A multi-agent methodology for multi-level modeling of

A multi-agent methodology for multi-level modeling of mechatronic systems a,

Moncef Hammadi

a Supmeca,

∗ a a , Jean-Yves Choley , Faïda Mhenni

3 rue Fernand Hainaut 93407 Saint-Ouen Cedex France

Abstract Mechatronic design aims to integrate the models developed during the mechatronic design process, in order to be able to optimize the overall mechatronic system performance. A lot of work has been done in the last few years by researchers and software developers to achieve this objective. However, the level of integration does not yet meet the purposes of mechatronic system designers, particularly when dealing with modeling changes. Therefore, new methodologies are required to manage the multiview complexity of mechatronic design.

In this paper, we propose a multi-agent

methodology for the multi-abstraction modeling issue of mechatronic systems. The major contribution deals with proposing a new method for the decomposition of the multi-level design into agents linked with relationships. Each agent is representing an abstraction level and both agent and relationships are managed with rules. By considering an application to a piezoelectric energy harvesting system, we show how we associate agents, rules and inter-level relationships to multi-abstraction modeling. We also show how modeling errors are identied using this approach.

Keywords:

Mechatronic Design, Multi-Agent System, Multi-Level Modeling.

1. Introduction Mechatronic design is the integrated design of mechanical-based systems and their embedded control systems [1]. The mechanical system may eventually be considered as a multi-physics system coupling the mechanical structure with other multi-physics

∗

Principal Corresponding author

Email addresses: [email protected] (Moncef Hammadi), [email protected] (Jean-Yves Choley), [email protected] (Faïda Mhenni)

Preprint submitted to Advanced Engineering Informatics

June 6, 2014

sub-systems: uid, thermal, electrical, electromagnetic, etc. This means that the optimal mechatronic system performance can be obtained when the overall mechatronic system is designed and optimized in an integrated way. However, integrating models coming from diversied disciplines, with dierent levels of abstraction and using heterogeneous modeling tools is not an easy task. Because, despite the eort deployed until now, it is still dicult to identify and track errors coming from changes in models during the mechatronic design process. Some software vendors claim to oer a solution that encompasses the entire mechatronic design process, although practical experiences showed that it is still far from satisfactory. One choice might be to apply the 'divide and conquer' technique [2] to manage design complexity, rather than looking to integrate the complex mechatronic design process in one platform or language. Based on this idea, multi-agent system approach has been successfully applied to several industrial elds: intelligent manufacturing [3], process control [4], telecommunications [5], air trac control and transportation systems [6], information management and health care [7], etc. For that purpose, several multi-agent methodologies have been developed in the last few years.

However, most of them are designated to specic systems or architec-

tures [8]. Therefore, a multi-agent methodology needs to be correctly specied and properly applied in order to reduce the distance between the dierent viewpoints of mechatronic design integration. In this paper, we propose a multi-agent technique that meets mechatronic design integration issues.

In the second section we begin by giving an overview of the

mechatronic design issues for multi-view integration.

Then, in the third section,

we present some previous works about multi-view modeling based on multi-agent techniques. In section four we detail our methodology to deal with one design issue which is the multi-level of abstractions. A piezoelectric energy harvesting system is treated as an application of our approach in the fth section. A conclusion and the future work to extend the approach to other mechatronic design issues are given in the section six.

2. What are the mechatronic design issues? Integrated design of mechatronic systems has been addressed by several researchers during the last years, this has led to the fact that integration issues depend on the

2

design viewpoints considered [1, 9, 10]. Accordingly, mechatronic design issues are dependent on each other and the following design viewpoints are the most investigated in literature:

• Multi-disciplinary domain :

Mechatronics include a combination of engineering

domains, mainly mechanical, electrical, control and computer engineering. For this reason, one important issue is how to ensure integration between models coming form dierent disciplines.

• Multi-activities :

The design process of mechatronic systems include dierent

modeling activities such as task clarication and detail design.

As a result,

performing these activities in a continuous way is a problem that must be dealt with.

• Multi-physics simulation :

Beyond the control system, the operative parts of a

mechatronic system have in most of cases multi-physics behavior with a coupling between structure and other physics such as thermal, uid, electric, electromagnetic and piezoelectric eect.

Thence, dierent simulation techniques

such as nite element method (FEM), nite dierence method or nite volume method should be correctly coupled to simulate the behavior of the overall system.

• Multi-level of abstraction :

Abstraction is dened as the process of simplifying

a model of the system that maintains some of the details or properties, while suppressing others [11]. Consequently, depending on modeling purposes, dierent models may represent the same object with dierent granularity of detail, and the question is how to ovoid inconsistency between dierent abstraction levels.

• Multi-tool design :

Related to previous viewpoints, modeling tasks in mecha-

tronic design could not be performed with one unique design tool or formal1

ism. For instance, some languages and formalisms such as SysML , Modelica 3

or Bond Graphs

2

are advocated in high abstraction level. While, other tools

and methods such as CAD software and FEM are recommended in a more detail level. Therefore, dening consistent strategies of interoperability between dierent modeling tools is an other problem to overcome.

