Anatol Pashkevich

Damien Chablat

Philippe Wenger

Roman Gomolitsky

Chapter No 11 Calibration of 3-d.o.f. Translational Parallel Manipulators Using Leg Observations

Keywords:

    Calibration of 3-d.o.f.

Introduction

Parallel kinematic machines (PKM) are commonly claimed as appealing solutions in many industrial applications due to their inherent structural rigidity, good payload-to-weight ratio, high dynamic capacities and high accuracy [START_REF] Tlusty | Fundamental comparison of the use of serial and parallel kinematics for machine tools[END_REF][START_REF] Tsai | Robot analysis: the mechanics of serial and parallel manipulators[END_REF][START_REF] Merlet | Parallel Robots[END_REF][START_REF] Wenger | Comparative study of parallel kinematic architectures for machining applications[END_REF]. However, while PKM usually exhibit a much better repeatability compared to serial mechanisms, they may not necessarily possess a better accuracy that is limited by manufacturing/assembling errors in numerous links and passive joints [START_REF] Wang | On the accuracy of a Stewart platform -Part I: The effect of manufacturing tolerances[END_REF]. Thus, the PKM accuracy highly relies on an accurate kinematic model, which must be carefully tuned (calibrated) for each manipulator individually. Similar to serial manipulators, PKM calibration techniques are based on the minimization of a parameter-dependent error function, which incorporates residuals of the kinematic equations [START_REF] Schröer | Calibration applied to quality control in robot production[END_REF][START_REF] Wampler | An implicit loop method for kinematic calibration and its application to closed chain mechanisms[END_REF][START_REF] Fassi | Calibration of Serial Manipulators: Theory and Applications[END_REF][START_REF] Legnani | Calibration of Parallel Kinematic Machines: theory and applications[END_REF]. For parallel manipulators, the inverse kinematic equations are considered computationally more efficient, contrary to the direct kinematics, which is usually analytically unsolvable for PKM. But the main difficulty with this technique is the full-pose measurement requirement, which is very hard to implement [START_REF] Innocenti | Algorithms for kinematic calibration of fully-parallel manipulators In: Computational Kinematics[END_REF][START_REF] Iurascu | Geometric algorithm for kinematic calibration of robots containing closed loops[END_REF][START_REF] Daney | Kinematic Calibration of the Gough platform[END_REF][START_REF] Jeong | Kinematic calibration of redundantly actuated parallel mechanisms[END_REF][START_REF] Huang | A general and novel approach for parameter identification of 6-dof parallel kinematic machines[END_REF]. Hence, a number of studies have been directed at using the subset of the pose measurement data, which however creates another problem, the identifiability of the model parameters [START_REF] Khalil | Self calibration of Stewart-Gough parallel robots without extra sensors[END_REF][START_REF] Daney | Robust parallel robot calibration with partial information[END_REF][START_REF] Besnard | Identifiable parameters for parallel robots kinematic calibration[END_REF][START_REF] Rauf | Complete parameter identification of parallel manipulators with partial pose information using a new measurements device[END_REF][START_REF] Rauf | Experimental results on kinematic calibration of parallel manipulators using a partial pose measurement device[END_REF]. Popular approaches in parallel robot calibration deal with one-dimensional pose errors using a double-ball-bar system or other measuring devices as well as imposing mechanical constraints on some elements of the manipulator [START_REF] Zhuang | Robot calibration with planar constraints[END_REF]Thomas et al., 2003;[START_REF] Daney | Self calibration of Gough platform using leg mobility constraints[END_REF]. However, in spite of hypothetical simplicity, it is hard to implement in practice since an accurate extra mechanism is required to impose these constraints. Additionally, such methods reduce the workspace size and the identification efficiency. Another category of calibration methods, the self-or autonomous calibration, is implemented by minimizing the residuals between the computed and measured values of the active and/or redundant joint sensors [START_REF] Hesselbach | Passive-joint sensors for parallel robots[END_REF]. Adding extra sensors at usually unmeasured joints is very attractive from a computational point of view, since it allows getting the data in the whole workspace and potentially reduces impact of the measurement noise. However, only a partial set of the parameters may be identified in this way, since the internal sensing is unable to provide sufficient information for the robot endeffector absolute location [START_REF] Zhuang | Self-calibration of parallel mechanisms with a case study on Stewart platforms[END_REF][START_REF] Williams | Kinematic error calibration of the gantry-tau parallel manipulator[END_REF]. More recently, several hybrid calibration methods were proposed that utilize intrinsic properties of a particular parallel machine allowing extracting the full set of the model parameters (or the most essential of them) from a minimum set of measurements. It worth mentioning an innovative approach developed by [START_REF] Renaud | Combining end-effector and legs observation for kinematic calibration of parallel mechanisms[END_REF][START_REF] Renaud | Kinematic and dynamic identification of parallel mechanisms[END_REF] who applied the vision-based measurement technique for the parallel manipulators calibration from the leg observations. In this approach, the source data are extracted from the leg images, without any strict assumptions on the end-effector poses. The only assumption is related to the manipulator architecture (the mechanism is actuated by linear drives located on the base). However, current accuracy of the camera-based measurements is not high enough yet to apply this method in industrial environment. This chapter summarises the authors' results in the area of parallel robotics [START_REF] Pashkevich | Design strategies for the geometric synthesis of Orthoglide-type mechanisms[END_REF](Pashkevich et al., , 2006) ) and focuses on the calibration of the Orthoglide-type mechanisms, which is also actuated by linear drives located on the manipulator base and admits technique of [START_REF] Renaud | Combining end-effector and legs observation for kinematic calibration of parallel mechanisms[END_REF][START_REF] Renaud | Kinematic calibration of parallel mechanisms: a novel approach using legs observation[END_REF]. But, in contrast to the known works, our approach assumes that the leg location is observed for specific manipulator postures, when the tool centre point (TCP) moves along the Cartesian axes. For these postures and for the nominal Orthoglide geometry, the legs are strictly parallel to the corresponding Cartesian planes. So, the deviations of the manipulator geometry influence on the leg parallelism that gives the source data for the parameter identification. The main advantage of this approach is the simplicity of the measuring system that can avoid using computer vision and is composed of standard comparator indicators, which are common in industry.

Orthoglide Mechanism

Manipulator architecture

The Orthoglide is a three d.o.f. parallel manipulator actuated by linear drives with mutually orthogonal axes. Its kinematic architecture is presented in Fig. 1 and includes three identical parallel chains that will be further referred to as "legs". Each manipulator leg is formally described as PRPaR -chain, where P, R and Pa denote the prismatic, revolute, and parallelogram joints respectively. The output machinery (with a tool mounting flange) is connected to the legs in such a manner that the tool moves in the Cartesian space with fixed orientation (i.e. restricted to translational motions). The Orthoglide workspace has a regular, quasi-cubic shape. The input/output equations are simple and the velocity transmission factors are equal to one along the x, y and z direction at the isotropic configuration, like in a conventional serial PPP machine. The latter is an essential advantage for machining applications [START_REF] Wenger | Kinematic analysis of a new parallel machine-tool: the Orthoglide[END_REF][START_REF] Chablat | Architecture Optimization of a 3-DOF Parallel Mechanism for Machining Applications, the Orthoglide[END_REF]. Another specific feature of the Orthoglide mechanism, which will be further used for calibration, is displayed during the end-effector motions along the Cartesian axes. For example, for the x-axis motion, the sides of the x-leg parallelogram must retain strictly parallel to the x-axis. Hence, the observed deviation may be a data source for calibration. For a small-scale Orthoglide prototype used for the calibration experiments, the workspace size is approximately equal to 200200200 mm 3 with the velocity transmission factors bounded between 1/2 and 2 [START_REF] Chablat | Architecture Optimization of a 3-DOF Parallel Mechanism for Machining Applications, the Orthoglide[END_REF]. The legs nominal geometry is defined by the following parameters: L = 310.25 mm, d = 80 mm, r = 31 mm where L, d are the parallelogram length and width, and r is the distance between the points C i and the tool centre point P (see Fig. 1b).

Modelling assumptions

Following previous studies on the PKM accuracy (Wang & Massory, 1993;[START_REF] Renaud | Combining end-effector and legs observation for kinematic calibration of parallel mechanisms[END_REF][START_REF] Caro | Sensitivity Analysis of the Orthoglide, a 3-DOF Translational Parallel Kinematic Machine[END_REF], the influence of the joint defects is assumed negligible compared to the encoder offsets and the link length deviations. This validates the following modelling assumptions: (i) the manipulator parts are supposed to be rigid bodies connected by perfect joints;

(ii) the manipulator legs (composed of a prismatic joint, a parallelogram, and two revolute joints) generate a four degrees-of-freedom motions; (iii) the articulated parallelograms are assumed to be perfect but non-identical; (iv) the linear actuator axes are mutually orthogonal and are intersected in a single point to ensure a translational movement of the end-effector; (v) the actuator encoders are perfect but located with some errors (offsets). Using these assumptions, calibration equations will be derived based on the observation of the parallel motions of the manipulator legs.

