Kinematic calibration of Orthoglide-type mechanisms from observation of parallel leg motions

Transcription

1 Pashkevich A, Chablat D. et Wenger P., Kinematic calibration of Orthoglide-tpe mechanisms from observation of parallel leg motions, Mechatronics, Vol. 19(4), June 29, pp Kinematic calibration of Orthoglide-tpe mechanisms from observation of parallel leg motions Anatol Pashkevich a,b, Damien Chablat b, Philippe Wenger b a École des Mines de Nantes 4, rue Alfred-Kastler, 4437 Nantes Cede 3, France anatol.pashkevich@emn.fr b Institut de Recherche en Communications et Cbernétique de Nantes 1, rue de la Noë B.P. 6597, Nantes Cede 3, France e-mals: {Damien.Chablat, Philippe.Wenger }@irccn.ec-nantes.fr Abstract The paper proposes a new calibration method for parallel manipulators that allows efficient identification of the joint offsets using observations of the manipulator leg parallelism with respect to the base surface. The method emplos a simple and low-cost measuring sstem, which evaluates deviation of the leg location during motions that are assumed to preserve the leg parallelism for the nominal values of the manipulator parameters. Using the measured deviations, the developed algorithm estimates the joint offsets that are treated as the most essential parameters to be identified. The validit of the proposed calibration method and efficienc of the developed numerical algorithms are confirmed b eperimental results. The sensitivit of the measurement methods and the calibration accurac are also studied. Kewords: parallel robots, kinematic calibration, model identification, joint offsets, error compensation. *Corresponding author: Prof. A.Pashkevich Department of Automatics and Production Sstems École des Mines de Nantes 4, rue Alfred-Kastler BP 2722 tel.: + 33 () fa: + 33 () anatol.pashkevich@emn.fr,

2 A.Pashkevich et al. Kinematic calibration of Orthoglide-tpe mechanisms from observation of parallel leg motions 2 1. Introduction Parallel kinematic machines (PKM) are commonl claimed to offer several advantages over serial manipulators, such as high structural rigidit, better paload-to-weight ratio, high dnamic capacities and high accurac (Tlust et al., 1999; Merlet, 2; Wenger et al., 21). At present, the conventional serial kinematic structures have alread achieved their performance limits, which are bounded b high component stiffness required to support sequential joints, links and actuators (Tsai, 1999). Thus, the PKM are prudentl considered as promising alternatives to their serial counterparts that offer faster, more fleible, less costl and more accurate solutions. However, while the PKM usuall ehibit a much better repeatabilit as compared to serial mechanisms, the ma not necessaril posses a better accurac, which is limited b manufacturing/assembling errors in numerous links and passive joints (Wang and Masor, 1993; Dane, 23; Renaud et al., 26; Fassi et al., 27; Legnani et al., 27). Besides, for non-cartesian parallel architectures, some kinematic parameters (such as the encoder offsets) cannot be determined b direct measurement. These motivate intensive research on PKM calibration, which recentl attracted attention of both academic and industrial eperts. Similar to the serial manipulators (Schröer et al., 1995), the PKM calibration procedures are based on the minimiation of a parameter-dependent error function, which incorporates residuals of the kinematic equations (i.e. differences between the measured and computed values of the sensor readings). For the parallel manipulators, the inverse kinematic equations are considered computationall more efficient, since most PKMs admit a closed-form solution of their inverse kinematics (contrar to the direct kinematics, which is analticall solvable for the serial machines but is usuall unsolvable in a closed-form for the PKM) (Innocenti, 1995; Iurascu & Park, 23; Jeong et al., 24; Huang et al., 25). But the main difficult with the inverse-kinematics-based calibration is the full-pose measurement requirement (position and orientation of the end-effector), which is ver hard to implement accuratel (Thomas et al., 25). Hence, a number of studies have been directed at using the subset of the pose measurement data (Dane & Emiris, 21), which, however, creates another problem: the identifiabilit of the model parameters (Besnard & Khalil, 21). Popular approaches in the parallel robot calibration deal with one-dimensional pose errors using a double-ballbar sstem or other measuring devices (Rauf et al., 24, 26; Williams, 26) as well as imposing mechanical constraints on some elements of the manipulator (Dane, 1999). However, in spite of hpothetical simplicit (joint measurements are needed onl), it is hard to implement in practice since an accurate etra mechanism is required to impose these constraints. Additionall, such methods reduce the workspace sie and consequentl the identification efficienc (Zhuang et al., 1999).

