Marc Budinger
email: marc.budinger@insa-toulouse.fr

Toufic Aurelien Reysset

El Halabi

Catalin Vasiliu

Jean-Charles Maré

Aurélien Reysset

Toufic El Halabi

Optimal preliminary design of electromechanical actuators

Keywords: design exploration, electromechanical actuator, flight control, inverse problem, inverse simulation, Modelica, Meta-Models, Response Surface Methodology, scaling laws, spoiler

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NOTATION

Efficiency of component (with comp the name of component)

π i
dimensionless parameters

CONTEXT

The current technical developments in aviation aim at making aircraft more competitive, greener and safer. The more electric aircraft (MEA) offers interesting perspectives in terms of performance, maintenance, integration, reconfiguration, ease of operation and management of power [START_REF] Ford | More-electric aircraft[END_REF] [START_REF] Weimer | The role of electric machines and drives in the more electric aircraft[END_REF].Using electricity as the prime source of energy for non-propulsive embedded power systems is considered by aircraft makers as one of the most promising means to achieve the above mentioned goals. Actuation for landing gears and flight controls is particularly concerned as it is one of the main energy consumers. This explains why, during recent years, a great effort has been put into the development of Power-by-Wire (PbW) actuators at research level (e.g. POA, MOET, DRESS and ACTUATION 2015 European projects). This recently enabled PbW actuators to be introduced in the new generation of commercial aircraft [3] [4], in replacement of conventional servo-hydraulic ones (e.g. Boeing B787 EMA brake or spoiler, Airbus A380 EHA and EBHA). On their side, space launchers are following the same trend for thrust vector control (TVC) as illustrated by the European VEGA project [START_REF] Dée | An electrical thrust vector control system with dynamic force feedback[END_REF] and NASA projects [START_REF] Cowan | Design and test of electromechanical actuators for thrust vector control[END_REF]. However, these technological step changes induce new challenges, especially for the preliminary design process for actuation systems and components which cannot simply duplicate former practices.

Section 2 sums up the main requirements, design drivers and design choices for EMA design. This section also describes the different types of models needed. The different possible associations for those models are then presented, and the most efficient and flexible process will be described in details. Section 3 illustrates the different steps, models and tools for the presented methodology with the case study of a spoiler EMA.

MODEL BASED DESIGN FOR ELECTROMECHANICAL ACTUATION SYSTEMS

Figure 1 shows the V-cycle design [START_REF]Design methodology for mechatronic systems[END_REF] of a mechatronic system such as an EMA (architecture is described in part 3.1). The described methodology is dedicated to the descending branch of the V-cycle corresponding to preliminary optimal sizing and device parts features synthesis. The inputs are the objectives and design constraints coming from the specifications document or the chosen architecture. As output components (rod end, roller or ball screw, gear reducer, brushless motor ...) specifications are generated in order to obtain an assembled actuation system which fully meets upper requirements.

Requirements and key design drivers of actuation systems

As contributors to the safety of embedded critical functions, aerospace actuators are subjected to numerous specific design requirements. These requirements are translated into design constraints which the designer must meet by discerning choices. Based on recent examples [START_REF] Dée | An electrical thrust vector control system with dynamic force feedback[END_REF] to [START_REF] Chevalier | More electrical landing gear actuation systems[END_REF]. Table 1 summarizes these design constraints, design degrees of freedom and their possible interactions. One can notice that the designer choices are numerous; their impact coupled and they should address multiple domains. The study and comparison of different possible architectures must thus be supported by efficient design methodologies.

Models needs for electromechanical preliminary design

To accelerate the design process, a general trend is to extend the role of modelling in design and specification [START_REF]Design methodology for mechatronic systems[END_REF] [START_REF]INCOSE, Systems engineering handbook[END_REF]. At preliminary steps of system design, different types of models are needed in order to get a complete sizing: simulation, estimation and evaluation models.

The objective of simulation models is to calculate all the variables of power, energy (force, speed, temperature ...) useful for the components selection. The transient simulation of an EMA is becoming mandatory because, unlike hydraulic actuators, some strong coupling exists between the transient power demand, the thermal behavior of motors, the inertia loads and the fatigue of mechanical components. These simulations are usually based on lumped parameters models [START_REF] Cellier | Continuous System Modeling[END_REF], also called 0D-1D models, run either in direct or inverse mode [START_REF] Murray-Smith | The inverse simulation approach: a focused review of methods and applications[END_REF] [START_REF] Liscouet | Modelling approach for the simulation-based preliminary design of power transmissions[END_REF] in environments such as Matlab/Simulink, Dymola [START_REF] Mattsson | Physical system modeling with Modelica[END_REF], AMESim [START_REF] Auweraer | Virtual engineering at work: the challenges for designing mechatronic products[END_REF]... The paper [START_REF] Liscouet | Modelling approach for the simulation-based preliminary design of power transmissions[END_REF] describes the use of inverse simulation with Modelica for actuators sizing. Yet, to run this kind of models, which only suits for analysis and not for validation, components parameters may be known, that is why some other types are needed.

Estimation models allow to estimate from a reduced set of independent parameters (ex: max load, stroke) all the dependent parameters/component characteristics: simulations (ex: inertia, efficiency…), integration (ex: mass, dimensions) and validation of safe operating area (ex: max speed, degradation parameters…). This is a way to replace components datasheets when they are not available (often the case for aerospace application) or to speed up the design exploration (analysis of different scenarios, optimization…). The reference [START_REF] Budinger | Estimation models for the preliminary design of electromechanical actuators[END_REF] describes some estimation models based on scaling laws for the main components of an electromechanical actuator.

The last model is the evaluation model whose goal is to check the ability of a component to operate in its safe operating area for the required lifetime and reliability. The paper [START_REF] Liscouët | Design for Reliability of Electromechanical Actuators[END_REF] describes the possible models and ways of implementing them in system simulation environments. Figure 2 describes the information exchanges between those different models and defines the structure suitable for model-based design of mechatronic systems. Decision making can be performed either manually as in [START_REF] Liscouet | Modelling approach for the simulation-based preliminary design of power transmissions[END_REF] or automatically as described in the following paragraph.

Models implementation and possible associations

These defined models can be implemented in different types of environment. Decision making can be achieved by different optimization methods. Figure 3 illustrates three different options:

1. All the models are implemented in a computer algebra environment (Matlab, Excel). The design and optimization are done in the same environment. This solution does not address time transient phenomena problematic. However, it has the advantage suit well taking into account multiple design criteria and exhibit fast optimization. Examples of such implementation were presented in [START_REF] Budinger | Optimal preliminary design of electromechanical actuation systems[END_REF] [ [START_REF] Messine | Optimal design of electromechanical actuators: a new method based on global optimization[END_REF] and [START_REF] Roos | Towards a methodology for integrated design of mechatronic servo systems[END_REF] ;

2. All the models are implemented in a simulation environment for algebraic differential equations (e.g. Modelica) and the optimization is done by external MDO software [START_REF] Sobieszczanski-Sobieski | Multidisciplinary aerospace design optimization: survey of recent developments[END_REF]. This solution allows managing fine simulation models with time transients. Yet, one main drawback is the optimization times which can become very long. Examples of such implementations are given in [START_REF] Rottach | Fault-tolerant electrical machine design within a Rotorcraft Actuation Drive System optimisation[END_REF] [29] ;

3. A mix of the first two solutions can be achieved with a compact representation of time-consuming temporal simulations using meta-modeling technics [START_REF] Simpson | Metamodels for Computer-Based Engineering Design: Survey and Recommendations[END_REF]. This solution allows to easily manage multiple design criteria and to obtain reasonable optimization time. However special care should be addressed to the accuracy of meta-models.

