Shamim Ahmed
email: shamimshawon@yahoo.com

Christophe Reboud

Pierre-Emile Lhuillier

Pierre Calmon

Roberto Miorelli

An Adaptive Sampling Strategy for Quasi Real Time Crack Characterization on Eddy Current Testing Signals

Keywords: Eddy Current Testing, Real Time Inversion, Learning By Examples, Partial Least Squares, Output-Space-Filling, Support Vector Regression, Kernel Ridge Regression, Relevance Vector Regression, Augmented Radial Basis Function

This work describes quasi real time flaw(s) characterization in conductive plate(s) through the inversion of eddy current testing (ECT) signals using learning by examples (LBE) paradigm. Within the framework of LBE, a fast and accurate learning model is fitted on an optimal training set based on simulated eddy current testing data and the corresponding set of parameters during a preliminary offline phase. More specially, the optimal training set has been generated in the offline phase by adopting an adaptive sampling strategy through exploiting Partial Least Squares (PLS) feature extraction and output space filling (OSF). Subsequently, a non linear model is fitted on the training set and used to predict the set of parameters associated to an unknown (possibly large) test set during the so-called online phase. Different models, i.e., learning algorithms, such as Support Vector Regression (SVR), Kernel Ridge Regression (KRR), Relevance Vector Regression (RVR) and

Introduction

Nowadays, machine learning and statistical methods are becoming pervasive data analysis tools in many scientific fields ranging from signal and image processing, chemometrics, biophysics to biomedical imaging and many more. In the very recent years, non-destructive testing scientific community is starting also to profit of the most advanced machine learning methods in order to enhance the capability of already established procedures. That is, automatic classification and regression tasks can be performed in a very efficient way by enabling almost real-time feedback to on field operators during inspection campaigns. In this context, the coming machine learning tools could be applied in order to enhance the accuracy in inspections and mitigate, to some extents, decision of errors due to the so called human factor.

In this paper, we studied and implemented a machine learning methodology targeting regression tasks in view of crack characterization. Inverse problem solutions in electromagnetic non-destructive testing (E-NDT) can be solved in iterative [START_REF] Salucci | Multifrequency particle swarm optimization for enhanced multiresolution gpr microwave imaging[END_REF], [START_REF] Salucci | Gpr prospecting through an inversescattering frequency-hopping multifocusing approach[END_REF] and non-iterative [START_REF] Henriksson | Non-iterative music-type algorithm for eddy-currentusa nondestructive evaluation of metal plates[END_REF], [START_REF] Tamburrino | Monotonicity imaging method for flaw detection in aeronautical applications studies in applied[END_REF], [START_REF] Massa | Learningby-examples techniques as applied to electromagnetics[END_REF] approaches. Due to higher computational complexity of the iterative solutions for minimizing certain cost function related to the discrepancy between simulation and observation, non-iterative approaches are gaining more interest for real time inversion. Application of machine learning algorithms started to appear in E-NDT almost three decades ago. Authors in [START_REF] Udpa | Eddy current defect characterization using neural networks: Materials evaluation[END_REF] showed eddy current testing (ECT) signals interpretation dealing with classification problem. Longitudinal and transverse surface-breaking notches detection and characterization are shown in [START_REF] Rao | An artificial neural network for eddy current testing of austenitic stainless steel welds[END_REF] by using time-domain parameters as the functions of digitized in-phase and quadrature components of probe impedance. Reconstruction of stress map in the strained metallic plates by using electromagnetic measures have been exploited in [START_REF] Cacciola | Advanced use of soft computing and eddy current test to evaluate mechanical integrity of metallic plates[END_REF] through support vector regression (SVR). Combination of principal component analysis (PCA) feature extraction and k-mean algorithms is used [START_REF] Kim | Classification of pulsed eddy current gmr data on aircraft structures[END_REF] for cracks detection and classification by applying pulse eddy current testing. Defect localization, and classification tasks are explored in [START_REF] Chen | Study on defect classification in multi-layer structures based on fisher linear discriminate analysis by using pulsed eddy current technique[END_REF] by applying Fisher linear discriminant analysis (FLDA) feature extraction with support vector machine (SVM) by applying pulse eddy current testing. In [START_REF] Pasadas | 2d surface defect images applying tikhonov regularized inversion and ect[END_REF], Tikhonov regularization has been applied as an inversion algorithm for visualization of the geometric profile and size of the cracks through the determination of the 2D image of the eddy current density.

In this work, an adaptive sampling strategy has been addressed for ECT crack characterization problem in order to increase the prediction ability of machine learning algorithms (refers as learning algorithms hereafter) within the framework of Learning by Examples (LBE) [START_REF] Massa | Learningby-examples techniques as applied to electromagnetics[END_REF], [START_REF] Ahmed | Real-time flaw characterization through learning-by-examples techniques: A comparative study applied to ect[END_REF], [START_REF] Salucci | Real-time brain stroke detection through a learning-by-examples technique-an experimental assessment[END_REF] paradigm. LBE can be formulated into two phases approach. A fast and accurate model is built on a training set of input-output (I/O) pairs at the preliminary offline phase by regression/prediction techniques (i.e., learning algorithms). During the second phase (online phase), the developed model from offline phase is used to predict the output associated to an unknown test sample. The performance of different learning techniques for crack detection, localization and characterization depends on the particular choice of the training set generation. Especially, the choice of optimal features selection/extraction from ECT signal space and selection of the corresponding crack parameters from parameter space have impacts on inversion accuracy. Moreover, in case of very time consuming simulations/experiments, one needs to face limitations of available data (e.g., samples). Towards this end, this work describes an adaptive-sampling strategy which combines Partial Least Squares (PLS) [START_REF] Wold | Pls-regression: a basic tool of chemometrics[END_REF] feature extraction and modified version of output space filling (OSF) [START_REF] Bilicz | Kriging-based generation of optimal databases as forward and inverse surrogate models[END_REF]. This strategy aims to uniformly explore the extracted feature space that collects enough information during training set generation phase [START_REF] Salucci | An adaptive learning-by-examples strategy for efficient eddy current testing of conductive structures[END_REF], [START_REF] Salucci | Real-time ndt-nde through an innovative adaptive partial least squares svr inversion approach[END_REF], [START_REF] Ahmed | Real time groove characterization combining partial least squares and svr strategies: application to eddy current testing[END_REF].

Different learning algorithms such as Support Vector Regression (SVR) [START_REF] Vapnik | The Nature of Statistical Learning Theory[END_REF], Kernel Ridge Regression (KRR) [START_REF] Saunders | Ridge regression learning algorithm in dual variables[END_REF], Relevance Vector Regression (RVR) [START_REF] Tipping | Sparse bayesian learning and the relevance vector machine[END_REF] and Augmented Radial Basis Function (A-RBF) [START_REF] Sui | An efficient global optimization algorithm based on augmented radial basis function[END_REF] have been utilized for accurate training model creation and real time flaw characterization by the corresponding trained model. Rather than finding the optimal learning algorithm, the main objective and contribution of this work is to show the impact and robustness of the mentioned PLS-OSF adaptive sampling approach on inversion by applying different learning algorithms in presence of corrupted (additive synthetic noise) test sets as well as experimental data.

Therefore, initially the performance evaluation of the mentioned prediction approaches on the data set obtained by PLS-OSF sampling strategy will be explored in presence of noisy test set and compared with more standard sampling approach (i.e., full factorial grid (GRID) [START_REF] Ahmed | Real-time flaw characterization through learning-by-examples techniques: A comparative study applied to ect[END_REF]). Finally, a set of unknown samples consisting in real experimental data are utilized for evaluating the performance of crack characterization. The paper is organized as follows.

In Section 2, a brief summary on the mathematical formulation of the eddy current signal modelling is provided. Thus an exhaustive description of the adaptive sampling strategy and the addressed learning methods employed in the paper are described. In Section 3, training and test set configurations are provided for both numerical and experimental cases. Moreover, a comparative analysis of the treated LBE solutions on numerical and experimental data are detailed in Section 4. Concluding remarks and further research directions are outlined in Section 5.

Mathematical formulation

Problem definition

Let us consider a homogeneous conductive specimen inspected by two coils working in emitting-receiving mode as the one shown in Fig. 1. The presence discontinuities within the inspected medium (e.g., flaws, voids, inclusions, etc.) perturb the eddy current flow compared to the flawless specimen.

The differential signal between eddy currents in healthy and flawed medium represents the so called crack signature or ECT signal signature. Magnitude and phase of the ECT signals are changed in function of the crack parameters e.g., crack geometry, conductivity, position in the specimen, probe geome-try, specimen characteristics etc. Numerical modelling of ECT signals aims at simulating inspection problems in which all these quantities vary. For sake of simplicity we introduce the modelling based on the inspection of a plate structure affected by a three dimensional crack with rectangular section. The extension of the model described hereafter has been provided for planar multi-layered structures affected by multiple cracks elsewhere [START_REF] Miorelli | Efficient modeling of ect signals for realistic cracks in layered halfspace[END_REF].

Let us describe the presence of the crack as a local variation of conductivity described as δσ (r) = σ c (r) -σ, where σ stands for the plate conductivity and σ c (r) stands for the crack conductivity at the point r = (x, y) which represents a coordinate in a three dimension Cartesian space. The eddy currents are induced within the specimen by a driven coil and having the incident electric field within the specimen in absence of flaw(s) described by E inc (r). Considering the Volume Integral Method (VIM) formalism [START_REF] Lesselier | Eddy current scattering and inverse scattering, Green's integral and variational formulations[END_REF], one can show that the total field within the crack zone E tot (r) is obtained in terms of the following Fredholm equation of second kind

E tot (r) = E inc (r) + iωµ 0 V f G (r, r ) δσ (r ) E tot (r ) dV f . (1)
In equation (1), V f stands for the the volume occupied by the flaw and G (r, r )

is the electric-electric dyadic Green function which is employed to describe the effects of a electric dipole source located at r on the electric field at the position r. Equation ( 1) is solved by using the method of moments thus the unknown of the problem i.e., the equivalent electric dipole density in the crack zone d (r) = E tot (r ) δσ (r ), is retrieved. Consequently, the impedance variation on the receiving coil at the k-th scanning position with k = 1, ..., K, can be calculated via the reciprocity theorem as

∆Z k = 1 I 2 V f δσ (r k ) E inc (r k ) • E tot (r k ) dV f . (2)
In this paper, in order to describe the different crack configurations, we have chosen to define the volume V f with a set of Q = 3 descriptors defined through the vector p = (l c , h c , w c ) where the different components represent the crack length, height and width, respectively. Crack center and direction are assumed to be a priori for the described problem. However, in case of other problems (e.g., crack localization or crack characterization and localization), crack position parameters can be defined by considering them within vector p. For sake of brevity, hereafter we refer to equation (2) as ∆Z k = F r k , p where F stands for the employed black box forward solver.

