Robotik manipülatörlerin konfigürasyon bağımsız hata parametreleri ile kalibrasyonu

Abstract

Manipülatörün yapımı sonunda ortaya çıkan mekanik sistem konstrüksiyon aşamasında düşünülen ideal sistemden çeşitli nedenlerle farklıdır. Çalışma uzayında uç elemanın belirli bir yönlenme (orientation) ile ulaştığı konum, sistemdeki konum sensörlerinden gelen işaretlere göre hesaplanan ve kontrolörün ekranından okunabilen konumla aynı değildir. Aralarındaki fark konum hatasıdır ve birbirin den farklı iki büyüklükle tarif edilir. Bunlardan birisi tekrarlayabilirlik (repeatability) hatası diğeri ise doğruluk(accuracy) hatasıdır. Doğruluk hatası tekrarlayabilirlik hatasını da içerir. Bir robotik manipülatörün kullanılması sırasında takip edilen yörüngenin düğüm noktaları öğretme(teaching) ile belirleniyorsa doğruluk hatasının kullanım acısından pek önemi olmamasına rağmen, görme(vision) esaslı veya CAD-CAM destekli uygulamalar doğruluk hatasının küçük tutulmasını gerektirir. Robotik kalıbrasyonun amacı, robotu oluşturan açık zincir deki ardışık uzuvların birbirine göre konumunu belirleyen parametre lerde mevcut olan hataların yani, ideal geometriden olan farklılıkların belirlenmesidir. Bu çalışmada 1. Bölümde(Giriş bölümü) öncelikle, Manipü latörlerin kalibrasyonuna yönelik olarak bugüne kadar yapılmış çalışmalar derlenmiştir. 2. Bölümde, robot kinematiğinde kullanılan nomojen koordinat dönüşümü ve jakobyen(diferansiyel) ilişki verilmiştir. 3. Bölümde ise kalibrasyon amacıyla kurulan hata modelleri, kullanılan ölçme sistemleri ve hataların gözlenebilirhği özetlenmiştir. 4. Bölümde, bu çalışmada gerçekleştirilen kalibrasyon simülas- yonlarma yer verilmiştir. Hata modelinin kurulmasında, iki eksen takımı arasındaki ilişkiyi ortaya koymak üzere 6 parametrenin varlığı föz önüne alınmış, bu 6 parametredeki hataları içerecek şekilde genel ir hata modeli kurulmuştur. Ölçme bilgisinin üretilmesi sırasında konfigürasyon bağımsız hata parametrelerinin yamsıra konfigürasyon bağımlı esneme hataları da göz önüne alınmıştır. Uç elemanın konumunun hesaplanmasında ayrıca, kodlayıcı rezolusyonundan ve ölçme sisteminden kaynaklanan hatalar da dikkate alınmıştır. Hata modelinde, yalnızca konfigürasyon bağımlı hatalar içerilmektedir. Bu hataların konfigürasyon bağımlı hataları da temsil edebilme yetenekleri çeşitli seviyede esneme hataları içerilen simülasyonlarla ortaya konmaya çalışılmıştır. Ölçme noktası sayısının ve ölçme hatasının kalibrasyona etkisi araştırılmıştır. Ayrıca, uygun ölçme konfigürasyon- larının belirlenmesi amacıyla daha önceki çalışmalarda teklif edilen bazı parametrelerin de geçerliliği ortaya konmaya çahşılrmştır. Son olarak, yapılan çalışmada elde edilen sonuçlar yorumlanarak verilmiştir. vı

Robotic manipulators are open-loop mechanical structures which consist of a certain number of links connected together by actuated joints. Mechanical structure which is obtained by manufacturing is different from the ideal structure expected(Figure 1). The differences between the manufactured and ideal manipulator, cause deviations on the location of end-effector in the working space of manipulator from nominal location. In other words, real position and orientation of end- effector differ from the ones computed by the joint sensor readings. Difference between actual and ideal positions is being expressed by repeatability and accuracy of manipulator. Repeatability is the measure oi the radius of sphere which involves positions of the end-effector obtained by commanding robot to the same location and orientation over and over. Accuracy is the difference between the real and expected positions of the end-effector. It includes repeatability. ',L j I, ' ',-L-J- L ' Actual /////////}////>///)////////////// Nominal Nominal Actual (a) (b) Figure 1 Deviations at ideal geometry a) Deviation from perpendicularity at shoulder b) Deviation from parallelism at elbow Industrial robots at present show satisfactory repeatability but poor accuracy. Repeatability is adequate if the robot is being programmed on-line, where it is required to show the robot desired motion(teaching robot). However, with the advent of computer integrated manufacturing systems and vision based systems, robots must be programmed off-line, using task level languages that draw on vuinformation from a CAD-CAM data base or using the information obtained from vision system simultaneously to tell the robot its motions. This type of positioning based on commanded motions requires high degree of robot accuracy. Calibration of the robotic manipulators aims at reducing the accuracy error of manipulator as close as possible to the repeatability error. Firstly it requires to define the errors in a robotic system. Errors which contributes the inaccuracy can be classified on the basis of type and source. Some of them are constant at joint level through working space. These type of errors are called Configuration Independent Errors. Some have random nature and some vary from one configuration to