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L 2 ! q 2 !L 2 !8q 2 !q 2 )zp 2 L 2 L 2 ]q 2 L 2 ( q 2  L 2 Bq 2  q 2  q 2  q 2 qSymbol- 2 = 2 t+ 2  + 2 != 2 !\+ 2 ! + 2 = 2 + 2 + 2  + 2  = 2  `- 2  + 2  + 2 =Times New Roman- 2 cos~p 2 sinpP 2 } sinpP 2 cos~p 2 ycos~p 2 !sinpP 2 !sinpP 2 !e sinpP 2 !sinpP 2 !@cos~p 2 cos~p 2  cos~p 2 sinpP 2  ([ 2  )[ Times New Roman@- 2 C1p 2 C22p 2 C2p 2 C 3p 2 Cj 23pp 2 CU4p 2 C1p 2 C234ppp 2 i1p 2 i2p 2 i2p 2 i 3p 2 iR 23pp 2 i=4p 2 i1p 2 iW234ppp 2 B,1p 2 BA2p 2 B2p 2 B 3p 2 B 23pp 2 BG4p 2 B234ppp 2 J 52p 2 J D3p 2 J E 4p 2  E5pTimes New Roman- 2  90Symbol- 2  j 2 y MT Extra- 2 oY & "System-ࡱ FMicrosoft Equation 2.0 DS Equation Equation.29qࡱ;ࡱ;  lJ@qJmJ Directkinematicequationsx=cosq 1 L 2 sinq 2 +L 3 sinq 23 ()+L 4 cosq 1 cosq 234 y=sinq 1 L 2 sinq 2 +L 3 sinq 23 ()+L 4 sinq 1 cosq 234 z=L 1 +L 2 cosq 2 +L 3 cosq 23 +L 4 sinq 234 j=90 o -(q 2 +q 3 +q 4 )y=q 52  Oh+'0 D h   @d 19316,19378,19450,19488,19787,19C:\WINWORD\TEMPLATE\NORMAL.DOT"7TH INTERNATIONAL DAAAM SYMPOSIUMBranko KATALINICFSB-Automatika@ ^@%Hi@)ma@0%Microsoft Word 6.0759th DAAAM INTERNATIONAL SYMPOSIUM Technical University Cluj - Napoca, Cluj - Napoca , Romania 22-24th October 1998 COLLISION SPACE MODELLING FOR RM501 ROBOT Crnekovi, M., itum, . Abstract: Problem of the collision space modeling for RM501 robܥe3 de.)X%&88DD$D$D$D$FD FFFF"FFWF:$G$G$G$G2G2G2GIIII II@JXTkXS^J) $D2G$G2G2G2G^J2GDD$GF2G2G2G2GD$$G$D$GIHDVD6DDDD2GI2G2G9th DAAAM INTERNATIONAL SYMPOSIUM Technical University Cluj - Napoca, Cluj - Napoca , Romania 22-24th October 1998 COLLISION SPACE MODELLING FOR RM501 ROBOT Crnekovi, M., itum, . Abstract: Problem of the collision space modeling for RM501 robot has been research. Collision space model is prerequisite for the robot trajectory calculation in the presence of obstacles. Instead of model building in the configuration space, modeling in the external coordinates has been used. This approach provides acceptable model length and time for its calculation. Calculated trajectory is similar to the human like solution faced with the same problem. Key words: path planning, obstacle avoiding, collision-space modeling. 1. Introduction If any problem occupies robotics several last years this certainly is problem of robot movement in space with obstacles. Why this problem and its solution are so important? Because if you know the solution of this problem you can make robot "intelligent" behavior. It means you can explain robot task (i.e., programming a robot) by the human language, with no specially explanation how to avoid obstacles during the task execution. This kind of program will be short and clear understood (Crnekovi 1995). The problem exists booth for mobile robots and for fixed robot arm. Solution methods are similar and always are compromise between execution time and accuracy/reliability. In this article obstacle space is modeling for the RM501 robot (Heine 1991). 2. ROBOT SPACE - MODELLING Robot Mitsubishi RM501 has 5 rotational degrees of freedom, fig. 1. For robot trajectory building it is necessary to know its kinematics. Internal and external robot coordinates are connected by the following equations:  EMBED Equation.2   EMBED Equation.2   EMBED Equation.2  Each internal coordinate is limited from qi.min to qi.max. These limits are well known and could be take into account in the final trajectory solution. Behind that, each robot link dimension should be determined for later collision calculation. Fig. 1. Robot Mitsubishi RM501 