A Govin
email: govin@emse.fr

A Peschard

E Fredon

R Guyonnet

New insights into wood and cement interaction ◊

Keywords: Alkaline degradation, Electrical conductivity, Extractives, Hydration of cement, Setting retardation, Wood-cement composites

This work deals with the influence of poplar extractives and poplar alkaline degradation products on the hydration of cement. The wood chemical composition was characterized to determine the impact of soft alkaline conditions on wood-fiber. Some of the constituents are hydrolyzed and converted into carboxylic acids which are identified and quantified. The monitoring by electrical conductivity clearly showed that wood extractives are less effective than wood alkaline degradation products to inhibit the hydration of cement.

Introduction

Many works have been devoted to the use of vegetable fibers as aggregates in cement. Woodcement composites were specifically studied since these aggregates enhance soundproofing and thermal insulation properties. However, several drawbacks have hindered the development of these composites. The main difficulty is the chemical incompatibility between wood and ordinary Portland cement (OPC) which retards cement setting and hardening. The chemical structure of wood is complex. It is composed of cellulose, hemicelluloses, lignins and extractives. [START_REF] Biblis | Effect on the setting of southern pine-cement mixture[END_REF] reported that among the extractives simple sugars were powerful retarders. However other compounds were also found to adversely affect the hydration of cement reactions such as sugar acids (Ficher et al. 1974), hemicelluloses [START_REF] Simatupang | Abbaureaktionen von Glucose, Cellobiose und Holz unter dem Einfluss von Portlandzementmörtel[END_REF]) and phenolic compounds such as tannins [START_REF] Miller | Wood-cement composites[END_REF][START_REF] Miller | Wood-cement composites: Effect of model compounds on hydration characteristics and tensile strength[END_REF].

Most of the previous works deal with the effect of several wood species or wood constituents on cement setting or hardening by means of adiabatic or isothermal calorimetry. However, the mechanism of the wood-cement interaction is not yet fully understood. The present study aims at the investigation of the effect of wood components on the hydration of cement and the behavior of wood in such systems. Precisely, the effect of cement and a soft alkaline medium was studied on poplar wood by quantifying the content of sugar, lignin and extractives in the leaching and analyzing the degradation products by gas chromatography (GC). The influence of extractives and alkaline degradation products on the hydration of cement was explored by means of electrical conductivity.

Materials and Methods

Raw materials

A gray Portland cement CPA CEM I 52.5 was used in this study. The designation follows the French standard NF P 15-301. Chemical composition and mineralogical phases of the cement were calculated by the Bogue approximation [START_REF] Taylor | Cement Chemistry[END_REF]) and these are listed in Table 1. Poplar (Populus hybrid I214) powder was prepared by milling wood and screening through a 150 and a 200 µm mesh. The powder between the two screens was used as wood sample and was stored at 20 °C and 60 % of relative humidity. Each experiment was conducted in triplicate.

Wood soaks preparation

The soaks were prepared by mixing wood (3 g) either with 150 ml of deionized water (Millipore, mQ, USA), either with lime solution (1 g.l -1 ) or with cement suspension (7.5 g in 150 ml of water, water to cement weight ratio W/C=20 (w/w)), at 25 °C under stirring. This two-phase solution was filtered at different lengths of time ranging from 1 to 24 h. Then the wood was dried at 103 °C for 48 h and weighed. The weight loss was then calculated.

Electrical conductivity

Electrical conductivity measurement was performed in dilute suspensions with a W/C ratio of 20 (w/w).

Conductivity equipment consisted of a 25 °C thermostated reactor, two stainless steel annular electrodes and a conductimeter (Tacussel CD810, France). The constant cell was first calibrated with a 0.1 M KCl solution.

Cement (5 g) was mixed with 100 ml of the previously described wood soaks. The reference used for lime soaks was a 0.2 g.l -1 lime solution. It corresponded to the chemical composition of the lime solution after 1 h of leaching with poplar. After 24 h of leaching, the lime soaks corresponded to a 0.06 g.l -1 lime solution. However, the conductivity curve was identical. For experiments requiring both cement and wood, wood was introduced 30 seconds before the cement with a wood to cement weight ratio (wood/C) equal to 0.4 (w/w).

Determination of wood chemical composition

The amount of poplar extractives were quantified according to the ASTM D1105-56 standard. The content of Klason lignin and each sugar were obtained by acid hydrolysis of poplar according to the ASTM D1106-56 standard. The hydrolyzates were neutralized by Ba(OH) 2 , then centrifuged and filtered. The sugar concentrations were determined by means of HPLC. The apparatus was equipped with a pump (WATERS 916, USA) and a refractometer (WATERS 410, USA). A specific column for sugar analysis (SUPELCO, Supelcogel Pb 7.8 mm x 30 cm, USA), conditioned in furnace at 80 °C, was used. Deionized water, in-line degassed, was used as eluent and the flow was maintained at 0.5 ml.min -1 . Calibration of the apparatus was performed by the injection of HPLC grade sugar samples. The acid soluble lignin contents, in acid hydrolyzates, were estimated by the determination of the total phenolic compounds by colorimetry [START_REF] Singleton | Colorimetry of total phenolics with phosphomolybdicphosphotungstic acid reagents[END_REF][START_REF] Singleton | Colorimetry of total phenolics with phosphomolybdicphosphotungstic acid reagents[END_REF][START_REF] Alberto | Influence des extractibles d'essences du Mozambique sur l'hydratation du ciment[END_REF]. The catechin was used as standard and the results were expressed by mg of catechin.l -1 .g -1 of wood. The crystallinity index of cellulose was determined by X-ray diffraction according to the method proposed by [START_REF] Segal | An empirical method for estimating the degree of crystallinity of native cellulose using the X-ray diffractometer[END_REF]Fan et al. (1980).

Trimethylsilylation of nonvolatile carboxylic acids, Gas Chromatography-Mass Spectrometry (GC-MS) analysis

The previous soaks (100 ml) were evaporated down to 2 ml by a rotary evaporator. A 0.2 ml sample of the liquors was then taken to be analyzed for nonvolatile carboxylic acids. The mixture was then converted into ammonium salts, evaporated and dried. Xylitol (0.06 mg) was introduced as an internal standard. The trimethylsilylation reaction was realized by introducing 0.9 ml of pyridine and 0.1 ml of BSTFA under nitrogen atmosphere and by mixing the constituents for 4 h. A volume of 1.5 µl was injected and analyzed by GC-MS.

Separation was carried out on a gas chromatograph (SHIMADZU GC-17A) equipped with a MS detector (SHIMADZU QP-5000) at 70 eV. The temperature program began with a step at 100 °C for 1 min followed by a heat rate of 20 °C.min -1 up to 240 °C and finally a step at 240 °C for 2 min. The injector as well as the detector temperature were set to 260 °C. The concentration of acetic and formic acids were measured out by ionic chromatography (DIONEX GP50 pump, DIONEX CD20 detector and DIONEX AS11HC column).

Results and discussion

Influence of leaching conditions on wood mass loss

The kinetic of poplar mass loss induced by the extraction in water and in lime solution for time ranging from 1 to 24 h is presented in Figure 1. The mass losses, equal to 2.15 % in water, are independent of the leaching time, contrary to those associated with the leaching in lime solution. These values are higher than those obtained in water and ranging from 2.2 to 3.9 %. Consequently, the amount and the origin of molecules solubilized in solution are controlled by the leaching conditions and will be assigned later. Their respective effects on the hydration of cement were followed by conductivity measurement.

Measurement on cement hydration by electrical conductivity

The electrical conductivity measurement is a quite simple and reproducible method to follow up the hydration of cement compared to isothermal calorimetry. [START_REF] Damidot | Mechanism of enzymatic hydrolysis of cellulose : Effect of major structural features of cellulose on enzymatic hydrolysis[END_REF] and Nonat et al. (1997) reported that this method allows to investigate the influence of admixtures on the nucleation and growth of cement hydrates and to simulate cement paste conditions with the help of saturated lime solutions. We interpreted the electrical conductivity curves according to the explanation proposed by [START_REF] Comparet | Chemical interaction of di-phosphonate terminated monofunctional polyoxethylene superplasticizer with hydrating tricalcium silicate[END_REF]. As soon as cement was introduced into the cell, the electrical conductivity raised as a consequence of cement dissolution (Figure 2).

Hydrates then nucleated on the grain surface, which led to a lower increase in the conductivity. Conductivity then accelerated more quickly due to the nuclei growth.

Meanwhile, cement dissolution still continued. The conductivity rose up to the critical supersaturation with respect to portlandite and decreased rapidly due to its precipitation.

The hydration of cement by water eluates is shown in Figure 3. During the first minutes, the conductivity reached is similar to that of neat cement. The dissolution of cement is thus not affected by the chemical composition of water soaks. The duration of the plateau is slightly extended. The growth of hydrates is a bit slowed down by the chemical composition of water soaks since the slope of the conductivity curve is weaker than for neat cement. Consequently, the portlandite precipitation is delayed of about 1.8 h. Moreover, the effect of water wood eluates on the hydration of cement is independent of the leaching time. This result can be related to the constant mass loss obtained in water.

