Modeling and Simulation of Short Range 3D Triangulation-Based Laser Scanning System

Transcription

1 Modeling nd Simultion of Short Rnge 3D Tringultion-Bsed Lser Scnning System Theodor Borngiu Anmri Dogr Alexndru Dumitrche April 14, 2008 Abstrct In this pper, simultion environment for short rnge 3D lser scnning system tht uses tringultion is presented. The simultion is used for integrting lser scnning probe tht uses line lser nd two cmers, with verticl rticulted robotic rm with 6 degrees of freedom nd rotry tble. The opticl subsystem is simulted using POV-Ry ry trcing softwre, nd the imge processing for tringultion, together with the robotic rm kinemtics nd the user interfce, re implemented in MATLAB. Keywords: 3D Lser Scnning, Simultion, Ry Trcing. 1 Introduction This work is prt of bigger project, whose gol is to integrte short rnge 3D lser scnning probe with verticl rticulted robotic rm with 6 degrees of freedom nd rotry tble. The lser probe is ble to mesure distnces from 70 to 250 millimetres, chieving 30 µm ccurcy. The robotic rm will move round the workpiece being scnned by using computer-generted dptive scnning pths, which re computed in rel-time while the scnner is discovering the workpiece fetures. The scnned 3D models will be then reproduced on CNC milling mchine. The simultor presented in this pper is designed to be development tool nd test bench for the dptive scnning lgorithms which will be developed. 2 Motivtion Using simultion environment for designing the scnning lgorithms hs severl dvntges: The possibility of collisions between the robotic rm, lser probe nd workpiece is eliminted. As the lser probe is n expensive device, collision voidnce is very importnt point to consider; The system cn be nlyzed in idel conditions, with no surfce reflections, externl light sources or perturbtions in the mesurements; The prmeters of the scnning system components, like cmer loction, focl length, opticl sensor resolution, lser bem width, rotry tble size nd loction, cn be freely chnged, nd the influence of these chnges cn be nlyzed thoroughly; Scnning of objects with complex shpes nd different surfce properties, tht my not be physiclly vilble, is possible. There re lso number of disdvntges, the biggest one being the computtionl power involved for ccurte simultions in rel-time, nd the second one being the difficulty for simulting the less-thnidel conditions of the rel system. In the uthors opinion, the dvntges outweigh the shortcomings involved here. This work is funded by the Ntionl Council for Scientific University Reserch, in the frmework of the Ntionl Pln for Reserch, Development nd Innovtion, grnt 69/2007

2 3 Problem description The lser scnning system tht is to be simulted consists of three min components: The robotic rm The lser probe The rotry tble For the robotic rm, the simultor should llow displying the robot in ny user-defined position. The user should be ble to control either the joint ngles for ech rticultion, or the Crtesin position together with the orienttion given by ZYZ Euler ngles. The user should lso be ble to specify the tool trnsformtion. For the lser probe, the softwre should simulte the interction of the lser bem with ny userdefined workpiece, hving vrious surfce properties. The two cmers which re integrted into the lser probe should lso be simulted, nd the imge which would be cptured by them should be displyed to the user. Furthermore, the lser bem should be detected in the imges from the two cmers, nd the tringultion equtions should be pplied to them in order to compute point cloud. The point clouds obtined from simulting the scnning process from different viewing ngles should be trnsformed into fixed coordinte system, tht will be ttched to the workpiece being scnned. As for the rotry tble, its rottion hve to be simulted, nd the workpiece tht sits on the tble should rotte synchronously with the tble. Behviors such s inertil movement due to fst tble movements do not hve to be simulted; the workpiece should be considered ttched to the tble. 4 Proposed Solution The first decision is to choose which softwre environments re suitble for development of this simultor. Since it is needed to model the interction between the lser bem nd n rbitrry surfce, possible solution is to compute the intersections between the lser rys nd the given object. This is not trivil tsk, nd there re mny 3D renderers tht use the ry trcing method, for which there is good description in [3]. Exmples of ry trcing softwre include the free POV-Ry, Ryshde, Rdince, nd the commercil softwre, such s 3D Studio Mx, My or Cti. For this simultor, POV-Ry ws chosen for modeling the lser bem. POV-Ry uses Scene Description Lnguge [4], which is used to describe the objects, lights nd cmers tht interct in virtul environment by mens of ASCII-bsed input files. This cpbility llows esy interfcing with mny progrmming environments. Furthermore, there is commnd-line version of POV-Ry, tht simplifies integrtion of POV-Ry with other softwre progrms. For the computtionl side of the simultor, which involves 3D mtrix multiplictions nd trigonometric opertions, the progrmming environment ws chosen to be Mtlb. Its dvntges include n interpreted lnguge with syntx oriented for mtrix opertions nd mthemticl functions, lso hving n Imge Processing Toolbox nd built-in 2D/3D grphics cpbilities. 4.1 Robot rm nd rotry tble modeling For modeling the robot rm, there re two min points to consider: computing positions nd orienttions for every link of the robot, including direct nd inverse kinemtics, nd displying the robot to the user in given configurtion. The World coordinte system used here, X 0 Y 0 Z 0, is right-hnded, with the X 0 Y 0 plne being horizontl, X 0 xis pointing forwrd, Z 0 xis pointing upwrds, nd the origin being t the bse of the robotic rm. The rotry tble hs the sme reference frme s the workpiece, X R Y R Z R, nd the Z R xis points in the sme direction s Z Direct nd Inverse Kinemtics The direct kinemtics for the robot rm function is obtined by using the Denvit-Hrtenberg [1] convention. The first step is to ssign individul reference frmes to ech link from the kinemtic chin,

