ON THE ACCURACY, REPEATABILITY, AND DEGREE OF INFLUENCE OF KINEMATICS PARAMETERS FOR INDUSTRIAL ROBOTS

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1 Internatonal Journal of Modellng and Smulaton, Vol. 22, No. 3, 2002 ON THE ACCURACY, REPEATABILITY, AND DEGREE OF INFLUENCE OF KINEMATICS PARAMETERS FOR INDUSTRIAL ROBOTS P.S. Shakolas, K.L. Conrad, and T.C. Yh Abstract In ths artcle, we dscuss ndustral robot characterstcs of accuracy and repeatablty. The factors that affect these characterstcs are dentfed, and an error tree s developed. Subsequently, the accuracy and repeatablty are nvestgated utlzng the Denavt-Hartenberg knematcs parameters, the homogeneous transformaton matrx, and the dfferental transformaton matrx theory, and correspondng measures are developed. The formulaton ndcates that the nfluence matrces assocated wth jont varables are constant. A new measure called degree of nfluence s establshed that qualtatvely assesses the relatve contrbuton of each knematc parameter varaton to the accuracy and repeatablty of rgd manpulators. The developed formulaton provdes for easy evaluaton of the degree of nfluence measures for rgd manpulators n ether numercal or symbolc form. A numercal example s ncluded n whch the degree of nfluence of the knematcs parameters for an artculated manpulator, PUMA 560, are evaluated and analysed. Key Words Robotc error, accuracy, repeatablty, degree of nfluence 1. Introducton Robotc applcatons have been expandng snce the ntroducton of robots. These applcatons stll nclude the tradtonal manufacturng, but n recent years applcatons n other felds, such as the medcal communty, have been ncreasng. These new applcatons, such as medcal surgery, requre robots that are both accurate and repeatable. Also, manufacturng applcatons requrements are changng n an effort to address qualty control ssues, thus pushng the envelope of robot capabltes. Therefore, better robots are requred. But what makes a better robot? Ths s a very ambguous queston that depends on the applcaton. At the same tme, can we utlze new technology that wll allow one to mprove the Mechancal and Aerospace Engneerng Department, Automaton and Robotcs Research Insttute, Unversty of Texas at Arlngton, Arlngton, TX 76019, USA; e-mal: shakolas@ uta.edu Recommended by Dr. I.N. Tansel (paper no ) 1 performance characterstcs of exstng robots as t relates to the postonng accuracy and repeatablty? A technologcal barrer n the robotcs ndustry has been the reducton or elmnaton of the error between the tool frame and the goal frame, as shown n Fg. 1. The sources of ths error were readly dentfed as beng due to manpulator modellng dfferences and to hardware fxturng. The major contrbuton to the error between the robot base and the tool frame s attrbuted to modellng dfferences between the controller and the robot. Inaccurate fxturng and manufacturng processes account for the dfferences between the base frame and the goal frame. The defnton of the tool and goal frames s depcted n Fg. 1 [1]. Solutons such as buldng a better robot, buldng more rgd and repeatable fxtures, and mprovng manufacturng processes that could help n mprovng ths problem are often not feasble due to the requred or unavalable resources. Compensaton for ths error through an n-process feedback mechansm s a much more attractve alternatve. As n any control system, the process parameters wll defne the level of requred sophstcaton. If the requrement s to mprove upon the resoluton or absolute accuracy of the robot, then a precse metrology system s needed to perform these measurements. The dentfed parameters relatng to robot postonal performance are accuracy, repeatablty, and resoluton. Camera Base frame Tool frame Pn Wrst frame Staton frame Fgure 1. Standard robot frames [1]. Goal frame

2 Each of these depends on the varous components used n constructng the robot (lnks, motors, encoders, etc.), the constructon procedure, and the capabltes of the drvng actuators and the controller. The resoluton s defned through the control system used to power the manpulator, but s also affected by the constructon procedure, manpulator stffness, (structural flexblty), encoders, and so on. Resoluton s defned as the smallest ncremental move that the robot can physcally produce. Repeatablty s a measure of the ablty of the robot to move back to the same poston and orentaton over and over agan. Accuracy s defned as the ablty of the robot to precsely move to a desred poston n 3-D space. These concepts are shown graphcally n Fg. 2. Poor Accuracy Poor Repeatablty Poor Accuracy Good Repeatablty Good Accuracy Poor Repeatablty Good Accuracy Good Repeatablty Fgure 2. Accuracy versus repeatablty. Absolute accuracy and repeatablty. Absolute