3 A.Pashkevich et al. Kinematic calibration of Orthoglide-tpe mechanisms from observation of parallel leg motions 3 Another categor of the methods, the self- or autonomous calibration (Khalil & Besnard, 1999; Wampler et al., 1995; Zhuang, 1997; Hesselbach, 25), is implemented b minimiing the residuals between the computed and measured values of the active and/or redundant joint sensors. Adding etra sensors at the usuall unmeasured joints is ver attractive from a computational point of view, since it allows getting the data in the whole workspace and potentiall reduces impact of the measurement noise. However, onl a partial set of the parameters ma be identified in this wa since the internal sensing is unable to provide sufficient information for the robot end-effector absolute location. Besides, in practice, these methods are not alwas economicall and technologicall feasible because usuall it is hard to add these etra sensors to an eisting mechanism. More recentl, several hbrid calibration methods were proposed that utilie intrinsic properties of a particular parallel machine allowing one to etract the full set of the model parameters (or the most essential of them) from a minimum set of measurements. An innovative approach was developed b Renaud et al. (24, 25) who applied the vision-based measurement sstem for the parallel manipulators calibration from the leg observations. In this technique, the primar data (manipulator leg poses) are etracted from the image, without an strict assumptions on the leg locations or on the corresponding end-effector poses (onl leg observabilit is needed). While defining advantages of this method, the authors stress that the legs can be observed more easil than the end-effector and the use of a camera does not impl an modification of the mechanism. The onl assumption is related to the manipulator architecture (the mechanism is actuated b linear drives located on the base). However, current accurac of the camera-based measurements is not high enough et to widel appl this method in industrial environment. This paper focuses on the identification of the most essential subset of geometrical parameters (joints offsets) for the Orthoglide-tpe mechanisms. These mechanisms are actuated b linear drives located on the manipulator base and therefore admits technique of Renaud et al. (24, 25) for calibration from the leg observations. But, in contrast to the known works, our approach assumes that the leg location is observed for specific manipulator postures, when the tool-center-point moves along the Cartesian aes. For these postures and the nominal geometrical parameters, the legs are strictl parallel to the corresponding Cartesian planes. So, the deviation of the manipulator parameters influences on the leg parallelism that gives the source data for the parameter identification. The main advantage of this approach is the simplicit and low cost of the measuring sstem that can avoid using computer vision. It is composed of standard comparator indicators attached to the universal magnetic stands. It is obvious that such hardware perfectl suits industrial requirements. The remainder of the paper is organied as follows. Section 2 describes the manipulator geometr, its inverse and direct kinematics, and also contains the sensitivit analsis of the leg parallelism at the eamined postures with respect to the joint encoder offsets. Section 3 focuses on the parameter identification, with particular emphasis on

4 A.Pashkevich et al. Kinematic calibration of Orthoglide-tpe mechanisms from observation of parallel leg motions 4 the calibration accurac under the measurement noise and selection the best set of the calibration equations. Section 4 contains eperimental results that validate the proposed technique, while Section 5 summaries the main results and contribution of the paper. 2. Kinematic modelling 2.1. Manipulator geometr The Orthoglide is a three degrees-of-freedom parallel manipulator actuated b linear drives with mutuall orthogonal aes. Its kinematic architecture is presented in Fig. 1 and includes three identical parallel chains, which will be further referred as legs. Kinematicall, each leg is formall described as PRPaR - chain, where P, R and Pa denote the prismatic, revolute, and parallelogram joints respectivel (Fig.2). The output machiner (with a tool mounting flange) is connected to the legs in such a manner that the tool moves in the Cartesian -- space with fied orientation (translational motions). (a) (b) Fig. 1. The Orthoglide mechanism - kinematic architecture (a) and general view (b). A i i i i j i B i d L C i P r Fig 2. Kinematics of the Orthoglide leg.

5 A.Pashkevich et al. Kinematic calibration of Orthoglide-tpe mechanisms from observation of parallel leg motions 5 In Figs. 1, 2, the base points A 1, A 2 and A 3 are fied on the ith linear ais such that A 1 A 2 = A 1 A 3 = A 1 A 2, the point B i is at the intersection of the first revolute ais i i and the second revolute ais j i of the ith parallelogram, and the point C i is at the intersection of the last two revolute joints of the ith parallelogram. When each B i C i is aligned with the linear joint ais A i B i, the Orthoglide is in an isotropic configuration and the tool centre point P is located at the intersection of the linear joint aes. In this posture, the base points A 1, A 2 and A 3 are equall distant from P. The smmetric design and the simplicit of the kinematic chains (all joints have onl one degree of freedom) contribute to lower the Orthoglide manufacturing cost. The Orthoglide is free of singularities and self-collisions. Its workspace has a regular, quasi-cubic shape. The input/output equations are simple and the velocit transmission factors are equal to one along the, and direction at the isotropic configuration, like in a serial PPP machine (Wenger et al., 2). The latter is an essential advantage of the Orthoglide architecture with respect to the machining applications. Another specific feature of the Orthoglide mechanism, which will be further used for calibration, is displaed during the end-effector motions along the Cartesian aes. For eample, for the -ais motion in the Cartesian space, the sides of the -leg parallelogram must also retain strictl parallel to the -ais. Hence, the observed deviation of the mentioned parallelism ma be used as the data source for the calibration algorithms. For a small-scale Orthoglide prototpe used in for the eperimental part of the paper, the workspace sie is approimatel equal to 222 mm 3 with the velocit transmission factors bounded between 1/2 and 2 (Chablat & Wenger, 23). The legs nominal geometr is defined b the following parameters: L = mm, d = 8 mm, r = 31 mm where L, d are the parallelogram length and width, and r is the distance between the points C i and the tool centre point P (see Fig. 2). Within the workspace, the manipulator is able to reach the Cartesian velocit of 1.2 m/s and the acceleration of 17 m/s 2 while carring a paload of 4 kg Modelling assumptions Following previous studies on the parallel mechanism accurac (Wang & Massor, 1993; Renaud et al., 24, Caro et al., 26), the influence of the joint/link defects is assumed relativel small compared to the joint positioning errors that are mainl caused b the encoder offsets. The latter is also justified b the authors eperience with the Orthoglide prototpe, where manufacturing tolerances.1 mm for the links and joints were achieved relativel easil, using common commerciall available equipment. However, usual assembling techniques produced the joint offset errors about.5 mm and motivated development of dedicated calibration method that are presented in this paper. These methods are based on the following modelling assumptions that are partiall validated during the eperimental stud (see Section 4):