Kinematic model

Since the kinematic parallelograms are admitted to be non-identical, the kinematic model developed in our previous works [START_REF] Pashkevich | Design strategies for the geometric synthesis of Orthoglide-type mechanisms[END_REF](Pashkevich et al., , 2006) ) should be extended to describe the manipulator with different length leg parameters. Under the adopted assumptions, similar to the equal-leg case, the articulated parallelograms may be replaced by the kinematically equivalent bar links. Besides, a simple transformation of the Cartesian coordinates (shifted by the vector (r, r, r) T , see Fig. 1b) allows us to eliminate the tool offset. Hence, the Orthoglide geometry can be described by a simplified model, which consists of three rigid links connected by spherical joints to the tool centre point at one side and to the allied prismatic joints at another side (Fig. 2). Corresponding formal definition of each leg can be presented as PSS, where P and S denote the actuated prismatic joint and the passive spherical joint respectively. Thus, if the origin of a reference frame is located at the intersection of the prismatic joint axes and the x, y, z-axes are directed along them (see Fig. 2), the manipulator kinematics may be described by the following equations

                x x x x x x x x x x L L e L        sin cos sin cos cos ) ( p (1a) sin ( ) c o s c o s sin cos y y y y y y y y y y L L e L                       p (1b)                 e L L L z z z z z z z z z z        cos cos ) ( sin cos sin p ( 1c
)
where p = (p x , p y , p z ) T is the output vector of the TCP position,  = ( x ,  y ,  z ) T is the input vector of the prismatic joints variables,  = ( x ,  y ,  z ) T is the encoder offset vector,  i ,  i , i{x, y, z} are the parallelogram orientation angles (internal variables), and L i are the length of the corresponding leg.

After elimination of the internal variables  i ,  i , the kinematic model ( 1) can be reduced to three equations

  2 2 2 2 ) ( i k j i i i L p p p         (2)
which includes components of the input and output vectors p and  only. Here, the subscripts

} , , { , , z y x k j i  , k j i  
are used in all combinations, and the joint variables  i are obeyed the prescribed limits

max min      i
defined in the control software (for the Orthoglide prototype,  min = -100 mm and  max = +60 mm).

It should be noted that, for the case

0       z y x    and L L L L z y x   
, the nominal ''mechanical-zero'' posture of the manipulator corresponds to the Cartesian coordinates p 0 = (0, 0, 0) T and to the joints variables  0 = (L, L, L). Moreover, in this posture, the x-, y-and z-legs are oriented strictly parallel to the corresponding Cartesian axes. But the joint offsets and the leg length differences cause the deviation of the "zero" TCP location and corresponding deviation of the leg parallelism, which may be measured and used for the calibration. Hence, six parameters ( x ,  y ,  z , L x , L y , L z ) define the manipulator geometry and are in the focus of the proposed calibration technique.

Inverse and direct kinematics

The inverse kinematic relations are derived from the equations (2) in a straightforward way and only slightly differ from the "nominal" case:

i k j i i i i p p L s p         2 2 2 (3)
where sx, sy, sz { ±1} are the configuration indices defined for the "nominal" geometry as the signs of x -px , y -py, z -pz, respectively. It is obvious that expressions (3) give eight different solutions, however the Orthoglide prototype assembling mode and the joint limits reduce this set to a single case corresponding to sx = sy = sz = 1.

For the direct kinematics, equations ( 2) can be subtracted pair-to-pair that gives linear relations between the unknowns px, py, pz, which may be expressed in the parametric form as

, ) ( 2 2 2 i i i i i i i i L t p                ( 4
)
where t is an auxiliary scalar variable. This reduces the direct kinematics to the solution of a quadratic equation

0 2    C Bt At with the coefficients ; ) ( ) ( 2 2        j i j j i i A     ; ) ( ) ( ) ( 2 2 2 2 k k j j k j i i i i i L B                   2 2 2 4 2 2 ( ) ( ) /4 2 ( ) ( ) /4 i i i i i i j j k k i i i j k i C L L                             where } , , { , , z y x k j i  .
From two possible solutions that give the quadratic formula, the Orthoglide prototype (see Fig. 1) admits a single one

A AC B B t 2 / ) 4 ( 2    
corresponding to the selected manipulator assembling mode.

Differential relations

To obtain the calibration equations, let us derive first the differential relations for the TCP deviation for three types of the Orthoglide postures:

(i) "maximum displacement" postures for the directions x, y, z (Fig. 3a);

(ii) "mechanical zero" or the isotropic posture (Fig. 3b);

(iii) "minimum displacement" postures for the directions x, y, z (Fig. 3c);

These postures are of particular interest for the calibration since, in the "nominal" case, a corresponding leg is parallel to the relevant pair of the Cartesian planes. The manipulator Jacobian with respect to the parameters  =( x ,  y ,  z ) and L = ( L x , L y , L z ) can be derived by straightforward differentiating of the kinematic equations ( 2), which yields

0 0 0 0 0 0 x x y z x x x y y z y y x y z z z z p ρ p p p ρ p p ρ p p ρ p p p ρ p ρ                               p ρ ; 0 0 0 0 0 0 x x y z x x y y z y x y z z z p ρ p p L p p ρ p L p p p ρ L                            p L .
Thus, after the matrix inversions and multiplications, the desired Jacobian can be written as

  ) , ( ); , ( ) , ( ρ p J ρ p J ρ p J L  
(5)

where

1 1 (.) 1 1 y z x x x x x z y y y y y x z z z z p p p ρ p ρ p p p ρ p ρ p p p ρ p ρ                              J 1 (.) y x x z x x x y y x z L y y y y x z z z z z p p ρ p L L L p ρ p p L L L p p p ρ L L L                         

J

It should be noted that, for the sake of computing convenience, the above expression includes both the Cartesian coordinates p x , p y , p z and the joint coordinates  x ,  y ,  z , but only one of these sets may be treated as independent taking into account the kinematic equations.

For the "Zero" posture, the differential relations are derived in the neighbourhood of the point {p 0 = (0, 0, 0) ;  0 = (L, L, L)}, which after substitution to (5) gives the Jacobian matrix

              1 0 0 1 0 0 0 1 0 0 1 0 0 0 1 0 0 1 0 J (6)
Hence, in this case, the TCP displacement is related to the joint offsets and the leg lengths variations L i by trivial equations

; i i i L p       } , , { z y x i  . (7)
For the "XMax" posture, the Jacobian is computed in the neighbourhood of the point

{ ) 0 , 0 , (  LS  p ; ) , , (    LC LC LS L  ρ
}, where  is the angle between the y-, z-legs and the X-axes:

) / sin( max L a    ; ) ( sin    S , ) cos(   C
. This gives the Jacobian

                   1 1 0 1 0 0 0 1 0 0 1 0 0 1       C T T C T T x J (8)
where

) tan(   T
. Hence, the differential equations for the TCP displacement may be written as

x x x L p       y x y x y L C L T T p          1      z x z x z L C L T T p          1      (9)
It can be proved that similar results are valid for the YMax and ZMax postures (differing by the indices only), and also for the XMin, YMin, ZMin postures. In the latter case, the angle  should be computed as min asin( / ) L    .

Calibration Method

Measurement technique

To identify the Orthoglide kinematic parameters specified in the previous section, two approaches can be used, which employ different measurement techniques to evaluate the leg-to-surface parallelism. The first of them (Fig. 4a) assumes two measurements for the same leg posture (to assess distances from both leg ends to the base surface). The second technique (Fig. 4b) assumes a fixed location of the measuring device but two distinct leg postures, which are assumed to be parallel to each other in the nominal case. It is obvious that, for the perfectly calibrated manipulator, both methods give zero differences for each measurement pair. In contrasts, the non-zero differences contain source information for the parameter identification. However, the first method involves absolute measurements that require essential implementation efforts; besides it allows evaluating parallelism only for the X-and Y-legs with respect to the XY-plane. So, the second method will be used here. For this method, which employs the relative measurements and allows assessing the leg parallelism with respect to both relevant planes (XY-and XZ-planes for the X-leg, for instance), the calibration experiment may be arranged in the following way:

Step 1. Move the manipulator to the Zero posture; locate two gauges in the middle of the Xleg (orthogonal to the leg and parallel to the axes Y and Z); get their readings.

Step 2. Move the manipulator sequentially to the XMax and XMin postures, get the gauge readings, and compute the differences

+ x y  , + x z  , x y  , x z
 with respect to the "Zero" posture values.

Step 3+. Repeat steps 1, 2 for the Y-and Z-legs and compute the differences

+ y x  , + y z  , y x  , y z
 , and In the above description, the variable following the symbol  denotes the measurement direction (x, y or z), the subscript defines the manipulator leg, and the superscript indicates the manipulator posture ('+' for XMax and '-' for XMin). For example,

+ z x  , + z y  , z x  ,
+ x z 
denotes the zcoordinate deviation of the X-leg for the XMax posture with respect to Zero location.

