echniques fo Robotic Foce Senso Calibation D. Baun Institute fo Pocess Contol and Robotics Kalsuhe Institute fo echnolog (KI) Kalsuhe, Gean e-ail: daniel.baun@kit.edu H. Wön Institute fo Pocess Contol and Robotics Kalsuhe Institute fo echnolog (KI) Kalsuhe, Gean e-ail: woen@kit.edu Abstact Calibation of foce toque sensos with high pecision is a tie consuing and theeb costl pocess. Futheoe the ecalibation of these devices akes a disassebl of the sste, whee the senso is used, necessa. In this pape ethods and a set-up fo foce toque sensos in obotics is descibed, whee the calibation can be accoplished with inial effot and high pecision. he adaptation of a calibation algoith and an outlook to futhe ipoveent in the ange of intelligent sensos is also povided in this epot.. Intoduction Multidiensional foce and toque sensos ae a inceasingl coon coponent in oden obot sstes. he aea of application fo such sensos is widespead and eaches fo sensos that ae attached to the flange of the obot to senso sstes copletel ebedded in the obot anipulato. In classic anufactuing obots in industial pocesses the ostl pla a ole as a senso in pecision assebl opeations. Hee the obot sste not onl has to eet high positional pecision, but the successful copletion of the opeation also deands fo a specified foce to be applied duing the opeation. In this case the foce and / o toque sensos ae ostl pat of the obotic tool o ounted between the obot flange and the tool. Most of the sensos used do not have full 6 degees of feedo (DoF), but ae tailoed to the application. Anothe application of foce toque sensos is the aea of light weight and high pecision obots. Hee the sensos ae ebedded into the obot stuctue and the joints of the obot. he pecision of these obots does not onl coe fo thei stiffness as in classic obot anipulato design, but it is a esult of an intelligent contolle and senso setup. he intenal foces of the obot, which ae caused b the ass of the anipulato pats as well as etenal foces applied to the anipulato ae detected b the senso sste and the contolle can theeb aintain the oveall pecision of the anipulato. In this wa it becoes possible to constuct obot anipulatos, which ae even capable of caing thei own weight. Yet anothe ae of application fo foce toque sensos in obotics is the field of sevice obotics and huan obot inteaction. Hee it is not onl necessa to detect an collision between the obot and the huan inteaction patne, but it is also ve ipotant to etact ve pecise senso eading, when collaboating with a huan patne in a shaed task. In these applications 6DoF foce toque sensos with high pecision ae found. he standad and widel used sensos pinciple is the stain gauge senso, in which the foces on the senso ae tansfoed into a phsical length vaiation of the senso bod. hese sall length vaiations can then be easued b esistive stain gauges. Moe senso pinciples like capacitive length easueent with pieoceaics o optical length easueent sstes eist in special applications. he coon poble in all the foce easueent sstes is to povide a coect calibation. Even if the sstes ae anufactued with the highest pecision thee is a substantial deviation in the chaacteistics of the specific unit fo all othe units and each one has to be calibated fo itself. heefoe pecise and efficient stategies have to be used fo the solution of this poble.. Standad calibation appoach he basic question in eve calibation pocedue fo foce toque sensos is the souce of a efeence foce o efeence toque. In the ajoit of calibation appoaches a pecision weight o set of pecision weights is used to poduce a efeence foce unde the pull of the gavitational foce. he efeence toque can be established b using a known leve and the efeence foce. Fo special applications it a be necessa to find the actual value of the gavitational constant at the location of the calibation equipent. Poceedings of the 3 th intenational wokshop on copute science and infoation technologies CSI, Gaisch-Patenkichen Gean, echniques fo Robotic Foce Senso Calibation 8