6 A.Pashkevich et al. Kinematic calibration of Orthoglide-tpe mechanisms from observation of parallel leg motions 6 (i) (ii) the manipulator parts are supposed to be rigid-bodies connected b perfect joints, without clearances; the articulated parallelograms are assumed to be identical and perfect, which insure that their sides sta parallel in pares for an motions; (iii) the manipulator legs (composed of one prismatic joint, one parallelogram, and two revolute joints) are identical and generate a four degree-of-freedom motion each; (iv) the linear actuator aes are mutuall orthogonal and intersected in a single point to insure a translational three degree-of-freedom movement of the end-effector; (v) The actuator encoders are assumed to be perfect but their location (ero position) is defined with some errors that are treated as the offsets to be estimated. Using these assumptions, an efficient calibration technique will be developed based on the observation of the parallel motions of the manipulator legs Kinematic model Let us first briefl present the Orthoglide kinematic model, which is described in details in the previous papers (Chablat & Wenger, 23; Pashkevich et al., 26). Under the adopted assumptions, the articulated parallelograms ma be replaced b kinematicall equivalent single rods of the same length. Besides, a simple transformation of the Cartesian coordinates (the shift b the vector (r, r, r) T ) allows to eliminate the tool offset. Hence, the Orthoglide geometr can be described b a simplified model, which consists of three rigid links connected b spherical joints to the tool centre point (TCP) at one side and to the allied prismatic joints at another side (Fig. 3). Corresponding formal definition of each leg can be presented as PSS, where P and S denote the actuated prismatic joint and the passive spherical joint respectivel. (a) (b) (,, ) = L (p, p, p ) O (,, ) p= O = L (,, ) = L Fig. 3. Orthoglide simplified model (a) and its isotropic configuration (b). Thus, if the origin of the reference frame is located at the intersection of the prismatic joint aes and the,, - aes are directed along them, the manipulator geometr ma be described b the following equations

7 A.Pashkevich et al. Kinematic calibration of Orthoglide-tpe mechanisms from observation of parallel leg motions p ( ) p p L ( ) p p p L ( ) 2 p p p L (1) where p = (p, p, p ) is the output position vector, = (,, ) is the input vector of the prismatic joints variables, = (,, ) is the encoder offset vector, and L is the length of the parallelogram principal links. Besides, we assume that the joint variables satisf the following prescribed joint limits ;,, min i ma i (2) defined in the control software (for the Orthoglide prototpe studied here, the were set as min =-1 mm and min =+6 mm). It should be noted that, for this convention and for the case = (,, ), the nominal isotropic posture of the manipulator corresponds to the Cartesian coordinates p = (,, ) and to the joints variables = (L, L, L), see Fig. 3b. In this posture, moreover, the -, and -legs are oriented strictl parallel to the Cartesian plane XY. But the joint offsets cause the deviation of the TCP location and corresponding deviation of the parallelism, which ma be computed appling the direct kinematic algorithm for the joint variables = (L+, L+, L+ ). On the other hand, in the calibration eperiments, this deviation can be detected b evaluating the parallelism of the - and -legs with respect to the manipulator base surface (-plane). This can be easil done b measuring distances from the leg ends to the base surface and computing the difference. However, the capabilit of this technique is limited b evaluating the offset of the -ais encoder onl, since the Orthoglide mechanical design does not allow making similar measurements for the remaining pairs of the legs, with respect to the - and -planes. Hence, within the adopted model, four parameters (,,, L) define the manipulator geometr, but because of the rather tough manufacturing tolerances used for the prototpe, the leg link is assumed to be known and onl the joint offsets (,, ) are in the focus of the proposed calibration technique Inverse and direct kinematics To derive calibration equations, first let us epand some previous results on the Orthoglide kinematics (Pashkevich et al., 26) taking into account the encoder offsets. The inverse kinematic relations are derived from the equations (1) in a straightforward wa and onl slightl differ from the nominal case p s L p p p s L p p (3) p s L p p 2 2 2

8 A.Pashkevich et al. Kinematic calibration of Orthoglide-tpe mechanisms from observation of parallel leg motions 8 where s, s, s { ±1} are the configuration indices defined for the nominal manipulator as signs of p, p, p, respectivel. It is obvious that epressions (3) define eight different solutions to the inverse kinematics, however the Orthoglide assembling and joint limits reduce this set for a single case corresponding to the s = s = s = 1. For the direct kinematics, the equations (1) can be subtracted pair-to-pair that gives the following epression for the unknowns p, p, p (for details, see Pashkevich et al., 25) i i t pi ; i,, 2 i i (4) where t is an auiliar scalar variable. This reduces the direct kinematics to the solution of a quadratic equation At 2 + Bt + BC = with coefficients A ( )( ) ( )( ) ( )( ); ( ) ( ) ( ) ; C L B ( ) ( ) ( ) 4 4. Of the two possible solutions t B m B ABC A 2 ( 4 ) (2 ), 1 m of the quadratic formula, onl the one corresponding to m=+1 is admitted b the orthoglide prototpe (because of the selected assembl mode) Sensitivit analsis To evaluate the encoder offset influence on the legs parallelism with respect to the Cartesian planes XY, YZ, and YZ, let us derive first the differential relations for the TCP deviation for three tpes of the Orthoglide postures: (i) maimum displacement postures for the directions,, (Fig. 4a); (ii) isotropic posture in the middle of the workspace (Fig. 4b); (iii) minimum displacement postures for the directions,, (Fig. 4c); XMa posture Isotropic posture XMin posture = L cos = L = L cos p O = L + L sin = L cos p = L O = L p = L cos O = L - L sin Fig. 4. Specific postures of the Orthoglide manipulator (corresponding to the -leg leg motion along the Cartesian ais X )