Calibration equations

The system of calibration equations can be derived in two steps. First, it is required to define the gauges' initial locations that are assumed to be positioned at the leg middle at the Zero posture, i.e. at the points ( )/2

i  p r , { , , } i x y z 
where the vectors r i define the prismatic joints centres:

( ;0 ;0 ) T x x L    r ; (0; ; 0) T y y L    r ; (0; 0; ) T z z L    r
. Hence, using the equation ( 7), the gauge initial locations can be expressed as 0 (

) / 2 ; ( ) / 2; ( ) / 2

T x x x y y z z L L L L               g 0 ( ) / 2; ( ) / 2 ; ( ) / 2 T y x x y y z z L L L L                   g (10) 0 ( ) / 2; ( ) / 2; ( ) / 2 T z x x y y z z L L L L               g
Afterwards, for the XMax, YMax, ZMax postures, the leg spatial location is also defined by two points, namely, (i) the tool centre point p, and (ii) the centre of the prismatic joint r i . For example, for the XMax posture, the TCP position is max ( ; ; )

x x x LS L         p
, the prismatic joint position is max ( ;0 ;0 )

x x L LS      r
. So, the leg is located along the line

  max max ( ) (1 ) ; 0;1 x x x           s p r ,
Since the x-coordinate of the gauge is kept constant (for X-leg measurements), the parameter  may be obtained from the equation

0 [ ( )] [ ] x x x x
  s g , which yields:

0.5 / x S S L L        ,
Hence, after some transformations, the deviations of the X-leg measurements (between the XMax and Zero postures) may be expressed as

1 (0.5 ) (0.5 ) ((0.5 ) 0.5) x x y x y y S T S S T L S C L                         1 (0.5 ) (0.5 ) ((0.5 ) 0.5) x x z x z z S T S S T L S C L                        
A similar approach may be applied to the XMin posture, as well as to the corresponding postures for the Y-and Z-legs. This gives the system of twelve linear equations in six unknowns:

                                                                                                                                                                     x
2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (11) where sin( ); i i a   (0.5 sin( )) tan( ); i i i b     (0.5 sin( )) / cos( ) 0.5; i i i c      {1, 2} i 
, and

1 m a x asin( / ) 0; L     2 m i n asin( / ) 0 L     .
This system can be solved using the pseudoinverse of Moore-Penrose, which ensures the minimum of the residual square sum for corresponding linear approximation of the kinematic equations that is valid for small values of , , ,

x y y z L L       
. Otherwise, it is prudent to apply straightforward numerical optimisation, which fits the experimental data to the manipulator kinematic model (1).

Calibration accuracy

Because of the measurement noise, the developed technique may produce some errors in estimates of the model parameters. Thus, for practical applications, it is worth to evaluate the statistical properties of the calibration errors. Within the linear calibration equations ( 11), the impact of the measurement noise may be evaluated using general techniques from the identification theory, under the standard assumptions concerning the primary measurement errors (.)

 (zero-mean independent and identically distributed Gaussian random variables with the standard deviation ). For these assumptions, the covariance matrix of the estimated parameters is written as [START_REF] Ljung | System identification : theory for the user[END_REF])

1 1 ( , ) ( ) ( ) ( ) T T T T            V ρ L J J J E s s J J J (12)
where E(.) denotes the mathematical expectation, J is the identification Jacobian, and s is the vector of the measurement errors in the right-hand side of the system (11). However, in contrast to the standard technique, the vector s includes some statistically-dependent components because the same measurement values, corresponding to the Zero position, are subtracted from those corresponding to the Max and Min postures. In particular,

0 0 0 ( ) ( ), ( ) ( ), ( ) ( ) T y y x x x x x x y y z z                   s  , (13)
where the index sequence strictly corresponds to (11). Thus, the covariance ( )

T
  E s s is the 1212 non-identity matrix that after relevant transformations may be expressed as

  2 12 12 2 0 1 0 0 2 0 1 ; 1 0 2 0 0 1 0 2 T                            G 0 0 E s s 0 G 0 G 0 0 G (14)
Hence, using expressions ( 12), ( 14) it is possible to evaluate the identification accuracy (via the covariance matrix (12)) for the set of parameters { , , , }

Experimental results

Experimental setup

For experimental verification of the developed technique, we used the measuring system composed of standard comparator indicators with resolution of 10.0 m. The indicators were attached to universal magnetic stands that allow fixing them on the manipulator base.

This system is sequentially used for measuring the X-, Y-, and Z-leg parallelism while the manipulator moves between the Max, Min and Zero postures. (It is obvious that for industrial applications it is worth using more sophisticated digital indicators with the resolution of 1.0 m or less, which yield more accurate calibration results.) During experiments, the legs were moved sequentially via the following postures: Zero  Max  Min  Zero … . To reduce the measurement errors, the measurements were repeated three times for each leg. Then, the results were averaged and used for the parameter identification. It should be noted that the measurements demonstrated very high repeatability compared to the encoder resolution (dissimilarity was less than 20.0 m).

Calibration results and their analysis

The experimental study included three types of experiments targeted to the following objectives: (#1) validation of modelling assumptions; (#2) obtaining source data for the parameter identification; and (#3) verification of the calibration results.

Experiment #1. The first calibration experiment demonstrated rather high parallelism deviation for the legs at the Max and Min postures, up to 2.37 mm as shown in Table1. This indicated low accuracy of the nominal kinematic model and motivated necessity of the calibration. On the other hand, the milling accuracy evaluated in separate tests was quite good. However, this is not an indicator of high absolute accuracy but just a proof of the Orthoglide architecture advantages (the milling tests were perfect just because of the high homogeneity of the manipulator workspace in the neighbourhood of the isotropic location).

The straightforward application of the proposed calibration algorithm to the data set #1 was not optimistic: in the frames of the adopted kinematic model the root-mean-square (r.m.s.) deviation for the legs can be reduced down from 1.19 mm to 0.74 mm only (see Table 1 where , etc.). Besides, the statistical estimation of the measurement noise parameter  (based on the residual analysis) also yielded unrealistic result compared to the encoder resolutions (0.01 mm). This impelled to conclude that some modelling assumptions are not valid and the manipulator mechanics required more careful tuning, especially orientation of the linear actuator axes (that are assumed to be mutually orthogonal and to intersect in a single point). Thus, the manipulator mechanics was retuned, in particular spatial locations of the actuator axes were adjusted. 1), which is about twice better than in the first experiment. Besides, the expected residual reduction was also essential (0.20 mm) that justified validity of the modelling assumptions. For these data, the developed calibration algorithm was applied for three sets of the model parameters: for the full set {, L} and for the reduced sets {}, {L}. As follows from the identification results (Tables 2,3), the calibration algorithm is able to identify simultaneously both the joint offsets and  and the link lengths L. However, both  and L (separately) demonstrate roughly the same influence on the residual reduction, from 0.32 mm to 0.14 mm (in terms of the deviations + ,

+ y y y x x x     , + z z z x x x    
y y x x   + , , z z
x x   , ), while the full set {, L} gives further residual reduction down to 0.12 mm only. This motivates considering  as the most essential parameters to be calibrated. Accordingly, the identified vales of joint offsets  x ,  y ,  z were incorporated in the Orthoglide control software. Experiment #3. The third experiment was targeted to the validation of the calibration results, i.e. assessing the leg parallelism while using the kinematic model with the parameters identified from the data set #2. This experiment demonstrated very good agreement with the expected values of + ,

y y x x   + , , z z x x   , .
In particular, the maximum deviation reduced down from 0.62 mm to 0.24 mm, and the root-mean-square value decreased down from 0.32 mm to 0.15 mm (expected value is 0.14 mm). On the other hand, further fitting of the kinematic model to the third data set gives both negligible improvement in the deviations and very small alteration of the model parameters. It is evident that further reduction of the parallelism deviation is bounded by the manufacturing errors and, by non-geometric reasons. Resume. Hence, the calibration results confirm validity of the proposed identification technique and its ability to tune the joint offsets and link lengths from observations of the leg parallelism. However, for these partucular experiments, combined influence of the parameters {, L} may be roughly decribed by the diffrence { -L} that allows us to simplify modifications of the kinematic model included in the control software. Another conclusion is related to the modelling assumption: for further accuracy improvement it is prudent to generalize the manipulator model by including parameters describing orientation of the prismatic joint axes, which is equavalet to relaxing some modelling assumption.

Residuals

Conclusions

Recent advances in parallel robot architectures encourage related research on kinematic calibration of parallel mechanisms. This paper proposes a novel calibration approach based on observations of manipulator leg parallelism with respect to the Cartesian planes. Presented for the Orthoglide-type mechanisms, this approach may be also applied to other manipulator architectures that admit parallel leg motions (along the Cartesian axes) or, in more general case, allow locating the leg in several postures with a common intersection point.

The proposed calibration technique employs a simple and low-cost measuring system composed of standard comparator indicators attached to the universal magnetic stands. They are sequentially used for measuring the deviation of the relevant leg location while manipulator moves the tool-center-point in the directions x, y and z. From the measured differences, the calibration algorithm estimates the joint offsets and link lengths that are treated as the most essential parameters that are difficult to identify by other methods.

The presented theoretical derivations deal with the sensitivity analysis of the proposed measurement method and also with the calibration accuracy. The validity of the proposed approach and efficiency of the developed numerical algorithm were confirmed by the calibration experiments with the Orthoglide prototype, which allowed dividing the residual root-mean-square by three.

To increase the calibration precision, future work will focus on the development of the specific assembling fixture ensuring proper location of the linear actuators and also on the expanding the set of the identified model parameters and compensation of the nongeometric errors that are not identified within the frames of the adopted model.

Fig. 1 .

 1 Fig. 1. The architecture of Orthoglide manipulator (a) and kinematics of its leg (b) (© CNRS Photothèque / CARLSON Leif)

Fig. 2 .

 2 Fig. 2. Orthoglide simplified model (a) and its ''zero'' configuration (b)

Fig. 3 .

 3 Fig. 3. Specific postures of the Orthoglide (for the x-leg motion along the Cartesian axis X)

Fig. 4 .

 4 Fig. 4. Measuring the leg parallelism with respect to the base plane

  error parameter  is known. For instance, for the Orthoglide prototype described in sub-section 2.1 and the Max/Min posture characteristic angles 1

Fig. 5 .

 5 Fig. 5. Experimental setup for calibration experimentsFor each measurement, the indicators are located on the mechanism base in such a manner that a corresponding leg is admissible for the gauge contact for all intermediate postures (Fig.5). The Min and Max postures are constrained by the software limits and defined as min

Table 1 .