This last solution will be developed in the paper. The use of meta-optimization is common for studies with finite elements methods in various fields [START_REF] Keane | Computational Approaches for Aerospace Design: The Pursuit of Excellence[END_REF]. But here, the meta-models applied to 3D multi-body and 0D/1D simulations and the estimation models based on scaling laws are used to speed up the optimization of the actuator.

Proposed methodology for optimal design of EMAs

As defined by [START_REF] Estefan | Survey of Model-Based Systems Engineering (MBSE)[END_REF] [33], a methodology is a collection of related process, methods and tools where:

-A process is a logical sequence of tasks performed to achieve a particular objective ("WHATs"); -A method consists of techniques for performing a task ("HOWs"); -A tool is an instrument that, when applied to a particular method, can enhance the efficiency of the task (supports the "HOWs"). Table 2 presents those three aspects. The main functional requirements and design degrees of freedom are given as inputs for this process. The actuator architecture and combination of power transmission components are also defined to meet the application functional requirements. -Verification matrix 2 Determination of the active design drivers 2.1 Perform initial power sizing (not optimized) using a "sizing wave" procedure and complementary simulations.

In-house Modelica library : -Inverse simulation (Modelica).

-Estimation models (scaling (section 0) laws) .

3 Sizing procedure setting-up and optimization problem definition (section 0)

3.1 Determining the calculation steps order for the sizing procedure.

-N² diagram.

3.2 Define the optimization problem.

-Influence diagram.

4 Creation of meta- models (section 3.5)

4.1 Non explicit calculations from step 3 may be replaced by meta-models.

-RSM models.

5 Design optimization (section 0)

5.1 Identify important parameters with quick exploration and reduce design space.

-Excel table.

5.2 Optimize in the reduced space.

-Excel solver 6 Validate « optimal » solution (section 3.7)

6.1 Repeat sizing performed during step 2 to the result obtained in step 5.

In-house Modelica library : -Backward simulation (Modelica). -Estimation models (scaling laws).

SPOILER ACTUATOR CASE STUDY

Case study and actuator architecture presentation

It is proposed in this section to illustrate the design methodology using the case of a linear EMA. This actuator is actuating a spoiler/airbrake with typical specifications for a business jet. The so called spoiler is only performing airbrake functions. Thereby possible primary flight control functions of commercial aircrafts are not taken here into consideration: control surfaces are thus controlled symmetrically with a given displacement curve.

Main functional requirement

The main functional requirement for the actuator is to move the mechanical load to 3 predetermined positions (0. 20 and 50 °) against mainly the aerodynamic forces. The retracted position (0°) has to be hold without energy consumption. Anchorage points on the structure and the control surface are not imposed by the aircraft maker and therefore, can be a degree of freedom for the design process.

The severe failure rate requirements at aircraft level are achieved with resort to multiple control surfaces. Thus the requested failure rate of one actuator is compatible with a simplex architecture. However minimizing mass is an objective. The architecture should also be safe regarding different critical cases :

-no dissymmetric deflection during a power loss : the actuator should thus fail-freeze; -no degradation of the surface control during gust : in order to avoid mechanical damage, a load limitation must be performed by folding down the control surface; -no degradation of the actuator or the surface control due to a jamming of the control surface during the actuator motion.

Actuator architecture

The physical architecture of the actuator is based on the Moog linear actuator presented in [START_REF] Davis | High performance electromechanical servoactuation using brushless DC motors[END_REF]. This architecture should be also close to the EMA actuator actuating several spoilers of the commercial aircraft Boeing 787 [START_REF] Estefan | Survey of Model-Based Systems Engineering (MBSE)[END_REF]. Figure 4 presents a picture of the Boeing 787 actuator prototype and describes an association of component meeting the requirements of section 3.1.1 where:

-The selected technologies, brushless motor and screw, meet the main function with a minimum mass compared to a direct drive solution. The main drawback of both screw and reducer is the jamming failure mode. This failure mode is however not an issue for the fail-freeze specification. Even if the 787 spoiler actuator uses a ball screw, a roller screw will be used instead during the sizing and optimization of this specific application.

-A one stage spur-gear reducer with intermediary wheel allows at the same time to adapt the brushless motor speed and the parallel axes distance (distance between motor and roller screw axes).

-A power-off brake can hold the spoiler without consuming any energy (normal mode or power failure). To meet the functions described in the previous paragraph, a control architecture with a force feedback sensor can be used. Figure 5 gives the principle of this last one. Main loops for typical control architecture are: current, speed and position loops. Load control can also be implemented like in the examples of aerospace actuators described in [START_REF] Dée | An electrical thrust vector control system with dynamic force feedback[END_REF] and [START_REF] Schaefer | Flight control actuation system for the B-2 advanced technology bomber[END_REF] to limit external perturbations (high frequency vibration, gust load…). In this case the load control is realized through a force feedback to torque control of the motor. The activation of the load control also leads to decrease of the authority of the main position loop to allow the drop of the surface in presence of excessive air loads.

Definition of design drivers and induced design drivers

The objective of task 1 is to represent by a set of scenarios the degradation phenomena and functions of the actuator. Those scenarios form the basis for the calculations to be carried out during the design.

The degradations of the components can be of two different types, depending on the phenomena dynamics:

-What is called rapid degradation (e.g. permanent deformation, fracture, maximum insulator temperature…) on transient power demand. They must be avoided to ensure maximum performance of the actuator.

-What is called gradual degradation (e.g. mechanical fatigue, insulator ageing…). They must be such as to ensure endurance and reliable operation of the actuator throughout its life.

Additionally, the components present some imperfections which increase stresses on themselves (e.g. inertia) or on other components (e.g. friction, copper losses…) and can create new critical cases (e.g. high stress induced by jamming of inertial loads). Those imperfections are called here induced design drivers.

Design drivers determination

The determination of the components design drivers can be assisted by checklists for the different technologies or components.

Table 3 is a possible representation of those lists. Some references are quoted to allow further study or justify the importance of listed items. The paper [START_REF] Budinger | Estimation models for the preliminary design of electromechanical actuators[END_REF] also recalls those points.

Sizing scenarios and corresponding mission profiles

They should permit to validate the fact that the solution meets all the functions and associated performance of section 3.1.1 and is not submitted to inappropriate damages listed in section 3.2.2. In the case of EMAs, those scenarios are mainly characterized by finite time dependent mission profiles.

For the present case study, the following examples of scenarios can be defined: e) To take into account the vibratory environment: a lateral sinusoidal acceleration of 10g between 5 and 2000 Hz [START_REF]Environmental Conditions and Test Procedures For Airborne Equipment, standard for environmental test of avionics hardware[END_REF] (validation of screw and housing resonant frequencies and induced stresses).

Scenario a) to d) can be described by transient simulations of 0D-1D models. Figure 6 describes spoiler angle time evolution for the a) mechanical mission profile. The paper [START_REF]Moog -Navy league:Sea, Air and Space. Special Edition[END_REF] explains how to take into account scenario e) during the preliminary design phase using analytical models.

