EP2321750A1 - Method for producing a flexible mechatronic system - Google Patents

Abstract

The method of the invention comprises at least: a step of modelling the system using a meshing including a given combination of elementary blocks (61, 62, 63, 64), each block being formed of a predetermined assembly of segments representing elementary beams, said meshing including at least one active block that can be controlled by a control signal; the step of simulating the behaviour of a terminal node (68) of the model in open-loop response to a control signal; the step of characterising said response by at least one static mechanical criterion (dx, Fx) and at least one digital criterion J1 k representative of the decrease of the resonance peaks (43, 44) of said response based on the frequency. The previous steps can be repeated. Said steps are then followed by the step of selecting a design based on the criteria defined in the characterisation step, the system being produced from the selected model.

Description

 METHOD FOR PRODUCING A MECATRONIC SYSTEM

FLEXIBLE

The present invention relates to a method for producing a flexible mechatronic system.

Mechatronics is a discipline that is at the crossroads of mechanics and automation. Many systems are thus mechatronics. This is the case, for example, with robotic systems. The dimensions involved may be in the field of miniaturization, typically the field of micro robotics, or may be of the order of several meters as in the case of large poly-articulated robots.

If we consider for example the case of micro robotics, a designer must understand both problems, the problem of miniaturization and the problem of control of these systems. The realization of micro robots by miniaturization of traditional robots comes up against technological and physical barriers. The miniaturization efforts must be applied simultaneously in several areas, particularly in the design of transmission structures and actuators for the application of forces and displacements in volumes of the order of one cubic centimeter. Other means of actuation and measurement must be studied to achieve the desired mechanical performance, controllability and observability. This scale effect, as well as the nature of the designed systems, both specific to micro robotics, make the analysis and writing of dynamic and kinematic models difficult. The movements are then difficult to predict. Dynamic behaviors are strongly non-linear and the control laws are consequently complicated.

There is therefore a need to solve this double problem encountered in the design of a system related to mechatronics, that is to say, to meet both the mechanical performance and control requirements. More generally, there is a need to address the problem of the optimal overall design of mechatronic structures, especially flexible, from a dual point of view: mechanical and automatic. Indeed, the use of only mechanical criteria, usually static but also sometimes dynamic, can lead to a set of solutions among which many are not able to be controlled in practice. In the case in particular of flexible micro robotic structures, solutions are known. Most often, the flexible micro robotic structures are made active by piezoelectric actuation, an active structure being a structure that integrates its actuation. Indeed, the piezoelectric actuators have interesting properties of force resolution and bandwidth. On the other hand, their small deformation, of the order of 0.1%, does not allow large strokes and may limit the performance of the system actuated in certain cases. Many studies have dealt with the parametric optimization of piezoelectric actuators, this optimization being oriented from the point of view of static mechanics. Existing designs are essentially based on the intuition and / or experience of the designer and the optimization method only allows the values of the dimensional parameters of a predefined form to be adjusted. Most optimizations concern the improvement of the mechanical performances of a basic actuator, generally its deflections. A basic actuator may be in particular a single beam morphe, bi-morph or a multilayer actuator. There are also implemented methods that make it possible to optimize the shape of a passive amplifying structure of a basic piezoelectric actuator to optimize the resulting movement of the assembly, as described in particular in the M.Frecker article. and S.Canfield "Optimal design and experimental validation of compilant mechanical amplifiers for piezoceramic stack actuators", Journal of Intelligent Materials Systems and Structures, Vol. 1 1, pages 360-369, 2000. Another general approach to optimize the design of piezo-actuated structures is to simultaneously or separately optimize the size of the actuator, but not its shape. H.Maddisetty and M.Frecker's article "Dynamic topology optimization of compiler mechanisms and piezoceramic actuators" ASME Journal of Mechanical Design, Vol. 126, Iss. 6, pages 975-983, 2004, describes a solution where the size of the actuator is optimized separately, but neither its shape nor its position. The article by M. Abdalla et al "Design of a piezoelectric actuator and compiler mechanism combination for maximum energy efficiency", Smart Matehal and Structures, Vol. 14 pages 1421-1430, 2005, describes a solution where dynamic loading is performed at a given frequency and simultaneously optimizes the size of the actuator.

In contrast to these methods where the piezoelectric elements are placed a priori on the structure, several other works deal with the optimal location of actuators on a given structure, as described in particular in R.Barboni et al. [0003] Finally, some studies consider optimizing the shape of the piezoelectric actuator as shown, for example, by EC NeIIi Silva and N. .Kikuchi "Design of piezoelectric transducers using topology optimization", Smart Material and Structures, Vol. 8, pages 350-365, USA 1999. All these optimizations do not really take into account the simultaneous optimization of the structure and the actuator, let alone the complete optimization of the system including its boundary conditions. Other known forms of optimization take into account only the mechanical aspect. For example, some micro robots move using the so-called "stick-slip" technique as described in particular in S.Fablbusch et al "Flexible Microbotic System MINIMAN: Design, Actuation Principle and Control", Proc. of IEEE / ASME International Conference on Intelligent Advanced Mechatronics, pages 156-161, Atlanta, Georgia USA, 1999 and exploit the high bandwidth of piezoelectric actuators, which allow for high dynamic range motions and very high frequency utilization. fast. However, a large part of the existing work in optimizing flexible mechanisms called "compilers" is only concerned with the case of quasi-static applications, which can reveal sub-optimal structures in dynamic regime, or even inappropriate, resulting for example the appearance of phase shifts or passive damping. Methods that include the analysis and optimization of frequency responses of active structures very often focus on maximizing, or minimizing in the case of application of active damping, the amplification of deflections or forces at one time. selected harmonic loading, the actuator being used at a chosen frequency, static performance being ignored, see the article by H.Maddisetty and M.Frecker supra. Finally, these methods rarely envisage optimizing the input / output relationships over a wider frequency band, especially for damping applications.