 1 Experimental data and expected improvements of accuracy via calibration[mm]

	Data Source	x y	x z	y x	y z	z x	z y	r.m.s.
	Experiment #1 (before mechanical tuning and before calibration)
	Measurements #1	+0.52	+1.58	+2.37	-0.25	-0.57	-0.04	1.19
	Expected improvement	-0.94	+0.63	+1.07	-0.84	-0.27	+0.35	0.74
	Experiment #2 (after mechanical tuning, before calibration)
	Measurements #2	-0.43	-0.37	+0.42	-0.18	-1.14	-0.70	0.62
	Expected improvement	-0.28	+0.25	+0.21	-0.14	-0.13	+0.09	0.20
		Experiment #3 (after calibration and adjusting of )
	Measurements #3	-0.23	+0.27	+0.34	-0.10	-0.09	+0.11	0.21
	Expected improvement	-0.29	+0.23	+0.25	-0.17	-0.10	+0.08	0.20
	Experiment #2. The second calibration experiment (after mechanical tuning) yielded lower
	parallelism deviations, less than 0.62 mm in terms of the deviations	y  , x

z x  , … (see Table

Table 2 .

 2 Residual compensation using different sets of kinematic parameters[mm]

				Experiment 2			Experiment 3
		Exper.	Expected improvement	Exper.	Expected improvement
			data	{, L}	{}	{L}	data	{, L}	{}	{L}
	y x  	-0.19	-0.09	-0.03	-0.03	-0.07	0.02		0.04	0.04
	x y  	0.08	0.12		0.03	0.04	0.02	0.04		0.02	0.02
	y x  	0.22	0.09		0.13	0.12	0.10	-0.07		-0.06	-0.06
		x y 	-0.34	-0.10	-0.13	-0.13	-0.24	0.01		0.00	0.00
		z x 	-0.29	-0.41	-0.32	-0.33	0.01	-0.02		-0.01	0.00
		x z 	-0.52	-0.45	-0.39	0.42	0.11	-0.02		-0.03	0.04
	z x  	0.08	0.23		0.26	0.26	-0.19	-0.05		-0.04	-0.04
	x z  	0.62	0.55		0.57	0.56	-0.03	0.10		0.09	0.09
	z y  	0.02	-0.04	-0.13	-0.12	0.07	-0.03		-0.05	-0.05
		y z 	-0.24	0.29	-0.26	-0.27	-0.21	-0.05		-0.07	-0.07
		z y 	0.20	-0.03		0.06	-0.06	0.17	0.04		0.03	0.03
		y z 	0.45	0.48		0.51	0.50	0.27	0.11		0.10	0.10
	Average	0.32	0.12		0.14	0.14	0.15	0.13		0.14	0.14
	Set of parameters		 x	 y	Identified values [mm]  z L x	L y	L z	Residuals
	{, L}		4.66	-5.36	1.46		5.20	-5.96	3.16	0.12
		{}		-0.48	0.49	-1.67	-	-	-	0.14
		{L}		-	-	-		0.50	-0.52	1.69	0.14
	Table3. Calibration results for parameters  and L


Dr David Vesseur

Marte S M Holmsen

Didier Bourissou
email: didier.bourissou@univ-tlse3.fr

Dr. M. S D M Holmsen

Bourissou

Dr. M. S M Holmsen

Au(III) -Allyl Complexes: Synthesis, Structure, Reactivity, and Catalytic Applications

-Allyl complexes of transition metals are key species in organometallic chemistry and homogeneous catalysis. Palladium(II) -allyl complexes in particular, have gained a lot of attention, but their isoelectronic Gold(III) counterparts long remained elusive. However, this situation changed during the last few years. This concept article describes the preparative routes, characterization, structure and reactivity of such species, together with their catalytic applications. The influence of the ancillary ligand at gold, either (P,C)/(N,C)cyclometalated or (P,N)-hemilabile, is analysed in detail. The Au(III) and Pd(II) -allyl complexes are also compared to highlight the similarities and differences.

Scheme 1. Preparation of Au(III) π-allyl complexes 3 and 6 (OAc F = OCOCF3) upon halide abstraction from σ-allyl complexes 2 and 5. The yields given for complex 2 and 3 are for R = iPr and R ' = H. No yield reported for 6 as it starts to decompose at the same time as it forms.

Introduction

Despite the spectacular progress achieved over the past ten years concerning Au(III) complexes featuring carbon-based -ligands (alkenes, alkynes, arenes), [1][2][3] Au(III) -allyl complexes remained elusive until very recently. [START_REF] Nakanishi | For computational and mass spectrometry studies dealing with Me2Au(-allyl)[END_REF] This is all the more surprising that the related Pd(II) -allyl complexes are key species in organometallic chemistry, both in terms of bonding/coordination and reactivity/catalytic applications (cf. the Tsuji-Trost reaction). [5][6][START_REF] Hartwig | Organotransition Metal Chemistry: From Bonding to Catalysis[END_REF] It is only a few years ago that the first Au(III) -allyl complexes could be isolated. They were reported simultaneously by our group and Tilset's group, using (P,C) and (N,C)-cyclometalated ligands, respectively. [START_REF] Rodriguez | [END_REF]9] From there on, the chemistry of Au(III) allyl complexes was explored systematically and thoroughly, [10][11][12] ranging from fundamental aspects such as their synthesis, characterization and geometric/electronic structure, to their reactivity towards nucleophiles, and finally their catalytic applications (Figure 1). These studies are summarized and analyzed herein, with special emphasis on the role and impact of the ancillary ligand at gold. Particular attention is given to the specific features of the Au(III) -allyl complexes when compared to their Pd(II) counterparts.

Figure 1. Au(III) π-allyl complexes: from their preparation and characterization, to their reactivity and catalytic application.

Preparation of Au(III) -allyl complexes

In 2020, we reported the preparation of (P,C)-cyclometalated Au(III) π-allyl complexes 3 by halide abstraction from their respective Au(III) -allyl complexes 2 thereby allowing for the η 3 coordination of the allyl moiety (Scheme 1a). [START_REF] Rodriguez | [END_REF] The (P,C) Au(III) σ-allyl complexes were conveniently prepared by reaction of (P,C)AuI2 complexes 1 with the appropriate allyl Grignard reagent (Scheme 1a). Both the parent Au(III) π-allyl complex (3, R' = H) and the closely related methallyl complex (3, R' = Me) were prepared. Simultaneously, Tilset and co-workers prepared the (N,C)-cyclometalated Au(III) π-allyl complex 6 by a closely related method (Scheme 1b). [9] Reaction of (N,C)Au(OAc F )2 (OAc F = OCOCF3) 4 with allyl magnesium bromide followed by halide abstraction of 5 with AgNTf2 and concomitant η 3 coordination of the π-allyl moiety led to the formation of complex 6. Although prepared similarly, Au(III) π-allyl complexes 3 and 6 showed a striking difference in terms of stability. Whereas complexes 3 are stable at room temperature for weeks, complex 6 rapidly decomposes at room temperature. The bonding of the allyl moiety in complexes 3 and 6 turned out to be quite different, and we have chosen to draw the complexes in their most representative form (symmetric  3 -allyl versus asymmetric +-coordination, as discussed below).

With the first two cyclometalated Au(III) -allyl complexes being prepared via a redox neutral halide abstraction from the allyl precursors, the question arose if they could be formed via oxidation from Au(I) to Au(III) as is often the case for their isoelectronic Pd(II) -allyl analogues. In other words, can Au(III) -allyl complexes be prepared from allyl electrophiles and Au(I), akin to the Pd(0)/Pd(II) elementary step in the Tsuji-Trost reaction? While Au(I) is infamous for its reluctance to undergo oxidative addition, recent efforts have shown that carefully selecting the ligand allows for the oxidative addition to proceed. [13] In particular, the hemilabile (P,N) type ligand Me-Dalphos has been demonstrated to be very efficient in triggering Au(I)/Au(III) oxidation. [14] Due to the sterically demanding adamantyl substituents on the phosphorus atom, the nitrogen lone pair points towards the Au center, resulting in a lowering of the barrier for oxidative addition. [13c] With this in mind, allyl acetate was reacted with the in situ formed (P,N)Au(I) + (Scheme 2). [11] Unlike its Pd(0) analogues, the Au(I) to Au(III) oxidation did not proceed spontaneously and instead, the Au(I) -alkene complex 7 was formed quantitatively. Despite the enhanced π-backdonating ability of gold induced by bending, the (P,N)Au + fragment was reluctant to undergo oxidation. However, addition of the strong Lewis acid tris(pentafluorophenyl)borane (BCF) resulted instantaneously in abstraction of the acetate. The ensuing Au(III) -allyl complex 8 is stable up to -30 °C and was fully characterized using multinuclear NMR.

Scheme 2. Formation of the Au(I) -alkene complex 7 and subsequent Lewis acid-assisted oxidation to give the Au(III) -allyl complex 8. SbF6 -and AcO-BCF - counter-anions for 8.