1 SysML:

Systems Modeling Language (www.sysml.org)

2 www.modelica.org

3 http://www.bondgraph.com 3

Analyzing the aforementioned viewpoints, one can remark that the integration issues in mechatronic design are not only a research topic but also concern other contributors such as modeling tool vendors and institutions of standards for the exchange of data.

Several research papers were interested only in one or few of the

viewpoints previously presented. For instance, Buur [12] and Isermann [13] reported two ways of integration in mechatronic systems: spatial integration (or hardware integration) and integration by information (or software integration). While the spatial integration was not well focused on, [13] has detailed information processing and control.

A multi-level approach for integrated design of control systems was pre-

sented by the [13] to characterize multi-level control functions from low-level control through supervision to overall control. Likewise, Groothuis and Broenink [14] proposed a multi-view methodology based on views, multidisciplinary core models and correctness preserving code generation to facilitate the multidisciplinary design of embedded control software. For the multi-activity viewpoint, many authors consider that the design process of any technical product is dened by four phases: task clarication, conceptual design, embodiment design and detail design [15]. The task clarication phase involves identifying and formulating design requirements and constraints. Conceptual design phase includes abstracting to nd essential design problems, establishing function structures, searching for working structures and selecting a suitable working structure.

Embodiment design is concerned with producing a denitive layout of the

proposed technical system or product. It should be noted that embodiment design is also called preliminary design by other authors [16]. In the detail design phase, all of the individual parts of the system are precisely specied: dimensions, forms, arrangement, materials, drawings and production documents[15]. Every design phase has its objectives and requires dierent engineering skills, such as systems engineering at conceptual level, architecture analysis at embodiment level and experts in modeling and analysis of components in detail level. One key issue of mechatronic multi-activity design is consistency between models developed during mechatronic design process.

This subject has been addressed in

previous works by several researchers. For instance, Hehenberger et al. [17] proposed to dene rules for the checking of design consistency as a solution to identify and track modeling errors when models are modied. For multi-physics simulation, several multi-tool approaches and frameworks have been proposed in previous published papers for coupling dierent modeling and simulation tools, either in sequential way or by co-simulation. For example, Hammadi et

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al. [18] proposed an approach for integrating dierent modeling and simulation tools for the optimization process of power electronic converters. The method proposed allows to automate the exchange of data between models with order reduction of FEM models during the optimization process. For multi-level modeling, several authors focused on relations between abstraction levels of modeling. For example, Scheidl and Winkler [19] proposed to dene relations between conceptual level models and detail level models. These relations are dened mathematically as projection functions between the two levels. Other approaches, such as Multi-Paradigm Modeling (MPM) and Model-Driven Engineering (MDE), have also been considered to address the integration issues when designing complex systems, taking into account the dependency between several viewpoints. For instance, Hardebolle and Boulanger [20] have presented a comparison between MPM techniques based on four design purposes: application domain, abstraction level, design activity and modeling view. Authors in [20] concluded that there is no single type of technique which solves all the categories of problems, since dierent modeling paradigms are used for dierent modeling goals. Lauder et al. [21] considered ve dimensions for the conceptual description of the models and information used for integration in the context of MDE approach. These ve dimensions are related to: relationships between element belonging to dierent disciplines, meta-modeling, domain customization, abstraction level and evolution of the model versions. Nevertheless, problems were also encountered when integrating tools in the complex domain of automation engineering. In their conclusion, authors in [21] reported that more ideas need to be explored in order to automate the change propagation between the heterogeneous models according to the ve dimensions previously mentioned. From the previous overview, we can conclude that despite the several works that have been interested in integrated design of complex systems, the integration issues have been partially fullled. Several integration solutions based mainly on MPM and MDE techniques were applied to domains such as electric design, software engineering and embedded control, but mechanical domain was not well covered due to the complexity related to 3D geometry.

Therefore, a global paradigm is still required

to respond in the best way to mechatronic design requirements. In the next section we explain why the multi-agent techniques are especially adapted for the multi-view modeling of mechatronic systems.

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3. Multi-view modeling with multi-agent techniques Recent research trends in systems engineering [16] focused on integration of multiagent paradigm with simulation technologies [22].

The Multi-agent approach can

be seen as an emergent development of a combination of trends including articial intelligence, object-oriented programming and concurrent object-based systems [23, 24]. Ören and Yilmaz [25] presented three complementary categories of integration between agent techniques and simulation: simulation for agent or agent simulation, agent-based simulation and agent-supported simulation. Agent simulation concerns the use of simulation techniques to simulate the behavior of agent systems. The agent system may be any complex system, such as a mechatronic system or a group of mechatronic systems which communicate between each other. Agent-based simulation is the use of agent techniques to generate simulation models. Agents are viewed in this case as intelligent systems of design that generate models and develop scenarios of simulation. Agent-supported simulation deals with the use of intelligent agents to enrich simulation environments.

For more details about the three categories of integration

between agent and simulation one should refer to [25]. More recently a methodology based on multi-agent technique was proposed by Treur [26].