3 ) Reference frmes for robot rm links, lser probe nd rotry tble b) Denvit-Hrtenberg prmeter tble Figure 1: Reference frme ssignment nd Denvit-Hrtenberg prmeter tble which includes the six robot rms nd the lser probe (Fig. 1). The direct kinemtics function is the product of the individul homogeneous trnsformtions [2] for ech link i = An individul mtrix, clled T i 1 i, is the trnsformtion from the (i 1) th link reference frme to the i th link reference frme. The 0 th link is the robot bse, nd the 7 th link is the lser probe. TL 6 is the trnsformtion from the 6th joint (robot flnge) to the lser probe reference frme, nd it is constnt mtrix, since the lser probe is rigidly ttched to the robot. Knowing the the Denvit-Hrtenberg prmeters i, d i, α i nd θ i for ech joint i = 1..6, with θ i being the joint vribles, the individul trnsformtions Ti i 1 cn be written: T i 1 i = R Z (θ i ) T ( i, 0, d i ) R X (α i ) (1) where T (x, y, z) is the homogeneous trnsltion, nd R A (φ) is the homogeneous rottion round xis A with ngle φ. All the distnces re expressed in millimetres, nd ll the ngles re expressed in rdins. The direct kinemtics trnsforms re 4 4 mtrices: TDK = T6 0 = T1 0 T2 1 T3 2 T4 3 T5 4 T6 5 (2) T DK = T7 0 = TDK TL 6 (3) The mtrix TDK my be used for trnsforming ny object ttched to the robot flnge, in this cse, the lser probe. The mtrix T DK expresses the position of the lser reference frme, with respect to the robot bse, nd will be used to convert the lser probe mesurements into n unique coordinte system. The inverse kinemtics ws implemented using Peter Corke s Robotic Toolbox, function ikine, which uses the pseudo-inverse of jcobin method [6]. The simultor is lso ble to connect to n existing robot controller vi TCP/IP nd use its internl inverse kinemtics routine, which hs been found to hve slightly different behvior, depending on the initil estimtion. Considering the rotry tble s 7-th link of the kinemtic chin, nd moving the reference frme to the center of the tble X R Y R Z R, the tble is modelled s new link, pplied before the 6 links of the robot (Fig. 1b), nd the trnsformtions required to convert from the robot bse reference frme to the rotry tble reference frme nd vicevers re: Rendering T R 0 = R Z ( θ R ) T ( R, 0, d R ) = R Z ( θ R ) T ( 500, 0, 200) (4) TR 0 = ( T0 R ) 1 = T ( R, 0, d R ) R Z (θ R ) = T (500, 0, 200) R Z (θ R ) (5) The robot is displyed by rendering it using POV-Ry, from tringle mesh model obtined from the CAD model for the robot rm. For the robot rm used in this project, the CAD model is vilble for

4 downlod from the mnufcturer s web site [5]. Ech link of the robotic rm is modeled s tringle mesh, nd cn be freely moved in the scene. Using the joint vlues computed from Inverse Kinemtics routine, the links re plced in the correct position using the individul trnsformtions T i 1 i from Eq. 1, nd the scene is rendered, with the result being shown in Fig. 4 from Section 5. A similr pproch ws used for rendering the lser probe nd the rotry tble, by creting POV-Ry models nd pplying the trnsformtions TDK nd T R 0 from Eq. 2 nd Eq Lser probe modeling The reference frme of the lser probe is X L Y L Z L, from Fig. 1, nd it will be clled XYZ in this section in order to simplify the nottions Lser bem modeling The lser probe emits lser bem focused into plne, such s when it intersects surfce, it csts line or curve. This lser bem my be modelled s point light source, which is constrined to pss to nrrow opening (Fig. 2). In the exmple from Fig. 2b, the lser rys strt from origin, is projected in the positive direction of the Z xis, the plne of the lser rys is YZ nd the nrrow opening hs the dimensions L (lrge edge) nd W (smll edge) nd is locted t distnce D from the origin. ) Lser bem modeling principle b) POV-Ry exmple Figure 2: Lser bem modeling using POV-Ry Cmer modeling nd tringultion The cmers used in the lser probe re be modeled s two stndrd perspective cmers, which my be implemented in POV-Ry by entering their prmeters such s position, orienttion nd focl length. In the following text, only one of the two cmers will be described, s the other one is identicl nd symmetricl to the first one. Let XYZ be the reference frme of the lser probe (Fig. 3b nd c), nd let xyz be the reference frme of the CCD rry from the cmer (Fig. 3 nd b). Referring to Fig. 3b, the cmer position nd orienttion with respect to the lser device is given by the prmeters, b nd φ. Using these nottions, let P = (P X ; P Y ; P Z ) the point of reflection of lser ry, in the XYZ reference frme, nd let p = (p x ; p y ) be the coordinte of the pixel t which the ry ws detected on the CCD mtrix, in xy reference frme. Knowing the pixel coordintes p, the loction of the point P cn be expressed using the tringultion equtions (6): P X = 0 P Y = ( f sin φ rctn p ) p x P Z = y f ( f tn φ rctn p ) + b (6) y f where f = H if the unit length is considered to be 1 pixel, i.e. the distnce between two djcent 2 tnγ pixels on the CCD rry. The re in the plne determined by the lser rys, i.e. YZ plne in Fig. 3c, is trpezoid, nd its limits re z min, z mx, y min nd y mx, whose expressions re given in Eq. 7. The scnning rnge is thus given by z min nd z mx, nd the length of the lser line L L tht is effectively being nlyzed is dependent of the distnce of the scnned object with respect to the lser probe, nd vries from L Lmin = 2 y min

