Patrice Russo
email: patrice.russo@cpe.fr

Gael Pillonnet
email: gael.pillonnet@cpe.fr

Nacer Abouchi
email: abouchi@cpe.fr

Sophie Taupin
email: sophie.taupin@st.com

Frederic Goutti
email: frederic.goutti@st.com

Efficiency Optimization of Class G Amplifiers: Impact of the Input Signals

Class G amplifiers are an effective solution to increase the audio efficiency for headphone applications, but realistic operating conditions have to be taken into account to predict and optimize power efficiency. In fact, power supply tracking, which is a key factor for high efficiency, is poorly optimized with the classical design method, because the stimulus used is very different from a real audio signal. Here, a methodology has been proposed to find class G nominal conditions. By using relevant stimuli and nominal output power, the simulation and test of the class G amplifier are closer to the real conditions. Moreover, a novel simulator is used to quickly evaluate the efficiency with these long duration stimuli, i.e. ten seconds instead of a few milliseconds. This allows longer transient simulation for an accurate efficiency and audio quality evaluation by averaging the Class G behavior. Based on this simulator, this paper indicates the limitations of the well-established test setup. Real efficiencies vary up to ±50% from the classical methods. Finally, the study underlines the need to use real audio signals to optimize the supply voltage tracking of Class G amplifiers in order to achieve a maximal efficiency in nominal operation.

1.

INTRODUCTION

Class AB amplifiers allow excellent audio signal reproduction and are widely used over a large range of applications, such as smartphones, home theater or automotive applications. However, Class AB amplifiers suffer from a limited efficiency expressed as:

dd out AB V V   4   (1)
where V OUT is the output signal amplitude and V DD is the power supply voltage. This expression is valid for a pure sinusoidal output signal and for a negligible quiescent current.

The Class AB efficiency reaches 78.5% for a full scale output signal (V OUT =V DD ) but decreases linearly with the signal amplitude. At a nominal listening level, the efficiency is only a few percent because the output signal amplitude needed is more than one decade lower than the power supply. To overcome its poor efficiency, other topologies have been studied for headphone applications: switched and hybrid amplifiers. Switched amplifiers (Class D) have a relatively higher efficiency over a large range of output power but suffer from lower audio quality reproduction (linearity, power supply rejection) [START_REF] Cellier | A Fully Differential Digital Input Class D with EMI Spreading Method For Mobile Application[END_REF][START_REF] Huffenus | A Class D Headphone Amplifier with DC Coupled Outputs and 1.2mA Quiescent Current[END_REF] and are therefore most often used for speaker applications. Furthermore, a Class D requires a costly and bulky external filter to avoid unpredictable electromagnetic interference with the neighboring circuits. In order to combine the advantages of switched architectures (smallest conduction losses) and linear ones (highest linearity), both hybrid topologies have been introduced. Parallel hybrid amplifiers (Class K) are not used because they show high static consumption at the desired output level [START_REF] Van Der Zee | A High Efficiency Class D + Class AB Audio Power Bridge Amplifier[END_REF]. The series hybrid amplifier (Class G) offers a better current trade-off between audio quality and efficiency. Therefore, the following study focuses on this topology.

AES 133rd

The Class G principle is to dynamically supply a linear amplifier in order to keep its power supply as close as possible to the output signal. In this way (V DD ≈V OUT ), the amplifier efficiency reaches its maximal value (78.5%) according to equation [START_REF] Cellier | A Fully Differential Digital Input Class D with EMI Spreading Method For Mobile Application[END_REF].

In embedded applications, conventional Class G includes a Class AB amplifier supplied by two integrated DC-DC converters as shown in Figure 1. The negative voltage regulator is used to cancel the output common mode in order to remove the bulky output coupling capacitor. The negative V SS and positive V DD Class AB supplies are controlled by a level detector to track them as close as possible to the audio signal. The difference between the output signal and the power supply is directly linked to the conducted Class AB losses. The tracking algorithm implemented in the level detector is therefore a key factor in increasing Class G power efficiency. Work by Raab [START_REF] Raab | Average Efficiency of Class-G Power Amplifier[END_REF] proposes a guideline to design an optimal voltage tracking. While these analyses work with a pure sinusoid input signal, an analysis in nominal conditions (with a real input signal) is not available in the literature. To overcome this problem, our paper gives a method for predicting the voltage tracking strategy effect on efficiency and audio quality.

Firstly, this paper investigates the use of audio amplifiers in realistic situations to select appropriate input stimuli and amplitude. Secondly, it presents an overview of the proposed simulator to predict Class G efficiency. Finally, the proposed stimulus and simulator are used to study the nominal Class G efficiency and an optimization using different stimuli is performed and compared.

DEFINITION OF REALISTIC OPERATION

The classical method for efficiency prediction consists in simulating the Class G at the transistor level with a pure input sine wave. Unfortunately, this stimulus is different from the input signal applied in nominal conditions. Thus, in this paper, we show how to choose an input as close as possible to the realistic operating state.

Stimuli selection

Audio signals selected in this paper are summarized in Table 1. First, a pure sine wave is chosen because it is conventionally used to simulate and test audio amplifiers. Secondly, random variation signals are selected: i) pink noise shaped because of its power spectral density which is suitable for audio power testing, ii) a noise according to IEC 268-1 which is well established for modeling the audio signal.

Concerning the audio signals, four were selected with different characteristics. The latter will be presented in part 2.3. Each signal is assigned a number and will be referred to by its number in the following sections.

Signal

Efficiency measurements of Class-G amplifiers

The current industrial Class G amplifiers [10], [START_REF]TPA6140 datasheet[END_REF] only specify their efficiency when a pure sine wave (#1) is applied to their inputs. Unfortunately, the efficiency in nominal operation is different from these specified values. Figure 2 (top) shows the efficiency measured at 2mW output power using a Class G from [10] and the signals from Table I. The number of switchings between the lower and upper power supply voltages is also presented in Figure 2 (bottom). In [10] only two different voltages are used (V1 and V2 where V1<V2) to modulate the power supply of the linear amplifier.

The efficiency variation is almost 20% between the sine wave and #4. In other configurations, (see the following section) efficiency could be as much as 50% less depending on the applied signal.

The conventional signal (#1) overestimates the efficiency compared to signals with a Crest Factor (CF)>10dB and a kurtosis>4 (see Part 2.3 for definitions and Table 2 for values). When a sine wave is applied, the amplifier power supply stays at the lowest value V1 (as shown in Figure 2), while the other signals cause the amplifier to modulate the power supply. Signals #2 and #7 (noise shaped signal) underestimate the efficiency because the power supply always stays at its highest value V2. Only the audio signals allow the level detector to switch, except for signal #3 in this configuration. Indeed, signal #3, which has a low CF, operates the amplifier in the same way as a pure sine wave. Therefore, #1, #2 and #7 cannot mimic the real operating conditions. The audio signals are needed for accurate efficiency testing. The following paragraphs show how to carefully select a few audio signals to summarize the infinite input possibility.