There are several methods how robot space could be modeled. One of the common and wide accepted is configuration space method (Crnekovi 1992, Crnekovi 1996) or C-space. In the C- 896. For each cell (i.e., set of x, y, z) collision by any obstacle is calculated. At that point we will found out that result is not well defined, because it is possible more robot orientation ((,() in the same point (x, y, z). The cell (x, y, z) will set free if exist at least one set of orientation ((,() for which there is no collision between robot, working port and any obstacle around the robot. In such a space, occupied by free and full cells, it is possible to find the robot trajectory that is the shortest from a start to a goal position. It the robot moving along the trajectory there will be no robot collision by any obstacle. It is obviously that calculated trajectory will not be optimal in global sense (like it will be if it was calculated in C-space), but it will be designed in much less, acceptable time. After all, "intuition" told us that it will not be far away from optimal. 3. COLLISION CALCULATION Each obstacle, working part and robot can be presented by set of cubes. Obstacles are presented by static cubes, while robot cubes depends on robot position (working part could be consider as a part of the robot). The most often is that only one cube is necessary for each robot link, obstacle and working part presentation, but if a part has complex geometry more than one cube can be used. The robot consists of three links (some part of robot is neglected because they never enter the working space) and the working part, which are presented by five cubes. Cubes position and orientation depends on robot position (x, y, z) and orientation ((,(). Thus a problem of robot collision by any obstacle could be stated as: for a given position and orientation, the robot is in collision if any of its cube collides by any obstacle cube. Now it is necessary to developed an algorithm for cube to cube collision detection. Although there are several methods to calculate this collision, we need some which is enough fast and reliable (it has to answer only with yes or no). This is because calculation is necessary for all of 12 896 cells in working space. ......ggg..xxx..gxxx........... .......ggg..................... ....gogg.ggg................... ....oogog..g.....ggg........... ..gggogggog......gggg.......... ..goggggggog.....oooog......... .gggoggggggg.....ggggg......... ggooggggggg......gggog......... ggooooggoog......gggog......... ggggooooggg......gggog......... ggggggooog.......oooog......... gggggggggg.......gggggg........ gggggggggg.......gggggg........ ggggggggg........ggggggg....... xxgggggggg.......ggggggg.....xx xxxgggggg........ggggggggg..xxx Fig. 2. Slice of the collision space model at level z=40 mm (not in scale). Two obstacles are in the working space. Cube To Cube (CTC) collision algorithm is divided in several steps. Because each cube consists of six areas, CTC collision problem is transformed in Area To Area (ATA) intersection algorithm. That means: if any robot cube/area is intersected by any obstacle cube/area that cell is not free. Because each area consists of four lines, ATA algorithm is transformed to Area To Line (ATL) intersection algorithm. Thus a final rule for the robot collision by the environment is: if any robot cube/area is intersected by any obstacle cube/area/line (or vice versa) that cell is not free. It means that ATL intersection algorithm is the