Modelling and traceability for computationally-intensive precision engineering and metrology

Transcription

1 Modellng and traceablty for computatonally-ntensve precson engneerng and metrology J.M. Lnares, G. Goch, A. Forbes, J.M. Sprauel, Clément Audbert, F. Haertg, W. Gao To cte ths verson: J.M. Lnares, G. Goch, A. Forbes, J.M. Sprauel, Clément Audbert, et al.. Modellng and traceablty for computatonally-ntensve precson engneerng and metrology. CIRP Annals - Manufacturng Technology, Elsever, 2018, 67 (2), pp < /j.crp >. <hal > HAL Id: hal Submtted on 18 Oct 2018 HAL s a mult-dscplnary open access archve for the depost and dssemnaton of scentfc research documents, whether they are publshed or not. The documents may come from teachng and research nsttutons n France or abroad, or from publc or prvate research centers. L archve ouverte plurdscplnare HAL, est destnée au dépôt et à la dffuson de documents scentfques de nveau recherche, publés ou non, émanant des établssements d ensegnement et de recherche franças ou étrangers, des laboratores publcs ou prvés.

2 Modellng and traceablty for computatonally-ntensve precson engneerng and metrology J.M. Lnares(1) a, G. Goch(1) b, A. Forbes c, J.M. Sprauel a, A. Clément(1) d, F. Haertg e, W. Gao(1) f a Ax Marselle Unv, CNRS, ISM, Marselle, France b Unversty of North Carolna, 9201 Unversty Cty Blvd, Charlotte, NC, Unted States c Natonal Physcal Laboratory, Hampton Road, Teddngton, Mddlesex, TW11 0LW, Unted Kngdom d Dassault Systèmes, France e Department of Coordnate Metrology, Physkalsch-Technsche Bundesanstalt,D Braunschweg, Germany f Tohoku Unversty, Japan In contrast to measurements of the dmensons of machned parts realzed by machne tools and characterzed by CMMs, software results are not fully traceable and certfed. Indeed, a computer s not a perfect machne and bnary encodng of real numbers leads to roundng of successve ntermedate calculatons that may lead to globally false results. Ths s the case for poor mplementatons and poorly condtoned algorthms. Therefore, accurate geometrc modellng and mplementatons wll be detaled. Based on the works of Natonal Metrology Insttutes, the problem of software traceablty wll also be dscussed. Some prospects for ths complex task wll fnally be suggested. Keywords: Geometrc modellng, Error, Computatonal accuracy 1. Introducton Humanty s facng very bg challenges such as global warmng, depleton of natural resources, and sustanable development, for example. At the same tme, the busness world tres to mprove ndustral compettveness accountng for these envronmental constrants. Most ndustralzed countres have thus launched development programs to answer these challenges (USA: Advanced Manufacturng, Manufacturng renassance, Natonal Network for Manufacturng Innovaton, Inda: Make n Inda, Japan: Innovaton 25 program, Chna: Intellgent Manufacturng, Made n Chna 2025, European countres: Horzon 2020, Factores of the Future, Industry 4.0 ). All these programs are based on the development of computerzaton and networkng n ndustral systems. Ths s the convergence of the physcal and vrtual worlds to Cyberspace. The German Natonal Academy of Scence and Engneerng defnes ths evoluton as a 4th generaton ndustral revoluton based on Cyber-Physcal Systems (CPS) [84,122,143,172]. In ths new envronment, nformaton technology (IT) wll allow decson makers to do more over tme to mprove product qualty and customsaton, productvty and customer satsfacton. Ths global expanson of the use of computers n ndustry brngs to the forefront the need for traceablty and certfcaton of ndustral software. Scentfc calculatons have ndeed become central ssues n desgn, manufacturng, precson engneerng and metrology software. The fundamental bnary code together wth all basc arthmetc operatons were developed by Lebnz n 1697 [123]. The prncple of modern programmable computers was frst proposed by Alan Turng n hs 1937 paper: "On Computable Numbers wth an Applcaton to the Entschedungsproblem" [167]. The Turng machne s the frst unversal programmable computer. It nvented the concepts of programmng and program. The constructon of Plot ACE (Automatc Computng Engne) based on Turng's desgns was completed by the Natonal Physcal Laboratory n the early 1950's. In 1946, the frst archtecture of electronc computers was proposed: the ENIAC (Electronc Numercal Integrator and Computer, usng the vacuum tube 1 technology. Fgure 1 shows a pcture of ths computer. The second generaton of computers was based on the nventon of the transstor n Despte the use of transstors and prnted crcuts, the computers were stll bulky and only used by unverstes, governments and large companes. The thrd generaton of computers (around 1959) was based on electronc chps. In 1971, Intel revealed the frst commercal mcroprocessor, the It dd not acheve more than 60,000 operatons per second. Today, standard desktop computers have a much bgger processng capacty (for example, an Intel Core 2 Duo processor at 2.4 GHz can execute around 2 bllon operatons per second). Mcroprocessors nclude most of the computng components (except for the clock and the memory) on a sngle chp. The computer s now the daly companon of people both at the offce and n prvate lfe. For the average user, the computer remans however a black box that just provdes results from the data that were entered. These outcomes are generally consdered above any suspcon. For future ndustral systems, based on cybernetcs, wll the results suppled by computers be really traceable and valdated numercally? To answer ths queston, technologcal ncdents could be reconsdered that have happened n recent hstory and had a computer error as source. In computng, an nteger overflow s a condton that occurs when a mathematcal operaton produces a numercal value larger than the greatest number that can be represented by the set of bts (bnary dgts) of the mplemented varable. Perhaps the best-known consequence of such an error s the selfdestructon of the Arane 5 rocket durng ts frst launch on June 4th, The Inertal Reference System (IRS) of Arane 5 was derved from that of Arane 4 but the new launcher had hgh ntal acceleratons and a trajectory that resulted n horzontal veloctes fve tmes larger than those of the prevous rocket. These hgh speeds were captured by the sensors of the nertal platform but exceeded the maxmum value that could be processed by the navgaton program. It resulted n an nteger overflow excepton n the IRS software and the shutdown of the computers caused by the converson from a 64-bt real number to a 16-bt nteger [125]. False flght data led than to erroneous

3 Interface Interface Interface correctons of the launcher trajectory and fnally to the selfdestructon of the rocket. algorthms. The thrd secton of the paper wll therefore deal wth detaled smart mplementatons of geometrc modellng. Based on the works of Natonal Metrology Insttutes, the problem of software certfcaton and traceablty wll also be dscussed n the fourth secton. Some prospects about these dfferent subjects wll fnally be suggested. 2. Intrnsc performances of computer hardware and software Fgure 1: ENIAC (Electronc Numercal Integrator and Computer) [95] An ntrnsc feature of numercal computng s that real numbers are represented n fnte precson and ths means nearly all real numbers have to be rounded to be represented. The accuracy of the roundng operaton can have great nfluence on calculated results. In 1992, at Dhahran n Saud Araba, a Patrot battery faled to track and to destroy a Scud mssle [64]. Ths ncdent was caused by a software problem n the system's weapon control computer due to an naccurate calculaton of tme and consequently of the trackng trajectory. The precson of calculatons of on-board computers often depends on the number of bts of ts regsters. Patrot s clock system was performng some arthmetc operatons usng a 24-bt fxed pont regster. Ths hardware lmtaton led to a drft between the tmes elapsed snce last boot, as measured by the system's nternal clock, and the real delays. Table 1 shows the evoluton of ths tme drft (naccuracy)and the estmated shft n the range gate that Patrot tracked. Table 1:Precson of a computer s calculatons Hours Seconds Calculed tme (s) Inaccuracy (s) Shft n range gate (m) After 100 hours of montorng, the elapsed tme calculated by Patrot s clock system drfted by approxmately 0.3 s. In connecton wth the speed of the tracked Scud rocket, the resultng error of the calculated ntercepton pont was estmated to be about 700 m. Patrot s battery therefore faled to destroy the mssle. The sze of the regsters of current generaton of computers s now at least 64 bts, permttng calculatons wth greater precson but roundng effects are stll nevtable. Can such problems arse n precson engneerng and metrology? Whereas there s an nfrastructure to provde traceablty of dmensons of machned parts realzed by machne tools and characterzed by CMMs, the results of software calculaton are usually not fully traceable and certfed. Indeed, a computer s not a perfect machne and bnary encodng of real numbers leads to roundng of successve ntermedate calculatons that may lead to globally false results for poorly constructed calculatons. To understand these calculaton lmts, the second secton of the paper wll be dedcated to the ntrnsc performances of computer hardware and software. False computaton results are often due to poor software mplementatons and badly condtoned or numercally unstable As dscussed n ntroducton, the hardware (number of bts, number of processors ) and software (converson effects, roundng effects, cancellaton effects ) of computers, have a great nfluence on the accuracy of the calculated results. These topcs wll therefore be dscussed now. The logcal structure and functonal characterstcs of computers are shown n Fgure 2. A computer s bult around one or more mcroprocessors wth each mcroprocessor have one or more cores. The processor (named CPU for Central Processng Unt) s an electronc crcut clocked at the rate of an nternal clock. A processor has nternal regsters of a fxed number of bts(now usually 64 bts) used to encode and manpulate the processed values. Several processor clock pulses are generally necessary to perform an elementary acton called an nstructon. The ndcator, Cycles Per Instructon (CPI), characterzes the mean number of clock cycles requred to execute a basc nstructon on a mcroprocessor. It s about four for most current mcroprocessors. The CPU power can thus be characterzed by the number of nstructons processed per second and s often expressed n unts of mllons of nstructons per second (MIPS) and corresponds to the frequency of the processor dvded by the CPI. The CPU ncludes one or several Arthmetc Logc Unts (ALU) that provde the basc functons of arthmetc calculatons and logcal operatons on ntegers, and a Floatng-Pont Unt (FPU) to perform operatons on floatng pont numbers. The processor employs cache memores (buffers) to reduce the tme to exchange nformaton between the man computer memory, ts Random Access Memory (RAM), and the nternal data regsters. Computer mcroprocessors only manpulate bnary nstructons and data. The encodng of such nformaton requres two states 0 and 1. In a smplfed way: ether the electrcal current flows through an elementary crcut or t does not. A bnary machne language encodes the set of basc nstructons mplemented n the mcroprocessor hardware to perform the avalable elementary operatons such as addton, subtracton, multplcaton, dvson, comparson, etc. Clock Speed Mcroprocessor Processor number Core number Bts number Computer's cache memory Sze Computer's central memory RAM Sze BUS Screen Hard dscs Computer network Fgure 2: Logcal structure and functonal characterstcs of a computer The qualty of numercal results n scentfc calculatons performed by a computer wll depend on both the hardware and the software. In the remander of ths secton, the hardware 2