Mean characteristics of the audio signals

To easily choose relevant stimuli over a large range of audio signals, some relevant parameters will be identified. By comparing them, the most representative signals can be chosen. First, the Root Mean Square (RMS) voltage value of the audio signal (see (eq 2)) is necessary to know how much power is delivered into the loudspeaker (thus the listening level). Note that some studies have compared the voltage vs. driving current of a loudspeaker [START_REF] Sturtzer | Comparison Between Voltage and Current Driving Methods of a Micro-Speaker[END_REF].

dt t s T V T t t rms     0 2 ) ( 1 (2)
The second parameter is the signal frequency composition. Assuming that the audio signal is periodic, the Fourier transform can be used to find its Frequency Decomposition (FD). This is given by:

      dt e t s f S itf  2 ) ( ) ( (3)
where s(t) and S(f) are the signal in the time and frequency domains, respectively.

The third parameter is the Crest Factor (CF) which defines the dynamic of the audio signal. Equation 3gives its definition:

          rms peak V V dB CF log 20 ) ( (4)
where V peak is the maximal value of the signal. Notice that the CF of a pure sine wave is equal to 3dB.

The fourth parameter is the Probability Distribution (PD) of the signal level. A histogram is used to provide a graphical representation. If the PD is uniform, it is equal to zero when the value of the sample is outside the interval [a,b], otherwise the PD is given by [START_REF] Sturtzer | Comparison Between Voltage and Current Driving Methods of a Micro-Speaker[END_REF]. For a Gaussian PD following a normal law, the equation can be expressed by [START_REF] Mijic | Statistical Properties of Music Signals[END_REF]. • the variance characterizes the dispersion around the mean. This moment is also known as the standard deviation (σ) which is the square root of the variance.

 ,

;b a x   a b x p X   1 ) ( (5)              2 2 2 exp 2 1 ) (    O X x x x p
• the skewness (S K ) is a parameter which gives a value to evaluate the symmetry of the PD. If the signal is totally symmetric the skewness is equal to 0.

• the kurtosis (K) measures the shape of the PD. The kurtosis of a Gaussian is equal to 3 and if the kurtosis is greater, this means that the PD is sharper.

It is important to point out that the first moment (mean) which is equal to zero in audio signal is not relevant.

The signals given in Table 1 were carefully investigated to choose a few appropriate audio signals from a large audio database [START_REF] Mijic | Statistical Properties of Music Signals[END_REF] (from classical to rock styles) to provide various characteristics (CF, FD, PD, skewness, Kurtosis and Variance). The values of these characteristics are listed in Table 2. We observe that each audio signal lasts ten seconds (taken after the first 25sec) to allow statistic analysis. Indeed, this length is the best compromise between time simulations and real condition testing. 3, the pure sine wave (#1) has a 3dB CF and contains only one frequency (1 kHz in this paper). Its PD is almost a uniform signal (kurtosis of 1.5 compared to 1.8 for an ideal uniform distribution). Signal #7 has a Gaussian PD (kurtosis of 3, as for an ideal Gaussian PD represented in red in Figure 3). Its spectrum decreases with frequency (as it is a filtered pink noise). The other characteristics are given in Table 2. The audio signals #3 and #6 are shown in Figure 4 (with the most differences in the music selected). We observe that their spectra are limited up to 16 kHz due to MP3 compression. Their parameters are quite different: i) CF=7 and 18dB, respectively, ii) kurtosis=2.39 compared to 5.64. However, they also have similarities. The two signals show several periods of silence between successive peaks of different magnitude which is one of the major factors causing power supply switching of Class G amplifiers.

Audio signal level

The audio signal level is related to the output power needed to reach the desired acoustic level. If the loudspeaker (electro-acoustic transducer) is assumed to be a pure resistive load R l , the relationship between the RMS input signal V i and the acoustic output power P a is:

k P R V a l i     (6)
where η, k are the electro-acoustical efficiency of the loudspeaker and overall gain of audio amplifier, respectively.

Experiments have been done to characterize the output power with respect to the desired acoustic power [START_REF] Eckert | Audio Level Safety Limits and their Impact on Personal Music Player[END_REF]. The test bench shown in Figure 3 is composed of a Head and Torso simulator (Bruel & Kjaer 4128C) and a Rohde & Schwarz UPD audio analyzer. The measurements have been realized in an anechoic chamber, using several types of headphone and a 1kHz input sinusoidal signal.

Figure 4 Test bench equipment

Figure 4 shows the output power versus the acoustical power received by ears. For a nominal listening level (75dB SPL), the output power varies between 2µW to 2mW which mainly depends on the electro-acoustical efficiency.

Figure 6 Sound pressure level Vs output power

Conclusion on stimuli selection

Seven audio signals have been selected to represent the large range of audio signal: from classical music to rock with various crest factors and large frequency and level compositions. They exhibit significant differences between each other and from a sine wave. The supply tracking algorithm implemented in the level detector will not respond in the same way to these complex audio signals as to a sine wave. To study the effect of these new stimuli on efficiency and audio quality, the next section proposes a simulator based on a novel model.

MODELING APPROACH FOR EFFICIENCY AND AUDIO QUALITY EVALUATION

The classical method (with a pure sine wave) requires only one signal period (usually 1ms) to be simulated to evaluate the Class G efficiency and audio quality. This simulation takes approximately one hour due to the circuit complexity including two switched converters, one linear amplifier and numerous analog blocks. By using much longer signals (10sec), the designer is faced with too long a calculation time to evaluate Class G performance. Moreover, multi-frequency and nonperiodic stimuli prevent any classical frequency analysis based on Fourier transform, such as harmonic distortion.

The authors have already suggested an original approach [START_REF] Russo | Switching Optimization for Class-G Audio Amplifiers with Two Power Supplies[END_REF] to avoid the expansive calculation. The main advantage of this model is the reduction of the calculation time step by averaging and simplifying Class G behavior. Furthermore, this approach uses the Perceptual Evaluation of Audio Quality (PEAQ) method to evaluate audio quality [START_REF] Salovarda | Estimating Perceptual Audio System Quality Using PEAQ Algorithm[END_REF]. This evaluation is needed because the power tracking algorithm can introduced some clipping effects on the audio signal. In fact, the Class AB can saturate if the power supply is lower than the output signal.