Gold has been shown to display rich reactivity in combination with strained carbocycles, [15] and recently the ring-opening of alkylidene cyclopropanes by Au(III) was reported. [16] Therefore we wondered if Au(III) -allyl complexes could be formed via a redox neutral rearrangement from their isomeric -cyclopropyl, driven by the release of strain energy. [12] To test this synthetic approach, the (P,C)-cyclometalated Au(III) -cyclopropyl complex 9 and the (P,N)-chelated Au(III) cyclopropyl complex 10 were prepared (Scheme 3). For the rearrangement to proceed, a coordination site had to be liberated, which could be done by halide abstraction with AgSbF6. In both cases, the reaction led cleanly to the ring-opened product. However, the temperature at which the rearrangement occurred depended strongly on the ligand used. For the (P,C)cyclometalated Au(III) -cyclopropyl 9, the rearrangement occurred already at -50 o C. Whereas the chelated biscationic (P,N)chelated Au(III) -cyclopropyl complex 10 undergoes rearrangement already at -80 o C and is complete when -60 o C is reached. Changing the phenyl substituents on 9 to the more electron-donating cyclohexyl did not noticeably change the temperature at which the ring-opening proceeded. The ring-opening process was investigated in-depth using DFT calculations. The free energy profile for the σ-cyclopropyl to π-allyl rearrangement for complex 9-Ph is given in Figure 2. The lowest barriers were found for the ring-opening where the distal (CC) bond was cleaved. For the (P,C) Au(III) -cyclopropyl complex, a barrier of 19.8 kcal/mol was found while for the (P,N) Au(III) -cyclopropyl complex, a lower activation barrier of 15.7 kcal/mol was found matching the experimental trend. Several other mechanistic pathways were considered, e.g. -H elimination and insertion of Au into the (CC) bond, but these were found to not be energetically competitive.

A deeper look into the ring-opening process showed that the rotation of the two CH2 moieties along the Intrinsic Reaction Coordinate (IRC) proceeds via a disrotatory ring-opening process. For analysis of the electron flow associated with the rearrangement, the intrinsic bond orbitals (IBO) were inspected. Here, the most noteworthy observation was the progressive conversion of the distal (CC) bond into a 3-center -orbital (Figure 2), in line with an electrocyclic ring-opening. Upon halide from complexes 9 and 10, a cationic coordinatively unsaturated Au(III) intermediate is formed. This type of complexes can engage in weak interactions as agostic interactions with surrounding ligands or substrates. [17] In the case of the -cyclopropyl complexes, it can be envisioned that the (CC) bond engages in a donor-acceptor interaction with the gold center resulting in an agostic interaction. While C-H agostic interactions have been described before with gold, this is not the case for C-C agostic interactions. [17a] Cyclopropyl has high ring strain resulting in bent banana bonds that are more prone to agostic interactions. [START_REF] Jaffart | For experimentally authenticated C-C agostic cyclopropyl complexes (M = Nb, Y, Sc and Rh)[END_REF] Natural Bond Orbital (NBO) analysis on the cationic 3-coordinate (P,C) Au(III) -cyclopropyl complex 9-Ph + showed donor-acceptor interactions with weak donations from the proximal and distal (CC) bonds to Au (2.9 and 4.6 kcal/mol, respectively). For the dicationic (P,N) Au(III) -cyclopropyl complex, similar agostic interactions were found, with donor-acceptor interactions of 6.9 kcal/mol between the proximal (CC) bond and gold, and 6.4 kcal/mol between the distal (CC) bond and gold.

To demonstrate the versatility of the -cyclopropyl to -allyl rearrangement, (N,C) cyclometalated complexes 12 were included in the study (Scheme 2). Like complexes 9 and 10, they undergo ring-opening upon liberation of a coordination site at gold. However, the rearrangement does not start before room temperature is reached, at which point the -allyl decomposes, preventing a clean transformation. The rigidity of the backbone of the ancillary ligand (tolyl-pyridine versus benzoquinoline) did not influence the ring-opening to any significant degree, this observation was in good agreement with the calculated barriers for the ring-opening (27.1 and 26.4 kcal/mol for tolyl-pyridine and benzoquinoline, respectively).

Characterization and structure of Au(III) -allyl complexes

Complexes 3, 6 and 11 were unambiguously authenticated by multinuclear NMR spectroscopy. [START_REF] Rodriguez | [END_REF]9,11] Complexes 3 and 6 were also characterized by single crystal X-ray diffraction analysis. In the case of 6, connectivity was established, but twinning and disorder in the crystals led to poor data quality, thus the optimized structure for complex 6 will be used herein when comparing with the other -allyl complexes.

The ancillary ligands employed in complexes 3, 6 and 11 have major differences in their electronic properties with the (P,C) ligand being close to electronically symmetric, whereas the (N,C) and (P,N) ligands are highly asymmetric due to P and C being significantly more donating than N. This difference is reflected in the bonding of the π-allyl moiety and thus the stability of the ensuing complexes (Table 1). Complex 3-iPr is stable at room temperature, whereas complexes 6 and 11 rapidly decompose at this temperature.

Table 1. Selected NMR, XRD and DFT data for complexes 3-iPr, 6 and 11 illustrating their striking differences in bonding. [START_REF] Rodriguez | [END_REF]9,11] With the naphthyl-based (P,C) ligand employed in complexes 3, bench stable complexes were obtained featuring a quasisymmetric η 3 coordination of the allyl moiety. [START_REF] Rodriguez | [END_REF] The symmetric coordination of the allyl in complex 3-iPr is illustrated by the equalization of the 13 C NMR chemical shifts of the two terminal carbons C a and C c of the π-allyl moiety ( 83/76 versus  48/110 in the corresponding σ-allyl complex 2-iPr) together with the similar bond lengths found for C a -C b and C b -C c and the small variation found in the distance between Au and the three carbon atoms of the π-allyl moiety. In complexes 6 and 11, the picture is very different. [9,11,12] Here, the terminal carbons C a and C c are found at significantly different 13 C NMR chemical shifts ( 50/106 for 6 and  67/115 for 11) indicating σ+π coordination of the allyl moiety where the C a remains bonded in a σ-fashion, similarly to the σ-allyl precursors, and the double bond of the allyl moiety is π-coordinated to gold. [START_REF] Wang | The allyl ligand in 6 is more asymmetrically bonded than that seen in a related (N,C)-cyclometalated Pd(II) π-allyl complex, see for examples: a)[END_REF] This is in line with the larger difference in trans influence (C > P >> N) of the two ends of the chelating ligand in complexes 6 and 11 compared to 3-iPr and the following thermodynamic preference of having the higher trans influence σ-end of the allyl moiety trans to the lower trans influence end of the ancillary ligand (aryl-N and ArMe2N in complexes 6 and 11, respectively). The asymmetric coordination of the allyl ligand was also observed in the optimized structures of complexes 6 and 11. In both cases C a is bound significantly tighter to gold compared to C b and C c , indicating significant σ-character of the Au-C a bond. Moreover, the bonding situation in complexes 3-iPr, 6 and 11 was analyzed by the atoms-in-molecules (AIM) approach showing noticeable differences between the different complexes (Table 2). For complex 3-iPr, bond critical points (BCP) are found between both the terminal carbons of the allyl ligand and gold, whereas for complexes 6 and 11, BCP is only found between gold and the terminal carbon of the allyl moiety trans to N. Furthermore, in complex 3-iPr, the associated electron densities (ρ) and the Bader's delocalization indexes (δ) are very similar, indicating a highly delocalized allyl moiety. In contrast, in complexes 6 and 11, very different values are obtained when comparing C a -C b and C b -C c , showing a significant double bond character for the latter bond.

The dynamic properties of π-allyl complexes play an important role when they are employed in catalysis. [START_REF] Hartwig | Organotransition Metal Chemistry: From Bonding to Catalysis[END_REF] The large electronic dissymmetry provided by the ancillary ligand and the following σ+π coordination of the allyl moiety observed in complexes 6 and 11 may point towards a higher fluxionality in these complexes, but this was only observed experimentally for complex 6. [9,11,12] DFT calculations showed that for complex 6, the η 1 forms are ca 12 kcal mol -1 higher in energy than the η 3 form, and the free energy barriers for the involved coordination, rotation and re-coordination processes are not too high (<18 kcal/mol). The dynamic behavior of complex 6 was evident from its 1 H NMR spectrum. The two protons H a are found as one resonance in contrast to what is normally observed in π-allyl complexes (including that of 3 and 11, see Table 1). This phenomenon indicates that double bond decoordination followed by rotation around the AuCH2-CHCH2 and re-coordination occurs at the NMR timescale leading to the coalescence of the two H a resonances. The same phenomenon was not observed for the two H c , indicating that the η 3 -η 1 -η 3 interconversion only occurs trans to Aryl-C in line with the asymmetric σ+π bonding of the allyl moiety with the thermodynamic preference of having the σ-character trans to Aryl-N. Lowering the temperature to ca -80 o C, and thereby slowing down the η 3 -η 1 -η 3 interconversion process, led to the observation of the coexisting η 3 (with the resonances of the two H a now decoalesced) and η 1 forms of complex 6. In contrast, for complex 11, the difference in free energy between the η 3 and η 1 forms (∆G = 36.5 kcal mol -1 ) is three times as large as that of 6, explaining why no fluxionality was observed for complex 11.

Table 2. Atoms-in-molecules (AIM) molecular graphes for complexes 3-iPr, 6 and 11 with relevant bond paths, bond critical points (BCP, green circles), electron densities (ρ in e bohr -3 ) and delocalization indexes (δ).