4 aspects wll be treated frst, and then the software aspects wll be presented. 2.1 Techncal advances of computer hardware For use n scentfc calculatons, the performance of a computer greatly depends on the number of processors and cores, the clock frequency, the number of bts of the CPU regsters and the sze of the cache memores. The consderable mprovements of computers over the past decades are manly lnked to a great ncrease n the number of the elementary components ntegrated n the processors. The emprcal Moore's law [1], largely verfed snce 1971, establshed that the number of transstors n a densely ntegrated crcut s doubled every two years (Fgure 3). Ths law s unlkely to be satsfed n the near future because of the physcal lmtatons of the actual slcon technology beng reached quckly. In fact, due to quantum tunnellng effects, the smallest sze of elementary transstors s at present lmted to 20 nm [163]. Nevertheless, Intel beleves that ths sze may be reduced to 7 nm by 2020, even f t wll perhaps requre usng materals other than slcon, such as Indum Gallum Arsende [89]. The computng performance also depends on the length of the data words manpulated by the computer. The sze of the nternal regsters and of the data bus of the frst Intel 4004 processor was lmted to 4 bts. It has grown snce the 1970 s to reach 64 bts on current processors. 4 Intel Pentum 32 Moore s Law? Intel Core 3/5/7 64 bts Fgure 3: Moore's law Year Other technologes are also beng developed to replace current hardware based on electronc transstors. In 1982, Rchard Feynman suggested that smple controllable quantum systems could be used to smulate the quantum dynamcs of problems that cannot be modelled by a conventonal computer [56]. In 2012, the Nobel Prze for Physcs was awarded for scentfc research on ground-breakng expermental methods that enable the measurement and manpulaton of ndvdual quantum systems [148]. The quantum propertes of matter, such as superposton and entanglement, provde the framework for the development of quantum computers. Fgure 4 shows the possble values of the state of a bt and a qubt (quantum bt). A standard computer s based on bnary data: a bt has only two ndependent values 0 or 1. The quantum computer s workng wth qubts that can have multple states [42,170]. These possble values (states) can be represented usng the Bloch sphere graph where α and β are complex numbers and probablty ampltudes. Ths property allows quantum processors to perform multple operatons n parallel. Research on quantum computers s very actve, wth around 8000 publcatons snce A quantum computer exstng today has several hundred qubts. Quantum processors need external coolng down to a temperature of about K (around -273 C, very close to absolute zero). The development of quantum nformatcs based on superconductng crcuts also requres accurate readout devces to gather the qubt states [132]. The quantum computer s well-adapted for combnatoral calculatons and uses smulated annealng algorthms for global optmzaton problems. Some large companes [112] have begun usng quantum computers prncpally n cryptography and optmzaton problems usng smulatons (Grover s search algorthm) [42]. Two ndependent electroncally states bt 0 qubt І0> Lnear superposton of the 2 bass states = = 0 OR = wth : + = 1 1 І1> Fgure 4: Values of states for bt and qubt A second set of technologes reles on the fact that slcon s transparent to nfrared lght, so that optcal fbres can be used to nterconnect computer elements or components nsde the processor core. Ths opens a new way to buld optcal computers [146]. For that purpose, an optcal nanowre swtch was frst desgned [147] and, by combnng two optcal swtches, a NAND logc gate was then developed. Smlarly, n non-lnear optcs, the property of some materals to change ther refractve ndex under an electrc feld (Kerr effect) was studed. Ths permtted creatng logc gates (AND, OR, NOR...)[121]. Such components may be employed n the desgn of future optcal processors and computers. In addton, photons do not produce magnetc nterferences wth the envronment and the heat generated by an optcal system remans very low. Optcal transstors can work at frequences much hgher than those of conventonal electronc devces. Optcal computers could thus be more powerful than current conventonal computers. The man dsadvantage of ths technque s the nablty to store photons and lght. Furthermore, as photons propagate n a straght lne, buldng nterconnectons causes major dffcultes n a reduced space. Another scentfc track to replace current processors s based on the outcome of molecular bology. The prncple of DNA computers (computng technology based on molecular bology), enuncated by Leonard Adleman n 1994, s to encode an nstance of the problem wth DNA strands and manpulate t by molecular bology to smulate operatons that wll solate the expected soluton of the problem [2,3]. As for quantum technology, DNA computers wll be specalzed n computng complex problems lke non-determnstc algorthms n polynomal tme because DNA strands can produce bllons of potental answers smultaneously. However, the process of synthess and reacton s very slow Techncal propertes of nternal computer devces Current processors work on the scheme of classcal Turng machne and are constraned to perform calculatons n sequence. The consequence s that t s less promsng to deal wth, n a gven tme, a large number of nstances of computatonal problems of hgh numercal complexty. As shown n Fgure 2, the performance of a computer s related to the number of processors and cores, the number of bts of the nternal regsters and the data bus, the clock and bus frequences and the sze of the cache memory. A global metrc was proposed to measure the theoretcal computer performance n scentfc calculatons that use floatngpont operatons: floatng-pont operatons per second (FLOPS). Equaton 1 shows FLOPS formula. 3

5 FLOPs FLOPS = N N f (1) processor core/processor processor clock cycle Wth: Nprocessor: Number of processors n mcroprocessor unt, Ncore/processor: Number of cores n processor, fprocessor clock: Frequency of processor clock, FLOPs 4 cycle wth actual processors. For a scentfc computer wth 2 processors contanng 12 cores each and workng at a clock frequency of 2.9 GHz, the theoretcal number of floatng-pont operatons per second s GgaFLOPS. A laptop wth a sngle-core 2.5 GHz processor has a theoretcal performance of 10 GgaFLOPS. Ths metrc s sometmes dvded by the electrcal power (FLOPS/watt) to analyse the energy effcency of the computer. The number of FLOPS s used to compare the theoretcal performance of computers but does not account for specfc tasks of the computaton and for the real load rate of each processor and core. New benchmarks are therefore proposed by the Standard Performance Evaluaton Corporaton, launched n 2000, to compare computer performances. These benchmarks are based on specfc procedures appled to test the computer behavour when runnng next-generaton of ndustral software (Dassault Systèmes: CATIA and Soldworks, Pro Eng: CREO, Semens: NX...), to stress a system's processor, etc... [83]. These tests hghlght the processor ablty to process a set of operatons n a lmted tme or to gve global nformaton on the computer's behavour. But these tests do not gve an ndcaton of the traceablty or the qualty of the numercal results provded by the computer. x Speed Up Fgure 5: Amdahl's law Parallelsm proporton Processor number Equaton 1 shows that the number of processors and cores nfluences the performance of calculaton. Multprocessor computers allow the program to complete several arthmetc operatons smultaneously, thus ncreasng the processng capacty. Ths technque s called parallelsm. The technque of parallelsm can be used nsde a processor to address the cores, between processors or for a ppelne technque. In the thrd case, the processor can start executng a new nstructon wthout watng for the prevous one to be completed. To reduce the nput/output bottleneck of nstructons, a vector processor was developed wth specfc nstructons optmzed for fast handlng of tables and quck matrx calculatons. Theoretcally, t s expected to halve the processng tme by sharng calculatons between two processors wth sngle core, to quarter the processng tme by usng 4 processors, etc. Unfortunately, not all scentfc operatons can be parallelzed effectvely. P=Nprocessor Ncore/processor The emprcal Amdahl's Law (Eq.2)can be used to defne an upper lmt to the parallelzaton contrbuton of software or hardware archtectures [4,88,145]. It assumes a constant amount of data to be processed. Fgure 5 shows the speedup of calculatons versus the processors number used and the proporton of parallelzed computer code. In Fgure 6, another emprcal law (Eq.3) s shown, known as the Gustafson Barss' relatonshp [78,96,113]. It s more optmstc than the prevous one and reflects the fact that more data can be processed at the same tme by ncreasng the number of processors. x Speed Up Fgure 6: Gustafson Barss' law Parallelsm proporton Processor number Speedup = 1/ [(1- α) + α / P] (2) Wth: : proporton of parallelsm Speedup = P - (1- α)( P -1) (3) Wth: : proporton of parallelsm P=Nprocessor Ncore/processor Supercomputers and LINPACK/LAPACK benchmarks A supercomputer s a computer desgned to acheve the hghest performances at the date of ts manufacture. Its use s targeted to Hgh-Performance Computng (HPC). These supercomputers have thousands of processors and hardware archtectures allowng them to use the benefts of parallelsm. To classfy the most effcent machnes n scentfc computng, a TOP500 classfcaton project was created. The LINPACK benchmark s used to test and to rank supercomputers for the TOP500 lst [41,45,92]. It measures the tme taken to solve a dense lnear system of n equatons n n unknowns, the soluton beng obtaned by a partal Gaussan elmnaton, by 2/3 n² + n³ floatng pont operatons [73]. The performance s then calculated by dvdng the number of operatons by the calculaton tme. To complement the FLOPS metrc, two other ndcators were created for ths benchmark: - Rmax: maxmum performance n LINPACK Gga FLOPS for the bggest computable mathematcal problem on the computer, - Nmax: sze of the mathematcal problem gvng Rmax computable on the machne. - Rpeak represents the theoretcal performance n GgaFLOPS of the computer. To extend the use of LINPACK packages on computers usng shared-memory vectors and parallel calculatons, a new LINPACK package was ntroduced n ths benchmark [7,29,44]. Ths software pack s beng constantly mproved, partcularly n terms of accuracy and performance [46]. Table 2 presents the ten frst supercomputers of the world. After three consecutve years as the world s No. 1 system of the Top500 rankng, Tanhe-2 of Natonal Super Computer Center has been exceeded n performance by Sunway Super Computer. Ths computer has 10,649,600 cores and needs 15,371 kw of electrcal power to obtan ts best calculaton performance. Table 2: TOP500 rankng n June