The previous results have validated the reliability of our model and allow the following simulations to be performed. The model has been validated in practice [START_REF] Russo | Switching Optimization for Class-G Audio Amplifiers with Two Power Supplies[END_REF] and the relative error is less than 5%. Therefore, this model enables longer simulation allowing the use of the audio signals selected for this study.

The classical performance, such as distortion, noise, idle current, has to be determined separately to check the design validity at transistor level by using the classical method. Here, the objective is to check the behavior of the power supply tracking algorithm on the overall Class G performance.

EFFECTS OF REALISTIC STIMULI ON EFFICIENCY

This section underlines the effect of realistic stimuli on Class G efficiency and on the choice of power tracking strategy. The results obtained are based on simulation of an industrial circuit [10]. In this integrated Class G amplifier, two discrete power supply levels are used to track the audio signal (1.2V and 1.9V). The other parameters are given in [START_REF] Russo | Switching Optimization for Class-G Audio Amplifiers with Two Power Supplies[END_REF].

Efficiency versus output power

At the same output power, the efficiency depends on the input stimulus as described in the previous paragraph but also over a large range of output power. Figure 7 shows the Class G efficiency compared to the selected signals over 20µW to 20mW (maximal output power). Below 0.8mW, the efficiency does not depend on the stimuli. In fact, the lower power supply level is always applied due to the very small signal amplitude. When the signal amplitude reaches the threshold voltage to switch the power supply to the upper level, the power tracking strategy has a strong impact on efficiency.

Here, the threshold voltage is around 1V [10] and the highest CF signal reaches this value around 8mW. After this limit, the sine wave over-or underestimates the efficiency.

Figure 7 Class-G efficiency vs. output power By using an optimized power tracking strategy [START_REF] Russo | Switching Optimization for Class-G Audio Amplifiers with Two Power Supplies[END_REF], the results are similar. We can see from Figure 8 a variation of ε=26% @1mW and a variation of ε=42% @2mW of output power. In addition, we see that the sine wave is not a relevant candidate for the prediction of efficiency.

Figure 8 Class-G efficiency with optimized power tracking

Voltage tracking behavior under realistic stimuli

The power supply modulation strongly depends on the input stimulus. Figure 9 shows the power supplies of a linear amplifier when signal #6 and a pure sine wave are applied at 2mW output power. For the audio signal, the supplies switch continuously between the lower and upper supply levels compared to the signal amplitude. With a pure sine wave, the supply level always stays at the lowest supply voltage. We propose 3ms of signal in Figure 9 but no switching appears, even with 10sec of signal due to the periodicity of the signal. It clearly shows that the chosen signal strongly influences the power supply tracking and thus overall efficiency.

As the input signal is random, the supply rising has to be anticipated to avoid unwanted clipping. This problem needs carefully study to find the better trade-off between efficiency and audio quality. The previous supply tracking algorithms do not introduce any clipping but they could be optimized to reduce their power consumption. previously proposed a method to optimize the power tracking strategy. The article [START_REF] Russo | Switching Optimization for Class-G Audio Amplifiers with Two Power Supplies[END_REF] gives the parameters to reach optimal efficiency in nominal operating condition based on audio stimuli described in this paper. To highlight the need for optimizing Class-G amplifiers with realistic input stimuli, an efficiency comparison is presented below.

The optimization has been done three times with a pure sine wave (#1), noise (#7) and the audio signal (#5). The method and parameters are the same as those described in [START_REF] Russo | Switching Optimization for Class-G Audio Amplifiers with Two Power Supplies[END_REF]. Based on a behavioral modeling of Class G amplifiers, a hybrid optimization using a global search algorithm (Genetic Algorithm) followed by a local search algorithm (Pattern Search) has been performed. The optimization has been realized for three output powers (0.1, 0.5 and 1mW) for audio signals, and 0.5, 1 and 10mW for signals #1 and 7 in order to apply an output power strong enough to modulate the power supply voltage. Parameter values of the switching algorithm are summarized in Table 3.

In the case of an optimization using signal #1 or #7, the lowest supply voltage V1 is placed relatively high compared to the signal #5. This leads to earlier modulation of the supply voltage. The decay time is the smallest with #5 which allows a faster switch down after an input peak. However, the parameters α and β that allow decisions to switch up/down the power supply are almost the same regardless of the signal used for optimization.

Parameters

CONCLUSION

This paper studies the audio Class G amplifier efficiency under nominal condition. To be closer to realistic operation, a few input stimuli have been chosen from a large database after identifying relevant key parameters to select them. From experiments, we also have determined the average power delivered by the amplifier for a nominal listening level. Based on existing Class G amplifiers, a dedicated model predicts efficiency and audio quality with the selected test pattern. Our paper clearly shows the limitation of the classical approach (with a pure sine wave) as it either over-or underestimates the real power consumption by up to 50%. Moreover, the proposed analysis helps to optimize the supply voltage tracking in an audio Class G amplifier to reach the maximal tradeoff between audio quality and efficiency. The use of the selected stimuli improves the efficiency by 20% without losing audio quality by redefining the voltage tracking algorithm.

6.

  Convention, San Francisco, CA, USA, 2012 October 26-29 Page 2 of 8

Figure 1

 1 Figure 1 Class-G block diagram and transient signal In a current industrial Class G, only two (or three) discrete levels are used to modulate the power supply.The difference between the output signal and the power supply is directly linked to the conducted Class AB losses. The tracking algorithm implemented in the level detector is therefore a key factor in increasing Class G

Figure 2

 2 Figure 2 Efficiency and switching behavior of selected signals at 2mW

( 6 )

 6 where xℝ, x O represents the mean value and σ the standard deviation. Three other well known parameters have been used corresponding to the moments of degrees 2, 3 and 4, respectively, for statistical signal analysis: AES 133rd Convention, San Francisco, CA, USA, 2012 October 26-29 Page 4 of 8

  Figures 3 and 4 show the main characteristics of four signals: transient plot, statistical distribution and frequency spectrum in the audio band (Fast Fourier Transform or spectrogram with the Short Time Fourier Transform by using the spectrogram function in Matlab).

