RobotikTrainingPPT课件

1、Contents Basic Mathematics Coordinates and Frames Robots Robot Vision - 2D and 3D Calibration第1页/共48页Intelligent Systems Robotics and AutomationISRA VISION SYSTEMS GROUPRobot Vision - Basic Training CourseBasic Mathematics第2页/共48页Vectors and MatricesabcvectormatrixABCDEFGHIrowcolumn addition and sub

2、tractiona db ec fdefabc=A JB KC LD M E NF OG PH QI RA B CD E FG H IJK LM N OP Q R=第3页/共48页Vectors and Matrices - Multiplicationa d + b e + c fdefabc=A*J+B*M+C*P A*K+B*N+C*QA*L+B*O+C*RD*J+E*M+F*P D*K+E*N+F*QD*L+E*O+F*RG*J+H*M+I*PG*K+H*N+I*QG*L+H*O+I*RA B CD E FG H IJK LM N OP Q R*=defabcx=b*f-e*ca*f-

3、d*ca*e-b*dABCaution: A * B B * A第4页/共48页Matrix Math - Basic Rules A * B B * A A * B only possible if number of rows of A = number of columns of B and number of columns of B = number of rows of A number of rows of A*B = number of rows of Anumber of columns of A*B = number of columns of B A * A-1 = 1A

4、 * 1 = A第5页/共48页Matrix EquationsEquationA * B * C=D * E * Fmultiplication of both sides with the same matrix from the same sidewith C-1 from the right sideA * B * C * C-1=D * E * F * C-1A * B * 1=D * E * F * C-1A * B=D * E * F * C-1or with A-1 from the left side A-1 * A * B * C=A-1 * D * E * FB * C=

5、A-1 * D * E * F第6页/共48页Intelligent Systems Robotics and AutomationISRA VISION SYSTEMS GROUPRobot Vision - Basic Training CourseCoordinates and Frames第7页/共48页Some Expressions Coordinate System (Coordinate) Frame Coordinate system attached to an object or a specific point Pose Describes the position a

6、nd orientation of a frame with respect to a coordinate system (or frame) Transformation Describes the relation between two frames Transforms a pose from one frame to anotherxyz第8页/共48页3D - 2D 3D: 6 degrees of freedom (d.o.f.)3 translations along and 3 orientations around the 3 cartesian axes 2D: 3 d

7、.o.f.2 translations along the 2 cartesian axes and 1 orientation in the plane 2D is a special case of 3D All 3D mathematics is also valid for 2D. One translation and two angles are always zero in 2D第9页/共48页Description of a point in spacexyzCPCyPCxPCzPCCxPCxP = (CxP, CyP, CzP)T = Position in frame C

8、of point Pcoordinate x in frame C of point PCxPCyPCzP第10页/共48页Description of a pose/frame in space - TranslationxyzCyACxACzACCxPCxA = (CxA, CyA, CzA)T = Position in frame C of origin of frame/pose Aonly translation of the frame/poseCxACyACzAAxyz第11页/共48页Description of a pose/frame in space - Rotatio

9、n / OrientationxyzCAxyzRot-xRot-yRot-zvector of 3 angles:Rot-xRot-yRot-zABCWPR3 angles which describe the orientation ofpose/frame A with respect to frame CCAUTION: different definitions exist and areused in different robot controllersnxoxaxnyoyaynzozazrotation matrixPossible representations of orie

10、ntationsmatrix which contains the 3 unit vectors n,o,a of frame A expressed in frame Cnznxny1quaternionq0q1q2q3angle andaxis in C which describerotation of A in a normalized formq0“q1“q2“q3“= CRACQA = CACCCAAA第12页/共48页Description of a pose/frame in space - Translation and OrientationxyzCAxyzRot-xRot

11、-yRot-zCxACyACzAfirst translatethen rotate第13页/共48页Representation of Translation and Rotation 6 dimensional vector T-Matrix Quaternion and 3 dimensional vectorxyzRot-xRot-yRot-znxoxaxpxnyoyaypynzozazpz0001q0q1q2q3xyzorientationtranslationorientationorientationtranslationtranslation=第14页/共48页Definiti

12、on of the Orientation Vector used by ISRAxyzxyzxyzxyzRot-xRot-yRot-zorder of rotationsrotate aroundrotated axesfirst rotateRot-z around zthen rotateRot-y around rotated ythen rotateRot-x aroundrotated xright hand rulepositive direction of axis =direction of thumbdirection of fingers =positive direct

13、ion of anglethumbfingeraxis第15页/共48页Other Definitions of the Orientation Vector Other order of notation Other order of rotation Rotation around fixed axes (Euler-angles)The different definitions are not compatible.The conversion from each of those defintions to a rotation matrix or quaternion and vi

14、ce versa is different.In general it is not possible to perform calculations with the 3 dimensional orientation vector. It is only usedfor the representation in interfaces, especially the user interface, as it is very difficult to have an imagination of the meaning of a rotation-matrix or a quaternio

15、n. However all calcualtions in a system are performed internally either in rotation matrices / T-matrices or in quaternions and translational vectors.第16页/共48页Common Robot Interfaces and Definitions FANUC Representation of orientations:rotation around fixed axesorder of rotations: x-y-znotation: W =

16、 Rot-x, P = Rot-y, R = Rot-z InterfaceVersion 1 (common in the US): 6 dimensional vectorVersion 2 (common in Europe and Asia): T-matrix ABB Representation of orientations:rotation around rotated axes (as ISRA)order of rotations: z-y-x (as ISRA)notation: A = Rot-x, B = Rot-y, C = Rot-z InterfaceVersi