MECHANICAL COMPONENTS

Due to force transmission:

• Maximum stress (rapid)

• Fatigue/wear (gradual) [START_REF] Liscouet | Modelling approach for the simulation-based preliminary design of power transmissions[END_REF] Due to vibratory environment:

• Stress at resonance frequency (gradual) [START_REF] Grand | Electromechanical actuators design for thrust vector control[END_REF] Induced design drivers:

• Mechanical losses and friction which increase force/torque especially at low speed [START_REF] Maré | Friction Modelling and simulation at system level -A practical view for the designer[END_REF] Rod end • Insulating ageing [START_REF] Schinstock | Controller Design for EMA in TVC Incorporating Force Feedback[END_REF] (gradual)

Induced design drivers:

• Rotor inertia which increases electromagnetic torque for high dynamic application [START_REF] Schinstock | Controller Design for EMA in TVC Incorporating Force Feedback[END_REF] or when jamming or brutal stop [START_REF] Atallah | A Brushless Permanent Magnet Motor With Integrated Torque-Limiter[END_REF] • Rotor inertia : bandwidth effect [

Task 2 : Initial sizing and active design drivers

In the considered case study, the initial sizing aims at the identification of the active design criteria and helps simplifying the models used for the selection of the actuator components. This first design is made for a given set of anchorage and transmission points that will be modified during the optimization phase coming later on. But, even for this first design, different types of simulation models are used.

Sizing wave

An in-house Modelica library for the preliminary design of EMAs, shown in Figure 7, has been developed to implement all the components models presented in Section 3.1.2 according to the structure of Figure 2. Object orientated and class inheritance properties of Modelica language facilitate the implementation of such a structure. The models are thus structured in a way that they can combine inverse simulation together with parameters estimation (using scaling laws). Components from the created Modelica library can be combined to represent various EMAs' architectures. The studied architecture has been sized up with a few number of simulations by following the "sizing wave" process described in paper [START_REF] Liscouet | Modelling approach for the simulation-based preliminary design of power transmissions[END_REF] where inverse simulation means to size successively each component propagating power from the load to the power source. This step is usually called power sizing and components selection, with respect to previously defined scenarios a) and b). From this initial power sizing, it is interesting to note that:

-The motor inertia does not impact the maximum electromagnetic torque because the load is mainly aerodynamic and can be considered as a nonlinear stiffness depending on airspeed and surface steering;

-The friction models of mechanical components have to be taken into account for the motor sizing;

-Thermal aspect is really important for motor selection;

-The requested lifetime and reliability are not critical for this case study and can be neglected.

Critical cases

For selected components, the needed parameters to simulate scenarios c) and d) can be calculated using the scaling laws. The type of simulation used for c) and d) scenarios is direct simulation. Components can be oversized to meet the requirements issued from all those scenarios.

For the considered case study:

-According to the reduction ratio, the mechanical components may be oversized in order to bear the stress induced by the jamming of the spoiler running at full speed.

-The maximum motor torque must allow a retraction fast enough to limit extreme load occurring during a gust.

For scenario e), the length of the actuator is relatively short that is why the vibratory environment has no significant impact on the housing induced stress. Therefore, the main design driver will be the minimal machining thickness constraint.

Initial sizing overview

Table 5 gives an overview of the masses, estimated thanks to scaling laws, for the various components of the EMA and the related criteria leading to their selection, criteria which should be considered during optimization (task 3).

The process leading to those results is the following:

-Choose first the anchorage points on structure farthest of the hinge axis to facilitate actuator integration;

-Select average values for all other parameters (defined in part 3.4.3).

-If motor thermal constraint is not validated, increase motor size (i.e. thermal over-sizing coefficient).

-If jamming induced stress constraint is not validated, increase mechanical safety coefficient. The objective of this step is to define the sizing problem as an optimization problem which can be formulated mathematically as follows:

Component Active design drivers Mass

Minimize objective function f(x)

Subject to equality and inequality constraints h(x) = 0, g(x) < 0

By action on the parameters vector in the range x low <x<x upp

Where:

-The goal, here the global mass, is the objective function f;

-Design alternatives are expressed by a set of values assigned to the design variables x within a design domain;

-Constraints (h & g) limit the number of alternatives to those satisfying physical principles and design specifications, that is to say feasible design.

The functions (f, g, h) can be explicit or implicit, algebraic or realized by subroutines that solve iteratively systems of differential equations. The goal of this task is to define those functions within a sizing procedure. The calculation steps of the sizing procedure should be time efficient, and to do so, only algebraic explicit functions without iteration loops are to be implemented. The number of design parameters x and constraints g and h should also be minimized to reduce design search space and thus, optimization time.

Sizing procedure definition

The sizing procedure is the sequence of calculation steps to be carried out for defining the actuator components.

As mentioned before, step 2 allows to determine the minimum number of design criteria to consider in order to have a correct but lighter optimization problem. This procedure can be represented as an algorithm or flowchart. The Design-Structure-Matrix (DSM) [START_REF] Browning | Applying the design structure matrix to system decomposition and integration problems: a review and new directions[END_REF] or N² diagrams [START_REF]Techniques of Functional Analysis[END_REF] can also represent this sequence. All those representations are useful for the assessment of the models needed and exchanged quantities and for the visualization of the sequencing quality. Figure 8 recalls the principle of representation by N² diagrams. The objective is here to avoid any information feedback/loop thanks to a good choice of calculations order or by introducing additional constraints and design parameters being managed by the optimization algorithm. In the case of DSM matrix representation, the objective is to obtain a triangular matrix. The optimization problem can be defined only when the sizing procedure is known. To represent it, an influence diagram [START_REF] Malak | Compositional modelling of fluid power systems using predictive tradeoff models[END_REF], Figure 10, can be used. Here:

-The main design parameters are internal reduction ratios (roller screw pitch p and reducer ratio N), integration parameters (anchorage to airframe x a , y a and transmission to load x t , y t points), shape of the displacement curve (α, β displacement curve parameters as described in Figure 6), oversizing coefficient (jamming coefficient k jam , brushless motor thermal coefficient k therm and screw vibration coefficient k vib );

-The main objective is the total mass obtained by the sum of the masses of all the components estimated thanks to scaling laws;

-The constraints are used to represent some design drivers which cannot be addressed in a direct way by the sizing procedure. Oversizing coefficients enable to take care of these constraints during optimization. To facilitate and accelerate the design exploration during optimization, it is more suitable to use models with only static parameters and without time dependent variables. The meta-modeling techniques [START_REF] Simpson | Metamodels for Computer-Based Engineering Design: Survey and Recommendations[END_REF] enable to extract a response surface from a set of simulation results. The calculated meta-models have here a polynomial form to facilitate their easy implementation into a calculation sheets (e.g. in the MS Excels environment) (Task 5, section 3.6). The generation of a meta-model follows the process shown Figure 11 and is performed here with the Optimus software [START_REF] Poles | Multiobjective Optimization Software[END_REF]. The meta-models representing multi-body simulation link the effect of actuator geometric integration (anchorage x a , y a and transmission x t , y t points parameters, Figure 12) and the displacement curve form (α, β displacement curve parameters) to useful sizing criteria as stroke, maximum speed, maximum load, RMS and RMC load. The identification of the main factors is done through screening. This screening step allows to conclude that for this application case, parameters α and β have only an influence on the maximum speed.