In this work we have employed the CIVA software [START_REF] Sabatier | CIVA: Simulation and Analysis for NDT[END_REF] in order to carry out the simulation campaign. Subsequently, we propose an adaptive algorithm called PLS-OSF aiming at homogeneously fill the the projected feature space. We will show that this strategy leads to an uneven sample filling of the parameter space. The samples locations obtained by the adaptive algorithm are such that an higher emphasis is provided on the set of parameters "carrying" more information content i.e., the ones for which the outputs vary more.

Adaptive sampling through feature extraction

The sampling strategy apply, as a first step, the PLS feature extraction on the ECT signal space N Z to reduce the dimension of N Z . Subsequently, an adaptive sampling strategy is performed directly in the reduced extracted feature space N T in order to retrieve the lowest number of training samples during training phase. The following steps describe the iterative procedure of PLS-OSF sampling approach summed-up by the flow chart in Fig. 2. It is worth to be mentioned that this preliminary and mandatory stage, also known as offline phase, enable to build suitable I/O pairs for building optimal training model.

i Initialization-Generate N 0 number of initial samples by using a uniform GRID (i.e., full factorial grid) sampling [START_REF] Ahmed | Real-time flaw characterization through learning-by-examples techniques: A comparative study applied to ect[END_REF] within the parameter space N p . Thus, a matrix of defect parameters p = (p (n) ; n = 1, ..., N 0 ) having

(N 0 × Q) dimension is formed, where p (n) is the n-th row of p. By using F{.}, generate the associated ECT coil signals having (N 0 × 2K) feature matrix Z = Z (n) ; n = 1, ..., N 0 , where n-th signal is represented by Z (n) . The n-th signal vector of 2K ECT features in the N Z , is represented by Z (n) = Z (n) k ; Z (n) k ; k = 1, ..., K .
ii PLS Feature Extraction-In this step, the 2K-dimension ECT features associated to the ECT signal are reduced to J-dimension extracted features where J << 2K. Considering a data set composed by N 0 samples, we refer to the predictors as the set of whole ECT signals

Z * = Z , Z with Z * ∈ R N 0 ×2K
and to the responses as the set of p * parameters having size R N 0 ×Q . Z * and p * are the zero mean versions of Z and p, respectively. That is, each of the columns of Z and p has been centered by subtracting to the mean value of that column. The underlying concept in PLS algorithm assume that predictors and responses can be properly described by a linear combination of a subset of J variables. These variables are also known as latent variables or extracted features which maximize the covariance between the predictors and responses. PLS decomposes the predictors Z * and responses p * such that [START_REF] Wold | Pls-regression: a basic tool of chemometrics[END_REF]:

Z * = TP + E p * = TC + F , (3)
where P ∈ R 2K×J is the predictors loading matrix and E ∈ R N 0 ×2K is the matrices of residuals associated to the predictors. C ∈ R Q×J and F ∈ R N 0 ×Q the response loading and the matrices of residuals associated responses, respectively. Finally, T ∈ R N 0 ×J is the score matrix and superscript " " stands for transpose operation. One can show that the score matrix can be defined as T = Z * × W with W ∈ R N 0 ×J is the so-called weights matrix and it is calculated such that each column of T is orthogonal to each other. The columns of T represent the latent variables associated to a given ECT signal. T is obtained by multiplying Z * with W that maximizes the covariance between Z * and p * . In this sense, T can be seen as the projection of Z * signal having 2K-dimensions onto a J-dimensional space. In this paper the weights have been calculated by using SIMPLS algorithm [START_REF] Jong | Simpls: An alternative approach to partial least squares regression[END_REF] as detailed in Annex 1. Assign the number of training sample N iterative = N 0 and construct an initial training set

DN iterative = T (n) ; p (n) ; n = 1, ..., N iterative for the adaptive step.
iii Adaptive Sampling-Generate V candidate samples within the parameter space (i.e., N p ) by p (v) cand = p (v) cand,q ; q = 1, ..., Q through Latin Hypercube Sampling (LHS) strategy where v = 1, ..., V . An estimation of the J-dimensional set of extracted features corresponding to each v-th candidate, T(v) cand is retrieved by applying a multi-dimensional lin-ear interpolator on DN iterative . Select the optimal v = v opt candidate (i.e., p

cand ) from V such that the minimum distance between the obtained extracted features T(vopt) cand and all the available extracted features

T (n) (n = 1, ..., N iterative ) within DN iterative is maximized [i.e., v opt = arg (max v=1,..., V {min n=1,..., N [d vn ]})]. d vn is the Euclidean distance be- tween T(v) cand (v = 1, ..., V ) and T (n) (n = 1, ..., N iterative ), which can be described by d vn = J j=1 T (v) cand,j -T (n) j 2
. The ECT features ; n = 1, ..., N on the generated training set for each q-th parameter (q = 1, ..., Q) of the crack.

Z (vopt) cand = Z (vopt) cand,k ; Z (vopt) cand,k ; k = 1, ...,
It is worth to mention that the test set containing 2K ECT features associ-ated to previously-unseen crack parameter configurations are also projected to the J-dimensional PLS-extracted feature space. Thus, the m-th unknown test sample Z (m) having 2K ECT features for the defect parameters p (m) is projected to

N T through W [i.e., T (m) = Z (m) * × W]. Z (m) *
is obtained by using the mean values obtained from the already described PLS Feature Extraction step (i.e., from offline phase). Finally, T (m) is given as input to the corresponding trained model in order to estimate the q-th parameter of the crack, p(m) q (q = 1, ..., Q).

Learning algorithms

Given a data set DN,q = T (n) ; p

(n) q
through adaptive sampling detailed in the previous section (Subsec. iii), the aim of the learning algorithm is to find a function Θq {.} that estimates the relationship between T (n) and p

(n) q
and provides a predicted/estimated output which is close to

p (n) q
for all n. The obtained Θq {.} is then used to estimate the q-th crack parameter, p(m) q = Θq (T (m) ) for an unknown m-th test sample T (m) where, q = 1, ..., Q; m = 1, ..., M . This section briefly describes the formulation of different learning algorithms that are treated within the scope of LBE.

Augmented Radial Basis Function (A-RBF)

A-RBF is the extended version of actual Radial Basis Function (RBF)

interpolator, where a polynomial term is added with the standard RBF interpolator. The prediction of the p q -th crack parameter by applying A-RBF can be formulated by p(m)

q = S s=1 z s (T (n) )ρ s + N n=1 κ(T (n) , T (m) )ω (n) q . ( 4
)
where z s (T (n) ) = 1, T n 1 , ..., T n J , (T n 1 ) 2 , (T n 1 T n 2 ) , ..., (T n J ) 2 , ... defines the polynomial function with polynomial coefficient ρ s . ω (n) q is the RBF weight for the n-th sample that is obtained during training model development. κ is the kernel function whose value depends on the radial distance between T (n) and T (m) . Different types of kernel function such as linear, Gaussian, thin plate spline, multi-quadratic etc. can be used as a kernel function. To be consistent with all other learning methods, we have applied Gaussian kernel that can be expressed by

κ(T (n) , T (m) ) = exp -γ q T (n) -T (m) 2 (5)
where γ q is the Gaussian coefficient that needs to be calibrated to obtain optimal training model in offline phase.

Support Vector Regression (SVR)

Analogously to [START_REF] Ahmed | Real-time flaw characterization through learning-by-examples techniques: A comparative study applied to ect[END_REF][START_REF] Caorsi | A comparative study of nn and svm-based electromagnetic inverse scattering approaches to on-line detection of buried objects[END_REF], by employing ε-SVR, prediction of p q -th parameter for the m-th unknown test sample can be derived by p(m)

q = N n=1 (ς (n) q -ι (n) q )κ(T (n) , T (m) ) + b q . (6)
where ς

(n) q , ι (n) q
∈ (0, C q ) are Lagrange multipliers that are determined by solving dual optimization problem [START_REF] Caorsi | A comparative study of nn and svm-based electromagnetic inverse scattering approaches to on-line detection of buried objects[END_REF]. C q is the user defined constant parameter that controls error larger than ε q for N number of training samples.

b q is the bias coefficient that is computed by fulfilling Karush-Kuhn-Tucker (KKT) conditions [START_REF] Cortes | Support-vector networks[END_REF]. Gaussian kernel κ has been utilized which is controlled by Gaussian coefficient γ q . Thus, the learning ability of SVR depends on the appropriate value of ε q , C q and γ q .

Kernel Ridge Regression (KRR)

Kernel ridge regression combines the well known ridge regression with kernel trick [START_REF] Saunders | Ridge regression learning algorithm in dual variables[END_REF]. Through applying KRR, the prediction of p q -th parameter for the m-th unknown test sample can be obtained by p(m)

q = p q κ T (n) , T (o) + a q I -1 κ T (n) , T (m) (7)
where n = 1, ..., N ; o = 1, ..., N . a q is the fixed positive constant that is also known as regularization parameter. For using Gaussian kernel κ, accurate KRR model development is depended on the appropriate choice of Gaussian coefficient γ q and user defined parameter a q .

Relevance Vector Regression (RVR)

Relevance vector regression adopts Bayesian probabilistic framework for developing regression model. Having the model parameters, for an unknown m-th test sample, the p q -th parameter can be obtained through the posterior predictive density [START_REF] Tipping | Sparse bayesian learning and the relevance vector machine[END_REF] p p(m)

q | p q , α M L , β M L = N p(m) q | y (m) q , (σ (m) ) 2 (8)
where N specifies a Gaussian distribution over p(m)

q with y (m) q = µ q ϕ T (m)
is the posterior predictive mean that is obtained by the basis vector weighted by a sparse vector of mean weights µ q . α M L and β M L define a vector of hyperparameters and the precision on the noise, respectively that are computed by

maximum likelihood estimation. (σ (m) ) 2 = β -1 M L + ϕ T (m) Σϕ T (m)
is the variance on the predictions. The posterior co-variance matrix is denoted by Σ = βφ φ + A -1 with [N ×N +1] design matrix φ = ϕ T (1) , ..., ϕ T (N ) .

A = diag(α 0 , α 1 , ..., α N ) is the precision vector on the weight vector w. The basis function vector ϕ T (m) = 1, κ T (m) , T (1) , ..., κ T (m) , T (N ) is defined by the input observations through kernel function κ. Due to the use of Gaussian kernel, the accurate training model depends on the choice of Gaussian coefficient γ q which must be properly selected.