.. Senso easueent equation and calibation poble he constuction of ost foce toque senso is caied out in such a wa, that the easueent equation becoes a set of linea equations and can be witten down as with the vecto of the econstucted applied loads, the vecto of the easueent signals and the calibation ati C. Basicall the senso tansfos the vecto of applied loads (foces and toques) to a signal vecto which can subsequentl be used to econstuct the applied loads (fig. ). Fig.. Scheatic View of the Senso and Calibation Basicall a valid econstuction of the calibation ati fo an n-diensional senso would equie n easueents. Howeve this is not possible in a eal application due to easueent noise, eaining nonlineaities of the senso and fo eliinating ssteatic eos duing the easueent pocess. heefoe it is desiable to collect a lage aount of (, ) pais fo the calculation of an accuate calibation ati. On the othe hand it is a costl pocess to suppl pecise efeence loads, because the senso sste has to be aligned with high pecision. Futheoe the easueent has to be caied out with geat cae, because eve ssteatic eo duing the easueent pocedue leads to eoneous esults in the calibation ati. Afte the input data collection the standad pocedue fo the calculation of the calibation ati is a ati invesion in the least squaes sense. Fo this, the easueent equation () is tansposed and subsequentl the load and signal vectos ae epanded to atices of all n data pais. C n C n Fo this equation the calibation ati can be easil coputed b using a pseudo-invese of the ati of easueent values. C ( Z Z ) Z M C ( Z M ) Anothe easueent effect is the bias in the easueent values. he bias is assued to be a constant () (3) offset of the easueent values towads the ideal values. In the easueent equation the bias can be denoted as an additional constant to the signal vecto: ( ) C'. ' C (4) hee ae now diffeent was of eliinating the bias fo the easueent data and stoing it fo late use in senso C usage. () A ve siple ethod would be easuing the tae values the senso befoe beginning the calibation. Howeve this is not an optial appoach, because the sensos own weight o soe attachents fo load application duing calibation cannot easil be eliinated fo the senso eadings this wa. he would late intefee with the acquisition of load/signal pais. A bette wa to deteine the bias eading is the application of loads duing calibation, which have a ean value of ove all easueent points. Afte the collection of input data, the ean value ove all signal vectos can be coputed and the esult is diectl the bias vecto which can then be subtacted fo all signal values befoe enteing the subsequent steps. When using the least squaes solution fo the calculation of the calibation ati thee is et a oe convenient wa to deteine the bias, even fo easueent data collections without an oveall ean value of eo: at fist the etended easueent equation () can be ewitten to C M S Z. (5) with the shape ati S as the pseudo-invese of the tansposed calibation ati. With equation (4) eve eleent j of the easueent vecto gets an additional bias coponent j which can be ebedded into the shape ati as well. Fo this pupose the ati of applied loads M is etended b a ow of constant ones and the bias vecto is added as an additional ow to the shape ati: n S n (6) With soe new abbeviations equation (6) now be solved fo the etended shape ati S' M' S' Z' ( M' M' ) M' Z'. S' (7) hus the calculation of the calibation ati is alteed to the following steps: fist the etended shape ati S' is coputed with (7), in the second step, the bias vecto is etacted and the shape ati S can be (pseudo-)inveted and tansposed to finall get the calibation ati C. With these steps the calculation of both, the calibation ati and the bias vecto fo a given senso is possible Wokshop on copute science and infoation technologies CSI, Gaisch-Patenkichen Gean, 9
fo a given set of signals and applied loads. Basicall thee ae now diffeent was to gathe this data. he two ethods descibed in the following ae anual application of efeence foces and toques with a igid calibation fitue and efeence weights and the autoated data acquisition using a obot and a econfiguable calibation bod... Data acquisition b efeence weights he basic idea of the data acquisition appoach with efeence weights is to appl weights to a specific leve connected to the senso and to easue the signal values which coespond to the known applied foces and toques. A calibation fitue connected to the senso helps to educe the ovehead caused b the need to appl vaious loads to the senso. he scheatic view of an eaple fo such a calibation fitue is shown in (fig. ). Fig. 3. Foce and toque of a calibation bod unde gavitational pull With a suitable pan/tilt unit o a obot, a vaiet of foces and toques can be applied to a senso with onl one well defined load just b positioning the load and the senso in diffeent angles to the gavitational pull (fig. 3). he foce toque sensos sees the foce F and the toque in its coodinate sste. he foce F itself is onl the effect of the igid bod ass unde the gavitational pull (vecto g in the base coodinate sste) otated b the alignent of the senso coodinate