The meta-model building is realized here with a Latin Hypercube DOE of 1000 experiments. It is important to note that the model quality depends on the mathematical function used to represent the behavior [START_REF] Montgomery | Applied Statistics and Probability for Engineers[END_REF] : for example assuming 1/F max has a polynomial form better suits than expressing directly F max as a polynomial function. With a second order development, the meta-model provides a regression coefficient R better than 0.999 for all calculated values. The meta-model described here will be used to get the maximal temperature of the motor winding during scenario b) (Cf section 3.2.3). For this example, the main factors' identification is done by dimensional analysis [START_REF] Vignaux | Theory & Methods: Simplifying Regression Models Using Dimensional Analysis[END_REF] [48] which will reduce the number of parameters to be studied without any loss of information. The thermal model of a motor given in Figure 13, distinguish the temperature of the winding from the temperature of the yoke by using a two thermal bodies model.

The relationship to define is characterized by 6 parameters:

-The heat capacity of the winding C th1 ;

-The heat capacity of the yoke C th2 ;

-The thermal resistance between the winding and the yoke R th1 ;

-The thermal resistance between the yoke and the ambient air R th2 ;

-The thermal losses P th ;

-The temperature increase θ of the winding after a given time t.

Buckingham theorem [START_REF] Vignaux | Theory & Methods: Simplifying Regression Models Using Dimensional Analysis[END_REF] [48] can be used to reduce this number to 4 dimensionless parameters:

) , , ( With a Latin Hypercube DOE of 100 experiments, a third order development provides a coefficient of regression R better than 0.99 for this function. The desired function could also be derived from the direct solution of differential equations of the lumped parameters model shown Figure 13. The aim here is to show that the designer can however get an algebraic expression from any numerical models. The meta-models thus allow him to focus more on the sequencing of calculations than on the solving of each calculation. The optimization process has been implemented using spreadsheet in the MS Excel environment taking benefit of the work done at previous steps: structuring the problem (task 3) and synthetizing meta-models (task 4). The design procedure is thus explicit and easily adaptable to any solvers or exploration functions (e.g. data tables, scenarios). Table 6 summarizes the optimal results obtained after 20 minutes of computation on a standard PC (Intel core i5 processor) with nonlinear GRG solver and Multi-Start option. The optimization has been carried out for different integer values screw pitches in millimeters to take into account manufacturing constraints and to evaluate the effect of this important parameter. Figure 14 gives the mass distribution for the optimal solution obtained using a 5mm/rev screw pitch. It is important to remark that the proposed process generates a gain of 33% on the mass in comparison with the initial sizing of section 0. The initial sizing has been done choosing an anchorage point on the structure which facilitates the geometrical integration and taking average values for lever arm. At this point, a major part of the actuator mass was devoted to the brushless motor (almost 50%) due to a low global reduction ratio.

Parameters

The parameters obtained for the optimal solutions highlight that:

-As inertial torque due to the motor inertia has limited impact on the maximal electromagnetic torque, the S curve parameter, β (constant speed ratio), has been maximized in order to minimize motor maximal speed.

The conclusion can change if accelerations are higher and/or load features are different as for TVC actuators [START_REF] Grand | Electromechanical actuators design for thrust vector control[END_REF].

-The lever arm has been maximized to minimize load and component masses. The conclusion can be different if the actuator is longer, as for a landing gear extension/retraction actuator, where the housing mass, representing the major part of actuator mass, increases significantly with its length in order to resist to transverse vibratory stress [START_REF]Moog -Navy league:Sea, Air and Space. Special Edition[END_REF].

-The global reduction ratio is limited here by the jamming requirement. This ratio can be different if the expected lifetime is greater (as for an aileron actuator) or for a different type of aircraft.

The different pitch values can be compared through design explorations around the optimal points. The graphs given in Figure 15 illustrate the evolution of the overall mass and main design constraints according to the two most influent parameters, i.e. the location the transmission point (parameters x t and y t) . The other design parameters are set to the optimal values of Table 6. Each constraint is represented with dotted lines and defines a feasible solution search area with upper and lower bounds. For a 2 mm pitch, Figure 15a), the feasible design field is clearly limited by all these constraints. Therefore, the designer has a poor design flexibility. While for a 5 mm pitch, Figure 15b) the feasible search space is much bigger. The designer can thus choose a slightly different solution without adding significant weight to the actuator, while increasing robustness in regards to the assembly and jamming constraints. For validation, it is particularly important to check that for the optimal parameters:

• The response surface approximation, established during task 4 (section 3.5), is sufficiently accurate. Otherwise, it is necessary to construct the response surface again, refining mathematical formula around the point of interest; • The simplification assumptions established during task 2 (section 0), remain effective. If a design driver defined as inactive during the initial design becomes dominant for the optimal configuration, this new constraint must be integrated and the optimization launched again.

In the case study considered here, time domain simulations validate the results obtained within spreadsheets. It may also be interesting to validate the estimation models results (extrapolation of selected industrial components features) comparing them to real industrial components extracted from manufacturer datasheets. Table 7 compares optimal results for a 5mm/rev screw pitch using scaling (column 1) to real industrial components (column 2). The mass deviation gives an idea of the mass penalty which may be due to components standardization and discretization of a product range. Yet, the scaling laws, in the case of a specific design as typically encountered till now in aerospace domain, allow specifying or estimating new components features. Solution using standard components does not differ a lot from obtained "optimal" configuration, with a difference lower than 20% on the overall mass. This difference is mainly due to weak manufacturer product choices when considering thrust bearing and roller-screw components.

Component definition

It may be noted that the overall methodology is well suited to collaborative work and development of multidomain systems:

-The rationalization of the sizing scenarios determination in step 1 is well suited to group meetings;

-The distribution of work is intrinsic to the methodology. The system engineer can define the sizing procedure (step 2) and achieve the final optimization in excel (step 5). Experts of different technical areas (kinematics and mechanical power transmission, thermal of electrical motors. ...) can establish meta-models;

-Thanks to distinction between simulation models and estimation models (scaling laws), the management of supplier's data are facilitated. The updating of scaling laws references enables to take into account technical developments or to compare some technologies;

-Meta-modeling takes into account multiple design criteria within multiple domains;

-Verification / validation are done through steps 1 and 6.

CONCLUSION

A technology shift towards more electric solutions is emerging in aerospace actuation. New technology brings new challenges, especially for the preliminary design process of actuation systems and components that cannot simply duplicate former practices. In this way, it is no longer meaningful to proceed to static sizing by typical operating points as was possible for hydraulic actuators that do not induce strong couplings in design. Instead, it is necessary to take into account transient mission profiles covering all possible sizing drivers (e.g. maximum speed and torque, fatigue, thermal behaviour, vibrations and shocks) and addressing various design criteria (e.g. geometrical envelope, weight and reliability). For this purpose, appropriate models and methodologies should be developed to support an efficient optimal preliminary design. In this paper, a structure for component model has been presented in order to transform the system simulation tools into design tools. This model structure is based on three different types of models: estimation, simulation and evaluation models. This paper also mainly describes a six step design methodology to define the sizing scenarios to consider and gradually transform the corresponding transient simulations into design models combining judiciously scaling laws and response surface approximation into selection procedures established using N2 diagram. This decomposition facilitates the optimization model, the exploration of the design space, taking into account multiple design constraints. This methodology fits well in a collaborative engineering work for the decomposition of tasks and studies; verification / validation process, the rapid updating of technological references; and the optimization and knowledge management process. This methodology has been illustrated with a study case of an EMA dedicated to spoiler actuation which is realistic and representative of the number of functions, the multiple points of views for design and the different technological domains to take into account in order to achieve preliminary design of an EMA. This case study has illustrated the interest of the methodology and has enabled to describe all the tasks. This methodology is applicable to other configurations (TVC, landing gear actuator…), other technologies (hydraulic, power electronics, control) and even other technical systems.