Training and test set configuration

This section describes the numerical and experimental configurations that are addressed in this paper. We consider a homogeneous plate of thickness 1.27 mm and conductivity 1.02 MS/m affected by a single sub-surface crack (crack is placed opposite side of the inspecting coil probe) as shown in Fig. 1. The plate is inspected by 2 coils in emitting and receiving mode working at a frequency of f = 200 kHz. More details on coil parameters can be found in [START_REF] Mistral | Array probe implementation (smx) on edf steam generator tubes[END_REF]. Q = 3 crack parameters (length l c , height h c , and width w c ) will be investigated for the rest of the paper. CIVA simulator has been utilized as forward operator F{.} for obtaining ECT signals. Real life inspections suffer for various types of noise sources (e.g., set up noise, man-made noise etc.).

Therefore, before dealing with real experimental data, it is a good practice to validate the trained models on noisy test set. To partially consider noise effects, the test set response signal collected by the coil is blurred by an Additive White Gaussian Noise (AWGN) with a signal-to-noise ratio (SN R),

where SN R = 10 log 10 {

Σ K k=1 | Z k | 2 Σ K k=1 |ζ k | 2 }.
ζ k denotes the complex valued additive noise which corrupts the ECT signal Z k at k-th measurement point. The performance of the inversion scheme is analyzed by computing the normalized mean error,

N M E q = 1 M M m=1 p (m) q -pq (m) /p (m) q
and relative prediction error, ξ

(m) q = p (m) q -pq (m) /p (m) q
for numerical and experimental evalua-tion, respectively. p (m) q and p(m) q are the actual and predicted q-th parameter

Parameters selection

Dealing with feature extraction and learning algorithms, it is essential and fare to use their own parameter in their optimal state. Our main goal is to show comparative analysis instead of showing superiority of each learning algorithm by utilizing as much as lower number of tuning parameters. As a consequence, the choice of optimal number of features (i.e., J) to be extracted from actual ECT signal space is a crucial issue. There is no particular criteria available for selecting optimal J value. Nevertheless, it is well known in machine learning framework that the lower the number of features to be learned the more robust learning model can be obtained. That is, the number of extracted features should be kept as low as possible to minimize the complexity of the learning algorithm in training phase. However, it should be large enough to ensure that all the essential information from ECT data is compressed within the extracted features. Within this guideline, it is worth pointing out that the PLS technique ranks the features from the most to the less important ones accordingly to the corresponding amount of information on the unknown quantity to be predicted [START_REF] Wold | Pls-regression: a basic tool of chemometrics[END_REF]. So, an effective and easy-toimplement strategy for choosing the optimal value of J is to build a training set with a maximum number of extracted features J max . Then, iteratively removing the higher-order (i.e., less informative) ones such that an optimum error is found. Figure 4 shows the behavior of the error index for predicting each crack parameter for J ∈ [2, 20] with a step size 1, while processing noisy test data (i,e., SN R = 10 [dB]) for applying P LS as a regression technique [START_REF] Wold | Pls-regression: a basic tool of chemometrics[END_REF]. By averaging the optimal values obtained for each defect parameter, J opt = 5 has been chosen.

Using ε-SVR from LibSVM [START_REF] Chang | LIBSVM: A library for support vector machines[END_REF], ε is fixed for 10 -2 . Whereas, Dlib [START_REF] King | Dlib-ml: A machine learning toolkit[END_REF] has been adopted for KRR regression. For initialization, α = 10 12 has been assigned as par the default value while using the RVR library described in [START_REF] Thayananthan | Pose estimation and tracking using multivariate regression[END_REF], which is updated during training model development. All learning techniques are exploited by Gaussian kernel. Thus, the choice of the kernel hyper-parameter γ q for all the learning algorithms and C q for SVR are obtained by performing 5-folds cross-validation during offline phase. Table 1 reports the optimal parameters obtained for different learning techniques for the training sets at N = 512 obtained by both PLS-OSF and GRID sampling strategies.

Experimental set up

A homogeneous conductive plate having the same properties described in Sec. 3 is affected by an external crack for 6 different crack configurations (Fig. 5). The actual dimensions of these cracks are indicated in Tab. (e) (f )

Results and discussion

Numerical assessment

As a reference and the further discussion, let us first analyze the prediction accuracy of the learning algorithms by applying GRID sampling strategy. In case of higher number of ECT features (i.e., F = 2K) while dealing with GRID sampling, higher N is required for solving COD problem. Hence, prediction accuracy improves significantly by each learning algorithm with we can also remark that all prediction techniques take lower prediction time, ranging from 0.03s -0.16s. This also confirms that all predictors are suitable for quasi real time inversion.

(a) (b) (c) (d) (e) (d)
(a)

(b) (c) (d) (e) (f ) (g) (h) (i) (j) (k) (l)

Experimental assessment

Referring to Tab. 2, Fig. 10 describes the experimental assessment of crack parameters (l c , h c and w c ) estimation through different learning techniques using PLS-OSF sampling strategy by means of relative error, ξ q (q = 1, ..., Q). Among all the prediction techniques, SVR shows highest prediction accuracy for predicting crack 4 (largest crack with l c = 12 mm) length and height in terms of lowest error with relative error ξ wc(P LS-OSF/RV R) ≈ 11%), similar to numerical evaluation, other learning algorithms have shown higher prediction error for w c estimation. Let's consider the prediction performance on the smallest crack (i.e., crack 1). KRR and RVR depict higher prediction accuracy with ξ hc(P LS-OSF/RV R) ≈ 9%). The training models have also been treated on some corrupted experiential test samples (e.g., crack 2, 5, 6). Considering the most corrupted crack (i.e., crack 5) shown in Fig. 5, we can observe that almost all learning methods can predict crack height for relative error ≈ 10% (Fig. 10 (e)). However, due to corrupted signal, crack length estimation has been highly affected. In this case, we can also observe that SVR is showing lowest error (ξ

Conclusion

In PR;

The j-th column of W denoted by W j = (W jk ; k = 1, ..., 2K), is set equal to the first left singular vector, where W = W j ; j = 1, ..., J .

The j-th Z-score denoted by T j = Z * W j is computed while, T = T j ; j = 1, ..., J , T j = (T nj ; n = 1, ..., N ) and W j = (W jk ; k = 1, ..., 2K).

Compute the j-th row of P such that P j = Z * T j / T j T j

Figure 1 :

 1 Figure 1: Example of the studied plate geometry.

Figure 2 :

 2 Figure 2: Flow chart representation of PLS-OSF sampling.

Figure 3

 3 Figure 3 represents the exploration of adaptive training samples generation in parameter space as well as in extracted feature space. In Fig. 3, feature space is represented for the first two extracted features (out of five) based on the obtained training set. At this stage, different predictions techniques have been utilized to train separately q-th set of I/O pairs DN,q = T (n) ; p (n) q

(q = 1 ,

 1 ..., Q) of the m-th (m = 1, ..., M ) unknown test sample respectively.

Figure 3 : 3 . 1 .

 331 Figure 3: Disposition of training samples obtained by PLS-OSF in (a)(c)(e) crack parameter space and (b)(d)(f ) extracted feature space for (a)(b) N = 27, (c)(d) N = 125, (e)(f ) N = 512.

Figure 4 :

 4 Figure 4: Prediction error, N M E q vs. number of extracted features J representation while performing inversion on the test set at SN R = 10 dB using PLS regression technique for N = 512 and M = 1000 obtained by PLS-OSF sampling strategy.

Parameters C opt γ opt γ opt γ opt γ opt l c 10 Parameters C opt γ opt γ opt γ opt γ opt l c 10 4 10 - 2 10 -2 10 -2 10 -5 h c 10 4 10 - 3 10 -2 10 -3 10 -4 w c 10 4

 1010210103104 10 -2 10 -1 10 -5 10 -4 Training models are obtained separately by different learning algorithms on the obtained training sets with the corresponding calibration parameters in offline phase. Finally, the estimation of crack parameters on the unknown test set is evaluated (online phase) by the obtained trained models.

  2. All other inspection parameters remain unchanged. ECT response signals are collected from K = 644 inspected points. As a consistency check, Fig. 6 compares the real and the imaginary parts of experimental ECT signals with the CIVA simulated signals along the x-axis of the cracks. For proper adjustment with the simulated data, a calibration factor [ = -0.0049; = 0.0135]and bias factor [ = 0.004; = 0.005] have been applied on the experimental data. This is worthy to mention that the discrepancy between experimental and simulated data may encounter due to the presence of uncertainties between normal and real parameters (e.g., probe lift-off, probe tilt/pitch, plate thickness etc.) employed in both cases. Moreover, non negligible noise components can also corrupt the realistic measurement. Fig. 5 (b) and Fig. 5 (e) contain the signal maps of two cracks which are partially corrupted during real experiment. Similar to the numerical test set, J opt = 5 optimal features are extracted for each test sample (i.e., ECT signal map for each crack) for applying PLS-OSF sampling. The obtained training models for N = 512 samples by PLS-OSF sampling with the corresponding optimal parameters (Tab. 1) of the learning algorithms are applied for predicting the experimental cracks parameters.

Figure 5 :

 5 Figure 5: Referring to Table 2, ECT signal maps of (a) Crack 1, (b) Crack 2, (c) Crack 3, (d) Crack 4, (e) Crack 5 and (f ) Crack 6.

Figure 6 :

 6 Figure 6: Referring to Tab. 2, real and imaginary parts of the coil voltage variation experimental/CIVA simulated for the coil position along x direction at y = 15 [mm].

Figure 7 :

 7 Figure 7: Prediction error, NME versus training size N representation at (a)(b) p 1 = l c , (c)(d) p 2 = h c and (e)(f ) p 3 = w c when processing noiseless and noisy test set (SN R = 20 [dB]) by GRID and PLS-OSF.

Figure 8 :

 8 Figure 8: Prediction error, NME versus SN R representation at (a) p 1 = l c , (b) p 2 = h c and (c) p 3 = w c when processing noiseless and noisy test set by GRID and PLS-OSF.

Figure 9 :

 9 Figure 9: Actual versus predicted values of p 1 = l c at (a)(d)(g)(j), p 2 = h c at (b)(e)(h)(k), and p 3 = w c at (c)(f )(i)(l) when processing noisy test set (SN R = 20 [dB]) for training size N = 512 through (a) -(c) PLS-OSF/SVR, (d) -(f ) PLS-OSF/KRR, (g) -(i) PLS-OSF/RVR and (j) -(l) PLS-OSF/A-RBF.