He suggested a unied specication format for interlevel relations be-

tween agent models in multi-abstraction dimensions. For this reason, [26] used the interpretation mapping technique to dene relations between a higher abstraction level and a lower abstraction level according to three abstraction dimensions which are process abstraction, temporal abstraction, and agent-cluster abstraction dimension. However, the unied specication format for interlevel relations presented by [26] is a high level format and need to be adapted to be applied to concrete cases of engineering systems. Compared to other techniques, Iglesias and al. [27] have illustrated in their survey about multi-agent methodologies that several emerging agent-oriented methodologies are mainly extensions to known object-oriented or knowledge engineering methodologies. This oers to the multi-agent method more exibility and autonomy of modeling and simulation under concurrent and cooperative design constraints. Therefore, multi-agent techniques are well suited to representing systems that have multi-problem solving methods, with multi-interacting entities and multi-abstraction levels.

Thus, multi-agent approach is a privileged way to multi-view modeling of

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mechatronic systems. Since it is impossible to cover in one paper all the viewpoints previously presented concerning mechatronic design, we will limit the scope in next section to multi-level modeling issue. At the end of the paper we will briey elaborate how the agent-based approach can deal with the other issues.

4. Description of the multi-agent methodology for multi-level modeling Our multi-agent approach is based on dening a multi-agent system with three views: Agent (A), Relationships (R) and Rules (R). An Agent is a representation of a mechatronic entity and its attributes to characterize every abstraction level. Relationships are used to describe relations between agents. Rules are used to handle knowledge representation related to both agent and relationships. The three views of the

ARR multi-agent methodology, as it will be detailed in this section, are sucient

to manage the multi-abstraction modeling of mechatronic systems.

4.1. Agent view The agent view is itself described with three dimensions:

agent-model (AM),

space dimension (SD) and temporal dimension (TD). Hence the abstraction level of any mechatronic entity model can be dened according to these three dimensions.

4.1.1. Agent-model An agent-model describes single agents representing mechatronic entities, their internal properties (materials, physical parameters, etc) and their design variables. Agent model describes also the behavior of a mechatronic entity in relation with its initial states. Moreover, in agent-model we can dene environment elements by dening connection interfaces, boundary conditions and environment parameters. Furthermore, an agent-model is specied with a formalism and a modeling tool. A formalism is considered here as a rigorous mathematical framework, such as, analytic models, dierential equations, state-spaces, transfer functions, etc. Modeling tools are for example MCAD/ECAD

4

tools, Modelica language, SysML language, FEM

software, etc. Depending on the abstraction level, several agent models may represent the same mechatronic entity. The behavior of the mechatronic entity represented by the agent model can (or not) be dependent on time (temporal dimension) or space (space dimension).

4 MCAD/ECAD:

Mechanical/Electrical Computer Aided-Design 7

4.1.2. Space-dimension An agent model can be dened with a space dimension

sd ∈ {0D, 1D, 2D, 3D}.

The space dimension of an agent-model depends on the discipline considered, the accuracy and the computing time required. For illustration, 0D models such as analytical or lumped parameter models are usually sucient for a global rapid evaluation of a required performance. Whereas, other problems require 1D, 2D or 3D models, particularly in mechanical design. In electric domain however, 0D models are commonly used, but also 2D models are required for example to design Printed Circuit Boards (PCB). Control design and software modeling do not need more than 0D models.

4.1.3. Temporal-dimension An agent model can be dened as time-dependent or steady-state. Time-dependent models are required for dynamic behavior modeling of agent-models with discreteevent, continuous or hybrid discrete-continuous simulations. Time-independent category is used to describe static or steady-state agent-models.

4.2. Relationship view Relationship view describes how interaction between agents take place, either at the same level of abstraction or in inter-level relationships. For the latter case, two types of relations can be described: Top-Down and Bottom-up relationships. Top-Down relationships are relations of simple abstract models to more complex and detailed models. The method of decreasing abstraction can be used for this purpose [28, 29]. Relations can be described for example by using mathematical equations relating parameters of higher abstraction level to those of lower abstraction level. Bottom-up relationships are relations of lower abstraction level to higher abstraction level.

For example, geometric shape abstraction techniques can be used for mod-

eling problems based on geometric shapes [30]. Similarly, the method of temporal abstraction [31] or the generalized averaging method [32] can also be used for timedependent problems. Furthermore, techniques of model order reduction (MOR) can be used for problems based on dierential equations [33, 34, 35]. To illustrate the Top-Down relationship view. We adapt the formalism used in [26] to the mechatronic design case and we consider dierent abstraction levels of a mechatronic entity as shown in gure 1. Every abstraction level is represented by an agent-model. Assuming

AM1

and

AM2

are the two agent-models corresponding respectively to the highest and the lowest abstraction levels. The interlevel relationships between the two agent-models

8

AM1

a d a (AM1) a d (AM1)

d

d(AM1)

AM1 a

d

a (AM1)

t

t t

t a

d

d t (AM1) t d (AM1)

AM2 = at d (AM1) = a d t (AM1) = t a d (AM1) = t d a (AM1) = d a t (AM1) = d t a (AM1)

t(AM1) a

d

a t (AM1) t a (AM1)

Figure 1: Interlevel relationships of agent models with dierent abstraction levels and AM2 are dened by the arrows drawn in gure 1. δ : AM1 → δ(AM1 ): represents a relationship between two dimensional abstraction levels. For example in a conceptual level AM1 , 1D Euler-Bernoulli beam theory is used to model the beam deection.

