Muhammed R Pac

Micky Rakotondrabe

Sofiane Khadraoui

Dan O Popa
email: popa@uta.edu

Philippe Lutz

GUARANTEED MANIPULATOR PRECISION VIA INTERVAL ANALYSIS OF INVERSE KINEMATICS

The paper presents a new methodology for solving the inverse problem of manipulator precision design. Such design problems are often encountered when the end-effector uncertainty bounds are given, but it is not clear how to allocate precision bounds on individual robot axes. The approach presented in this paper uses interval analysis as a tool for uncertainty modelling and computational analysis. In prior work, the exponential formulation of the forward kinematics map was extended to intervals. Here, we use this result as an inclusion function in the computation of solutions to set-valued inverse kinematic problems. Simulation results are presented in two case studies to illustrate how we can go from an uncertainty interval at the end-effector to a design domain of allowable uncertainties at individual joints and links. The proposed method can be used to determine the level of precision needed in the design of a manipulator such that a predefined end-effector precision can be guaranteed. Also, the approach is general as such it can be easily extended to any degree-of-freedom and kinematic configuration.

INTRODUCTION

The success of automated assembly by robotic manipulators is highly dependent on the precision of the positioning mechanisms employed in the kinematic design of the robot. The importance of precision becomes more prominent when the desired operational accuracy is in micro/nano scale as in micro-assembly and nano-manipulation applications. Most of the parametric uncertainties that are negligible in conventional robotics become the predominant error sources in micro/nano applications. With the emergence of micro/nano-robotics in the last decade, there is a growing demand for design guidelines describing how to build these robots based on application specific precision criteria. Since the current methodology have not addressed this problem well, we propose a new approach to the kinematic analysis and design of robots using interval analysis.

In late 80's and early 90's, modelling of errors in robot kinematics was a popular topic [START_REF] Benhabib | Computer-aided joint error analysis of robots[END_REF][START_REF] Lin | Direct volumetric error evaluation for multi-axis machines[END_REF][START_REF] Shin | A statistical analysis of positional errors of a multiaxis machine tool[END_REF][START_REF] Soons | Modeling the errors of multi-axis machines: a general methodology[END_REF][START_REF] Vaishnav | A general procedure to evaluate robot positioning errors[END_REF][START_REF] Veitschegger | Robot accuracy analysis based on kinematics[END_REF]. However, these works are limited to analysing the effects of general error transformations and do not address how they come about or how to choose or design the parameters of a manipulator for a given end-effector preci-sion. On the other hand, mechanical design of multi-axis machines based on the accuracy required at the tool tip has been investigated for precision machine design. A concept called error budget was proposed to account for the effect of each error source on the tool accuracy [7, p. 61], [START_REF] Frey | Robotics and Automation Handbook[END_REF]. Using first order and small angle approximations to simplify the homogeneous transformation matrices describing geometric errors in the mechanism of a machine, an overall kinematics map can be created to analyse the effects of error terms on the tool accuracy. The results of this analysis provide the designer with insight into how to allocate mechanical tolerances to the individual system components. However, there is no systematic way of doing this allocation due to the fact that this type of an inverse problem is difficult to pose and solve.

A recent work in [START_REF] Das | Precision evaluation of modular multiscale robots for peg-in-hole microassembly tasks[END_REF] studies modular robotic chains made up of individual axes with link/joint uncertainties for several different configurations. It was shown that some of the kinematic configurations provide more successful operation. Also, joint position uncertainties were shown to manifest themselves as end-effector inaccuracy in different magnitudes depending on the kinematic design. The analysis of error propagation in this work was done using Monte Carlo simulations which suffer from inability to provide guaranteed closed form solutions since Monte Carlo method can only solve for sample points and produce sets of points in the solution space.

Finding guaranteed solutions to a set of equations with uncertain data is possible with interval analysis. Interval analysis is a mathematical tool for computation of rigorous bounds on solutions to ideal model equations when the input arguments of the model are represented as intervals instead of point values. It extends the model equations to the interval domain and allows for analytical and computational handling of uncertain data without having to assume a distribution for it or to sample it. It also helps avoid the complex mathematical formulations involving distribution functions [START_REF] Moore | Introduction to Interval Analysis[END_REF].

Intervals were used in [START_REF] Wu | Uncertainty analysis and allocation of joint tolerances in robot manipulators based on interval analysis[END_REF] to model uncertain physical parameters of a robot and to find its forward kinematics map using Denavit-Hartenberg (D-H) notation. Optimization of the D-H parameters was done to minimize volumetric end-effector error while optimizing the cost of precision. While this work nicely approaches the problem from a mechanism design perspective, it fails to consider the fact that a mechanism design cannot be optimized practically without considering the uncertainty in joint positions. Interval analysis was also used in [START_REF] Merlet | Solving the forward kinematics of a goughtype parallel manipulator with interval analysis[END_REF] to find the multiple solutions to the forward kinematics problem of parallel robots. Then, [START_REF] Merlet | Interval analysis for certified numerical solution of problems in robotics[END_REF] extends the method to finding the robot parameters that guarantee a singularity-free workspace.

In our prior work [START_REF] Pac | Interval analysis for robot precision evaluation[END_REF], the exponential formulation of the forward kinematics map for serial manipulators was extended to intervals. This makes it possible to use interval analysis to find guaranteed precision bounds on the end-effector pose given the uncertainty of the kinematic parameters. The contribution of the current paper is that we now propose a new method that can solve the inverse problem of bounding the allowable uncertainty in kinematic parameters of a manipulator based on given end-effector precision specifications. Besides precision machine designers, this method bears an importance for those roboticists who have to design a manipulator using elementary building blocks. For instance, custom design of multi-axis precision manipulators using individual single-axis stages is a common practice in the micro-assembly area [START_REF] Dechev | Microassembly of 3-d microstructures using a compliant, passive microgripper[END_REF][START_REF] Probst | A microassembly system for the flexible assembly of hybrid robotic mems devices[END_REF][START_REF] Das | A multiscale assembly and packaging system for manufacturing of complex micro-nano devices[END_REF]. The cost of such stages increase significantly with the increase in motion precision. For a given application, therefore, determining the level of precision required in each axis is an important yet insufficiently addressed consideration.

Next section provides an overview of [START_REF] Pac | Interval analysis for robot precision evaluation[END_REF], the forward kinematics problem with uncertain parameters. Then, we discuss how the precision manipulator design problem can be posed via interval analysis of inverse kinematics. We present simulation results illustrating the use of our method for a simple 2-link manipulator and a 3-DOF PPR stage, a commonly used robot in precision assembly. Finally, the paper is concluded with references at the end.

FORWARD KINEMATICS WITH UNCERTAIN JOINT PA-RAMETERS

Kinematic description of a robot involves certain parameters such as link lengths and joint axis vectors. In an ideal robot model, these are all assumed to be perfectly precise values. A practical implementation, however, requires calibrating the robot to minimize the uncertainty in the knowledge of such constants. Even with calibration, identification of kinematic parameters can only be done with a limited precision. Robot kinematics also involves joint position variables which are known to a limited precision as well. Therefore, there needs to be a kinematic model that can handle these uncertainties with ease and efficiency.

Using the screw-based exponential formulation [START_REF] Murray | A Mathematical Introduction to Robotic Manipulation[END_REF] and interval analysis [START_REF] Moore | Introduction to Interval Analysis[END_REF], we formulated the interval extension of forward kinematics map in [START_REF] Pac | Interval analysis for robot precision evaluation[END_REF]. The advantage of the exponential formulation over D-H parameters is that the joint parameters can be described with respect to the world reference frame which is the frame of reference used in the description of precision metrics. We model uncertain parameters and variables of a robot with intervals. For example, a (real) interval such as x [START_REF] Benhabib | Computer-aided joint error analysis of robots[END_REF][START_REF] Lin | Direct volumetric error evaluation for multi-axis machines[END_REF] is a closed set of real numbers represented by the two boundary values, 1 is called the left endpoint and 2 is called the right endpoint. In this paper, we follow the notation in [START_REF] Jaulin | Applied Interval Analysis[END_REF] and denote intervals with square bracketed letters.