( 4 )( 4 )

 44 lc(P LS-OSF/SV R) ≈ 0.7% and ξ hc(P LS-OSF/SV R) ≈ 1%, respectively where, RVR also shows lower error for ξ (4) lc(P LS-OSF/RV R) ≈ 4% and ξ

( 4 )

 4 hc(P LS-OSF/RV R) ≈ 5%. KRR and A-RBF exhibits reasonable prediction accuracy on l c and h c estimation on the same crack. Though RVR shows highest prediction accuracy (ξ[START_REF] Tamburrino | Monotonicity imaging method for flaw detection in aeronautical applications studies in applied[END_REF]

( 1 )

 1 lc(P LS-OSF/KRR) ≈ 1% and ξ (1) hc(P LS-OSF/RV R) ≈ 6% respectively. Having regularization capability, SVR also shows lower prediction error for both l c and h c estimation (ξ (1) lc(P LS-OSF/SV R) ≈ 10% and ξ (1)

( 5 )

 5 lc(P LS-OSF/SV R) ≈ 27%) for crack length estimation. Similarly, SVR outperforms for l c estimation on crack 2 and 6 with estimation error ξ (2) lc(P LS-OSF/SV R) ≈ 16% and ξ (6) lc(P LS-OSF/SV R) ≈ 2%. Whereas, KRR and RVR have lowest estimation error on h c estimation for crack 2 and crack 6, respectively (ξ (2) hc(P LS-OSF/KRR) ≈ 11%, ξ

( 6 )

 6 hc(P LS-OSF/RV R) ≈ 8%). Being an interpolator A-RBF fails to predict crack length of all the experimental crack which are effected by noise. Finally, it is quite visible that, through obtaining optimal training set by PLS-OSF sampling, regression methods (i.e., SVR, KRR, RVR) are able to deal with noisy experimental data for crack characterization. Sparser, regression models (SVR, RVR) are preferable for CPU efficiency. Nevertheless, we can refer that SVR has promising inversion accuracy of experimental data in presence of noise, while RVR shows better accuracy for smaller crack.

Figure 10 :

 10 Figure 10: Referring to Tab. 2, relative error, ξ on crack parameters estimation of (a) Crack 1, (b) Crack 2, (c) Crack 3, (d) Crack 4, (e) Crack 5 and (f ) Crack 6 for N = 512, F = 1288 J = 5 processed by PLS-OSF sampling through A-RBF, RVR, KRR and SVR algorithms.

  this paper, we have shown an adaptive sampling strategy, called PLS-OSF, within the framework of LBE for enhancing the learning abilities of different learning algorithms (e.g., SVR, KRR, RVR and A-RBF). A test case, consisting a single crack in a conductive plate has been treated for solving crack characterization problem by means of inversion. Comparative analysis of learning algorithms for performing inversion has been numerically performed on the training set obtained by PLS-OSF sampling and compared with more standard GRID sampling approach. Quasi-real time inversion has also been demonstrated by predicting M = 1000 test samples within 0.03s to 0.16s during online phase on a standard laptop. By applying PLS-OSF sampling strategy, an optimal training set can be retrieved which implies on higher prediction accuracy in presence of noisy data. Moreover, PLS-OSF sampling strategy increases the learning capability of all prediction methods. This strategy also demonstrates higher prediction accuracy for crack dimension estimation on real experimental data. Due to the generalization capability SVR, RVR and KRR exhibit higher prediction accuracy during experimental assessment. Other linear feature extraction techniques such as principle component analysis, linear discriminant analysis, canonical correlation analysis and non linear extensions based on the use of kernels can be straightforwardly applied within the framework of LBE. As different learning algorithms can be useful for estimating different crack parameters by means of optimal prediction accuracy, multi-regression by combining the treated regression techniques, i.e. the framework known under the name of ensemble learning in the machine learning community, may improve the overall prediction accuracy during crack characterization. In authors' opinion, the presented LBE strategy can also be extended to other NdT methods such as ultrasounds testing (UT) and infra-red thermography testing. In particular, due to the characteristics of the treated data, it would be interesting to apply PLS-OSF on the UT signals issued by post processing (i.e. scattering matrix), collected by ultrasonic arrays[START_REF] Bai | Characterization of defects using ultrasonic arrays: a dynamic classifier approach[END_REF].AcknowledgmentThis work has been partially supported by the SIRENA project (2014-2017) funded by DIGITEO (France) under the Call for Chairs 2014, the French project ANR-ByPASS. Annex 1: SIMPLS Algorithm 1. A matrix R = ( Z * ) p * is computed. 2. Run a for loop over J-For j = 1, ..., J Apply the Singular Value Decomposition to R. If j > 1, R = R -P P P -1

Table 1 :

 1 Optimal setup for the calibration parameters

Table 2 :

 2 Crack dimension in experimental test set

	Crack No. 1	2	3	4	5	6
	l c [mm]	6	7	8	12	9	9
	h c [mm] 0.508 0.508 0.508 0.508 0.254 0.381
	w c [mm] 0.2 0.2 0.2 0.2 0.2 0.2

Preprint submitted to NDT&E International December 5, 2023

racy has shown higher prediction accuracy (Fig. 8) for PLS-OSF sampling compare to GRID sampling by applying all the learning algorithms.


Member, IEEE Micky Rakotondrabe

Joël Agnus

Kanty Rabenorosoa

Member, IEEE Nicolas Chaillet

Characterization, modeling and robust control of a nonlinear 2-dof piezocantilever for micromanipulation/microassembly

come

Characterization, modeling and robust control of a nonlinear 2-dof piezocantilever for micromanipulation / microassembly.

I. Introduction

Face to the need of small products that have several functions (sensing, actuation, ...) and natures (mechanical, electrical, optical, ...), micromanipulation/microassembly tasks become more and more studied and used in laboratories nowadays. To perform such tasks, systems with high performances should indeniably be utilized. In most of cases, these systems are based on active materials because hinges based elements derive frictions and then imprecisions. Piezoelectric materials are among the most utilized materials. While they offer both high resolution and low response time, they can be used for sensing and for actuation. One of their actuation applications is the piezoelectric microgripper [START_REF] Haddab | A microgripper using smart piezoelectric actuators[END_REF][2][3] [START_REF] Agnus | Overview of microgrippers and design of a micro-manipulation station based on a MMOC microgripper[END_REF].

A piezogripper (piezoelectric microgripper) is composed of two piezoecantilevers (piezoelectric cantilevers) working in deflection (Fig. 1). The first piezocantilever is used to position the manipulated micro-object whilst FEMTO-st Institute, UMR CNRS-6174 / UFC / ENSMM / UTBM Automatic Control and Micro-Mechatronic Systems department (AS2M department) 25000 Besançon -France corresponding author: mrakoton@femto-st.fr the second one is used to control the force, both along the y-axis [START_REF] Rakotondrabe | Modelling and robust position/force control of a piezoelectric microgripper[END_REF]. On the one hand, despite the difficulty to measure or estimate the force, the force control in a piezocatilever is being emerging [START_REF] Rakotondrabe | Modelling and H∞ force control of a nonlinear piezoelectric cantilever[END_REF]. On the other hand, the deflection control of a piezocantilever is nowadays overcome, [START_REF] Devasia | A survey of control issues in nanopositioning[END_REF] gives a survey of several applied techniques. If the development and control of 1-dof piezocantilevers based piezogrippers are enough advanced, the tasks that they can performed are limited to either rotation or translation but not the combination of both. Hence, Agnus et al. developed a two-fingered piezoelectric microgripper (Fig. 2) [START_REF] Lit | A four-degreeof-freedom microprehensile microrobot on chip[END_REF], each finger being able to move independently from the other in two orthogonal directions: in plane and out of plane deflections. A finger is made up of a 2-dof piezocantilever and eventually a tool. Here, in plane deflection is defined as the deflection in the same plane than a piezocantilever electrodes. This microgripper has therefore four articular dof able to grip, hold and release submillimetric-sized objects in the y-axis. The fingers can also move up and down (zaxis) for fine motions like micro-insertion tasks. Finally, they are able to orientate (around x-axis) micro-objects by combining opposite out-of-plane motions of the fingers. The microgripper characteristics are very useful in micromanipulation/microassembly and have proved its utilities in micromanipulation of watch screws or in microspectrometers alignment tasks [START_REF] Pérez | Modeling, fabrication and validation of a high-performance 2dof piezactuator for micromanipulation[END_REF]. Notwithstanding, when applying a reference deflection along y-axis (resp. z-axis), a residual deflection is obtained in the z-axis (resp. y-axis). This is due to mechanical imperfection of the structure and particularly to the misalignment of the electrodes. This coupling makes very inaccurate the system because unwanted movements appear during micromanipulation/microassembly tasks. In addition, hysteresis and creep phenomena become non-negligible when the applied electrical field is large. The applied voltage should effectively be relatively enough in order to cover the required deflection ranges. This paper presents the deflection control of the 2-dof piezocantilever in taking into account the axis coupling and the nonlinearities. A robust H ∞ controller is used because model uncertainty due to the hysteresis must be taken into account and specifications relative to performances required for tasks should be ensured. The paper is organized as follows. First, the 2-dof piezocantilever is presented. After that, we present its characteristics, notably the hysteresis and coupling parts. Then, the modeling and robust control follow the section. Finally, experimental results end the paper.

II. Presentation of the 2-dof piezocantilever

The functioning principle of one finger of the microgripper is based on a piezoelectric cantilever with local electrodes, called 'duo-bimorph', for which a static modelling has been established [START_REF] Agnus | Description and performances of a four-degrees-of-freedom pizoelectrique microgripperp[END_REF]. It can be assimilated to a cantilever beam clamped at one end. The two dof are obtained with a judicious application of voltages on the piezoelectric plates. Fig. 3 pictures the functioning of the actuator with cross section views. The structure at rest is presented with solid area and the deformed actuator with dashed lines. P indicates the polarisation of the piezoelectric material (Fig. 3-a). In the first configuration (Fig. 3-b), the four electrodes are set to the same voltage V Z > 0. The upper layer stretches along x while the lower layer contracts, which leads to a flexion along -z, as the layers are glued one to each other. In the second situation (Fig. 3-c), the voltages on two adjacent and two opposite electrodes are V Y and -V Y (V Y > 0); the electric fields are in the same direction as the polarisation in the left half-part of the beam and in the opposite direction in the right half-part of the same beam. Thus, the left part stretches while the right part contracts, which leads to a deflexion along y. Finally, in the last configuration (Fig. 3-d), the electrodes are set at voltages V Z +V Y and V Z -V Y , which yields a deflexion in y and z directions. The four articular dof (two per finger) of the gripper are useful to perform gripping, insertion or rotation tasks. They also permit to correct a possible misalignment of the tools. In this paper, we are interested by the modeling and control of one finger (piezocantilever).