Reactivity and catalytic applications of Au(III) -allyl complexes

Nucleophilic addition to π-allyl complexes is a key elementary step in transition metal catalysis, most notably in the well-known Pd-catalysed Tsuji-Trost reaction. [START_REF] Hartwig | Organotransition Metal Chemistry: From Bonding to Catalysis[END_REF] Most common is the outer-sphere nucleophilic addition to one of the terminal positions of the π-allyl (Scheme 3a, path I) leading to π-alkene complexes and ultimately the allylation products. The nucleophilic addition can also occur at the metal itself leading to σ-allyl complexes which can then undergo reductive elimination to release the allylation product (Scheme 3a, path II). Less common, but present in the literature, is the nucleophilic addition to the central position of the π-allyl leading to the formation of a metallacyclobutane (Scheme 3, path III). [20,21] The reactivity of Au(III) -allyl complexes 3 towards β-diketo enolates 14 was thoroughly investigated. [START_REF] Rodriguez | [END_REF]10] In all cases studied, the desired allylation products 15 originating from nucleophilic addition to the terminal position of the π-allyl were ultimately obtained in good yields (71-99%) together with the Au(I) dimer 16 (Scheme 3b). However, after close inspection of these reactions both experimentally and computationally, they turned out to be much more complex than first anticipated. Nucleophilic addition to all sites (paths I, II and III) turned out to be possible and competitive. Four combinations of Au(III) π-allyl complexes 3 and nucleophiles 14 were studied in detail by multinuclear low temperature NMR spectroscopy and DFT calculations. It was found that, depending on the combination of Au(III) π-allyl complex and nucleophile, additions to all the different positions were indeed possible and complexes 17-19 could all be authenticated in solution by NMR spectroscopy. Complexes 17 and 19 were also isolated and characterized crystallographically and by high resolution mass spectrometry (HRMS). NMR spectroscopy together with DFT calculations showed that the nucleophilic additions to the central position of the π-allyl and to gold itself were reversible, explaining why ultimately, the coupling products 15 originating from the nucleophilic addition to the terminal position is observed in all cases studied. The reaction of the Au(III) π-allyl complex 3-iPr-Me with the β-diketo enolate 14a was the most complex, but also the most fascinating. In this case, nucleophilic addition to all possible sites could be authenticated within the same reaction (Scheme 4). Initially, the formation of σ-allyl complex 18 and auracyclobutane 20 (in a 3/2 ratio) was observed. Warming the reaction mixture to 25 o C for ca. 30 min before cooling back to -20 o C led to the quantitative formation of Au(I) π-alkene complex 21. Finally, the C-allylated product 15a was released from the Au(I) π-alkene complex 21 at room temperature with concomitant formation of Au(I) dimer 16. Scheme 4. Reaction of the π-allyl Au(III) complex 3-iPr-Me with the β-diketo enolate 14a.

As mentioned previously, nucleophilic addition to the terminal position of Pd(II) π-allyl complexes usually dominates, and nucleophilic addition to the central position remains a curiosity. [20,21] To rationalize the different reactivity of Pd(II) and Au(III) πallyl complexes, the Molecular Orbitals relevant for the nucleophilic additions to [(P,C)Au(π-allyl)] + 3-iPr and the related [(Ph3P)2Pd(II)(π-allyl)] + complex 3-Pd were computed (see Figure 2 for selected data). [10] Significant differences between the Au(III) and Pd(II) π-allyl complexes were discovered. While in both cases, the LUMO has large contributions on the terminal carbons (Cterm) and the metal itself, only in 3-iPr vacant orbitals with significant contributions of the central carbon (Ccent, 15.5 %) can be found at accessible energies (LUMO+2, only 0.6 eV above the LUMO). In contrast, for complex 3-Pd, it is not before LUMO+15, lying 2.2 eV above the LUMO, where a significant participation of Ccent is found. This explains why nucleophilic attack to the central position of the -allyl and formation of auracyclobutanes is so much more favourable than palladacyclobutanes. The limited stability of (N,C) Au(III) π-allyl complex 6 prevented its reactivity towards nucleophiles to be studied in detail. However, attempts at obtaining single crystals of a gem-dimethylallyl complex led to the authentication of a water adduct which may result from nucleophilic addition to the central position of the π-allyl followed by protodeauration. [22] The stoichiometric reaction of (P,C) Au -allyl complexes 3 with carbon based nucleophiles demonstrated the feasibility of C-C bond formation via nucleophilic attack to Au(III) -allyls. While being fundamentally interesting, it was not a synthetically relevant method for C-C bond formation due to the use of stoichiometric amounts of gold. In order to achieve catalytic allylation via Au(III) -allyl intermediates, a Au(I)/Au(III) redox cycle was envisioned similar to Ir, Ru and Pd catalysts. Here, (P,N) Au(III) allyl complexes were considered as good candidates as the (P,N) ligand enable to cycle between Au(I)/Au(III) as has been shown to great success in catalytic C-C, C-N and C-S coupling reactions. [14] To demonstrate the feasibility of C-C coupling, the (P,N) Au(III) -allyl complex 11 was reacted stoichiometrically with Nphenyl indole as a synthetically relevant nucleophile (Scheme 5). [11] The C3 allylated product was obtained as the only product, as expected with the more nucleophilic nature of the C3 position of indoles. The ability of the Me-Dalphos ligated Au complex to cycle through Au(I)/Au(III) catalytic cycles combined with the formation of a single product in the stoichiometric reaction made it a good candidate for catalytic allylation. After optimization of the reaction conditions, it was found that the allylation could proceed with catalytic amounts of (P,N)AuCl (Scheme 6). For the catalytic transformation, AgSbF6 was used to carry out the functions of both halide abstraction from the initial (P,N)AuCl and to replace BCF as a Lewis acid to promote the oxidation of the -alkene complex. While the transformation worked well with a variety of allyl substrates (alcohols, acetates, carbonates, phosphates and halides), the investigation of the scope was achieved on allyl acetates and alcohols which are the most valuable synthetically. Control experiments showed that it was indeed the gold catalyst that promoted the reaction and only traces of the allylation product were observed without it. Scheme 6. Ligand-enabled Au(I)/Au(III) catalytic allylation of indoles with allyl electrophiles.

To provide further proof for the involvement of Au(III) -allyl intermediates in the catalytic reaction,  and  substituted allyl alcohols and acetates were investigated (Scheme 6). It was expected that the same ratio for linear and branched products is obtained if the reaction proceeds via -allyl intermediates as is observed with Pd, Ir, Rh catalysts. [23] Subjecting the  and  substituted allyls to the catalytic conditions indeed resulted in the formation of the same branched C3 allylated indoles independently of whether the  or  position was substituted on the initial allyl. All tested substrates were completely selective for the branched isomer, with the exception of ethyl allyl acetate where a 95:5 ratio was obtained. To further probe the generality of the transformation, substituted  phenyl allyl acetates were tried with indoles containing different substitutions on the nitrogen and indole backbone. In all cases, complete selectivity for the C3 allylated branched product was observed with good to excellent yields (62-92%). Only when the C3 position of the indole was sterically hindered or the nucleophilicity lowered with electron withdrawing groups were lowered yields observed (35-56%). Good functional group compatibility was observed with the reaction tolerating C-F, C-Br and C-Si bonds. Notably when the C2 position of the indole was methylated, the allylated product was still obtained in high yields (76%) while for the previously reported arylation of indoles with the same (P,N)AuCl catalyst, no product was obtained. [14a] The selectivity for the branched product could be elucidated when the Au(III) -allyl intermediate was investigated computationally. The most stable optimized structure for the methyl substituted -allyl was found with the Me group in the pseudoequatorial position, trans to the phosphorus (Figure 3). The LUMO showed a higher contribution to this carbon (20.7%) compared to the carbon cis to the phosphorus (13.9%). The higher contribution of the LUMO combined with the easier accessibility due to the less sterically demanding NMe2 groups compared to the PAd2 group offer an explanation to the observed selectivity for the branched product. When comparing these results with earlier reports on the allylation of indoles via transition metal -allyls, the complementary nature of Au becomes clear. In the allylation of indoles with Pd catalysts, the linear product is obtained, while with Ir, usually branched products are observed. As for Ru catalysts, the selectivity is highly dependent on the ligand, with selectivities in favour of both the linear and branched product being reported as well as the formation of mixtures. [23]

Summary and outlook

During the last three years, Au(III) -allyl complexes have emerged as a very interesting and attractive class of compounds. The use of (P,C), (N,C) and (P,N) bidentate ligands was shown to impart enough stability to enable the characterization or even the isolation of such complexes. Various synthetic routes, with or without oxidation of gold, have been developed. Detailed analyses of their structure revealed either symmetric  3 or asymmetric +-coordination of the allyl moiety to gold, depending on the ancillary ligand. Rich reactivity towards nucleophiles was also substantiated, with possible additions to the terminal and central positions of the -allyl, as well as to gold itself. The possibility to achieve catalytic transformation via Au(III) -allyl complexes has also been demonstrated. Most noteworthy are the specific features of the Au(III) -allyl complexes, compared to the isoelectronic Pd(II) species, making them valuable tools.

These first achievements are very exciting and for sure, much remains to be discovered and developed when it comes to Au(III) -allyl complexes. The variety of ancillary ligands at Au(III) can certainly be further expanded, enabling to finely tune the structure and reactivity of the -allyl moiety at Au(III). [13] Other catalytic transformations than the allylation of indoles are also most probably achievable via Au(III) -allyl complexes. Selectivities complementary to the other transition metals may be encountered and transformations involving nucleophilic attack to the central position of the -allyl may be developed. Recent pioneering contributions on enantioselective Au(I)/Au(III) catalysis also give credence to the possibility to achieve asymmetric gold-catalyzed allylation reactions. [24][25][START_REF] Rodriguez | For reviews dealing with asymmetric gold catalysis[END_REF]

Scheme 3 .

 3 Scheme 3. Formation of Au(III) -allyl complexes 3 and 11 from their corresponding Au(III) -cyclopropyl compelexes 9 and 10 by halide abstraction.

Figure 2 .