For a revolute joint, the interval extension of the homogeneous transformation matrix, [T], can be represented with the function in [START_REF] Benhabib | Computer-aided joint error analysis of robots[END_REF] where [ω] is the joint axis vector, [ ω] is the skewsymmetric matrix of [ω], [θ ] is the joint displacement, and [q] is a point along the joint axis. Note that in exponential formula-tion, joint transformations are with respect to the base reference frame.

[T]([ω],[θ ],[q]) e [ ω][θ ] (I -e [ ω][θ ] )[q] 0 1 (1)
In equation ( 1)

, [T]([ω],[θ ],[q]
) is called an inclusion function since it covers all possible homogeneous transformation matrices T(ω,θ ,q) of the special Euclidean group SE(3) that can be obtained by varying the arguments ω, θ , and q in the respective ranges of [ω], [θ ], and

[q].
Similarly for a prismatic joint, the interval extension for the joint transformation is represented with the inclusion function in [START_REF] Lin | Direct volumetric error evaluation for multi-axis machines[END_REF] where [d] is the joint displacement and [v] is the joint axis vector. The rotation part of the transformation denoted with [R]is ideally a 3 ×3 identity matrix. In an actual linear axis, however, tilt error of the stage may also play an important role. Hence, this part can be modelled as a general rotation transformation

[R] = e [ ω][θ ] . [T]([d],[v]) [R] [d][v] 0 1 (2)
Finally, interval extension of the forward kinematics map of a serial manipulator with n joints can be written as

[f]([x]) = [T 1 ]•••[T n ]•[f]([x 0 ]) (3)
where [f] represents the inclusion function for the forward kinematics map and [x]is the vector of joint parameters and variables.

For further details on the forward kinematics map, the reader is referred to [START_REF] Pac | Interval analysis for robot precision evaluation[END_REF].

INTERVAL ANALYSIS OF INVERSE KINEMATICS

Computing the joint angles of a manipulator for a given end-effector pose is called inverse kinematics. There are two main types of solution to this problem: closed-form solutions and numerical solutions [20, p. 106]. Closed-form solutions are based on analytical expressions of the inverse relationship between joint angles and end-effector pose. Due to the complex nonlinear nature of robot kinematic equations, finding a closedform solution is difficult in general. On the other hand, numerical solutions rely on the description of forward kinematics map and repeated evaluation of approximate joint angles until the desired end-effector configuration is sufficiently approached.

Inverse Kinematics with Joint Parameter Uncertainty: An Example

Consider the simple example shown in Fig. 1 where a twolink planar manipulator with revolute joints is depicted. The

l 1 ±δl 1 θ 1 ±δθ 1 θ 2 ±δθ 2 l 2 ±δl 2 B x B y (x E , y E ) FIGURE 1: TWO-LINK MANIPULATOR WITH UNCERTAIN LINK LENGTHS AND JOINT ANGLES.
knowledge of the link lengths and joint positions are assumed to be uncertain to the extent denoted by δ l i and δ θ i for i = 1,2, respectively. Given the nominal link lengths l i and joint positions θ i , the corresponding interval arguments can be expressed as

[x]      [l 1 ] [θ 1 ] [l 2 ] [θ 2 ]           [l 1 -δ l 1 ,l 1 +δ l 1 ] [θ 1 -δ θ 1 ,θ 1 +δ θ 1 ] [l 2 -δ l 2 ,l 2 +δ l 2 ] [θ 2 -δ θ 2 ,θ 2 +δ θ 2 ]      . ( 4
)
It was shown in [START_REF] Pac | Interval analysis for robot precision evaluation[END_REF] that the rotation part of the revolute joint transformation in (1) reduces to the interval extension of the Rodrigues' formula when the joint axis is a real (point-valued) vector as in the case of this example. That is,

R(ω,[θ ]) e ω[θ ] = I + ω sin([θ ]) + ω2 (1 -cos([θ ])) (5)
where ω = (0,0,1) T is a real vector for the manipulator in Fig. 1. Then, the inclusion function for the transformation of the end effector pose can be written as

[f]([x]) = [T](ω 1 ,[θ 1 ],[q 1 ]) •[T](ω 2 ,[θ 2 ],[q 2 ]) •[f]([x 0 ]) (6)
where

[T](ω i ,[θ i ],[q i ]) = R(ω i ,[θ i ]) (I -R(ω i ,[θ i ]))[q i ] 0 1 . (7)
Then, the inverse problem for this mapping can be posed as follows: For a given interval of end-effector position

[y] = [f]([x]
), what should be the manipulator parameters

[x] = ([l 1 ],[θ 1 ],[l 2 ],[θ 2 ]) T ?
Note that the answer to the above question can provide not only the range of [θ i ] for the desired interval of end-effector position but also the maximum allowable uncertainty in [θ i ] and [l i ]. Therefore, this methodology can be used as a tool for precision [START_REF] Shin | A statistical analysis of positional errors of a multiaxis machine tool[END_REF] Copyright © 2013 by ASME design of manipulators. That is, determination of the required joint encoder resolution and of the manufacturing tolerance for the robot mechanism can be done based on the results of this inverse kinematic analysis.

When the kinematic mechanism is as simple as the one in Fig. 1, the point-valued inverse kinematics problem can be solved analytically as in [START_REF] Frey | Robotics and Automation Handbook[END_REF] [20, p. 112]. These equations provide a couple of solutions for θ 1 and θ 2 for each end-effector position (x,y). However, as the number of joints increases, the forward kinematics map becomes non-invertible. Also, incorporation of mechanism errors and parameter uncertainties introduces additional degrees of freedom to the formulation. Then, finding the inverse solution with interval arguments requires a set-based computational method. In order to address this problem, next we will introduce a set inversion algorithm that is based on interval analysis.

θ 2 = ±cos -1 ( x 2 +y 2 -l 2 1 -l 2 2 2l 1 l 2
)

θ 1 = atan2(y,x) cos -1 ( x 2 +y 2 +l 2 1 -l 2 2 2l 1 √ x 2 +y 2 ) (8)

Set Inversion via Interval Analysis

Given a nonlinear function f from R n to R m and a set Y in R m , finding X described as

X = {x ∈R n | f(x) ∈Y }= f -1 (Y ) (9)
is the set inversion problem [19, p. 55]. Jaulin et al. addressed this problem via an algorithm called SIVIA (Set Inversion Via Interval Analysis) [START_REF] Jaulin | Set inversion via interval analysis for nonlinear bounded-error estimation[END_REF]. For a given Y ∈R m , X can be bounded arbitrarily closely with a lower bound X and an upper bound X such that X ⊂ X ⊂ X provided that an inclusion function [f] can be found for f. Note that X is guaranteed to be a solution set whereas X may contain non-solution points. The process of finding X using SIVIA is illustrated in Fig. 2 for 2-dimensional x and y spaces. SIVIA starts with an initial search domain [x 0 ] that is guaranteed to contain X. Then, the following 4-step procedure is applied:

If the mapping [f]([x]
) results in an interval (box) in the yspace that intersects with Y without being fully enclosed by Y as in Fig. 2(a), then [x] is said to contain part of the solution set X but regarded as undetermined. If the width of [x] is greater than a predetermined resolution parameter ε, then it needs to be bisected along the longest side and the procedure needs to be repeated recursively on each sub-box. Finally, if [x] is undetermined and width([x]) < ε as in Fig. 2(d), then the procedure stops for [x] and it is added to the upper bound X of X.