Figure 1 :

 1 Figure 1: V-cycle design for a mechatronic system

Figure 2 :

 2 Figure 2: Component models structure for model-based preliminary design

Figure 3 :

 3 Figure 3: Models implementations and associations

Figure 4 :

 4 Figure 4: Physical architecture

Figure 5 :

 5 Figure 5: Control architecture 3.2. Task 1 : Sizing scenarios definition

  a) To take into account the mechanical stresses a mission profile representative of the maximum speed. For maximum load and fatigue phenomena: a succession of extension / retraction for different speeds of the aircraft; b) To represent the thermal stresses: holding of the control surface opened at 50 ° and submitted to full load for 3 minutes within an ambient temperature of 30 ° C; c) To take into account the effect of the motor inertia: a jamming of the control surface occurring at full speed ; d) To ensure adequate dynamics during load limitation: an active retraction of the spoiler under the effect of a gust ;

  Maximum stress (rapid) Roller Screw -Maximum stress (rapid) -Fatigue stress (gradual) -Vibratory stress (gradual) -Mechanical losses and friction (induced) Thrust bearing -Maximum stress (rapid) -Fatigue stress (gradual) -Mechanical losses and friction (induced) Pinions -Maximum stress (rapid and gradual) -Mechanical losses and friction (

Figure 6

 6 Figure 6: S Curve displacement for mechanical profileTable 4: Verification matrix Checking points Scenarios Type Description

  a) Library structure b) Components model from the library c) Spoiler actuator model

Figure 7 :

 7 Figure 7: In-house Modelica library for the preliminary design of EMAs

•

  Maximal torque due to magnetic saturation at teeth (rapid) • Mean torque and maximal temperature due to thermal stress (gradual) • Maximal speed (rapid)

Figure 8 :

 8 Figure 8: N² diagram principle

Figure 9 :

 9 Figure 9: N² diagram of the sizing procedure

Figure 10 : 3 . 5 .

 1035 Figure 10: Influence diagram of the optimization problem

Figure 11 :

 11 Figure 11: Meta model synthesis process

Figure 12 :

 12 Figure 12: Anchorage and transmission points

Figure 13 :

 13 Figure 13: Thermal model of the motor

Figure 14 :

 14 Figure 14: Mass distribution of an optimal sizing (pitch of 5 mm/rev)

a) 2

 2 mm/rev pitch b) 5 mm/rev pitch

Figure 15 :and constraints exploration 3 . 7 .

 1537 Figure 15: Mass and constraints exploration

Table 1 : Main design drivers and designer choices during EMA design

 1

		Designer choices and/or degree of freedom
	Architecture	Technologies	Component	specifications	Displacement curve	Control parameters	Kinematic and	transmission ratios	Design models	and tools	Reliability	allocation

Table 2 : Synthesis of the proposed methodology Process (« WHATs ») Methods (« HOWs ») Tools (supports the « HOWs »)

 2

	1 Sizing scenarios	1.1 Determine components design drivers.	Checklist by fields
	definition (section		(mechanical. Electrical,
	0)

thermal ...) or components. 1.2 Determine sizing scenarios to verify the design drivers and the actuator functions.

  Table 4 is a verification and validation matrix to check that all design features or drivers are covered by those five scenarios.

Table 3 : Main design drivers of EMA components TECHNOLOGY Design drivers Component Key design drivers selected for the application

 3

Table 6 : Optimum configurations

 6

		Range	Initial sizing	Optimal sizing (discrete screw pitch)
	Pitch p [mm]	[2;5]	5	2	3	4	5
	Reducer ratio N [-]	[1;5]	3.00	2.44	4.40	5.00	4.99
	Transmission x t [mm]	[-50;0]	-25.0	-33.4	-33.7	-37.0	-50.0
	Transmission x t [mm]	[-80;-30]	-55.0	-80.0	-80.0	-80.0	-80.0
	Anchorage x a .[mm]	[-400;-300]	-400.0	-400.0	-400.0	-396.3	-400.0
	Anchorage x a [mm]	[-150;50]	-150.0	-149.8	-149.7	-148.0	-150.0
	S curve coefficient β [-]	[0.05;04]	0.25	0.4	0.4	0.4	0.4
	Jamming oversizing k jam [-]	[1;4]	1.02	1.47	1.48	1.43	1.44
	Thermal oversizing k ther [-]	[1;4]	1.10	1.18	1.21	1.20	1.19
	Vibration oversizing k vib [-]	[1;4]	1	1	1	1	1
	Total mass [kg]		8.94	6.42	6.01	5.99	5.92

Table 7 : Scaling laws and on the shell components comparison

 7

ACKNOWLEDGMENTS

The work presented was partly funded by the European ACTUATION 2015 project (Modular Electro Mechanical Actuators for ACARE 2020 aircraft and helicopters).


Aurelien

Aurélien Reysset

Marc Budinger

Jean-Charles Maré

Computer aided definition of sizing procedures and optimization problems of mechatronic systems

Keywords: Mechatronics, system design, optimized sizing procedure, graph theory, matching-ordering algorithm

Computer-aided definition of sizing procedures and optimization problems of mechatronic systems

From models to optimal preliminary design of mechatronics systems

Preliminary design of mechatronic systems A mechatronic system [START_REF] Auslander | What is mechatronics?[END_REF] expands the capabilities of conventional mechanical systems through the integration of different technological areas around a power transmission part and an information processing part. The references [START_REF] Van Der Auweraer | Virtual engineering at work: the challenges for designing mechatronic products[END_REF] [START_REF] Hehenberger | Hierarchical design models in the mechatronic product development process of synchronous machines[END_REF] highlight the fact that the design of such multi-domain systems requires different modelling layers as represented in Figure 1: 1. A mechatronic system layer, to take into account the functional and physical coupling between components. This level of modelling is usually done using 0D-1D models also called lumped parameter models represented by algebraic equations, ordinary differential equations (ODE) or differential algebraic equations (DAE) [START_REF] Cellier | Continuous System Modeling[END_REF]. During preliminary design, these models represent sizing scenarios. Metamodelling techniques often enable to represent those models by algebraic equations for a quicker optimization [START_REF] Budinger | Optimal preliminary design of electromechanical actuators[END_REF].

2. A specific domain/component layer, to describe the performance limits and parameters necessary in the previous layer, based on a geometric representation. The specific domain phenomena are generally represented through partial differential equations (PDE). This level of modelling can be achieved, for simplified geometries by using analytical models or, for complex 2D and 3D geometries, by using numerical approximations like the finite element method (FEM) for instance. During preliminary design and optimization, this layer can be represented by analytical models, meta-models [START_REF] Budinger | Scaling-law-based metamodels for the sizing of mechatronic systems[END_REF] or scaling laws [START_REF] Budinger | Estimation models for the preliminary design of electromechanical actuators[END_REF].