17、on 1: 6 dimensional vectorVersion 2 : quaternion and 3 dimensional translation vectorWPRSurprizingly this results in the same angles as we use, although the definition is different - can be treated as the sameABCdepends on controller version第17页/共48页Common Robot Interfaces and Definitions (ctd.) KUK

18、A Representation of orientations:rotation around rotated axes (as ISRA)order of rotations: z-y-x (as ISRA)notation: A = Rot-z, B = Rot-y, C = Rot-x (= ISRA backwards) Interface6 dimensional vectorABC第18页/共48页Transformation of PointsxyzAxyzCPCyPCxPCzPCCxPATC (= APC)CxPposition of point P described in

19、 frame CAxPposition of point P described in frame AAPCdescription of frame A in C (position and orientation)ATPtransformation from A to CAxPThe T-matrix which describes the position and orientation of frame A in frame C transforms the description of a point in A to a description of the same point in

20、 CAxPAzPAyP第19页/共48页Transformation of Points - Math using T-matrices:Cx=CTA*Ax using quaternions (not used in ISRA -systems):use the quaternion-calculusnxoxaxpxnyoyaypynzozazpz0001xyz1xyz1CCAA*=第20页/共48页Transformation of FramesxyzByRBxyzAxyzCAPC (= ATC)BPC (= BTC)ATB (= APB)APBposition and orientati

21、on of frame B described in frame ABPCposition and orientation of frame C described in frame BAPC position and orientation of frame C described in frame AATBtransformation from B to AThe T-matrix which describes the position and orientation of frame B in frame A transforms the description of a frame

22、described in B to a description of the same frame in A第21页/共48页Transformation of Frames - Math using T-matrices: APC =ATB* BPC using quaternions (not used in ISRA -systems):use the quaternion-calculusnxoxaxpxnyoyaypynzozazpz0001AB*=nxoxaxpxnyoyaypynzozazpz0001BCnxoxaxpxnyoyaypynzozazpz0001ACAs the n

23、otations of poses and transformations are the same it is very common to name both T and call both transformations第22页/共48页Inverse TransformationxyzByRBxyzAxyzC ATC BTCATB BTC = ATB-1 * ATC Inverse transformation- transformation in the oposite direction第23页/共48页Inverse Transformation - Mathnxoxaxpxny

24、oyaypynzozazpz0001ABATB = ATB-1 = nxoxaxpxnyoyaypynzozazpz0001AB-1= BTAATB * ATB-1 = ATB-1 * ATB = 1 = 1000010000100001- no transformation第24页/共48页Multiple TransformationsxyzByRBxyzAxyzCATEBTDATBxyzDxyzEETCCTDATCATC = ATB * BTD * CTD-1 ATC = ATE * ETCATC * CTD = ATB * BTDATB * BTD * CTD-1 * ETC-1 *

25、ATE-1 = 1 第25页/共48页2D Transformation as special case of 3Dxy000Rot-znxox0pxnyoy0py00100001orientation: only rotation around ztranslation: no z-translationstandard case: x-y plane6 dimensional vectorT-matrix第26页/共48页Transformations - ExamplesxyzBRByRBxRBzRBxyzAxyzCPCyPCxPCzPCCxPAxRAxPATC = APCCTB = C

26、PBATB = APB第27页/共48页Intelligent Systems Robotics and AutomationISRA VISION SYSTEMS GROUPRobot Vision - Basic Training CourseRobots第28页/共48页Robot - Expressionsbasehandtool / gripperloadlinkjoint / axis第29页/共48页Robot - Frames and Transformationsbase frame / world frameuser base framehand frameuser too

27、l framebase offsettool offsetposition of tool in worldposition of tool in user frame第30页/共48页Example User Base Frameworldframeuser base framemove thescreen ina frameattachedto thescreenopening第31页/共48页Robot - Possibilities to Drive the RobotxyzzxyxyzA1A2A3A4A5A6worldbasetoolCommon frames todrive in:

28、- world- base (= user base)- tool- jointThe joint frame“ is notreally a frame.In the joint frame the robotdrives its axes separately.jointtool center point TCP第32页/共48页Robot MotionsTCPstart point of motionend point of motionlinear motion LINpoint to point motion PTPlinear motion:The TCP moves on a l

29、inear pathfrom the starting point to the end pointpoint to point motion:all axes are driven in a way that they arriveat the same time at their end location.The path of the TCP is not defined!第33页/共48页Intelligent Systems Robotics and AutomationISRA VISION SYSTEMS GROUPRobot Vision - Basic Training Co

30、urseRobot Vision - 2D and 3D第34页/共48页Robot Vision - from 2D to 3D 2Dfind an object in a planeORIS 2 1/2 Dfind an object in a plane and determineORISthe distance to the object 3D monocularfind an object in space with one cameraCAPMESper view 3D stereofind an object in space with twoVIAMEScameras per

31、view 3D selective d.o.f.sfind and object in space but determineVIAMESonly part of the d.o.f.s第35页/共48页Examples 2Ddepaletizing of cylinder heads2 1/2 D:find the layer in which the cylinder head is located andfind the position and orientation in the layerORIStake boxes from a conveyor2D moving line:fi

32、nd the box on the conveyor with respect to a movingframe attached to the conveyorORIS第36页/共48页Examples 3Dlocate car body in spray painting line3Dmonocularfixed camerasCAPMESlocate body parts in a rack for unloading3Dstereomoving cameras on robotrestricted d.o.f.sVIAMES第37页/共48页Intelligent Systems Ro

33、botics and AutomationISRA VISION SYSTEMS GROUPRobot Vision - Basic Training CourseCalibration第38页/共48页Purpose of Camera - Calibration Purpose: Determination of camera position in space x, y, z. Determination of camera parameters picture size, lense influences Determination of common coordinates for robot and vision Method: Measurement of known points in space by the cameras Calculation of paramet