If [f]([x]) ∩Y = / 0 as in Fig. 2(b), then [x] is not part of X hence can be discarded. If [f]([x]) ⊂ Y as in Fig. 2(c), then [x] ⊂ X and [x] ∈X and [x] ∈X. x 1 x 2 [x 0 ] y 1 y 2 X [f] Y x 1 x 2 [x 1 ] X [f] y 1 y 2 Y (a) (b) x 1 x 2 [x n ] X [f] y 1 y 2 Y (d) x 1 x 2 [x n ] X [f] y 1 y 2 Y (c)

General Case of Inverse Kinematics via Set Inversion

Let f be the forward kinematics map of a serial manipulator from R n to SE [START_REF] Shin | A statistical analysis of positional errors of a multiaxis machine tool[END_REF] where n is the total number of joint variables and parameters that are uncertain. Also, let Y be a subset of SE(3). Then, calculating X in ( 9) is the inverse kinematics problem in presence of joint parameter uncertainties for a given set of end-effector configurations. This is inherently a set inversion problem and can be addressed using SIVIA.

Application of SIVIA to a general inverse kinematics problem requires comparing [f]([x]) in (3) with Y and determining whether or not they intersect or one includes the other. This is relatively simpler if Y can be represented as an interval so that both [f]([x]) and Y are 4×4 interval matrices. Then, the comparison can be done element-wise. Otherwise, each member of Y has to be compared with [f]([x]) one by one.

SIMULATIONS

In this section, we present some example simulation results that show how the proposed method can be implemented. We carried out interval calculations using MATLAB and a toolbox called INTLAB developed by S. M. Rump [START_REF] Rump | Intlab interval laboratory[END_REF]. INTLAB supports many operations with real and complex interval scalars, vectors, and matrices. It provides efficient functions for basic operations in algebra, trigonometry, etc.

In order to verify the results obtained using interval analysis, we also performed Monte Carlo analysis. When an analytical expression for inverse kinematics is available, it can directly be used to go from the configuration space to the parameter space. For instance, by evaluating equation ( 8) for various samples of

1: TWO-LINK MANIPULATOR SIMULATION PA- RAMETERS. l 1 δ l 1 l 2 δ l 2 [θ 1 (0)] [θ 2 (0)] 1 0.001 1 0.001 [0, π / 2] [-π / 2, π / 2] the input arguments [l 1 ], [l 2 ], [x]
, and [y], one can obtain a set of points in the joint space whose convex hull approximately provides the inverse solution X. We refer to this as the Monte Carlo solution in the next part.

Precision Design of the Two-Link Manipulator

In this part, we present the solution to the inverse kinematics problem of the two-link manipulator discussed in . This serves as a validation example such that we verify the interval analysis results with Monte Carlo simulation of inverse kinematic equations. The results will enable us to determine the minimum joint encoder resolution required to achieve a given end-effector precision.

For the manipulator parameters given in Table 1 and for a sample end-effector position ([x],[y]) = (1.4 ±0.01,1.2 ±0.01), the solutions of the SIVIA algorithm for a stopping criterion of ε = π / 1800 is shown in Fig. The picture on the left shows the 7018 subpavings (i.e. bisected arguments) of θ 1 and θ 2 each of which represent an interval sample from the joint space. SIVIA produces increasing concentration around the two solution regions as it converges by bisecting initial 'undetermined'intervals. The exploded view of one of those regions on the right shows that the remaining undetermined region X (yellow) found by SIVIA properly encloses the inverse Monte Carlo solution X (blue line) which in turn encloses the lower bound X (green region) as suggested by SIVIA.

The result in Fig. 3 shows that if the joint positions can be addressed as precisely as to fit in the lower bound (green region, X), then it can be guaranteed that the end-effector can be positioned in the given interval

([x],[y]) = (1.4 ± 0.01,1.2 ± 0.01).
The question is then how to quantify the precision based on this result.

Fig. 4(a) shows X in more detail where an interval of (θ 1 ,θ 2 ) with the largest overlapping area with X is encircled.

If the length of this interval along θ 1 is A and that along θ 2 is B, then the proper design choice for the joint resolution of the manipulator about these two axes can be described as in equation [START_REF] Moore | Introduction to Interval Analysis[END_REF]. This makes sure that the actual joint positions

[θ 1 ] [θ 1 -δ θ 1 ,θ 1 + δ θ 1 ] and [θ 2 ] [θ 2 -δ θ 2 ,θ 2 + δ θ 2 ]
can be commanded to be inside X for a certain value of θ 1 and θ 2 . This is explained pictorially in Fig. 4(b) with an arbitrarily positioned grid of addressable intervals each of which represents the set of actual joint positions for a given position command. When the condition in ( 10) is satisfied, there exists at least one interval of ([θ 1 ],[θ 2 ]) that completely overlaps with X. In Fig. 4(b), there are five such possible joint positions shown by boxes with the slash pattern. Therefore, δ θ * 1 and δ θ * 2 are the coarsest resolution values for the joint encoders that guarantee the given end-effector precision. In this case, (A,B) is measured to be ( π / 900, π / 450) which correspond to (δ θ * 1 ,δ θ * 2 ) = (0.05°,0.1°). For this particular end-effector position, it can be seen that the resolution requirement for the first joint is higher as it can introduce more Abbe error due to its distance from the end-effector.

δ θ 1 ≤ δ θ * 1 A / 4 , δ θ 2 ≤ δ θ * 2 B / 4 (10)
The lower bound X can always be improved in expense of computational time by reducing the value of ε. Fig. 5(a) compares the previous result in Fig. [START_REF] Shin | A statistical analysis of positional errors of a multiaxis machine tool[END_REF] where ε was π / 1800 with the one in Fig. 5(b) where ε is π / 3600. It can be seen that a finer bisection resolution improves the lower bound X towards the Monte Carlo result. The amount of improvement in this case is from (A,B) = ( π / 900, π / 450) to (A,B) = ( 7π / 3600, π / 400) which means that the δ θ * 1 and δ θ * 2 are now 0.0875°and 0.1125°, respectively. The total number of subpavings processed in this case is 57590 which is significantly higher than the previous value 7018. Indeed, SIVIA terminates after generating less than ( width([x 0 ]) / ε +1) n bisections while the computing time increases exponentially with the dimension of [x] [START_REF] Jaulin | Guaranteed nonlinear parameter estimation from bounded-error data via interval analysis[END_REF] which is propor- tional to the number of degrees of freedom of a manipulator. However, using bisections is one of the most basic techniques of interval analysis in terms of computational efficiency. There are efficient solvers that combine use of bisections, contractors, inclusion tests, and local optimization procedures to reduce the computational complexity to polynomial order [START_REF] Jaulin | Applied Interval Analysis[END_REF]. Application of these solvers to the inverse kinematics problem is not covered in this paper but will be addressed in the future.

Allocation of Mechanism Tolerances in a 3-DOF Precision Stage

In this part, we will demonstrate how the presented method can also be used to allocate tolerances to the mechanical design of a 3-DOF PPR manipulator as in Fig. 6 such that a given endeffector precision can be achieved. For clarity, we will focus only on some of the parametric uncertainties of the manipulator such as prismatic joint axes and rotary joint position vectors. The presented method will enable us to bound the allowable misalignment in these vectors. The vectors [v x ], [v y ], and [q] that parametrize the manipulator are represented as shown in [START_REF] Wu | Uncertainty analysis and allocation of joint tolerances in robot manipulators based on interval analysis[END_REF] with error bounds denoted by [δ v y ] for the misalignment of [v x ] along y-axis, [δ v x ] for the misalignment of [v y ] along x-axis, and [δ q] for the misalignment of [q] along x and y axes. In a practical scenario, for instance, these terms can represent the errors introduced into the geometry of the manipulator during its manufacturing or assembly. [

v x ] =   1 [δ v y ] 0   [v y ] =   [δ v x ] 1 0   [q] =   [δ q] [δ q] 0   (11)
The objective in this simulation is to find how large [δ v y ], [δ v x ], [δ q x ], and [δ q y ] can be for a given stage positioning precision. For example, assume a configuration attached to the rotary stage needs to be driven from an initial pose p i to a final pose p f through the manipulator transformation that can be described as

[p f ] = I d x [v x ] 0 1 I d y [v y ] 0 1 e ωθ (I -e ωθ )[q] 0 1 p i ( 12
)
where d x , d y , and θ are the desired displacements of the axes. We chose the initial end-effector position to be (10mm,0mm,0°) and final joint position to be (200mm,400mm,45°). Then, we solved for [δ v y ], [δ v x ], and [δ q] that guarantee a maximum deviation of 30µm from the desired final position. Fig. 7(a) shows the lower bound solution of this simulation. Similar to the previous case, the largest interval box that can be fit into the lower bound has approximate dimensions of (60,140,13) µm. Then from [START_REF] Moore | Introduction to Interval Analysis[END_REF], the allowable maximum uncertainty in the axes vectors can be expressed as shows the distribution of end-effector position by sampling the intervals in (13) using Monte Carlo method. It can be seen that the results all lie inside the allowed boundary for final end-effector position hence (13) is a guaranteed tolerance allocation. It can also be seen in ( 13) that the amount of allowable uncertainty is different for different parameters. This depends on the given initial and final positions, desired stage positioning precision, and the kinematic design. Hence, the result can also be used to do sensitivity analysis to find out which parameter is likely to cause more deviation at the end-effector.