III. Open-loop analysis

In this section, we characterize the piezocantilever. The experimental setup is based on:

• one 2-dof piezocantilever,

• two optical sensors to measure the deflections of the two axis (Fig. 4-a). Each sensor (Keyence-2420) has 10nm of resolution, • a computer-DSpace hardware and the Matlab-Simulink c software for the data-acquisition and control, • and a voltage amplifier. The piezocantilever can be considered as a MIMO system with two-inputs and two ouptuts (Fig. 4-b). In the sequel we analyze the nonlinearities (hysteresis and creep) and the vibration of the system. It has been shown that for piezocantilevers, the hysteresis only affect the static part while the creep can be considered as an additional disturbance [START_REF] Rakotondrabe | Quadrilateral modelling and robust control of a nonlinear piezoelectric cantilever[END_REF]. Moreover the hysteresis, the vibration during the transient part and the creep can be separately analyzed and identified. The vibration and the creep will be characterized using a step input. The former is the transient part of the response while the latter is the drift that appears after the end of the transient part. The hysteresis will be characterized using a sine input voltage.

The frequency of the sine input is chosen such as it is not too low (to avoid the creep effect) and not too high (to avoid the dynamic part effect).

A. Dynamic analysis and vibration compensation

First, we analyze the transient part of the system. The analysis is performed as follows. A step voltage V y = 20V is first input while V z = 0. The transient part of the obtained deflection Y is plotted. At the same time, its effect on the deflection Z is also plotted. After that, a step voltage V z = 20V is applied and V y is reset to zero. Thus, the direct deflection Z and the resulting coupling Y are plotted. The four results are pictured in the Fig. 5.

Fig. 5-a and Fig. 5-d respectively indicate that the transient parts of the direct transfers V y → Y and V z → Z are very oscillating. In micromanipulation/microassembly tasks, vibrations are unwanted. In fact, a microgripper with oscillating piezocantilevers may destroy a breakable manipulated micro-object if the overshots are high. Elsewhere, the coupling effect V z → Y is also oscillating (Fig. 5-b). Despite that, the coupling effect V y → Z is low (less than 3µm) (Fig. 5-c). In closed-loop technique, it is very difficult to synthesize controllers for oscillating systems if the specified settlingtime is largely low relative to the settling-time of the vibration. So, before synthesizing a closed-loop controller, we propose to attenuate the vibration. We choose the input-shaping Zero-Vibration technique [START_REF] Singer | Shaping command inputs to minimize unwanted dynamics[END_REF] because of its ease of implementation. Because of the limited length of the paper, this part will not be detailed in this paper. However, the readers can see more detail on its computation and application for one dof piezocantilever in our previous work [START_REF] Rakotondrabe | Hysteresis and vibration compensation in a nonlinear unimorph piezocantilever[END_REF]. In this paper, to compensate the vibration, we do not consider the coupling between the axis:

• one compensator is used to compensate the vibration of the transfer V y → Y . The compensator computation is based on the step response pictured in Fig. 5-a, • another compensator is used for the transfer V z → Z. It is computed using the Fig. 5-d. Fig. 6 presents the bloc-scheme of the piezocantilever with the vibration compensators. In the figure, V y and V z are the input voltages while V y and V z are the new inputs of the piezocantilever. The experimental results are interesting and are pictured in the Fig. 7. When a step input V y = 20V is applied, the obtained deflection Y is henceforth without vibration (Fig. 7-a)). Similarly, when a step input V z = 20V is applied the initial vibration of the deflection Z is attenuated (Fig. 7-d). Finally, the vibration of the coupling transfer has been attenuated (see Fig. 7-b and-c respectively).

B. Static analysis

Here, we analyze the static characteristics, i.e. the behaviors in the voltages-deflections plane, of the piezocantilever. For that, we use sine input voltages. For the experimented piezocantilever, various experiments lead us to choose a frequency of 0.5Hz. The results are pictured in Fig. 8.

Fig. 8-a shows the deflection Y vs a sine applied voltage V y and for three constant values of V z . We first remark that the direct transfer V y → Y is hysteretic. Secondly, it appears that this transfer is influenced by the constant input V z . Indeed, not only the hysteresis Transient part characteristics when using vibrations compensation: experimental results.

obtained with V z = 20V is slightly angled relative to others but the three curves are shifted each other. To well understand the coupling and more evaluate the shift, we apply a sine voltage V z and plot the resulting deflection Y (Fig. 8-b). It appears that the coupling is nonlinear and the effect of V z on Y ranges between ±5µm for V y = 0.

Similarly to the previous experiments, Fig. 8-d shows the deflection Z vs a sine applied voltage V z and for three constant values of V y . Like V y → Y , we also remark that the direct transfer V z → Z is hysteretic. However, it seems that whatever an applied constant V y is, the hysteresis curves are neither shifted nor angled themselves: the coupling seems negligible. To more analyze that, we apply a sine voltage V y and check its influence on the deflection Z. The results are plotted in the Fig. 8-c and clearly indicate that the coupling is negligible. Indeed, from the curves, we a priori have a low value of ∂Z ∂Vy .

C. Creep analysis

Now let us analyze the creep phenomenon that may behave the piezocantilever. The creep is defined as the slow drift that appears after the transient part when a step voltage is applied. The amplitude of the creep depends on the amplitude of the input. In general, the transient part of piezocantilevers has a settling-time less than 100ms whilst the creep has more than 3min.

The experiments are performed with a step V y = 20V while V z = 0V and then V y = 0V while V z = 20V . To well observe the drift, the duration of the acquisition data is 600s. The results lead to the following remarks.

• Fig. 9-a: the creep in the transfer V y → Y is considerable: more than 2.5µm (i.e. 27% = 2.5 9 , where 9µm is the final value before creep).

• Fig. 9-d: despite the noise, the creep in the transfer V z → Z is also considerable: more than 10µm (i.e. 40% = 10 25 ).

IV. Modeling

In order to synthesize a controller, we give a model of the system in this section. The considered system is now the 2-dof piezocantilever including the vibration compensators as shown in Fig. 6.

Consider the following multivariable nonlinear model:

Y Z = Γ (V y , V z , s) ⇔ Y = Γ y (V y , V z , s) Z = Γ z (V y , V z , s) (1)
where Γ i (V y , V z , s) represents a nonlinear operator dependent at the same time on the input V y and V z . The reliance of Γ i on the Laplace variable s means that it also includes dynamic parts. According to [START_REF] Rakotondrabe | Quadrilateral modelling and robust control of a nonlinear piezoelectric cantilever[END_REF], nonlinear 1dof piezocantilevers can be described by a static hysteresis followed by a linear dynamic part. Moreover, the creep can be considered as an additional disturbance. This is expressed as follow:

δ = Γ (V, s) = H(V ) • D(s) + C(V, s) ( 2
)
where δ is the output, H(V) is the static hysteresis part that depends on the amplitude of V , D(s) is the linear dynamic part and C(V, s) is the creep part that depends on the amplitude of V and has its own dynamic part.

In some applications, the dynamic part inside C(V, s) is approximated by a linear transfer function [START_REF] Croft | Creep, hysteresis and vibration compensation for piezoactuators: atomic force microscopy application[END_REF]. The result given by (equ 2) makes easy the analysis because we can first model the static part.

According to the Fig. 8-a, the hysteresis curves of V y → Y are similar but only angled and shifted themselves when using different values of V z . Because of the importance of the shift and the weakness of the tilt of the curves, we neglect the latter. However it will be considered during the controller synthesis. From these remarks, we can write:

Y = H y (V y ) + T V z→Y (V z ) ( 3 )
where H y (V y ) is the static hysteresis of (V y → Y ) at V z = 0 and T V z→y (V z ) represents a nonlinear operator modeling the coupling (V z → Y ). The latter can be expressed using one of the curves of Fig. 8-b.

Based on Fig. 8-c and Fig. 8-d, the similar remarks can be applied to the deflection Z. We have:

Z = H z (V z ) + T V y→Z (V y ) ( 4 )
Fig. 8-c shows that T V y→Z (V y ) may be approximated by a linear function with low value of ∂Z ∂Vy . However, in order to maintain the generality, we keep the notation T V y→Z (V y ).

According to the four figures of Fig. 7, each transfer has its own transient part.Based on (equ 2), we can write:

           Y = H y (V y ) • D y (s) + T V z→Y (V z , s) + C V y→Y (V y , s) +C V z→Y (V z , s) Z = H z (V z ) • D z (s) + T V y→Z (V y , s) + C V y→Z (V y , s) +C V z→Z (V z , s)
(5) where:

• D y (s) and D z (s) are respectively the dynamic parts of the direct transfers and can be identified using Fig. 7-a and Fig. 7-d, • T V z→Y (V z , s) and T V y→Z (V y , s) are respectively the coupling transfers of V z → Y and V y → Z.

Each of these transfers include the static linear or nonlinear part (pictured in Fig. 8-b and Fig. 8-c) and the dynamic part (pictured in Fig. 7-b and Fig. 7-c),

• C V y→Y (V y , s) and C V z→Y •(V z , s) are the creeps that appear in Y when V y and V z are respectively applied, • C V y→Z (V y , s) and C V z→Z V z , s) are the creeps that appear in Z when V y and V z are respectively applied. During a micromanipulation/microassembly task, a piezogripper applies a force to the manipulated microobject. This manipulation force disturbs the functioning of each piezocantilever and may decreases its performances. So it is careful to account this force during the modeling and controller synthesis of the piezocantilever. As the force can be considered as an external disturbance, it can be easily introduced into (equ 5). We obtain:

           Y = H y (V y ) • D y (s) + T V z→Y (V z , s) + C V y→Y (V y , s) +C V z→Y (V z , p) + s Fy • F y Z = H z (V z ) • D z (s) + T V y→Z (V y , s) + C V y→Z (V y , s) +C V z→Z (V z , s) + s Fz • F z (6)
where F y and F z are the components of the external manipulation force respectively along y-axis and z-axis. The coefficients s Fy and s Fz are the compliance of the piezocantilever.

Assembling the coupling, the creep and the force into one disturbance, (equ 6) becomes:

   Y = H y (V y ) • D y (s) + d y Z = H z (V z ) • D z (s) + d z (7)
with d y and d z the disturbances respectively along y-axis and z-axis.