Figure 3

 3 Figure 3 Characteristics of #1 (left) & #7 (right)

Figure 4

 4 Figure 4 Characteristics of #6 (left) & #3 (right)

Figure 9

 9 Figure 9 Signal #1 and #6 at 2mW output power 4.3. Toward a better optimization of power tracking The previous results shows that the designer needs to take into account real stimuli to design and optimize the supply voltage tracking which is one of the key factors to improve the Class G efficiency. The authors have

Table 3

 3 Parameters of the power tracking algorithmThe parameters in Table3were then used to evaluate the efficiency of the Class-G amplifier with an audio signal not used by the optimizer. The signal (interpreted by the Red Hot Chilli Peppers) has a CF of 17dB, a standard deviation of 0.05, a skewness of 0 and a kurtosis of 4.42. The results are presented in Table4for four output powers. It is clear that the best efficiency is obtained with the supply voltage algorithm optimized from audio signal #5 over the power range. This result confirms the need to use audio signals during the design and the optimization to obtain the best efficiency performance. For example, the efficiency has been improved by 20%, without losing audio quality, by redefining the voltage tracking algorithm with signal #5.

		#1	#7	#5
	V1 (V)	0.93	0.84	0.7
	Α	7/8|V SS |	7/8|V SS | 7.2/8|V SS |
	Β	6.4/8|V SS | 6.6/8|V SS | 6.5/8|V SS |
	Decay Time (ms)	6	3.2	0.1
	Attack Time (ms)	0	0	0

ACKNOWLEDGEMENTS

The authors would like to thank the Region Rhône-Alpes for their financial support. They would also like to express their gratitude to STMicroelectronics for industrial feedback.

7.


Jérome Bachmann
email: bachmann@iut.univ.aix.fr

Jean Marc Linares

Jean Michel Sprauel

Pierre Bourdet

Aide in decision-making: contribution to uncertainties in three-dimensional measurement ˆ

Keywords: Uncertainties, Propagation, ISO tolerances

The authorities of the standards organization International Organization of Standardization (ISO) advocate mastering any uncertainties in all parts of the industrialization process. In the three-dimensional (3D) measurement process, uncertainty is usually obtained at the end of a battery of tests. It is defined as a whole because it includes several types of errors, known systematic components, unknown systematic components and random components. Automated calculations of uncertainty can be made based on statistics. This method is based on statistical concepts, which are in accordance with "The Guide to the expression of the uncertainty in measurement" (GUM). It also enables us to generate uncertainties on the verification of ISO specifications (or specs in the ISO directives). In the course of this article, a usage will be presented that takes the knowledge of uncertainties into account: this usage will help the operator to take a decision on the conformance of a mechanical part in reference to its conformance to geometric tolerance.

Introduction

One of the concerns of the ISO/TC 213 [START_REF] Srinivasan | An integrated view of Geometrical Product Specification and Verification[END_REF] work group is to take uncertainties into account in the global industrialization process. It is a question of mastering the propagation of the uncertainty due to best-fit calculations and geometric constructions that the metrologist makes while checking specifications. This objective must be met while respecting the duality principle according to which it is difficult to specify a geometric function without integrating verification procedures [START_REF] Dovmark | New interesting concepts from ISO/TC 213[END_REF].

In the past few years, the group has attempted to reduce any deficiencies and contradictions which affected dimensional and geometric tolerance. Proposals were then adapted to the functions of the part. The tools proposed could not be applied to mechanical specifications. Up to now, the explanation of standards has been illustrated by a certain number of unique cases. Today, ISO/TC 213 is moving towards a univocal and generic mathematical description of the problems of geometric specifications [START_REF] Ballu | Global view of geometrical specifications[END_REF].

On a larger scale, The Geometric Product Specification (GPS; ISO/TC 14638) matrix has been defined, and it aims at creating a coherent system for the elaboration of standards for the industrialization of a product. It can coherently define a set of standards in relation to a particular problem and to the geometric characteristics of the element. Among the standards suggested, GPS language provides a transverse response to the problems of checking and specifying. However, some deficiencies are observed in tolerance formalization. In this context, ISO/TC 213 proposes a concept based onamathematical formalism, which can express any dimensional and geometric demand onamechanical part inaunivocal manner.

ISO/TC 213 has taken the responsibility of developing a language and methods of verification associated to each specification acting inafunction in the product. Within this framework, the notion of the generalized uncertainty principle has appeared. It is based on the fact that generalized uncertainty includes:

-Correlation uncertainty: it arises from the fact that the intended functionality and the controlled geometric may not be perfectly correlated. -Specification uncertainty: it results from incorrect or incomplete application of geometric specifications. -The measurement uncertainties in a verification process.

Concerning the expression of measurement uncertainties, the GUM [START_REF] Bipm | IUPAP, OIML. Guide to the expression of the uncertainty in measurement[END_REF] and associated standards offer methods to de-terminate their origins and of quantifying the latter (physical and chemical phenomena, mathematical modeling of the understood phenomena, interactions between materials and operating conditions).

In conformance with the new directives taken by ISO, the setting up of calculations of uncertainties in the software of Coordinate Measuring Machines (CMM) is proposed. This consists in being able to automatically display the uncertainty associated with the result.

The range of normalized specifications generally leads to the calculation of a set of distances when no modifier of the envelope condition, maximum or minimum material condition is specified. These situational characteristics are of the following type:

-Point/point distance -Point/plane distance -Point/line distance
In the course of this article, the concept of the Statistical Confidence Boundary (SCB) will be presented. This concept enables the user to visualize the uncertainties in normal surfaces. Secondly, the uncertainties in the case of the study of a coaxial specification will be propagated, conforming to the directives of ISO/TC 213. The use of a computerized model will allow the metrologist to provide extra information on the acceptance of the specification. This tool which aides in decision-making helps to define corrective actions on the range of measurements.

3D uncertainties: optimization and representation

searchers. A number of studies refer to the GUM [START_REF] Bipm | IUPAP, OIML. Guide to the expression of the uncertainty in measurement[END_REF], which shows the statistical bases that must be used.

Different directions, which lead to the calculation of 3D uncertainties, are distinguished. One of them concerns the study and best-fit of a set of measured points coming directly from CMM. A number of projects related to the study of a set of measured points were carried out by Henke and Summerhays [START_REF] Summerhays | Optimizing discrete point sample patterns and measurement data analysis on internal cylindrical surfaces with systematic form deviations[END_REF][START_REF] Henke | Methods for evaluation of systematic geometric deviations in machined parts and their relationships to process variables[END_REF]. They suggest methods of the evaluation of geometric errors obtained on the parts where the production process is known. The best-fit methods used take into account the characteristics of the process in order to find the best model. The least-squares criterion has been retained as best-fit method. It can calculate the extended zone model (EZN) coefficients for each of the geometric defects in the retained production method. In the same way, Kurfess and Banks [START_REF] Kurfess | Statistical verification of conformance to geometric tolerance[END_REF] propose a strategy for sampling the surface of the most efficient part and a more adequate optimization with the functions of the part in order not to reject good parts or accept bad ones. Uncertainty is obtained through the knowledge of the covariance matrix. Best-fit methods are carried out numerically by algorithms which allow for different kinds of errors generated by each process type.