CONCLUSIONS

In this paper we propose a methodology for solving manipulator design problems with precision bounds. The simulation results show that the proposed method is effective in determining the level of precision needed in the design of a manipulator for a given end-effector precision. Solution of the inverse kinematics problem using interval analysis not only provides the range of joint space variables for a desired end-effector position but also allows calculation of the maximum allowable uncertainty in the joint sensor (encoder) feedback and the mechanism geometry tolerances. Therefore, the method can be used as a design aid in robotic applications that involve designing, building, or choosing individual axes of a multi-degree-of-freedom manipulator. Since we use standard interval analysis tools such as SIVIA algorithm, our approach can easily be applied to any kinematic configuration. Future work will include using heuristics to speed up the computational efficiency of the algorithm and exercising this method on the design of a modular micro-assembly robot.

FIGURE 2 :

 2 FIGURE 2: SIVIA ALGORITHM PROCEDURE (BASED ON [19, p. 57]).

FIGURE 3 :

 3 FIGURE 3: BOUNDS ON θ 1 AND θ 2 OF THE TWO-LINK MANIPULATOR FOR ([x],[y]) = (1.4 ±0.01,1.2 ±0.01).

FIGURE 4 :

 4 FIGURE 4: (a) THE INTERVAL OVERLAPPING WITH THE LOWER BOUND X WITH MAXIMUM AREA OF A × B (b) AN ARBITRARILY POSITIONED GRID OF ADDRESS-ABLE INTERVALS FOR δ θ 1 = A / 4 AND δ θ 2 = B / 4.

FIGURE 5 :

 5 FIGURE 5: UPPER (X) AND LOWER (X) BOUNDS FOUND USING SIVIA AND (X) FOUND USING MONTE CARLO FOR (a) ε = π / 1800 (b) ε = π / 3600.

FIGURE 6 :

 6 FIGURE 6: 3D MODEL OF A 3-DOF PPR PRECISION MO-TION STAGE.

FIGURE 7 :

 7 FIGURE 7: (a) BOUNDING OF ERROR TERMS [δ v y ], [δ v x ], and [δ q] (b) END-EFFECTOR POSITION BY MONTE CARLO SAMPLING OF (13).

[ 13 )Fig. 7 (

 137 Fig.7(b) shows the distribution of end-effector position by sampling the intervals in (13) using Monte Carlo method. It can be seen that the results all lie inside the allowed boundary for final end-effector position hence (13) is a guaranteed tolerance allocation. It can also be seen in (13) that the amount of allowable uncertainty is different for different parameters. This depends on the given initial and final positions, desired stage positioning precision, and the kinematic design. Hence, the result can also be used to do sensitivity analysis to find out which parameter is likely to cause more deviation at the end-effector.

TABLE


Ivan Nourdin

Guillaume Poly

An invariance principle under the total variation distance

Keywords: Convergence in law, convergence in total variation, absolute continuity, invariance principle

 to hold under the total variation distance. There are many works about CLT under various metrics in the literature, but the present one seems to be the first attempt to deal with homogeneous polynomials in the X i with degree strictly greater than one.

Introduction and main results

Let X 1 , X 2 , . . . be independent copies of a random variable with mean zero and variance one. According to the central limit theorem, the normalized sums

W n = X 1 + . . . + X n √ n (1.1)
converge in distribution to the standard normal law N ∼ N (0, 1). In fact, using, e.g., the second Dini's theorem it is straightforward to prove a much stronger result, namely that W n converges to N in the Kolmogorov distance:

lim n→∞ d Kol (W n , N ) = 0, (1.2)
where

d Kol (U, V ) = sup x∈R |P (U x) -P (V x)|.
In (1.2), can we replace the Kolmogorov distance d Kol by the total variation distance d T V , defined as d T V (U, V ) = sup A∈B(R) |P (U ∈ A) -P (V ∈ A)|? In other words, do we also have

lim n→∞ d T V (W n , N ) = 0 (1.3)
for W n defined by (1.1)? The right answer is provided by an old and celebrated result of Prohorov [START_REF] Prohorov | On a local limit theorem for densities[END_REF]. To formulate it, first let us introduce the Lebesgue decomposition (of the distribution) of a random variable. As is well known, each cumulative distribution function (cdf) F may indeed be represented in the form:

F (x) = u x -∞ g(y)dy + (1 -u)G(x), x ∈ R, (1.4)
where u ∈ [0, 1], g : R → [0, ∞) satisfies R g(y)dy = 1 and G is a singular cdf (corresponding to a distribution concentrated on a set of zero Lebesgue measure) with G ′ (x) = 0 for almost all x. The real number u ∈ [0, 1] is uniquely determined by F ; the density function g is uniquely determined (up to a set of measure zero) if and only if u > 0.

Definition 1.1 When X is a random variable with cdf F , we say that X is singular if u = 0 in (1.4). If u > 0, we say that X has an absolutely continuous component with density g.

We can now state Prohorov's theorem [START_REF] Prohorov | On a local limit theorem for densities[END_REF]. A proof will be given in Section 2.4, to illustrate a possible use of our forthcoming results. Theorem 1.2 (Prohorov) The convergence (1.3) takes place if and only if there exists n 0 1 such that the random variable W n 0 has an absolutely continuous component.

Prohorov's theorem has been the starting point of a fruitful line of research around the validity of the central limit theorem under various metrics and the estimation of their associated rates of convergence. Let us only give a small sample of references dealing with this rich and well studied topic. Convergence of densities in L ∞ are studied by Gnedenko and Kolmogorov [START_REF] Gnedenko | Limit distributions for sums of independent random variables[END_REF]. On their side, Mamatov and Halikov [START_REF] Mamatov | Global limit theorems for distribution functions in the higher-dimensional case[END_REF] dealt with the multivariate CLT in total variation. Barron [2] studied the convergence in relative entropy, whereas Shimizu [START_REF] Shimizu | On Fisher's amount of information for location family[END_REF] and Johnson and Barron [START_REF] Johnson | Fisher information inequalities and the central limit theorem[END_REF] studied the convergence in Fisher information. As far as rates of convergence are concerned, one can quote Mamatov and Sirazdinov [START_REF] Sirazdinov | On mean convergence for densities[END_REF] for the total variation distance and, very recently, Bobkov, Chistyakov and Götze for bounds in entropy [START_REF] Bobkov | Berry-Esseen bounds in the entropic central limit theorem[END_REF], in Fisher information [START_REF] Bobkov | Fisher information and the central limit theorem[END_REF] and for Edgeworth-type expansions in the entropic central limit theorem [START_REF] Bobkov | Rate of convergence and Edgeworthtype expansion in the entropic central limit theorem[END_REF]. Finally we mention [START_REF] Cacoullos | Variational inequalities with examples and an application to the central limit theorem[END_REF][START_REF] Cacoullos | Another characterization of the normal law and a proof of the central limit theorem connected with it[END_REF] for a variational approach of these issues with some variance bounds.

