Robot-Arm-Kinematics=DH-intro：机器人手臂运动学=-DH-intro课件

1、More details and examples on robot arms and kinematicsDenavit-Hartenberg NotationMore details and examples on rINTRODUCTIONForward Kinematics: to determine where the robots hand is? (If all joint variables are known)Inverse Kinematics: to calculate what each joint variable is? (If we desire that the

2、 hand be located at a particular point) INTRODUCTIONForward KinematicDirect KinematicsDirect KinematicsDirect Kinematics with no matricesWhere is my hand?Direct Kinematics:HERE!Direct Kinematics with no matrDirect KinematicsPosition of tip in (x,y) coordinatesDirect KinematicsPosition of tDirect Kin

3、ematics Algorithm1) Draw sketch2) Number links. Base=0, Last link = n3) Identify and number robot joints4) Draw axis Zi for joint i5) Determine joint length ai-1 between Zi-1 and Zi6) Draw axis Xi-17) Determine joint twist i-1 measured around Xi-18) Determine the joint offset di9) Determine joint an

4、gle i around Zi10+11) Write link transformation and concatenateOften sufficient for 2DDirect Kinematics Algorithm1)Kinematic Problems for ManipulationReliably position the tip - go from one position to another position Dont hit anything, avoid obstacles Make smooth motions at reasonable speeds and a

5、t reasonable accelerations Adjust to changing conditions - i.e. when something is picked up respond to the change in weightKinematic Problems for ManipulROBOTS AS MECHANISMsROBOTS AS MECHANISMsRobot Kinematics: ROBOTS AS MECHANISMFig. 2.1 A one-degree-of-freedom closed-loop four-bar mechanismMultipl

6、e type robot have multiple DOF. (3 Dimensional, open loop, chain mechanisms)Fig. 2.2 (a) Closed-loop versus (b) open-loop mechanism Robot Kinematics: ROBOTS AS MChapter 2Robot Kinematics: Position AnalysisFig. 2.3 Representation of a point in space A point P in space : 3 coordinates relative to a re

7、ference frameRepresentation of a Point in SpaceChapter 2Robot Kinematics: PoChapter 2Robot Kinematics: Position AnalysisFig. 2.4 Representation of a vector in space A Vector P in space : 3 coordinates of its tail and of its headRepresentation of a Vector in SpaceChapter 2Robot Kinematics: PoChapter

8、2Robot Kinematics: Position AnalysisFig. 2.5 Representation of a frame at the origin of the reference frame Each Unit Vector is mutually perpendicular. : normal, orientation, approach vectorRepresentation of a Frame at the Origin of a Fixed-Reference FrameChapter 2Robot Kinematics: PoChapter 2Robot

9、Kinematics: Position AnalysisFig. 2.6 Representation of a frame in a frame Each Unit Vector is mutually perpendicular. : normal, orientation, approach vectorRepresentation of a Frame in a Fixed Reference Frame The same as last slideChapter 2Robot Kinematics: PoChapter 2Robot Kinematics: Position Ana

10、lysisFig. 2.8 Representation of an object in space An object can be represented in space by attaching a frame to it and representing the frame in space. Representation of a Rigid BodyChapter 2Robot Kinematics: PoChapter 2Robot Kinematics: Position AnalysisA transformation matrices must be in square

11、form. It is much easier to calculate the inverse of square matrices. To multiply two matrices, their dimensions must match.HOMOGENEOUS TRANSFORMATION MATRICESChapter 2Robot Kinematics: PoRepresentation of Transformations of rigid objects in 3D space Representation of TransformatChapter 2Robot Kinema

12、tics: Position AnalysisFig. 2.9 Representation of an pure translation in space A transformation is defined as making a movement in space. A pure translation. A pure rotation about an axis. A combination of translation or rotations.Representation of a Pure Translation identitySame value aChapter 2Rob

13、ot Kinematics: PoChapter 2Robot Kinematics: Position AnalysisFig. 2.10 Coordinates of a point in a rotating frame before and after rotation around axis x. Assumption : The frame is at the origin of the reference frame and parallel to it.Fig. 2.11 Coordinates of a point relative to the reference fram

14、e and rotating frame as viewed from the x-axis. Representation of a Pure Rotation about an Axis Projections as seen from x axisx,y,z n, o, aChapter 2Robot Kinematics: PoFig. 2.13 Effects of three successive transformations A number of successive translations and rotations.Representation of Combined

15、Transformations Order is importantx,y,z n, o, anioi aiT1T2T3Fig. 2.13 Effects of three sucFig. 2.14 Changing the order of transformations will change the final result Order of Transformations is important x,y,z n, o, atranslationFig. 2.14 Changing the order oChapter 2Robot Kinematics: Position Analy

16、sisFig. 2.15 Transformations relative to the current frames. Example 2.8Transformations Relative to the Rotating FrametranslationrotationChapter 2Robot Kinematics: PoMATRICES FORFORWARD AND INVERSE KINEMATICS OF ROBOTSFor positionFor orientationMATRICES FORFORWARD AND INVERChapter 2Robot Kinematics:

17、 Position AnalysisFig. 2.17 The hand frame of the robot relative to the reference frame. Forward Kinematics Analysis: Calculating the position and orientation of the hand of the robot. If all robot joint variables are known, one can calculate where the robot is at any instant. . FORWARD AND INVERSE

18、KINEMATICS OF ROBOTSChapter 2Robot Kinematics: PoChapter 2Robot Kinematics: Position AnalysisForward Kinematics and Inverse Kinematics equation for position analysis : (a) Cartesian (gantry, rectangular) coordinates. (b) Cylindrical coordinates. (c) Spherical coordinates. (d) Articulated (anthropomo

19、rphic, or all-revolute) coordinates. Forward and Inverse Kinematics Equations for PositionChapter 2Robot Kinematics: PoChapter 2Robot Kinematics: Position AnalysisIBM 7565 robot All actuator is linear. A gantry robot is a Cartesian robot. Fig. 2.18 Cartesian Coordinates.Forward and Inverse Kinematic

20、s Equations for Position (a) Cartesian (Gantry, Rectangular) CoordinatesChapter 2Robot Kinematics: PoChapter 2Robot Kinematics: Position Analysis2 Linear translations and 1 rotation translation of r along the x-axis rotation of about the z-axis translation of l along the z-axis Fig. 2.19 Cylindrical

21、 Coordinates.Forward and Inverse Kinematics Equations for Position:Cylindrical CoordinatescosinesineChapter 2Robot Kinematics: PoChapter 2Robot Kinematics: Position Analysis2 Linear translations and 1 rotation translation of r along the z-axis rotation of about the y-axis rotation of along the z-axi

22、s Fig. 2.20 Spherical Coordinates.Forward and Inverse Kinematics Equations for Position (c) Spherical CoordinatesChapter 2Robot Kinematics: PoChapter 2Robot Kinematics: Position Analysis3 rotations - Denavit-Hartenberg representation Fig. 2.21 Articulated Coordinates.Forward and Inverse Kinematics E

23、quations for Position (d) Articulated CoordinatesChapter 2Robot Kinematics: PoChapter 2Robot Kinematics: Position Analysis Roll, Pitch, Yaw (RPY) angles Euler angles Articulated joints Forward and Inverse Kinematics Equations for OrientationChapter 2Robot Kinematics: PoChapter 2Robot Kinematics: Pos

24、ition AnalysisRoll: Rotation of about -axis (z-axis of the moving frame)Pitch: Rotation of about -axis (y-axis of the moving frame)Yaw: Rotation of about -axis (x-axis of the moving frame)Fig. 2.22 RPY rotations about the current axes.Forward and Inverse Kinematics Equations for Orientation (a) Roll

25、, Pitch, Yaw(RPY) AnglesChapter 2Robot Kinematics: PoChapter 2Robot Kinematics: Position AnalysisFig. 2.24 Euler rotations about the current axes.Rotation of about -axis (z-axis of the moving frame) followed byRotation of about -axis (y-axis of the moving frame) followed byRotation of about -axis (z

26、-axis of the moving frame).Forward and Inverse Kinematics Equations for Orientation (b) Euler AnglesChapter 2Robot Kinematics: PoChapter 2Robot Kinematics: Position Analysis Assumption : Robot is made of a Cartesian and an RPY set of joints. Assumption : Robot is made of a Spherical Coordinate and a

27、n Euler angle.Another Combination can be possibleDenavit-Hartenberg RepresentationForward and Inverse Kinematics Equations for OrientationRoll, Pitch, Yaw(RPY) AnglesChapter 2Robot Kinematics: PoForward and Inverse Transformations for robot armsForward and Inverse TransformaFig. 2.16 The Universe, r

28、obot, hand, part, and end effecter frames. Steps of calculation of an Inverse matrix: Calculate the determinant of the matrix. Transpose the matrix. Replace each element of the transposed matrix by its own minor (adjoint matrix). Divide the converted matrix by the determinant.INVERSE OF TRANSFORMATI

29、ON MATRICESFig. 2.16 The Universe, robot,Identity TransformationsIdentity TransformationsWe often need to calculate INVERSE MATRICESIt is good to reduce the number of such operationsWe need to do these calculations fastWe often need to calculate INVHow to find an Inverse Matrix B of matrix A?How to

30、find an Inverse Matrix Inverse Homogeneous TransformationInverse Homogeneous TransformaHomogeneous CoordinatesHomogeneous coordinates: embed 3D vectors into 4D by adding a “1”More generally, the transformation matrix T has the form:a11 a12 a13 b1a21 a22 a23 b2a31 a32 a33 b3c1 c2 c3 sfIt is presented

31、 in more detail on the WWW!Homogeneous CoordinatesHomogenFor various types of robots we have different maneuvering spacesFor various types of robots weFor various types of robots we calculate different forward and inverse transformationsFor various types of robots weFor various types of robots we so