sste to the base coodinate sste. F R, senso g. (8) Fig.. Calibation fitue fo attaching efeence weights his pat is connected to one side of the foce toque senso, the othe side is connected to a tilt and pan unit, which can be positioned a few fied positions. With these echanical pats and a set of 5 efeence weights good setical dataset fo the calibation of a 6DoF foce toque senso can be acquied. Fo eaple 6 diffeent signal/load pais ae easued with onl 5 epositioning opeations of the senso. he alignent of the senso is hee ve siple, as it onl points into the diections of the coodinate aes. his is also the data acquisition pocedue, which is coonl found in the calibation potocols of high pecision iniatue sensos, fo eaple sensos fo AI echnologies. Although this pocedue esults in suitable data with good pecision, the calibation pocess is tie consuing and not easil autoatable. Futheoe the acquisition of oe data leads to a popotionall oe effot..3. Autoated data acquisition he answe to the copleit pobles discussed in the section above can be patl given b a odified appoach fo data acquisition. he following discussion concentates on a 6DoF senso, all points ae also valid fo sensos with less diensions, in these cases soe of the consideations ae easie. he toque is also easil defined b the coss poduct of F and the location of the cente of gavit of the calibation bod: F. (9) With the foces and toques cobined to the easueent vecto and the coesponding signal values, the acquisition of a lot of data points can be accoplished easil b just changing the alignent of the senso and the calculation of the data with the alignent and calibation bod infoation. Howeve thee is a poble with 6DoF sensos, because thee ae dependencies between all foces and toques when using onl one calibation bod. hese dependencies lead to a deficient ank in the load and signal atices, which in tun can then not be inveted fo the calculation of the calibation ati. he poble of the deficient ank can be solved b using oe than one calibation bod o one econfiguable calibation bod. In the eaple given in (fig. 4) it is possible to attach ods to diffeent sides of the cental cube of the calibation bod. he geoetical design (cf.[]) akes it eas to calculate the pecise cente of gavit of the assebled bod and on the othe hand, the assebl of vaious configuations can caied out in ve shot tie. With the shown level of autoation the calibation pocess can be uch less tie consuing, because the duation fo easuing one data point is onl the aount of tie necessa fo stopping the oveent fo one point to anothe. On the othe hand a geat aount of data can be acquied with ve little inteaction, which akes echniques fo Robotic Foce Senso Calibation
a ve pecise calibation affodable. he echanical actuation can be eithe caied out b a standad obot o pan tilt unit with appopiate angula easueent pecision. Yet thee is still oo fo ipoveent, because the whole calibation pocedue needs high pecision oientation easueent in eve step. Data acquisition duing slow oveent o without pope oientation easueent leads to low pecision o fault calibation esults. 3. Shape fo otion fo calibation he shape fo otion appoach sued up hee has been descibed in [] and subsequent aticles, the notation used hee goes along with the notation in this publication. he basic idea of the shape fo otion appoach fo the calibation of foce toque sensos is the obsevation, that too uch high pecision data is used duing the LS pocedue descibed above. It should be possible to etact infoation about the behaviou of the senso just b looking at data gatheed in a calibation easueent situation with soe basic known facts, such as the echanical popeties of a calibation bod. 3.. Calibation algoith he calibation stateg stats again at (5) with the (n ) load ati M, the (n p) signal ati Z and the shape ati S. he signal ati is now analsed with singula value decoposition (SVD) [] Z U Σ V, () which poduces the othogonal atices: U(n n), V (p p) and the eo-padded diagonal ati (n p) with the singula values in descending ode on the diagonal. With the ank S of the easueent ati this can be ewitten to Z U Σ V, () whee U * consists of the fist S coluns of U, *is the diagonal ati of the S fist singula values and V * consists of the fist S ows of V. Because Z * is the best pojection of Z into an S -space, it is possible to cobine (5) and () into Z U Σ V M S, () fo which estiates fo M and S can be ade (acc. []): M U S / ( Σ ) / ( Σ ) V (3) hese estiates ae not now the eal otion (load) and shape atices, but the ae indeteinate b an indefinite tansfoation A: M S M A