Figure 4 depicts the evolution of conductivity measured out in lime soaks. The alkaline wood eluates significantly modify the hydration. After the constant dissolution of cement, a conductivity plateau appears which varies from 3 to 6.5 h for leaching time ranging from 1 to 24 h. According to [START_REF] Comparet | Chemical interaction of di-phosphonate terminated monofunctional polyoxethylene superplasticizer with hydrating tricalcium silicate[END_REF], the duration of this step is inversely proportional to the number of hydrate nuclei that precipitated during the initial hydration. Wood soaks in lime solution seems to limit the number of precipitated hydrates. The slope of the curve is equal to 1.3 mS.cm -1 .h -1 , 1 mS.cm -1 .h -1 and 0.9 mS.cm -1 .h -1 for the reference, the wood soak leached for 1 h and 24 h in lime solution respectively. The growth of hydrates is then slowed down by the chemical composition of the solutions. The modifications of hydration cause a strong delay in portlandite precipitation which depends on the leaching time. The supersaturation with respect to portlandite is reached for hydration times ranging from 9.6 h (1 h of leaching) to 13.8 h (24 h of leaching). The kinetic of the cement hydration is, then, inversely proportional to the leaching time and to the mass loss.

The effects of the 24 hours-extraction in water and in lime on the mass loss and the retardation of portlandite precipitation are listed in table 2. The mass loss, equal to 2.2 % in water, was assigned to the amount of poplar extractives solubilized. It induces a portlandite precipitation delay of about 1.8 h. In alkaline solution, the mass loss is equal to 3.9 % and the delay of portlandite precipitation is extended to 7.8 h. These values were allocated to the global effect of extractives and alkaline degradation products. Therefore, the difference between leaching in water and in lime solution was attributed to alkaline degradation products. Although the difference in mass loss is only equal to 1.7 %, the portlandite precipitation is delayed of about 6 h. It point out that degradation products are more efficient than extractives to inhibit the cement hydration. The chemical composition of poplar was analyzed after 24 h of leaching to confirm the allocation of the mass loss.

Influence of leaching conditions on wood chemical composition

The chemical composition of wood was examined according to the ASTM D1105-56 and ASTM D1106-56 standards on natural and 24 hours-leached poplar. The alcohol/alcoholbenzene extraction reveals a decrease of about 2.2 % in the extractive content after leaching (Table 3). Moreover, the amount of extractives solubilized in the soak is independent of the solvent (water or alkaline media). The Klason lignin content and the acid soluble lignin content are not affected significantly by deionized water or alkaline conditions. The breakdown of lignin is mainly the result of cleavage of various ether bonds, especially at the α and β position, but requires drastic conditions (Lai 1991). The effect of alkaline media is enhanced for carbohydrates except for the cellobiose content that is constant. As expected for most hardwoods, arabinose and galactose contents are negligible and will not be discussed.

The average content of xylose in natural poplar, equal to 12.4 %, remains unchanged after the water leaching. However, this rate falls to 10.7 and to 9.8 % for leaching in lime and in cement suspension respectively. This sugar, being representative of glucuronoxylan content in poplar, indicates that hemicelluloses are unchanged in water and hydrolyzed under soft alkaline conditions. The decrease in the mannose and glucose contents, which are representative of glucomannans, confirms this result. However, the crystallinity index of cellulose, determined by X-ray diffraction, raises with alkaline treatment. According to [START_REF] Knill | Degradation of cellulose under alkaline conditions[END_REF], the glycosidic (1-4)-linkage between the β-D-glucopyranose units in cellulose is alkali stable. Nevertheless the dissolution of short-chain units detached from the reducing ends of the cellulose molecules can occur. It leads to an additional reduction of the glucose rate. These results highlight that in water the mass loss is only due to the solubilization of a part of wood extractives. But under alkaline conditions such as cement, hemicelluloses and amorphous cellulose are hydrolyzed besides the solubilization of extractives. Cement environment is slightly more aggressive towards polysaccharides than a 1 g.l -1 lime solution. The wood composition, after leaching in water and in lime, allows to assign the difference of mass loss to the degradation of polysaccharides by alkaline media.

Consequently, the alkaline degradation products are more effective than the poplar extractives solubilized in the eluates to inhibit the cement hydration.

In order to confirm it, natural poplar and poplar leached in water and lime solution for 24 h were directly introduced into the conductivity cell with deionized water. The critical supersaturation with respect to portlandite, reached by the system "natural polar-cement", is retarded of 5.1 h (Figure 5). When poplar was previously leached by water or lime, this delay is reduced to 3.5 h because of the absence of extractives from wood. The acceleration of the hydration (1.6 h) was conformed to the retardation produced by the water soaks containing the extractives. The delay of about 3.5 h was mainly allocated to alkaline degradation products of wood since wood extractives were previously removed by water or lime. Consequently, the poplar extractives are twice less efficient than hydrolysis products. These results are consistent with those obtained with wood soaks and those found by [START_REF] Young | The influence of sugars on the hydration of tricalcium aluminate[END_REF]. Indeed, he proposed that products from alkaline degradation of sugar were more efficient as retarding compounds than sugars themselves on C 3 A hydration. The compositions of wood soaks were analyzed to identify the active molecules.

Chemical composition of soaks

The HPLC analysis (method described earlier) carried out on lime soaks did not reveal the presence of sugars in solution. This result could be explained by the soft alkaline conditions which were strong enough to decompose the water-soluble hemicelluloses into acidic compounds. The degradation mechanism of polysaccharides under alkaline conditions was intensively studied in the pulping industry. According to several authors, polysaccharides were converted into carboxylic acids by alkaline treatment [START_REF] Whistler | Alkaline degradation of polysaccharides[END_REF][START_REF] Sjöström | The behavior of wood polysaccharides during alkaline pulping processes[END_REF][START_REF] Alen | Gas-liquid chromatographic separation of hydroxy monocarboxylic acids and dicarboxylic acids on a fused-silica capillary column[END_REF]Niemelä and Sjöström 1986). The chemical composition of eluates was monitored by GC-MS according to the method described earlier (Figure 6). Most of the 18 molecules detected resulted from polysaccharides degradation by peeling reaction.

The results of the analysis at two leaching times and in two solvents are shown in Table 4.

The soak, carried out in water for 1 h, is consisted of many carboxylic acids with a concentration lower than 1 mg.l -1 excepted for formic (4.2 mg.l -1 ), acetic (18 mg.l -1 ), malic (2.5 mg.l -1 ), benzoic (2.7 mg.l -1 ) and p-salicylic (5.6 mg.l -1 ) acids. According to [START_REF] Hillis | The role of wood characteristics in high temperature drying[END_REF], the degradation of polysaccharides might occur during wood drying and produces carboxylic acids. When leaching time is extended to 24 h, the content of carboxylic acids is unchanged as the conductivity measurements and the mass loss. After 1 h of leaching in lime, the concentration of acids grows compared to those determined in water (1 h). This increase is enhanced after 24 h of leaching since the concentration of acetic, glycolic, pyruvic, glyceric, malic, benzoic, hydrocinnamic, o-and p-salicylic acids are multiplied by 38, 13, 9.5, 3.5, 3, 3.8, 13, 6 and 5 respectively. The higher concentration of carboxylic acids in lime confirms the decreasing content in wood sugars due to the alkaline degradation of polysaccharides and the rise in mass loss. However, neither glucoisosaccharinic acid nor xyloisosaccharinic acid, the main degradation acids of hexosans and pentosans [START_REF] Sjöström | The behavior of wood polysaccharides during alkaline pulping processes[END_REF][START_REF] Alen | Gas-liquid chromatographic separation of hydroxy monocarboxylic acids and dicarboxylic acids on a fused-silica capillary column[END_REF]Niemelä and Sjöström 1986), were detected. This result might be due to softer alkaline conditions than those applied in these study. In spite of its high concentration in lime solution, the acetic acid, resulting from the cleavage of O-acetyl groups and glucuronoxylan chain, is considered as a poor accelerator at low concentration and a poor retarder at higher concentration [START_REF] Young | A review of the mechanisms of set-retardation in Portland cement pastes containing organic admixtures[END_REF]. A solution of 1 g.l -1 of acetic acid, inducing a delay lower than 0.5 h can not explain the experimental retardation of about 7.8 h. Several authors referred to the retarding effect of some of the detected organic molecules. [START_REF] Taplin | Some chemical addition and admixtures in cement paste and concrete[END_REF] supposed that the active group was HO-C-C=O. He identified glycolic, pyruvic and malic acids which have this chemical function as very efficient retarders. Other authors also obtained these results [START_REF] Wilding | A classification of inorganic and organic admixtures by conduction calorimetry[END_REF][START_REF] Ramachandran | Conduction calorimetric investigation of the effect of retarders on the hydration of Portland cement[END_REF]. Wilding proposed the following classification of benzoic acids: o-salicylic acid > benzoic acid > phenol. He suggested that osalicylic acid is active on cement as well as malic or pyruvic acids.

Conclusion

The experiments concerning the chemical composition of the eluates and poplar show that wood polysaccharides are indeed hydrolyzed under alkaline conditions such as in lime solution with a concentration of 1 g.l -1 and in cement suspension. These polymers are converted into numerous carboxylic acids, which are known to be powerful retarders. The electrical conductivity measurement demonstrate that the effect induced by wood extractives is three time less efficient than alkaline degradation products on the hydration of cement.

Accordingly, the hydrolysis products must be taken into account to understand the wood and cement interaction.

Caption of Tables

Table 1 Chemical composition of cement and potential phases as determined by Bogue approximation.

Table 2 Wood chemical composition according to ASTM standards for natural poplar, leached poplar in water, in lime solution and in cement suspension for 24 h (in percent of initial dry wood weight).

Table 3 Chemical composition of natural poplar soaks in water and lime solution for 1 and 24 h.

Caption of Figures

Fig. 1 Wood mass loss / initial dry wood weight ratio associated with leaching time in water and lime solution .