All these above-mentioned references have in common to 'only' deal with sums of independent random variables. In the present paper, in contrast, we will consider highly non-linear functionals of independent random variables. It is arguably a much harder framework to work with, precisely because all the nice properties enjoyed by sums of independent variables are no longer valid in this context (in particular, the use of characteristic functions is not appropriate).

Let us now turn to the details of the situation we are considering in the present article. Fix a degree of multilinearity d 1 (d = 1 for linear, d = 2 for quadratic, etc.) and, for any n 1, consider a homogeneous polynomial Q n : R Nn → R of the form

Q n (x) = Nn i 1 ,...,i d =1 a n (i 1 , . . . , i d )x i 1 . . . x i d , x = (x 1 , . . . , x Nn ) ∈ R Nn .
(1.5)

In (1.5), it is implicitely supposed that N n → ∞ and also that a n (i 1 , . . . , i d ) are real numbers vanishing on diagonals and symmetric in the indices. We further assume, in all what follows, that Q n is properly normalized:

d! Nn i 1 ,...,i d =1 a n (i 1 , . . . , i d ) 2 = 1, n 1, (1.6)
and that all the terms of Q n are asymptotically negligible, meaning in our context that

lim n→∞ max 1 i 1 n Nn i 2 ,...,i d =1 a n (i 1 , . . . , i d ) 2 = 0. (1.7)
As anticipated, property (1.7) will play a crucial role in the sequel. It is also the key to obtain the following invariance principle, due to Mossel, O'Donnel and Oleszkiewicz [START_REF] Mossel | Noise stability of functions with low influences: Variance and optimality[END_REF].

Theorem 1.3 (Mossel, O'Donnel, Oleszkiewicz) Fix an integer d 1, and let Q n be a sequence of homogeneous polynomials satisfying both (1.5), (1.6) and (1.7). Let X = (X 1 , X 2 , . . .) be a sequence of independent random variables with mean zero and variance one, belonging to L 2+ǫ (Ω) for some ǫ > 0 (the same ǫ for each X i ). Assume the same for Y = (Y 1 , Y 2 , . . .). Then

lim n→∞ d Kol (Q n (X), Q n (Y)) = 0. (1.8)
Observe that one recovers (1.2) by considering, in (1.8

), d = 1, a n (i) = 1 √ n , 1 i n (which satisfies (1.7)) and Y 1 ∼ N (0, 1) (which leads to Q n (Y) ∼ N (0, 1) for any n).
In the light of the aforementioned results, it seemed natural to us to ask under which assumption the convergence (1.8) may be strenghtened to the total variation distance as follows:

lim n→∞ d T V (Q n (X), Q n (Y)) = 0.
(1.9)

Before detailing our answer, let us first do a quick digression. As anticipated, a main aspect of our approach will consist in introducing the following class of random variables. Definition 1.4 For any p ∈]0, 1] and α > 0, the class C(p, α) is the set of real random variables X satisfying

X law = ε(αU + x 0 ) + (1 -ε)V, (1.10)
where x 0 ∈ R is a real number, and U ∼ U [-1,1] , ε ∼ B(p) and V (with no specified distribution) are three independent random variables.

Since it will play a crucial role in the sequel, let us first try to catch the meaning of (1.10). To this aim, we introduce yet another class of random variables. Definition 1.5 For any c, α > 0, the class G(c, α) is the set of real random variables X having an absolutely continuous component and whose density g, see (1.4), satisfies g(x) c for all x ∈ [x 0 -α, x 0 + α] for some x 0 ∈ R.

The following result compares the two classes C(p, α) and G(c, α). Roughly speaking, it asserts that the class of random variables with an absolutely continous component (that is, exactly the kind of random variables appearing in Prohorov's Theorem 1.2) coincides with ∪ p∈]0,1],α>0 C p,α . Observe also that G(c, α) is not empty if and only if 2cα 1.

Proposition 1.6 Fix c, α > 0 and p ∈]0, 1]. One has G(c, α) ⊂ C(2cα, α). Moreover, any random variable belonging to C(p, α) has an absolutely continuous part.

In Lemma 2.2 below, we state two further important properties of C(p, α). Firstly, the sum of two independent random variables having an absolutely continuous component belong to ∪ c,α>0 G(c, α) ⊂ ∪ p∈]0,1],α>0 C(p, α). Secondly, if 0 < q p 1 and 0 < β α, then C(p, α) ⊂ C(qβ/α, β). Now C(p, α) has been introduced and is arguably well-understood, let us give a name to the set of sequences of independent and normalized random variables we will deal with throughout the sequel.

Definition 1.7 Let α > 0, p ∈]0, 1] and ǫ > 0. A sequence X = (X 1 , X 2 , . . .) of random variables belongs to D(α, p, 2 + ǫ) if the X i are independent, satisfy sup i E|X i | 2+ǫ < ∞ and if, for each i, E[X i ] = 0, E[X 2 i ] = 1 and X i ∈ C(p, α).
We are now in a position to state the main result of the present paper.

Theorem 1.8 Fix an integer d 1, and let Q n be a sequence of homogeneous polynomials satisfying both (1.5), (1.6 and (1.7). Let X and Y belong to D(α, p, 2 + ǫ) for some ǫ, α > 0 and p ∈]0, 1]. Then (1.9) holds true.

A noticeable corollary of Theorem 1.8 is a new proof of Prohorov's Theorem 1.2, see Section 2.4. Another one is the following result.

Corollary 1.9 Fix an integer d 2, and let Q n be a sequence of homogeneous polynomials satisfying

(1.5)-(1.6)-(1.7). Let X belong to D(α, p, 2 + ǫ) for some ǫ, α > 0 and p ∈]0, 1]. If Q n (X)
converges in law to W , then W has a density and Q n (X) converges to W in total variation.

The statement of Corollary 1.9 would be clearly wrong without assuming (1.7). Consider, e.g., Q n (x) = x 1 , n 1 with X 1 singular. Another interesting consequence of Theorem 1.8 is provided by the next theorem.

Theorem 1.10 Let {a n (i 1 , . . . , i d )} 1 i 1 ,...,i d Nn be an array of real numbers vanishing on diagonals, symmetric in the indices and satisfying (1.6). (We do not suppose (1.7).) Let G = (G 1 , G 2 , . . .) be a sequence of independent N (0, 1) random variables, and let N ∼ N (0, 1). Then, the following four assertions are equivalent as n → ∞.

(a) Nn i 1 ,i 2 ,••• ,i d =1 a n (i 1 , i 2 , • • • , i d )G i 1 G i 2 • • • G i d law → N . (b) d T V   N, Nn i 1 ,i 2 ,••• ,i d =1 a n (i 1 , i 2 , • • • , i d )G i 1 G i 2 • • • G i d   → 0. (c) For all X = (X 1 , X 2 , . . .) belonging to D(α, p, 2 + ǫ) for some ǫ, α > 0 and p ∈]0, 1], we have Nn i 1 ,i 2 ,••• ,i d =1 a n (i 1 , i 2 , • • • , i d )X i 1 X i 2 • • • X i d law → N.
(d) For all X = (X 1 , X 2 , . . .) belonging to D(α, p, 2 + ǫ) for some ǫ, α > 0 and p ∈]0, 1], we have

d T V   N, Nn i 1 ,i 2 ,••• ,i d =1 a n (i 1 , i 2 , • • • , i d )X i 1 X i 2 • • • X i d   → 0.
The rest of our paper is organised as follows. In Section 2 we prove all the results that are stated in this Introduction, except Theorem 1.8; in particular, we give a new proof of Prohorov's Theorem 1.2 in Section 2.4. Finally, the proof of our main result, namely Theorem 1.8, is provided in Section 3.