another. These are defined as Configuration Dependent Errors. But it must be expressed that the effects of configuration independent errors on the accuracy of the manipulator will be configuration dependent although they are constant at joint level. If the errors in a robotic system are classified on the base of source, they can be divided into two groups. On of them is Mechanical Structure Errors and the other is Control System Errors. Modelling of errors necessitates taking them into account at joint level For this reason, classification of errors as Configuration Independent and Configuration Dependent is more important. Some researchers named the first as Geometric Errors and the latter as Non-Geometric Errors. Configuration Independent Errors; -Errors in link dimensions, caused by manufacturing and measuring errors. -Deflections on the geometry of system components in limits of manufacturing tolerances. -Errors at encoder offsets. Configuration Dependent Errors; -Joint Compliance caused by the elastic deflections on the elements located between motor shaft and joint -Link Deflections. -Gear transmission errors(Spacing errors and eccentricity). -Backlash in gear and screw mechanisms. -Accumulated spacing errors at encoder readings. -Deflections at bearings. -Dimensional changes caused by temperature deviations. -Steady-state errors at control system. -Errors caused by numerical computations. -Errors in the derivation of inverse solution. Calibration procedures vary widely in their complexity. Calibration of robotic manipulators can be divided into three levels m complexity. First level is called joint level calibration. This level of calibration's goal is to determine the true relationship between joint sensor reading and actual joint displacement. This usually involves calibration of the kinematics of the joint drive mechanism and joint sensor mechanism. At many applications, first level of calibration is viiitaken into account in the design step of manipulator. Second level of calibration is defined as the entire robot kinematic calibration. The purpose of this level of calibration is to determine the correct kinematic parameters of the manipulator. Level 2 calibration includes first level of calibration. Third level of calibration is called non-kinematic calibration. Non-kinematic errors in positioning of the end-effector of a robot are due to effects such as joint compliance, link compliance, backlash as well as control system errors. Also, if the robot is under dynamic control rather than kinematic control, then corrections for changes in the dynamic model of the robot constitutes a level 3 calibration. In general, the calibration process at any level has some basic steps. These steps are as follows. -Modelling step, selection of parameter error model. -End-effector position or position and orientation measurement. -Computation of nominal position and orientation of end-effector. -Estimation of the difference between nominal and actual positions. -Solving the equations obtained from the model using the position differences. In this study, both configuration independent and configuration dependent errors have been modelled by means of configuration independent error parameters. Generally, there are six parameters in the transformation representing the relationship between two consecutive coordinate systems. So, it can be said that there will be six error parameters in this transformation(three translational and three rotational error parameters). So, six error parameters model has been used in the study. To some extent, kinematic notation used is similar to Shape Matrix notation. Shape of the links and joint parameter have been taken into account individually. Global transformation for a link has been obtained by the multiplication of shape transformation and joint parameter transformation. These transformations have been shown at Figure 2. Here A^, Am = cos (3 cosy -cos P sin y sin (3 a sin a sin P cos y + cos asin y cos a cos y- sin a sin Ş sin y - sin a cos P b sin a sin y - cos asin P cos y sin a cos y + cos asin P sin y cos a cos P c 0 0 0 1 (D andAe i+l» A9 i+l cos8j+1 -sin0i+1 0 0 sin8i+ı cos8j+i 0 0 0 0 10 0 0 0 1 (2) IXAhi=AsAhiAei+1 Figure 2. Coordinate tranf ormations f or nominal and actual geometry AgjAşiü can be expressed as sum of A^ and a small transformation matris. Thus, AsjAjjiu Agi + dAjj (3) By the help of a similar approach AM can be defined as A^ plus a small transformation. Aw = A^ + dA-i (4) here dAi is the transformation of the error parameters at i.th joint. When the global nominal transformation from base to end-effector is written, TN = AsOA01AslA92 As5A96A s6 (5) is obtained, then global transformation for the actual transformation TH =(AsOAshOAei)(AslAshlA82) (As5Ash5Ae6)(As6Ash6) (6) is written with similar approach. Here, one can use Aq7 as an imaginery transformation for 07=0, so that formulation of transformations can be expressed in a simple way. Small transformation which shows diffe