6 Top 500 Rmax Year Name Total Cores Country Nmax Power (kw) Sunway 1,1E+07 Chna Tanhe Chna Pz Dant Swtzerland Ttan Unted States Sequoa Unted States Cor Unted States Oakforest Japan Rken Japan Mra Unted States Trnty Unted States In 2005 and n lne wth the new challenges of ths world (global warmng, resource reducton, sustanable development), a new rankng of supercomputers has been set up: Green500 [91]. It ncorporates the calculaton concepts used n the TOP500 rankng, but t s based on a new metrc for supercomputer rankng: the power-performance defned by the number of FLOPS per Watt (FLOPS/W). Green500 proposes a rankng of the most energy-effcent supercomputers n the world. Table 3 presents the ten supercomputers ranked usng ths new metrc. At present, the Tsubame 3.0 heterogeneous supercomputer (Tokyo Insttute of Technology)obtans the top spot n the Green500 lst and currently clams the ttle of the most energy-effcent (or greenest) supercomputer n the world. The Tsubame 3.0 heterogeneous supercomputer surpassed the fourteen ggaflops/watt mlestone [91]. To maxmze the powerperformance metrc, computer manufacturers use specalzed cards named Hgh Performance Computng (HPC) ncludng many-core accelerators, on whch parts of the computatons are subcontracted. These new many-core accelerators are coupled to the CPU wth an energy-effcent software desgn. For example, computer wth HPC cards can treats large data up to 10 tmes faster than a sngle CPU. Table 3: Green500 rankng n June 2017 Green 500 Top5 00 MFLOP S/W Year Name Total Cores Country Rmax TSUBAME Japan kuka Japan AIST AI Cloud Japan RAIDEN GPU Japan Wlkes UK Pz Dant Swtzerland Gyoukou 3E+06 Japan Res. Comp. Faclty Japan Facebook US DGX Saturn V US 3307 Wthout fundamental change n the desgn of supercomputng systems, the computer performance advances wll not contnue at ther current pace [55,151] Sub concluson The mprovement of computer performance s now mpacted by the constrants of sustanable development. The tests conducted under Green500 benchmark show that the manufacturers drect the development of ther supercomputer to heterogeneous machnes usng Hgh Performance Computng accelerators. These new hybrd systems, although energetcally optmzed, are stll based on electroncs. Ths technology s now well under control, but the resstvty of the crcuts causes sgnfcant energy loss as heat. For example, the data can requre up to 80% of the power consumed by a mcroprocessor. To solve ths problem, new technologes have been proposed based on quantum physcs, optcs or molecular bology. The systems derved from these technologes are however specalzed. They have functonaltes smlar to the accelerators used n actual heterogeneous computers. Such new devces wll therefore surely be coupled to the machnes desgned wth current technology, thus allowng contnuous mprovement of the computer performance. Fgure 7 summarzes the hardware tems necessary to enhance and to optmze the computng performance of computers. However, the performance of computers not only depends on the hardware, but also on the manner n whch data processng s mplemented,.e. the software. For most scentfc calculatons (outsde grand challenges) computatonal tme s usually not a lmtng factor, and solutons to a suffcent accuracy can be determned n an acceptable amount of tme. The precson of calculated results manly depends on the qualty of the data processng, n partcular on the way the software s mplemented. Processor Clock Ppelne treatment Parallelsm ntra processor Parallelsm Multprocessors Hardware Scentfc calculaton performances Memory Complaton Data management Software Implementaton (ppelne, vector, access parallelzaton) Fgure 7: Parameters nfluencng scentfc calculaton performances 2.2 Techncal advances of computer software Transstor gates of current computer processors can only handle bnary nformaton, n two dstnct states 0 and 1, called bt. At the hardware level, n bts are then gathered to buld words that are transmtted to the nternal regsters of the processors and cores. They are treated as nstructons or operands for arthmetc and logc operatons. The set of nstructons of the machne language drectly acheves basc mathematcal operatons on nteger numbers (addton, subtracton, multplcaton, dvson, modulus ). A floatng-pont unt s also generally embedded n the processor (or connected to t)realzng arthmetc calculatons on real numbers and computng classcal mathematcal functons (sne, cosne, exponental, power, square root, ). These operatons, performed at the hardware level, are very effcent, but the range of the manpulated data and the accuracy of some calculaton results are lmted by the sze of the nternal processor regsters. If a greater range or precson s requred, the number of bts used to encode a number has to be ncreased. A software layer wll then be added to perform the operatons. The frst electronc computer, the ENIAC, used decmal arthmetc. In most current computers, the use of bnary encodng has however been generalzed, prncpally for ts calculaton smplcty and ts coherence wth the hardware [37]. If another encodng base (decmal, hexadecmal...) s chosen to represent a number, t wll be necessary to add a software layer to acheve encrypton and calculatons. IEEE 754 standard provdes the rules of encodng real numbers for bnary floatngpont arthmetc. However, only ts last revson n 2008 defnes the encodng of numbers n bnary coded decmals. For a better understandng, the effect of type and base of encodng and the number of bts on the accuracy of calculatons n precson engneerng and metrology has to be explaned Implementaton propertes Computer hardware s desgned wth regsters of fxed sze, thus leadng to a lmted accuracy when performng arthmetc operatons wth the floatng-pont processor unts. However, usng programmng methods, the number of bts used to represent numbers can be ncreased. Nevertheless, at the same tme, the sze requred to store ths nformaton grows and may exceed the maxmum avalable capacty of memory. 5

7 Computer lmtaton Calculatons wth floatng numbers Calculaton based on floatng pont encodng s used the most n computng, but problems may arse when only a few bts represent the real numbers because of the lmted accuracy. Memory sze Bt number Fgure 8: Lmtaton of computer encodng Fgure 8 shows that wth a fnte memory capacty, the software cannot decrease the gap between two successve real numbers (the precson of the floatng-pont arthmetc) to zero. Therefore, all real numbers cannot be represented by a computer. Ths s the frst lmt of computer calculatons. Two phenomena are nduced by ths lmt: roundng and cancellaton [72]. In ths context, floatng pont calculatons can gve entrely naccurate results when no partcular precauton s taken. To llustrate ths, the computaton of the smple functon f of the varable M mght be consdered, presented n Equaton 4 [140,19]. f(m)= M + M M - M 2 gves 0 for M>10 8 (4) For large M values (n ths case when M exceeds about 10 8 ) the value returned by software mplementng IEEE arthmetc [100] s 0, whle the true result s of course 1. The usual computer addton process s sequental as shown n Fgure 9. M M 2 1 -M -M 2 M + 2 M + 2 M Fgure 9: Usual computer addton process [19] To compensate ths bas, more effcent calculaton methods have been developed but they are not automatcally mplemented n all usual programs. For example, computer calculatons usng a Two Sum algorthm (TS) [116] gve true results (Fgure 10). M M 2 1 -M -M 2 M 2 M 2 M 2 0 TS TS TS TS M 1 -M 0 M 0 0 TS TS TS TS TS TS TS TS Fgure 10: Two Sum algorthm [19] If the order of operatons n Equaton 4 s changed (Fgure 11), the rght calculaton result s also obtaned. Therefore, the way the equatons are mplemented n the computer affects the qualty of the result, whch s called the mplementaton effect. M 2 -M 2 M -M 1 0 M Fgure 11: Implementaton effect IEEE standard For most computers, the representaton of numbers n bnary and decmal floatng-pont calculatons s based on IEEE standard [100,133,140], where bnary floatng pont s commonly used n the computer feld. Ths standard also defnes specal values Base 10 Base 2 Normalzed x 2-4 Exponent + Offset = = 123 IEEE Exponent Wth Offset = 2 e 1 1 Example: e=8 Offset=127 Sgn Shfted exponent Sgnfcant 1bt + e bts + s bts Lmted number of bts (21bts n example) Fgure 12: IEEE standard (floatng pont number) The encrypton of numbers and mplct conversons between the decmal and bnary system durng data nput/output are the frst source of roundng. Ths process s shown n Fgure 12. In consequence, not all the decmal numbers (.e., ratonal numbers, the ratos of ntegers) can be expressed exactly as bnary floatngpont numbers. As shown n Equaton 5, a real number wrtten usng IEEE standard s defned by three ntegers: sgn, shfted exponent and sgnfcant. ( shfted exponent- offset) Encoded number = sgn 1sgnfcant 2 (5) wth offset=2 e 1 1, e beng the bt number of the exponent The data s represented n a scentfc form. The most sgnfcant bt gves the sgn, set to 0 when the number s postve and 1 f t s negatve. The next e-bts defne the exponent, shfted by a fxed offset to avod negatve values. Fnally, the last bnary dgts defne the normalzed sgnfcant, truncated to the avalable numbers of remanng dgts. Because the most sgnfcant bt s always 1 for a normalzed number, ths bt s not stored n the mantssa and s called the hdden bt. The expresson of the exponent offset and the whole formula permttng the defnton of the real number are presented n Equaton 5. In Fgure 12 ths representaton was used to encode the real number 0.1 usng a regster of 21 bts. If ths floatng number s reconverted to base 10, the decmal value s obtaned. It clearly shows the roundng effects due to the lmted accuracy of bnary floatng numbers. In double precson (encodng wth 64 bts), the smallest postve and greatest negatve normalzed number dfferent from zero are: ± ± As the mathematcal nfnte and the mathematcal zero cannot be encoded, three specfc exceptons are consdered n IEEE standard: - the exponent offset and mantssa are both zero: the real number s ± 0 (accordng to the sgn bt), - the exponent offset s equal to 2 e -1, and the mantssa s zero: the number s ± nfnty (accordng to the sgn bt), - the exponent offset s 2 e -1, but the mantssa s not zero: the number s NaN (not a number). To lmt the roundng effects of numbers translated nto base 2, the followng roundng procedures have been ncluded n IEEE standard[133,140,38]. The four roundng modes [100] are: -Round toward : RD(x) s the largest floatng-pont number less than or equal to x, -Round toward + : RU(x) s the smallest floatng-pont number greater than or equal to x, -Round to nearest: RN(x) s the floatng-pont number that s the closest to x, 6

8 - Round toward zero: RZ(x) s the closest floatng-pont number to x that s no greater n magntude than x. It s equal to RD(x) f x 0, and to RU(x) f x 0. Many studes have been conducted on the effect of roundng n the calculaton of basc functons (-, +, x, /, exp, log...) [40,75,110, 118,119,135,176]. In calculatons wth floatng pont numbers, the mplementaton of arthmetc operatons [141]: addton [140], subtracton, multplcaton, dvson, square root [111], fused-multplcaton-addton [128] are the bass of scentfc calculatons. A lot of research work has been carred out to make these functons more precse, stable and robust despte the naccuracy nduced by the encodng of real numbers. A Graphcs Processng Unt (GPU) s an electronc crcut on a graphcs card or CPU. It performs the computatons requred by the dsplay functons. The GPU has a parallel structure. It s desgned for the great number of calculatons requred by the realzaton of 3D renderng, for example. The manufacturers of these components had to develop effectve and precse algorthms to perform such computatons. Classcal basc arthmetc functons, vector operatons (dot product, cross product...) and matrx calculaton algorthms have thus been developed for graphcal dsplay and have been ncorporated n the GPU hardware. These effcent procedures are accessble to software developers and are sometmes used to mprove the performance of programs [174]. The gudelnes of the IEEE standard must be ncluded n the programmng of busness software used n metrology and precson engneerng. However, most tradtonal software has not yet mplemented these gudelnes Calculaton wth Decmal Floatng-Pont usng Bnary Coded Decmal The last revson of the IEEE-754 standard ntroduced Decmal Floatng-Pont (DFP) formats to encode real numbers n scentfc calculatons [36,142]. The frst domans of applcaton were fnancal analyss, tax calculaton and others, where accurate calculaton s sought to avod mstakes causng fnancal losses. The numercal value of a number s gven by sgn sgnfcant 10^Exponent. The mantssa and the exponent can be descrbed n several ways: both values n Bnary Coded Decmals (BCD), sgnfcant n BCD and exponent n bnary or more complex compressed formats. Two methods were proposed by Intel [34] and IBM[35]: The frst named bnary nteger decmal (BID), encodes the sgnfcant of the DFP number as an unsgned bnary nteger. The second, named densely packed decmal (DPD), encodes the sgnfcant as a compressed bnary coded decmal nteger. 13 Decmal BCD Fgure 13: Bnary Coded Decmal (BCD) Fnally, dfferent ways exst to encode the same number n decmal floatng-pont. One way to code a number based on the ntrnsc propertes of a bnary computer s to use the BCD method. The numbers are then represented by decmal dgts, each of them encoded on four or eght bts (Fgure 13). The basc arthmetc operatons and mathematcal functons have then been developed to make ths DFP encodng method more accessble [8,30,51,53,74]. DFP s not yet used on all hardware/software products. Therefore, only bnary coded floatng-pont numbers wll be consdered n the remander of ths secton Numercal effects of -approxmaton and mplementaton A floatng-pont number often results from a set of operatons performed on a real machne that tres to compute a gven functon F(X) wth the best possble approxmaton. The noton of calculablty was frst ntroduced by Turng [167] as follows: a real number X s "computable", f there s an algorthm that takes > 0 as nput and produces an approxmaton x of X wth x-x <. To realze that, an exact real number X s converted and rounded by an algorthm to obtan a floatng-pont number approxmaton x. The error between these two numercal values manly depends on the codng base, the number of bts used n codng (truncaton error) and the roundng error. In the followng, the computng of a gven functon result Y=F(X) s nvestgated. In such a procedure, the exact argument X frst needs to be stored n the computer memory and therefore must be converted to ts floatng-pont approxmaton x, thus ntroducng an nput error X=x-X. The floatng-pont number x s next entered n a sequence of software nstructons that are mplemented to generate the desred functon F. However, due to successve truncaton and roundng errors and perhaps naccurate algorthms, the functon really realzed by the calculator, denoted as f, may sgnfcantly dffer from F. Therefore, the floatng-pont result ӯ=f(x) calculated by the computer may tself devate from F(x). To be dsplayed or prnted, ths number s fnally rounded and formatted to the decmal approxmaton y delvered to the user (Fgure 14). The absolute computng error (output error) s then characterzed by the dfference Y between the value y gven by the machne and the exact mathematcal result Y=F(X),.e. Y=y-Y. It s dvded by the true exact value Y to defne the relatve error Y.e. Y=(y-Y)/Y. Fgure 14 shows the effect on the absolute error Y of the mplementaton of the functon F n f. Wth poor mplementaton, the absolute error Y ncreases. Ths phenomenon s called data senstvty. Encoded and rounded data x=x+x ΔX ɛ Input data: X Computed and rounded output data y= ỹ +Δ ỹ Δỹ ɛ ỹ ΔY Y Mathematcal output data Accurate mplementaton f(x)=f(x) Encoded and rounded data x=x+x ΔX ɛ Input data: X Fgure 14: Computng error Computed and rounded output data y=ӯ+δ ӯ Δ ӯ ɛ ӯ ỹ Y Mathematcal output data Poor-mplementaton f(x) F(X) In case of poor mplementaton, the absolute error Y can be dvded n three man error contrbutors (Fgure 14): - The computatonal error (Y-ỹ) where ỹ=f(x): It s only senstve to the nput error X and the computer algorthm stablty. X s caused by the truncaton or roundng of the exact real number X durng encodng as floatng-pont. - The Implementaton error (ỹ-ӯ) where ӯ=f(x): The nput error X s amplfed by errors of modellng and mplementaton. It must be ponted out that ths error s not related to measurement uncertantes, even f poor and unstable programmng may also amplfy the dspersons of the measurement data entered n a ΔY 7