Choi and Kurfess [START_REF] Choi | Uncertainty of extreme fit evaluation for three-dimensional measurement data analysis[END_REF] put forward a method of estimating the uncertainty of the measurement on envelope surfaces. They adopt a statistical approach in considering the measurement uncertainty as the stochastic noise of a point sample. This technique is similar to the one used in the evaluation of extreme fit. Results are given in a particular case where surfaces are theoretically built using a random sample method.

A number of methods, which are sometimes divergent, are used to associate a shape to a point sample. An analysis of these problems was presented by Weckenmann et al. [START_REF] Weckenmann | Functionality-oriented evaluation and sampling strategy in coordinate metrology[END_REF]. He clearly demonstrated that the association of a surface to a set of measured points assumed the knowledge of the surface in question completely. A criterion of evaluation must be chosen in relation to the function of each surface in the part being checked. However, the least-square criterion remains the one most used because of its robustness especially when the points are sparse in the sample. Without pre-judging the distribution of the sample points (study of the roughness of a surface). Mestre and Abou-Kandil [START_REF] Mestre | Measuring the errors of form of industrial surfaces: prediction and optimisation[END_REF] demonstrated that a best-fit with the least-squares method sometimes causes a non-negligible error in the result. They put methods of Bayesian prediction to use for surface metrology.

Determining uncertainties using a vectorial modeling of surfaces based on the same model as Yau is proposed [START_REF] Yau | Evaluation and uncertainty analysis of vectorial tolerances[END_REF]. The displacement matrix obtained will allow us to use a non-linear model and will allow us to set up an analytical model for ascertaining uncertainties.

The geometric representation of uncertainties was the subject of a study done by Hernla [START_REF] Hernla | Calculation of measuring uncertainty with CMMs under industrial conditions[END_REF] in 1993. He presents the foundations of the representation of uncertainties in 2D cases. The hypothesis retained only makes references to models where the variables are not correlated. In this case, the author wonders about the geometric shape that the confidence Determining uncertainties is a great concern for re-boundaries of a point can have when it is the result of two points having an interval of confidence. He thus approaches the question of the propagation of uncertainties. The concept of Statistical Confidence Boundaries will be presented below.

It is based on the same principal but no longer requires any particular hypothesis (3D cases and correlated variables).

Uncertainties : the concept of Statistical Confidence Boundary (SCB)

Geometric approach in surface association

This is the approach, which is used the most. It consists in determining a theoretical surface which best passes from a set of points following a criterion: infinite norm (Tchebichev norm), norm 1 (least absolute deviation LAD) or norm 2 (least-squares norm). The best surface obtained is the one, which minimizes a criterion. If the sum of the squares of the distances of points to the plane under research is the criterion retained, then, the criterion of the least-squares method is used. The result of the association enables us to evaluate mean values of the parameters of the associated surfaces. As a general rule, linearizing displacement to the first order is sufficient. This amounts to using a translation and a rotation of the local 3D basis. The drawback to this approach resides in the fact that the uncertainty on the estimated parameters can only be obtained with an approximation. Fromanumeric point of view, providing initial parameters, which are close to the final solution, is indispensable [START_REF] Kruth | Parameter identification of geometric elements from digitized data of coordinate measuring machines[END_REF]. Moreover, the geometric modeling of certain surfaces requires making a transfer. This is the case in the association ofastraight line in space, which presently needs to be projected on two perpendicular planes (characteristics of the local 3D basis). Therefore, it is easier to use information technology to calculate each of the coefficients of the two straight lines found in this manner.

Statistical approach in surface association

All measurements, however careful and scientific, are always tainted by uncertainty. Their estimation through different studies simply allows us to evaluate them, to get a rough estimate of their value, in order to reduce them if necessary [START_REF] Arri | On the application, Guide to the expression of the uncertainty in measurements to measuring instruments[END_REF]. This evaluation rests on the notion of inference: it consists in stating results (e.g. dimension) related to a phenomenon without knowing it entirely. This notion must be differentiated from the notion of deduction [START_REF] Estler | Measurement as inference: fundamental ideas[END_REF]. The latter requires setting up hypotheses, which will serve in obtaining a result. The strong hypothesis that must be verified concerns the distribution of the distances d i , which must be compatible with the criterion, used for the association. This approach is based fundamentally on statistical mathematics (Fig. 1).

Practically speaking, the studied parameter must be considered as the population to be characterized. Determining moments in a statistical sense can do this. It is, in fact, one of the characteristics of the population. Generally, only the mathematical expectation, or first statistical moment is determined [START_REF] Sprauel | Contribution of nonlinear optimization to the determination of measurement uncertainties[END_REF].

As the surface is real, it is made up of an infinite number of points. When an operator takes a measurement, he is in fact taking a sampling of the real surface, thus creating a sampling, made up ofaset of n points. From these n points, CMM software associates the surfaces. Statistical theory makes the calculation of the set of all moments serving to character-ize the population. The surface characteristics (point, vector, scalar) are estimated by their mathematical expectation and by their variance. The statistical approach of surface association and the ascertainment of their uncertainty are compatible with the GUM proposals [START_REF] Bipm | IUPAP, OIML. Guide to the expression of the uncertainty in measurement[END_REF].

A set of digitized measured points contains a certain amount of information that must be in order to express the second statistical moment. Only in making an adequate model can automatic calculations of this information be had [START_REF] Sprauel | Contribution of nonlinear optimization to the determination of measurement uncertainties[END_REF][START_REF] Goch | Universal algorithm for the alignment of sculptured surfaces[END_REF]. The uncertainty obtained is a convolution of several state variables: including machine variability, geometric error by the machine, along with the signature of the process.

Vectorial description and uncertainties

In the above paragraph, it has been shown that the notion of statistical vectors is present in 3D metrology. The following paragraph will define the properties of this statistical vector.

A statistical vector a of dimension n belonging to R n is the vector [a 1 , a 2 , . . . , a n ] T . The a i components are statistical variables. Its cumulative distribution function F of R n to R is defined by:

F : a i → Prob(a i < u i ), ∀ (a i ) ∈ R n    F(a i ) = ... f(u i ) du i
With f probability density associated with vector a if the statistical vector has continuous components. The classic properties in 1D statistics are still valid:

-The density of probability f has values in R+.

-The cumulative distribution function has a value of:

   F(a i ) = ... f(u i ) du i = 1
It is assumed that the statistical vector a has continuous components with a density of probability f which allows for moments.