2 Proofs of all stated results except Theorem 1.8

Some useful lemmas

The following lemma will be used several times in the sequel. Lemma 2.1 Fix q ∈ [0, 1], and let Y, Z be two random variables satisfying

E[f (Y )] qE[f (Z)]
for all positive bounded function f . Then there exists two independent random variable W and ζ ∼ B(q), independent from Z, such that

Y law = ζZ + (1 -ζ)W.
(2.11)

Proof of Lemma 2.1. Our assumption ensures that the linear form

f → E[f (Y )] -qE[f (Z)] is positive.
From the Riesz representation theorem, one deduces the existence of a positive finite Radon measure ν such that

E[f (Y )] = qE[f (Z)] + R f (x)dν(x).
(2.12)

Choosing f ≡ 1 in (2.12) gives ν(R) = 1 -q. If ν(R) = 0 then q = 1 and the proof of (2.11) is established. Otherwise, ν(R) > 0 and one can consider W ∼ 1 ν(R) dν(x), implying in turn (2.11). 2

In the following lemma, we gather useful properties of the classes C(p, α) and G(c, α).

Lemma 2.2

The following properties take place.

1. If 0 < q p 1 and if 0 < β α, then C(p, α) ⊂ C(qβ/α, β). In particular, C(p, α) ⊂ C(q, α).
2. If X and Y both have an absolutely continuous component and if X is independent from Y , then there exists c, α > 0 such that X + Y ∈ G(c, α).

Proof. 1. Fix 0 < q p 1 and 0 < β α, and consider X ∈ C(p, α). According to (1.10), we have, for any positive f ,

E[f (X)] = p R f (x) 1 2α 1 [x 0 -α,x 0 +α] (x)dx + (1 -p)E[f (V )] qβ α R f (x) 1 2β 1 [x 0 -β,x 0 +β] (x)dx.
The conclusion follows from Lemma 2.1.

2. Consider the decomposition (1.4) of the cdf F of X. This settles u ∈]0, 1] and g : R + → R in a unique way. Settle similarly v ∈]0, 1] and h : R + → R for Y . For any Borel set A, one has P (X ∈ A) u A g(x)dx and the same for Y . We deduce

P (X + Y ∈ A) uv A (g ⋆ h)(x)dx,
with ⋆ denoting the usual convolution. Besides, g ⋆ h = lim M →∞ g ⋆ inf(h, M ) and the limit is increasing by positivity of g. Finally, we note that, since g ∈ L 1 and inf(h, M ) ∈ L ∞ , the convolution g⋆inf(h, M ) is continuous. Let x 0 ∈ R and M > 0 be such that (g⋆inf(h, M ))(x 0 ) > 0.

(Such a pair (x 0 , M ) necessarily exists, otherwise we would have g ⋆ h ≡ 0 by taking the large M limit.) By continuity, there exists c > 0 and α > 0 such that, for any

x ∈]x 0 -α, x 0 + α[, (g ⋆ h)(x) (g ⋆ inf(h, M ))(x) ≥ c. That is, X + Y belongs to G(c, α).
3. We have

X law = ε(αU + x 0 ) + (1 -ε)V , with x 0 ∈ R a real number, and U ∼ U [-1,1]
, ε ∼ B(p) and V (with no specified distribution) three independent random variables. On the other hand, one can write Y law = εY + (1 -ε)Z, with Z having the same law than Y and independent from Y, U, ε, V . Thus,

X + Y law = ε(αU + x 0 + Y ) + (1 -ε)(V + Z).
The random variable αU + x 0 + Y has a density g given by

g(v) = R 1 2α 1 [x 0 -α,x 0 +α] (v -y)dP Y (y) = 1 2α P (Y ∈ [v -x 0 -α, v -x 0 + α]).
As a matter of fact, g is a regulated function, since it is the difference of two increasing functions.

In particular, the set E of its discontinuous points is countable. As a consequence, Leb(E) = 0, implying in turn 1 = R g(v)dv = R\E g(v)dv, so that there exists x 1 ∈ E satisfying g(x 1 ) > 0.

Since g is continuous at x 1 , there exists r > 0 such that g(v)

1 2 g(x 1 ) for all v ∈ [x 1 -r, x 1 + r]. By Lemma 2.1, it comes that αU + x 0 + Y law = η(rU + x 1 ) + (1 -η)T, where η ∼ B(p ′ ) for some p ′ ∈]0, 1], U ∼ U [-1,1] and T are independent. Hence X + Y law = εη(rU + x 1 ) + ε(1 -η)T + (1 -ε)(V + Z).
As a result, for any bounded positive function,

E[f (X + Y )] = pp ′ E[f (rU + x 1 )] + p(1 -p ′ )E[f (T )] + (1 -p)E[f (V + Z)] pp ′ E[f (rU + x 1 )].
Finally, one deduces that X + Y belongs to C(pp ′ , r) by Lemma 2.1. 4. Obvious. 2

Proof of Proposition 1.6

Let X be an element of G(c, α). Let F denote its cdf, and consider g, G and u as in (1.4). Let Y ∼ g(x)dx, Z ∼ dG(x) and η ∼ B(u) be three independent random variables. Then

X law = ηY + (1 -η)Z.
Using the assumption made on g, one obtains, for any positive function f ,

E[f (Y )] 2cα E [f (αU + x 0 )] .
Observe that 0 < 2cα 1 necessarily. We then deduce that X ∈ C(2cα, α) from Lemma 2.1. Consider now a random variable X belonging to C(p, α). We have, for any positive function f and according to the decomposition (1.10),

E[f (X)] = pE[f (αU + x 0 )] + (1 -p)E[f (V )].
(2.13)

Let us consider the Lebesgue decomposition (u, g, G) of V , see (1.4):

E[f (V )] = u R f (x)g(x)dx + (1 -u) R f (x)dG(x).
Plugging into (2.13) yields

E[f (X)] = R f (x) p 2α 1 [x 0 -α,x 0 +α] (x) + (1 -p)u g(x) dx + (1 -p)(1 -u) R f (x)dG(x),
from which we deduce that X has an absolutely continuous component. 2

Proof of Corollary 1.9

Let the assumption of Corollary 1.9 prevail and consider a sequence G = (G

T V (Q n (G), W ) → 0 as n → ∞. But d T V (Q n (X), Q n (G)) → 0 by our Theorem 1.8, so d T V (Q n (X), W ) d T V (Q n (X), Q n (G)) + d T V (Q n (G), W ) → 0 as n → ∞.
This concludes the proof of Corollary 1.9. 2

Proof of Theorem 1.2

We want to use Corollary 1.9 but the problem is that X is not assumed to belong to D in the statement of Theorem 1.2. To overcome this issue we shall need Lemma 2.2.

Let the assumptions and notation of Theorem 1.2 prevail. First, when W n is singular, then there exists a Borel set A such that P (N ∈ A) = 0 and P (W n ∈ A) = 1; in particular, d T V (W n , N ) = 1. Hence, if W n is singular for all n, then (1.3) cannot hold. Now, assume that W n 0 has an absolutely continuous component for some n 0 . To prove (1.3) is obviously equivalent to prove that, for all k ∈ {0, . . . , 2n 0 -1}, lim n→∞ d T V (W 2n 0 n+k , N ) = 0.

(2.14)

So, fix k ∈ {0, . . . , 2n 0 -1} and consider a sequence (Y 2 , Y 3 , . . .) of independent copies of W 2n 0 . By Lemma 2.2 (points 2 and 4), observe that each Y i belongs to C(p, α), for some p ∈]0, 1] and α > 0 (the same p and the same α for all i 2); also, we have

E[Y i ] = 0 and E[Y 2 i ] = 1.
On the other hand, let Y 1 be independent of Y 2 , Y 3 , . . . and have the same law than W 2n 0 +k . By Lemma 2.2 (points 3 and 4), Y 1 belongs to C(q, β) for some q ∈]0, 1] and β > 0; also, we have E[Y 1 ] = 0 and E[Y 2 1 ] = 1. In fact, thanks to Lemma 2.2 (point 1), one may and will choose the same p and the same α for each Y i , without making a difference between i = 1 and i 2.