9 metrology or precson engneerng software. Propagaton of uncertantes n software s a topc that has already been consdered n many research works [14,16,127,130,131]. Ths topc has also been treated n a fundamental keynote paper enttled "Measurement as nference" [52]. Measurement uncertantes are therefore not consdered sgnfcantly n ths paper. Error propagaton s a research feld ntensvely treated n laboratores of mathematcs and computer scence. Generally, the understandng of the accuracy of floatngpont programs s based on the estmaton of the condton number. It s an ndcator that permts to know the algorthm behavour when t s calculated at each program lne. Condton number wll be detaled n subsecton (Masterng of forward error). - The error (ӯ-y) generated by the converson of the computed result to the decmal number dsplayed or prnted to the user. These numercal effects wll be dscussed n detal n the next subsectons Loss of specfc propertes of arthmetc operatons Due to cancellaton, roundng, overflow and underflow effects, careful mplementaton s requred n the codng of sets of arthmetc operatons. In fact, the commutatve, assocatve and dstrbutve propertes of the operators may be lost n calculatons performed wth floatng numbers and coded wth a lmted number of bts,.e. (A+B)+C A+(B+C), (AB)C A(BC), (A+B)C AC+BC, [33,93,94] Numercal effects of cancellaton or overflow The subtracton of two nearly equal floatng-pont numbers may lead to a sgnfcant loss of relatve accuracy. Ths phenomenon s named catastrophc cancellaton. f = 21bb 2aa + 55bbbb 10aabb + a / 2b (6) wth : a = and b = To demonstrate the effect on the cancellaton of the number of bts used to encode floatng-pont numbers, the computng of Equaton 6 represents a good example [152]. f 1, ,5. 0-0,5. -1 Usual MATLAB 7.10 (Double) Dgt nb Fgure 15: Effect of the number of bts on calculaton Ths functon was computed wth floatng-pont numbers of ncreasng bt length. The result s shown n Fgure 15. As soon as the number of bts exceeds a sze that enables the dsplay of 19 decmal dgts, an accurate value s returned by the software. Below ths, the calculaton result s false. In concluson, to reduce the numercal effect of the cancellaton, the number of bts must be ncreased. Arthmetc operatons can also undergo overflow and underflow Backward and forward errors analyses Forward and backward error analyses are two paradgms used to study error analyss and data senstvty [28,32,87,175,176,181]. The backward error s defned as the estmated nput error that would lead to a gven fxed absolute error Y after computaton. Its calculaton process s shown n Fgure 16. The forward error s the absolute error Y calculated for a fxed nput error X. ΔX ˆ Mathematcal calculaton Y = F ( X ) = 1 X X = 3 Y =1/ 3 X ˆ = F 1 ( X + ΔXˆ ) ΔX = X X ΔY = y Y Backward error Forward error ΔXˆ = ΔY = x = X + ΔX y = 0.34 Computatonal y = f ( x) = 1 + f ( x) x ΔX : Inputerror calculaton f( x) :Implementaton error Fgure 16: Calculaton processes of forward and backward errors The analyss of the change n backward error n relaton to the forward error allows studyng the qualty of algorthms. Ths connecton can dffer from one geometrcal problem to another. It can be summarzed by the classfcaton of Fgure 17. Backward error Large Small Small Accurate mplementaton Accuracy nsenstve to mplementaton Stable process Forward error Large Ill-condtoned problem Reducng backward error Unstable process Fgure 17: Forward error versus backward error A large forward error of a computaton can have the followng sources: - the amplfcaton of a known nput naccuracy onto the output error. It can be characterzed by the condton number. The condton number s a mathematcal property of an algebrac functon F and s therefore ndependent of the algorthm or the computer used to evaluate an expresson. - the amplfcaton of some truncaton and roundng errors generated by the algorthm used to compute the desred functon. It depends on the number of bts used to encode the floatng-ponts and can be reduced by mplementng mult-precson calculatons. Ths effect s called: stablty or nstablty of the calculaton process. To mnmse the numercal effect of -approxmaton and mplementaton, the backward error, the condton number and the process stablty must be managed wth cauton Reducton of nput error Wth floatng-pont numbers, the nput error manly depends on the qualty of the encodng and roundng. Increasng the number of bts s therefore an obvous soluton to reduce backward errors n floatng-pont calculatons. Some modern commercal floatngpont computng software provde functons (such as the functon eps() of Matlab that computes the floatng-pont relatve accuracy) that allows the estmaton the error. Fgure 18 shows the result of the computaton of constant (p) usng Matlab software. The number was calculated wth standard double precson floatng-ponts and was dsplayed wth 15 dgts after the decmal pont. Its relatve and absolute errors were also defned. The same constant was also computed wth Excel spreadsheet. The result: was obtaned. >> p 8

10 ans = >> eps(p) ans = e-016 >> eps(p)*p ans = e-015 Fgure 18: Numercal value of constant, relatve and absolute errors n case of calculatons wth double precson floatng-pont numbers One dgt was thus lost n comparson wth Matlab. Multple precson toolboxes are also avalable n some software systems to perform calculatons wth hgh fxed arbtrary precson. Quadruple precson floatng-pont numbers, complant wth IEEE standard, are also now progressvely ntroduced n programmng tools and hardware. It wll permt computatons wth a precson of about 34 decmal dgts. Fgure 19 fnally shows the result obtaned wth 300 decmal dgt precson [90]. Such multple precson computatons can be realsed wth Maple or Mathematca software wthout any specfc lbrary. They requre however enough memory space to encode the real numbers (Fgure 8) and may lead to large computng tmes. >> mp.dgts(300) >> mp('p') ans= Fgure 19: Numercal value of constant wth 300 decmal dgts Fgure 20 shows the ncrease of the calculaton tme needed to compute the constant as a functon of the number of encodng decmal dgts. Varous C lbrares dedcated to multple-precson floatng-pont computatons wth accurate roundng have already been developed (GNU-GMP, GNU-MPFR, FLINT, MPIR). Most of them are open-source and permt calculatons wth an arbtrary multple precson. Lnks to such lbrares are avalable n [168]. Seconds 4 E-3 3 E-3 2 E-3 1 E-3 Dgt Nb Fgure 20: Calculaton tme of n multple precson The calculaton tmes must also be evaluated for computatons requrng complex numercal operatons (matrx nverson, least squares optmzatons ). The U.S. Natonal Insttute of Standards and Technology (NIST) publshes a set of statstcal reference datasets usng multple precson calculatons wth an accuracy of 500 decmals. Another soluton to reduce the nput error s to use BCD encoded numbers. However, ndustral software currently does not offer toolkts to develop n BCD. Nevertheless, the potentaltes of the actual object-orented programmng languages, such as C++, permts the overload of all the mathematcal operators and most arthmetc functons. Thus, generc algorthms developed wth classcal floatng-pont numbers can be reused. Ths wll permt evolvng towards BCD processng [21] Masterng of forward error Equaton 7 presents the demonstraton of the condton number C n the case of a sngle-varable nonlnear functon F. The relatve forward error s the result of the multplcaton of the condton number C by the relatve nput error X. ΔY y Y f ( x) F( X ) f ( X + ΔX ) F( X ) δy = = = = Y Y F( X ) F( X ) wth :f ( X + ΔX ) F( X + ΔX ) F( X ) + ΔX F'( X ) δy ΔX X X F'( X ) X F'( X ) δx C δx F( X ) F( X ) In these expressons, X s the exact nput argument of the mathematcal functon F to be calculated; F s the dervatve of F; f(x) represents the result provded by the computer; X s the absolute nput error; X s the relatve nput error; Y s the relatve forward error and C s the condton number. It must be ponted out that the approxmaton f(x+x) F(X+X) s only vald n case of accurate and stable software mplementatons. The Taylor expanson used n Equaton 7 requres also a lmted nput error X. The condton number C represents an ntrnsc property of the mathematcal problem and has nothng to do wth the computer. A computng condton number c can therefore also be defned. It takes the floatng-pont approxmaton x of X as nput argument and s based on the result f(x) really provded by the program mplemented n the calculator. Equaton 8 shows such defnton and demonstrates that the mathematcal meanng of the condton number may lead to naccurate estmatons of computng errors because t does not account for devatons caused by naccurate or unstable mplementaton. X F'( X ) x f '( x) C = c = (8) F( X ) f ( x) Equaton 9 shows the symbolc defnton of the condton number C. It s the rato between the relatve output error (change n output) and the relatve nput error (change n nput). Y C = (9) X C C=X tan(x) Fgure 21: Behavour of the condton number of the Cosne functon Estmatng the condton number C (or c) s very mportant n understandng the accuracy of floatng pont software. When C (or c) s large, the relatve nput error s amplfed and the accuracy of the computng results becomes poor. A problem characterzed by a low condton number s called well-condtoned, otherwse t s named ll-condtoned. The condton number C may greatly depend on the nput argument X. Thus, t can take large values up to nfnty and ths even for smple mathematcal functons (cosne, sne, tangent...), ths amplfes errors n scentfc computaton. As an example, Fgure 21 shows the behavour of the condton number calculated for the functon cosne. In case of a nonlnear dfferentable functon F of multple varables X, the condton number s defned by equaton 10, 2 X (7) 9