The latter is defined by:

k    m ki = ... u i i f(u i ) du i
With k i ∈ N designating the first statistical moment k of the variable is of vector a. If the a i (statistical) components of the vector have an expected value, the centered moments can be calculated by:

   µ ki = ... [u i -E(u i )] ki f(u i ) du i
For example, in the case of an equation of a plane u • x + v • y + w • z + h = 0, the statistical vector a has the following components:

a = [u, v, w, h] T
The first and second statistical moments can thus be defined as:

m 1,0,0,0 = E(u), m 0,1,0,0 = E(v), m 0,0,1,0 = E(w), m 0,0,0,1 = E(h) µ 2,0,0,0 = Var(u), µ 0,0,2,0 = Var(w),
The moments µ 1,1,0,0 ; µ 1,0,1,0 ; µ 1,0,0,1 ; µ 0,1,1,0 ; µ 0,1,0,1 ; µ 0,0,1,1 define the terms in covariance between the variables of the statistical vector.

µ 1,0,1,0 = Cov(u, w), µ 1,1,0,0 = Cov(u, v), µ 1,0,0,1 = Cov(u, h) µ 0,1,1,0 = Cov(v, w), µ 0,1,0,1 = Cov(v, h), µ 0,0,1,1 = Cov(w, h)            .
The notion of covariance matrix is useful in representing the second statistical moments.

The above paragraph defines the different moments of the statistical vector under the existence hypothesis. Starting from these fundamental notions of multidimensional statistics, it is therefore possible to define the expected value and the covariance matrix of a statistical vector.

When a,an dimension statistical vector and of components [a 1 , a 2 , . . . , a n ] T has first statistical moments, its expected value is defined as

 E(a 1 )  E(a 2 ) E(a) =  .  E(a n )
Given a,astatistical vector of dimension n and of components [a 1 , a 2 , . . . , a n ] T which admits second statistical moments, its covariance matrix is defined as:

Cov(a)      Var(a 1 ) Cov(a 1 , a 2 ) • • Cov(a 1 , a n )       Cov(a 2 , a 1 ) • • • •  =  • • • • •   • • • • •  Cov(a n , a 1 ) • • • Var(a n )
The covariance matrix is thus a square matrix of side n. For example, in the case of a plane, the expected value and the covariance matrix are: Cov(a)

       E(u) E(a) =  E(v) E(w)  E( h) C (Point) Cx +/-UCx Cy +/-UCy Cz +/-UCz V (Line) Vx +/-UVx Vy +/-UVy Vz +/-UVz
=   Var(u)  Cov(v, u)  Cov(w, u ) Cov(h, u) Cov(u, v) Var(v) Cov(w, v) Cov(h, v) Cov(u, w) Cov(v, w) Var(w) Cov(h, w)  Cov(u, h)  Cov(v, h)  Cov(w, h ) 
Var(h) The first and second statistical moments can be calculated, in other words, the expected value and the covariance matrix of each of the characteristic entities. The least-squares criterion is used. The modeling used is a vectorial modeling of surfaces (C is the characteristic point on the surface, V the characteristic vector, R and α two scalars or intrinsic parameters) (Fig. 2). Determining these four entities enables us to calculate the set of normal surfaces in 3D metrology in conformance with the vectorial description.

Knowing the covariance matrix facilitates the identification of the uncertainty, which was used in estimating the associated derived element. This, in turn, enables us to define the volume in which the associated derived element with given risk is found. The limit of this volume is defined with the SCB.

Definition of a SCB:

Theoretical envelope obtained from first and second statistical moments of the parameters of position and orientation of the best-fitted surface to a given risk α.

The dimensions of the SCB of the surface in a local 3D basis are defined by eigenanalysis of the covariance matrix. Let us take the example of a plane surface described using vectorial geometry by a vector and a point. The vector and point entities are characterized by three components, which are, in a 3D basis (x-abscissa, y-ordinate, z-coordinate) (Fig. 3). Each component is estimated using a numerical value (mean value). The covariance matrix links these statistical components. It is used to characterize the uncertainty associated with each component.

Visualizing the results

With this concept the surface is no longer considered as uniquely determined but rather as belonging toaspacing zone limited by a defined envelope with a given risk. In the fol- lowing sections, the example of a simple numerical will be considered. A plane has been drawn with 16 randomly set points using a measuring machine equipped with a Renishaw type TP2 touch trigger probe, in the measuring conditions defined by the manufacturer. The characteristics of the asso- ciated surface are shown in Table 1 (the six components are linked statistically). Table 1 displays the results of the associated surface with a confidence ratio of k = 2 which is necessary in framing mean values. The table is made up of two columns:

• The first one proposes an estimate of the mean (mm). The results are typical and conform to those found with the MMT software. • The second one presents the uncertainty with a confidence ratio of k = 2. This corresponds to a new piece of information, which is not currently offered.

The limits of the zones of this surface are calculated from the covariance matrix. This space can be visualized from data points. For the point, the envelope is elliptical. For the plane, there are two symmetric envelopes in relation to the mean surface. It is noted that uncertainty rises with the distance of the point considered at the centroid (Fig. 4).

To make a representation in SCB form, the covariance matrix (order 2 tensor) is used. This tensor can be characterized byaconical whose features do not depend on an axis system. For instance, the point position resolution can be characterized by an elliptical of the following equation:

X Y Z 2 2 2 1/σ X + 1/σ Y + 1/σ Z = 1
The standards deviation σ X , σ Y , and σ Z are obtained from the covariance matrix expressed in a local 3D axis (matrix eigenanalysis).

The SCB characterizes the doubt with which the mean is estimated. For the plane and the line, the associated surface (or mean) is the functional surface. This is not the case for the other surfaces in general:

• For the cylinder or the cone, the mean value characterizes the axis. • For the sphere and the circle, the mean value characterizes the center. Indeed, for the sphere, SCB limits the point positions, which are characteristic of the associated surface and in no way the associated surface.

The propagation of uncertainties

From the SCB to the Statistical Generated Object (SGO). During the verification phase, a process measurement will be established. When calculating each associated derived feature, an SCB can be determined. In order to be able to estimate the influence of these SCB on the resulting SCB, mathematical formalism must be put in place. It is suggested to use the SGO to this effect. The SCB are the geometric representation of the zone of uncertainty calculated from a covariance matrix. In order to have a synthetic object, the SGO will be defined.

For random vector of three components, the SGO is a matrix of dimension [START_REF] Henke | Methods for evaluation of systematic geometric deviations in machined parts and their relationships to process variables[END_REF][START_REF] Bipm | IUPAP, OIML. Guide to the expression of the uncertainty in measurement[END_REF], which gathers the first and second statistical moments.