accuracy and repeatablty descrbe the ablty of a robot to move to a desred locaton wthout any devaton. Dynamc accuracy and repeatablty descrbe the ablty of a robot to follow a desred trajectory wth lttle or no varance. Addtonally, n all robotc applcatons zero overshoot s a necessty to avod dsastrous collsons wth other parts n the work-cell. Therefore, controller desgn ntroduces another dmenson to the problem, as we would lke to maxmze the stffness and bandwdth whle mnmzng the response tme. Ideally, both the absolute and dynamc accuracy and repeatablty could be mnmzed to the attanable resoluton. The largest effect on the robot accuracy s attrbuted to the robot lnks. Manufacturng or machnng of these lnks nevtably ntroduces some varaton n ther dmensons from one robot to the next, as well as some varaton n the orentaton of the jonts. The manufacturng varatons are attrbuted to the defned machnng tolerances. The dfferences between the physcal jont zero poston reported by the robot controller and the actual physcal jont zero poston usually has the second bggest effect on the accuracy of the robot. For a standard PUMA-type artculated sx-dof robot, errors n the zero poston of the 2 wast (1), shoulder (2), and elbow (3) jonts wll have a larger effect on the robot postonal error than that of the wrst jonts (ptch 4, roll 5, and yaw 6). Although jonts 1, 2, and 3 contrbute prmarly to the poston of the tool centre pont frame, the man contrbuton for jonts 4, 5, and 6 s to the orentaton of ths frame. The jont zero poston error s often responsble for the robot postonal error on the order of 90% as dentfed n the analyss to be presented n ths study. A mathematcal model wthn each robot controller assumes that the lnks on one robot are the same length as the lnks on another robot of the same model and type. Addtonally, the same model also assumes that the relatve orentatons of the jonts on one robot are the same as on another robot of the same type. Unfortunately, ths assumpton s not true due to manufacturng and assembly varatons. Therefore, the controller wll ncorrectly estmate the robot endpont gven a set of jont angles. The next most sgnfcant factor n the robot postonal error s jont complance. Ths may be thought of as a factor representng the elastcty of each jont caused by the effects of gravty, payload, and nerta. Each of the followng robot characterstcs accuracy, repeatablty, and resoluton depends upon many factors that nclude, but are not lmted to, frcton, temperature, loadng, and manufacturng tolerances. Of the three robot characterstcs, hgh accuracy s the most dffcult to accomplsh. Dfferences between the modelled, as-desgned components and the actual, as-bult components wll affect the accuracy. The controller software can be desgned to account for dscrepances between the mathematcal controller model and the actual bult part. However, ths approach s not cost-effectve for mass-produced robots because consderable effort must be expended to dentfy the characterstcs of the varous robot components wthn the desred accuracy usng complex, expensve, and tme-consumng technques. The major error contrbutons can easly be subdvded nto structural, knematc, and dynamc. A theoretcal error tree llustratng factors that contrbute to the postonal accuracy and repeatablty of a robot s presented n Fg. 3. Ths work examnes the concepts of accuracy and repeatablty. A lterature survey on accuracy and repeatablty s presented. The accuracy and repeatablty prncples are framed wthn the context of the homogeneous and dfferental transformaton. Subsequently, an error analyss and the degree of nfluence of the DHparameters on the accuracy and repeatablty are evaluated and dscussed wthn the framework of the developed methodology. A numercal example usng publshed values for the PUMA 560 robot focused on the aforementoned concepts s presented and dscussed. 2. Lterature Survey There are many avalable solutons on the market that could be mplemented to mprove the ablty of a robot to