11 where J(X) represents the Jacoban matrx of F and s a gven chosen matrx norm (usually the Eucldan norm). C = X J( X ) F( X ) (10) For non-sngular systems of lnear equatons, gven n the form A.X=Y, the condton number (A) s computed as follows: 1 κ ( A ) = A A (11) Its value thus depends on the choce of norm. Wth the Eucldan norm, the condton number C s the rato of the largest to the smallest sngular value n the sngular value decomposton (SVD) of a matrx. The effects of the nput error propagaton can be mtgated by reducng the number of floatng-pont operatons (FLOPs) performed n a computaton. Matrx computaton s ntensvely used for the mplementaton of approxmaton methods n metrology and precson engneerng software. In the case of lnear equaton systems, the Gauss pvotng method s often used to obtan the soluton. In a problem wth n equatons and n unknowns, [n(n+1)]/2 dvsons and [(n(n-1)(2n+5)]/6 addtons and multplcatons are needed to obtan the result. The Gauss pvotng method should therefore only be used for lnear systems of lmted sze (less than a thousand of unknowns). For large systems (e.g. for Fnte Element calculatons) specfc algorthms are to be used. f 3,00E-13. 1,00E ,00E ,00E ,00E-13. 3, , , ,0001 Fgure 22: Stablty or nstablty of computaton Calculaton process stablty To llustrate the stablty or nstablty of the calculaton process [19], Fgure 22 presents the behavour of two mplementatons: f1 and f2 (Equaton12) of the same mathematcal functon F, when the nput argument X tends to 4. The results plotted n blue are obtaned from relaton f1, whle the red curve results from mplementaton f2. These computatons were performed usng MATLAB 7.10 software and standard double floatng-pont precson. f1= X 4-16X X - 256X f2 = ( X - 4)( X - 4)( X - 4)( X - 4) 2. X (12) The frst mplementaton f1 s affected by the roundng errors of the floatng-pont calculatons. Ths mpacts the stablty of the calculaton process. A computng process wthout subtractve cancellaton s usually stable, especally when a small number of numercal operatons s used. A few gudelnes can be found n [87]. Backward error calculatons can be used to test the stablty of the calculaton process (method) appled to solve lnear equaton systems [28,87]. If the backward error s small, t means that the result y found by the computer s close to the true soluton Y of the mathematcal problem. To mprove the stablty of a numercal process, a scalng of the data can lower the condton number. In lnear equaton systems, a nearly optmal strategy s to equlbrate the rows or columns of the assocated matrx Interval and ball arthmetc The prncple of nterval arthmetc (IA) s to encode a real value by an nterval provded by the computer. Ths nterval evolved to shapes n N dmensons. Snce the 1960s, ths topc was ntensvely studed [48, 86, 114, 120, 137]. The nput nterval x can be represented by ts lower and upper endponts (nterval arthmetc) or as a centre xc and a radus rx (ball arthmetc). The IEEE Standard for IA defnes basc IA operatons of the commonly used mathematcal nterval models (Equaton 13). IA estmates the upper and lower lmts of an output, calculated from a set of nputs bounded by ntervals. x + y = [ x, x] + [ y, y] = [ x + y, x + y] x y = [ x, x] [ y, y] = [ x y, x y] x y = [ x, x][ y, y] x y = [mn( x y, x y, x y, x y), max( x y, x y, x y, x y)] 1 1 x/ y = [ x, x], f y 0 and y 0 y y X X (13) The dependency problem that may lead to large overestmatons of computaton errors, s a major dffculty n the applcaton of IA. Very early, the wrappng effect of nterval arthmetc was brought to the forefront. Ths effect s well ntroduced n the presentaton of the one-dmensonal problem detaled n [177]. 2 y = a. x + b. x + c wth : a = 0.001, b = 300andc = Poor mplementaton Robust mplementaton 2 b b 4. a. c 2 root1 = b + b 4. a. c 2. a root1 = Sgn(b). 2. a 2 b + b 4. a. c root2 = c root2 = 2. a a.root1 Relatve forward error Poor mplementaton ntval root1 = 1.0e+005 * < , > ntval root2 = 1.0e-007 * < , > -4.26e-04 Robuste mplementaton ntval root1 = 1.0e+005 * < , > ntval root2 = 1.0e-007 * < , > -3e-14 Fgure 23: Self-valdatng mplementaton wth ball arthmetc, appled to roots of a sngle-varable quadratc equaton To reduce ts mpact, coordnate transformatons can be used. In N-dmenson domans, new shapes for nterval boundares were also chosen as polytopes or ellpsods [169]. For dynamc problems, Chebyshev or Interval Newton methods can be appled to solve nonlnear functons wth ntervals. Ball arthmetc seems to partally solve the over-estmaton of computaton errors. Many lbrares for nterval arthmetc (GNU Octave) or ball arthmetc (Mathemagx [169]) have already been developed. Matlab lbrary INTLAB [153] also proposes tools to perform IA calculatons. In addton, arthmetc ntervals are also handled by Computer algebra systems, such as Mathematca or Maple. To llustrate the use of nterval arthmetc for the self-valdaton of an algorthm, Fgure 23 presents the results of the calculaton of the roots of a sngle-varable quadratc equaton. 10

12 2.2.4 Sub concluson In scentfc computaton, the lmts of the use of floatng-pont numbers were ntensvely studed durng the last ffty years (wth actvty ntensfed durng the run up to the year 2000 due to antcpated problems wth mllennum bugs ). Computatonal errors and algorthm nstabltes are lnked to truncaton and roundng errors, generated by the encodng of the handled numbers nto bnary data of lmted sze, naccuraces of the mplemented mathematcal basc functons, cancellaton effects and overflows or underflows. The frst way to avod these phenomena s to ncrease the number of bts used n the converson of exact real numbers nto floatng-ponts. In fact, computng precson s closely lnked to the number of bts assgned to store the floatng-pont sgnfcant. In addton, the range of numbers that can be encoded and handled s related to the number of bts assgned to the exponent. Quadruple precson calculatons conformng to IEEE wll soon be avalable for software developers and wll provde outputs wth 34 decmals. Many mult-precson lbrares are now also avalable to engneers or researchers n precson engneerng or metrology. However, the ncrease of the number of dgts mproves the computng accuracy at the expense of computaton tme. Therefore, only the routnes that perform ntensve scentfc calculatons are generally programmed wth mult-precson lbrares. But a careful handlng of nputs and outputs s requred to avod roundng and truncaton errors generated by data conversons between program modules of dfferent types. The second way to avod computatonal errors and algorthm nstabltes s to perform scentfc calculatons wth decmal numbers. Ths s realzed n pocket calculators and some supercomputers that have dedcated hardware, but not n laptops or desktops. Software solutons are avalable based on the BCD codng. However, these solutons currently reman reserved for IT developers who mplement ther own codes. The qualty of scentfc calculatons s lnked to the qualty of the software mplementaton [5]. The adjectve "well" or "ll" condtoned refers to the algebrac expresson of a gven functon F. On the other hand, the adjectve "stable or unstable" refers to the algorthm and the numercal results assocated wth a machne. When the algebrac expresson s well condtoned, n prncple one can always fnd a stable process to evaluate t. When the algebrac expresson s very poorly condtoned, t s dffcult to fnd a stable process to evaluate t. Combnng an mproperly condtoned algebrac expresson wth an unstable process s generally a recpe to obtan poor result. In exact arthmetc calculatons performed wth computer algebra systems (Mathematca, Maple, ), only ratonal numbers are mplemented, thus lmtng the nstabltes of algorthms. In floatng-pont calculatons, a numercal certfcaton of results can be realzed by usng nterval or ball arthmetc. In ths secton, the ntrnsc performances of computer hardware and software were only hghlghted. The numercal result provded by a metrology or precson engneerng software also depends on the qualty of the model descrbng the physcal problem and on ts mplementaton. Ths s the subject of Secton Modellng and mplementaton The physcal problems faced by researchers or engneers n precson engneerng or metrology manly deal wth the quantfcaton of measures (scalar quanttes) (e.g. parameters of geometrcal models) used to descrbe geometrcal features of the measured object, to calbrate machne-tools, to compensate measurng devces, etc. In ths secton, the propertes of the mathematcal models used to descrbe the physcal problem are dscussed. The choces made n modellng have a sgnfcant mpact on the qualty of the result. Numercal mplementaton of the mathematcal approach smultaneously requres a sutable defnton of the nomnal geometrc model, the devatons from the nomnal features, and the solvng method. All ths modellng should be realzed at the same level of qualty. The global performance of the process wll n fact be mposed by the software component of lowest qualty. On the other hand, approprate choces may mprove the qualty of numercal results even f calculatons are based on a lmted number of dgts. NOMINAL DEVIATION SOLVING 3D model z x z Δ M A V Δ = z z Iteratve algorthm y Equaton root A: Nomnal pont V: Nomnal unt vector M: Measured pont : Devaton z d: Plumb lne dstance M dstance d = AM V x z x z V XY y A XZ Egenvalues Egenvectors 2x2D models A XY V XY Numercal soluton: Solver, y Covarance matrx Var Cov Covxz x xy Cov Var Cov yz xy y Cov Cov Var z xz yz SVD, Analytcal soluton Fgure 24: Nomnal model, devatons, solvng methods mplemented to brng to the fore ther mpacts on the calculaton accuracy To show the effect of modellng, dfferent least squares optmzaton algorthms were mplemented n a spreadsheet applcaton (Mcrosoft Excel) to approxmate a straght lne based on a set of acqured coordnates. Reference data provded by the Natonal Metrology Insttute of Germany (PTB) was used for that purpose. Ths reference data set ncludes 8 ponts. It s well known, n least squares optmzaton, that the barycentre of the measured coordnates les on the approxmatng lne. The problem thus comes down to the determnaton of the three components of the unt vector defnng the lne drecton. These vector components Vest were evaluated by several solvng methods and compared to the results VPTB certfed by PTB. The error of each calculaton process was thus defned by the norm of the dfference of the two vectors (Equaton 14). All these calculatons were performed wth 64-bt floatng pont numbers. Error = V V est PTB wth : V PTB (14) Fgure 24 detals the dfferent nomnal models, the defntons of devatons (.e. the dstances between nomnal and actual ponts) and the solvng methods that were mplemented. The resultng errors are summarzed n Table 4. The choce of the nomnal model, the defnton of devatons (dstances) or devaton functons and solvng methods have a great nfluence on the qualty of the obtaned result. In the case of a 3D lne, calculatng the egenvectors of the coordnate covarance matrx or ts Sngular Value Decomposton (SVD) leads to the best precson. These algorthms are also the optmzed solvng solutons to be used n the case of a plane. Choce 8 (3D nomnal model, 3D devaton functon, descrpton of the lne unt vector by 2 ndependent angles) also gves results very close to the certfed values. But n ths case, the solver ntegrated n the spreadsheet applcaton s appled, workng as a black box. Ths does not allow a fne tunng of the optmzaton process. The three tems: Nomnal model, devaton functon, and solvng method wll be further detaled n ths secton. 11