5 ) CCD sensor nd its reference frme b) Side view of the lser probe c) Front view of the lser probe Figure 3: Tringultion when the workpiece is close to the probe, to L Lmx = 2 y mx when the workpiece is fr from the lser probe. y min = tn γ sin(φ + γ) y mx = tn γ sin(φ γ) z min = tn(φ + γ) + b (7) z mx = tn(φ γ) + b Creting the point cloud There re three min steps in computing point cloud by processing the imge obtined from the simulted cmer. In the first plce, one needs to identify the lser bem in the imge. The simplest nd fstest method is to use gry workpiece nd red lser, nd pply thresholding opertion on the sturtion component of the imge. The second step is to pply Eq. 6 to every pixel on the lser bem, obtining cloud point in the lser probe s locl reference frme. The third step is to trnsform the coordintes of the points in n reference frme tht is ttched to the workpiece. The homogeneous trnsformtion required here is computed by first expressing the point cloud coordintes in robot s World reference frme, nd then expressing them in the rotry tble s reference frme, using the mtrices defined in Eq. 3 nd Eq. 4: T PC = T 0 R T DK (8) A more dvnced imge processing strtegy would be ble to chieve sub-pixel ccurcy, nd my lso ddress the reflections, mbient light nd other error sources.

6 5 Simultion exmple The screen shot of the simultion softwre is displyed in Fig. 4. The user cn control either the position nd orienttion of the robot in Crtesin mode, reltive to either robot bse or rotry tble, or the individul joint ngles. The softwre my simulte continuous movement of the lser probe over the workpiece, compute the imges tht would be seen by the two cmers, nd generte point cloud from processing them (Fig. 4b-d). ) Screen shot of the simultor b) Schemtic view of lser probe c) Imges from the two cmers d) Computed point cloud Figure 4: Lser probe simultor 6 Summry nd Conclusions A computer simultor of 3D lser scnning system ws presented, long with the theoreticl spects involved. The softwre simultes the kinemtics of the robot rm moving synchronously with rotry tble, nd the interction of the lser probe, which uses lser bem nd two cmers, with virtul workpiece. A point cloud representing the workpiece cn be generted by processing the rendered imges. Other types of robot rms my be simulted by providing their Denvit-Hrtenberg prmeters nd 3D mesh model for displying. The bottleneck in the simultion is the rendering speed, which is currently performed by ry trcing. Therefore, rel-time simultions re not currently possible. On n Intel Pentium 4-M processor t 2.0 GHz, the simultor is ble to render nd process round 3 frmes per second by simulting sensor, while the speed of rel lser probe my be of the order of 100 frmes per second, s described in the technicl specifictions. References [1] Mrk W. Spong et.l., Robot Modeling nd Control, John Wiley nd Sons, Inc., pp , 2005 [2] Tom Dvis, Homogeneous coordintes nd computer grphics, 2001 [3] Andrew S. Glssner, An Introduction to Ry Trcing, Morgn Kufmnn Publishers, Inc., [4] Persistence of Vision Rystrcer Pty. Ltd., POV-Ry Online Documenttion [5] Adept Technology, Inc. Web Site, [6] P.I. Corke, A robotics toolbox for MATLAB. IEEE Robotics nd Automtion Mgzine, Vol. 3, No. 1, pp , Mrch Theodor Borngiu, Anmri Dogr, Alexndru Dumitrche University Politehnic of Buchrest Deprtment of Automtic Control nd Industril Informtics Spliul Independentei 313, Buchrest, Romni E-mil: borngiu@cimr.pub.ro, dogr@cimr.pub.ro, lex@cimr.pub.ro Received: April 14, 2008