Fig. 2

 2 Fig. 2 Interpretation of the electrical conductivity curve of cement.

Fig. 2

 2 Fig. 2 Interpretation of the electrical conductivity curve of cement.

Fig. 3

 3 Fig. 3 Effect of poplar leaching time in water on cement hydration.

Fig. 5

 5 Fig. 5 Effect on time to reach supersaturation with respect to portlandite of natural poplar,

Fig. 6

 6 Fig. 6 Chromatogram of the trimethylsilylated soaks obtained by leaching poplar in lime

Table 1

 1 Chemical composition of cement and potential phases as determined by Bogue approximation.

	Chemical composition Weight %
	SiO 2	19.98
	TiO 2	0.29
	Al 2 O 3	5.47
	Fe 2 O 3	2.72
	CaO	63.03
	MnO	0.03
	MgO	2.32
	Na 2 O	0.46
	K 2 O	0.93
	P 2 O 5	0.26
	LOI	0.42
	SO 3	5.14
	Bogue approximation
	C 3 S	64
	C 2 S	9
	C 3 A	10
	C 4 AF	8

  Effect of poplar leaching time in lime solution on cement hydration.

	0 2 4 6 8 10 12 14 16 Conductivity (mS.cm -1 ) 12 14 16 -1 ) Fig. 4 0 2 4 6 8 10 Conductivity (mS.cm						Neat cement Leaching 15 min Leaching 1h Leaching 2h Leaching 4h Leaching 8h Leaching 24h Reference Leaching 15 min Leaching 1h Leaching 2h Leaching 4h Leaching 8h Leaching 24h
	0 0	2 2	4 4	6 6	8 8 Time (h) 10 10 Time (h)	12 12	14 14	16 16	18 18

Table 2

 2 Effect of a 24 hours-extraction in water and in lime on mass loss and retardation of portlandite precipitation. * due to extractives; ** due to extractives and alkaline degradation products.

	Leaching	Mass loss	Precipitation
		(%)	delay (h)
	In water	2.2 *	1.8
	In lime	3.9 **	7.8
	Difference 1.7	6.0

Table 3

 3 Wood chemical composition according to ASTM standards for natural poplar, leached poplar in water, in lime solution and in cement suspension for 24 h (in percent of initial dry wood weight). Effect on time to reach supersaturation with respect to portlandite of natural poplar, leached poplar in water (24 h), leached poplar in lime solution (24 h).

			Leached poplar in	Leached poplar in	Leached poplar in
		Poplar
			water	lime solution	cement solution
	Alcohol/alcohol-benzene	5.7 ± 0.3	3.3 ± 0.5	3.3 ± 0.7	3.7 ± 0.2
	extractives (%)
	Klason lignin (%)	23.5 ± 0.6	24.2 ± 1.1	22.0 ± 0.4	22.2 ± 1.1
	Acid soluble lignin (mg	32.8 ± 1.5	34.7 ± 2.3	32.8 ± 1.0	34.0 ± 1.3
	of catechin.l -1 .g -1 wood)
	Cellobiose (%)	4.8 ± 0.1	4.5 ± 0.5	4.5 ± 0.2	5.1 ± 0.2
	Glucose (%)	35.7 ± 0.1	35.6 ± 0.5	34.1 ± 0.5	30.7 ± 0.7
	Xylose (%)	12.4 ± 0.1	12.2 ± 0.3	10.7 ± 0.5	9.8 ± 0.2
	Mannose (%)	3.9 ± 0.6	3.9 ± 0.3	2.8 ± 0.3	2.5 ± 0.3
	Galactose (%)	0.17 ± 0.02	0.25 ± 0.09	0.09 ± 0.04	0.31 ± 0.03
	Arabinose (%)	0.022 ± 0.01	n.d.	0.06 ± 0.04	0.06 ± 0.006
	Crystallinity Index (%) 71.0 ± 1.1	70.5 ± 0.6	80.6 ± 0.5	75.3 ± 0.8

ether; 5: a. malonic; 6: diethylene glycol; 7: a. benzoic; 8: glycerol; 9: a. succinic; 10: a. glyceric; 11: a. maleic; 12: a. hydrocinnamic; 13: a. malic; 14: a. o-salicylic; 15: a. p-salicylic; 16: a. vanillic.

Table 4

 4 Chemical composition of natural poplar soaks in water and lime solution for 1 and 24 h.

	Class	Molecule name	1 h in water (mg.l -1 )	24 h in water (mg.l -1 )	1 h in lime solution (mg.l -1 )	24 h in lime solution (mg.l -1 )
		Diethylene glycol ethyl ether	1.2 ± 0.1	1.5 ± 0.3	1.4 ± 0.1	2.2 ± 0.1
	Alcohols	Diethylene glycol	1.9 ± 0.4	2.4 ± 0.4	15.2 ± 2.1	21.1 ± 1.3
		Glycerol	0.4 ± 0.1	0.5 ± 0.1	0.7 ± 0.1	1.1 ± 0.1
		Formic acid	4.2 ± 0.5	5.17 ± 0.7	6.5 ± 0.3	9.0 ± 0.3
		Acetic acid	18.0 ± 2	19.6 ± 2	662 ± 12	773 ± 10
		Lactic acid	1.2 ± 0.1	1.6 ± 0.1	2.7 ± 0.2	3.7 ± 0.5
		Glycolic acid	0.6 ± 0.1	0.8 ± 0.1	2.8 ± 0.2	10.2 ± 1.0
	Aliphatic	Pyruvic acid	0.2 ± 0.01	0.2 ± 0.04	0.53 ± 0.03 1.9 ± 0.1
	acids	Malonic acid	0.3 ± 0.04	0.4 ± 0.1	0.5 ± 0.1	1.2 ± 0.1
		Succinic acid	0.7 ± 0.1	1.2 ± 0.1	1.1 ± 0.1	1.4 ± 0.02
		Glyceric acid	0.3 ± 0.04	0.4 ± 0.03	0.78 ± 0.1	1.4 ± 0.02
		Maleic acid	0.3 ± 0.04	0.4 ± 0.02	0.4 ± 0.06	0.8 ± 0.08
		Malic acid	2.5 ± 0.2	2.2 ± 0.1	2.8 ± 0.2	6.9 ± 0.2
		Benzoic acid	2.7 ± 0.3	3.6 ± 0.2	5.7 ± 0.3	13.7 ± 1
		Hydrocinnamic acid 0.2 ± 0.0	0.2 ± 0.04	1.6 ± 0.4	2.6 ± 0.1
	Benzoic acids	o-Salicylic acid	0.8 ± 0.1	1.0 ± 0.1	1.8 ± 0.1	5.8 ± 0.2
		p-Salicylic acid	5.6 ± 0.4	7.3 ± 0.3	11.8 ± 0.4	35.6 ± 0.6
		Vanillic acid	1.1 ± 0.1	1.5 ± 0.1	1.7 ± 0.2	2.4 ± 0.1


A Goldsztejn
email: alexandre.goldsztejn@irccyn.ec-nantes.fr

S Caro

G Chabert
email: gilles.chabert@mines-nantes.fr

A New Methodology for Tolerance Synthesis of Parallel Manipulators

Keywords: Certified tolerance synthesis, parallel manipulators, Kantorovich Theorem

Computing the maximal pose error given an upper bound on perturbations is challenging for parallel robots, mainly because the direct kinematic problem has several solutions, which become unstable near or at parallel singularities. In this paper, we propose a local uniqueness hypothesis that will allow safely computing pose error upper bounds using nonlinear optimization. This hypothesis, together with a corresponding maximal allowed perturbation domain and a certified pose error upper bound valid over the complete workspace, will be proved numerically using a parametric version of Kantorovich theorem and certified nonlinear global optimization. We will then show how to synthesize tolerances reaching a prescribed maximal pose error over a workspace using approximate linearizations. This approximate tolerance synthesis will finally be checked using the certified pose error upper bound we propose. Preliminary experiments on a RPRPR and a 3RPR with fixed orientation parallel manipulators are presented.

I. Introduction

For two decades, parallel manipulators have attracted the attention of more and more researchers who consider them as valuable alternative design for robotic mechanisms. As stated by some authors, conventional serial kinematic machines have already reached their dynamic performance limits, which are bounded by high stiffness of the machine components required to support sequential joints, links and actuators. Thus, while having good operating characteristics (large workspace, high flexibility and maneuverability), serial manipulators have disadvantages of low stiffness and low power. Conversely, Parallel Kinematics Machines (PKM) offer essential advantages over their serial counterparts (lower moving masses, higher stiffness and payload-to-weight ratio, higher natural frequencies, better accuracy, simpler modular mechanical construction, possibility to locate actuators on the fixed base).

However, PKM are not necessarily more accurate than their serial counterparts. Indeed, even if the dimensional variations can be compensated with PKM, they can also be amplified contrary to with their serial counterparts. Wang et al. [START_REF] Wang | On the accuracy of a stewart platform -part i, the effect of manufacturing tolerances[END_REF] studied the effect of manufacturing tolerances on the accuracy of a Stewart platform. Kim et al. [START_REF] Kim | The kinematic error bound analysis of the stewart platform[END_REF] used a forward error bound analysis to find the error bound of the end-effector of a Stewart platform when the error bounds of the joints are given, and an inverse error bound analysis to determine those of the joints for the given error bound of the end-effector. Kim and Tsai [START_REF] Kim | Design optimization of a cartesian parallel manipulator[END_REF] studied the effect of misalignment of linear actuators of a 3-DOF translational parallel manipulator on the motion of its moving platform. Han et al. [START_REF] Han | Kinematic sensitivity analysis of the 3-upu parallel mechanism[END_REF] used a kinematic sensitivity analysis method to explain the gross motions of a 3-UPU parallel mechanism, and showed that it is highly sensitive to certain minute clearances. Fan et al. [START_REF] Fan | Sensitivity analysis of the 3-prs parallel kinematic spindle platform of a serial-parallel machine tool[END_REF] analyzed the sensitivity of the 3-PRS parallel kinematic spindle platform of a serial-parallel machine tool. Verner et al. [START_REF] Verner | Optimal calibration of parallel kinematic machines[END_REF] presented a new method for optimal calibration of PKM based on the exploitation of the least error sensitive regions in their workspace and geometric parameters space. As a matter of fact, they used a Monte Carlo simulation to determine and map the sensitivities to geometric parameters. Moreover, Caro et al. [START_REF] Caro | Tolerance synthesis of mechanisms : A robust design approach[END_REF] developed a tolerance synthesis method for mechanisms based on a robust design approach.