Whereas, a 3D FEM model can be used in a

δ(AM1 ) for a more accurate modeling of the same τ : AM1 → τ (AM1 ): represents a relationship between two detail level

beam deection. temporal abstraction

levels. For example, the same case of a beam deection can be modeled with steadystate model in the conceptual level and a continuous time-dependent model in a more detail level.

α : AM1 → α(AM1 ):

represents a relationship between two agent-models, represent-

ing the behavior of the same object with two abstraction levels. For example, the agent-model

AM1

is dened with a simple formalism represented with an analytical

model. The formalism describing the agent model

α(AM1 ),

however, can be a par-

tial dierential equation combined with MCAD tool and FEM method for simulation. The relationships between dierent abstractions levels are mathematically illustrated for the case of a piezoelectric energy harvesting system treated in the next section.

9

4.3. Rule view The rule view is the set of knowledge related to the design process of the mechatronic system. They are established by designers and experts involved in the design process according to the design requirements.

In multi-abstraction level modeling

context, we dene two types of rules, namely modeling rules and checking rules. Modeling rules are used in the upstream of the modeling process to dene assumption conditions and constraints, required at each abstraction level, to apply eciently the formalisms and approximation techniques to agent-models. Therefore, modeling rules can be described by the following representation:

Rule : Assumption condition/constraint ⇒ F ormalism/Approximation method For example, we dene the modeling rule R1 as: RC1 ⇒ F1 , where RC1 designates a modeling condition or constraint and F1 designates a formalism or an approximation method to be applied. Moreover, modeling rules are also used to describe how interlevel relations are applied. These rules describe the logical conditions to be respected when applying the mathematical equations relating variables belonging to dierent abstraction levels. Furthermore, modeling rules are utilized to specify how to verify the models used. In this case verication metrics should be preliminarily dened. In addition, the level of acceptance should also be specied to measure the level of satisfaction of the model. Checking rules can be used either in the upstream or downstream of the modeling process for inconsistency checking. Inconsistency is dened as the situation in which prescribed modeling rules are violated. logical rules of type if ...

then ....

In this case checking rules can be simple

Therefore, checking rules allow designers to

check the consistency of the formalisms applied and the interlevel relations dened. Conjointly checking rules help designers to verify if the design objectives are met. However, rules should be structured and implemented in knowledge-based software to be able to automate the checking task.

The structuring of rules can be

performed by relating generic rules of high abstraction level to more detailed rules of lower abstraction levels. This allows designers detecting and tracking modeling inconsistencies during the design process, where changes could frequently take place. To conclude this section, relationships between agents are dened by designers with respect to modeling rules, and both agent models and relationship models can be veried with checking rules.

So that, identifying modeling errors and tracking

them can be automated.

10

For illustration, the methodology presented here will be applied in the next section to the case of the multi-level modeling of a piezoelectric energy harvesting system.

5. Application to a piezoelectric energy harvesting system

5.1. Introduction Piezoelectric energy harvesting is a multidisciplinary domain attracting several engineering disciplines including mechanical, electrical, aerospace, civil, materials as well as the mechatronic eld, for purposes related to environment and in-situ power generation from ambient vibration energy. Piezoelectricity is a form of coupling between the mechanical and electrical eect of piezoelectric materials. When a piezoelectric material is mechanically strained, an electrical polarization is produced, this is known as the direct piezoelectric eect. When the same material is subjected to an electric polarization, it becomes mechanically strained, this is called the converse or inverse piezoelectric eect [36].

For

instance, piezoelectric actuators are based on the inverse eect, whereas the direct piezoelectric eect is utilized in energy harvesting. The most common way to proceed in piezoelectric energy harvesting is by using a cantilever beam with integrated piezoelectric ceramics. Designers of piezoelectric energy harvesting systems (PEHSs) are interested in optimizing mass, geometry and materials of piezoelectric device to maximize the generated electric power at dierent excitation frequencies and acceleration amplitudes. The power generated by a PEHS is maximum when the natural frequency of the device matches the frequency of excitation.

Therefore, a proof mass is commonly

used to reduce the natural frequency. The gure 2 shows four possible congurations of PEHSs with one active piezoelectric layer (Unimorph), two active piezoelectric layers (Bimorph), with and without a proof mass. For both unimorph and bimorph congurations, a passive exible layer is required as a mechanical support for the PEHS.

The objective of this study is to apply the multi-agent

ARR

methodology to

the case of multi-level modeling of a PEHS. Therefore, three agents and

AM 3

AM 1, AM 2

corresponding to three levels of abstraction will be described in the next

section.

11

x2

x2

x1

a)

b)

x2

x1

x2

x1

c)

d)

x1

Figure 2: Dierent geometric congurations of piezoelectric harvesters: a) unimorph without mass, b) unimorph with mass, c) bimorph without mass and d) bimorph with mass.

5.2. Dening a high abstraction level agent-model: AM1 ARR

As it has been described in the

multi-agent methodology, modeling rules

are used to precise modeling assumptions that help designers to dene formalisms. Hence, for dening the agent-model

AM 1,

we will be limited to the following four

assumptions which associated to four modeling rules[36]:

• R11 :

The length-to-thickness aspect ratio is usually high enough to neglect the

shear deformation and rotary inertia eects (Euler-Bernoulli beam assumptions [37]).