3 Fgure 3. Postonal accuracy and repeatablty error tree. accurately and repeatedly poston tself. These solutons range from calbraton technques that requre robot remasterng to software compensaton technques, as well as many combnatons of them, or to other process-based custom methods. For the majorty of ndustral processes, the elmnaton of all error s not a realstc goal [2 10]. However, mprovng the accuracy and repeatablty of exstng hardware to meet desred process parameters s always the prmary focus. An overvew of robot calbraton technques and the dentfcaton of three calbraton levels are ntroduced and dscussed n [11]. Level 1 s the jont level calbraton, level 2 s the entre robot knematc model calbraton, and level 3 s the nonknematc (nongeometrc) calbraton. In ths work, the lmts on mprovng robot accuracy are attrbuted to the lmts of robot repeatablty and accuracy and to the accuracy of the measurng system. The extensve use of off-lne-programmng (OLP) systems for robot programmng and path plannng and the mportance of calbraton are descrbed n [12]. In ths work, a technque for mprovng the accuracy and dentfyng the knematcs parameters of the PUMA 560 robot was presented and expermental results were evaluated. The development and use of a tumblng technque for drectdrve robots and the dentfcaton of geometrc and encoder errors are dscussed n [13]. The authors observe that after the calbraton the mproved accuracy of the three-dof robot s very close to ts repeatablty. A method focused on the use of wre potentometers attached to the robot tool s ntroduced n [14], where the authors use a PUMA 560 manpulator for expermental verfcaton of the proposed method and dentfcaton of the knematc parameters of the robot through an automated dentfcaton algorthm. The authors compared two dfferent calbraton technques, the wre potentometer and a CMM, and drew conclusons. A neural network-based technque for onlne dentfcaton of the relatve poston and orentaton of robotc manpulators s presented n [15]. In ths work, a vson system and a 3D force/torque sensor were used along wth modfcatons n the control system usng neural networks. The expermental results ndcate that ths method does not need the mathematcal model of the robot and s smpler than other calbraton procedures for dentfyng the robot 3 parameters. Robot geometrc errors and ther dentfcaton usng varous technques have been extensvely dscussed n the lterature. The effect of nongeometrc errors such as complance and thermal s dscussed n [16]. In ths work, an approach for dentfyng geometrc and nongeometrc errors s presented and expermentally verfed usng a 6-dof robot. The robot accuracy mproved by an order of magntude after calbraton. In the surveyed lterature, the focus s on the development of technques for the dentfcaton of the geometrc or nongeometrc robot parameters. Once the parameters are dentfed, technques to ncorporate them for use n the robot model to mprove accuracy were developed. However, n ths lterature the degree of nfluence of these errors on the overall robot repeatablty and accuracy was not addressed. Ths s the premse of ths work: the dentfcaton of the degree of nfluence of the knematc geometrc error parameters. 3. Accuracy and Repeatablty Analyss The frst step n attemptng to mprove the accuracy and repeatablty of ndustral robots s to examne the current state of the robotcs technology and other related technologes, such as metrology systems. It s mportant to understand that robot manufacturers whose precson clams cannot stand up to the world s most sophstcated measurement systems have no place n the ndustry. 3.1 Robotcs Technology Robot manufacturers, as an ndustry standard, publsh the repeatablty specfcatons of each robot. These specfcatons are determned by performng strngent experments n accordance wth ISO 9283 [17]. As a general rule of thumb, larger robots have larger errors n repeatablty. The publshed repeatablty values for the RX-seres robots produced by Stäubl Corporaton are tabulated n Table 1 [18]. The knematcally smallest robot, RX 60, has less than half the repeatablty error of the largest, RX 130, ndcatng that the knematc parameters of smlar model robots drectly nfluence the robot repeatablty. In most ndustral applcatons the selected robot repeatablty values are smaller than the process requrements. In those few nstances when ths s not true, other solutons must be found. A common approach has been Table 1 Repeatablty for Stäubl RX-Seres Manpulators Manpulator Model Repeatablty Value mm nch RX RX RX

4 to start from the base of the manpulator and determne f mprovements can be made n each homogeneous transformaton between lnks n order to develop a more accurate knematcs model. The homogeneous transformatons between the lnks, however, depend not only on the resoluton of the actuators for each jont, but also on the knematc parameters of the actual machned lnk. Therefore, the assembly and machnng accuracy drectly affect the accuracy of the homogeneous transformatons. There has been a wealth of research and nformaton n the lterature dscussng technques and approaches for calbraton and dentfcaton of the knematc parameters of robotc manpulators. 