13 Table 4: Comparson of 9 computaton processes Choce Nomnal Devaton Resoluton Error 1 2x2D Pt/lne n Y and Z Solver 2,54E x2D Pt/lne n Y and Z Analytc 2,15E D Dstance Pt/lne Solver (6 dependant parameters) 1,66E x2D Pt/lne n Y and Z Solver and reducng para. 1,52E x2D Dstance Pt/lnes Analytc 1,05E x2D Egenvalue/vector Analytc 1,05E D Dstance Pt/Lne Solver (3 dependant parameters) 9,34E D Dstance Pt/Lne Solver (2 ndependant parameters) 7,46E D Egenvalue/vector Analytc 7,22E Gudelnes to a smart mplementaton of a nomnal model Dfferent general prncples exst to gude researchers or engneers n modellng physcal problems. One such basc rule s Occam's Razor (OR), attrbuted to an Englsh Francscan frar, Wllam of Ockham ( ). It s also called the Law of Parsmony (LP) and may be formulated n Latn as follows: Pluraltas non est ponenda sne necesstate (enttes should not be multpled unnecessarly) [164]. In scence, Occam's razor s used as a heurstc to gude scentsts n developng theoretcal models [150, 159]. In precson engneerng or metrology, ths powerful rule leads researchers or engneers to use a lmted number of parameters to explan a physcal phenomenon. An addtonal consequence of the applcaton of ths prncple s that t permts defnng the mnmum number of parameters requred to characterze a model. Ths allows usng varables that are statstcally ndependent and thus smplfes uncertanty evaluaton and propagaton. vx V (15) = vywth V = 1 3 parameters and1constrant equaton vz var vx cov( vx, vy) cov( vx, vz) VAR ( p ) = cov( vx, vy) var vy cov(vy, vz) cov( vx, vz) cov( vy, vz) var vz Sn θ.cos var V = Sn θ.sn 2 ndependent parameters VAR ( p ) = θ 0 Cos 0 varθ Equaton 15 detals the parametrsaton of the unt drecton vector for a straght lne, correspondng to choces 7 and 8 of Table 4. Table 4 already hghlghted that the choce of two ndependent parameters (two angles) gves a better estmaton of the 3D lne drecton vector than a modellng by three dependent components. Another aspect to pont out s the orthogonalty of the coordnate bass that enables the characterzaton of a devaton. Fgure 25 shows the calculaton of a dstance d, defned by the two components p1 and p2 of a vector n a 2D plane. It llustrates the effect of non-orthogonalty of the coordnate bass on the descrpton of the same devaton. p2 Devaton: d=4,472 d=ǁa1 p1+a2 p2ǁ a2 p 2 Orthogonal p1, p2 a1=4 a2=2 p 1, p 2 a 1= Non orthogonal a 2= p1=p 1 a 1 a1 Fgure 25:Orthogonal or non orthogonal parameters descrbng a devaton d In Fgure 25, the data plotted n green are descrbed by an orthogonal coordnate system. It s non-orthogonal for the blue tems. When the angle α approaches 90, the coordnates a 1 and a 2 tend to nfnte. In classcal model of an aspherc shape [59], potental numercal nstabltes nduced by such effect led the authors to propose a new mathematcal defnton usng an orthogonal bass of the parameters. The geometrc characterzaton of surfaces, parts or products s, generally, based on the measurement of two-pont dstances, or angles between tems (Fgure 26). Ths data s scalar. In the 1970 s, the development of CMM s enabled capturng the coordnates of a pont n a reference frame. These coordnates are dstances acqured n three orthogonal drectons. A Cartesan approach of geometry s then used n modellng. Snce the structures of real devces are not perfect, geometrc models of CMMs or CNC machne structures were substantally mproved, gvng rse to the currently appled calbraton and error compensaton methods for three-dmensonal measurng or manufacturng systems. The nomnal mathematcal models of calbraton and measurement processes are characterzed by a set of parameters (dstances, angles and ntrnsc parameters). The frst type of nomnal models ams thus to dentfy the geometrc errors of machne structures and to compensate for these defects afterwards. The goal of the second knd of nomnal models s to characterze the geometry of a surface area or an entre measurng object durng an nspecton process. Some parameters of a nomnal model defne the poston and orentaton of a geometrc entty wth respect to other geometrc elements or a reference coordnate system derved from dfferent features. Nomnal parameters p dstances, angles or ntrnsc parameters wth =1 to n Scalar measures Example Example angle Calbraton model Each captured pont needs: CMM: 3 dstances 1 measured pont Laser Tracker: 1 dstance + 2 angles 1 measured pont Measurng arm: 3 angles 1 measured pont Laser Tracer: (Pos.Pts) dstances Pts measured ponts Wth: Pts=(4.Pos-6)/(Pos-3) Pos: Nb of Laser Tracer postons Machne Tool: 3 dstances 1 measured pont 3 dstances + 1 angle or 3 dstances + 2 angles 1 measured pont and orentaton X Z p3 π/2 π/2 p1 π/2 Fgure 26: Lnks between measures and models Metrologcal model Measured set of ponts p6 Measured ponts p4 p2 p7 p5 Y Localzaton and orentaton of geometrcal features Other ntrnsc parameters (dmensons, angles, curvatures, etc) defne the shape of the geometrcal elements. A mnmum of 6 parameters s requred to locate and orent a geometrcal tem n a 3D space (3 translatons and 3 rotatons). The mathematcal models that descrbe the rotaton of a geometrcal entty are generally based on Euler matrx transformatons, Roll-Ptch-Yaw matrces (or the smplfed lnear representatons: Small Screw Dsplacement) or Rodrgues' rotatons. Euler s angles can descrbe transformatons wth large angles, but they degenerate for small rotatons. Roll-Ptch-Yaw representatons are well adapted to small rotatons, but they cause problems for angles close to π/2. However, these two transformatons use the mnmum number of parameters (.e. three) requred to defne any 3D rotaton. Rodrgues rotaton has no angular lmtaton, but t requres an addtonal parameter. Ths representaton s not mnmal. The nomnal geometrc models for the calbraton and error compensaton of CMMs, CNC machne tools or other measurng devces were descrbed n many papers [6,20,27,43, 50,99,155,171] and dfferent CIRP Keynotes [65,156]. These 12

14 models wll not be dscussed here, but the general rules mentoned above can be appled to them, too. n C M n M M th C M th M C Mth n M th M th C M M n V n V n,j M M th,,j,k n,j,k M th,,j M Drect Contnuous Model Dscrete Model STL STEP Sgned Dstance Explct Implct Explct Fgure 27:Model typology Level of complexty Data processng The nomnal mathematcal models, used n metrology to descrbe surface areas or geometrcal features, can be subdvded nto two complementary forms: contnuous models and dscrete models (Fgure 27). Contnuous models are used n the descrpton of basc surfaces (spheres, cylnders, aspheres, B- splne surfaces, gear flanks...). C Model M th Parametrc model X ( u, v) = f( u, v) = X + R Cos ( u) Sn( v) Y( u, v) = g( u, v) = Y + R Sn( u) Sn( v) Z( u, v) = h( u, v) = Z + R Cos( v) f( u, v) g( u u, v) = u u,t( h(, v) u T ( u) v) Normal equaton: n or N X M d t n or N Am: Sgned dstance calculaton = d Z Sphere Y f( u, v) g( u v, v) = T h( u v ( u) T( v) t ( u) =, t (v) = T( u) T( v), v) v N = T( u) T( v), n = t ( u) t ( M th M Solve ( CM N )( CM N ) 0 CM = M M Implct model F(, Y, Z) = ( X - X )² + ( Y -Y )² + ( Z - Z )² - R² = 0 X Normal equaton: n or N v) th f x f N N =, n = y N f z = Fgure 28: Plumb lne dstance calculaton based on a parametrc and mplct model The mathematcal defnton for the surface of such a contnuous model can be expressed n two ways: - By an mplct equaton: the 3D coordnates (X, Y, Z) of all ponts on the nomnal surface are gven by an equaton n the form F (X, Y, Z) = 0. As an example, the mplct equaton of a sphere s presented n Fgure 28. Ths equaton defnes all those ponts, whose coordnates X, Y and Z fulfl the equaton. They are located on a sphere around the center (X0,Y0,Z0) wth the radus R. - By a parametrc equaton: the coordnates of all ponts on the surface are explctly wrtten as functons of two surface parameters u and v,.e. X(u, v), Y(u, v), Z(u, v). The parametrc equaton of a sphere s also shown n Fgure 28. The choce between these two models s generally made when choosng the devaton. Generally, t s the nomnal model gvng the smplest metrc equaton that wll be selected to lmt potental numercal dscrepances. The next paragraph, dedcated to devatons, wll provde further detal regardng ths aspect. When the topology becomes more complex (e.g. a car body door n Fgure 29), free form surfaces, free form shaped parts or full 3D masters [129, 130,154] are splt nto a set of elementary surfaces that can stll be descrbed by mplct or parametrc equatons (set of planes n STL fles, set of basc surfaces and B-splnes n STEP fles). Such models are named dscrete models. The accuracy of a full 3D master used n metrology s determned by the qualty of the process used to translate the CAD model nto a data exchange fle (STL: CAD models n stereo-lthography or sold freeform fabrcaton technologes, IGES: Intal Graphcs Exchange, ASME Y14.26M [102], VDAFS: Verband der Automoblndustre- Flächenschnttstelle or automotve ndustry assocaton surface data nterface [149], STEP: Standard for the Exchange of Product model data, ISO [13]). The natve model mplemented n a CAD system s the representaton used the most n dscrete models, snce t does not requre any translaton and thus leads to the best accuracy. The qualty of dscrete models greatly depends on the condtons of contnuty of the elementary surfaces: C0 (pont contnuty), C1 (slope contnuty) and C2 (curvature contnuty). The STL format transforms the CAD model nto a set of planes, delmted by three trangle vertces and ts normal. It does therefore not satsfy the contnuum n slope (C1) and curvature (C2). The IGES format descrbes a volumetrc geometrc element by a set of parametrc tles or basc surfaces. The STEP neutral fle presents the latest technologcal advances n the volume descrpton of complex or smple features. Complex surfaces are descrbed by a set of B-Splnes. The degree of these parametrc surfaces can guarantee the geometrcal contnuty n C0, C1 and C2. However, geometrc dscontnutes can stll be observed wth such a surface exchange format, dependng on the qualty of the translaton module. Fgure 29: Full 3D master of a car body door, gven n blue, and measured pont devatons 3.2 Devaton calculaton ISO part 1 [15,104] defnes the basc operatons avalable to verfy a dmensonal or geometrc specfcaton: partton, extracton, fltraton, collecton, assocaton and constructon. The surface model s defned n the partton operaton. Secton 2 summarzed some precautons to take n order to obtan a smart modellng of the studed metrologcal problem. In the assocaton operaton, a devaton quantty s requred to approxmate the measured coordnates to the nomnal model. d proj (Projected dstance) Z nomnal surface C n c M /proj M th n Mth M d Y d eucldean (Plumb lne dstance) X (Eucldean dstance) Fgure 30: Dstance defnton In order to handle all devatons n the same way,.e. to gve all measured ponts the same weght n approxmaton routnes, a 13