• R12 :

Deformations of beam are assumed to be small and its composite structure

is assumed to exhibit linear-elastic material behavior.

• R13 :

The piezoelectric (electromechanical) coupling eect is neglected at this

level of abstraction.

• R14 :

steady-state models are required at this level of modeling.

Regarding the aforementioned rules, simple analytic equations are enough to estimate the power generated by the piezoelectric harvester. Hence, the available vibrating power can be calculated in this case with the following equation[38];

P o1 = where and

Me

Me χe Y02 ( ωωn )3 ωn3 (1 − ( ωωn )2 )2 + (2 (χe + χm ) ( ωωn ))2

is the equivalent mass of the system,

Y0

(1)

is the amplitude of vibration,

ω

ωn are respectively the excitation frequency and the natural frequency of the χm and χe are the mechanical and electrical damping coecients, respectively.

beam,

12

The maximum available power is determined when

ω = ωn

and it can be calculated

with following equation;

P o1max =

M e χe Y02 wn3 4 (χe + χm )2

(2)

In the two previous equations dening the agent-model

AM 1,

few parameters are

used and they are of two types, internal property parameters (M e, and input parameters (Y0 and

ωn ,χm

and

χe )

ω ),

which are due to the interaction between the

AM 1

is a 0D steady-state model that complies the

system and its environment. Accordingly, the agent-model modeling rules.

5.3. Dening an intermediate abstraction level agent-model: AM2 Some assumption rules of in particular

R11

and

R12 .

AM 1

will also be considered in this level of modeling,

Thus, the following assumption rules will be taken into

consideration:

• R21 = R11 . • R22 = R12 . • R23 :

The piezoelectric (electromechanical) coupling eect is taken into account

at this level of abstraction.

• R24 :

Dynamic eect is taken into consideration.

• R25 :

Vibration is assumed to be harmonic.

The lumped-parameter time-dependent model represented by gure 3 was used in several works for modeling PEHSs [39].

The mathematical representation of the

lumped-parameter model that respects the set of rules mentioned above is given by the following system of ordinary dierential equations (ODEs).



Where

M

M z¨ + D z˙ + K z − α v = −M y¨ α z˙ + Cp v˙ = 0

α

D is the damper damping, K is the spring

is the equivalent mass of the piezoelectric harvester,

coecient that represents the mechanical and electrical stiness,

(3)

is a the force-voltage coupling factor and

Cp

is the internal piezoelectric

device capacitance. The input is the relative displacement

z(t)

dened with:

z(t) = x(t) − y(t) 13

(4)

K M

x(t)

PZT D

i(t) v(t)

y(t)

Figure 3: Lumped parameter model of resonant generator

y(t) is the displacement of the system due to vibration and x(t) is the absolute M. output is the voltage v(t) that can be, for instance, determined numerically by

Where

displacement of the mass The

solving the ODE system 3. The available vibrating power in the piezoelectric energy harvesting system can be given by the following equation.

P o2 (t) = M ω 2 y z˙ Where

ω

(5)

is the vibrating frequency. The electric current generated by the PEHS is

depending on the load used.

5.4. Dening a detail level agent-model: AM3 For this abstraction level, we consider a 3D model of a bimorph conguration with a proof mass as illustrated in gure 4. More rule assumptions are required at z

Lxm

Lym PZT

Mass hm

x

hpz h hpz

Lx Ly

Figure 4: 3D layout of a cantilever beam for piezoelectric energy harvesting this level of modeling.

14

• R31 = R11 . • R32 = R12 . • R33 = R23 • R34 = R24 • R35 = R25 • R36 =

The middle layer is considered grounded.

• R36 =

The poling direction and the 3D notation are described in gure 5.

• R37 =

The electromechanical coupling mode to be considered is the transverse

mode 31.

PZT

Central layer P PZT

Figure 5: Indices used in piezoelectric mathematical formalism for a poling direction P Piezoelectric eect depends on electromechanical coupling modes. coupling modes are '33' and '31'. poling in the same direction 3.

The most used

The '33' mode evokes a strain and an electric

Whereas strain direction is 1 and electric poling

direction is 3 for the '31' mode. Despite the better electromechanical coupling for the '33' mode, the '31' mode promotes vibrating at lower natural frequencies. Four forms of the piezoelectric constitutive equations are used in literature: StrainCharge from (S-D), Stress-Charge form (T-D), Strain-Voltage form (S-E) and StressVoltage form (T-E).



S = [sE ] T + [d]t E D = [d] T + [T ] E

 (6)

Strain − Charge

T = [cE ] S − [e]t E D = [e] S + [S ] E Stress − Charge

15

(7)



S = [sD ] T + [g]t D E = −[g] T + [T ]−1 D

 (8)

T = [cD ] S − [h]t D E = −[h] S + [S ]−1 D

Strain − V oltage

(9)

Stress − V oltage

The denitions of the variables used in the four forms are given in the table 1.