With regard to the control of flexible systems, a first difficulty encountered is the prior identification of all the dominant modes. This necessary step allows the construction of an appropriate reduced model. However, it must meet a dual objective, firstly be of a reasonable order not to complicate a posteriori the synthesis of the regulator and secondly be as close as possible to the dynamic behavior of the real flexible system, which is usually strongly resonant. A number of approaches to model reduction exist, including the technique of balanced realization. This technique is generally based on a reduction order of the complete model which, in reality, is of almost infinite dimension. From this technique, it emerges the difficulty of choosing the order for the reduced model and consequently the accuracy of the latter with respect to the non-truncated model. A second difficulty lies in the choice to be adopted on the strategy and the actual synthesis of the control law against such systems with very oscillating behavior.

An object of the invention is to overcome the disadvantages of the aforementioned methods, allowing, for the realization of a flexible mechatronics system, to take into account very varied criteria to allow on the one hand to facilitate the calculation of a model reduced as precise as possible, and secondly to facilitate the control of the vibratory behavior of this type of structure. To this end, the subject of the invention is a method for producing a flexible mechatronic system, comprising at least: a step of modeling the system by a finite element mesh;

a step of simulating the behavior of a terminal node (68) of the model in open-loop response to a control signal;

a step of characterizing said response by at least one mechanical criterion and at least one numerical criterion J representative of the relative amplitude of the resonance peaks of said response as a function of the frequency, these resonance peaks being respectively chosen upstream and downstream of a predetermined mode number; the preceding steps being capable of being repeated, these steps being followed by a step of selecting a design obtained according to the criteria defined in the characterization step, the system being made from the selected design. The criterion J / is for example defined by the following relation:

where k is the number of the first modes to be dominant with respect to all the other modes of said frequency response and O ₁ , defined from the state representation which completely characterizes the input / output relation of the mechanism, is the i Hankel's Singular Value of the Flexible System. In one possible implementation of the method, the characterization step comprises another numerical criterion J ₂ ^k representative of the alternation of the resonances and antiresonances of the response in the frequency domain, in a selected frequency band. The criterion J ₂ ^k is for example defined by the following relation:

/ * = Σsigniçfr)

I = I where k is the number of first resonance modes of said frequency response and C ₁ and b _ι respectively represent the influence of the sensors and actuators on the input / output frequency response of the system. An embodiment is for example selected when the criterion J / is greater than a value chosen by the user / designer.

An embodiment is for example selected when the criterion J ₂ ^k is greater than a given value.

The mesh may be composed of a combination of elementary blocks to be determined, each authorized block being formed of a predefined assembly of segments representing elementary beams, said mesh comprising at least one active block controllable by means of a control signal.

In another possible embodiment, the mesh being composed of a combination of elementary blocks to be determined, each authorized block being formed of a predefined assembly of segments representing elementary beams, the mesh comprises at least one node controllable at means of a control signal.

The blocks are for example from a library of predefined blocks.

A command of an active block can be exerted by a deformation signal of at least one of its beams.

A static mechanical criterion is for example the displacement δ _x of the terminal node.

The terminal node being an effector, a mechanical criterion is the value of the force that it applies to the external environment F _x , for example the clamping force. Other mechanical criteria may be used, for example stresses, deformation energies or parasitic displacements.

The control signal is for example a voltage or electric current signal.

The system is for example a piezoelectric actuator, all the beams constituting an active block being controlled by a voltage signal.

Other features and advantages of the invention will become apparent from the description which follows, with reference to appended drawings which represent: FIG. 1, an example of a library of elementary flexible blocks used for reduced modeling of a flexible mechatronic system;

FIG. 2, an illustration of a piezoelectric beam, the elementary part constituting a piezoelectric system or actuator; FIG. 3, a curvilinear representation of the preceding piezoelectric beam represented by a segment and its orientation in a system of axes;

- Figure 4, a desired form, which can be considered ideal, frequency response of a flexible mechatronic system; - Figures 5a and 5b, the locations of the poles and zeros of the frequency response of a co-located system, respectively unamortized and slightly damped;

FIG. 6, an example of a mesh of a flexible mechatronic system whose elementary blocks constituting it are to be defined, with a view to the design of the system;

FIGS. 7a to 7d, examples of flexible system structures modeled by an assembly of blocks based on the previous grid; FIGS. 8 to 11 show illustrations of the open-loop frequency response of each of the preceding structures.