15 general functon vald for all captured ponts s requred called the devaton functon. Dependng on the nspecton task and geometrc restrctons, three types of devaton functons are commonly appled: the Eucldean dstance(cm norm), the projected dstance (CM/proj norm), and the plumb lne dstance (MthM norm). The two last dstances are Sgned Dstances (SD), calculated between a measured pont M and the nomnal geometrc element (Fgures 28 and 30). Ths nomnal geometry can ether be represented by a nomnal pont Mth or C correspondng to M and the unt normal vector n on the nomnal surface n the envronment of M [70], or by the mplct or parametrc nomnal surface (or a real sub-patch of t). The latter gves the plumb lne dstance, defned by the smallest-possble dstance magntude between M and the nomnal surface. Ths dstance vector crosses the nomnal surface perpendcularly,.e. ts drecton s gven by grad F(X,Y,Z) n mplct model defnton [67] or partal dervatves n parametrc model (Fgure 28). Its dstance value s defned postve when the pont M s located outsde the materal or when the dot product between the vector MMth and the normal vector n on the modelled surface or curve s postve, equal to zero when the measured pont les on the surface or curve (dot product equal to zero) and negatve otherwse (negatve dot product). By conventon, the normal vector on a surface s orented to the outsde of the materal. Usng one of the 3 devaton functon types defned before, other devaton functons (or measures of dstance) can be defned, but they must satsfy some condtons. For example, n an nspecton process for assessng the form or orentaton of a geometrc feature, complant wth the ISO 1101 standard, the devaton used to realze the assocaton are no longer drectly gven by sgned dstances, but ether by the dfference between ts maxmum and mnmum devaton or two tmes the maxmum dstance (mnmum zone crteron) [139]. The law of conservaton s a sutable gude for the choce of the devaton quantty. Several types of conservaton prncple are known n engneerng scence: conservaton of mass, and conservaton of energy, for example. In physcs, a conservaton law states that a measurable property of a system remans constant whle the system s state mght change. Ths defnton can be easly appled to the feld of precson engneerng and metrology. For the three types of devaton functons explaned before (see also Fgures 27,28 and 30), t s obvous that sgned dstances and egenvalues/vectors are ndependent of any change n the reference frame, whereas the undrectonal dstance depends on the selected drecton of computaton. Sgned dstances and egenvalues respect the law of conservaton and defne dmensonal quanttes that allow the locaton of the 3D lne n the space, ndependent of the reference system. If the devaton functon s based on a dstance (Fgures 27 and 30), ts computng leads to three basc confguratons: pont-topont dstances or Eucldean dstances (calbraton of Machnetools or Coordnate Measurng Machnes, Iteratve Closed Pont (ICP) algorthms, etc), pont-to-curve dstances (approxmaton of crcles or lnes n metrology, toolpath optmzaton n manufacturng, etc) and pont-to-surface dstances (approxmaton of basc or complex surfaces n metrology, control of geometrcal specfcatons wth full 3D masters, etc). The latter two confguratons could be based on projected dstances or plumb lne dstances (Fgure 30). In many cases, the topology of the geometrc element explctly provdes a sense for the normal vector. Ths s the case for the standard geometrc elements crcle, lne, plane, cylnder, sphere, cone and torus. Dscrete models usng STL format or meshed surfaces can be added to ths class. The sgned dstance s obtaned usng pont-pont, pont-lne and pont-plane dstance formulae. These cases are n the Explct boxes of Fgure 27. For ths frst class of standard geometrc elements the calculaton of devatons does not present any dffculty. In the case of other contnuous models (parabolod, ellpsod, asphere ) and dscrete models usng parametrc surfaces (B-splne, Bezer, Coons ), the calculaton of the devaton becomes much more dffcult, because then t s necessary to defne the mnmum dstance between each measured pont M and the approxmatng surface. Ths s generally acheved by the determnaton of plumb lne dstance. If the normal vector n the envronment of M s known, the projected dstance s a reasonable estmaton for the plumb lne dstance. Analytcally, the plumb lne dstance s gven by the orthogonal projecton Mth of pont M onto the surface. Fgure 28 summarzes the computatonal process that enables the determnaton of ths projecton and therefore the computng of the sgned dstance. These two subclasses are merged n the Implct box of Fgure 27. ( N ) = ( CM N )( CM N ) = 0 CM 0 (16) Equaton 16 shows an mplct expresson that can be used to determne the orthogonal projecton Mth n Fgure 28. The degree of ths equaton depends on the model used. Ths degree can quckly ncrease whch requres numercal teratve methods such as Gauss Newton or Levenberg Marquardt algorthms for ts soluton [138]. However, t must be ponted out that Equaton 16 may lead to more than one soluton f the normal vector lne N or n ntersects multple surface ponts. To obtan the plumb lne dstance, t s then necessary to select the result wth the smallest dstance. Best-ft crteron Least square Mnmum zone Mnmum crcumscrbed or Maxmum nscrbed Norm L2-Norm or Gaussan Norm T-Norm T-Norm PDF Prob(=d ) d k :SD number Equaton k d = 1 max( d ) mn( d ) ² max( d ) R MC MI fx fx Example = R + max( d ) R = R max( d ) Fgure 31: Objectve functons or approxmaton crterons In the case of least squares approxmaton there s no need for some geometrc elements such as planes or lnes to compute dstances between the measured coordnates and the theoretcal nomnal tem. In fact, t can be shown that the barycentre of the measured ponts les on the approxmatng element. Moreover, the vector whch completes the characterzaton of the approxmated feature (normal to the plane, drecton vector of the lne) can be deduced from the covarance matrx of the measured coordnates and corresponds to the egenvector of lowest egenvalue. It s ths method, usng an SVD factorsaton, that obtaned the best result n the test carred out n Fgure 24. Ths algorthm thus avods the teratons of the classcal methods and therefore prevents successve roundng and cancellaton errors of the computer. Ths case s labelled Drect n Fgure 27. Fgure 27 hghlghts two major dffcultes encountered n computng the devatons: the level of complexty n the calculaton of the dstances that greatly ncreases when mplct equatons are to be solved, and the volume of data to be processed. For explct models, no real numercal dffculty exsts. However, for 3D full masters, the number of devatons to be 14

16 processed ncreases, whch leads to the management of large data fles (bg data). 3.3 Soluton methods In prevous sub-sectons, the nomnal model and ts parameters were chosen. The types of devaton functons and the dstances between the measured ponts and the model were also defned. The sgns and magntudes of all dstances depend on ther relatve poston and orentaton wth respect to the poston and orentaton of the approxmatng geometrc element (whch s to be determned) and on the geometrc features characterzng the element lke cylnder radus or cone angle (whch are also to be determned. Ths means that startng from a gven cloud of captured measurng ponts, there s an nfnte number of possble approaches for the parameters of the geometrc element that represent, to a greater or lesser extent, a good approxmaton. To fnd the best-possble approxmaton, a crteron s requred to dfferentate a good from a better soluton of the approxmaton problem. These crtera are called objectve functons. Several types of objectve functons are used n producton metrology (see Secton 3.3.1), prmarly determned by the nspecton task and the defnton of tolerances. Mathematcally, an objectve functon s a functonal,.e. a mappng from a vector space (more specfcally: a space of functons) nto the space of real numbers. Ths objectve functon assgns each possble approxmaton soluton for the geometrc element to one correspondng scalar value. In other words: the objectve functon creates a rankng lst among the possble solutons, where the best-possble approxmaton can be determned unambguously by the mnmum scalar value of the functonal [67,68,69] Types of objectve functons The followng wll explan the types of objectve functons, predomnantly appled n producton metrology. The selecton of the approxmaton crteron s related to the solvng method (numercal or analytcal solvng, teratve computaton or root calculaton) to be used to fnd the optmal parameters of the model. Snce metrology and precson engneerng software are manly mplemented wth floatng pont numbers, ths subsecton wll focus on the numercal behavour of the crteron or the optmzaton method. The approxmaton methods used the most n metrology and precson engneerng are least squares optmzaton, mnmum zone evaluaton, and calculaton of mnmum crcumscrbed or maxmum nscrbed feature. They are all approxmated accordng to an objectve functon, whch s a norm of the devatons d between the measured ponts and the geometrcal element to be determned. Ths norm s called Lpnorm, wrtten as Lp Norm = k 1/ p p d = 1 wth p = 1to (17) where p s the degree of the norm, rangng between 1 and nfnty. And k s the number of measured ponts. Two specal cases of ths norm are manly used as approxmaton crteron (Fgure 31): the L2-Norm wth p = 2 [60,157] that leads to least squares optmzaton (LSQ), and the nfnte norm (L - Norm), where p tends to nfnte, also called Tschebyscheff-Norm (T-Norm) [9,69, 71,158,178]. As shown n [71], calculatons of mnmum crcumscrbed or maxmum nscrbed features can also be realzed by approxmatons accordng to the T-norm. The Probablty Densty Functon (PDF) assocated wth each approxmaton crteron s shown n Fgure 31 [10]. LSQ corresponds to the maxmum lkelhood estmaton for Gaussan nose. The L1-Norm wth p = 1 may also be used n specfc cases. An nfnte norm would requre the calculaton of the functonal n Equaton 17 for a degree p tendng to nfnty, but that cannot be acheved numercally. The case of p = nfnty corresponds to a mnmax problem, mnmsng the maxmum resdual dstance. It can be mplemented n a comparably smple way by selectng for p a value between 50 and 100 whch provdes a good estmaton of the T-norm. Hgher degree values (e.g. p=300 to 500) can mprove ths estmaton, but requre more decmal dgts and thus more calculaton tme [71]. [67] and [70] suggest an upper and lower bound for the T-norm, both based on the Lp-norm wth a fnte p. a) b) W = K = 1 d ² W = 0 p2 p2 Best soluton W = K = 1 d Best soluton? p1 p1 W = 0 p1 Fgure 32: Numercal effect of an ncomplete set of ponts: a) ponts dstrbuted over a large angular range b) ponts dstrbuted over a comparable small angular range [31] From the pont of vew of computatonal convenence, a desred property of an approxmaton crteron s to provde an objectve functon wth only one sngle mnmum. Stuatons of nonunqueness of the mnmum are however reported by [158] n case of approxmatons accordng to the T-Norm (form nspecton or maxmum-nscrbed crteron). In fact, t s possble to construct examples of approxmatng a plane accordng to the T- norm that have a number of local solutons that s almost the same as the number of data ponts. As shown Fgure 32a, the L2- norm (least squares optmzaton) should present, theoretcally, only one sngle mnmum. Ths s the case, n practce, when the acqured coordnates are dstrbuted over a large angular range of closed geometrc features (crcles, cylnders, cones, spheres, etc) or on a wde lateral extent of the measured surface. When the extent of the measured ponts s reduced, numercal problems are amplfed due to the presence of local mnma. These local mnma are added by successve roundng and cancellaton errors of floatng pont operatons (see Fgure 32b). It can lead to a poor parameter estmaton of the approxmatng element. In the case of a crcle, the orgn of these numercal problems s the cancellaton of hgh degree terms of the polynomal approxmaton used by the computer to calculate square roots [31]. Ths phenomenon of dgtal degeneracy can be observed for any type of surface. Ths shows the mportance of the choce of the ntal parameters (also called startng soluton) requred for an teratve numercal process. In the case of exstng local mnma, the algorthm wll converge to the nearest local mnmum and therefore not necessarly fnd the global optmum parameters. The use of floatng pont numbers n computer codes leads to ths phenomenon. Usng mult-precson lbrares or computer algebra systems (Mathematca, Maple, ) wll lmt the cancellaton effects. Factors such as the choce of the optmzaton crteron (objectve functon), the dstrbuton of the ponts measured on the geometrc element, the mathematcal model, etc, wll have an mpact on the success of a computaton method appled to fnd the approxmatng parameters. In followng subsecton, the behavour of the computaton methods wll be studed. ² p2 15