Sizing procedure vs. optimization

Generally the engineering models enable to determine the opposite relationship to what is desired during sizing, i.e. calculate the performance of the system (which is required by the specification) from the characteristics of components (which sizing activities want to define). All representative models have to be manipulated in order to solve this inverse problem. There are essentially two types of solutions:

1. Selection or sizing procedures, judicious succession of calculations and tests which determine the main characteristics of components to select.

2. Optimization algorithms that minimize the difference between the desired and achieved performances by changing the main characteristics of components.

In this paper, the goal is to make joint use of these two approaches, sizing procedures and optimization, combined to mechatronic systems and components layers representation of knowledge. The next section will present a state-of-the-art overview of the different ways of representing a sizing procedure. We will focus on their generation by using simple graphical means or by using algorithms from graph theory. The different approaches and issues will finally be illustrated with simple but typical examples.

State of the art on Sizing procedure representations

The sizing procedure can be considered as a major part of the whole preliminary sizing process. When the problem is complex, the methods that represent all the interactions between components from various fields (mechanical, electrical, etc.), become valuable. This section focuses on representation methods.

Causal diagram

The causal diagram is a graphical tool representing the causal relationships between variables. It is in fact an oriented graph representing a calculation scheme. Examples of causal diagrams can be found in [8] or [START_REF] Liscouet-Hanke | A Methodology for Systems Integration in Aircraft Conceptual Design -Estimation of Required Space[END_REF]. With a causal diagram, the couplings between the different entities may be difficult to visualize. An important problem is that of algebraic loops, which is named circular reference errors in spreadsheets, and is characterized by relations such as y=f(x) and x=g(y).

N² diagrams

The N² diagram formalism was introduced in the 70s by the system engineer Robert J. Lano [START_REF]Techniques of Functional Analysis[END_REF]. N gives the number of entities considered. This diagram is a causal diagram where the graphical entities (blocks) and interactions (arrows) are dictated by an organization convention as shown in Figure 2a. The inputs of the blocks are represented by vertical arrows while the outputs must be horizontal arrows. This display provides a quick visual summary of the organization of the system studied. In particular, the couplings and algebraic loops between blocks are easily identifiable: for example, Figure 2 highlights coupling between block "2" and block "3".

DSM matrix

A Design Structure Matrix or Dependency Structure Matrix (DSM) is the adjacency matrix of a N² diagram (an example is given in Figure 2b). The DSM, as its name says is a matrix representing links or information exchanges which can be used at various levels: working group, system level [START_REF] Shekar | Managing complexity in aircraft design using design structure matrix[END_REF], or component level focussing on modularity analysis [START_REF] Qiang | A product module identification approach based on axiomatic design and design structure matrix[END_REF]. In this representation, a perfect straightforward sequence is represented by a lower triangular matrix, while the upper non-empty box represents a feedback design loop.

Graphs

Finally, a sizing procedure is quite similar to a constraint network [START_REF] Friedman | Constraint theory, part I: fundaments[END_REF] [14] [START_REF] Gelle | Solving mixed and conditional constraint satisfication problems[END_REF], since sizing parameters represent variables, equalities and inequalities are viewed as constraints. Therefore, all the tools applied to the representation of constraints network and therefore graph theory can be applied. That is why various algorithms applied to graph theory can be integrated to a global calculation ordering method. In the following sub-parts, different graphical representations will be used. Some of them represent how equations will match parameters; in this case both equations (square node) and non-determined parameters (circular node) are represented in a non-oriented graph. Let's take the non-oriented, declarative equations:

  0 2 2 3 2 1 1     x x x x f (1)   0 1 3 3 2 2    x x x f (2)
A matching is generally represented in what is called a bi-partite graph, Figure 3a-b, with on one side all the equations nodes and on the other side all the non-determined parameters nodes. Once equations are oriented, i.e. one parameter is matched to the equation or calculated by the equation, the matching parameter/equation can be represented by a double-column, Figure 3b, or the information flow can be represented by oriented arrows, Figure 3c. The set of equations can also be represented by a dependency matrix (Figure 3).

Sizing procedure definition

Objectives

This section deals with the representation of a set of design calculations and the scheduling of information flows into a sizing procedure. In order to have a well-posed sizing procedure, easy to use in any computing environment, it must be explicit without internal numerical solver. For a complex problem gathering numerous equations and constraints, it might be not obvious to find a proper explicit sequence (no singularity and no loop), which is why the problem must be adapted.

Next sections will present: the typical inputs and outputs of such procedure which will be used in an optimization problem, the proposed methodology and process, how to set up manually a sizing procedure using graphical tools, a computer implementation using algorithms from graph theory, and finally how to identify the main typical problems that can occur with illustrations on elementary examples.

Inputs and outputs of an optimization problem

An optimization problem can be formulated mathematically as follows:

Minimize the objective function: f(x)

Subject to equality/inequality constraints: h(x)=0, g(x)<0

Acting on the parameters vector in the range: x low <x<x up Where:

1. The goal is the objective function f; 2. Design alternatives are expressed by a set of values assigned to the design variables x within a design domain;

3. Constraints (h & g) limit the number of alternatives to those satisfying physical principles and design specifications.

The functions (f, g, h) can be explicit or implicit, algebraic or achieved by subroutines that solve systems of differential equations iteratively. The goal here is to define those functions within a sizing procedure.

This sizing procedure should be preferably explicit and quick to evaluate.

Ordering methodology and process

The equations ordering methodology is organized into 3 main steps summed-up in Table 1.

This global method helps to reveal additional design parameters and to structure the sizing procedure which can be used by an optimization algorithm. This process of block lower triangulation may be applied to any type of knowledge represented by models.

However, in the case of a mechatronic system, it is possible to distinguish two levels of knowledge:

1. A component layer that represents the specific domains knowledge for each component. This knowledge is represented by parameters (definition, integration, simulation, evaluation) and equations (estimation models) linking the parameters of only one component. Each set of equations and parameters is component-independent and can thus be stored and reused in the form of a library.

2. A mechatronic system layer that represents the system-level interactions between components and specifications declined on all sizing scenarios. These models can link characteristics of multiple components from component layer freely.

"DesignGraph": sizing procedure synthesis thanks to graphical manipulations

For design problems characterized by a reduced set of equations, a skilled designer will be able to order quickly and intuitively design constraints of his area of knowledge. However, it is also possible to graphically represent and order these equations. This graphical representation, here called "Design Graph", can be used as a teaching tool with students or as an analysis tool for an engineer in the case of new design scenarios or new technology.

The problem definition, Task 1 of Table 1, is achieved with a bipartite graph for each design layer as represented in Figure 4a where:

1. The parameters are set at mechatronic system layer from requirements specifications (performances, constraints) and at components layer by listing the components of the architecture and their main characteristics.

2. The equations are then introduced to link the different parameters: at the mechatronic system layer level with equations representing design scenarios, at the components layer level with estimation models.

At this point the relationships between the parameters and equations are represented as non-oriented edges.