Figure 1 shows an example of a library of active and passive elementary flexible blocks used in a method of designing flexible mechanisms. This more general method is notably described in the following documents:

- Grossard M., Rotinat-Libersa C, N Chaillet, Perrot Y., "12th IFToMM World Congress, Besancon, France, 2007," Flexible building blocks method for the optimal design of compilant mechanisms using piezoelectric material ";

- Grossard M., Rotinat-Libersa C, N. Chaillet, "Gramian-based Optimal Design of a Dynamic Stroke Compiler-micro-mechanism", IEEE / RSJ International Conference on Robots and Systems, San Diego, USA, 2007; Bernardoni P., et al., "A new compiler mechanism design methodology based on flexible building blocks", Proceedings of Smart Structures - SPIE Modeling, Signal Processing and Control Conference, San Diego, USA, vol. 5383, pp. 244-254, USA, 2004 market; - Rotinat-Libersa C, Perrot Y., Friconneau J.-P., "Potentialities of optimal design methods and associated numerical tools for the development of new micro- and nano-intelligent Systems based on structural compliance - An example -", IARP - IEEE / RAS-EURON Joint Workshop on Micro and Nano Robotics, Paris, France, 2006. The library of FIG. 1 comprises, for example, 36 passive elements and 19 elements made active by piezoelectric effect 4. This library makes it possible to construct a large variety of topologies of structures. Each elementary block 1, 2, 4 is also constituted by a predefined assembly of elementary beams, or segment 3.

The specification of the problem of optimal design of flexible systems considers, in a given size, different boundary conditions: the connection points between the mechanism and the frame (position and list of degrees of freedom blocked), the inputs (actuators), the outputs (effector nodes), contacts. With regard to actuation, several principles can be envisaged:

• either actuators in force or displacement that can act in particular nodes of the structure,

• either active blocks from the corresponding library; these blocks may be made of piezoelectric material whose upper and lower surfaces are each covered with a conductive surface (called an electrode) and subjected to electrical potentials.

The advantage of using active flexible blocks of various topologies instead of simple piezoelectric beams is that it allows to couple the degrees of freedom, to generate complex movements in a small footprint.

A goal of the design stage is therefore to seek an optimal distribution of flexible blocks in a given space, as well as the optimal set of parameters (optimization variables: materials, dimensions, boundary conditions including actuators ...) which define flexible mechatronic structures whose performances (objectives to be optimized) are the closest to those specified in a given specification. Variables and optimization criteria are used. For optimization variables, several parameters can be used to define structures. Some of these parameters may or may not be optimized, including: o points fixed to the frame (number, location, degree of freedom concerned), o the topology of the structure (types of flexible blocks of the library, their nature -active and / or passive-and their arrangement), o the (or) matéhau (x) constituting the passive blocks and their thickness o l ' thickness and electrical potential difference between the upper electrode and the lower electrode (for active blocks, which are currently made of piezoelectric material) o unilateral contacts between the structure and the frame, or between two parts of the structure (number , degree of freedom concerned, play and location), o the actuators (number, type of actuation, action point, degree of freedom concerned and amplitude), o sensors (number and location only, at present) .

The mechanical criteria available, for a static model, include: o Free movement of the degrees of freedom (ddls) output o Geometric advantage, ie amplification of the ratio displacement output on displacement input o Blocking force of degrees of freedom of output o Advantage mechanical, ie amplification of the ratio of the force output on the input force o Inverse of the strain energy o Inverse of the orthogonal displacements to the output displacement o Manipulation o Mass o Maximum mechanical stress o Critical buckling load.

Other criteria related to mechanics are available for a dynamic model. These use a formulation specific to the field of automatics, based on the grammars of controllability and observability. These grammars are calculated from a state representation that has an input node driven, or commanded, and output a single node as described in particular in the document "Gramian-based optimal design of a dynamic stroke amplifying compiler micro-mechanism" above. For example, in this document, two criteria are:

• shift the first resonance of the system beyond a chosen limit frequency. This criterion considers in an absolute way all the first modes that are not very influential in the input-output behavior of the system, without being interested in the following modes. In addition, this method aims to minimize the influence of the first modes only.

• Avoid the presence of resonances in a range of frequencies chosen for example filter unwanted movements transmitted from the base to the output.

A stochastic optimization method used is for example a genetic algorithm inspired by the code described in Deb K., et al., "A fast elitist non-dominated sorting genetic algorithm for multi-objective optimization: Nsga-ll", Proc. of the 6th Int. Conf. on Parallel Problem Solving from Nature, pp. 849-858, France, 2000. The optimal design method uses this optimization principle which allows a real multicriteria optimization (without a priori weighting of one criterion compared to another criterion), and the use of discrete variables. In this method, we design a discrete parameterization adapted to the specification of flexible mechatronic structures. The structure of the algorithm is for example organized as follows:

• Given the specifications of the system to be designed, the designer informs the permissible values for the parameters (which are the variables to be optimized); • The algorithm synthesizes individuals and then evaluates them according to the criteria chosen by the designer;