9 A.Pashkevich et al. Kinematic calibration of Orthoglide-tpe mechanisms from observation of parallel leg motions 9 These postures are of particular interest for the calibration since in the nominal case (ero encoder offsets) the corresponding leg is parallel to the relevant pair of the Cartesian planes. On the other hand, the considered parallelism can be perturbed b the deviation of the TCP that defines location of points C i (see Fig. 2), while the opposite sides of the legs are mechanicall constrained b the actuator joint aes (points B i in Fig.2). The differential kinematical model ma be derived from the Orthoglide Jacobian, the inverse of which is obtained from (1) in a straightforward wa (see Pashkevich et al., 26 for details): J 1 p p 1 p p ρ p p ( p, ρ) 1 p p p p p 1 p p (5) It should be noted that, for computing convenience, the above epression includes both the Cartesian coordinates p p p and the joint coordinates,,, but onl one of these sets ma be treated as independent because of,, the inverse/direct kinematic relations. For the isotropic posture, the differential relations are computed in the neighbourhood of the point p = (,, ) and = (L, L, L), which after substitution to (5) gives the identit Jacobian matri Jp (, ρ) I (6) 33 It means that in this case the TCP displacement is related to the joint offsets b trivial equations p, i,,, (7) i i and each joint offset influences on the TCP deviation independentl and with the scaling factor of 1.. Taking into account the Orthoglide geometr, this deviation ma be estimated b evaluating parallelism of the legs with respect to the Cartesian planes (i.e. measuring difference of distances from the leg ends to the relevant plane). However, as mentioned in subsection 2.3, this technique is feasible for the -direction onl, hence it ma produce an estimation of merel. For the maimum displacement posture in the -direction (see Fig. 4a), the differential relations are derived in the neighbourhood of the point p ( Lsin,,) ; ρ ( L Lsin, Lcos, Lcos ) where is the angle between the -, -legs and corresponding Cartesian aes: asin( / L). After the ma substitution into (5), this gives the inverse Jacobian as a lower triangle matri, which admits analtical inverse ielding

10 A.Pashkevich et al. Kinematic calibration of Orthoglide-tpe mechanisms from observation of parallel leg motions 1 1 Jp( ), ρ ( ) T 1, (8) T 1 where tan( ). Hence, the differential equations for the TCP displacement ma be written as T p ; p T ; p T (9) and the joint offset influences on the TCP deviation is estimated b factors 1. and T. It is also worth mentioning that measurement of the -leg parallelism with respect to the XY-plane gives an equation for estimating the offset (provided that the offset has been obtained from the isotropic posture). Similar results are valid for the maimum displacement postures in the - and -directions (differing b the indices onl), and also for the minimum displacement postures. In the latter case, the angle should be computed from an equation asin( / L). min Table 1. Sensitivit of the TCP location for the representative Orthoglide postures Posture Leg Plane Deviation Tpical value * X XY 1. XZ 1. Isotropic Y XY 1. YZ 1. Z XZ 1. YZ 1. Ma / Min X-displacement Ma / Min Y-displacement Ma / Min Z-displacement X Y Z XY T XZ T XY T YZ T XZ T YZ T The results on the TCP sensitivit with respect to the joint offsets are summaried in Table 1 that gives also numerical values corresponding to the hpothetical joint offset = (1 mm, 1 mm, 1 mm) and to the angle = 2 that are tpical for the Orthoglide prototpe studied in the eperimental part of the paper. Analsis of these values allows concluding that the leg parallelism is rather sensitive to the joint offsets. Thus, relevant deviations p, p, p, ma be used for the offset identification.

11 A.Pashkevich et al. Kinematic calibration of Orthoglide-tpe mechanisms from observation of parallel leg motions Measurement techniques 3. Calibration methods To identif the Orthoglide kinematic parameters specified in the previous section, we propose two calibration methods, which emplo different measurement techniques for the leg/surface parallelism. The first of them (Fig. 5a) assumes two measurements for the same leg posture (to assess distances from both leg ends to the base surface). The second technique assumes a fied location of the measuring device but two distinct leg postures, which ensure positioning of the leg ends in the neighbourhood of the device. It is obvious that, for the perfectl calibrated manipulator, both methods give ero differences for each measurement pair. Conversel, the non-ero differences contain source information for the joint offset identification. The following sub-sections contain detailed descriptions of these measurement techniques and relevant identification procedures. In particular, sub-sections 3.2 and 3.3 introduce respectivel the single- and double-pose methods along with corresponding literalised calibration equations. Sub-section 3.4 describes a non-linear calibration routine that is based on the minimisation of the residual-square sum. Finall, sub-section 3.5 focuses on the calibration accurac and sensitivit to the measurement noise. (a) absolute measurements (b) relative measurements Posture #1 Manipulator legs d 1 Manipulator legs Base plane d1 + = d2 - d1 d2 Manipulator legs Posture #2 + = d 2 - d 1 Base plane d 2 Base plane Fig. 5. Measuring the leg/surface parallelism using single-posture-double-sensor (a) and double-posture-single-sensor (b) methods Calibration using single-posture measurements Using the single-posture measurements and taking into account the Orthoglide design limitations allowing locating gauges on the XY surface onl (i.e. for the -direction measurements), the calibration eperiment ma be arranged in the following wa. Step 1. Locate the manipulator in the isotropic posture and measure parallelism of the X- and Y-legs with respect to the XY-surface:,