3.2 Knematcs Analyss The knematc structure of robots s often represented mathematcally usng a compact representaton of the poston and orentaton of each jont relatve to the prevous jont. In our work, we employed the modfed Denavt-Hartenberg notaton as presented n [1]. The Denavt-Hartenberg notaton represents a set that unquely defnes the knematc parameters of the robot. The DHparameters, a 1, α 1, d, θ, between two successve jonts are shown n Fg. 4 [1]. modfed DH-parameters s gven n (1) [1]. cos θ sn θ 0 a 1 sn θ 1 cos α 1 cos θ cos α 1 sn α 1 d sn α 1 T sn θ sn α 1 cos θ cos α 1 cos α 1 d cos α The effect of the knematcs errors between an as-desgned and as-bult manpulator on ts overall postonng wll be examned. The transformaton that expresses the tool-centrepont frame, TCP frame, frame-n, relatve to the robot base frame, frame-0, s obtaned by concatenatng the ndvdual jont homogeneous transform matrces, as n (1). r 11 r 12 r 13 tpx 0 nt 0 1T 1 2T n 1 r nt 21 r 22 r 23 tpy r 31 r 32 r 33 tpz R 3 3 P 3 1 (2) (1) R 3 3 f(θ,α 1 ),1,..., 6 s the orentaton matrx, and P 3 1 f(θ,a,d ),1,..., 6 s the poston vector. In ths work, we address only the poston of the TCP frame. The analyss for the orentaton of the TCP frame s to be dscussed n future research. The equatons for the theoretcal poston vector, tp {tpx, tpy, tpz}, provde the bass for the analyss used n ths work. Usng the knematcs values n Table 2, the theoretcal component equatons for the PUMA 560 are evaluated and presented n (3). These equatons were derved usng the symbolc capabltes of MATLAB. cos θ 1 [a 2 cos θ 2 + a 3 cos(θ 2 + θ 3 ) Fgure 4. Graphcal representaton of DH-parameters [1]. d 4 sn(θ 2 + θ 3 )] d 3 sn θ 1 tp sn θ 1 [a 2 cos θ 2 + a 3 cos(θ 2 + θ 3 ) (3) The followng notaton apples: a 1 lnk offset, dstance from Z 1 to Z measured along X 1 α 1 twst angle from Z 1 to Z measured about X 1 d dstance from X 1 to X measured along Z, jont varable for prsmatc jonts θ rotaton angle from X 1 to X measured about Z, jont varable for revolute jonts The DH-parameters are combned nto a 4 4 matrx called the homogeneous transformaton matrx, 1 T, that descrbes the orentaton and poston of frame relatve to frame 1. The homogeneous transformaton usng 4 d 4 sn(θ 2 + θ 3 )] + d 3 cos θ 1 a 3 sn(θ 2 + θ 3 ) a 2 sn θ 2 d 4 cos(θ 2 + θ 3 ) 3.3 Error Analyss The accuracy and repeatablty of a gven manpulator can be derved from (2). The theoretcal zero poston for a PUMA 560 manpulator s calculated by substtutng the values n Table 2 for the DH-parameters and values for the jont values, θ. However, the actual poston of the robot s a functon not only of the DH-parameters, but also of ther correspondng ndvdual varatons. These varatons are manly attrbuted to machnng or manufacturng tolerances or component lmtatons (encoder resoluton). They

5 wll be ndcated as a delta ( ) value for each DHparameter. Note that n ths analyss we are addressng only the effect of lnear varatons. The error analyss s performed usng the noton of dfferental transformaton [19], where the dfferental transformaton s ntroduced and used n a dfferent perspectve. An example cted s that of a camera observng the poston of the tool frame of the manpulator and calculatng the dfferental changes n poston and orentaton n order to accomplsh a desred task [19]. In ths analyss, the dfferental transformaton theory s employed to obtan an estmate of the lnear dfferental change from the theoretcal poston of the robot tool frame, the last frame of a manpulator. The lnear dfferental error n the orentaton and poston of an arbtrary jont of the manpulator, T, s a functon of all the DH-parameters and ther respectve lnear varatons as shown n (4). T T θ θ + T a a + T α α + T d d (4) The actual orentaton and poston are gven as the sum/dfference (±) between the theoretcal, T, and the dfferental change, T as shown n (5). T,actual T ± T ( T T ± θ θ + T a a + T α α + T ) d d The varaton due to each DH-parameter could be expressed as a functon of the theoretcal transform, T, multplyng an nfluence matrx, Tj, j (θ, α, a, d). Ths representaton yelds a formulaton that dentfes the structure of the varatons as functons of the DH-parameters. The procedure for analysng one DH-parameter, the rotatonal jont angle θ, and dentfyng the correspondng nfluence matrx, Tθ, s presented. Let T / θ T Tθ, then, Tθ s evaluated as: Tθ T 1 T θ sn θ cos θ 0 0 T 1 cos θ cos α 1 sn θ cos α cos θ sn α 1 sn θ sn α (5) (6) The nfluence matrx, T, for each DH-parameter s gven for completeness Tθ T d (7) cosθ Ta sn θ 0 0 sn θ d sn θ Tα 0 0 cos θ d cos θ sn θ cos θ 0 0 The actual orentaton and poston of the tool frame, T actual, for a sx-dof manpulator s evaluated usng the noton of the nfluence matrx and (5) (8). T actual (T ± T ) {T ± (T Tθ θ + T Ta a + T Tα α + T Td d)} T {I 4 ± (Tθ θ + Ta a (8) + T α α + T d d)} (9) An nterestng observaton s that Tθ,T d, the nfluence matrces for the generalzed jont varables for any manpulator, are constants, where as the other two nfluence matrces Ta,Tα (generally related to the geometrc lnk propertes) are functons of the generalzed jont varables as shown n (7) and (8). Ths s an mportant result, especally f the dfferental transformaton or varaton nformaton s examned on a frame-by-frame bass startng from the base and endng at the TCP frame of the manpulator. For example, for an all-revolute manpulator wth equal varatons on the jont varables, the actual poston and orentaton are evaluated usng (5) (9). T actual T {I 4 ± Tθ θ} T 1 θ 0 0 ± θ (10) 5