• Stochastic operators, specific to genetic algorithms, modify the description of structures by manipulations on their discrete variables, so as to synthesize new structures, which are then evaluated as in the previous generation;

• A new cycle of synthesis and evaluation is carried out, and so on, until convergence of the algorithm (achievement of the global optimum); • When convergence is reached, the design algorithm provides a set of pseudo-optimal solutions, restituted on one (or more in the case of more than 2 criteria) two-dimensional Pareto fronts. Then, the designer chooses and interprets the resulting structures that respond optimally to his specifications. The optimization process is based on a stochastic optimization technique (genetic algorithm) in which the candidate solutions are considered as individuals of a population. They are parameterized by chromosomes, themselves made of genes. The entire population evolves with simplified operators inspired by genetics, such as breeding, mutation and crossbreeding. For example, the genotype of a flexible block assembly is encoded by an array of integers, each of the integers referring to the active and passive library blocks of FIG. 1. In an implementation example implemented, the blocks For example, assets are dominant on passive blocks, which are themselves dominant on an empty block.

A modeling of the piezoelectric elementary flexible blocks 2 can be implemented. For this modeling, only displacements and forces contained in the plane of the structure are considered. For example, it is assumed that the flexible mechanisms are subjected to structural deformations, mainly resulting from the bending of the beams constituting the blocks as illustrated for example in Figure 1. Thus, the model adopted for the blocks is obtained by considering Navier-Bernoulli's hypotheses. The parameters defining the blocks are the thickness, the material, the width and the height (and the electrical potential difference between the upper and lower electrodes for the active blocks). The characteristics of the material of each block are parameterized by its Young's modulus, its Poisson's ratio, its elastic limit, its density (and the piezoelectric coupling coefficients for the active blocks). Before obtaining the formulation of the piezoelectric block model, a simple beam model is first needed. FIG. 2 illustrates a piezoelectric beam 20 with electrodes 21, 22 perfectly distributed on its upper and lower faces, polarized with potentials φ-i, φ ₂ depending on the thickness, and oriented in an orthotropic reference (ei, e ₂ , e ₃ ). Under the effect of transverse piezoelectric coupling, a longitudinal strain Su is caused along the dimension L by the electric field E ₃ depending on the thickness. Considering the unidimensional shape of the piezoelectric equations along the longitudinal axis of the beam, the piezoelectric coupling d, electrical permittivity ε and compatibility s matrices are each represented by a scalar coefficient, d ₃₁ , ε ₃₃ and sn respectively. The constituent relationships of the piezoelectricity for the deformation Su and the electrical displacement D ₃ as a function of the stress T ₁₁ and of the electric field E ₃ are in this case reduced to:

FIG. 3 illustrates a curvilinear representation of the piezoelectric beam 20, represented by a segment AB, and its orientation in a global reference system (O, x, y, z) with the forces R _A , R _B and the nodal moments H _A , H _B at the ends of the beam. The displacement field on a piezoelectric beam element is described by its longitudinal components u, tangential i / and rotational w at the curvilinear abscissa x _p (FIG. 3). It is expressed relative to the corresponding nodal values of the beam, in the coordinate system of the beam R _P = (A, _{Xp, yp, Zp)}

The Hamilton principle, generalized to electromechanical systems, provides the model of the dynamic behavior of the piezoelectric beam described for example in A.Preumont "Mechatronics: Dynamics of Electromechanical and Piezoelectric Systems (Solid Mechanics and its Applications)", published by Springer, September 25, 2006,:

M _b x \ _b + K _b x \ _b = G _b Φ _b + Εr _b (2) The different matrices are calculated under the assumptions of a displacement field interpolated at the nodes of the beam, by the shape functions specific to the Navier-Bernoulli beams:

M _b denotes the mass matrix; K _b denotes the stiffness matrix;

- G _b denotes the electromechanical coupling matrix, which induces the piezoelectric loading to achieve contraction or elongation of the beam proportionally to the potential difference φi - φ ₂ ; - Φ _b = (φ-i, φ ₂₎ ^f is the vector of the electric potentials on the upper and lower faces of the piezoelectric beam;

- Fr ₆ = (R ^* R ^ H ^ R ^* R _B ^y H |) ^f is the vector of mechanical nodal forces.

As for the passive blocks, the mass matrices M _B , stiffness K _B and coupling G _B of each active block are obtained by matrix assembly of the mass matrices M _b ', stiffness K _b ', and coupling G _b 'of each constituent beam of the blocks. Each of these latter matrices is expressed in the global coordinate system R '= (0, x, y, z) according to the basic change formula by:

M ₆ = P ^f M ₆ P

K ₆ = PK ₆ P

G ₆ = PG, where P is a marker change matrix conventionally obtained.