12 A.Pashkevich et al. Kinematic calibration of Orthoglide-tpe mechanisms from observation of parallel leg motions 12 Step 2. Locate sequentiall the manipulator in the X-maimum and X-minimum postures and measure + parallelism of the X- legs with respect to the XY-surface:, Step 3. Locate sequentiall the manipulator in the Y-maimum and Y-minimum postures and measure + parallelism of the Y- legs with respect to the XY-surface:, In the above description, the variable following the -smbol denotes the measurement direction ( in all cases), the subscript defines the manipulator leg, and the superscript indicates the manipulator posture for this leg. For eample, + denotes the -direction deviation of the X-leg for the X-maimum posture. Using epressions from sub-section 2.5 presented in Table 1, the sstem of the calibration equations ma be written as follows 1 1 a1 1 a2 1 a1 1 a2 1 (1) where a1 T and 1 a2 T, which ma be also computed as 2 a L and ma ma a L min min For instance, for the Orthoglide prototpe (see subsection 2.1) a 1.2 and a This overdetermined sstem of si linear equations in three unknowns ma be solved in a straightforward wa, using the Moore-Penrose pseudoinverse. However, from the application point of view, it is worth to separate the equations for three pairs and sequentiall solve them for,, : this approach ields the following epressions for the joint offsets 2 a ( ) a ( ) a1 a2 a ( ) a ( ) a1 a2 which are computationall convenient but ma produce slightl higher residuals than the standard pseudoinverse. (11) However, the measurement procedure for this method is rather complicated in comparison with an alternative one, described in the following subsection. It should be stressed that the single-posture method requires separate measurements of d 1 and d 2 (see Fig. 5a) that are further used for computing the difference d 2 d 1, while the alternative technique directl evaluates this difference using a single measuring device. It is obvious that the first

13 A.Pashkevich et al. Kinematic calibration of Orthoglide-tpe mechanisms from observation of parallel leg motions 13 method is based on the absolute measurements that are ver sensitive to the gauge calibration, while the second approach (based on the relative measurements) does not require an calibration of the gauges Calibration using double-posture measurements Since in this case a single gauge is used onl, it is possible to assess the leg parallelism with respect to both relevant planes (XY and XZ for the X-leg, for instance). This advantage is charged however b using two legs postures, allowing sequentiall locating both leg ends close to the gauge. For this measuring technique, the calibration eperiment ma be arranged in the following wa: Step 1. Locate the manipulator in the isotropic posture and place two gauges in the middle of the X-leg ensuring required measurement directions (orthogonal to the leg and parallel to the Cartesian aes Y and Z); get the gauge readings. Step 2. Locate sequentiall the manipulator in the X-maimum and X-minimum postures, get the gauge readings, and compute differences +, +,, + + Step 3+. Repeat steps 1, 2 for the Y- and Z-legs and compute differences,,,, and +, +,,. The sstem of calibration equations can be also derived using epressions from Table 1, but in two steps. First, it is required to define the gauge location that is assumed to be positioned at the leg middle point in the isotropic posture. * Hence, for the X-leg for instance, it is the midpoint of the line segment bounded b the TCP (,, ) and the centre of the X-ais prismatic joint (L+,, ). This ields the following differential epressions for the leg midpoints: X - leg Gauges : Y - leg Gauges : Y - leg Gauges : ( L 2 ; 2 ; 2 ) ( 2 ; L 2 ; 2 ) ( 2 ; 2 ; L 2 ) Afterwards, in the X-maimum posture, the X-leg location is also defined b two points, namel, (i) the TCP, and (ii) the centre of the X-ais prismatic joint. Their coordinates are defined as follows (see Fig. 4a and Table 1) Tool centre point : X-joint centre: ( LS ; T ; T ) ( L LS ; ; ) Then, the equations of a straight-line passing along the X-leg ma be written as ( LS ) (1 )( LLS ) ( T ); ( T ) (12) * This assumption is not critical here because, as follows from relevant analsis, potential errors in the initial location of the gauge produce identification errors that are negligible as compared to the measurement noise.