Table 1: Variables used in the piezoelectric constitutive equations Symbol

Object name

Object type

Size

Unit

T S E D sD , sE cD , cE T , S d e g h

Stress

vector

6x1

Strain

vector

6x1

Electric eld

vector

3x1

Electric charge density displacement

vector

3x1

Compliance

matrix

6x6

Stiness

matrix

6x6

Electric permittivity

matrix

3x3

Piezoelectric coupling coecient for strain-charge form

matrix

3x6

Piezoelectric coupling coecient for stress-charge form

matrix

3x6

Piezoelectric coupling coecient for strain-voltage form

matrix

3x6

Piezoelectric coupling coecient for stress-voltage form

matrix

3x6

N/m2 m/m N/C C/m2 m2 /N N/m2 F/m C/N C/m2 m2 /C N/C

For more precision,

sE

and

sD

are the mechanical compliance matrix at constant E electric led and constant electric charge density, respectively. In the same way, c D and c are the mechanical stiness matrix at constant electric eld and constant S T electric charge density, respectively. As well as,  and  are the permittivity (dielectric constant) matrix at constant strain and constant stress. Two of the four formalisms are most used in FEM programs, namely strain-charge and stress-charge forms. The matrix representation of the the strain-charge form is given by the following two equations:

   E  E S1 s11 sE 0 0 0 12 s13 E S2  sE  sE 0 0 0 22 s13    12  E E E S3  s13 s13 s33 0  0 0  =  E S4   0  0 0 s 0 0 44     E  S5   0 0 0 0 s44 0 E E S6 0 0 0 0 0 2.(s11 − s12 )

    T1 0 0 d31 T2   0   0 d31      E1 T3   0 0 d33   +    T4   0 d15 0  E2     E3 T5  d15 0 0 T6 0 0 0 (10)

16

   D1 0 0 0 0 d15 D2  =  0 0 0 d24 0 D3 d31 d31 d33 0 0

  T1  T2   T     0   0 0 E1 11 T3   +  0 T11 0  E2  0  T4   0  0 0 T33 E3 T5  T6

(11)

In the same way, the matrix representation of the the stress-charge form is given by the following two equations:

   E E E c11 c12 c13 0 0 T1 E E T2  cE c c 0 0 22 13    12 E E E T3  c13 c13 c33 0 0  = E T4   0 0 0 c 0 44    T5   0 0 0 0 cE 44 T6 0 0 0 0 0

     S1 0 0 e31  S2   0   0 e31       E1  S3   0 0 e33    −     S4   0 e24 0  E2      E3  S5  e15 0 0 E cE 11 −c12 S6 0 0 0 2

(12)

  T1     T2   S  0 0 E1 0   11 T3   +  0 S11 0  E2  0  T4   0 0 S33 E3 0  T5  T6

(13)

0 0 0 0 0

   D1 0 0 0 0 e15 D2  =  0 0 0 e24 0 D3 e31 e31 e33 0 0

5.5. Dening the relationships Since we have three agent models corresponding to three abstraction levels, two types of relationships can be described; relationships between agent-models at the same abstraction level and relationships between agent-models of dierent abstraction levels.

5.5.1. Relationships at the same level of abstraction To illustrate interchangeability between agent-models at the same level of abstraction, we consider the agent-model

AM 3.

The following equations present 4 out

of the 6 possible combinations relating the dierent formalisms previously presented to dene agent-model

AM 3

[40]:

Strain-Charge to Stress-Charge transformation:

[cE ] = [sE ]−1 17

(14)

[e] = [d] [sE ]−1

(15)

[S ] = [T ] − [d] [sE ]−1 [d]t

(16)

Strain-Charge to Strain-Voltage transformation:

[sD ] = [sE ] − [d]t [T ]−1 [d]

(17)

[g] = [T ]−1 d

(18)

Stress-Charge to Stress-Voltage transformation:

[cD ] = [cE ] + [e]t [S ]−1 [e]

(19)

[h] = [S ]−1 [e]

(20)

Strain-Voltage to Stress-Voltage transformation:

[S ]−1 = [T ]−1 + [g] [sD ]−1 [g]t

(21)

[cD ] = [sD ]−1

(22)

[h] = [g] [sD ]−1

(23)

To sum up, if we choose the strain-charge as a formalism for

AM 3, the gure 6 shows

the modeling parameters allocated to the three agent-models and the relationships between the dierent abstraction levels.

We note that the more the abstraction

level is high the more the number of modeling parameters is reduced, therefore the complexity and interchangeability are better managed.

5.5.2. Inter-Level Relationships Relationships between agent-models

AM 1

andAM 2 concern the equivalent mass

which is the same in the two models;

Me = M In addition, the natural frequency of the piezoelectric harvester is the same in and

(24)

AM 1

AM 2. ωn = ω0 18

(25)

AM1

w0 , ccm,

Me , ce ,

AM2

m

Me , w0 , D , Mb , Mp , Ke , Ee , Ie , Cp

Lx, Ly, hpz, rpz, ce, d15, d24, d31, d33, eT11 eT33, sE11, sE12, sE13, sE22, sE33, sE44

Lx, Ly, h, rb, E, cm

AM3

Flexible layer

Lxm, Lym, hm, rm

PZT layer

Proof mass

Figure 6: Agent-model parameters in the three levels of abstraction Where

ω0

can be calculated with:

r ω0 = The damping ratios in

K M

(26)

AM 1 and the damping coecient D

the following equation:

D = 2 (χe + χm )

in

AM 2 are related with

√ MK

(27)

The relationship between the outputs of the two agent-models can be dened by the following equation;

P o1max = average(P o2 (t))

(28)

The last equation is an example of relationships between a time-dependent agentmodel and a steady-state agent-model,

AM 2

and

AM 1

in this case.