Before performing an optimization, the matrices mass, stiffness and electromechanical coupling of all the blocks of the library are calculated numerically, and this, considering all the possible combinations of the different discrete values authorized by a designer for the optimization variables (materials , block size ...). This calculation is done once. The results are kept in memory, which saves computing time during the iteration processes of the optimization steps. During the optimization, the evaluation of the different criteria specified by the designer uses the computation of the model of static or dynamic behavior of the structures, according to which are considered static mechanical criteria, dynamic or oriented automatic. In the dynamic and piezoelectric case (which is the most complete), the matrices (expressed in the reference R ') of all the constituent blocks of the structure are assembled to describe the conservative behavior of the mechanism according to: M, ή, + K, η, = G, Φ, + Fr, (3)

This assembly is performed during the optimization process for each individual and each generation. In relation (3), η _g refers to the nodal displacements of the lattice structure discretized into blocks, themselves discretized into beams. Φ _g denotes the electrical potentials applied to the upper and lower electrodes of each block, and Fr ₉ the external mechanical forces applied to the system.

In the case of the dynamic calculation of a purely passive structure, equation (3) is simplified according to:

^M A ^{+ κ} Λ = ^Fr , Actuator forces are considered this time as all external to the structure and included in the term Fr ₉ .

In the case of calculating the static behavior of a piezoelectric structure, the equation can be reduced to: K η _? = G Φ + Fr

Finally, in the case of the computation of the static behavior of a passive structure, the equation can be reduced to: K ^ = Fr ^.

According to the invention, two numerical criteria Ji and J ₂ are used to evaluate the dynamic performance of input-output transfer of flexible mechatronic systems. More particularly, these two criteria make it possible to define an optimization strategy with a view to the subsequent control of the flexible mechatronic systems to be produced. Criteria Ji and J ₂ relate to the open-loop response of these structures. In particular, they allow the frequency input-output response to respect certain templates and constraints on the controllability and observability of certain modes of vibration. The criterion Ji relates in particular to the amplitude of the open-loop frequency response of a system and the criterion J ₂ relates in particular to the phase of this frequency response. These two criteria Ji and J ₂ make it possible respectively to characterize the dynamic properties of a system model, in amplitude and in phase. A first difficulty encountered when ordering flexible systems is the prior identification of the dominant modes. This step, which is necessary, allows the construction of an appropriate reduced model which, ideally, must meet a double objective, on the one hand to be of a reasonable order so as not to complicate a posteriori the synthesis of the regulator and of on the other hand, to be as close as possible to the dynamic behavior of the real flexible system, which is generally strongly resonant.

A second difficulty lies in the choice to be adopted on the strategy and the actual synthesis of the control law against such systems with very oscillating behavior. Faced with this double problem, the two criteria Ji and J ₂ allow on the one hand to facilitate the calculation of a reduced model as accurate as possible and on the other hand to facilitate the control of the vibratory behavior of these structures. Before formulating these criteria Ji and J ₂ , we recall modal equations of the movement of flexible structures.

In accordance with the document KBLim et al, "Actuators and sensor placement for control of flexible structure," in Control and Dynamics Systems: Advances in Theory and Applications, ed. London, Academy Press, 1993, each synthesized flexible structure is defined as a linear system of finite dimension, controllable and observable, with weak conjugated complex conjugate poles. His conservative dynamic behavior is governed by the differential matrix equations of the ^2nd following order:

^M Λ ^{+ K} Λ ^{= E} , ^U ₍₄₎

Each element of u (respectively y), denotes an actuator (respectively sensor) whose degree of freedom is defined by a non-zero value in the corresponding column of E ₉ (respectively line of

Fg).

The modal decomposition of the system makes it possible to look for a solution in the form:

It consists of a linear combination of modal deformations Ψ _; . q is the vector of modal displacements of dimension pχ1. The matrix of eigenvectors Ψ = [Ψ _X ... Ψ _p ], and the associated eigenvalues are obtained as solutions of the following conservative problem:

It is assumed here that the damping has little influence on the frequency positioning of the resonances of the flexible structures. The eigenvalues are ranked in increasing order & f ≤ ... ≤ ω _p ² , and Ψ is chosen normalized with respect to the mass matrix M.

By replacing η _g by Ψq in relation (4) and multiplying on the left by Ψ ^f , the orthogonality relations induced by the modal form give: q + diag (^ ² ) q = Ψ ^f E ^ u

(5) y = F ^ Fq

To obtain a state model of the structures, one chooses, for example, the state vector x of dimension 2p x 1 defined by: x = (q _γ ύ \ _qι ... q _p ω _p q _p )

The triplet (A, B, C) of the state representation in modal space x = Ax + Bu y = Cχ takes the particular form

A =, B ₁ =. C ₁ = 0 - F ^ (6) ω 0 o OJ, ⁸

The evolution matrix A depends on the structure (natural frequencies and modal damping), the control matrix B (respectively C) depends on the location and nature of the actuators (respectively sensors). In the following, we note Ψ [E _g (of dimension Us): bi, and F ^ Ψ, (of dimension rχ1): Cj. In the frequency domain, the transfer matrix between the inputs u and the outputs y is seen as the sum of all the modal contributions: y (jω) = G {jω) u {jω) where the rxs transfer matrix is: c , b