14 A.Pashkevich et al. Kinematic calibration of Orthoglide-tpe mechanisms from observation of parallel leg motions 14 where S sin( ); T tan( ), and is a scalar parameter, [, 1]. Since the gauge -coordinate remains the same independentl of the current posture, the parameter ma be obtained from the equation L 2, which gives the following solution:.5 S. (13) Hence, the Y- and Z-gauge readings for the X-leg in the X-maimum posture are (.5 S ) T (.5 S ) (.5 S ) T (.5 S ) (14) and, finall, the deviations of the X-leg measurements while it changes its posture from the X-maimum to the isotropic one are (.5 S ) T S (.5 S ) T S (15) A similar approach ma be applied to the X-minimum posture, as well as to the equivalent postures for the Y- and Z-legs. This gives the following sstem of twelve linear equations in three unknowns b1 c1 c1 b1 b2 c2 c2 b2 b1 c 1 c1 b1 b2 c 2 c2 b 2 b1 c 1 c1 b1 b2 c2 c2 b 2 (16) where bi sin i; ci (.5 sin i) tan and i 1 asin( ma L) ; 2 asin( min L). For instance, for the Orthoglide prototpe (see subsection 2.1) b 1.19, c 1.14 and b , c 2.6. The reduced version of this sstem ma be obtained if one assesses the leg/plane parallelism b the difference between the maimum and minimum postures. The latter leads to the sstem of si linear equations in three unknowns

15 A.Pashkevich et al. Kinematic calibration of Orthoglide-tpe mechanisms from observation of parallel leg motions 15 b c c b b c (17) c b b c c b where bb1b2; cc1 c2 and ;, etc. For the Orthoglide prototpe this values are as follows: b.52, c.2. Both sstems (16) and (17) ma be solved using the pseudoinverse of Moore-Penrose, which ensures minimiing the residual square sum. But as follows from the simulation stud, for rather essential joint offsets (about 5 mm and more) the differential equations ma produce non-accurate results. For this reason, the net subsection focuses on the non-linear calibration equations and their solution through the straightforward minimiation of the square sum of the residuals Non-linear calibration equations From a general point of view, the considered calibration problem ma be presented as the fitting of the eperimental data to the Orthoglide kinematic model incorporating the joint offsets. Hence, it is necessar to obtain numerical algorithms that allow computing all the eamined deviations for an given offsets. To present relevant results in a concise form, let us introduce special notations for the direct and inverse kinematic models of the nominal Orthoglide (with ero offsets): p f ( ρ); ρ f ( p) if ρ (18) 1 Then, in the isotropic posture, the TCP position ma be epressed as p, p, p f L, L, L, (19) while epressions for the position of the prismatic joints remain the same: X - leg Prismatic Joint : Y - leg Prismatic Joint : Y - leg Prismatic Joint : ( L ) ( L ) ( L ) Hence, the leg midpoints defining the gauge locations ma be computed as follows: L p p p g g g 2 ( ) 2; 2; 2; p 2; L 2 ( p ) 2; p 2; (2) g g g p 2; p 2; L 2 ( p ) 2; g g g where the subscripts,, define the leg and the subscript g refers to the gauge. For the X-maimum posture, the TCP position is computed as

16 A.Pashkevich et al. Kinematic calibration of Orthoglide-tpe mechanisms from observation of parallel leg motions 16, p, p p f LLS, LC, LC, (21) where C cos( ); S sin( ), while the position of the X-link prismatic joints is described b the epression ( LLS ; ; ). Hence, the equations of a straight-line passing along the X-leg ma be written as p (1 ) ( LLS ) p ; p where is a scalar parameter, as above, which is determined b the -coordinate of the gauge this equation ields (22) g. Solution of L 2LS p 2 2 LLS p (23) that allows one to compute the Y- and Z-gauge readings for the X-leg as get the final epression for the desired deviations of the X-leg: and p respectivel and to p p p ( ρ) 2 p p ( ρ) 2 (24) where smbol (.) is used to distinguish functions of the joint offsets and the eperimental values, which are denoted b. A similar approach ma be applied to the X-minimum posture, as well as to the equivalent postures for the Y- and Z-legs. Relevant epressions are summaried in Table 2 where smbol stands for both the maimum and minimum postures and angle is defined b the joint limits: 1 asin( ma L) ; asin( L). 2 min The obtained epressions allow posing the following optimisation problem for the joint offset identification 2 2 F ( ρ) ( ρ) K min, (25) which gives the desired values of,,. It ma be also presented in the reduced form b replacing the pairs of the deviations (, ), (, ), etc. b their differences ;, etc. Both problems ma be solved numericall b means of the standard gradient search technique using the Jacobians from Eqs. 16 and 17. ρ

17 A.Pashkevich et al. Kinematic calibration of Orthoglide-tpe mechanisms from observation of parallel leg motions 17 Table 2 Epressions for the non-linear calibration model Content TCP locations Epressions p, p, p f L, L, L p, p, p f L LS, LC, LC p, p, p f LC, L LS, LC p p, p f LC, LC, L LS, Scaling factors L/2 LS p /2 / 2 L LS p L LS p /2 /2 / 2 L LS p L LS p /2 /2 / 2 L LS p Leg deviations ρ p p ρ p p ( ) 2; ( ) 2; ρ p p ρ p p ( ) 2; ( ) 2; ρ p p ρ p p ( ) 2; ( ) 2; 3.5. Calibration accurac Because of the measurement noise, the developed technique ma produce the biased estimates of the model parameters. Thus, for practical application, it is worth to evaluate the statistical properties of the calibration errors. Within the linear calibration equations, the impact of the measurement noise ma be evaluated using general techniques from the identification theor, under the standard assumptions concerning the measurement errors i : ero-mean independent and identicall distributed Gaussian random variables with the standard deviation. Let us consider separatel two cases corresponding to the si-equation and twelve-equation sstems (7), (8), since the differ in residual covariance. For both linear sstems (16) and (17), the variance-covariance matri of the identification parameters is written as (Ljung, 1999) T 1 1 ( ) ( ) T T T V ρ JJ J E( s s) J( JJ ) (26) where E(.) denotes the mathematical epectation, J is the Jacobian, and s is the vector of the measurement errors.