 2 Figure 2. Computed energy profile for the ring-opening of the (P,C) Au(III) -cyclopropyl complex 9-Ph to give the corresponding -allyl complex 3-Ph. G values in kcal/mol, main distances in Å. Profile computed in the gas phase at the B3PW91/SDD+f(Au), 6-31G** (other atoms) level of theory. And main IBOs accounting for the ring-opening of the distal (CC) bond along the reaction path (numbers relate to the fraction of electrons of the doubly occupied orbital assigned to each atom).

Scheme 2 .

 2 Scheme 2. Formation of the (N,C) Au(III) -allyl complexes 13 from their corresponding Au(III) -cyclopropyl complexes 12 by halide abstraction.

Scheme 3 .

 3 Scheme 3. (a) Possible reactive sites for the nucleophilic addition to transition metal -allyl complexes. (b) Reactivity of complexes 3 (R = iPr or Cy, R' = H or Me) towards β-diketo enolates 14 (R'' = OEt, OtBu or Ph). (c) Examples of characterized complexes (17-19) showing nucleophilic addition to the central and terminal position of the -allyl, as well as to gold.

Figure 2 .

 2 Figure 2. Key Molecular Orbitals (MO) computed for complexes 3-iPr and 3-Pd. Energy of the MOs in eV, cutoff: 0.05. Contribution of the main atoms in each MO in %.

Scheme 5 .

 5 Scheme 5. Stoichiometric reactivity of the (P,N) Au -allyl complex 11 with indole as the nucleophile.

Figure 3 .

 3 Figure 3. (a) Optimized geometry of the Au(III) π-allyl complex (P,N)Au[H2CC(H)CH(CH3)] 2+ at the SDM(CH2Cl2)-B3PW91/SDD + f(Au),6-31G** (other atoms) level of theory (distances in Å). Simplified perspective view as inset chart. (b) Plot of the LUMO (cutoff: 0.05) with key atomic contributions in %.

Acknowledgements

Financial support from the Centre National de la Recherche Scientifique, the Université de Toulouse and the Agence Nationale de la Recherche (ANR-19-CE07-0037) is gratefully acknowledged. This study is a part of the FRINATEK project 315004 funded by the Research Council of Norway (stipend to M. H.). Special gratitude is expressed to Karinne Miqueu for all computational studies and extremely stimulating collaboration. The following research associates are also warmly thanked for their invaluable contribution: J. Rodríguez, A. Tabey, G. Szalóki, E. Daiann Sosa Carrizo, Y. García-Rodeja, P. Lavedan, N. Saffon-Merceron and S. Mallet-Ladeira.


Guillaume Marthe
email: guillaume.marthe@insa-lyon.fr

Claire Goursaud
email: claire.goursaud@insa-lyon.fr

Romain Cazé
email: romain.caze@univ-lille.fr

Laurent Clavier
email: laurent.clavier@imt-nord-europe.fr

On exploiting the synaptic interaction properties to obtain frequency-specific neurons

Keywords: Low power design, Synaptic interactions, Temporal integration

Energy consumption remains the main limiting factors in many IoT applications. In particular, micro-controllers consume far too much power. In order to overcome this problem, new circuit designs have been proposed and the use of spiking neurons and analog computing has emerged as it allows a very significant consumption reduction. However, working in the analog domain brings difficulty to handle the sequential processing of incoming signals as is needed in many use cases. In this paper, we use a bio-inspired phenomenon called Interacting Synapses to produce a time filter, without using non-biological techniques such as synaptic delays. We propose a model of neuron and synapses that fire for a specific range of delays between two incoming spikes, but do not react when this Inter-Spike Timing is not in that range. We study the parameters of the model to understand how to choose them and adapt the Inter-Spike Timing. The originality of the paper is to propose a new way, in the analog domain, to deal with temporal sequences.

I. INTRODUCTION

The Internet of Things (IoT) brings a lot of promises for everyday life applications [START_REF] Mehta | Internet of Things: Vision, Applications and Challenges[END_REF]. One significant limiting issue remains the energy consumption of the devices: lifespans of several years or tens of years cannot be achieved as soon as a minimum of intelligence has to be deported into the devices. However, as most devices involved in the IoT only send data sporadically. Thus, to reduce this consumption a key solution is to let them sleep as much as possible and only wake them up when they need to interact. Nonetheless, it is still important to be able to address the devices (whether it is the network, i.e. a gateway, or a neighbouring device) at any time without depending on a global synchronisation of the system. This can be achieved through the use of wake-up receivers (WuR), circuits whose role is to continuously monitor the channel and wake up the main receiver only when a specific signal is received [START_REF] Demirkol | Wakeup receivers for wireless sensor networks: benefits and challenges[END_REF]. WuR leads to a decrease of the energy wasted between communications.

An essential condition for their use is that the listening consumption is extremely low, but many challenges remain unresolved [START_REF] Patrick | Low-Power RF Wake-Up Receivers: Analysis, Tradeoffs, and Design[END_REF]. Up to now, those WuR still rely on micro-controllers that can perform pattern recognition but consume power around 200µW [START_REF] Seyed | Performance Analysis and Energy Optimization of Wake-Up Receiver Schemes for Wireless Low-Power Applications[END_REF]. This is too high to allow the battery of the WuR to last the intended lifespan of the node [START_REF] Yetgin | A Survey of Network Lifetime Maximization Techniques in Wireless Sensor Networks[END_REF].

Meanwhile, Spiking Neural Networks (SNN) is a technology that has emerged in the field of signal processing in recent years as a low power consumption solution [START_REF] Hopkins | Spiking Neural Networks for Computer Vision[END_REF]. They are mainly used for image recognition [START_REF] Diehl | Unsupervised learning of digit recognition using spike-timing-dependent plasticity[END_REF] or to study model brain activity in neuroscience [START_REF] Zhang | Spike-Based Indirect Training of a Spiking Neural Network-Controlled Virtual Insect[END_REF]. This bio-inspired systems are based on transistors to mimic neuron behavior. They can compute as fast as actual devices but consume less [START_REF] Sourikopoulos | A 4-fJ/Spike Artificial Neuron in 65 nm CMOS Technology[END_REF].

To reduce a step further the energy consumption, we consider in this paper specific fully analog circuits that reproduce the neurons' behavior [START_REF] Danneville | A Sub-35 pW Axon-Hillock artificial neuron circuit[END_REF][START_REF] Marthe | Wake-up radio receiver based on Spiking Neurons for detecting activation sequence[END_REF]. Until now, these neurons have been used for combinatorial approaches that do not involve time sequences. They consume even less energy than traditional SNNs but are more limited on applications because they restrict the usage we can make of the neuron. For example, it is highly complex to use synaptic delay without a digital circuit. The sequential applications such as vision, video recognition or signal processing were up to now impossible to realise. In this paper, we show that it is possible to introduce this time dependency. For this, we propose to exploit the biological phenomenon of synaptic interaction and its memory effect [START_REF] Cazé | All neurons can perform linearly nonseparable computations[END_REF]. The phenomenon we use corresponds in biology to the effect of dendrites on the cortical pyramidal neurons [START_REF] Abrahamsson | Thin Dendrites of Cerebellar Interneurons Confer Sublinear Synaptic Integration and a Gradient of Short-Term Plasticity[END_REF]. We exploit this phenomenon to make the neuron reacting to a specific time pattern and evaluate the neuron parameters on the selectivity.

The paper is organized as follows: section II presents the neurons and system models and how they are simulated. Section III describes the neurons behavior and their implementation. The choice of the parameters and their impact are then studied in Section IV. Finally, Section V concludes the paper.

II. MODELS

A. LIF neuron with classical synapses

A biological neuron receives spikes from the incoming synapses, aggregate them within its membrane voltage, and then sends forward a spike if the voltage has reached a given threshold. An illustration of neurons and synapse can be found in Fig. 1.

A bio-inspired neuron is based on the same principle. It is modeled by its reaction according to the incoming electric spike train. The internal voltage varies depending on the incoming current. There exists several models to depict the relationship between the incoming current and the voltage. The Leaky Integrate and Fire (LIF) model is the reference, widely considered in the literature [START_REF] Hunsberger | Spiking Deep Networks with LIF Neurons[END_REF], it is expressed by :

C m dv dt = -g L (v(t) -v rest ) + I(t) (1)
with C m the membrane capacity, v(t) the membrane potential at time t, v rest = -65mV the resting membrane potential and g L the membrane conductance. I(t) models the timevarying synaptic input current, which can be written in case this incoming signal is a sequence of spikes as:

I(t) = i δ(t -t i ) (2)
with t i corresponding to the different spike times, and δ the Dirac function.

Each time the voltage reaches the threshold v th a spike is triggered and the voltage is reset to v rest . The gray curve in Fig. 2a represents the temporal evolution of the voltage over time when receiving a single spike. In a LIF, we observe an instantaneous growth at the reception of the spike, then an exponential decrease due to the leakage in the neuron.

B. Saturating LIF

The LIF model is a useful, yet simplified model. Indeed, there is an additional phenomenon that is usually neglected, but that we will exploit here. In this model, called Saturating LIF (SLIF), the synapses are now saturating synapses. This models the fact that the synapse parameters are temporary modified after receiving a spike. In this case, the current no longer depends only on the current input but also on what has happened before. Thus, I(t) (2) is now given by I s (t) such that:

I s (t) = g s (t)(E s -v(t)) (3)
The synaptic current depends on the difference between v(t) the neuron voltage and E s = 0 mV the synaptic reversal potential. g s (t) is the time-varying synapse conductance depending on the previous incoming spike pattern. It is bounded between 0 and the saturation value that we set at 100 pS for our simulations. Each time the synapse receives an incoming spike, g s (t) will rise to its maximal value, and decays exponentially following :

dg s (t) dt = -g s (t) τ s (4)
with τ s the synaptic time constant. This is called "Saturating Synapses" because we make g s (t) saturate to a specific value.