Bearing all the previous notation in mind, we can write

W n law = 2n 0 + k n Y 1 + 2n 0 n n k=2 Y k .
The convergence (2.14) is now a direct consequence of Theorem 1.8 applied to d = 1, a sequence

X = G = (G 1 , G 2 , . . .) of independent N (0, 1) variables, a n (1) = 2n 0 +k n and a n (i) = 2n 0 n , i = 2, . . . , n. Note that d F M (W n , N ) = d F M (Q n (Y), Q n (X)) → 0 as n → ∞
by the usual CLT, so that it is not necessary to rely on Theorem 1.3 to conclude the proof of Theorem 1.8 (see Step 7 of Section 3) and so to assume the existence of an absolute qth moment for X 1 with q strictly greater than 2.

F n := Nn i 1 ,i 2 ,••• ,i d =1 a n (i 1 , i 2 , • • • , i d )G i 1 G i 2 • • • G i d (which is normalized so that E[F 2 n ] = 1, see indeed (1.6)) satisfies E[F 4 n ] → 3.
Secondly, one has, according to [12, (11.4.7) and (11.4.8) pp. 192-193]:

max 1 i 1 n Nn i 2 ,...,i d =1 a n (i 1 , . . . , i d ) 2 1 dd! E[F 4 n ] -3.
These two facts together imply that, if (a) holds, then (1.7) is automatically satisfied. Thus, Theorem 1.8 implies the validity of assertion (d). 2

3 Proof of Theorem 1.8

Let the assumptions and notation of Theorem 1.8. Without loss of generality, for simplicity we assume that N n = n.

The proof is divided into several steps.

Step 1. In the definition of Q n (X) one may and will replace each X i by ε i (αU

i +x i )+(1-ε i )V i , where e = (ε 1 , ε 2 , . . .) is a sequence of independent Bernoulli random variables (ε i ∼ B(p i )), U = (U 1 , U 2 , . . .) is a sequence of independent [-1, 1]-uniformly distributed random variables and V = (V 1 , V 2 , . . .
) is a sequence of independent random variables; moreover, e, U and V are independent. That is,

Q n (X) = n i 1 ,...,i d =1 a n (i 1 , . . . , i d ){ε i 1 (α i 1 U i 1 +x i 1 )+(1-ε i 1 )V i 1 } . . . {ε i d (α i d U i d +x i d )+(1-ε i d )V i d }.
Now, let us expand everything, and then rewrite Q n (X) as a polynomial in the U i . We obtain

Q n (X) = A n + B n + C n , where A n = n i 1 ,...,i d =1 a n (i 1 , . . . , i d )α i 1 . . . α i d ε i 1 . . . ε i d U i 1 . . . U i d B n = Q n (X) -A n -C n C n = n i 1 ,...,i d =1 , a n (i 1 , . . . , i d ){ε i 1 x i 1 + (1 -ε i 1 )V i 1 } . . . {ε i d x i d + (1 -ε i d )V i d } satisfying E[A n B n |e, V] = E[A n C n |e, V] = E[B n C n |e, V] = E[A n |e, V] = E[B n |e, V] = 0. As a result, Var[Q n (X)|e, V] = E[A 2 n |e, V] + E[B 2 n |e, V] + Var[C n |e, V] E[A 2 n |e, V] α 2d 3 -d d! n i 1 ,...,i d =1 a n (i 1 , . . . , i d ) 2 ε i 1 . . . ε i d .
To go one step further, let us decompose ε i into (ε i -p) + p and use (1.6), so to obtain

Var[Q n (X)|e, V] d! α 2 p 3 d + α 2d 3 -d d! d k=1 d k p k n i 1 ,...,i d =1 a n (i 1 , . . . , i d ) 2 (ε i 1 -p) . . . (ε i k -p).
Using the assumptions (1.6) and (1.7), we can write, for any fixed k ∈ {1, . . . , d},

E     n i 1 ,...,i d =1 a n (i 1 , . . . , i d ) 2 (ε i 1 -p) . . . (ε i k -p)   2   = p k (1 -p) k n i 1 ,...,i k =1   n i k+1 ,...,i d =1 a n (i 1 , . . . , i d ) 2   2 n i 1 =1   n i 2 ,...,i d =1 a n (i 1 , . . . , i d ) 2   2 max 1 j 1 n n j 2 ,...,j d =1 a n (j 1 , . . . , j d ) 2 × n i 1 ,...,i d =1 a n (i 1 , . . . , i d ) 2 = 1 d! max 1 j 1 n n j 2 ,...,j d =1 a n (j 1 , . . . , j d ) 2 → 0 as n → ∞.
We deduce that, in probability,

lim inf n→∞ Var[Q n (X)|e, V] d! α 2 p 3 d . (3.15)
Convention. From now on, and since all the quantities we are dealing with are measurable with respect to e, U and V, we shall write E U (resp. E e,V ) to indicate the mathematical expectation with respect to U (resp. e and V) Note that E U coincides with the conditional expectation

E[•|e, V]. Step 2. Set p α (x) = 1 α √ 2π e -x 2 2α 2 , x ∈ R, 0 < α 1, and let φ ∈ C ∞ c be bounded by 1. It is immediately checked that φ ⋆ p α ∞ 1 1 α and (φ ⋆ p α ) ′ ∞ 1 α . (3.16)
We can write

|E[φ(Q n (X)] -E[φ(Q n (Y)]| |E[(φ -φ ⋆ p α )(Q n (X))]| + |E[(φ -φ ⋆ p α )(Q n (Y))]| + 1 α d F M (Q n (X), Q n (Y)),
where d F M stands for the Fortet-Mourier distance, which is known to metrize the convergence in law. Let us concentrate on the first two terms. We have, e.g., for the first term:

|E[(φ -φ ⋆ p α )(Q n (X))]| E[(φ -φ ⋆ p α )(Q n (X))1 {Var[Qn(X)|e,V]< d! 2 α 2 p 3 d } ] + E[(φ -φ ⋆ p α )(Q n (X))1 {E[Qn(X) 2 |e,V]>M } ] + E[(φ -φ ⋆ p α )(Q n (X))1 {Var[Qn(X)|e,V] d! 2 α 2 p 3 d , E[Qn(X) 2 |e,V] M } ] 2P Var[Q n (X)|e, V] < d! 2
α 2 p 3 d + 2P E[Q n (X) 2 |e, V] > M + E[(φ -φ ⋆ p α )(Q n (X))1 {Var[Qn(X)|e,V] d! 2 α 2 p 3 d , E[Qn(X) 2 |e,V] M } ] .
We have, using the Markov inequality,

P E[Q n (X) 2 |e, V] > M 1 M E[E[Q n (X) 2 |e, V]] = 1 M .
On the other hand,

E (φ -φ ⋆ p α )(Q n (X))1 {Var[Qn(X)|e,V] d! 2 α 2 p 3 d , E[Qn(X) 2 |e,V] M } E e,V E U [(φ -φ ⋆ p α )(Q n (X))] 1 {Var[Qn(X)|e,V] d! 2 α 2 p 3 d , E[Qn(X) 2 |e,V] M } .
Step 3. In this step, we shall introduce the framework we are going to use for the rest of the proof. We refer the reader to [START_REF] Bakry | Analysis and Geometry of Markov Diffusion Semigroups[END_REF] for the details and missing proofs. Fix an integer m and let µ denote the distribution of the random vector (X 1 , . . . , X m ), with X 1 , . . . , X m independent copies of U ∼ U [-1,1] , There exists a reversible Markov process on R m , with semigroup P t , equilibrium measure µ and generator L given by

Lf (x) = m i=1 (1 -x 2 i )∂ ii f -2x i ∂ i f , x ∈ R m . (3.17)
The operator L is selfadjoint and negative semidefinite. We define the carré du champ operator Γ as

Γ(f, g)(x) = 1 2 L(f g)(x) -f (x)Lg(x) -g(x)Lf (x) = m i=1 (1 -x 2 i )∂ i f (x)∂ i g(x). (3.18)
When f = g we simply write Γ(f ) instead of Γ(f, f ). An important property satisfied by Γ is that it is diffusive in the following sense:

Γ(φ(f ), g) = φ ′ (f )Γ(f, g). (3.19)
Besides, the eigenvalues of -L are given by Sp