During the early design process of engineering systems, the analysis of the performance sensitivity to uncertainties is an important task. High sensitivity to parameters that are inherently noisy can lead to poor, or unexpected performance. For that reason, it is important to analyze the sensitivity of their performance to variations in their geometric parameters and to determine the optimal dimensional tolerances.

To this end, some indices such as the dexterity and the manipulability have been used to evaluate the sensitivity of robots performance to variations in their actuated joints [START_REF] Yu | Geometric approach to the accuracy analysis of a class of 3-dof planar parallel robots[END_REF], [START_REF] Bai | Design and analysis of a 3-ppr planar robot with u-shape base[END_REF], [START_REF] Merlet | Parallel robots[END_REF]. However, they are not suitable for the evaluation of this sensitivity to other types of uncertainty such as variations in geometric parameters. Two indices were proposed in [START_REF] Caro | Sensitivity analysis of the orthoglide, a 3-dof translational parallel kinematic machine[END_REF] to evaluate the sensitivity of the endeffector pose (position + orientation) of the Orthoglide 3axis, a three Degree-of-Freedom (DOF) translational PKM, to variations in its design parameters. In the same vein, four 3-RPR planar parallel manipulators (PPMs) were compared in [START_REF] Caro | Sensitivity Analysis of 3-RPR Planar Parallel Manipulators[END_REF] based on the sensitivity of their performance to variations in their geometric parameters. In [START_REF] Tannous | Sensitivity analysis of parallel manipulators using an interval linearization method[END_REF], an interval linearization method is used for the sensitivity analysis of some parallel manipulators. However, the foregoing research works do not deal with the tolerance synthesis of parallel manipulators, which is a critical issue.

In the present paper, we overcome two lacks in the literature: First, a fully rigorous methodology is proposed to compute a certified upper bound for the pose error due to perturbations of the parameters of a PKM over a full workspace. Second, a methodology is proposed for tolerance synthesis of PKM, aiming synthesizing the largest tolerances keeping the pose error below a given limit.

The paper is organized as follows: In Section III, we first introduce a uniqueness hypothesis that allows computing a certified pose error upper bound over a given workspace by solving a nonlinear optimization problem. Second, we propose a parametric version of Kantorovich theorem, which provides both a maximal perturbation domain for which this uniqueness hypothesis holds, and a crude certified pose error upper bound valid inside this perturbation domain and the whole workspace. As a result, for perturbations within this maximal perturbation domain, some sharp pose error upper bounds can be computed using the nonlinear optimization problem previously introduced. In Section IV, we propose an approximate linearization of the maximal error in the workspace, which allows performing some approximate tolerance synthesis. These approximate tolerances can finally be corrected using the results of Section III. Preliminary experiments on a RPRPR and a 3RPR with fixed orientation parallel manipulators are presented in Section V, which show the potential and the limits of the approach.

Notations

We use the ∞-norm throughout the paper. Let B(x, ) be the open ball {y ∈ R k : y-x < }, B(x, ) be the closed ball {y ∈ R k : yx ≤ }, and for short B := B(0, ).

II. Basic notations and definitions

We consider a kinematic model f (x, q, p) of a nonredundant parallel manipulator, where f : R n × R n × R m → R n , x being the pose, q the actuated joint coordinates and p a perturbation vector, so that the nominal model is f (x, q, 0). We also denote this nominal model by f (x, q) := f (x, q, 0). We suppose that f is differentiable and has locally Lipschitz continuous first derivatives with respect to pose and perturbation, e.g., f is twice differentiable with respect to these variables. The Jacobian matrix of f with respect to variables x and p are denoted respectively as F x , called the kinematic parallel Jacobian matrix, and F p , called the sensitivity Jacobian matrix.

The nominal generalized workspace is defined by

G := {(x, q) ∈ R 2n : f (x, q) = 0, g(x, q) ≤ 0}, ( 1
)
where g is a set of inequalities that defines the generalized workspace of interest. We assume that G is bounded. We also require that G does not contain any parallel singularity, but this will be checked by the proposed method.

In some situations, the kinematic model can be solved for the pose coordinates inside G, giving rise to a direct model d : R n → R n that provides an explicit description of the nominal generalized workspace:

x = d(q) ∧ g(x, q) ≤ 0 ⇐⇒ (x, q) ∈ G. (2)
When perturbations are to be taken into account, we expect to have a direct model that depends of perturbations:

x = d(q, p). Such a direct model cannot be correct for arbitrarily large perturbations, and therefore has to be associated to a perturbation domain P for which it is valid (for simplicity, the perturbation domain P is supposed to contain 0 and to be convex). Although such a direct model with explicit dependence on perturbations naturally arises in the context of serial robots, it is usually quite difficult to obtain for parallel robots.

III. Upper bounds for maximal pose errors

In this section, we compute an upper bound on the distance between a nominal pose and its perturbed pose. In the first subsection, we introduce an uniqueness condition that allows associating the nominal and perturbed posed unambiguously, which is obviously required to compute the distance between them. In the second subsection, we show how Kantorovich theorem can be used in a parametric way within the whole generalized workspace and perturbation space in order to provide both a perturbation domain for which the above uniqueness condition holds, and a pose error upper bound valid within the whole workspace and for all considered perturbations.

A. General framework

When an explicit direct model x = d(q, p) is available, finding the maximal pose error over the workspace can be modeled by the following constrained optimization problem: max

(x,q)∈G,p∈P x =d(q,p) e(x, x ), (3)
see e.g. [START_REF] Berger | Finding the Maximal Pose Error in Robotic Mechanical Systems Using Constraint Programming[END_REF]. The variables are the joint coordinates q ∈ R n , the nominal pose x ∈ R n , the perturbed pose x ∈ R n and the perturbation p ∈ R m . The constraint (x, q) ∈ G states that (x, q) is a nominal configuration, while the constraint x = d(q, p) states that x is the perturbed pose corresponding to the same joint coordinates q. The cost function e(x, x ) is an error measurement (we consider here either the norm of the positioning error or the norm of the orientation error, so e(x, x ) = Π(xx ) where Π is a projection on a subset of the coordinates of the pose). The aim of this section is to generalize this approach to the case where the direct model with explicit dependence on perturbations is not available. The naive generalization max

(x,q)∈G, p∈P f (x ,q,p)=0 e(x, x ) (4)
is not correct because the direct kinematic problem may have several solutions or no solution at all for a given perturbation. This means that the maximum of e(x, x ), if defined, is likely to be reached for a x that does not corresponds to a perturbation of x, i.e., the optimization problem (4) makes no sense. Hence, the perturbation domain has to be small enough so that perturbing the nominal pose leads to a solution and a unique one. We formalize this restriction as follows: For given > 0, G ⊆ R n × R n and P ⊆ R m , we say that G is -safe with respect to P if

∀(x, q) ∈ G, ∀p ∈ P, ∃! x ∈ B(x, ), f (x , q, p) = 0. (5)
The existence of a unique perturbed pose within B(x, ) is crucial in this definition since it enforces a functional dependence between x and p ∈ P, which we denote by x = d x,q (p). It is easy to check that this function is continuous, and that d x,q (0) = x, which makes x = d x,q (p) the perturbed pose associated to the nominal pose x with no ambiguity. Therefore, under the hypothesis that G is -safe with respect to P, we can safely use the kinematic model f (x , q, p) = 0 in order to characterize the perturbed pose:

(x, q) ∈ G, p ∈ P x ∈ B(x, ) f (x , q, p) = 0 ⇐⇒ (x, q) ∈ G, p ∈ P x = d x,q (p) . (6)
As a consequence, even though no explicit expression of d x,q is available, the maximal error inside the nominal workspace is given by max

(x,q)∈G, p∈P x ∈B(x, ) f (x ,q,p)=0 e(x, x ). (7)
As a conclusion, the constrained optimization problem [START_REF] Chabert | Contractor Programming[END_REF] allows computing a sharp pose error upper bound, provided that the nominal generalized workspace G is proved to be -safe with respect to P. The next subsection deals with the determination of P and using Kantorovich theorem.

Remark 1: The error function in [START_REF] Chabert | Contractor Programming[END_REF] can be chosen to be either the norm of the position error or the norm of the orientation error, which is interseting since aggregating these errors is often difficult or irrelevant. However, they have to be aggregated in the norm condition x ∈ B(x, ) of ( 5), ( 6) and [START_REF] Chabert | Contractor Programming[END_REF]. The impact of this aggregation is on the size of the perturbation domain P (as it will be shown in the next section).