The SGO of each elementary surface or the result of an operation between two SGOs will be defined from basic SGOs corresponding to intrinsic parameters (vector and scalar element). Table 2 above represents the basic SGO of a point or a vector.

0 0 0 Cov(X, Y) Var(Y) Cov(Z, Y) 0 0 0 Cov(X, Z) Cov(Z, Y) Var(Z)
In the example of the associated-derived cylinder, the SGO is obtained during the association phase. It will be the assembling of two basic SGOs of the vector type and of a basic SGO of the scalar type.

Propagation

The propagation procedure uses the SGOs obtained during the association phase. This stage is essential for initiating propagation.

Definition of propagation

It is the possibility of creating a SGO from other SGOs. The SGO obtained in this way must be minimal (the determinant of the covariance matrix of the propagated parameters must be minimal).

The question is to determine the SGO resulting from the beginning SGO for each of the calculated surfaces, i.e. to be able to trace the SCB.

From a statistical point of view, the question is to determine the second statistical moment at the studied parameter in relation to the entry parameters. Fig. 5 illustrates propagation by placing a sensibility factor. The studied parameter is a scalar and only depends on a unique entry x variable. The second statistical moment of the exit parameter is a function of the local gradient calculated by the partial derivative of the function at the studied point [START_REF] Goch | Universal algorithm for the alignment of sculptured surfaces[END_REF]. This function is generally known. It can, however, be implicit but it is at any rate solvable thanks to numerical methods [START_REF] Cox | Measurement uncertainty and the propagation of distributions[END_REF].

For an exit variable depending on several entry variables, the propagation principle is identical [START_REF] Tuninski | Concept and rules in the guide to the expression of uncertainty[END_REF].

k ∂x   ∂y ∂x ∂y ∂x Var(y) =   ∂y 2 Var(x i ) i=1 i k-1 k   + 2 Cov(x i , x j ) i=1 j=i+1 i j
For a scalar parameter y depending on several variables of dimension n, variance can be obtained with:

Var(y) = J y Cov(A)J T y p
where Cov(A) represents the covariance matrix of n entries parameters x i and J y , the Jacobian matrix defined by:

∀ i ∈ {1 . . . n}, ∀ p ∈ {1 . . . n} ∀ x i ∈ R ∂y 1 ∂x 1 2 ∂x
∂e Jacobian matrix J(di) [J ] = ∂(OMi • v) j ∂e [J] = ∂(||OMi ∧ v||) j 1 ∂ [J] = ∂(||M e M2||)
i ej: Surface parameters ej: Surface parameters ej: Surface parameters 3 gives an example of propagation applied to the calculation of three fundamental distances. In 3D measurement, the studied parameter is a vector. Thus, the second statistical moment is homogeneous in a covariance matrix of dimension 3. This is the SGO of the studied parameter.

   Var(v) 0 Covariance matrix Var(X) Var(X) =  0 Var(O) 0 0 Var(Mi) 0  0    Var(X) =  Var(M1) 0 Var(M2) 0  di uncertainty U(di) = [J] Var(X)[J] T Table
Determining the final SGO requires setting up a process. Calculating a SGO from two SGOs is done via a set of propagation modulus, which are function of the constructions needed to check a specification.

Mathematical operations between SGOs can be carried out with these modules. This corresponds to geometric operations, which allow for the creation ofaSCB from other SCBs. For example, the Projected Point SCB module on Plane SCB enables us to create a SCB Point.

Example

Let us consider the case of a geometric tolerance of position: a coaxiality (Fig. 6).

The geometric elements considered are two non-ideal surfaces (S1) and (S2) which are nominally cylindrical and identified byapartition and filtration operation taken from a "skin model." a. A best-fitted surface of a cylindrical type (CY1) comes from a set of points (S1) with an objective of minimizing the sum of the squares of the distances from the S1 points to the cylinder (CY1). The axis of cylinder S1 is defined by a series of the following mathematical operations (Fig. 7). b. A set of ideal constructed i planes perpendicular to the CY1 axis is considered (Fig. 8).

7.1.1.1. Interpretation. The specified cylinder axis must be included in a cylindrical zone of diameter t, coaxial to the axis of the referenced surface. c. In each of these planes (PL1i), the nominally circular lines are considered.Abest-fitted operation must be carried out in order to obtain i circles (CE1i) in each of the planes (PL1i) (Fig. 9). d. Next a surface is best fitted whose reference isacylindrical type (CY2) from the surface (S2).

In order to meet specifications, the greatest distance value d i (CE1i, CY2) must be inferior to half the tolerance t (max(d i ) < t/2).

In this example, it is clearly illustrated that the calculations of distances come after a long series of operations, which is necessary in order to respect the directives (orders, instructions) of the ISO/TC 213 group.

Propagation of the uncertainties of any geometric specifications that follow the principle of independence (without modifiers) can be done via the software program. It is not linked to a set of particular operations.

The following paragraph will deal with this coaxiality in a numerical point of view. The CMM programming is initiated according to the range given above. The CMM is only used as a means of acquiring points on real surfaces. Any 3D measurement machine could have also been used such as laser Tracker or coaxial optical captor. The data used can be of any kind as long as it contains a series of three coordinates (X, Y, Z).

The result of the specification to be studied inherits the value of the uncertainties of the beginning digitized surfaces. Thus, first of all, the results of the uncertainties for each of the surfaces will be shown.

The specification requires a definition of a datum surface. In our example, this is the axis of the cylindrical surface (Fig. 10). The axis of the surface, after fitting, is defined by a characteristic point (three coordinates and their respective uncertainties) as well as the directing vector of the axis (three coordinates + their respective uncertainties): Table 4.

As for the specified surface, four circles have been digitized. Thus, an image of the real axis of the specified cylinder has been calculated, as well as the geometric characteristics of the fitted surfaces for each of these four circles. Table 5 groups the results of the four circles and their uncertainties.

Results in distance

In accordance with the above paragraph, interpreting specifications requires calculating distances. Table 6 gives the result of the four point/line distances.

If only the mean value distances are considered, the part will be declared in conformity for any interval of a tolerance greater than 4.37 mm. Columns 2 and 3 present the uncertainty and enlarged uncertainty with a confidence ratio of k = 2. This corresponds to a unilateral risk of 97.7%.