(-L) = {i 1 (i 1 -1) + • • • + i m (i m -1) | i 1 , . . . , i m ∈ N}.
It may be ordered as a countable sequence like 0 = λ 0 < λ 1 < λ 2 < • • • , with a corresponding sequence of orthonormal eigenfunctions u 0 , u 1 , u 2 , • • • where u 0 = 1; in addition, this sequence of eigenfunctions forms a complete orthogonal basis of L 2 (µ). Also, note that the first nonzero element of Sp(-L) is λ 1 = 1 > 0. Also, one can compute that, when λ ∈ Sp(-L), then Ker(L+λ I) is composed of those polynomial functions R(x 1 , . . . , x m ) having the form

R(x 1 , . . . , x m ) = i 1 (i 1 +1)+•••+im(im+1)=λ α(i 1 , • • • , i nm )J i 1 (x 1 ) • • • J im (x m ).
Here J i (X) is the ith Jacobi polynomial, defined as

J i (x) = (-1) i 2 i i! d i dx i (1 -x 2 ) i , x ∈ R.
To end up with this quick summary, we recal the following Poincaré inequality, that is immediate to prove by using the previous facts together with the decomposition L 2 (µ) = λ∈Sp(-L) Ker(L + λ I):

Var µ (f ) Γ(f )dµ. (3.20)
Step 4. We shall prove the existence of a constant κ > 0, depending on p, α and d but not on n, such that, for any δ > 0,

sup n 1 E U δ Γ(Q n )(X) + δ 1 {Var[Qn(X)|e,V] d! 2 α 2 p 3 d } κ δ 1 2d+1 . (3.21)
The proof of (3.21) will rely on the Poincaré inequality (3.20) which, here, takes the following form:

Var[Q n (X)|e, V] = Var U [Q n (X)] E U [Γ(Q n )(X)]. (3.22)
Another ingredient is the Carbery-Wright inequality, that we recall for sake of completeness.

Theorem 3.1 (see [START_REF] Carbery | Distributional and L q norm inequalities for polynomials over convex bodies in R n[END_REF]Theorem 8]) There exists an absolute constant c > 0 such that, if Q : R m → R is a polynomial of degree at most k and µ a log-concave probability measure on R m , then, for all α > 0,

Q 2 dµ 1 2k × µ{x ∈ R m : |Q(x)| α} c k α 1 k . (3.23)
Observe that the density of U is log-concave, as an indicator function of a convex set. Let us now proceed with the proof of (3.21). For any strictly positive u, and provided Var Step 5. We shall prove that

[Q n (X)|e, V] d! 2 α 2 p 3 d , one has E δ Γ(Q n )(X) + δ δ u + P (Γ(Q n )(X) u) δ u + c u 1 2d , ( 3
sup n 1 E U [Γ(Γ(Q n ))(X)] + E U (LQ n )(X) 1 {E[Qn(X) 2 |e,V] M } c(M ) (3.25)
where c(M ) is a constant only depending on M (whose value may change from one line to another within this step). First, relying on the results of Step 3 we have that, for any n,

Q n ∈ α λ 2d Ker(L + αI).
Since L is a bounded operator on the space α λ 2d Ker(L + αI), we deduce immediately that

sup n 1 E U [(LQ n )(X) 2 ]1 {E U [Q 2 n (X)] M } c(M ).
Besides, one has Γ = 1 2 (L + 2λI) on Ker(L + λI) and one deduces for the same reason as above that

sup n 1 E U [Γ(Γ(Q n ))(X)]1 {E U [Q 2 n (X)] M } c(M ).
The proof of (3.25) is complete.

Step 6. We shall prove that, for any n 1, any 0 < α 1, any δ > 0 and any M > 0,

E U (φ -φ ⋆ p α )(Q n (X)) Γ(Q n )(X) + δ Γ(Q n )(X) + δ 1 {Var[Qn(X)|e,V] d! 2 α 2 p 3 d ,E[Qn(X) 2 |e,V] M } 2 E U δ Γ(Q n )(X) + δ 1 {Var[Qn(X)|e,V] d! 2 α 2 p 3 d } + E U (φ -φ ⋆ p α )(Q n (X)) Γ(Q n )(X) Γ(Q n )(X) + δ 1 {E[Qn(X) 2 |e,V] M } 2 κ δ 1 2d+1 + E U (φ -φ ⋆ p α )(Q n (X)) Γ(Q n )(X) Γ(Q n )(X) + δ 1 {E[Qn(X) 2 |e,V] M } . (3.27)
Now, set Ψ(x) =

x -∞ φ(s)ds and let us apply (3.19). We obtain The desired conclusion (1.9) then follows by letting (in this order) α → 0, δ → 0 and M → ∞.

E U (φ -φ ⋆ p α )(Q n (X)) Γ(Q n )(X) Γ(Q n )(X) + δ = E U 1 Γ(Q n )(X) + δ Γ (Ψ -Ψ ⋆ p α ) • Q n , Q n (X) = E U ((Ψ -Ψ ⋆ p α ) • Q n )(X) Γ Q n , 1 Γ(Q n ) + δ (X) + (LQ n )(X) Γ(Q n )(X) + δ = E U ((Ψ -Ψ ⋆ p α ) • Q n )(X) - Γ(Q n , Γ(Q n ))(X) (Γ(Q n )(X) + δ) 2 + (LQ n )(X) Γ(Q n )(X) + δ 1 δ E U |((Ψ -Ψ ⋆ p α ) • Q n )(X)| × Γ(Γ(Q n ))(X) + (LQ n )(X) . ( 3

2 2. 5

 25 Proof of Theorem 1.10 Implication (a) ⇒ (b) is a reformulation of [12, Corollary 5.2.8]. Implications (b) ⇒ (a) and (d) ⇒ (c) are obvious. Implication (c) ⇒ (a) is because G belongs to D(α, p, 2 + ε) for some α, ε > 0 and p ∈]0, 1[ (use Proposition 1.6). Finally, implication (a) ⇒ (d) is a consequence of the following two facts. Firstly, if (a) takes place then, by a usual hypercontractivity argument, the sequence

  .24) where c > 0 denotes a constant only depending on d, α and p and where the last inequality follows from the Carbery-Wright inequality (3.23), the inequality (3.22) and the fact that Γ(Q n ) is a polynomial of order 2d, see (3.18). Finally, choosing u = δ 2d 2d+1 in (3.24) leads to the desired conclusion (3.21).

  |E U [(φ -φ ⋆ p α )(Q n (X))]| 1 {Var[Qn(X)|e,V]

Step 7 :

 7 .28) On the other hand, we have|Ψ(x) -Ψ ⋆ p α (x)| = R p α (y) x -∞ (φ(u) -φ(u -y)) du dy R p α (y) (3.26) now follows easily from (3.25), (3.27), (3.28) and (3.29). Concluding the proof. Combining the results of all the previous steps, we obtain, for any n 1, any 0 < α 1, any δ > 0 and any M > 0, supφ∈C ∞ c : φ ∞ 1 |E[φ(Q n (X)] -E[φ(Q n (Y)]| 1 α d F M (Q n (X), Q n (Y)) + 4 M + 2P Var[Q n (X)|e, V] 30)-(3.31), take the limit n → ∞. Due to (3.15) on one hand and Theorem 1.3 on the other hand (plus the fact that the Fortet-Mourier distance d F M metrizes the convergence in distribution), one obtains lim sup n→∞ sup φ∈C ∞ c : φ ∞ 1 |E[φ(Q n (X)] -E[φ(Q n (Y)

If X belongs to C(p, α) with α > 0 and p ∈]0, 1] and if Y is any random variable independent from X, then X + Y belongs to C(q, β) for some β > 0 and q ∈]0, 1].

If a = 0 and b are two real numbers and if X belongs to C(p, α) with α > 0 and p ∈]0, 1], then aX + b ∈ C(p, |a|α).

* supported in part by the (french) ANR grant 'Malliavin, Stein and Stochastic Equations