basically algorithm for the robot space modeling. Its basis is matrix of homogenous transformation. Applying the CTC, ATA and ATL algorithms, fig. 2 presents a slice of the collision space model for the robot RM501 at level z=40 mm. Symbols has meaning: ( cell is free ( position is out of the robot range o collision by the working part g collision by the robot gripper Because the robot and obstacles dimension are of the same order, the free space is relatively small. Thus, situations are possible where solution of collision free trajectory does not exist. For complete model building on the PC Pentium 166, 360 seconds is required. This time strongly depends on the number of obstacles and itspace method problem of the robot collision by obstacles is transformed in the robot internal coordinates. Each internal coordinate is discretized and for each set of qi collision by any obstacle is calculated. If there is no collision then qi set is free, otherwise it is occupied and could not be a potential point for trajectory solution. In the case of robot RM501 C-space is 5-dimension space. If we take into account each coordinate limit and discretization of only 10(, then total number of C-space will be 2 274 480 cells (sets of qi). It is obviously, that such large model is not recommended for starting the problem solving. So we will have to find some another way. Let us back to external coordinates (x, y, z, (, (). The idea is next: Let us select a cube in front of the robot, inside which all robot movement will be done. Let dimensions of the cube be length(width(high = 600(300(500 mm and discretization of each coordinate is 20 mm. In that case the total number of cells in working cube will be "only" 12Z"[""""###%%%%&&(&&&&&&&&&&''#'Q'T'V'y'''''( (_(j(p(q(()))))))))!)")&)')+)-).)**+++"+9+`+n+o+++++++++8888U]c U[]cu UV]c]ch]cV]cU[] JjV]c]c JyV]cO"^stu             ! " # %%%%%%%%% )s dimension, and could be significantly reduced as explained in the conclusion. As long as robot environment does not change, free space model is valid and there is no necessary for its new building. After the model of the free space has been build, some search strategy could be applied for trajectory calculation. One of the most fastness and reliable is greed search algorithm (GSA) improved by the grass fire algorithm. It finds the shortest trajectory for a few seconds. Unfortunately, this trajectory isOle @PIC  ALMETA C CompObjsfࡱ; L ] ࡱ;  ]>:  .1  & & MathType-  $ - $ U- ] Z Z -[-[ V V Z Z Times New Roman-2 Inverse^~~pp~2  kinematic~P~PP~2  equations~PPp 2 q 2 y~ 2 x~ 2 g7x~ 2 g2x~ 2 gL 2 giq 2 @y~ 2 Dy~ 2 L 2 Zq 2 Rzp 2 ;zp 2 L 2 1 [d 2 1 x~ 2 1 y~ 2 1 ~zp 2 1 p L 2 q 2 DL 2 M d 2 L 2 f L 2 s d 2 Ozp 2 AL 2 vx~ 2 y~ Times New Roman- 2 1p 2 3p 2 P4p 2 1p 2 h3p 2 hb4p 2 h1p 2 !3p 2 !G4p 2 y E3p 2 W2p 2 y i3p 2 {2p 2 y 3p 2 y 1p 2 2p 2 2p 2 1 2p 2 h 2p 2 h  2p 2 1 ' 3p 2 h  2p 2  2p 2 1 3p 2 1 1p 2 K3p 2 2p 2 K3p 2 ,2pTimes New Roman- 2 90 2  2Symbol- 2 7= 2 gZ= 2 g- 2 c= 2 - 2 c= 2 - 2 1 *= 2 1 + 2 1 ,+ 2 1 - 2 Z= 2 - 2 + 2 - 2 m 2 m 2 m 2 m 2  m 2 m 2 `- 2 `- 2 + 2  m 2 Cm 2 G m 2 rm 2  m 2 Cm 2 G m 2 rmTimes New Roman-2 arctan(p~P[ 2 =/P 2 )[ 2 gcos~p 2 gncos~p 2 sinpP 2 _cos~p 2 sinpP 2 1 ([ 2 1 )[2 arccosp~~p2 &arctanp~PSymbol- 2 g j 2  j 2 6j MT Extra- 2 < BoY & "System-<L%P (&(|ࡱ FMicrosoft Equation 2.0 DS Equation Equation.2ࡱ;ࡱ; *Π/?7Hx 7H Inversekinematicequations in the external coordinates (x, y, z) and tell us nothing about the robot orientation ((,() in the each trajectory point. To find out a set of the robot orientation, additional calculation is necessary. Its base is optimization technique in 1+1 space ((+(). 