In the same way, we can dene relationships between agent-models

AM 3.

For this end, to relate the equivalent mass

M

used in

AM 3, we rst use 33 M = Me = Mb + Mp 144

of the piezoelectric harvester used in

19

AM 2

AM 2

and

with dimensions

the following equation [41]; (29)

where

Mb

Then, we use the following equations

where

Mp is the proof for Mb and Mp ;

is the mass of the beam and

ρb , ρpz

and

ρm

mass.

Mb = ρb Lx Ly h + 2 ρpz Lx Ly hpz

(30)

Mp = ρm Lxm Lym hm

(31)

are material densities of middle layer, piezoelectric layers and

proof mass, respectively. In the same way, to relate the equivalent stiness

K = Ke

to the geometric dimen-

sions of the PEHS, we use the following equation;

Ke = where

Ee

is the equivalent Young's modulus, and

To calculate

Ee

E

and

(32)

Lx

is the beam length.

the following equation is used:

Ee = Where

3 Ee Ie L3x

Epz

E h + 2 Epz hpz h + 2 hpz

(33)

are respectively the Young's modules of the exible layer and the

piezoelectric material. The latter module, is given by the following equation:

Epz = The equivalent moment of inertia

Ie

Ie = I + Where

I

and

Ipz

1 sE 11

(34)

can be calculated using the parallel axes rule;

Ipz + hpz Ly (h + hpz )2 2

(35)

are the moments of inertia of the middle layer and one piezoelectric

layer, dened respectively by the two following equations;

I=

Ipz

Ly h3 12

Ly h3pz = 12

20

(36)

(37)

The internal piezoelectric capacitance

AM 2

Cp

and the coupling coecient

AM 3

can be expressed with parameters in

Cp =

utilized in

using the two following equations;

33 Lx Ly 2 hpz

α = n Epz d31

α

(38)

Ly (2 hpz ) Lx

(39)

Where n is the number of piezoelectric layers, n=2 in this case. Therefore, the modeling rules allow us to nd the simplications and then to dene the relationships between the piezoelectric parameters of the 3D agent-model

AM 3

AM 2. These relationships are illustrated by [d], [T ] and [cE ] to parameters d31 , T33 and sE 11 ,

and those of the 0D agent-model

reducing respectively the matrices as shown in the following relations;



 0 0 0 0 d15 0 0 0 0 d24 0 0 ⇔ d31 d31 d31 d33 0 0 0

(40)

 T11 0 0  0 T11 0  ⇔ T33 0 0 T33

(41)

  E E 0 0 0 s11 sE 12 s13 E  sE 0 0 0 sE 22 s13   12 E E  sE 0  ⇔ sE  13 s13 s33 0E 0 11  0 0 0 s 0 0 44   E  0 0 0 0 s44 0 E E 0 0 0 0 0 2 (s11 − s12 )

(42)



The last relationships are an example of transformation between abstraction levels of dierent space dimensions, 0D and 3D in this case.

5.6. Dening the checking rules Checking rules are an ecient way for identifying and tracking errors. For example, in the case of our application, we need to modify dimensions of the system to maximize its output power. following:  if

Lx /h ≥ Lh0

For this, if we add a checking rule dened as the

then the rule

R1

dening the limit of Bernoulli assumptions.

21

is violated, where

Lh0

is a threshold

During an optimization process, it is

prevalent that designers do not detect this error. But with the checking rule it is possible to program a message to indicate to designers at dierent abstraction levels that a modeling rule is no more respected. In that case designers either limits the dimensions to the intervals in which the rule

R1

is usually respected, or change

the model used to take into consideration the shear deformation and rotary inertia eects.

5.7. Results and discussion As an application, we consider the modeling case of a bimorph conguration of a PHES. The materials used are PZT-5H for the piezoelectric layers, steel for the middle layer and copper for the proof mass. Using the strain-charge formalism, the piezoelectric material parameters are given by the three following matrices;



 16.5 −4.78 −8.45 0 0 0 −4.78 16.5 −8.45 0 0 0    −8.45 −8.45 20.7  −12 m2 0 0 0 E   10 s = 0 0 43.5 0 0  N  0   0 0 0 0 43.5 0  0 0 0 0 0 42.6 

0 0 0 d= 0 −274 −274  3130 0 T  3130 = 0 0 0 0

 0 0 741 0 C 0 741 0 0 10−12 N 593 0 0 0  0 F 0  , 0 = 8.854 10−12 m 3400

(43)

(44)

(45)

In addition, the other modeling parameters are dened in the table 2. The values of the parameters utilized in the agent-model

AM 2

are summarized

in the table 3; The gure 7 shows the available vibrating power deduced using agent-models

AM 1

and

frequency

AM 2. The f0 = 21.4Hz ,

vibrating power for

AM 2

is determined for the resonant

which is calculated using the given parameters.