B. Parametric Kantorovich Theorem for -safety B.1 Informal presentation

Kantorovich theorem [START_REF] Ortega | The Newton-Kantorovich Theorem[END_REF], [START_REF] Gragg | Optimal Error Bounds for the Newton-Kantorovich Theorem[END_REF], [START_REF] Merlet | Parallel robots[END_REF], [START_REF] Ciarlet | On the Newton-Kantorovich Theorem[END_REF] is now applied in order to both provide a perturbation domain P and an error upper bound for which G is -safe with respect to P.

The basic idea is, for an arbitrary configuration (x, q) ∈ G and an arbitrary perturbation p ∈ P, to apply Kantorovich theorem for solving the perturbed direct kinematic problem f ( • , q, p) = 0, with the nominal pose x as an initial condition. The perturbation domain P is going to be defined so that the hypotheses of Kantorovich Theorem are satisfied for every perturbation it contains. As a consequence, the existence and uniqueness regions provided by Kantorovich theorem are going to enforce G to besafe with respect to P, for a given also provided by Kantorovich theorem.

The key feature of Kantorovich Theorem that allows applying it over the whole generalized workspace and perturbation space is that when the constants involved in Kantorovich Theorem are sharper, the size of the existence ball decreases and the size of the uniqueness ball increases. Therefore, worst case constants over G and P will provide a greatest existence domain and a smallest uniqueness domain valid over both G and P.

B.2 The main theorem

Choose an apriori maximal tolerance ∆ > 0 so that only perturbations satisfying p ∈ B ∆ will be considered 1 . Then define the constants κ, χ and γ (i) such that:

κ ≥ max (x,q)∈G p∈B ∆ f (x, q, p) (8)
χ ≥ max

(x,q)∈G p∈B ∆ F x (x, q, p) -1 (9)
γ (i) ≥ max (x,q)∈G p∈B ∆ F x (x, q, p) -1 F p (i) (x, q, 0) (10)
where p (i) form a partition of the perturbation vector p. Define r := 2κχ, [START_REF] Han | Kinematic sensitivity analysis of the 3-upu parallel mechanism[END_REF] and suppose the two following Lipschitz conditions hold for constants λ and µ

∀(x 0 , q) ∈ G, ∀p ∈ B ∆ , ∀x , x ∈ B(x 0 , r), F x (x , q, p) -F x (x , q, p) ≤ λ x -x , (12)
∀(x, q) ∈ G, ∀p, p ∈ B ∆ , F p (x, q, p) -F p (x, q, p ) ≤ µ p -p . ( 13
)
Theorem 1: Let ∆ > 0, and κ, χ, γ ≥ 0 be such that (8), ( 9) and ( 10) are satisfied. Consider λ, µ ≥ 0 such that [START_REF] Kim | The kinematic error bound analysis of the stewart platform[END_REF] and ( 13) are satisfied. Define r := 2κχ as in [START_REF] Han | Kinematic sensitivity analysis of the 3-upu parallel mechanism[END_REF],

η(p) := 2 i γ (i) p (i) +µχ p 2 , ( 14
)
and the perturbation domain

P = {p ∈ B ∆ : λχη(p) ≤ 1}. ( 15
)
Then G is -safe with respect to P for

= min{r, 1 χλ }. (16)
Furthermore, the distance between the nominal and the perturbed poses is at most

(p) := η(p) 1 + 1 -χλη(p) . ( 17
)
Proof: Consider an arbitrary nominal pose (x 0 , q) ∈ G. We just need to prove that for an arbitrary fixed p ∈ P there exists a unique solution x * to h(x) = 0, with h(x) := f (x, q, p), inside B(x 0 , ), and that this solution satisfies x 0 -x * ≤ (p). To this end, we apply Kantorovich theorem to the resolution of the system h(x) = 0, starting from the initial condition x 0 . Let D 0 := B(x 0 , r). Then by [START_REF] Kim | The kinematic error bound analysis of the stewart platform[END_REF],

H x (x) = F x (x, q, p) is Lipschitz with constant λ inside D 0 . By (9), Γ 0 := H x (x 0 ) -1 is defined and Γ 0 ≤ χ. We prove now that Γ 0 h(x 0 ) ≤ 1 2 min{r, η(p)} =: δ. (18)
First, Γ 0 h(x 0 ) ≤ Γ 0 h(x 0 ) ≤ κχ = r 2 , the second inequality resulting of ( 8) and [START_REF] Fan | Sensitivity analysis of the 3-prs parallel kinematic spindle platform of a serial-parallel machine tool[END_REF]. Second, defining M := F p (x 0 , q, 0), M (i) := F p (i) (x 0 , q, 0) and z := h(x 0 ) -Mp,

Γ 0 h(x 0 ) = Γ 0 (Mp + z) (19)
= Γ 0 i M (i) p (i) + z (20)
≤ i Γ 0 M (i) p (i) + Γ 0 z (21) ≤ i γ (i) p (i) +χ z , (22)
where the last inequality follows from ( 9) and [START_REF] Gragg | Optimal Error Bounds for the Newton-Kantorovich Theorem[END_REF]. Finally, following the classical argument given in [START_REF] Ortega | The Newton-Kantorovich Theorem[END_REF], [START_REF] Ciarlet | On the Newton-Kantorovich Theorem[END_REF], and using (13), we have z ≤ 1 2 µ p 2 , which proves [START_REF] Verner | Optimal calibration of parallel kinematic machines[END_REF]. Now, it remains to prove that h := 2χλδ ≤ 1 and that B(x 0 , t * ) ⊆ D 0 , where t * := 2δ(1-

√ 1-h) h . First, h = 2χλδ ≤ (18) λχη(p) ≤ (15) 1, (23)
Second, we have

t * ≤ (57) 2δ ≤ (18)
r.

Therefore, B(x 0 , t * ) ⊆ B(x 0 , r) = D 0 .

Therefore we can apply Kantorovich theorem [START_REF] Gragg | Optimal Error Bounds for the Newton-Kantorovich Theorem[END_REF] which proves that h(x) has a solution x * inside B(x 0 , t * ) with H x (x * ) nonsingular, and that this solution is unique inside B(x 0 , ) ⊇ B(x 0 , t * ): On the one hand, if h ∈ [0, 1) then Kantorovich theorem proves that it is unique inside B(x 0 , t ) ∩ D 0 , while

B(x 0 , t ) ∩ D 0 ⊇ (56) B(x 0 , 1 χλ ) ∩ D 0 = (16) B(x 0 , ). (25)
On the other hand, if h = 1 then Kantorovich theorem proves that it is unique inside B(x 0 , t ) , while

B(x 0 , t ) = (55) B(x 0 , 1 χλ ) ⊇ (16)
B(x 0 , ).

Finally, using 2δ ≤ η(p), which holds by ( 18), we have h = 2χλδ ≤ χλη(p), and eventually

t * = 2δ 1 + √ 1 -h ≤ (p), (27)
therefore x 0 -x * ≤ (p).
Remark 2: The perturbation domain P provided by Theorem 1 is convex. Indeed, it has the form a T p + b p 2 ≤ 1, with b > 0, so a T p + b p 2 is the sum of two convex functions.

Remark 3: Kantorovich Theorem also proves

∀(x, q) ∈ G, ∀p ∈ P, ∀x ∈ B(x, ), det F x (x , q, p) = 0, (28)
which is of great practical interest, since for the perturbation domain considered, it proves that no perturbed pose leads to any parallel singularity.

B.3 Asymptotic analysis

It can be seen from the definition (14

) that η(p) ≈ 2 i γ (i) p (i) provided that p 2 2 µχ i γ (i) p (i) , (29)
which is satisfied as soon as p is small enough. If furthermore χλη(p) ≈ 2χλ i γ (i) p (i) is small with respect to 1 then (p) ≈ η(p)

2 = i γ (i) p (i)
. Figure 1 shows the typical behavior of η(p), (p), and γ 0 p in the case where p is not split (i.e., i γ (i) p (i) = γ p ).

Another interesting observation is that, under the validity of the approximation η(p) ≈ 2 i γ (i) p (i) , the size of the perturbation domain {p ∈ B ∆ : λχη(p) ≤ 1} is proportional to the inverse of χ, λ and γ (i) .

IV. Tolerance synthesis

Let P be the perturbation domain provided by Theorem 1. Our aim is to determine a vector of tolerances ∆ = (∆ i ) such that

S ∆ := {p ∈ R m : p (i) ≤ ∆ i } (30)
is contained inside P, and that the error e(x, x ) is less than a given threshold e for all perturbations in S ∆ .

In Subsection IV-A, we propose a non-rigorous linear approximation of the maximal error in the workspace. In Subsection IV-B, we formulate the Tolerance synthesis as a multi-objective optimization problem, aiming maximizing the different tolerances ∆ i . Finally, a certified upper bound for the pose error corresponding to the tolerances synthesized by this process is computed by solving the optimization problem described Section III.

A. Approximate linearization of the pose error

Theorem 1 involves several overestimations, which leads to an overestimated error upper bound for the considered perturbation domain. This allows making the assumption that a linear approximation is going to be accurate within the perturbation domain provided by Theorem 1. Therefore, we make the assumption that the following linear approximation holds:

xx ≈ F x (x, q, 0) -1 F p (x, q, 0)p.

(31)

We obtain an approximate workspace worst case error e(x, x ) = Π(xx ) in the following way: Using (31), we obtain that ( 7) is approximately max

(x,q)∈G p∈S ∆ ΠF x (x, q, 0) -1 F p (x, q, 0)p , (32)
which we approximate by max

(x,q)∈G p∈S ∆ ΠF x (x, q, 0) -1 F p (x, q, 0) p . ( 33
)
We finally approximate (33) by

max p∈S ∆ i γ (i) 0 p (i) = i γ (i) 0 ∆ i , (34)
where

γ (i) 0 = max (x,q)∈G ΠF x (x, q, 0) -1 F p (i) (x, q, 0) . ( 35
)
Although it is difficult to assess the accuracy of this linear approximation in general, it turns out to be very accurate in the preliminary experiments presented in Section V.