AS. (4) So the task at hand is to find an invetible ati A( ) which fulfils M M A S A S (5) he fist equation in (5) is not solved diectl, but it can be solved b the knowledge about the easueent situation as entioned at the beginning of this section. he esulting shape ati is not now the eal shape ati, but it is abitail scaled and otated. heefoe a sall nube of easueents with known loads has to be intoduced. Fo this loads a scaling and otation ati can be obtained, which can then be used fo the coputation of the shape ati. he est of the calibation sipl consists in inveting A and calculating the shape ati S which can in tun be inveted and tansposed to obtain the calibation ati C S ( SS ) ( S ) (6) he so intoduced shape fo otion appoach fo the calibation of foce toque sensos futhe siplifies the pocess of the data acquisition unde the cost of inceasing the coputational ovehead. As the data acquisition pocess has to be caied out fo eve single unit and this pocess is ve tie consuing, the cost of coputational ovehead is negligible. Afte the initial adjustent of the algoith to a class of sensos and the calibation bod (degees of feedo and geoetical data) thee ae no changes o additional decisions needed. he algoith can be just fed with an nube of data points and eveals the calibation ati without futhe inteaction. On aveage pesonal coputes the tie needed fo the calculations is in the ange of a few seconds. 3.. Calibation of a 6DoF senso he data acquisition fo the calibation with the shape fo otion algoith uses a calibation bod as descibed above fo the autoated data acquisition pocess. Hence the cobined foce toque vecto of the loads becoes with (8, 9) [ F ] F (7) Unfotunatel this eans, that the esulting load ati M fo (5) will onl have ank 3, whee ank 6 would be Wokshop on copute science and infoation technologies CSI, Gaisch-Patenkichen Gean,
desiable to obtain the shape ati. When using new notations (cf. []) fo the final shape ati S and the full load ati M, which oiginates fo (7), the following equation can be witten down. MS M S MS (8) Fo (8) it becoes clea, that the oent a becoes ebedded into the shape ati S, which can be coputed b the shape fo otion algoith: S [ IX ]S (9) o eove the oent a in ode to find the desied final shape ati it is necessa to ca out thee data acquisition tials, because it is not possible to sipl invet the ati [I X]. Each of these tials is caied out constucting a diffeent oent a b econfiguation of the calibation bod. he ank fo one tial in () is S 3 and it is theeb possible to copute M in (3). o now deteine a coect tansfoation ati A (5) it is necessa to appl knowledge about the foces in M. A single vecto in M is in this special tial case is i [ f f f ] () and this ake it possible to see one siple constaint, which can be used fo finding an appopiate tansfoation: the length of all these vectos is the sae. his can also be bought futhe to the assuption, that the length is, because scaling and oienting the shapes has to be done in the following steps anwa. heefoe it is possible to state i i! A () which leads to a sste of n equations that can be solved in steps (the following lines use a ij fo the eleents of A - and ij fo the eleents of M ): i ( a a a ) ( a a a ) ( a a a ) 3 i i i i i3 i3 i ( aa aa a3a3 ) ( aa3 aa3 a3a33 ) ( a a a a a a ) 3 3! 3 3 3 i3 33 3 3 33 () he fist step is to solve fo the 6 backeted tes in the least squaes sense [4]. In the second step a nueical solution is coputed fo those epessions [5]. he eaining degees of feedo can be used to give the ati A a good shape (tiangula o setical). Afte finding the solution a solution fo the shape ati S is found b using (5). his shape is then scaled and oiented with soe pecise easueents with known loads and signals. Afte evaluating all thee tials and coputing thei shape atices, the final shape ati fo all 6 diensions has to be found. he solution fo this poble can be found looking at (9) and the solutions of the 3 tials S, S and S 3. B cobining all oent a ebedding atices and shapes an equation fo the etaction of the final shape ati is evealed: S S S 3 And (3) can be efoulated to S [ IX] [ IX ] [ IX ] 3 S [ IX] S [ IX ] [ ] S IX S 3 3 (3) (4) in ode to get to the final shape ati and theeb being able to copute the calibation ati. In sua the steps to adjust the calibation algoith fo a given tpe of 6 diension foce toque senso consist of the selection of a pope calibation bod, so that the cobined ati in (4) is invetible and choosing an a nueical solve fo () in ode to obtain the tansfoation ati A. All othe steps of the shape fo otion algoith eain the sae as the ae fo eve othe senso. 