4. CONCLUSION Although presented algorithm for the robot collision space modeling is not optimal in the global sense, it is simple and fast for running on the PC computers. Calculated trajectory is similar to the human-like solution faced with the same problem. It is possible to improve procedure of trajectory finding by shortening of collision space modeling. The main idea is not to build the whole collision space. Instead of that, process immediately begin by the trajectory searching in unmodeled collision space. Collision space is modeled only for those cells that requires searching process. This technique will be research and described in further works. 5. References R. Heine, T. Schnare, (1991). Kollisionsfreie Bahnplanung fr Roboter, Robotersyq 1 =arctan(y/x)x 3 =x-L 4 cosq 1 cosjy 3 =y-L 4 sinq 1 cosjz 3 =z-L 4 sinjd= x 32 +y 32 +(z 3 -L 1 ) 2 q 2 =90 o -arccosL 22 +d 2 -L 32 2L 2 d()-arctanz 3 -L 1  x 32 +y 32 ()&D0Jࡱ; L lࡱ; PIC LMETA CompObjfObjInfo O  .1   &  & MathType -0 Times New Roman- 2 bq 2 d 2 L 2  L 2 -L 2 -L 2 q 2 q 2 'q 2 9q Times New Roman- 2 3p 2 M2p 2 q2p 2 h2p 2  3p 2  2p 2 u&2p 2 uN 3p 2 4p 2 2p 2 3p 2 5pTimes New Roman- 2 -2 2  90Symbol- 2 bL= 2 - 2 " - 2 1m 2 xm 2 \m 2 1G m 2 xG m 2 \G m 2 Z= 2 - 2 a+ 2 \+ 2 9E=Times New Roman-2 barccosp~~p 2 ([ 2  )[ MT Extra- 2 BoYSymbol- 2  j 2 9 y & "System-p 2 ;zp 2 L ࡱ;  FMicrosoft Equation 2.0 DS Equation Equation.2ࡱ;ࡱ; * /?7Hl7H q 3 =arccosd 2 -L 22 -L 32 2L 2 L 3 ()q 4 =90 o -(q 2 +q 3 +j)q 5 =ycAfeesteme 7, pp 17-21 M. Crnekovi, (1992). Configuration Space and Network of Equidistant Path - Methods for Robot Global Path Planning, 3rd DAAAM Symposium, pp 43-44, Budapest M. Crnekovi, (1995). Task Oriented Language for RM501 Robot, Strojarstvo 37(1995) , pp 31-35 M. Crnekovi, (1996). Trajectory Planning for SCARA robot in C-space, 7th DAAAM Symposium, pp 89-90, Wiena Author: Assistant Professor Mladen CRNEKOVI, Faculty of Mechanical Engineering and Naval Architecture Zagreb, Automation Department, I. Luia 1, 10000 Zagreb, Croatia, Phone: +385 1 61 68 435, e-mail: mladen.crnekovic@ fsb.hr Assistant eljko ITUM, Faculty of Mechanical Engineering and Naval Architecture Zagreb. Research is supported by the Croatian Ministry of Science and Technology, No. 120008. y x -300 300 0 200 500 .Annnn #.Annnn ࡱd0Z@&u ( ,!Q(~,!Q&%!Q(,!Q(,!Q(,!Q(,!Q(9H,!QCOLLISION SPACE MODELlem of the collision space modeledbehavior. The problem exists bon this article obstacle space has been 2. ROBOT SPACE lated. At that point we will fim a start to a goal position. Ifime. After all, "intuition" tell solutionsthe are neglected because into "u           J K P T U Z չեՑقق]chuDl8]ceKuDl8]avKuDp8]cefKuDp8]avKuD 8]ceKuD 8]avKuD]c]ccU[]]cV]c UV]c U[]c]cU]U] U[]c]]c3Z , K klqr|}")1234 '()*9:$%UV @APQ"%"X"Y"Z" J]cU]c]]cuD]aU] J]c JyV]c JjV]c J]c]ch UV]c]cV]c]cF# $ % & ' ( ) * + , K L 9$%Hk=`2UV*    7)?Ou ####%%%%&& 'y'z'_(()))))))))!)")&)')+),)-).)o+  mmrm'K@Normal]a "A@"Default Paragraph Font "@ CaptionxxU O u 7]cNow it is necessary to develop of 01 at level z=40 mm. 