A. Identification of the dynamic part

The dynamic part D i (s) can be easily identified using the Fig. 7-a, Fig. 7-d and an ARMAX method. D i should be normalized, i.e. D i (0) = 1, because the gain is expressed by the static hysteresis. The identified models and the experimental results are presented in the Fig. 10a (for V y → Y ) and Fig. 10-b (for V z → Z). Because the gains of the models are equal to one, the simulation pictured in the figures have been scaled in order to allow the comparison. We have:

D y (s) = (3333) 2 (s + 3333) 2 (8)
and

D z (s) = 111 s + 1.07 × 10 4 s 2 -3.3s + 5.2 × 10 4 (s 2 + 5.8s + 4.7 × 10 4 ) (s 2 + 2023s + 1.4 × 10 6 ) (9)

B. Approximation of the hysteresis

In the literature, there are various models of hysteresis that have been applied to piezocantilevers, as exemple: the Prandtl-Ishlinskii model [START_REF] Rakotondrabe | Hysteresis and vibration compensation in a nonlinear unimorph piezocantilever[END_REF], the Preisach model [START_REF] Croft | Creep, hysteresis and vibration compensation for piezoactuators: atomic force microscopy application[END_REF], the Bouc-Wen model [START_REF] Low | Modeling of a three-layer piezoelectric bimorph beam with hysteresis[END_REF]. Despite the accuracy of these models, the synthesized controllers are often complex and nonlinear. Therefore, in order to use a linear controller, in our previous work [START_REF] Rakotondrabe | Quadrilateral modelling and robust control of a nonlinear piezoelectric cantilever[END_REF], we proposed to approximate the 1dof static hysteresis by a linear expression with a nominal gain α subjected to a direct multiplicative input uncertainty ∆ • W e . In this approximation, the hysteresis curve is approximated by a quadrilateral (Fig. 11). The gain α is the slope of its axis. While ∆ ∈ R and |∆| 1, the weighting W e ∈ R is chosen to cover the radius For V z → Z, the coefficients are easily computed using one of the three curves of Fig. 8-d. We have: α z = 1.5 µm V and W ez = 0.15 µm V . For V y → Y , we must take into account the tilt between the three hysteresis curves of Fig. 8-a. Hence, the maximal and minimal slopes, and then the weighting function W ey , are computed using the three curves. However, α is only computed with one curve: we choose the hysteresis curve corresponding to V z = 0V . We obtain: α y = 0.45 µm V and W ez = 0.075 µm V .

C. Final model

According to (equ 7), the initial coupled model becomes two independent models because the coupling parts were input inside the disturbance d i . From the previous sub-section, each model is linear but subjected to an uncertainty. So, the controllers synthesis is based on the models presented in the Fig. [START_REF] Singer | Shaping command inputs to minimize unwanted dynamics[END_REF]. In the figure, i denotes for y or for z and δ represents the deflection Y or Z. W ei is the previous weighting function. V. H ∞ control In this section we present the synthesis of a H ∞ controller for each axis. Such a controller has been chosen to ensure the stability and the required performances despite the uncertainty and the coupling effects.

A. Scheme and problem formulation

During the controllers synthesis, the following points are taken into account:

• the stability should be ensured despite the presence of uncertainty on the static gain α i , • the disturbance d i should be rejected,

• and the performances required for micromanipulation/microassembly tasks must be ensured. For each system, the Fig. 13-a shows the closed-loop system with the weighting functions. From it, the standard scheme can be formed (Fig. 13-b). In the figure: δ ref is the reference input, K i (s) is the controller to be synthesized, W 1i is the weighting for the pereformances specifications, W 2i is the weighting for the disturbance rejection and W ei is the previous weighting for uncertainty.

Using the two bloc-schemes of Fig. 13 and applying the standard H ∞ problem [START_REF] Balas | µ-analysis and synthesis toolbox[END_REF], the problem consists in finding the corrector K i and an optimal value of γ such as: [START_REF] Pérez | Modeling, fabrication and validation of a high-performance 2dof piezactuator for micromanipulation[END_REF] where G i = α i D i (s) is the nominal system and S i = 1 1+KiGi is the sentitivity function. B. Weighting functions 1) For the y-axis: the following specifications are used to compute the weighting functions.

|S i | < γi |W1i| , |S i | < γi |W1iW2i| , |S i | < γi |W2iW3i| , |S i G i K i | < γi |W2iW3i|
• the settling time is less than 40ms, • the accuracy is better than 1% and the overshot is nul, 2) For the z-axis: we use the following specifications.

di bi o1i o2i Vi Vi δref δ αi .D(s) + + + - Wei Ki(s) Ki W2i W1i Pi (a) (b) i ε i ε ref i b δ       1 2 i i o o      
• the settling time is less than 30ms,

• the accuracy is better than 1% and the overshot is nul, • the settling time for the disturbance rejection is 20ms, • when an external force 4mN , voltages of V y = 20V and V z = 20V are applied, they generate an elastic deflection, a creep and a coupling of more than 11µm in open-loop. So we want that the maximal deflection in closed-loop due to them is less than 1µm. From these, we choose:

       W 1z = s + 100 s + 1 W 2z = 1.25 (s + 1) (s + 0.18) (s + 100) (s + 0.013) (12)

C. Computation of the controllers

The controllers K y and K z have been computed using the Glover-Doyle algorithme [START_REF] Glover | State-space formulae for all stabilizing controllers that satisfy an H∞-norm bound and relations to risk sensivity[END_REF] [START_REF] Doyle | State-space solutions to standard H 2 and H∞ control problems[END_REF]. Each of the corrector has respectively an order equal to 5 and 7 and have been reduced into 2 and 5 using the balanced realization technique [START_REF] Moore | Principal component analysis in linear systems: controllability, observability and model reduction[END_REF]

VI. Experimental results

The two controllers have been implemented in a computer-Dspace material. The first experiments consist in applying a series of steps reference Y ref = 10µm and Z ref = 30µm and observing the coupling. The results show that the influence of an input Z ref on the Y -axis is rapidly rejected (zoom of Fig. 14-a) while the influence of Y ref on the Z-axis is negligible (zoom of Fig. 14-b). In the step responses, the settling time of Y is lower than 40ms and the overshot is null so that the wanted specifications are satisfied (Fig. 15-a). However, while the response of Z presents a slight overshot, its settling time is nearly 60ms (Fig. 15-b). The difference relative to the specifications is due to γ = 1.7. Despite that, these results are well suited for micromanipulation/microassembly tasks.

Finally, a harmonic analysis is performed. For that, a sine input Y ref with 10µm of amplitude is applied.

VII. Conclusion

The scope of this paper is the characterization, modeling and control of a 2-dof piezocantilever dedicated to micromanipulation/microassembly tasks. In addition to the hysteresis and the creep, the piezocantilever presents a coupling in the two axis that can compromise accurate tasks. Hence, a model was developed and a controller was synthesized. In order to account the hysteresis, the creep, the coupling and the performances specifications, a H ∞ robust controller was applied. The experimental results show the efficiency of the method and indicate that the obtained performances are well suited to the micromanipulation/microassembly requirements.

Fig. 1 .

 1 Fig. 1. Scheme of a piezoelectric microgripper made up of two piezocantilevers.

Fig. 2 .

 2 Fig. 2. Picture of the microgripper.

Fig. 3 .

 3 Fig. 3. Cross section of a gripper finger (Y-Z plane): working principle of the actuator.

Fig. 4 .

 4 Fig. 4. The setup.

Fig. 5 .

 5 Fig. 5. Transient part characteristics: experimental results.

Fig. 6 .

 6 Fig.6. Scheme of the piezocantilever with vibrations compensation.

  Fig. 7.Transient part characteristics when using vibrations compensation: experimental results.

Fig. 8 .Fig. 9 .

 89 Fig. 8. Static characteristics: experimental results.

Fig. 10 .

 10 Fig.10.Experimental result and simulation of the identified transient part.

Fig. 11 .

 11 Fig. 11. Quadrilateral approximation of the hysteresis[START_REF] Rakotondrabe | Quadrilateral modelling and robust control of a nonlinear piezoelectric cantilever[END_REF].

Fig. 12 .

 12 Fig. 12. Scheme of the models.

Fig. 13 .

 13 Fig. 13. (a): scheme of the closed-loop with weighting functions. (b) standard scheme.

Fig. 14 .

 14 Fig. 14. Experimental coupling analysis of the closed-loop piezocantilever.

Fig. 15 .

 15 Fig. 15. Experimental steps response of the closed-loop piezocantilever: zoom.

Fig. 16 .

 16 Fig.[START_REF] Low | Modeling of a three-layer piezoelectric bimorph beam with hysteresis[END_REF]. Experimental harmonic analysis of the closed-loop piezocantilever.

Acknowledgment

This work is partially supported by the French research project NANOROL (ANR N o PSIROB07 184846).


Mario Amendola
email: mamendola@genethon.fr

Mégane Brusson

Annarita Miccio
email: annarita.miccio@institutimagine.org

CRISPRthripsis: The Risk of CRISPR/Cas9-induced Chromothripsis in Gene Therapy

Keywords: genome editing, CRISPR/Cas9, chromothripsis, gene therapy, genotoxicity, micronuclei, chromosomal instability. Graphical

The Clustered Regularly Interspaced Short Palindromic Repeats (CRISPR)/Cas9 nuclease system has allowed the generation of disease models and the development of therapeutic approaches for many genetic and non-genetic disorders. However, the generation of large genomic rearrangements has raised safety concerns for the clinical application of CRISPR/Cas9 nuclease approaches. Among these events, the formation of micronuclei and chromosome bridges due to chromosomal truncations can lead to massive genomic rearrangements localized to one or few chromosomes. This phenomenon, known as chromothripsis, was originally described in cancer cells, where it is believed to be caused by defective chromosome segregation during mitosis or DNA double-strand breaks. Here, we will discuss the factors influencing CRISPR/ Cas9-induced chromothripsis, hereafter termed CRISPRthripsis, and its outcomes, the tools to characterize these events and strategies to minimize them.

Introduction

The CRISPR/Cas9 system has revolutionized the field of gene therapy for genetic and non-genetic diseases allowing the generation of disease models and the development of effective treatments for numerous disorders. However, safety concerns on the use of CRISPR/Cas9 for clinical applications have emerged. Here, we will discuss recent findings on unanticipated catastrophic DNA rearrangements induced by CRISPR/ Cas9 and their implications for gene therapy approaches.

Gene Therapy Using Designer Nucleases

Gene therapy was originally devised as a therapeutic replacement approach for monogenic disorders based on the delivery to the cells of a functional gene capable of compensating for the defective gene. Nowadays, the uses and indications are much broader, with most clinical trials concerning cancer treatment. There has been considerable diversification in the techniques, which are based on various corrective strategies, vectors, and methods including genome editing. Genome editing approaches use designer nucleases, such as the CRISPR/Cas9 nuclease system to induce DNA double-strand breaks (DSBs) via a guide RNA (gRNA) complementary to the genomic target (Fig. 1). The DSB can be repaired via homology-directed repair (HDR) by providing a donor DNA template containing the wild type sequence, allowing direct gene correction of the mutation. However, in the absence of integration of the DNA donor template, the DSB is simply repaired by the nonhomologous end joining (NHEJ) pathway that usually generates short insertions or deletions (InDel). Other genome editing approaches leverage the NHEJ pathway to inactivate genes or cis-regulatory regions, eg, the enhancer of the BCL11A gene encoding a master transcriptional repressor of fetal hemoglobin (HbF) expression with the aim of reactivating HbF expression in adulthood and cure diseases characterized by deficient adult hemoglobin expression (namely beta-hemoglobinopathies).