17 3.3.2 Calculaton methods In metrology or precson engneerng software, two types of computaton methods are used: - Numercal and/or teratve computaton methods, - Symbolc computaton methods. The frst one s strongly nfluenced by the use of floatng pont numbers and ts lmtatons, whereas the second one, n theory, s not nfluenced by them Numercal and/or Iteratve computaton The mathematcal problems met n the feld of precson engneerng or metrology generally correspond to the optmzaton of objectve functons wth specfc characterstcs. Fgure 33 detals these characterstcs,.e.: what s the number of estmated parameters; s the objectve functon lnear, quadratc or nonlnear, wth or wthout constrants? Are specalzed mathematcal methods or algorthms used? The objectve functon may be determnstc or stochastc and may or may not requre the calculaton of dervatves. As wrtten n Secton 2.2.4, the precson to whch a numercally stable algorthm can solve an ll-condtoned problem s lmted by the accuracy of the data. However, a numercally unstable algorthm can produce bad solutons even for well-condtoned problems. Ths means that an unstable algorthm can yeld solutons that are less precse than theoretcally achevable from the gven data [11,66,162]. Number of parameters Constraned or unconstraned objectve functon Global or local method Mathematcal method choce to derve optmal result Lnear, quadratc, Determnstc Methods wth nonlnear objectve or stochastc dervatve functon method calculaton or not Fgure 33: Characterstcs of mathematcal problem To avod these problems, numercal computaton must respect three basc rules: - the nverse problem used n parameter model approxmaton must be well condtoned, - the appled algorthms must be numercally stable n order to acheve results wth a gven fnte arthmetc precson, - the software requres a careful mplementaton of the algorthms. The stablty of the optmzaton method wth respect to roundng-off errors s a fundamental characterstc to obtan accurate numercal results. Intal approxmaton Convergence Parameters of optmzaton method Stablty Stoppng crteron Fgure 34: Parameters of optmzaton method Fgure 34 summarzes the parameters whch nfluence the precson of the obtaned result: stablty, ntal approxmaton, convergence, and stoppng crteron. Newton's method s the bass for many optmzaton routnes or root search programs (Fgure 35). Optmzaton algorthms, generally, requre computng dervatves of the frst (Gradent or Jacoban) and often second order of the functon (Hessan). The propertes of the dfferent mathematcal methods (advantages and nconvenences) are summarzed n Fgure 35. The solvng of the equatons used n least-squares methods, can be performed by specfc calculatons (Cholesky, QR factorsaton [73]). Usng the QR factorsaton of the Jacoban matrx, for example, s more numercally stable than fndng the Cholesky factorsaton of the product of the transposed Jacoban matrx wth tself. The condton number of the product s the square of the condton number of the Jacoban matrx; there wll also be a loss of precson smply by formng the product. The use of a sngular value decomposton of the Jacoban matrx s also numercally stable [73]. Adv Inv Steepest descent methods (SDM) Optmal, fxed or varable steps Conjugate gradents methods (lnear prob.) [12,57,58,85] Conjugate gradent method s smpler to code and requres less storage space Local method Successve determnaton of search drectons and step lengths Precondtonng requred Adv Inv Least squares method (LSM) Newton's method (NM) Newton -Raphson method Lnear: Moore-Penrose Non lnear: Gauss Newton Levenberg Marquardt (LM) [49,124,134, 136,138,161] Gauss Newton (GN) Normal equatons are used Levenberg Marquardt Far from the soluton, t reacts lke a SDM, close as GN. More stable than GN For some very regular functons LM can converge slghtly slower than GN Quas Newton's method (QNM) BFGS method L-BFGS method [17-18] [49,126,136,144] If the cost functon s quadratc, the global mnmum s reached n 1 teraton Newton method s convergence s quadratc Hessan nverse s computed at each teraton Wth many parameters, calculaton are long and expensve n storage Rsk of dvergence Approxmaton of the Hessan whch s locally re-estmated at each teraton L-BFGS method s able to handle large memory problems Less nformaton about cost functon form BFGS s the best Quas -Newton methods Intellgence orented algo. Genetc algorthms Swarm colony optmzaton Bees algorthm Partcle swarm [97,173,179,180] Smple algorthm Good flexblty Search for mnmum or maxmum overall faclty for functons wth mn or max local Method does not guarantee the true extreme dscovery Fgure 35: Propertes of mathematcal methods Quas-Newton methods attempt to buld an approxmaton of the Hessan matrx (or ts nverse) that ncorporates second order nformaton by ncorporatng frst order nformaton as the optmsaton proceeds. The Broyden Fletcher Goldfarb Shanno Algorthm (BFGS) [49,126,136,144] s one of the most famous quas-newton algorthms for unconstraned optmzaton. Movng away from determnstc algorthms, ntellgence-orented algorthms (Genetc algorthms, Swarm algorthms) [97,173,179,180] wth ther smplcty are another way to search the soluton of extreme problems wth many local mnma. An optmzaton toolbox has been mplemented n Matlab software for solvng complex optmzaton problems. It automatcally selects the most effcent algorthm for the computed mathematcal problem. Matlab uses several algorthms dependng on the type of problem to be solved: nteror reflectve Newton 16

18 method, trust-regon-dogleg, trust-regon-reflectve, Levenberg- Marquardt, smplex, BFGS, MnMax, and so on Symbolc computaton In Secton , numercal methods were nvestgated performng calculatons related to problems of precson engneerng and metrology. As an alternatve, symbolc calculaton s offered today to researchers and engneers [39]. Iteratons Numercal process Step1 R1= numercal ntermedatecalculaton Step R =Functon (R -1 ) Fnal step Model Numercal data Symbolc computaton wth Computer Algebra System (CAS) Numercal computaton Numercal result Fgure 36: Numercal or symbolc computaton Fgure 36 llustrates the dfference between these two manners of handlng a computatonal problem. In numercal methods, all the data s handled n a numercal form, usually as floatng-pont numbers, extendng from the begnnng of the computatonal process to ts end. Roundng and cancellaton errors may thus be generated at any ntermedate calculaton. Careful mplementaton of each step of the computatonal process s therefore requred to obtan a correct result. Symbolc calculaton, on the contrary, gves a formal soluton of a mathematcal problem. The numercal applcaton s therefore performed at the end of the calculaton process, whch reduces the roundng and cancellaton effects that arse wth floatngpont calculatons. To avod numercal degeneracy, floatng-pont numbers are not permtted n symbolc calculus. Decmal numbers are thus treated as ratonals (rato of two ntegers). Polynome degree Methods Computaton 1rd Classcal method Symbolc 2th Classcal method Symbolc 3th Grolamo Cardano method Symbolc 4th Lodovco de Ferrar method Symbolc Abel Theorem & Galos Theorem 5th Sturm's theorems Numercal 6th Sturm's theorems Numercal Sturm's theorems Numercal Fgure 37: Root research of unvarate polynomal Symbolc calculaton s based on exact calculatons and equatons ncludng parameters or numbers n arbtrary precson. Unfortunately, all mathematcal problems cannot be processed n symbolc computaton. The dfferentaton or ntegraton of functons, the manpulaton of polynomals, vectors or matrces (lnear equatons) are treated n symbolc calculaton. The resoluton of polynomal systems and systems of nonlnear multvarate equatons [22,23,24,54] are also avalable n symbolc computaton. Formal calculaton solutons are offered by commercal computer algebra systems (Maple, Mathematca, ) and open source software (GAP, Maxma, SAGE, ). The calculaton of the devatons (dstances) between the nomnal model and the measured ponts often requres fndng the roots of a polynomal. For example, n the case of a parabolod, a 5 th degree polynomal equaton has to be solved. For an ellpsod, the equaton s of degree 6. Smlarly, for the approxmaton of a plane usng the SVD method, t s necessary to determne the egenvalues. They result from the roots of a 3 rd order characterstc polynomal. The computaton of roots of unvarate polynomals s thus one core problem to be solved n metrology and precson engneerng. Any non-constant real polynomal can be factored as a product of rreducble real polynomals of degrees 1 or 2. y θ1? θ2? x x= Reflector Fgure 38: Robot calbraton θ3? Laser Ths theorem does not provde any explct decomposton algorthm. It only predcts what the fnal form of the result should be. Consequently, ths rases the problem to () ascertan the exstence of real roots and (), f they exst, to evaluate them wth a certfed precson. The drect method to prove the exstence of roots s to formally exhbt them (when possble). One can fnd an explct formula - usng radcals - for each root of a polynomal of 1 st, 2 nd, 3 rd and 4 th degree. But from the 5th degree on, there s an nsurmountable dffculty. The work of N.H. Abel and E. Galos [63] has hghlghted, n fact, that the roots of polynomals of a degree greater than 4 cannot, n the general case, be expressed wth radcals (Fgure 37). It s therefore mpossble to obtan an explct formula determnng the roots and, consequently, t s necessary to mplement numercal methods wth all of ther wellknown weaknesses. Sturm's sequence or Sturm's theorems can however be appled to ths problem to defne the number of roots exstng n a gven real range [160]. Dchotomc search algorthms can thus be used to fnd the roots wth the desred precson. Notatons: Sn(θ)=s, Cos(θ)=c wth =1 to 3 Maple symbolc calculatons: wth(groebner); P[1]:=-392*c2*s1-(9475/100)*c2*c3*s1-392*c1*s2-(9475/100)*c1*c3*s2- (9475/100)*c1*c2*s3+(9475/100)*s2*s3*s1-425*s1-778; P[2]:=-(9475/100)*c1*s2*s3-(9475/100)*c3*s1*s2- (9475/100)*c2*s1*s3+392*c2*c1+(9475/100)*c2*c3*c1+425*c1-392*s1*s2; P[3]:=c1*c2*c3-c1*c2*s3-c1*c3*s2-c1*s2*s3-c2*c3*s1-c2*s1*s3-c3*s1*s2 +s1*s2*s3; P[4]:=-c2*c3*s1-c1*c3*s2+c3*s1*s2-c1*c2*s3+c2*s1*s3+c1*s2*s3- c1*c2*c3+s1*s2*s3-sqrt(2); P[5]:=c1^2+s1^2-1; P[6]:=c2^2+s2^2-1; P[7]:=c3^2+s3^2-1; Eqs:= [P[1], P[2], P[3], P[4], P[5], P[6], P[7]]; Base_Eq:= Bass(Eqs, plex(c1, c2, c3, s1, s2, s3)); Groebner polynomal bass: EQ1: *sqrt(2) *s3^2+( *sqrt(2) )*s3 EQ2: *sqrt(2) ( *sqrt(2) )*s *s2 EQ3: *sqrt(2) ( *sqrt(2) )*s *s1 EQ4: *sqrt(2) ( *sqrt(2) )*s *c3 EQ5: *sqrt(2) *c2 EQ6: *sqrt(2) ( *sqrt(2) )*s *c1 Fgure 39: Symbolc calculaton for the example shown n Fgure 38 17