18 A.Pashkevich et al. Kinematic calibration of Orthoglide-tpe mechanisms from observation of parallel leg motions 18 In the si-equation case, the vector s consists of the statisticall independent components corresponding to the deviations,, K and is epressed through differences of the measurement errors at the min/ma leg postures: s (6),, K,. (27) where the subscripts and the superscripts are defined similar to subsection 3.4. Hence, the covariance is the 66 identit matri T and the epression (26) is reduced to T 2 2 E s s I (28) (6) (6) 66 T 1 2 V( ρ) 2( J(6) J (6)) (29) However, in the twelve-equation case, the vector s includes some dependent components s (6),,, K,, (3) corresponding to the pairs (, ), (, ) (, ), since each leg deviations are measured twice (for the Ma/Min postures) but with respect to the same isotropic location. So, the covariance is the 1212 nonidentit matri T 2 (12) (12) 1212 E s s G (31) epressed as 2 1 G G ; G G G Consequentl, the covariance (26) is presented as V( ρ) ( J J ) J G J ( J J ) (32) T 1 T T 1 2 (12) (12) (12) (12) (12) (12) These epressions allow us to compute a scalar performance measure for the calibration accurac that ma be T defined as the square-averaged standard deviation of the calibration errors for the joint offsets,, 1 ( ) 3 trace V ρ (33) where the subscript is used for distinguishing with the standard deviation of the measurement noise. For the Orthoglide prototpe described in subsection 2.1, the latter epression ields 2.6 in the case of twelve equations and 1.98 in the si-equation case. This justifies using the si-equation method because of simplicit and slightl higher identification accurac in comparison with the twelve-equation technique.

19 A.Pashkevich et al. Kinematic calibration of Orthoglide-tpe mechanisms from observation of parallel leg motions 19 While confirming this conclusion theoreticall, it is worth mentioning that reduction of the equation number from 12 to 6 usuall increases the calibration error b the factor 2. However, using the deviations,, K (measured between the Ma and Min postures) instead of,, K (measured between the isotropic and Ma/Min postures) increases the deviation measurement sensitivit that gives reduction of trace ma T 1 JJ. In particular, for the case stud, ma L.19 ( ) min and min L.32 while ( ) L.52. It means that the sensitivit increase compensates reduction of the equation number. For the non-linear calibration equations (see subsection 3.4), the impact of the measurement errors was investigated using the Monte-Carlo method. The simulation results (obtained for 2 replications with 1 runs for =.1 mm and two values of ) are presented in Table 3. The coincide with the above linear-approimation epressions and also justif advantages of the si-equation method for the practical applications. Table 3 Simulation results on impact of the measurement errors for =.1 mm Calibration technique std() (offset.1 mm) std() (offset 1. mm) Si-equation method.198 mm (.3).199 mm (.2) Twelve-equation method.27 mm (.3).27 mm (.4) 4.1. Eperimental setup 4. Eperimental results The measuring sstem is composed of standard comparator indicators attached to the universal magnetic stands allowing fiing them on the manipulator bases. The indicators have a resolution of 1 m and are sequentiall used for measuring the X-, Y-, and Z-leg parallelism while the manipulator moves between the Ma, Min and isotropic postures (it is obvious that for industrial applications, it is better to use more sophisticated, high precision digital indicators with the resolution of 1 m or less, which ield more accurate calibration results). For each measurement, the indicators are located on the mechanism base in such a manner that a corresponding leg is admissible for the gauge contact for all intermediate posters (Fig. 6). The Min and Ma postures are

20 A.Pashkevich et al. Kinematic calibration of Orthoglide-tpe mechanisms from observation of parallel leg motions 2 constrained b the software joint limits and defined as min =-1 mm and ma = 6 mm respectivel. The initial position of the indicator corresponds to the leg middle for the manipulator isotropic posture. Fig. 6. Eperimental Setup. During eperiments, the legs were moved sequentiall via the following postures: Isotropic Ma Min Isotropic. To reduce the measurement errors, the measurements were repeated three times for each leg. Then, the results were averaged and used for the parameter identification. It should be noted that the measurements demonstrated ver high repeatabilit compared to the encoder resolution (dissimilarit was less than.2 mm) Calibration results and their analsis To validate the developed calibration technique and the adopted modelling assumptions, we carried out three eperiments targeted to the following objectives: Eperiment #1: validation of modelling assumptions (it lead to the mechanical retuning ) Eperiment #2: collecting eperimental data used for the parameter identification; Eperiment #3: validation of calibration results using the identified model parameters. Eperiment #1. The first calibration eperiment produced rather high parallelism deviation, up to 2.37 mm as shown in Table 4. It was unepected since the Orthoglide demonstrated quite good qualit and accurac of milling in previous tests. However, the milling tests were perfect just because of the high uniformit of the Orhoglide workspace due to the advantages of the manipulator architecture. The straightforward application of the proposed calibration algorithm to this data set was not optimistic: in the frames of the adopted kinematic model, the root-mean-square (r.m.s.) deviation for the legs can be reduced down from 1.19 mm to 1.7 mm onl (see Table 4). On the other hand, the statistical estimation of the measurement noise parameter (based on the residual analsis) also ielded an unrealistic result: 1. mm. It impels to conclude that the manipulator mechanics requires more careful tuning, especiall location of the linear actuator aes that are