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Oh+'0 D h   @d 19316,19378,19450,19488,19787,19C:\WINWORD\TEMPLATE\NORMAL.DOT"7TH INTERNATIONAL DAAAM SYMPOSIUMBranko KATALINICFSB-Automatika@ ^@%Hi@hq)ma@L%Microsoft Word 6.074386649,386681,386682,386687,386698,386710,386714,386742,386744,386745,386747,386763,386779,386782,386792,386795,386817,386820,386868,386889,386893,386899,386907,386912,386920,386927,386929,386945,386949,386951,386952,386955,386959,386973,386997-386999,3870ܥe3 Te.)X%&88DD$D$D$D$FD FFFF"FFcWF:$G$G$G$G2G2G2GIIII II@JWTGXS^J $D2G$G2G2G2G^J2GDD$GF2G2G2G2GD$$G$D$GIHDVD6DDDD2GI2G2G9th DAAAM INTERNATIONAL SYMPOSIUM Technical University Cluj - Napoca, Cluj - Napoca , Romania 22-24th October 1998 COLLISION SPACE MODELLING FOR RM501 ROBOT Crnekovi, M., itum, . Abstract: Problem of the collision space modeling for RM501 robot has been research. Collision space model is prerequisite for the robot trajectory calculation in the presence of obstacles. Instead of model building in the configuration space, modeling in the external coordinates has been used. This approach provides acceptable model length and time for its calculation. Calculated trajectory is similar to the human like solution faced with the same problem. Key words: path planning, obstacle avoiding, collision-space modeling. 1. Introduction If any problem occupies robotics several last years this certainly is problem of robot movement in space with obstacles. Why this problem and its solution are so important? Because if you know the solution of this problem you can make robot "intelligent" behavior. It means you can explain robot task (i.e., programming a robot) by the human language, with no specially explanation how to avoid obstacles during the task execution. This kind of program will be short and clear understood (Crnekovi 1995). The problem exists booth for mobile robots and for fixed robot arm. Solution methods are similar and always are compromise between execution time and accuracy/reliability. In this article obstacle space is modeling for the RM501 robot (Heine 1991). 2. ROBOT SPACE - MODELLING Robot Mitsubishi RM501 has 5 rotational degrees of freedom, fig. 1. For robot trajectory building it is necessary to know its kinematics. Internal and external robot coordinates are connected by the following equations:  EMBED Equation.2   EMBED Equation.2   EMBED Equation.2  Each internal coordinate is limited from qi.min to qi.max. These limits are well known and could be take into account in the final trajectory solution. Behind that, each robot link dimension should be determined for later collision calculation. Fig. 1. Robot Mitsubishi RM501 There are several methods how robot space could be modeled. One of the common and wide accepted is configuration space method (Crnekovi 1992, Crnekovi 1996) or C-space. In the C-space method problem of the robot collision by obstacles is transformed in the robot internal coordinates. Each internal coordinate is discretized and for each set of qi collision by any obstacle is calculated. If there is no collision then qi set is free, otherwise it is occupied and could not be a potential point for trajectory solution. In the case of robot RM501 C-space is 5-dimension space. If we take into account each coordinate limit and discretization of only 10(, then total number of C-space will be 2 274 480 cells (sets of qi). It is obviously, that such large model is not recommended for starting the problem solving. So we will have to find some another way. Let us back to external coordinates (x, y, z, (, (). The idea is next: Let us select a cube in front of the robot, inside which all robot movement will be done. Let dimensions of the cube be length(width(high = 600(300(500 mm and discretization of each coordinate is 20 mm. 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