6 Usng the developed approach, the actual orentaton and poston of the TCP frame can be found n symbolc and numercal form. Numercal solutons can only be evaluated f the varaton of each DH-parameter s known or provded by the robot manufacturer. Even though robot manufacturers do not release ths type of nformaton, one can obtan an estmate of the errors and the level of nfluence or contrbuton of each DH-parameter usng reasonable manufacturng varaton (tolerances) values for machned components or resoluton for encoders. A numercal example demonstratng ths formulaton wll be presented below. 3.4 Accuracy and Repeatablty Evaluaton The bounds of the actual poston vector, ap {apx, apy, apz}, of the robot tool frame can be obtaned from the theoretcal poston vector f one assumes that each DHparameter (jont parameter) s assocated wth a varaton as shown n (10). The actual poston vector s the P 3 1 vector n T actual of (10). The poston error, e, for each axs s gven as the dfference between the actual and theoretcal postons as n (11). The total poston error, totpos_err, s defned as the norm of the maxmum ndvdual axes poston error as shown n (12), thus provdng the maxmum estmate for the total error. e ap ± tp, {x, y, z} (11) totpos_err norm(max(e x ), max(e y ), max(e z )) [max(apx ± tpx) 2 + max(apy ± tpy) 2 + max(apz ± tpz) 2 ] 1/2 (12) An estmate of the accuracy s calculated as the postonal error when varatons exst n all the DH-parameters. Repeatablty, on the other hand, s calculated by assumng that varatons exst only for the jont varables. Ths postulate s due to the fact that once a manpulator s assembled, the only dynamc (changng) components are the actuators, the jont varables. All the other knematcs parameters are dmensonally statc and do not change durng operaton. Once a lnk s manufactured, ts length wll not change durng operaton, assumng no envronmental changes such as temperature or deflectons due to loadng are present. If the dmensonally statc noton s extrapolated from the lnk to the manpulator, varatons that exsted durng manufacturng and assembly wll not affect the repeatablty as these varatons wll always be present n the assembled manpulator. It s mportant to note that structural and jont flexblty are present n manpulators, although these effects are beyond the scope of the current analyss. Ths artcle addresses accuracy and repeatablty from a pure knematcs perspectve due to lnear varatons on the DH-parameters only. The repeatablty and accuracy poston errors are defned as functons of the DH-parameters and ther varatons. The analyss that follows assumes a manpulator 6 wth all revolute jonts. Usng the dfferental change defntons n (9), the repeatablty and accuracy are evaluated as shown n (13) and (14), respectvely: repeatablty; pos_err f(α,a,d,θ, θ ) (T ± T ) accuracy; pos_err f(α, α,a, a,d, d,θ, θ ) T (I 4 ± Tθ θ) (13) (T ± T ) T {I 4 ± (Tθ θ + Ta a + Tα α + Td d)} [T {I 4 ± (Ta a + Tα α + Td d)} ± repeatablty; pos_err] (14) The accuracy s a functon of the statc (α,a,d ) and dynamc/jont varable (θ ) DH-parameters. The statc varatons ( α, a, d ) are attrbuted to machnng and assembly tolerances, where the dynamc varaton s attrbuted to the jont varable resoluton ( θ ) for all revolute jonts. The knematcs model of the robot resdng n the robot controller and used for the nverse knematcs analyss uses the nomnal values for the statc parameters n estmatng the jont dynamc value for a partcular poston. However, the varatons n the statc parameters are not accounted for n the nverse knematcs as they are not known. Even f the values of the varatons are known, they are just varatons, manufacturng or assembly tolerances, whch could be postve, negatve, or any number wthn a range from the nomnal value. For example, assumng a machnng tolerance of mm (0.005 nch) and usng the values for the PUMA 560, a ± mm. Ths means that any value n the range between ( ) mm and ( ) mm s an acceptable value for a 2. Therefore, t s mpossble for the controller to account and compensate for these varatons. On the other hand, repeatablty s easer to address and analyse. The repeatablty of a robot ndcates the robot s ablty to return to the same taught poston. The bass of ths statement s the fact that robots return to taught postons. Therefore, any varatons n the statc parameters ( α, a, d ) do not affect the repeatablty. The robot s already constructed no matter what the varatons are, and nomnal values are assgned to the statc varables (α,a,d ). When a poston s taught to the robot, the controller wll only need to remember the current state of the dynamc varables, jont varables, and s not concerned wth the values of the statc varables. The only nformaton that the robot uses to return to the taught poston s the jont varable values. Ths leads to the concluson that the only varaton that wll drectly