The orientation of the equations is the next step and corresponds to Task 2 from Table 1. The parameters with value imposed by the specifications are selected as inputs for the sizing procedure and can be removed from the matching process. Some parameters in the specification may take the form of outputs if the wording of the requirement defines them as an objective to minimize or maximize or as a maximum or minimum limit constraint to respect. Figure 4b summarizes the graphical notation that can be chosen to represent these parameters in a form approaching an influence diagram [START_REF] Malak | Compositional modelling of fluid power systems using predictive tradeoff models[END_REF] in order to prepare the implementation of the optimization problem.

As an equation can generate only one output, the edges have to be oriented in order to have only one output per calculation node (Figure 4c). If the same equation remains undetermined for several parameters (Figure 4d), some design assumptions have to be done by stating that some parameters are known. These new design parameters can be included in the optimization problem and their optimal value can be determined via the optimization algorithm. A preferred choice for these new design parameters is that of parameters participating in a large number of equations (such as gear reduction ratio) and values defined into a known range [min; max]. These assumptions allow to balance the number of equations available and the number of parameters to be determined by the calculation procedure. The first example of the last section illustrates this case of under-constrained problem.

If several equations of design scenarios have the same output, Figure 4d, the problem may be over-constrained. Two solutions can be applied: by adding a safety factor and managing one of the equations as a constraint in the optimization problem or by giving the assignment to the most restrictive equation when direct comparison is possible. The second example of the last section illustrates this case of over-constrained problem.

The treatment of possible algebraic loop is the last step and corresponds to Task 3 in Table 1. An algebraic loop can be typically found in the selection of a component involving the use of characteristics of this same component. This case is represented graphically by a loop involving a design scenario equation and an estimation model equation as shown in Figure 4d. Adding once again a new design parameter, such as an oversizing coefficient, and a constraint to be checked by the optimization algorithm makes the sizing procedure explicit. The last example of the last section illustrates this case.

At the end of this graphical treatment, the final sequence of equations can be represented as an upper triangular N² diagram or as a lower triangular matrix (Figure 2) in order to check that the sequence of equations is explicit.

"GraphSize": sizing procedure synthesis through algorithmic manipulations

In the case of the design of a multi-domain product characterized by a large set of equations, it may be difficult for an engineer to write and order manually the equations properly in order to establish an explicit and straightforward design procedure. A computer tool, that can help him do so, then becomes valuable.

Here, an in-house Matlab implementation of such software, that we have called "GraphSize", is presented.

For the problem definition, Task 1 in Table 1, "GraphSize" represents the equations and parameters in both components and mechatronic system layers, defined in the previous two paragraphs. This decomposition facilitates the reuse of models through component libraries (Component layer) or sizing scenario libraries (Mechatronic system layer). The relations, called "rules", may take the form of equalities or inequalities.

The tool automatically lists the parameters from the equations. These rules can be declared in an equation section, and will then be analysed to be ordered, or in a constraint section, and will then be used as constraints only by the optimization algorithm remaining out of the ordering process.

The problem is thus defined by a given number of non-oriented equalities/inequalities. Most of the time, optimal values are obtained for a solution around the constraint. For example, for a motor where the design driver is the coil maximum temperature T max , the designer may check that T coil ≤T max , leading to an oversized solution. But if in addition to that inequality, the motor mass has to be minimized, the final result will tend to an equality Tcoil=Tmax. Therefore, at first, inequalities in the equations section will be considered as equalities and only if the problem is over-constrained, some adaptation will be made adding over-sizing parameters and moving inequalities to the constraints section in order to decrease over-constraint order. So, having a problem of 'n' equations functions of 'm' unknown parameters, it will be first interesting to assign one output parameter to each equation in order to orient the problem. The orientation of the equations corresponds to Task 2 in Table 1.

This means that the equations will be transformed as follows:

    m m x x x f x x x x f ,..., , 0 ,..., , 3 1 ' 1 2 2 1 1    (3)
To ensure that the assignment number is as high as possible, an augmenting path method will be applied. This method is based on the matching Hopcroft & Karp algorithm [START_REF] Duff | Remarks on implementations of O(n1/2 τ) assignment algorithms[END_REF]. Before applying the H&K algorithm, a heuristic method is applied to match equations (rows) and parameters (columns). It consists in matching each equation to the first free column encountered that is linked to the row, repeating the principle from first to last row. Then, if the matching is lower than the maximum possible assignment (lowest matrix dimension), H&K is applied to an assigned matrix. The algorithm presented in [START_REF] Duff | Remarks on implementations of O(n1/2 τ) assignment algorithms[END_REF], consists in two steps: a breadth first search (BFS) and a depth first search (DFS), repeated till the assignment does not increase.

It must be applied to a graph that is not disjoint (i.e. all the nodes are reachable by performing a DFS or BFS with a single starting node):

1. Generate the graph by applying a BFS starting from all currently free row nodes (equations) and stopping when a first free column node (parameter) is encountered. During this step, while descending the graph tree, a parameter node is linked to its matching equation node, while an equation is linked to the remaining parameters. During the BFS, dependency matrix crosses are replaced by level orders so as to identify the path orientation and simplify the following step.

2. Starting from a free column node, a DFS is applied crossing decreasing indices of the dependency matrix. Then, when an augmenting path is encountered, matching along the path is modified and the algorithm returns to step 1.

The next step will identify parts of the problem presenting under-constrained or over-constrained singularities [START_REF] Pothen | Computing the block triangular form of a sparse matrix[END_REF]. Of course, if n≠m, the problem is singular, but we will see that n=m does not necessarily mean the problem is not singular. This step may help the designer to fix parameters as inputs and move inequalities from equations to constraints (handled by the optimization upper-layer).

When the equation orientation has been done, the dependency matrix, as shown in Figure 3, can be drafted, feedback loops and strongly connected components (SCC) will be extracted thanks to a Tarjan algorithm [START_REF] Allain | Capitalisation et traitement des modèles pour la conception en génie électrique[END_REF] [START_REF] Helary | Algorithmique des graphes[END_REF]. And here comes the last step of the method (Task 3 of Table 1). For each SCC, a heuristic algorithm finds the minimal number of internal parameters to fix in order to determine all the other parameters encountered in the SCC [START_REF] Buckley | Design Sheet: An Environment for Facilitating Flexible Trade Studies During Conceptual Design[END_REF] [START_REF] Reddy | Constraint management methodology for conceptual design tradeoff studies[END_REF]. Normally, these particular parameters are used as initial inputs to the numerical iteration loops, and then all other parameters are obtained and will lead to a new estimation of the inputs values, looping until the error between fixed and estimated value falls below an acceptable threshold. In our case, the SCC input parameters will be extracted at input level and defined within a range, then the value calculated within SCC will be compared to the input through an additional constraint handled by the optimization layer. These problem solving choices were driven by the wish to use an independent solver generating a standard code which can be imported into as many calculation/optimization environments as possible.

Typical problems

Global concepts were presented in previous parts. The idea is now to present a few examples to highlight the value of such methods especially regarding singularity and loop issues. The examples discussed here are deliberately simple to facilitate the demonstration but deal with each typical issue encountered in the implementation of sizing procedure. They are based on servo-hydrostatic (SHA) and electromechanical actuators (EMA) sizing (Figure 5).