21 A.Pashkevich et al. Kinematic calibration of Orthoglide-tpe mechanisms from observation of parallel leg motions 21 assumed to be mutuall orthogonal and intersected in a single point (see subsection 2.2). Thus, the manipulator mechanics was retuned, in particular the locations of the actuator aes were adjusted mechanicall using the singlepose measurement technique described in subsection 3.2. Eperiment #2. The second calibration eperiment (after mechanical tuning) ielded lower parallelism deviations, less than.7 mm (see Table 4), which is on average twice better than in the first eperiment. For these data, the developed calibration algorithm ielded the joint offsets that are epected to reduce the root-mean-square deviation down from.62 mm to.28 mm, i.e. b three times. Besides, the estimated value of.28 mm is more realistic taking into account both the measurement accurac and the manufacturing/assembling tolerances. Accordingl, the identified values of the joint offsets = -.53 mm, =.59 mm, = mm were incorporated in the Orthoglide control software. Eperiment #3. The third eperiment was targeted to the validation of the calibration results, i.e. assessing the leg parallelism while using the model parameters identified from the second data set. It demonstrated ver good agreement with the epected values of,,. In particular, the maimum deviation reduced down to.34 mm (epected.28 mm), and the root-mean-square value decreased down to.21 mm (epected.2 mm). On the other hand, further adjusting of the kinematic model to the third data set gives both negligible improvement of the deviations and ver small alteration of the model parameters (see Tables 4 and 5). It is evident that further reduction of the parallelism deviation is bounded b the manufacturing and assembling errors or, probabl, the non-geometric errors. Discussion. As follows from the above analsis, the calibration eperiments confirm validit of the proposed identification technique and its abilit to tune the joint offsets from observations of the leg parallelism. The achieved accurac coincides with the qualit of the Orthoglide prototpe manufacturing and assembling. Another related conclusion deals with the comparison of the si-equation and twelve-equation identification methods (see subsections 3.4 and 3.5) using real data sets, which do not necessar follow the classical assumptions on the measurement errors (Gaussian ero-mean random variables). As follows from Table 5, both methods produced roughl the same values of the model parameters, however the si-equation method is more computationall attractive and, thus, more suitable for the practice.

22 A.Pashkevich et al. Kinematic calibration of Orthoglide-tpe mechanisms from observation of parallel leg motions 22 Table 4 Eperimental data and epected improvements of accurac Data Source r.m.s. mm mm mm mm mm mm mm Initial settings (before mechanical tuning and calibration) Eperiment # Epected improvement After mechanical tuning (before calibration) Eperiment # Epected improvement After calibration Eperiment # Epected improvement Table 5 Model parameters obtained using the si- and twelve equation methods Eperiment Calibration method mm Model parameters mm mm Residual r.m.s. mm Eperiment #1 Si-equation Twelve-equation Eperiment #2 Si-equation Twelve-equation Eperiment #3 Si-equation Twelve-equation

23 A.Pashkevich et al. Kinematic calibration of Orthoglide-tpe mechanisms from observation of parallel leg motions Conclusions Recent advances in parallel robot architectures encourage related research on kinematic calibration of parallel mechanisms. This paper proposes a new calibration method for parallel manipulators, which allows efficient identification of the joint offsets using observations of the manipulator leg parallelism with respect to the base surface. Presented for the Orthoglide-tpe mechanisms, this approach ma be also applied to other manipulator architectures that admit parallel leg motions (along the Cartesian aes) or, in more general cases, that allow locating the leg in several postures with a common intersection point. The proposed calibration technique emplos a simple and low-cost measuring sstem composed of standard comparator indicators attached to the universal magnetic stands. The are sequentiall used for measuring the deviation of the relevant leg location while the manipulator moves the tool-centre-point in the directions, and. From the measured differences, the calibration algorithm estimates the joint offsets that are treated as the most essential parameters that are difficult to identif b other methods. The presented theoretical derivations deal with the sensitivit analsis of the proposed measurement method, selecting the best set of the calibration equation, and also with the calibration accurac. It has been proved that the highest accurac is achieved for the measuring the leg parallelism at the etreme leg postures, while additional measurements at the isotropic posture does not reduce the identification error. The validit of the proposed approach and the efficienc of the developed numerical algorithm were confirmed b the calibration eperiments with the Orthoglide prototpe, which allowed reducing the residual root-mean-square b three times. To increase the calibration precision, future work will focus on the development of the specific assembling fiture ensuring proper location of the linear actuators and also on the epanding the set of the identified model parameters and compensation of the non-geometric errors that are not compensated within the frames of the adopted model. References Besnard, S., Khalil, W. (21). Identifiable parameters for parallel robots kinematic calibration. In IEEE International Conference on Robotics and Automation (pp ), Seoul, Korea. Caro, S., Wenger, Ph., Bennis, F. & Chablat, D. (26). Sensitivit Analsis of the Orthoglide, a 3-DOF Translational Parallel Kinematic Machine. ASME Journal of Mechanical Design, 128 (2), Chablat, D., Wenger, Ph. (23). Architecture Optimiation of a 3-DOF Parallel Mechanism for Machining Applications, the Orthoglide. IEEE Transactions on Robotics and Automation, 19(3), Dane, D. (1999). Self calibration of Gough platform using leg mobilit constraints. In World Congress on the Theor of Machine and Mechanisms (pp ), Oulu, Finland.

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