7 affect the repeatablty s the varaton attrbuted to the jont varables ( θ ). Table 2 DH-Parameters for PUMA 560 α 1 a 1 d θ 3.5 Degree of Influence In ths work a new measure, called the degree of nfluence, s defned as the relatve contrbuton of a DH-parameter varaton to the base accuracy of a rgd manpulator. The degree of nfluence s a qualtatve and not a quanttatve measure. Ths defnton allows for easy dentfcaton of the DH-parameter that contrbutes the most to the overall accuracy measure. For example, the accuracy due to the statc parameters s evaluated due to all DH-parameter varatons beng nonzero. Then, only one DH-parameter s set to a nonzero varaton and the accuracy measure s re-evaluated. The relatve contrbuton of ths nonzero DH-parameter to the total accuracy on a percentage bass s called the degree of nfluence. The same concept s also extended to nclude combnatons of the statc DH-parameters wth nonzero varaton values, thus provdng the degree of nfluence for these combnatons. Identfyng and understandng the degree of nfluence concept s very mportant. A robot manufacturer could possbly acheve sgnfcant mprovements on the accuracy of a robot by mprovng the tolerance (varaton) of only one of the DH-parameters. Ths qualtatve measure wll be dscussed n the numercal example, where the degreeof-nfluence analyss s performed for two cases. For the frst case a tolerance of nch and for the second case a tolerance of nch (a consderable mprovement over the frst case) was assumed on the length parameters. The rotatonal postonal tolerance remaned the same for both cases. 4. Numercal Example In ths secton a numercal example wll be presented to corroborate the theory and concepts presented n ths work. In ths example accuracy, repeatablty, and degree of nfluence measures for the PUMA 560 manpulator wll be evaluated and examned. The PUMA 560 robot was chosen prmarly because so much has been publshed about t for varous robotcs concepts such as knematcs, dynamcs, and control [1, 19]. The DH-parameters of the PUMA 560 to be used n the numercal example to demonstrate the uncertanty n accurately and repeatedly postonng the tool frame are shown n Table 2 [1, 19]. The repeatablty and accuracy for the PUMA 560 are evaluated usng zero for the jont varables, a rotatonal/angular varaton of ±0.005, a length/lnear varaton of ±0.005 nch, and the DH-parameter values n Table 2. The rotatonal varaton corresponds manly to the encoder resoluton and was chosen based on hardware n the laboratory, and the lnear varaton corresponds to machnng or manufacturng tolerance and was chosen based on standard ndustry practce. All the calculatons θ θ a 2 d 3 θ a 3 d 4 θ θ θ 6 a mm, a mm, d mm, and d mm. Ths s an all-revolute robot; therefore the vector θ, 1,...,6, represents the jont varables. presented n ths work were performed usng the software package MATLAB. If the need arses, the numercal developed m-fles could easly be modfed to generate symbolc expressons through the symbolc toolbox of MATLAB. The repeatablty and accuracy values are evaluated to be: Repeatablty ± mm ± nch Accuracy Base ± Repeatablty ( ± ) mm ( ± ) nch The degree of nfluence of each DH-parameter varaton and ther combnatons for the PUMA 560 are presented n Table 3. Note that the repeatablty of ths manpulator wll always be the same as t does not depend on the statc varables, and s thus not ncluded n ths table. In Table 3, the frst column ndcates the DHparameter or combnaton that s set to a nonzero varaton value n the analyss. The second and thrd columns represent the evaluated accuracy (mm) for the respectve nonzero varaton and the degree of nfluence as a percentage of the accuracy value obtaned when all statc varatons are gven a nonzero value. The same explanaton apples to the thrd and fourth columns. All the revolute jonts (jont varables) are set to zero, and the varaton of the angular varables ( θ, α) s set to The analyss s performed for two dfferent lnear varaton (tolerance) values n order to assess the effect of the lnear varaton change n both qualtatve and quanttatve terms. The frst varaton (results n columns 2 and 3) s defned to be mm (0.005 nch), and the second varaton (results n columns 4 and 5) s defned to be half of the frst, mm ( nch). The numercal results for accuracy and degree of nfluence are presented n Table 3. The results presented n Table 3 provde not only the accuracy expected by a robot wth the gven tolerance specfcatons, but the degree of nfluence of each and combnatons of the statc DH-parameters on the accuracy