Aide in decision-making

Mechanical parts have always obtained acceptance through the knowledge of mean values. The capability of the procedure is only generally known for CMMs, by methods based on the experience of metrologists and on a repetition of measurements. This uncertainty is applied in an empirical way and is systematically a function of the precise formula of a CMM of the form: +/-(a + bL). Knowledge of propagated uncertainties on a result enables us to better define the limits of acceptance.

The following paragraph gives evidence of the different types of results that can be obtained on the acceptance of a part. When having an estimation of measurement quality is not possible, the major difficulty is in accepting defective parts and refusing correct parts.

Let's consider a tolerance interval defined by a greater tolerance Ts and a lesser tolerance Ti. The measure is accepted when it is situated in this tolerance interval (Fig. 11). If the uncertainty is estimated, it can be represented graphically using a Gaussian law [START_REF] De Vicente | A generalization of the law of propagation of uncertainty. An example: determining screw thread uncertainty[END_REF]. Indeed, the convolution of a large number of entry parameters characterized by a given distribution induces a Gaussian distribution of the studied parameter (Fig. 12).

Several cases (Fig. 13) can thus be presented as a function of the value of typical difference which characterizes the law. When the measure is contained in the tolerance interval, the part is generally declared in conformance. The metrologist cannot simply make a decision as to the acceptance of the part if uncertainty is not estimated.

When the measure is not contained in the tolerance interval, the part is usually declared in non-conformance.

The model offers an aide to the operator. From the files of digitized surfaces, the set of distances, which influence decisions on conformance, can be calculated. The uncertainty associated with each distance can be represented graphically which gives information about the chosen risk. Fig. 14 presents the results from the aide in decision-making software program. The two right and left limits define the area of the tolerance zone. Four distances are calculated. For each of these, the position of the mean value (d 1 , d 2 , d 3 , d 4 ) is depicted. In view of the chosen interval tolerance, the coaxiality specification is respected with regard to the mean values. The user truncates the functions of distribution at a chosen risk. (For our example, a unilateral risk of 2.5% was chosen.) The unilateral risk is obtained with a form defect law when the mean value of the distance is less than three times the uncertainty of the distance. Its cumulative density remains, however, the same than the Gaussian law's one if the unilateral condition is assumed. The latter can thus give the operator quantitative information as to exceeding the interval tolerance (Fig. 14). The amount exceeding the upper limits of the tolerance interval is calculated as a percentage. Thus, d 4 does not pose a problem for acceptance. However, for value d 1 there is a 31.09% risk that the mean value goes beyond the upper limit of the tolerance interval.

The operator can thus make his decision in consequence, taking corrective action aiming at reducing the obtained uncertainty. He will be able to act directly on the part checking program, modifying it, or increasing the number of points of the concerned surface in order to reduce uncertainty. The software accompanies the operator in his decision. This depends in fact on a number of parameters (price of the part, cost of measuring on the CMM, software possibilities).

Conclusion

The article has shown how the knowledge of uncertainty can be used. This method is based on statistical concepts, which are in accordance with "The Guide to the expression of the uncertainty in measurement" (GUM). It also enables us to generate uncertainties on the verification of ISO specifications (or specs in the ISO directives). The final result of the specification is a direct consequence of the actions taken by the CMM operator. Mastering the risk of acceptance or refusal enables us to inform the operator in a quantitative way. He can thus take one or more corrective actions relating to the characteristics of the product and of the enterprise. This software aide lies in the scope of a process of improving quality and conforms to the directives of the authorities of the standards organization.

Fig. 1 .

 1 Fig. 1. Normal distribution.

Fig. 2 .Fig. 3 .

 23 Fig. 2. Characteristic elements.

Fig. 4 .

 4 Fig. 4. Presentation of the SCB point, plane and line.

Fig. 5 .

 5 Fig. 5. Propagation.

Fig. 6 .Fig. 8 .

 68 Fig.6. Example of coaxiality specification.

Fig. 10 .

 10 Fig. 9. Third step.

Fig. 12 .

 12 Fig.12. Distribution of entries and exit frequency.

Fig. 11 .

 11 Fig. 11. Characteristics of a tolerance interval.

Fig. 13 .Fig. 14 .

 1314 Fig. 13. Different cases of acceptance risk or refusal.

Table 1

 1 Results (confidence ratio k= 2)

	Ox	-4.240E+01	+/-	1.818E-06
	Oy	6.285E-02	+/-	7563E-06
	Oz	1.511E-02	+/-	5062E-03
	Nx	3.592E-04	+/-	8976E-05
	Ny	1.494E-03	+/-	1077E-04
	Nz	1.000E+00	+/-	1639E-07

Table 2

 2

	Statistical generated object
	Ex	0
	Ey	0
	Ez	0
	0	Var(X)
	0	Cov(X, Y)
	0	Cov(X, Z)

Table 3

 3 Propagation on three fundamental distances

i: numerous point; j : 1 to 3

Table 4

 4

	Characteristics of the datum surface
		Datum (mm)
	Ox	-192.474	+/-	2.602E-05
	Oy	-225.970	+/-	1.644E-03
	Oz	66.740	+/-	1.661E-03
	Nx	0.9999	+/-	4.052E-07
	Ny	-7.256E-03	+/-	5.581E-05
	Nz	2.287E-04	+/-	5.774E-05

Table 5

 5 Characteristics of the circles representing the specified surface

	Circle 1 (mm)
	Ox	-249.4645	+/-	3.476E-05
	Oy	-225.5520	+/-	4.641E-03
	Oz	66.7315	+/-	4.640E-03
	Nx	0.9999	+/-	1.491E-06
	Ny	-7.475E-03	+/-	1.992E-04
	Nz	4.244E-04	+/-	1.991E-04
	Circle 2 (mm)
	Ox	-229.465	+/-	3.761E-05
	Oy	-225.702	+/-	5.047E-03
	Oz	66.738	+/-	5.046E-03
	Nx	0.999	+/-	1.700E-06
	Ny	-7.440E-03	+/-	2.281E-04
	Nz	4.229E-04	+/-	2.280E-04
	Circle 3 (mm)
	Ox	-209.466	+/-	3.016E-05
	Oy	-225.847	+/-	4.048E-03
	Oz	66.742	+/-	4.047E-03
	Nx	0.999	+/-	1.678E-06
	Ny	-7.440E-03	+/-	2.2517E-04
	Nz	4.170E-04	+/-	2.251E-04
	Circle 4 (mm)
	Ox	-169.467	+/-	2.833E-05
	Oy	-226.139	+/-	3.814E-03
	Oz	66.751	+/-	3.813E-03
	Nx	0.999	+/-	1.877E-06
	Ny	-7.416E-03	+/-	2.528E-04
	Nz	4.086E-04	+/-	2.527E-04