G (jω) = Σ _Gι (jω) = Σ t _ι o ^ - ω ² + 2jξ _ι ω _ι ω ^' FIG. 4 shows a desired shape for the open-loop frequency response of a flexible system, for example a structure consisting of a set of piezoelectric beams. This form is represented by a curve representative of the amplitude, expressed in dB, with respect to the pulsation, expressed in radians / seconds. To increase the control authority over the different modes, the amplitudes of the resonances must be maximized in the frequency band [0, ω _c ], or bandwidth 42, as illustrated by the major peaks 43 of the curve 41 to FIG. In contrast, the amplitudes of the resonance peaks 44 after the cutoff pulse ω _c must be minimized in order to limit the gain margin and to limit the so-called "spillover" phenomena. The criterion Ji defined subsequently makes it possible, in particular, to evaluate whether the response of a system model approaches the ideal case of FIG. 4. In other words, the criterion Ji makes it possible to evaluate the reduction capacity of the system. model and its precision. The interpretation that can be made of the HL standard of the transfer matrix makes it possible to write such a criterion for obtaining this form of frequency response. This standard characterizes the maximum amplifications that the system can produce on the input signals. Especially in the case of a SISO function (only one input, one output), the value of the maximum amplitude Gi of the i-th mode of the frequency response can be approximated, under the assumption of small depreciation, by :

In addition, Hankel singular values (HSV) associated with the state representation of the relation (6) are the values of the balanced grammar. This grammar is a positive diagonal matrix defined by r = diag (σ _! ), Which characterizes the degree of controllability and observability joined, of each component of the state vector. In the present case of nodal representation, each HSV thus characterizes here the degree of controllability and observability of the vibratory mode of the system in question. In the case of low depreciation, HSVs are defined by: c, b.

O, =

It follows then that IGJ ₀₀ = I ^.

For the first k resonant modes, where k <p, to be dominant, they must guarantee important HSVs, for good controllability / observability. The model reduced to the first k modes, G ₁ -, is defined according to the following relation:

G _Γ (J ") = ΣG, (J") ι = l

This model G _r reflects the behavior of the complete system G with a difference, relative to HSV omitted, defined by the following relation:

ι = k +

To make the first k dominant modes and simultaneously improve the precision of the truncated model to these first k modes, the optimization criterion J ₁ , evaluation function, is written thus:

G ₁ representing the Hankel singular values (HSV) associated with the state representation of the relation (6) for the particular eigenmode No. i. This criterion has an indicator function, rating, so that more than ₁ J ^κ is the big plus has good controllability and observability system simultaneously for the first k modes and bad controllability and observability simultaneously for other modes. In practice, this criterion will favor systems for which only the first k modes are dominant.

Figures 5a and 5b illustrate the places of the poles, represented by x, and those of zeros, represented by o, of the frequency response of a colocalised respectively undamped and slightly damped system, in the complex plane where the abscissa represent the real response Re (s) and the ordinates represent the imaginary response lm (s). A second criterion used by the method according to the invention, J ₂ , characterizes the co-location behavior of flexible systems. One property of these systems, advantageously exploited by the invention, is the alternation of the poles and zeros in the complex half-plane. Such systems are at a minimum of phase and the phase response oscillates continuously between 0 ° and 180 ° respectively corresponding to the zeros and the poles of the response. Collocated systems are known for these interesting properties, which make it possible to ensure a stability of their closed-loop control. According to the invention, an evaluation function can be used to estimate whether the behavior of a given structure can be considered as being close to a collocated system, thus with simple stability criteria. This evaluation function is the criterion J ₂ , defined for the first k modes by the following relation:

J ^k - Σsignicfr) where SIgH (C ^ ₁ ) = +1 or -1 according to whether the sign of the argument of the complex response is respectively positive or negative, signicfa) being equal to 0 if the argument is null. The sum on the index i concerns all the modes contained in the frequency spectrum of the first k modes.

C ₁ and b _x respectively represent the influence of the sensors and actuators on the frequency response input / output of the system. In the SISO case in particular, the maximization of the criterion J ₂ makes it possible to favor the structures which show an alternation of poles and zeros. This criterion therefore favors the structures whose first k static gains of Gj are of the same sign. In practice, this amounts to favoring systems with alternating poles and zeros for the first k modes.

FIG. 6 illustrates the mesh of the symmetrical part of a micro-actuator with the boundary conditions, with four blocks 61, 62, 63, 64 to be determined and nodes 65, 66, 67 admissible for blocking, these nodes being the three nodes at the bottom. An output node 68 representing the so-called node effector, forming the terminal member of the actuator. By way of example, the global optimization carried out by the method according to the invention is applied to design this micro-actuator, more particularly to design a piezoelectric monolithic structure, with integrated actuator, for micromanipulation tasks. The specification of the problem here takes into account four optimization criteria simultaneously:

two static mechanical criteria, the maximizations of the displacement δ _x and the force F _x of the effector node 68 of the structure;

the two criteria Ji and J ₂ characterizing the transfer between the integrated actuation and the effector node 68.

In this example, we choose to take k = 2, that is to say to consider only the first two modes of resonance, for the determination of criteria Ji and J ₂ . In this case, the criterion J ₂ can take only two discrete values. The maximum value is 2, case of collocation, or 0, case of non-localization.