8 Nonzero DH-Parameter Table 3 Degree of Influence of DH-Parameters for PUMA 560 Rotatonal/Angular Varaton, θ, α Lnear Varaton: d, a mm (0.005 nch) Lnear Varaton: d, a mm ( nch) Accuracy (mm) Degree of Influence Accuracy (mm) Degree of Influence d a α d, a d, α a, α d, a, α of the manpulator beng examned. The calculated entres n Table 3 depend on the values of the DH-parameters and ther varatons. The repeatablty of the robot remaned the same for both cases as t depends only on the resoluton of the jont varables. However, the calculated accuracy of the robot showed sgnfcant mprovement n the second case. The accuracy mproved by nch, from to nch, correspondng to a 45% mprovement. Ths drastc change s acheved just by defnng strnger manufacturng and assembly tolerances. Usually, qualtatve measures lke the degree of nfluence are better vsualzed n a graphcal representaton. The graphcal representaton for the degree of nfluence n Table 3 for the two lnear varatons s presented n Fg. 5. These calculatons are performed wthout accountng for any other effects, such as deflectons due to external loadng (payload) or the mass of outer lnks of the manpulator. Ths assumes that the structural and jont flexblty are zero and rgd lnks. Usng the presented concept for accuracy and repeatablty, t has been shown that n ths nstance 88% of the postonal error s due to errors n the ntalzed poston of the robot. Ths s derved from the noton that by elmnatng any error assocated wth the statc varables, the servo system can poston the robot to wthn a volume of 10% of the total volume seen when these errors are present. Although the analyss and numercal example presented were performed for an all-revolute artculated manpulator, they could be extrapolated and easly appled to other robot topologes. Fgure 5. Qualtatve representaton of degree of nfluence versus nonzero DH-parameters. The degree of nfluence of the DH-parameters and ther combnatons, though, reman smlar between the two cases, as shown numercally n Table 3 and graphcally n Fg. 5. The results presented n Table 3 ndcate that for the PUMA 560 the parameter that has the least effect on the accuracy s the lnk twst, α (angular varaton ), wth a degree of nfluence of The sngle DH-parameter wth the hghest degree of nfluence on the accuracy s the lnk offset, a (lnear varaton mm), wth a degree of nfluence of almost of 80% Concluson In ths artcle an error tree wth sources that contrbute to the accuracy and repeatablty of manpulators s dentfed. Measures of accuracy and repeatablty were derved and calculated ndcatng that hgh repeatablty s more desrable than hgh accuracy n daly applcatons wth ndustral robots. In addton, usng the noton of dfferental transformaton, an error analyss technque was developed and related to accuracy and repeatablty measures. Ths technque dentfed the lnear varatons as functons of the DH-parameters and the jont varables. A formulaton for evaluatng the accuracy and repeatablty of a seral lnk manpulator was developed employng the newly defned noton of the nfluence matrx. The nfluence matrces for jont varables are constants where for geometrc varables they are not. A new quanttatve measure called the degree of nfluence was establshed as the relatve contrbuton of a knematc parameter varaton to the accuracy when all varatons are nonzero. Ths analyss ndcated the level of accuracy mprovement that could be acheved f varatons

9 on the DH-parameters are mproved. In addton, the degree of nfluence of the DH-parameters or ther combnatons on the accuracy of the robot was evaluated usng the developed theory. Acknowledgement The authors thank the revewers for ther nsghtful and helpful comments n mprovng the orgnal manuscrpt of ths artcle. References [1] J.J. Crag, Introducton to robotcs: Mechancs and control, 2nd ed. (Readng, Massachusetts: Addson-Wesley, 1989). [2] B. Greenway, On the money: The mportance of robot accuracy s ncreasng as applcatons become more sophstcated, Robotcs World, September/October [3] J. Owens, Robot calbraton questons and answers (Robot Smulatons Lmted, 1998). [4] J.F. Qunet, Calbraton for offlne programmng purpose and ts expectatons, Industral Robot: An Internatonal Journal, 22 (3), 1995, [5] ISO/TR Manpulatng