rence between the actual and nominal positions is dT = TH-TN (7) Subsituting (3)i4),(5) and (6) into (7), one can obtain difference trans formation as follows, dT = £ AnoAnl An(i-l)dAiAn(i+l) An5An6 (8) i=0dT can also be written as individual sums of each error parameter. dT = dTa + dip + dTy + dTa + dTb + dTc (9) and this individual parts can be written as follows, <rç* = ^AnoA-nl An(i-l)JaiAn(i+l) An5An6 i=0 Aa (10) dTp = S^ncAnl An(i-l)JpiAn(i+l) An5An6 i=0 AP (11) dTy = £AnOAnl An(i-l)JyiAn(i+l) An5An6 i=0 Ay (12) dTa = £AnOAnl An(i-l)JaiAn(i+l) An5An6 i=0 Aa (13) dTh = £AnOAnl An(i-l)JbiAn(i+l) An5An6 i=0 Ab (14) dTc = 2An0Anl An(i-l)JciAn(i+l) An5An6 i=0 Ac (15) In this study, only the cartesian position errors of the manipu lator has been used to compute the parameter errors. After the formulation outlined above, one can have error effect coefficiants of 42 error parameters. For each measurement configuration, three cartesian end-effector position error data can be collected. Finding error parameters necessitates solving the equation system comprised of error effect coefficiants and position errors. A. As = AS (16) A is a matrix which has the error effect coefficiants, AS vector of end- effector position errors, As vector of error parameters so that, XIAs = [to0 Apo Ay0 Aa0 Ab0 Ac0 Aa6 Ap6 Ay6 Aa6 Ab6 Ac6] (17) Errors would affect the accuracy of the manipulator have been modelled by configuration independent error parameters. Simulations have been achieved on a puma type robot. Its dimensions, joint para meter ranges and zero position parameters are as shown in Figure 1 Measurement data has been produced by taking into account configuration independent errors, jomt compliance, encoder noise and measurement noise. Configuration independent error parameters have Figure 3. ^fenipalator used in siirulations. been selected under consideration of manufacturing and assembling conditions between certain maximum and minimum values randomly. Algorithm has been tested for two random error set to confirm its success. It has been assumed that joint encoders have had (3600/2500) resolution and there has been 1/100 reduction ratio between motor and joint itself. Different measurement error values have been used in simulations to show the effects of measurement error on calibration. An important source of inaccuracy in manipulators is joint compliance. Some previous studies showed that joint compliance isfar more important than link deflections. So, in this study only the xujoint compliance has been taken into account. It has been assumed that there are Harmonic Drives at shoulder and elbow of the manipulator. They have been selected so that they are suitable for an industrial manipulator. Base bearing has been selected in a similar approach. Compliances have been computed according to characteristics given by the manufacturers. It has been proved that compliance affects calibration success strongly. In previous studies, joint compliance were not taken into account simulations showed good calibration results. But, it has been proved in this study that joint compliance can not be neglected. Table 1 shows the effects of compliance on calibration. Table 1 Effects of joint carpliance en manipulator's accuracy for first set of errors. Number of Measurement 80, MaasureiEnt error 0.050 ran. Encoder noise exists. It is obvious that six error parameters at joint level is redundant but a calibration algorithm should be able to make relation between manipulator accuracy and joint tolerances. So, in this study, one can obtain end-effector accuracy when the joint tolerances are given. This will cast light upon the manufacturing tolerances which must be achieved when a certain accuracy is proposed for the manipulator. Also, in the study optimum observability index and condition number for measurement configurations have been computed to help the selection of optimum configurations. In addition, effect of measurement number on calibration has been searched and convenient number of measurement has been tried to be estimated. Also, calibration simula tions achieved in a limited work space especially important for joint compliance, have been tested through work space and validity of computed parameters has been searched. xniCALIBRATION ALGORITHM Computation of new nom. end-effector pos, Modification of nominal parameters Generation of joint parameters Working Space definition Number of meas. points Computation of nominal end-effector positions Nominal manipulator parameters Computation of actual end-effector positions Conf. Independent Err.Pa^q. Manufacturing and Assembling conditions Encoder noise. Encoder and joint features Difference between actual and nominal positions Computation of Error Effect Coefficiants Singular Value Decomposition Actual Positions obtained with the comp. err. par. Difference between the first and the last actual positions Joint compliance L^». Bearing and joint features - ». Measurement system fatures Measurement noise XIV