Finally, the optimal synthesis is carried out in this example for a symmetrical mechanism made of a piezoelectric material. Figure 6 illustrates the problem specification. The fixed data are the dimensions of the symmetrical part, the modal damping and the electric potential difference V fixed for the active blocks. The modal damping is for example taken constant and equal to 1% for all resonance modes. The mesh of Figure 6 is made in a plane O, x, y. The symmetrical part has for dimensions L1 x L2, L1 being for example equal to 15mm and L2 to 9mm, for a thickness following z of 200μm. This part is formed of four blocks 61, 62, 63, 64 to be optimized, the blocks each having a dimension a which is specific to it according to the dimension L1 and a dimension b which is specific to it according to the dimension L2. The active blocks are those having electrodes powered by the voltage V equal for example to 200 volts, to exploit the piezoelectric effect. Passive blocks are made of the same material but without electrodes. The size of the blocks may vary for a between a maximum value a _max and a minimum value a _min such that the ratio a _max / a _m i _n may vary between 1 and 2. Similarly, the dimension b may vary between maximum value b _max and a minimum value b _min so that the ratio b _max / b _m i _n can vary between 1 and 2. In the example of FIG. 6, the mechanical parameters to be optimized are, for example, the displacement along the x-axis and the clamping force along the x-axis of the output node 68. In other words, when the structure is controlled voltage, the output node 68 should occur: - a displacement δ _x along the x axis the largest possible;

a clamping force F _x along the largest possible x-axis. The control may be an electrical voltage exerting a deformation of the piezoelectric beams.

FIGS. 7a, 7b, 7c and 7d illustrate four examples of solutions respectively called A, B, C and D, from the four-dimensional Pareto front (because here four optimization criteria are considered) obtained at the convergence of the procedure. optimization, to analyze the contribution of the use of the criteria Ji and J ₂ according to the invention during the optimization. The performances of the four structures are reported in the table below where δ _x represents the displacement of the effector 68 along the x axis and F _x its clamping force along the same axis.

Solutions Results of selected criteria δ _x F _x Ji J ₂

At 15.55 μm 1, 26 N 2.24 0

B 11, 74 μm 1, 26 N 21, 00 0

C 12.34 μm 0.63 N 0.28 2

D 10.69 μm 0.84 N 5842.35 2

Figures 7a, 7b, 7c and 7d are the geometric representations of the four piezoelectric actuators. The structures of these actuators are composed of an assembly of blocks, for example from the library of FIG. 1 and each formed of one or more segments 3 whose ends form nodes 70 in which the segments can be fixed to each other. to others.

The solid lines 71 represent the active blocks and the dashed lines 72 represent the passive blocks. In the example of FIG. 7a, the two left blocks 61, 63 are passive, the right blocks 62, 64 are active, the middle node 66 being blocked. In the example of FIG. 7b, the blocks at the top left 61 and at the bottom right 64 are passive, the blocks at the top right 62 and at the bottom left are active, the left nodes 65, 66 being blocked. In the example of FIG. 7c, the left block at the bottom 63 is passive, the right blocks 62, 64 are active, the right nodes 66, 67 being blocked. In the example of FIG. 7d, the left block at the bottom 63 is passive, all the other blocks 61, 62, 64 are active, the left nodes 65, 66 being blocked.

In the context of a conventional optimization, that is to say only with the two static mechanical criteria δ _x , F _x , only the solutions A and B would have been retained in a first step because responding to the optimization below. above, namely a displacement δ _x along the largest possible x axis and a maximum clamping force F _x possible. Solutions A and B maximize these two mechanical criteria. Solutions C and D would have been discarded and removed from the Pareto front due to their poorer static mechanical performance. But it turns out that the structure A has poor dynamic performance for subsequent control as shown in Figure 8 and that confirm the criteria Ji and J ₂ .

FIG. 8 presents, by a Bode diagram, the response of the displacement δ _x of the effector 68 following a voltage harmonic excitation for the structure A of FIG. 7a. The amplitude of the response is represented by a first curve 81 as a function of the frequency and the phase is represented by a second curve 82 as a function of frequency. The first curve departs from that of the ideal case expressed in Figure 4 for amplitude 81. The second curve illustrates the strongly non-minimum phase character. The phase curve 82 shows that the input-output transfer does not have the interesting property of alternating poles and zeros, which was illustrated in FIGS. 5a and 5b, expressed by continuous oscillations of the phase between 0 ° and -180. °, in the chosen spectrum. Indeed a discontinuity 83 appears to break the continuous succession of poles and zero, corresponding to the succession of two antiresonances between the first two resonances 84, 85 and effectively J ₂ is equal to 0 in the previous table. Moreover, the control authority over these first two resonances is weak compared to the other modes. Indeed the peaks 84, 85 which represent them are not maximized with respect to the other peaks 86 placed outside the frequency band of interest, and whose value amplitude is not reduced, corresponding to a lack of gain drop, which is confirmed by the low value of Ji.