B. Approximate tolerance synthesis and its validation

In order to be able to compute a rigorous pose error upper bound using Theorem (1), we need to choose ∆ i using (34) under the additional constraint that S ∆ is a subset of P. Therefore, admissible tolerances ∆ satisfy i γ (i)

0 ∆ i ≤ e ( 36
)
2 i γ (i) ∆ i + µχ ∆ 2 ≤ 1 λχ (37) ∆ ≤ ∆, (38)
where ( 37) and (38) encode S ∆ ⊆ P (the constraint (37) comes from ( 14) and ( 15), while (38) is the a priori maximal perturbation norm). As mentioned earlier, the domain (37) is convex, and so is the set of admissible tolerances defined by (36), (37) and (38). Finally, we want to select tolerances satisfying (36), (37) and (38) that maximizes each ∆ i in a multi-objective sense, i.e., that are Pareto optimal for these objectives. Since objectives and constraints are convex and almost linear, this multi-objective optimization problem is actually trivial to solve. Then, solving the optimization problem (7) provides a rigorous upper bound for the pose error over the workspace for the synthesized tolerances. Two cases arise: Either the rigorous upper bound ( 7) is close enough to the approximate one (34) so that the synthesized tolerances can be used, or the process can be repeated for neighbor tolerances in order to achieve better tolerances. In the experiments presented in the next section, the approximate upper bound (34) is very accurate, but this will have to be investigated on more models.

V. Preliminary experiments

In this section, we provide some first simulations to assess the usefulness of the proposed method. The upper bounds κ, χ, γ (i) and γ (i) 0 have been rigorously computed solving the optimization problems ( 8), ( 9), ( 10) and (35) using the global solver IBEX2 [START_REF] Chabert | Contractor Programming[END_REF], [START_REF] Trombettoni | Inner regions and interval linearizations for global optimization[END_REF], [START_REF] Araya | Upper Bounding in Inner Regions for Global Optimization Under InequalityConstraints[END_REF]. It allows comput- ing an arbitrarily sharp certified upper bound of these optimization problems, and we stopped the computations when the relative precision of the maximum is 1% (all presented upper bounds are therefore certified and accurate). All computations have been performed on an Intel i7 2.80GHz, the code has not been parallelized, and all timing are given in seconds (a 1 hour timeout has been enforced).

Two different robots are investigated: A RPRPR in Subsection V-A and a 3RPR, with constant orientation, in Subsection V-B. We consider no joint limits, i.e. q i ∈ [0, +∞). The generalized workspace of interest is defined by a rectangular domain in the cartesian space: x ∈ [x, x] =: W. Therefore,

G = {(x, q) ∈ R 2n : f (x, q) = 0, x ≤ x ≤ x}. (39)
Several workspaces W i are going to be investigated for each robot.

Remark 4: We formally inverse the matrix F x (x, q, p) in order to solve ( 9) and [START_REF] Gragg | Optimal Error Bounds for the Newton-Kantorovich Theorem[END_REF]. Tackling higher dimensional problems will require investigating some rigorous numerical inverse enclosure methods.

A. The RPRPR robot

In this subsection, we study the simple parallel robot RPRPR with 6 perturbations, 4 of them acting on the anchor points, and 2 of them on the commands. Its kinematic model F(x, q, p) is therefore

(1 + p 1 + x 1 ) 2 + (p 2 + x 2 ) 2 -(p 3 -q 1 ) 2 (40) (-1 + p 4 + x 1 ) 2 + (p 5 + x 2 ) 2 -(p 6 -q 2 ) 2 . ( 41
)
Since there is not orientation error, we use e(x, x ) = xx , i.e. Π is the identity matrix. Perturbations

F x (x, q(x), 0) -1 F p (x, q(x), 0) , ( 42
)
where q(x) is the inverse kinematic model. We expect from Figure 2 that W 1 is the best in term of sensitivity and tolerance synthesis. The kinematic model involves only quadratic constraints, for which we can easily obtain tight Lipschitz constants for the derivatives: λ = 2 and µ = 2.

A.1 Workspace W 1

The first workspace we consider is defined by -1 ≤ x 1 ≤ 0 and 1 ≤ x 2 ≤ 2. The constants upper bounds computed for this workspace using ∆ = 0.1 are given in the following table, as well as the time needed to compute them: The corresponding error upper bound given by Theorem 1 is = 0.79. The perturbation domain given by Theorem 1 is shown on Figure 3, as well as the isocontours of the linearized error

κ χ γ (1) (γ (1) 0 ) γ (2) (γ (2) 0 ) u.b.
e(∆ 1 , ∆ 2 ) ≈ 3.01∆ 1 + 2.43∆ 2 . (43)
The three points represent three different compromises of tolerance design, which maximize the tolerances inside the perturbation domain provided by Theorem 1. The following table shows for each of them the linearized error, the error upper bound obtained solving (7) using IBEX, as well as the solving time. We see that the linearized error is accurate inside the perturbation domain provided by Theorem 1. The tolerance synthesis can therefore be performed inside this perturbation domain using the linearized errors by solving the biobjective problem consisting of maximizing ∆ 1 and ∆ 2 subject to (36), (37) and (38), and eventually checking the chosen tolerances a posteriori solving [START_REF] Chabert | Contractor Programming[END_REF].

A.2 Workspace W 2

The first workspace we consider is defined by 1 ≤ x 1 ≤ 2 and 3 ≤ x 2 ≤ 4. The constants upper bounds computed for this workspace using ∆ = 0.1 are given in the following table, as well as the time needed to compute them: The corresponding error upper bound given by Theorem 1 is = 0.83. The perturbation domain given by Theorem 1 is shown on Figure 4, as well as the isocontours of the linearized error

κ χ γ (1) (γ (1) 0 ) γ (2) (γ (2) 0 ) u.b.
e(∆ 1 , ∆ 2 ) ≈ 6.01∆ 1 + 4.57∆ 2 .
(44) Fig. 5. In gray, the perturbation domain given by Theorem 1 for W 3 . Dashed lines represent the error isocontours obtained with the linearization approximation (34).

The three points represent three different compromises of tolerance design, which maximize the tolerances inside the perturbation domain provided by Theorem 1. The following table shows for each of them the linearized error, the error upper bound obtained solving (7) using IBEX, as well as the solving time. We see that the linearized error is accurate inside the perturbation domain provided by Theorem 1. The tolerance synthesis can therefore be performed inside this perturbation domain using the linearized errors by solving the biobjective problem consisting of maximizing ∆ 1 and ∆ 2 subject to (36), (37) and (38), and eventually checking the chosen tolerances a posteriori solving [START_REF] Chabert | Contractor Programming[END_REF].

When compared with W 1 , we see that the linearized error is approximately twice bigger, while the perturbation domain provided by Theorem 1 is twice smaller. The workspace W 1 is therefore better with respect to sensitivity and tolerance synthesis.

A.3 Workspace W 3

The first workspace we consider is defined by -0.5 ≤ x 1 ≤ 0.5 and 0.1 ≤ x 2 ≤ 1.1. The constants upper bounds computed for this workspace using ∆ = 0.0053 are given in the following table, as well as the time needed to compute them: The corresponding error upper bound given by Theorem 1 is = 0.1. The perturbation domain given by Theorem 1 is shown on Figure 5, as well as the linearized error e(∆ 1 , ∆ 2 ) ≈ 11.09∆ 1 + 10.07∆ 2 isocontours.

The three points represent three different compromises of tolerance design, which maximize the tolerances inside the perturbation domain provided by Theorem 1. The following table shows for each of them the linearized error, the error upper bound obtained solving (7) using IBEX, as well as the solving time. We see that the linearized error is accurate inside the perturbation domain provided by Theorem 1. The tolerance synthesis can therefore be performed inside this perturbation domain using the linearized errors by solving the biobjective problem consisting of maximizing ∆ 1 and ∆ 2 subject to (36), (37) and (38), and eventually checking the chosen tolerances a posteriori solving [START_REF] Chabert | Contractor Programming[END_REF].

When compared with W 1 , we see that the linearized error is now approximately 4 times bigger, while the perturbation domain provided by Theorem 1 is now 20 times smaller. The workspace W 3 is therefore the worst with respect to sensitivity and tolerance synthesis.

B. The 3RPR robot

We now study the parallel robot 3RPR with 9 perturbations, 6 of them acting on the anchor points, and 3 of them on the commands. Its kinematic model F(x, q, p) is

(d 1 -x 1 ) 2 + (d 2 -x 2 ) 2 -(d 3 + q 1 ) 2 (45) L+d 4 -x 1 -l sin( π 6 +x 3 ) 2 + d 5 -x 2 +l cos( π 6 +x 3 ) 2 -(d 6 + q 2 ) 2 (46) L 2 +d 7 -x 1 -l cos(x 3 ) 2 + L √ 3 2 +d 8 -x 2 -l sin(x 3 ) 2 -(d 9 + q 3 ) 2 , ( 47
)
with L = 1 and l = 0.5. We study both the position error e P (x, x ) = Π P (x-x ) = max{|x workspaces to be investigated are displayed on Figure 2, where only x 1 and x 2 are represented because the orientation is fixed to x 3 = 0 for each considered workspace. This figure also shows the level-sets of the sensitivity index ΠF x (x, q(x), 0) -1 F p (x, q(x), 0) ,

where q(x) is the inverse kinematic model, Π = Π P for the upper graphic, and Π = Π O for the lower graphic. We expect from Figure 2 that W 1 is the best in term of sensitivity and tolerance synthesis. The kinematic model is not quadratic anymore with respect to x, therefore we solve the following optimization problem, whose upper bound provides a second derivatives based Lipschitz constant satisfying [START_REF] Kim | The kinematic error bound analysis of the stewart platform[END_REF]: The kinematic model is quadratic with respect to p, and we can obtain the sharp Lipschitz constant µ = 2. IBEX resolution timings are quite higher than for the RPRPR robot, because more variables are involved and expressions are more complex. On the other hand, the standard version of IBEX has been used, and specific tuning may turn out to significantly reduce the resolution timings.