Initial concerns on the use of the CRISPR/Cas9 system were focused on the potential off-target activity leading to unwanted generation of InDels in genomic regions other than the on-target site. [START_REF] Fu | High-frequency off-target mutagenesis induced by CRISPR-Cas nucleases in human cells[END_REF] Fortunately, numerous studies to understand the underpinning mechanism and strategies to predict, detect and reduce this risk have been proposed. [START_REF] Tsai | Defining and improving the genomewide specificities of CRISPR-Cas9 nucleases[END_REF] Later, several groups have described that CRISPR/Cas9-induced DSB can be resolved in a complex and heterogeneous way, with the risk of inducing genomic rearrangements, such as large 1-to 50-kb deletions/inversions, translocations, chromosome loss, or chromosome truncations and combination of these rearrangements. 3,[START_REF] Teboul | Variability in genome editing outcomes: challenges for research reproducibility and clinical safety[END_REF] Overall, large deletions around the nuclease cutting site are the most common rearrangement and have been observed in mouse zygotes, [START_REF] Parikh | Detailed phenotypic and molecular analyses of genetically modified mice generated by CRISPR-Cas9-mediated editing[END_REF][START_REF] Shin | CRISPR/Cas9 targeting events cause complex deletions and insertions at 17 sites in the mouse genome[END_REF] human embryos, [START_REF] Alanis-Lobato | Frequent loss of heterozygosity in CRISPR-Cas9-edited early human embryos[END_REF][START_REF] Zuccaro | Allele-specific chromosome removal after Cas9 cleavage in human embryos[END_REF] human and mouse embryonic stem cells [START_REF] Kosicki | Repair of double-strand breaks induced by CRISPR-Cas9 leads to large deletions and complex rearrangements[END_REF] and human hematopoietic stem cells (HSCs). [START_REF] Wen | Effective control of large deletions after double-strand breaks by homology-directed repair and dsODN insertion[END_REF] Thanks to the use of long-read sequencing, in combination with long-range PCR and targeted sequencing, [START_REF] Gilpatrick | Targeted nanopore sequencing with Cas9-guided adapter ligation[END_REF][START_REF] Wallace | CaBagE: a Cas9based background elimination strategy for targeted, long-read DNA sequencing[END_REF][START_REF] Iyer | ACME: an affinity-based Cas9 mediated enrichment method for targeted nanopore sequencing[END_REF] this outcome can be quantified and characterized at the nucleotide level. Very recently, CRISPR/Cas9-induced chromothripsis (hereafter termed CRISPRthripsis; Fig. 1) has been described in cell lines. [START_REF] Leibowitz | Chromothripsis as an on-target consequence of CRISPR-Cas9 genome editing[END_REF] Unfortunately, unlike off-targets, these on-target risks cannot be reduced by more specific DSB approaches.

Chromoanagenesis or Genomic Chaos

Chromothripsis was firstly described in cancer cells as a process characterized by the occurrence of massive chromosomal rearrangements usually clustered on one or few chromosomes. [START_REF] Stephens | Massive genomic rearrangement acquired in a single catastrophic event during cancer development[END_REF] In fact, although these events can lead to cell death, they can also have a role in malignant transformation. Chromothripsis is observed with a frequency of >50% in several cancers [START_REF] Cortes-Ciriano | Comprehensive analysis of chromothripsis in 2,658 human cancers using whole-genome sequencing[END_REF] and is emerging as a predictor of negative clinical outcome. The mechanisms underlying chromothripsis are not fully elucidated. It is believed that DNA fragmentation is followed by the formation of micronuclei containing an entire chromosome or part of a chromosome that are further fragmented, reassembled, and eventually incorporated into the genome of the nucleus in the following mitoses [START_REF] Reimann | Long-term fate of etoposideinduced micronuclei and micronucleated cells in Hela-H2B-GFP cells[END_REF] (Fig. 2A). Chromothripsis can also occur as a consequence of the formation of chromosomal bridges during mitosis due to the fusion of the two sister chromatids with telomeric loss [START_REF] Umbreit | Mechanisms generating cancer genome complexity from a single cell division error[END_REF] (Fig. 2B). Finally, chromothripsis can generate double minute chromosomes, small circular acentric chromosomes that can be present at very high copy number and carry oncogenes, thus promoting tumor development. [START_REF] Storlazzi | Gene amplification as double minutes or homogeneously staining regions in solid tumors: origin and structure[END_REF] Nowadays, it is clear that the chromothripsis mechanism could not account for all the phenomena of rapid chromosomal rearrangements arising during single chaotic cellular events. In fact, these catastrophic phenomena include not only chromothripsis, but also chromoanasynthesis and chromoplexis, and are grouped under the name of "chromoanagenesis" (chromosome regeneration/evolution). [START_REF] Ostapinska | Insight into the molecular basis underlying chromothripsis[END_REF] Chromoanasynthesis (chromosome resynthesis) results in localized complex rearrangements with duplication and triplication of a single chromosome due to erroneous DNA replication, [START_REF] Liu | Chromosome catastrophes involve replication mechanisms generating complex genomic rearrangements[END_REF] and chromoplexy (chromosome restructuring) refers to the occurrence of multiple inter-and intra-chromosomal translocations and deletions with little or no copy number alterations. [START_REF] Baca | Punctuated evolution of prostate cancer genomes[END_REF]

Safety Concerns of Nuclease-based Editing Approaches: Micronuclei, Chromosomal Bridges, and CRISPRthripsis

Recently, CRISPR/Cas9-induced chromothripsis (CRISPR thripsis) has been described in cell lines [START_REF] Leibowitz | Chromothripsis as an on-target consequence of CRISPR-Cas9 genome editing[END_REF] (Fig. 1). In addition, the formation of micronuclei and chromosomal bridges due to the on-target cleavage has also been observed in primary cells including mouse embryos and human hematopoietic stem/progenitor cells (HSPCs), the target cell population in gene therapy approaches for many hematopoietic and non-hematopoietic disorders [START_REF] Leibowitz | Chromothripsis as an on-target consequence of CRISPR-Cas9 genome editing[END_REF][START_REF] Papathanasiou | Whole chromosome loss and genomic instability in mouse embryos after CRISPR-Cas9 genome editing[END_REF] (Fig. 1). In particular, the formation of micronuclei was observed in 20%-30% of mouse embryos and 2.5% of human HSPCs edited with Cas9 targeting the BCL11A enhancer, a proposed clinical target for beta-hemoglobinopathies [START_REF] Leibowitz | Chromothripsis as an on-target consequence of CRISPR-Cas9 genome editing[END_REF][START_REF] Frangoul | CRISPR-Cas9 gene editing for sickle cell disease and beta-thalassemia[END_REF] (NCT03655678 and NCT03745287). The analysis of micronuclei in the drug product used in these clinical trials [START_REF] Stephens | Massive genomic rearrangement acquired in a single catastrophic event during cancer development[END_REF] was not reported. However, CRISPRthripsis was observed neither in mouse embryos nor in human HSPCs, probably because only a small fraction of cells containing micronuclei undergo CRISPRthripsis and the number of cells analyzed in these studies was likely too low to observe this event. Furthermore, some cells can undergo cell death after CRISPRthripsis. However, even a rare event leading rather to a malignant transformation could be a concern in many gene therapy approaches where millions of cells are targeted. Fortunately, to date, no genotoxic events were reported in the NCT03655678 and NCT03745287 clinical trials.

Tools to Study Micronuclei and Chromosomal Bridges Formation and CRISPRthripsis

Micronuclei are defined as small-sized nuclei between 1/20th and 1/5th of the size of the main nucleus [START_REF] Schmid | The micronucleus test[END_REF] containing one/few chromosomes or chromosome fragments. [START_REF] Heng | Karyotype heterogeneity and unclassified chromosomal abnormalities[END_REF][START_REF] Ye | Micronuclei and genome chaos: changing the system inheritance[END_REF] Micronuclei differ from nuclei in terms of chromatin condensation, nuclear envelope composition, and the absence of proteasomes. [START_REF] Maass | Altered nuclear envelope structure and proteasome function of micronuclei[END_REF] Since micronuclei are considered a reliable readout of genomic instability, several approaches to quantify and characterize them have been described. In primis, nuclear and chromosomal alterations, such as micronuclei and chromothripsis, can be studied with classical and modern cytogenetics approaches, like GTG banding [START_REF] Seabright | rapid banding technique for human chromosomes[END_REF] and fluorescence in situ hybridization, [START_REF] Lee | Multicolor fluorescence in situ hybridization (FISH) approaches for simultaneous analysis of the entire human genome[END_REF] which enables a whole genomic view in a cost-efficient and single-cell oriented fashion. A special case of fluorescence hybridization is comparative genomic hybridization (CGH array), which allows genome copy number variation analysis. [START_REF] Kallioniemi | Comparative genomic hybridization for molecular cytogenetic analysis of solid tumors[END_REF] However, the resolution of these methods is limited to kilo-to mega-base pair. An exception is molecular combing, which consists in performing FISH after combing high molecular weight DNA on a glass surface. [START_REF] Schurra | Combing genomic DNA for structural and functional studies[END_REF] An overview of chromosomes' spatial organization and interactions can be obtained via high-throughput chromosome conformation capture approaches, which can provide both a multi-cell-based genomic view or at a single-cell level. [START_REF] Dekker | Mapping the 3D genome: aiming for consilience[END_REF][START_REF] Nagano | Single-cell Hi-C reveals cellto-cell variability in chromosome structure[END_REF][START_REF] Mallard | Hi-C detects genomic structural variants in peripheral blood of pediatric leukemia patients[END_REF] Numbers, shape, and size of micronuclei, and chromosome bridges can be studied via imaging using either microscopy [START_REF] Lepage | An automated, single cell quantitative imaging microscopy approach to assess micronucleus formation, genotoxicity and chromosome instability[END_REF][START_REF] Janssen | Current methods and pipelines for image-based quantitation of nuclear shape and nuclear envelope abnormalities[END_REF] or imaging flow cytometry. [START_REF] Wills | Inter-laboratory automation of the in vitro micronucleus assay using imaging flow cytometry and deep learning[END_REF][START_REF] Rodrigues | The in vitro micronucleus assay using imaging flow cytometry and deep learning[END_REF] Beyond frequency, size of micronuclei represents an important feature since there is a strong correlation between the size of the micronucleus and the presence of centromere, with centromere-positive micronuclei being larger than centromere-negative micronuclei. [START_REF] Hashimoto | An in vitro micronucleus assay with size-classified micronucleus counting to discriminate aneugens from clastogens[END_REF] Recent development of flow imaging will allow enrichment of rare micronuclei-containing cells for subsequent analysis. [START_REF] Schraivogel | High-speed fluorescence image-enabled cell sorting[END_REF] In addition, novel techniques Figure 1. CRISPR/Cas9-induced events. The CRISPR/Cas9 complex is driven to a specific genomic site thanks to the complementarity of the gRNA to the target DNA region. Then, Cas9 induces a DNA double-strand break (DSB, red double arrow) 3-4 nucleotides upstream of the protospacer adjacent motif (PAM). This DSB can be repaired by the homology-directed repair (HDR) in the presence of a donor DNA or by non-homologous end joining (NHEJ) to generate InDels (insertion and deletions). However, if the DSB is not correctly repaired, one of the possible outcomes is CRISPRthripsis, with the formation of micronuclei containing an acentric chromosome fragment (light red box), chromosome bridges, and chromothriptic chromosomes (colored blocks represent shuffled DNA) (see Fig. 2).

allowing purification of micronuclei and sequencing, at both population and single-cell levels, will provide a systematic approach to study genomic instability and reveal novel molecular details in the process. [START_REF] Pereira | Sequencing micronuclei reveals the landscape of chromosomal instability[END_REF] Finally, "Look-Seq" is a powerful technique for tracking cells containing micronuclei and subjecting their progeny to single-cell sequencing in order to understand the evolution and fate of micronuclei and cells bearing them. 14

Factors Influencing Micronuclei and Chromosomal Bridges Formation and CRISPRthripsis and Their Outcomes

There are several factors that can influence the formation of micronuclei and chromosome bridges, and CRISPRthripsis and determine their consequences.