Under-constrained singularity: hydraulic jack

The aim is to implement the selection procedure of a hydraulic servo-actuator (jack and servo-valve) to ensure a rated load F 0 and a rated speed V 0 for a useful stroke dx. The anchorage structure has a stiffness K and does not tolerate effort higher than Fmax. The maximum flow rate Qmax (no-load flow) required by the actuator should be minimized to reduce the mass of the hydraulic supply system.

The physical equations for such a system are summed up here and can be represented by using a DesignGraph, as shown in Figure 6.

net P S F . max  (4) 0 .F K dx C   (5) 0 0 .P S F  (6) S Q V / 0 0  (7)
  nom net nom dP P P Q Q 0 0 .   (8) nom net nom dP P Q Q . max  (9)
Equations ( 5) to [START_REF] Budinger | Estimation models for the preliminary design of electromechanical actuators[END_REF] refer to jack component and link the total stroke C, piston area S and rated flow Q0 to the specifications. Equations ( 8) and ( 9) refer to the servo-valve flow formula, where Q nom and dP nom are respectively nominal component flow and pressure drop features.

From the listed parameters, some will be determined as input by the customer's specification document, F0, V 0 , d x , P net and K; while others are objective, Q max which must be minimized, or constraint, F max ≤F structure limited by maximum allowable load of the structure (optimization upper-layer). With these choices the problem is reduced to 6 equations -8 parameters and is even undetermined.

The design graph allows to highlight the sub-constraints: calculation nodes 6 to 8 (Figure 6a-b) indeed require to assume that 2 parameters are known to allow the sequencing of computations. As dPnom may be given as a test definition by the valve technology, it may be taken as input and P 0 is easy to limit to a range equal to [0; Pnet], so it will be taken as second input (variable). A new sequence of equations, given Figure 6c-d, fixes and solves under-constrained singularity.

An optimization problem with P0 as design parameter, Qmax as objective and Fmax as constraint is thus obtained. The optimization enables to find a classical hydraulic design rule: the pressure P 0 must be equal to 2/3 of P net in order to minimize Q max .

Over-constrained singularity: rollers-screw

Here will be presented an over-constrained problem which is quite simple but which will allow to present the way to solve this kind of singularity. The objective is to select a roller screw subjected to two constraints: a static load F max and a dynamic load representing rolling fatigue F RMC .

The problem can be divided this time by using the two above-mentioned layers, beginning with the component layer and roller-screw scaling laws explained in [START_REF] Budinger | Estimation models for the preliminary design of electromechanical actuators[END_REF].

  9 , 0 , 0 0 , . ref ref d d F F F F  (10)   5 , 0 , 0 0 , . ref ref n n F F L L  (11)   ref ref s s F F m m , 0 0 , .  (12)
Where F 0 and F d are respectively rollers-screw static and dynamic definition load, L n the nut length and m s the screw mass per unit length. A 'ref' index refers to reference component features. The screw total mass Ms is therefore calculated using the formula:

s s s L m M
.  [START_REF] Friedman | Constraint theory, part I: fundaments[END_REF] Where the screw total length L s comes from the mechatronic system layer equation: The last two equations are constraints expressed at mechatronic system layer and insuring proper use of the rollers-screw:

max 0 F F  (15) RMC d F F  (16)
As reference component data (m s,ref , F 0,ref ,…) and specifications (F max and F RMC ) are available, and as thrust bearing will be considered to be sized first, then the problem is in fact 7 equations with 6 undetermined parameters (Figure 7a).

The obtained matrix (Figure 7b) clearly highlights that one of the last 3 equations must be removed to obtain non-singular problem. To do so, one of the inequalities must be displaced to constraints, equation [START_REF] Malak | Compositional modelling of fluid power systems using predictive tradeoff models[END_REF] for example.

Yet, without modifying the other equation (considered as: F 0 =F max ), if fatigue is the sizing criteria, the constraint will never be met. That is why a degree of freedom 'kc' must be introduced to the equation:

max 0 .F k F c  (17)
With k c a variable input here within the range [1; +∞]. Then the final result is the one presented in the Figure 7c, where k c will represent an over-sizing coefficient to validate dynamic sizing.

Algebraic loop: brushless electric motor

The purpose of the last example is to highlight algebraic loop within a sizing procedure. To do so, a brushless electric motor will be sized considering only the maximum transient torque it has to deliver T0. This torque is calculated at mechatronic system layer as the sum of transient application load demand T load and the additional inertia load, product of the motor inertia J and its acceleration rate dw:

w load d J T T . 0   (18)
Then, applying scaling laws of [START_REF] Budinger | Estimation models for the preliminary design of electromechanical actuators[END_REF], the following equations of inertia and mass M are obtained using reference properties:

  5 , 3 / 5 , 0 0 . ref ref T T J J  (19)   5 , 3 / 3 , 0 0 . ref ref T T M M  (20)
As reference components such as torque and acceleration demands are determined, the problem is composed of 3 equations and 3 unknowns.

In Figure 8a-b, an algebraic loop appears since T 0 and J computation are linked. To break the loop, an oversizing coefficient k m can be introduced to adapt equation [START_REF] Pothen | Computing the block triangular form of a sparse matrix[END_REF] and reuse the original one as a constraint:

load m T k T . 0  (21) w load d J T T . 0   (22)
Another way to solve it, this time automatically with a heuristic program, is by fixing as input T 0,f within the range [Tload; 10.Tload]. Then, the algebraic loop does not exist anymore. Equation ( 18) is used as an output estimator:

w load e d J T T . , 0   (23)
The acceptance error criterion is described as a constraint:

Conclusion

This paper has shown that preliminary design of mechatronic systems can be modeled by equations represented into 2 layers, the mechatronics and the components layer, which enable storage of knowledge as libraries of non-oriented equations. Whenever a dynamic simulation representing the behavior of a key design indicator is needed, some effort can be done to replace it by an explicit meta-model formulation.

Anticipated simulation and modelling have two advantages: gathering knowledge and quickening the whole optimization. The optimization problem corresponding to the preliminary design is function of the sizing procedure definition which results of the ordering of the calculation steps. This paper has proposed graphical representation, based on graphs representation and algorithms that can be applied to highlight the different problems the designer has to face (singularity, calculation loop) during the generation of this sizing procedure.

Finally the problem is formulated in an explicit way which can be implemented in as many solvers/optimization tools as possible. 2.3 Errors are fixed by the designer manually modifying parameters status (to determined/input) or moving inequalities to constraint while introducing safety factors.

3 Break problem calculation loops.

  in fact the sum of the useful actuator stroke C, the nut length and the thrust bearing length L b .

  new sizing procedure has no algebraic loop thanks to the introduction of a new design parameter and a new constraint for the optimization problem.

Figure 1 .

 1 Figure 1. Component layer and Mechatronic system layer

Figure 2 .Figure 3 2 . 2

 2322 Figure 2. N² diagram principle

3. 1 3 . 3 Figure 4 .Figure 5 .Figure 6 .Figure 7 .Figure 8 .

 13345678 Figure 4. Problem graphical representation and orientation

Funding

The research was carried out as part of a European Union funded 7th Framework Program project -ACTUATION 2015 (Modular Electro Mechanical Actuators for ACARE 2020 Aircraft and Helicopters) (grant number 284916).

Corresponding author:

Marc Budinger, INSA Toulouse, Clément Ader Institute, Toulouse, 31077, France. Email: marc.budinger@insa-toulouse.fr