λ ≥ max (x,q)∈G p∈B∆ max i jk ∂f i ∂x j x k (x, q, p) . (49)

B.1 Workspace W 1

The first workspace we consider is defined by -0.55 ≤ x 1 ≤ 0.05, 1.2 ≤ x 2 ≤ 1.8 and x 3 = 0. The constants upper bounds computed for this workspace using ∆ = 0.01 are given in the following table, as well as the time needed to compute them: The corresponding error upper bound given by Theorem 1 is = 0.03. The perturbation domain given by Theorem 1 is shown on Figure 7, as well as the isocontours of the linearized position (blue dashed lines) and orientation (green dashed lines) errors

κ χ λ γ (1) (γ (1) 0 ) γ (2) (γ (2) 0 ) u.b. ( pos
e P (∆ 1 , ∆ 2 ) ≈ 5.69∆ 1 + 4.31∆ 2 (50) e O (∆ 1 , ∆ 2 ) ≈ 5.68∆ 1 + 4.72∆ 2 . (51)
The three points represent three different compromises of tolerance design, which maximize the tolerances inside the perturbation domain provided by Theorem 1. The following table shows for each of them the linearized error, the error upper bound obtained solving [START_REF] Chabert | Contractor Programming[END_REF] We see that the linearized error is accurate inside the perturbation domain provided by Theorem 1. The tolerance synthesis can therefore be performed inside this perturbation domain using the linearized errors by solving the biobjective problem consisting of maximizing ∆ 1 and ∆ 2 subject to (36), (37) and (38), and eventually checking the chosen tolerances a posteriori solving [START_REF] Chabert | Contractor Programming[END_REF].

B.2 Workspace W 2

The first workspace we consider is defined by 0.45 ≤ x 1 ≤ 1.05, 1.2 ≤ x 2 ≤ 1.8 and x 3 = 0. The constants upper bounds computed for this workspace using ∆ = 0.01 are given in the following table, as well as the time needed to compute them: κ χ λ γ (1) (γ (1) 0 ) γ (2) (γ The corresponding error upper bound given by Theo-rem 1 is = 0.03. The perturbation domain given by Theorem 1 is shown on Figure 8, as well as the isocontours of the linearized position (blue dashed lines) and orientation (green dashed lines) errors e P (∆ 1 , ∆ 2 ) ≈ 4.7∆ 1 + 3.85∆ 2 (52) e O (∆ 1 , ∆ 2 ) ≈ 12.16∆ 1 + 9.13∆ 2 .

(53)

The three points represent three different compromises of tolerance design, which maximize the tolerances inside the perturbation domain provided by Theorem 1. The following table shows for each of them the linearized error, the error upper bound obtained solving [START_REF] Chabert | Contractor Programming[END_REF] We see that the linearized error is accurate inside the perturbation domain provided by Theorem 1. The tolerance synthesis can therefore be performed inside this perturbation domain using the linearized errors by solving the biobjective problem consisting of maximizing ∆ 1 and ∆ 2 subject to (36), (37) and (38), and eventually checking the chosen tolerances a posteriori solving [START_REF] Chabert | Contractor Programming[END_REF].

As predicted by Figure 6, we see that the position linearized error is approximately the same as W 1 (actually slightly better for W 2 ) but the orientation error is now twice worse. The perturbation domain provided by Theorem 1 is twice smaller as well. The workspace W 1 is therefore better with respect to sensitivity and tolerance synthesis.

Fig. 1 .

 1 Fig. 1. Typical upper bounds provided by Theorem 1, obtained with the dummy constants κ = χ = γ = λ = µ = 1 and γ 0 = 0.9.

Fig. 2 .

 2 Fig. 2.Sensitivity index and the three workspaces considered for the RPRPR robot (white curves are drawing artefacts due to the nondifferentiability of the sensitivity index).

Fig. 3 .

 3 Fig.3. In gray, the perturbation domain given by Theorem 1 for W 1 (note that it is truncated by ∆ ≤ ∆). Dashed lines represent the error isocontours obtained with the linearization approximation (34).

Fig. 4 .

 4 Fig. 4. In gray, the perturbation domain given by Theorem 1 for W 2 . Dashed lines represent the error isocontours obtained with the linearization approximation (34).

Fig. 6 .

 6 Fig.6. Sensitivity indices and the two workspaces considered for the 3RPR robot (the green circle represents parallel singularities; the white curves are drawing artefacts due to the non-differentiability of the sensitivity index). The upper graphic and the lower graphics correspond to position and orientation sensitivities respectively.

Fig. 7 .

 7 Fig. 7. In gray, the perturbation domain given by Theorem 1 for W 1 . Dashed lines represent the error isocontours obtained with the linearization approximation (34).

Fig. 8 .

 8 Fig. 8. In gray, the perturbation domain given by Theorem 1 for W 2 . Dashed lines represent the error isocontours obtained with the linearization approximation (34).

  1 -x 1 |, |x 2 -x 2 |} and the orientation error e

O (x, x ) = Π O (x-x ) = |x 3 -x 3 |.

Perturbations are partitioned in two classes: The geometric perturbations p

(1)

= (p 1 , p 2 , p 4 , p 5 , p 7 , p 8 ) and command perturbations p

(2)

= (p 3 , p 6 , p 9 ). The two different

  using IBEX, as well as the solving time.

	100∆ linearized certified	time
	(2,1)	0.012 0.012	0.013 0.012	606.7 657.4
	(1,3)	0.011 0.012	0.011 0.012	579.9 92.7
	(2,2)	0.011 0.012	0.012 0.012	621.7 47.1

  using IBEX, as well as the solving time.

	1000∆	linearized	certified	time
	(0.76,0.07)	0.0038 0.0097	0.0038 0.0098	2333 91
	(0.07,0.97)	0.004 0.0096	0.005 0.0097	2112 85
	(0.41,0.52)	0.004 0.0097	N.A. 0.0097	T.O. 79

Fixing the value of ∆ can be done with successive adjustments when necessary: Too large or too small values of ∆ will result in small perturbation domain.

Optimization problems to be solved are non-convex, non-smooth and equality and inequality constrained, and we require rigorous upper bounds. Up to our knowledge, IBEX is the only software available that is able to solve such problems. It can be downloaded from http://www. ibex-lib.org/.

This workspace is closer to parallel singularities, and sensibly larger ∆ leads to perturbed poses with parallel singularities.

Appendix I. Technical results

Lemma 1: With the notations of Kantorovich Theorem [START_REF] Gragg | Optimal Error Bounds for the Newton-Kantorovich Theorem[END_REF],

we have

Proof: Equation (55) is trivial, and (57) follows from the previous ones. Equation (56) remains to be proved. First note that

which is strictly increasing with respect to h. Therefore, h ∈ [0, 1) implies t * < 2δ. Second note that

which is strictly decreasing with respect to h. Therefore, h ∈ [0, 1) implies 1 χλ < t .


Bertrand Reeb

Guillaume Dubois

Gilles Proulx

PhD Ion Candel

Measuring water flow to generating units at the 1,520-MW Manic-5 powerhouse, with the goal of maximizing unit efficiency, is a monumental task but the long-term benefits could help Hydro-Quebec customers and establish industry-wide benchmarks.

By Bertrand Reeb, Guillaume Dubois, Gilles Proulx, Ion Candel and Cornel Ioana

In 2013, personnel with Canadian provincial utility Hydro-Quebec performed penstock intake flow comparison tests to measure characteristics of the flow to its turbines at the Manic-5 powerhouse, part of the Daniel-Johnson hydroelectric facility, constructed between 1959 and 1970. Daniel-Johnson Dam is a multiple-arch buttress dam on the Manicouagan River that creates the Manicouagan Reservoir, 133 miles north of Baie-Comeau in Quebec, Canada.

The facility produces hydroelectric power and supplies water to the Manic-5 and Manic-5-PA powerhouses that have a combined capacity of 2,596 MW. The dam is 702 feet tall by 4,311 feet in length making it the largest dam of its type in the world and one of the company's largest hydroelectric plants.

Information from the tests allow plant engineers to achieve optimal performance from Manic-5's eight 190-MW Francis turbines.

Measurement objectives

In their efforts to keep the total cost of the measurement tests as low as possible, engineers carefully considered flow measurement methods. Engineers consider the Manic-5 design intakes favorable and decided to use two types of measurements within the intake: the current meter (CM) and the acoustic scintillation flow meter (ASFM) methods.

Flow entering each penstock intake passes through two measurement systems after moving beyond the trashracks apparatus, but prior to reaching turbine-generators.

