb friction(50 kN)to four cases in which a jamming fault is triggered in a rising position(at time 0.18 s,black curve),in the overshoot domain(at time 0.24 s,red curve),in a steady position(at time 0.65 s,blue curve),and after the application of an aerodynamic load(at time 1.15 s,pink curve).The rod position is immediately locked as expected.Fig.31 presents the jamming case triggered at the rising position(at time 0.18 s).The huge magnitude of torque transmitted to the nut-screw is 4.5 times the rated torque due to the inertial effect of the motor rotor.The electromagnetic torque remains saturated on its side as position errors cease to evolve.

Fig.29 Faults to failure caused by backlash.

Fig.30 Faults to failure caused by jamming cases.

Fig.31 Torque responses with/without jamming.

6.Conclusions

This study aims to present best practices in system-level modeling and simulation to develop EMA models in order to support virtual prototyping of MEA subsystems with a systemlevel perspective of MBSE design.The proposed approach is based on the use of a Bond graph to establish the structure of the model.In comparison with a block diagram approach,as usual for control design,the consideration on power flows and the use of the Bond graph formalism provides two main advantages.Firstly,it makes the model easier to structure,to link with reality,and to extend or detail versus designing needs,especially when various physical effects with multilevel details are considered.Secondly,it anticipates potential numerical issues(algebraic loops and derivations)by addressing causalities as early as possible.Although emerging simulation codes and commercial software admit non-causal models,our experience has shown that addressing causalities with care makes a model more robust and reduces the time spent to fix numerical issues.Progressive models have been implemented in a causal commercial simulation environment by using available standard libraries as far as possible.The main advantages of the proposed EMA models and their numerical implementations are as follows:

(1)Distinction is clearly made among functional effect,parasitic effects(switching for power electronics,iron and copper losses for electric machines,inertia,friction,and compliance for mechanical transmission),and faults to failure(jamming and free-play).Functional,basic,advanced,and behavioral models have been developed based on the requirements of engineering tasks.These models intend to use object-oriented elements and interfaces from standard model libraries.

(2)Each lumped-parameter model of an EMA component can be balanced,particularly the MPT models,at both energy and mechanical levels.Conduction and switching losses have been progressively introduced into the PDE.Copper and iron losses are considered in the EM,and the MPT models are reproduced by basic/linear and advanced friction models(that consider sensitivity to thetransmitted force,operation temperature,and motion velocity).The advanced compliance model considers component design and service issues to illustrate preloading and backlash effects.

(3)Mechanical faults to failure have been considered for the development of the MPT models.The features of aging/wear and faults to failure(free-play and jamming)are modeled without increasing the complexity and dynamics of the models.These models can facilitate future activities related to the design of health and usage monitoring features of complete EMA actuation systems.

The proposed best practices have proven to provide considerable bene fits for improving EMA virtual prototyping by saving time and re-using models for various engineering activities.These practices are ready to be integrated into realistic virtual MEA.

Acknowledgments

The authors would like to acknowledge the supports of the China Scholarship Council(CSC)and the National Natural Science Foundation of China(No.51275021and No.61327807).

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