First, these events occur in dividing cells, therefore the genotoxic risk is theoretically minimized or abolished in quiescent and non-dividing cells. By way of example, HSPCs are a mixed population of short-term progenitors and HSCs, which will sustain in the long-term the hematopoietic system in the treated patients. HSCs are mostly quiescent and therefore the risk of such chromosomal rearrangements is low. HDR-based approaches for gene correction require a long stimulation of the cell cycle (72-96 hours) as HDR occurs only in dividing cells. Therefore, when possible, NHEJ-based approaches should be preferred as NHEJ occurs in all the phases of the cell cycle. Nevertheless, it is difficult to assess the cell cycle in bona fide HSCs at the time of transplantation as they are difficult to identify, and we cannot exclude that even a short-term treatment (typically 48 hours for NHEJ-based approaches) can induce the cell cycle in HSCs. Finally, HSPC-based gene therapy involves the transplantation of proliferating hematopoietic progenitors together with HSCs. If such chromosomal rearrangements occur in progenitor cells, we cannot exclude the development of malignant clones from this cell population.

Furthermore, an important consideration is that the occurrence and frequency of these rearrangements can vary across tissues and cell types according to their response to DNA DSBs, [START_REF] Meyenberg | Tissue specific DNA repair outcomes shape the landscape of genome editing[END_REF] eg, between HSCs and hematopoietic progenitors. [START_REF] Milyavsky | A distinctive DNA damage response in human hematopoietic stem cells reveals an apoptosis-independent role for p53 in self-renewal[END_REF] The probability of these events is also specific to each CRISPR/Cas9-based therapeutic strategy and the number of DSB that are generated per cell, eg, editing of multiple genes (gamma-45 or alpha-globin [START_REF] Pavani | Ex vivo editing of human hematopoietic stem cells for erythroid expression of therapeutic proteins[END_REF] genes) or performing genomic deletions (beta-47 or alpha-globin 48 genes) will introduce more than one DSB per chromosome.

Moreover, the chromatin context of the target site can also influence the choice of DNA repair pathway and thus the occurrence of these massive genomic rearrangements. By way of example, locus-specific differences in the chromatin status or in DNA repair efficiency [START_REF] Schep | Impact of chromatin context on Cas9-induced DNA double-strand break repair pathway balance[END_REF] can determine if DSBs are correctly repaired or if they lead to loss of part of a chromosome, and as a consequence formation of micronuclei, chromosome bridges, and CRISPRthripsis. Similarly, the chromosomal location of the target site can also affect the outcome of these events in terms of cell death or clonal expansion, depending on the number and nature of genes present in the target chromosomes (eg, oncogenes and tumor suppressor genes).

Finally, another important factor is the presence of TP53. In fact, while TP53 does not influence the formation of micronuclei, it inhibits the division of around 50% of micronucleated cells, thus potentially avoiding the occurrence of CRISPRthripsis in the following mitosis. [START_REF] Leibowitz | Chromothripsis as an on-target consequence of CRISPR-Cas9 genome editing[END_REF][START_REF] Soto | p53 prohibits propagation of chromosome segregation errors that produce structural aneuploidies[END_REF] Therefore, transient TP53 inhibition proposed to increase HDR efficiency by minimizing apoptosis of edited cells 51 should be considered in view of a possible increase in the occurrence of the CRISPRthripsis. [START_REF] Kloosterman | Chromothripsis in congenital disorders and cancer: similarities and differences[END_REF] Overall, the occurrence of these catastrophic events should be closely monitored in pre-clinical gene therapy studies. The recent studies described the occurrence of these events in an HSC-based gene therapy product potentially used for many different diseases. [START_REF] Kloosterman | Chromothripsis in congenital disorders and cancer: similarities and differences[END_REF] However, long-term in vitro and in vivo experiments in the specific target cell types using highly sensitive tools are required to assess the safety of CRISPR/Cas9 nuclease-based gene therapy approaches.

Strategies to Minimize Micronuclei and Chromosomal Bridges Formation and CRISPRthripsis

To reduce the possibility of such complex genomic rearrangements, gene therapy approaches should: (i) reduce the number of edited cells by stringent selection of target cells (eg, only real HSCs 53 ); (ii) reduce cell cycling or restrict editing to the G1 phase of the cell cycle [START_REF] Lomova | Improving gene editing outcomes in human hematopoietic stem and progenitor cells by temporal control of DNA repair[END_REF][START_REF] Gutschner | Post-translational regulation of Cas9 during G1 enhances homology-directed repair[END_REF][START_REF] Matsumoto | A cell cycle-dependent CRISPR-Cas9 activation system based on an anti-CRISPR protein shows improved genome editing accuracy[END_REF] ; (iii) reduce additional stress associated with cell manipulation [START_REF] Mantel | Enhancing hematopoietic stem cell transplantation efficacy by mitigating oxygen shock[END_REF][START_REF] Aerts-Kaya | Strategies to protect hematopoietic stem cells from culture-induced stress conditions[END_REF] ; (iv) target the "right" genomic harbor for each application, eg, to achieve high transgene expression with a limited number of edited cells [START_REF] Pavani | Corrigendum: targeted gene delivery: where to land[END_REF] or to avoid chromosomal regions enriched in oncogenes or oncosuppressors; (v) reduce large genomic rearrangements due to repeated DNA cleavage of seamless repaired DSB, by modulating DNA repair [START_REF] Yin | Cas9 exo-endonuclease eliminates chromosomal translocations during genome editing[END_REF] or by reducing exposure time to nucleases using protein delivery. [START_REF] Lattanzi | Optimization of CRISPR/ Cas9 delivery to human hematopoietic stem and progenitor cells for therapeutic genomic rearrangements[END_REF] In addition, approaches based on the use of dead Cas9 or Cas9 nickase (eg, base editing, prime editing, and epigenome editing) minimize the formation of DSB and, likely, the generation of chromosomal bridges and the occurrence of CRISPRthripsis. However, there is still a risk of DSB when using technologies using Cas9 nickase (eg, base and prime editing), therefore the occurrence of these rearrangements should be investigated when there is evidence of DSB occurring after Cas9 nickase treatment.

Summary/Conclusion

Chromotripsis is a major driver of extrachromosomal DNA, [START_REF] Shoshani | Chromothripsis drives the evolution of gene amplification in cancer[END_REF] thus representing an important safety issue for gene and cell therapy that needs to be carefully addressed. Although occurring at low frequency, it is a dangerous event considering the large number of cells that need to be modified to achieve a clinical benefit. In addition, chromothripsis is not the only genome catastrophe that can happen. According to in vitro models, chromothripsis makes up roughly <10% of all different types of chaotic genomes identified. [START_REF] Liu | Genome chaos: survival strategy during crisis[END_REF][START_REF] Ye | Experimental induction of genome chaos[END_REF] Besides the occurrence of chromoanasynthesis and chromoplexy, various types of cell death, including mitotic cell death, [START_REF] Stevens | Mitotic cell death by chromosome fragmentation[END_REF][START_REF] Stevens | Diverse system stresses: common mechanisms of chromosome fragmentation[END_REF][START_REF] Heng | Imaging genome abnormalities in cancer research[END_REF] apoptosis, [START_REF] Tang | Reversibility of apoptosis in cancer cells[END_REF][START_REF] Tang | Molecular signature of anastasis for reversal of apoptosis[END_REF] necroptosis-a programmed version of necrosis 70 and entosis 71 can all reverse their own process, causing genomic alterations in surviving cells. The newly available technologies and the sudden interest brought to the domain by the CRISPRthripsis phenomenon promise to elucidate in the near future the mechanism causing chromothripsis, micronuclei, and genomic chaos, and their consequences on the surviving cell population, the end product that really matters for gene therapy.

Figure 2 .

 2 Figure 2. Chromothripsis leads to massive genomic rearrangements. (A) A DNA double-strand break (DSB, red double arrow) can lead to the formation of an acentric chromosome fragment (light red box). After mitosis, this fragment can be enwrapped by a lipid membrane forming a micronucleus (red circle). After DNA condensation, the chromosome fragment is shattered generating multiple DNA fragments, which are eventually reassembled and reincorporated into the nuclear genome forming a chromotriptic chromosome. If not reincorporated in the genome, these fragments can be lost (deleted fragments) or form double-minute chromosomes. (B) A DNA DSB can lead to the formation of sister chromatids with shortened or absent telomeres that form a chromosomal bridge. During the first mitosis, cell division leads to breakage of the chromosomal bridge, which can induce local DNA fragmentation and chromothripsis. During the second mitosis, the broken chromosome missegregates, potentially leading to the formation of micronuclei, which also trigger chromothripsis (see panel A).

Acknowledgments

We apologize to all whose work was not cited because of space restrictions. M.A. and A.M. thank the members of their laboratory at the Imagine Institute and the Genethon Institute.

Funding

This work was supported by State funding from the Agence Nationale de la Recherche under "Investissements d'avenir" program (ANR-10-IAHU-01 to A.M.), "Appel à projets générique 2020" (ANR-20-CE17-0016-01 to M.A. and A.M.), and the European Research Council (865797 DITSB to A.M.).

Data Availability

No new data were generated or analyzed in support of this research.

Conflict of Interest

The authors declare no potential conflicts of interest.

Author Contributions

M.A., M.B., A.M.: conception and design, manuscript writing, final approval of manuscript.