FIG. 9 always shows, by a Bode diagram, the frequency response of the displacement δ _x of the effector 68 under the same conditions but for the structure B of FIG. 7b, by an amplitude curve 91 and a phase curve 92. remarks of the same order as those on structure A can be made. The control authority on the first two resonance modes is correct as shown by the significant drop in gain after the second resonance 93 on the amplitude curve 91, which is confirmed by a fairly good value of the criterion Ji in the table. previous. However, the character of continuous alternation between the poles and zeros is not observed in view of the phase curve 92 in the frequency domain of interest and J ₂ is effectively 0.

Figure 10 illustrates the case of structure C of Figure 7c. In this case where the criterion J ₂ is equal to 2, the frequency response has a resonance / antiresonance alternation that varies continuously up to the second resonance mode as shown by the amplitude curve 101 of the Bode diagram. On the other hand, there are higher frequency resonance modes that are as dominant as some modes of the frequency domain of interest. For example, the amplitude of the fourth resonance 103 is almost as large as the amplitude of the first 104, which is confirmed by the very low value of criterion Ji, so that these high dynamic resonance modes can not be reasonably neglected in the synthesis of the reduced model, which would generate a phase of identification and controller synthesis infeasible given the very large number of dynamics that should be considered to achieve a specific model.

Finally, FIG. 11 illustrates the case of the structure D of FIG. 7d. It turns out that this structure can present an interesting compromise. The form of its frequency response is the expected one, both in amplitude as illustrated by the curve 1 1 1 of the Bode diagram, where there is a sharp decrease after the second resonance 113, as compared with the continuous alternation of resonances and antiresonances in the frequency domain of interest. This makes this structure a mechatronic system whose truncated model (limited to the first two resonances) will be precise, easy to identify and adapted to an easy synthesis of its regulator in view of the criterion J1, and whose stability of its subsequent loop control closed will be guaranteed in view of criterion J2. The structure D appears as the one which presents the best compromise between the mechanical performances, related to the criteria δ _x and F _x and the dynamic performances, with a view to the control, related to the criteria Ji and J ₂ . The preceding examples show that the criteria Ji and J ₂ make it possible to take into account at the design stage certain characteristics of the dynamic response of a system, notably affecting its subsequent control. More specifically, they make it possible to design flexible systems that have frequency characteristics that are conducive to the implementation of simple, simple control laws and / or dedicated to flexible systems.

Claims

1. A method of producing a flexible mechatronic system, characterized in that it comprises at least:

a step of modeling the system by a finite element mesh; a step of simulating the behavior of a terminal node (68) of the model in open-loop response to a control signal;

a step of characterizing said response by at least one mechanical criterion (δ _x , F _x ) and at least one numerical criterion J / representative of the relative amplitude of the resonance peaks (43, 44) of said response as a function of the frequency, these resonance peaks being respectively selected upstream and downstream of a predetermined mode number; the preceding steps being capable of being repeated, these steps being followed by a step of selecting a design obtained according to the criteria defined in the characterization step, the system being made from the selected design.

2. Method according to claim 1, characterized in that the criterion J / is defined by the following relation:

k _ ι = l

^J l ^~ , i = Σ σ k + l

where k is the number of the first modes to be dominant (1 13) with respect to all other modes of said frequency response and σ, defined from the state representation which completely characterizes the input / output relation of the mechanism , is Hankel's i-th Singular Value of the flexible system.

3. Method according to any one of the preceding claims, characterized in that the characterization step comprises another numerical criterion J ₂ ^k representative of the alternation of the resonances and antiresonances of the frequency domain response in a selected frequency band.

4. Method according to claim 3, characterized in that the criterion J ₂ ^k is defined by the following relation:

where k is the number of first resonance modes (93) of said frequency response and C ₁ and b _x respectively represent the influence of the sensors and actuators on the input / output frequency response of the system.

5. Method according to any one of the preceding claims, characterized in that an embodiment is selected when the criterion Ji ^k is greater than a value chosen by the user / designer.

6. Method according to any one of the preceding claims, characterized in that an embodiment is selected when the criterion J ₂ ^k is greater than a given value.

7. Method according to any one of the preceding claims, characterized in that the mesh is composed of a combination of elementary blocks (61, 62, 63, 64) to be determined, each authorized block being formed of a predefined assembly of segments representing elementary beams (3), said mesh comprising at least one active block (71) controllable by means of a control signal.

8. Method according to any one of claims 1 to 6, characterized in that the mesh being composed of a combination of elementary blocks (61, 62, 63, 64) to be determined, each authorized block being formed of an assembly predefined segments representing elementary beams (3), said mesh comprises at least one node controllable by means of a control signal.

9. Method according to any one of claims 7 or 8, characterized in that the blocks are from a library of predefined blocks.

10. Method according to any one of claims 7 to 9, characterized in that a control of an active block is exerted by a deformation signal of at least one of its beams.

11. Method according to any one of the preceding claims, characterized in that a static mechanical criterion is the displacement δ _x of the terminal node (68).

12. Method according to any one of the preceding claims, characterized in that the terminal node (68) being an effector, a mechanical criterion is the value of the force it applies to the external environment F _x .

13. Method according to any one of the preceding claims, characterized in that the control signal is a voltage or electrical current signal.

14. Method according to any one of the preceding claims, characterized in that the system is a piezoelectric actuator, all the beams constituting an active block being controlled by a voltage signal.