ndustral robots nformatve gude on test equpment and metrology methods of operaton for robot performance evaluaton n accordance wth ISO 9283, [6] B. Shrnzadeh, Laser nterferometry-based trackng for dynamc measurements, Industral Robot: An Internatonal Journal, 25 (1), 1998, [7] H. Janocha & D. Bernd, ICAROS: Over-all-calbraton of ndustral robots, Industral Robot: An Internatonal Journal, 22 (3), 1995, [8] F. Hdalgo & P. Brunn, Robot metrology and calbraton systems a market revew, Industral Robot: An Internatonal Journal, 25 (1), 1998, [9] M.S. Woodward, Automaton gets smart and agle, Assembly, June [10] 1998 Robotcs Industry Trends Report, Robotcs Internatonal, Socety of Manufacturng Engneers, [11] Z.S. Roth, B.W. Moorng, & B. Ravan, An overvew of robot calbraton, IEEE Journal of Robotcs and Automaton, RA-3 (5), 1987, [12] P.S. Rocadas & R.S. McMaster, A robot cell calbraton algorthm and ts use wth a 3D measurng system, Proc. of the IEEE Internatonal Symp. on Industral Electroncs, 1, 1997, SS297 SS302. [13] M. Vncze, H. Gander, & J.P. Prennnger, A model of tumblng to mprove robot accuracy, Mechansm and Machne Theory, 30 (6), 1995, [14] M. Drels & W.E. Swayze, Automated partal pose measurement system for manpulator calbraton experments, IEEE Transactons on Robotcs and Automaton, 10 (4), 1994, [15] T. Lu & G.C.J. Ln, An on-lne relatve poston and orentaton error calbraton methodology for workcell robot operatons, Robotcs and Computer Integrated Manufacturng, 13 (2), 1997, [16] C. Gong, J. Yuan, & J. N, Non-geometrc error dentfcaton and compensaton for robotc system by nverse calbraton, Internatonal Journal of Machne Tools and Manufacture, 40, 2000, [17] ISO 9283, Manpulatng ndustral robots performance crtera and related test methods, Internatonal Standards Organzaton, [18] RX Robots, Stäubl Corporaton, [19] R.P. Paul, Robot manpulators mathematcs, programmng, and control, 7th Prntng (Cambrdge, Massachusettes: MIT Press, 1986). [20] P.I. Corke & B. Armstrong-Helouvry, A search for consensus among model parameters reported for the PUMA 560 Robot, IEEE Conf. on Robotcs and Automaton, San Dego, May 1994, Bographes Dr. Shakolas receved the B.S.M.E. and M.S.M.E. degrees from the Unversty of Texas at Austn n 1986 and 1988, respectvely, and the Ph.D.M.E. degree from the Unversty of Texas at Arlngton n He joned the Mechancal and Aerospace Engneerng Department of the Unversty of Texas at Arlngton as an Assstant Professor n fall He s also a member of the Automaton and Robotcs Research Insttute (ARRI). Pror to jonng UTA, he performed sponsored research n the areas of rapdly confgured robotc and manufacturng systems. He has publshed n numerous journals on the subject of computer-aded desgn/engneerng and has presented hs work at several nternatonal conferences, where he chared sectons on robotcs and robot control. Hs research nterests are n the areas of manufacturng processes and systems automaton, robot desgn, control and calbraton, modular robotcs technology, dynamc systems, controls, and magnetc bearng modellng and control. Dr. Yh receved hs B.Sc. degree n 1981 from the Natonal Unversty of Marne Scences and Technology, Tawan, hs M.S.M.E. n 1983, and hs Ph.D.M.E. n 1988 from the Catholc Unversty of Amerca. He has 18 years of teachng and research experence n the areas of desgn and analyss of robots and mechansms, computer-aded desgn and manufacturng, and boengneerng. He has been ntegratng hs expertse nto a new fronter BEMS/NEMS (bo/nano-electromechancal systems) n recent years. He has secured funded research grants and contracts from federal agences and ndustres. He was the recpent of three teachng awards (1989 department, 1996 unversty, 1997 state), one research award (1995 unversty), one servce award (1993 unversty), and awards from other organzatons. He has authored more than 90 refereed publcatons and has experence n consultng servce. 9

10 Mr. Kevn Conrad receved hs B.S.M.E. degree n 1998 from Rce Unversty n Houston, Texas, and hs M.S.M.E. degree from The Unversty of Texas at Arlngton n He s a 2001 graduate of the Lockheed Martn Operatons Leadershp Development Program. He s currently workng as a Program Manager of Unmanned Ground Combat Vehcles as part of the Autonomous Combat Systems group at Lockheed Martn Mssles and Fre Control Dallas. Pror to jonng the Autonomous Combat Systems group at Lockheed Martn, he worked n the ManTech organzaton researchng and developng flexble automaton cells n support of the Aerospace and Defense ndustry s low rate hgh mx producton programs. In hs current poston, hs nterests nclude robotc vehcle desgn, dynamc controls, postonal accuracy, machne vson, sensor and data fuson, artfcal ntellgence, and systems ntegraton. 10