AM 1 is 13mW . While the value of power AM 2 reaches a pick of 25mW . The average power

The available vibrating power given by of the transient model given by output of

AM 2 is 12mW .

Therfore, it can be remarked that the relationship between

the two power outputs dened by the equation 5 is respected. The gure 8 shows the variation of the voltage at resonant frequency given using the

22

Table 2: Piezoelectric design parameters PEHS length PEHS width Middle layer thickness Middle layer density Middle layer Young's module PZT layer thickness PZT layer density PZT layer Young's module Piezoelectric strain coecient Piezoelectric permittivity Proof mass length Proof mass width Proof mass thickness Proof mass density Mechanical damping ratio Electrical damping ratio

Lx Ly h ρb E hpz ρpz Epz d31 T33 Lxm Lym hm ρm ξm ξr

60.10−3 10.10−3 0.2.10−3

m

7850

kg.m−3 N.m−2

200.109 0.15.10−3 7500

60.9.109 −274.10−12 3.01.10−8 10.10−3 10.10−3 10.10−3 8800 0.02 0.02

m m

m

kg.m−3 N.m−2 m.V −1 F.m−1 m m m

kg.m−3 s−1 s−1

Table 3: Parameters calculated with relationship equations Beam mass Proof mass Equivalent mass Equivalent Young's module Equivalent modulus of elasticity Equivalent Stiness Damping coecient Natural system frequency Internal piezoelectric capacitance Force-voltage coupling factor

Mb Mp Me Ee Ie Ke D ω0 Cp α

0.0023 0.0088 0.0093 1.16.1011 1.04.10−13 168.3 0.0455 134.4

60.2.10−9 −0.00166

Kg Kg Kg N.m−2 m4 N.m−1 N.s.m−1 rad.s−1 F N.V −1

Po2

Power [W]

Po1

Time [s]

Figure 7: Available vibrating power: comparison between agent models AM 1 and AM 2 23

Output voltage (V)

Time [s]

Figure 8: Generated voltage by the PHES for the case of an open circuit using AM 2 agent-model

AM 2.

It is observed that the voltage reaches a maximum of

33V .

It should be mentioned that this voltage is simulated for an open serial circuit, with a negative port on the bottom of the system and a positive port at the top, and no load is considered in this case. The gure 9 shows the result of a frequency analysis using a FEM technique to

AM 3.

It can be observed that the rst res-

Voltage [V]

solve the strain-charge model dening

Frequency [Hz]

Figure 9: FEM frequency analysis of voltage using AM 3

24

onant frequency is

20.1Hz ,

which is close to value calculated using

AM 2.

We can

also remark that the voltage at the rst resonant frequency reaches a value about equal to

30V .

The acceptance level of this value can be specied as a checking rule.

One should also notice that the agent-model

AM 3

allows us to produce results for

more than one resonant frequency. For instance, the second resonant frequency is equal to

217Hz

and the corresponding output voltage is

not be produced using

AM 2

90V .

The latter result can

modeled with a system of ODE equations.

To conclude, this application illustrates how to apply the agent methodology to model a PEHS. Three agent-models are dened with respect to the modeling rules.

The relationships between agent models are mathematically illustrated and

an example of a checking rule is described. We have therefore showed that the use of rules in multi-agent approach allows designers to easily identify errors and track them when design modications occurs.

6. Conclusion and future work Multi-view modeling of mechatronic systems is an important issue to ensure modeling consistency during the mechatronic design process.

In this paper, the

mechatronic design issues were detailed to gather the dierent views of mechatronic modeling and apprehend the diculty related to the integration problem. Then, we described in detail an agent methodology responding to multi-level modeling issue of mechatronic design. Our major contribution is in proposing a new agent-based method that decomposes the multi-level design of a mechatronic system into agents, where each agent encapsulates an abstraction level. Links between abstraction levels are ensured using relationship models. Both agent models and relationship models are developed according to rule-based models which are also used for the identication of modeling errors. To validate it, the approach has been applied to the modeling case of a piezoelectric energy harvesting system based on a cantilever beam with attached piezoelectric layers. We explicitly highlighted the continuity and the coherence between the mathematical models developed for the three abstraction levels considered.

Thus,

the exibility of the presented methodology oers an interesting base for mechatronic multi-view modeling to shorten the design cycle and reduce modeling errors. Even though the approach we proposed is concerning the multi-level modeling of mechatronic systems, it can be extended to other issues of mechatronic design. Indeed, to include the multi-tool modeling issue, it is possible to implement an information system based on the relationship models and the rule-based models to share

25

information between modeling tools. It is also possible in this case to extend agents to represent the dierent disciplines of the mechatronic system by specifying the design objectives of every discipline. The interdisciplinary relationship models will be handled with a collaborative agent which tasks are identifying the design conicts and assisting designers in solving them using predened negotiating strategies and communication protocols. The forthcoming development for this approach is to dene a generic mathematical formulation for the relationships between abstraction levels of mechatronic multi-level modeling. The second step is to dene a prototyping framework enabling the denition of agent rules, agent models and relationships. And the ultimate objective is to automate the checking processes to identify and track modeling errors.

Acknowledgements This work has been partially supported by the Linz Center of Mechatronics (LCM) in the framework of the Austrian COMET-K2 programme.

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