The consequence of this saturation is that the dynamic of the tension is constrained and makes it slower to reach the target membrane voltage as we can see on the grey curve in Fig. 2b.

We can observe on this curve in Fig. 2 that the voltage increases smoothly, and then decreases until it reaches its resting potential. We are going to show how this can be used to create a short term memory for temporal computations.

III. PROPOSITION

Our objective is to use this neuron structure to spike only at the reception of a specific firing rate among any incoming spike train. This allows to spike only for a given range of interspike duration or, said differently, to filter a specific period of spikes. To do so, we want to reach the highest voltage if two incoming spikes arrive within a given Inter-Spike Timing (IST) range.

To clarify our proposal, we propose to study in more details the behavior of a neuron with this saturating synapse. To evaluate the phenomenon we send two consecutive spikes to the two types of neurons LIF and SLIF spaced by different IST which represent the time between the reception of those two spikes.

The results are shown in Fig. 2 with 4 curves each, corresponding to different configurations. The leftmost (grey) one When a classical LIF neuron receives a spike, its membrane potential suddenly rises. Then, its potential decays due to the leaky aspect of the neuron. When a second spike is sent, the potential rises again instantly of the same amplitude value. The highest amplitude is reached when the IST is minimal. Indeed, the potential leak is lower when it has less time to decay. On the opposite, if the second spike is sent a long time after the first one, the potential has decayed more, and then the amplitude is lower than in the first case. Thus, the amplitude decays when the IST increases.

The response of the SLIF is presented on Fig. 2b in the same conditions. With the saturating synapses, if a second spike is received right after the first one, the synapse is still saturated and this spike does not make the potential rise of the same quantity the first one did. We can observe that, contrarily to the LIF case, the increase in the amplitude of the membrane voltage after the reception of second spike is not constant and depends on the IST. For the second curve from the left, the black one, the spikes are too close and the synaptic saturation limits the impact of the second spike, which will have only a small impact on the membrane potential. For the rightmost one (green), the spikes are too far from each other and the leaky aspect of the neuron makes the potential drop significantly before receiving the second spike. This will reduce the highest reached amplitude of the membrane voltage.

In addition, we have plotted the amplitudes reached as a function of IST in Fig. 3. With classical synapses we have had an exponentially decreasing and monotonous function. For the , g L = 10 -5 S.cm -2 and τ s = 0.01ms SLIF, we can see that the maximal amplitude is reached for an IST of 3ms and that there are two phases. For an IST lower than 3ms, the saturation leads to a lower amplitude than the optimum, with a linear rise until the favorite IST, and then a decrease due to the leaky part of the neuron.

To further illustrates the impact of the saturating synapse, let us represent the value of the spiking threshold by the horizontal dashed line. We clearly understand that the neuron will spike only for a given range of IST so that the threshold can be used to discriminate a fine range of IST values. In this simulation, the neuron was spiking if the input spikes were spaced by an IST between 2 and 3.5 ms. The width of this range of ISTs will be called Timewidth (TW). Finally, we can note that the IST of 3 ms leads to the highest increase.

The shape of the curve in Fig. 3 (increase time, decrease time, maximum amplitude and when it occurs) are influenced by the parameters of the equations:

• C m , the membrane capacity which permits to scale the whole dynamic of the neuron. • g L , the conductance of the neuron which regulates the leaky phenomenon for the membrane voltage. The bigger g L is, the fastest the voltage will return to its resting value v rest . • g S , the synaptic conductance. Its value is impacted by the parameter τ S and modifies the speed of evolution of the membrane potential, when a spike is received. Therefore, τ S is the parameter which permits to choose how fast g S will return to its default value, here 0. In our model, we consider that g S value is clipped between 0 and 100pS.

By modifying the previous parameters, we can affect the neuron response. For example, in Fig. 4, the new parameter set permits to switch from a millisecond IST range to a microsecond range and thinned the curve, while keeping the amplitude constant.

We can see in Fig. 4 the 3 main parameters describing the neuron behavior: the amplitude, the position of the TW and its width. The amplitude corresponds to the difference between the maximum and minimum voltage reached for the considered ISTs. To measure the TW, we set a threshold 0.1 mV under the maximal amplitude, and we look at the range of IST that makes the membrane voltage exceeds this threshold.

The objective is now to optimize the parameter set. The goal is to obtain the highest amplitude range (so as to be more robust to perturbations), while reducing TW (so as to be more precise in the selected rates). The favorite IST to target depends on the frequency pattern chosen.

IV. PARAMETERS STUDY Our simulations were realised using the Python library Brian2 [START_REF] Stimberg | Brian 2, an intuitive and efficient neural simulator[END_REF]. We have seen that the SLIF is a model that permits to spike only for a specific range of IST values, depending of our parameters. Thus, we study here the impact of the parameters on both the amplitude and TW. To realize those studies, we have set the parameters at a reference set which is C m = 10 -4 µF.cm -2 , g L = 10 -4 S.cm -2 and τ s = 0.1ms. We then varied two parameters simultaneously between 0.1 to 10 times its original value.

As a reminder, we try to optimise the thinness of the TW and the maximal amplitude to distinguish better the responses to different ISTs.

A. Membrane capacity

As it directly impacts the dynamic of the neuron, we first analyze C m to determine the order of magnitude of the favorite ISTs and then the other parameters to adjust the amplitude value and the TW. C m is the membrane capacity and it is multiplied to the dynamics of the neuron potential dv dt . We can see in Fig. 5a and 5c that when C m increases, the maximal amplitude of the voltage rises.

For TW, we can see in Fig. 5b and 5d that if C m increases, TW is reduced. Thus increasing C m permits to have a more precise identification of the spike rate. Finally, Fig. 6a and6b show that the position of the favorite IST evolves in the same direction as C m . When it increases, the IST take larger values.

B. Membrane conductance

g L is the membrane conductance which defines the dynamics of the membrane potential leak. It is directly proportional to how fast the membrane voltage comes back to v rest after receiving a spike. We can see in Fig. 5a and 5e that when g L increases, the maximum amplitude decreases as intended. Indeed, the more g L rises, the more the leak phenomenon is strong, and so, the sooner we come back to the resting potential. Thus, the IST is reduced and the TW is thinner, as we can see respectively in Fig. 6a and6c, and Fig. 5b and5f.

C. Time constant

The value of g L is used to scale the maximal amplitude we want to reach. We can then modify τ S to adjust the TW more precisely.

τ S is the time constant of g s , which is the synaptic conductance. After receiving a spike, g s instantly rises until it reaches its upper bound. It will then decay to its lower bound. The speed of the decay is ruled by τ S . τ S is impacting g s which can be associated as a weight in the model. Thus, when τ s has a low value, g s varies faster and then get bigger variations. This enhances the effect of the second spike on the membrane potential, because g s saturates less than for a higher value of τ S as shown in Fig. 5c and5e. As it rules the membrane voltage gain from the second spike, τ S is influencing TW. The higher τ S gets, the wider the TW becomes. Indeed, g s varies less so the amplitude of potential decays slower. Thus, we will use τ S to adapt the range of IST making the neuron spike.

However, we can observe in Fig. 6b and 6c that the evolution of τ s has almost no impact of the favorite IST. That is another reason to scale it at the end, to only change the TW when the IST is set.

Finally, one may note that interestingly, there is no need to compromise between amplitude range and TW, improving one also improves the other.

V. CONCLUSION

In this paper, we studied a new bio-inspired spiking neuron model based on saturating synapses called SLIF. We showed that our neuron responds differently based on the Inter-Spike Timing of the incoming spikes. Unlike the LIF neuron with classical synapses, the SLIF allows us to reach a maximal membrane voltage amplitude for an IST that can be configured. We have exploited this natural phenomenon to transform a spiking neuron and its synapses into a temporal filter that emits a spike only for a chosen IST. Then, we studied the impact of the main parameters of our model on the amplitude of the membrane voltage and the timewidth of the IST that can represent the selectivity of the model. Future works will analyze how SLIF can allow to discriminate spike trains based on the delay between each spikes to develop an ultra lowpower wake-up radio.

Figure 1 :

 1 Figure 1: Diagram of a biological neuron

Figure 2 :

 2 Figure 2: Response of the neuron to different ISTs

Figure 3 :Figure 4 :

 34 Figure 3: Highest amplitude reached by a SLIF and a classical LIF neurons for different ISTs where C m = 10 -5 µF.cm -2 , g L = 5 • 10 -5 S.cm -2 and τ s = 0.9ms

  Maximum amplitude for the favorite IST and τs = 0Timewidth around the favorite IST and τs = 0.1ms Maximum amplitude for the favorite IST and gL = 10 -4 S.cm -2 Timewidth around the favorite IST and gL = 10 -4 S.cm -2 Maximum amplitude for the favorite IST and Cm = 10 -4 µF.cm -Timewidth around the favorite IST and Cm = 10 -4 µF.cm -2

Figure 5 :

 5 Figure 5: Evolution of Amplitude and TW according to C m , g L and τ s

  Favorite IST with gL = 10 -4 S.cm -2

  Favorite IST with Cm = 10 -4 µF.cm -2

Figure 6 :

 6 Figure 6: Evolution of Amplitude, IST and TW according to g L and τ s

ACKNOWLEDGMENT

This work was supported by the research project U-WAKE, financed by the French National Research Agency (ANR). BIBLIOGRAPHY