4. Ipleentation into a obot sste and evaluation of the calibation h calibation sste fo foce toque sensos which has been constucted fo the developent of hadwae and softwae stuctues has the following coponents: a KR-3 indust obot (KUKA Robote GbH), a senso with 3 foces 3 toques FD ini45 (AI Industial Autoation) connected to a PCI-ADC cad and a standad PC with Matlab installed. hee ae soe special featues in this sste, which should be entioned hee: the obot is diectl contolled b the PC using a eal-tie capable etension of Matlab and the obot softwae coponent KUKA. RobotSensoInteface which akes it possible to diectl influence the obot contolle. he second pat is the odula softwae stuctue which is heavil based on a object oiented odel of sensos and the vaious calibation algoiths. his stuctue akes it possible to echange the vaious pats easil and it also allows the integation of othe sensos with diffeent intefaces o the evaluation of algoiths and coputation odules. echniques fo Robotic Foce Senso Calibation
he odula stuctue akes is also possible to scale the eseach platfo down to a specific application. he following epeients have been caied out, using a econfiguable aluiniu calibation bod with detachable ods (fig. 4). Fig. 5. Recoveed Foce Vectos on a Sphee In anothe epeient, the length of the applied foce vecto has been tested. As the load is not changed duing one tial, all ecoveed foce vectos should lie on a sphee (fig. 5). his test also showed ve good esults and was bette than the esults obtained anufactues calibation ati. Fig. 4. Robot with ounted Calibation Bod he ods have been used to geneate vaious locations of the cente of gavit. he echanical popeties of the calibation bod ae: weight (g) diensions () base 65 677 od 4 35 As an eaple the 6 diension foce toque senso has been calibated with the calibation bod. As a data base 7 easueents pe tial have been ade in equall spaced diections of the esulting foce vecto. Due to secuit concens in the obot lab the speed of the obot was deceased, the easueents fo one tial took about 5 inutes. Counting one additional tial fo evaluation puposes and the tie needed fo ounting and econfiguing the calibation bod, the total tie fo the calibation is still below 3 inutes, which is an enoous speed up in copaison to the anual calibation pocedue. With a defined ated load of the senso of ±5N in each foce diension and ±N in the toque diensions, the achievable pecision easued duing evaluation was bette than.7% of the end value fo the foces and below.3% fo the toques. Unfotunatel a false eading in the sensos -Ais led to copaabl high eos in this ais aound %. hese values ae actuall bette than the values that can be obtained with the anufactue supplied calibation values. Howeve the diect copaison with those values suffes of the fact that the supplie calibation has been ade fo a uch boade ange of values and is theeb not this accuate fo a sall egion of the senso capacit. 6. Conclusion and futhe wok In the woks pesented vaious appoaches fo the calibation of foce toque sensos have been investigated and thei featues have been evaluated and discussed. he ost poising appoach to the solution of the calibation poble in obotic foce and toque sensos is the shape fo otion algoith adapted fo []. he futue ai is futhe investigation in a ethod to calibate foce toque sensos duing thei use and without disassebling the application. Anothe ve inteesting topic is the tansfe of the coplete calibation pocedue awa fo a pesonal copute diectl into the senso. So called intelligent sensos with copletel integated data pocessing capabilities ae alead available on the aket. Unlike a sste with an etenal inteface to the obot these sensos cannot diectl gathe infoation on the actual position o oientation. heefoe it is necessa fo an intelligent calibation to use as less as possible pecise easueents as possible. Refeences. Richad M.Voles, J., J. Daniel Moow, Padeep K. Khosla. Shape fo Motion Appoach to Rapid and Pecise Foce/oque Senso Calibation. In: Poc. of the ASME Dnaic Sstes and Contol Division, DSC-Vol. 57, San Fancisco, CA, 995.. Klea V. C., Laub A. J. he Singula Value Decoposition: Its Coputation and Soe Applications. In: IEEE ans. on Autoatic Contol. 98. Vol. 5, N. P. 64 76. 3. Fosthe G. E., Malcol M., Mole C. B. Copute Methods fo Matheatical Coputations, Pentice Hall, Englewood Cliffs, NJ., 977 4. D. Maquadt: An Algoith fo Least-Squaes Estiation of Nonlinea Paaetes, SIAM J. Appl. Math., 43 44, 963. Wokshop on copute science and infoation technologies CSI, Gaisch-Patenkichen Gean, 3