32、lve different forward and inverse kinematic problemsFor various types of robots weForward and Inverse Kinematics: Single Link ExampleForward and Inverse KinematicsForward and Inverse Kinematics: Single Link ExampleeasyForward and Inverse KinematicsRobot-Arm-Kinematics=DH-intro：机器人手臂运动学=-DH-intro课件De

33、navit Hartenberg ideaDenavit Hartenberg idea Denavit-Hartenberg Representation : Fig. 2.25 A D-H representation of a general-purpose joint-link combination Simple way of modeling robot links and joints for any robot configuration, regardless of its sequence or complexity. Transformations in any coor

34、dinates is possible. Any possible combinations of joints and links and all-revolute articulated robots can be represented. DENAVIT-HARTENBERG REPRESENTATION OF FORWARD KINEMATIC EQUATIONS OF ROBOT Denavit-Hartenberg RepresentaChapter 2Robot Kinematics: Position Analysis : A rotation angle between tw

35、o links, about the z-axis (revolute). d : The distance (offset) on the z-axis, between links (prismatic). a : The length of each common normal (Joint offset). : The “twist” angle between two successive z-axes (Joint twist) (revolute) Only and d are joint variables.DENAVIT-HARTENBERG REPRESENTATION S

36、ymbol Terminologies : Chapter 2Robot Kinematics: PoLinks are in 3D, any shape associated with Zi alwaysLinks are in 3D, any shape asOnly rotationOnly translationOnly offsetOnly offsetOnly rotationAxis alignmentOnly rotationOnly translationODENAVIT-HARTENBERG REPRESENTATION for each linkDENAVIT-HARTE

37、NBERG REPRESENTAT4 link parameters4 link parametersChapter 2Robot Kinematics: Position Analysis : A rotation angle between two links, about the z-axis (revolute). d : The distance (offset) on the z-axis, between links (prismatic). a : The length of each common normal (Joint offset). : The “twist” an

38、gle between two successive z-axes (Joint twist) (revolute) Only and d are joint variables.DENAVIT-HARTENBERG REPRESENTATION Symbol Terminologies : Chapter 2Robot Kinematics: PoExample with three Revolute JointsZ0X0Y0Z1X2Y1Z2X1Y2d2a0a1Denavit-Hartenberg Link Parameter TableThe DH Parameter TableApply

39、 firstApply lastExample with three Revolute JoDenavit-Hartenberg Representation of Joint-Link-Joint TransformationDenavit-Hartenberg RepresentatNotation for Denavit-Hartenberg Representation of Joint-Link-Joint TransformationAlpha applied firstNotation for Denavit-HartenberFour Transformations from

40、one Joint to the NextOrder of multiplication of matrices is inverse of order of applying themHere we show order of matricesJoint-Link-JointFour Transformations from one Denavit-Hartenberg Representation of Joint-Link-Joint TransformationAlpha is applied firstHow to create a single matrix A nDenavit-

41、Hartenberg RepresentatEXAMPLE: Denavit-Hartenberg Representation of Joint-Link-Joint Transformation for Type 1 LinkFinal matrix from previous slidesubstitutesubstituteNumeric or symbolic matricesEXAMPLE: Denavit-Hartenberg ReThe Denavit-Hartenberg Matrix for another link typeSimilarity to Homegeneou

42、s: Just like the Homogeneous Matrix, the Denavit-Hartenberg Matrix is a transformation matrix from one coordinate frame to the next. Using a series of D-H Matrix multiplications and the D-H Parameter table, the final result is a transformation matrix from some frame to your initial frame.Z(i - 1)X(i

43、 -1)Y(i -1)( i - 1)a(i - 1 )Z i Y i X i a i d i i Put the transformation here for every link The Denavit-Hartenberg Matrix In DENAVIT-HARTENBERG REPRESENTATION we must be able to find parameters for each linkSo we must know link typesIn DENAVIT-HARTENBERG REPRESENRobot-Arm-Kinematics=DH-intro：机器人手臂运

44、动学=-DH-intro课件Links between revolute jointsLinks between revolute jointsRobot-Arm-Kinematics=DH-intro：机器人手臂运动学=-DH-intro课件ln=0Type 3 LinkJoint n+1Joint ndn=0Link nxn-1xnln=0Type 3 LinkJoint n+1Joint ln=0dn=0Type 4 LinkOrigins coinciden-1Joint n+1Joint nPart of dn-1Link nxn-1yn-1xnnln=0Type 4 LinkOri

45、gins coincidLinks between prismatic jointsLinks between prismatic jointsRobot-Arm-Kinematics=DH-intro：机器人手臂运动学=-DH-intro课件Robot-Arm-Kinematics=DH-intro：机器人手臂运动学=-DH-intro课件Forward and Inverse Transformations on MatricesForward and Inverse TransformaRobot-Arm-Kinematics=DH-intro：机器人手臂运动学=-DH-intro课件Start point: Assign joint number n to the first shown joint. As