Hydro-Quebec's CM method uses trolleys that support a single row of meters. [START_REF] Lamy | A Different Approach in Measuring Individual Turbine Efficiencies in Multiple Unit Power Plants[END_REF] The meters move vertically, allowing velocity sample profiles over the entire height of the measurement section. This set-up reduces the installation time because it can be used for flow measurement for as many as four units at this location (during the test, only one of the four units was operated). Even though flow measurement is traditionally considered difficult to perform within intakes, especially in a short converging intake, the measurement is proving accurate in many situations such as shown in Kootenay Canal comparative tests2 and showing many advantages over penstock measurement. [START_REF] Proulx | Hydro-Québec Experience with Discharge Measurement in Short Converging Intake[END_REF] Within the penstock and aft of the CM measurement system, engineers used an ASFM to measure discharge as part of a three-year partnership between Electricite de France's General Technical Division (EDF DTG), Hydro-Quebec and the manufacturer of the ASFM, ASL AQFLow (the "SMASH" project). [START_REF] Candel | Electricité de France's Study of the Acoustic Scintillation Flow Meter Results in Expanding its Range and Sensitivity[END_REF] The main focus by partnership members studied ASFM in more detail and, depending on their findings, would possibly extend ASFM's range of usage. As a matter of fact, the ASFM measurements from the Manic-5 study now serve as benchmarks for normal measurement conditions. 07/12/2015 09:15

The power plant

The power plant has eight 190-MW Francis turbines operating with a net head of 141.8 m (465 feet). Two sets of conduits each provide water to four units and each conduit has two rectangular bays, which converge into one circular section downstream from the head gates. The measurement sections are 10.97 m by 6.1 m (36 feet by 20 feet). The CM measuring section was located about 10 m from the trash racks in the stoplog slots, while the ASFM measuring section was 3.4 m farther downstream in the gate slots (see Figure 1). With only one unit in operation, the average velocity was in the range of 1 to 2.5 m/sec, which falls within the usual range of both the CM and ASFM. 07/12/2015 09:15

The measurement conditions were considered good for both CM and ASFM. In effect, the smooth transition between the trashracks and the measurement sections is likely to produce smooth velocity profiles. The only concern were the trashracks cross members, which can generate large wakes. This has been taken into account in the selection of the number of ASFM measurement levels. The flow angle was expected to be near horizontal. ASFM ASFM uses a technique called acoustic scintillation drift to measure the flow velocity perpendicular to a number of acoustic paths established across the intake to the turbine. Short (16 µ sec) pulses of high-frequency sound (in the order of 307 kHz) are sent from transmitting arrays on one side to receiving arrays on the other, at a rate of about 250 pings/sec.5

Frame supporting the current meters. The frame had 14 current meters installed on the bottom of a simple trolley made of two end plates, profiled rods and steel cables.

Fluctuations (known as scintillations) in the amplitude of those acoustic pulses result from turbulence carried along by the flow.

ASFM measures scintillations and from them computes the lateral average (i.e., along the acoustic path) of the velocity perpendicular to each path. In its simplest form, two transmitters are placed on one side of the measurement section and two receivers at the other. The signal amplitude at the receivers varies randomly as the turbulence along the propagation paths changes with time and flow.

A steel frame, which is formed by two main vertical beams, supports 21 ASFM transducers.

If the two paths are sufficiently close (∆x), the turbulence remains embedded in the flow, and the pattern of these amplitude variations at the downstream receiver will be nearly identical to that at the upstream receiver, except for a time delay, ∆t. This time delay corresponds to the peak in the time-lagged crosscorrelation function calculated for Signal 1 and Signal 2. The mean velocity perpendicular to the acoustic paths is then ∆x/∆t. Using three transmitters and three receivers at each measurement level allows both the magnitude and inclination of the velocity to be measured. ASFM computes the flow through each bay of the intake by integrating the horizontal component of the velocity over the cross-sectional area of the intake. In a multi-bay intake, the flows through each bay are summed to compute the total flow. For the measurement, 21 pairs of transducers were installed on a steel frame. The number of transducers was chosen in order to resolve the possible oscillation of the velocity profile due to wake of the trash racks main cross members. All transducers were set horizontally because the expected flow angle was 07/12/2015 09:15 CM measurement set up CM measurement was done with 14 meters installed on the bottom of a simple trolley made of two endplates, profiled rods and steel cables (see photos on pages 72 and 73).

The CMs were equipped with Type A self-compensating propeller that compensates for up to 45-degree flow angle. The profiled rods are the same that are used for calibration of the CMs. Steel wheels help guide the trolley in the gate slot both laterally and longitudinally. The frame was moved by means of two hoists controlled by variable-frequency power drives, which allowed selection of the travel velocity. The two hoists were synchronized by means of encoders. Two displacement transducers measured the elevation of each end of the trolley.

The trolley velocity was set to 50 mm/sec except at the top and bottom of the measurement section, where it was set to 10 mm/sec for better defining the rapidly changing velocity profile. Total measurement time was 8 minutes for each run. The trolley velocity for the profiling method represents about 2% of the average velocity at the lowest flow. The data acquisition software developed by Hydro-Quebec allows recording the instantaneous rotational velocity of each current meter, (i.e., it records the time stamp of each revolution). Once the rotational velocities are recorded, it is relatively easy to calculate the mean value for different time intervals.

Results

Due to the presence of trashracks and a rigid and vibration-free support frame for the ASFM transducers, the conditions for ASFM testing were very close to ideal, meaning that all flow velocities were used in computing the flow (see Figure 2 on page 71). The velocity profile measured by ASFM shows small oscillations that are related to the trashracks cross members.

CM results show that the velocity profiles produced are similar to the ones from ASFM, with ASFM results slightly higher. The small difference (higher velocities in the bottom part of the measurement section) can result from the ASFM measurement section being farther downstream than the CM section, as the velocity profile tends to develop or flatten as the turbulence is mixing the different layers of the flow. A part of the difference can also be due to two other factors: the acquisition times for the CMs and ASFM are not concurrent and the sizes of the measurement sections of the two methods are marginally different.

The three-dimensional velocity profile (also Figure 2) from the CM measurement shows some asymmetry, especially in the bottom part of the section. It is likely due to the asymmetry of the intake upstream of the trashracks or the presence of some debris there. This asymmetry of the velocity profile in the bottom part of the measurement section can be the source of difference between the CM and ASFM methods.

Flow measurement comparison

The comparison of the flow measured by CM and ASFM (see Figure 6) includes the results for four units and 40 runs. Overall, the difference between the two methods is 1.4 %, with the ASFM flow being higher. The standard deviation is ± 0.8 % and includes the deviation of both methods. The random uncertainty of the regression line is ± 0.3 %, which means that there exists a statistical difference of 1.1% between the two methods. There is no significant variation of the difference between the two methods as a function of the flow. Both methods have an expected uncertainty of ± 1% to ± 1.5%. This means each method's uncertainty band overlaps. Thus, project engineers consider the agreement between the two methods as "good."

Reprocessing ASFM data Project personnel compared the standard time series computed by ASFM and the recomputed time series based on the recorded raw signals. They used a new algorithm to determine the amplitude of the acoustic pulses. This algorithm has already shown to improve the results of ASFM in case of very low turbulence3, which is not within the normal range of usage of the method. This new algorithm is not expected to have a significant effect on the calculated flow under normal stipulated conditions of usage of ASFM. From the eight runs reprocessed with the new algorithm, the difference from the original calculation is 0.35%, which is low. As expected, the difference for the present tests is not large, unlike what was found for low turbulence level tests. The new algorithm will be tested more thoroughly in future ASFM comparison tests as Hydro-Quebec, EDF and ASL AQFlow continue to collaborate on this matter.

Comparison of efficiency

Figure 7 (see page 79) shows the efficiency of Unit 4 (this unit represents an average one from the point of view of flow comparison between the CM and ASFM). The efficiency measured with the CM and ASFM in 2013, the thermodynamic method (Unit 8) measured in 1992 and the expected efficiency obtained from the model test step up are plotted as a function of the turbine output. At peak efficiency, all four curves are within 1.4%. Both the CM and ASFM intake measurement methods can be considered to have a measurement uncertainty of about ±1 to ± 1.5%.

At peak efficiency, the difference between the eight units reaches 0.3 % and will be taken into account for optimal dispatching. There is also some difference between the powers at peak efficiency, which will also be taken into account for turbine set up. In addition, for turbine output toward the maximum values, the difference of efficiency reaches 1.5% between the most and least efficient units. Again, knowing this fact will help Hydro-Quebec to properly operate its units, because they are often used during high load periods.

Conclusions

Hydro-Quebec has performed efficiency tests in the intake of one of its major power plant. The goal of those tests was to measure the efficiency of all eight units for optimal dispatching. It was also a good opportunity to perform comparative test between two methods, CM ASFM.

The results show that the flow measured by the CMs and ASFM are within 1.4% from each other, which is considered excellent. The results of the CMs and ASFM also compare well with the previous tests done with the thermodynamic method.

Significant differences exist between each unit at peak efficiency and near maximum power.

The results of those tests will help Hydro-Quebec to better operate this plant.

Notes

Figures 3 ,

 3 Figures 3, 4 and 5 (see pages 72, 73 and 74) show a very high resemblance between the two time series, which leads to close values for the total flow. The quality factor of the measurement, for each individual level5 was above 0.9, thus a high degree of confidence in the ASFM results.

horizontal, as it was later confirmed by the measurement.

This is a close up of horizontally installed ASFM transducers on a steel frame used for flow measurement.

In addition to the normal ASFM, a high-speed data acquisition system was used to record the raw acoustic signal for each element. In total, 12 signals were simultaneously recorded at a rate of 3.3 mega samples per second) for each 7-minute run. Those data were used after the tests for an alternate post-processing analysis.