1、Forward 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 hand be located at a particular point)Direct KinematicsDirect Kinematics Where is my hand?Direct Kinematics:HERE!Direct Kinema
2、tics Position of tip in(x,y)coordinatesDirect Kinematics 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
3、 joint offset di9)Determine joint angle i around Zi10+11)Write link transformation and concatenateOften sufficient for 2DKinematic Problems for Manipulation Reliably position the tip-go from one position to another position Dont hit anything,avoid obstacles Make smooth motions at reasonable speeds a
4、nd at reasonable accelerations Adjust to changing conditions-i.e.when something is picked up respond to the change in weightROBOTS AS MECHANISMFig.2.1 A one-degree-of-freedom closed-loop four-bar mechanismMultiple type robot have multiple DOF.(3 Dimensional,open loop,chain mechanisms)Fig.2.2(a)Close
5、d-loop versus(b)open-loop mechanism Chapter 2Robot Kinematics:Position AnalysisFig.2.3 Representation of a point in space A point P in space:3 coordinates relative to a reference framekcjbiaPzyxChapter 2Robot Kinematics:Position AnalysisFig.2.4 Representation of a vector in space A Vector P in space
6、:3 coordinates of its tail and of its head_kcjbiaPzyxwzyxP_Chapter 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 vectorzzzyyyxxxaonaonaonFChapter 2Robot Kinematics:Posi
7、tion AnalysisFig.2.6 Representation of a frame in a frame Each Unit Vector is mutually perpendicular.:normal,orientation,approach vector1000zzzzyyyyxxxxPaonPaonPaonFThe same as last slideChapter 2Robot Kinematics:Position AnalysisFig.2.8 Representation of an object in space An object can be represen
8、ted in space by attaching a frame to it and representing the frame in space.1000zzzzyyyyxxxxobjectPaonPaonPaonFChapter 2Robot Kinematics:Position AnalysisA transformation matrices must be in square form.It is the inverse of square matrices.To multiply two matrices,their dimensions must match.1000zzz
9、zyyyyxxxxPaonPaonPaonFChapter 2Robot Kinematics: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.1000100010001zyxdddTidentitySam
10、e value aChapter 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 frame and rotating
11、 frame as viewed from the x-axis.Projections as seen from x axisx,y,z n,o,aFig.2.13 Effects of transformations A number of successive translations and rotations.x,y,z n,o,anioi aiT1T2T3Fig.2.14 Changing the order of transformations will change the final result x,y,z n,o,atranslationChapter 2Robot Ki
12、nematics:Position AnalysisFig.2.15 Transformations relative to the current frames.Example 2.8translationrotation For position For orientationChapter 2Robot Kinematics:Position AnalysisFig.2.17 The hand frame of the robot relative to the reference frame.Forward Kinematics Analysis:Calculating the pos
13、ition 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.Chapter 2Robot Kinematics:Position AnalysisForward Kinematics and Inverse Kinematics equation for position analysis:(a)Cartesian(gantry,rectangular)coordinates.(b
14、)Cylindrical coordinates.(c)Spherical coordinates.(d)Articulated(anthropomorphic,or all-revolute)coordinates.Chapter 2Robot Kinematics:Position AnalysisIBM 7565 robot All actuator is linear.A gantry robot is a Cartesian robot.Fig.2.18 Cartesian Coordinates.1000100010001zyxcartPRPPPTTChapter 2Robot K
15、inematics: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 Coordinates.100010000lrSCSrCSCTTcylPR,0,0)Trans(,)Rot(Trans(0,0,),(rzllrTTcylPRcosinesineChapter 2Robot Kinematics:Pos
16、ition Analysis2 Linear translations and 1 rotation translation of r along the z-axis rotation of about the y-axis rotation of along the z-axis Fig.2.20 Spherical Coordinates.10000rCCSSrSSSCSCCrSCSSCCTTsphPR)Trans()Rot(Rot()(0,0,yzlrsphPRTTChapter 2Robot Kinematics:Position Analysis3 rotations-Denavi
17、t-Hartenberg representation Fig.2.21 Articulated Coordinates.Chapter 2Robot Kinematics:Position Analysis Roll,Pitch,Yaw(RPY)angles Euler angles Articulated joints Forward and Inverse Kinematics Equations for OrientationChapter 2Robot Kinematics:Position AnalysisRoll:Rotation of about -axis(z-axis of
18、 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)aaononFig.2.22 RPY rotations about the current axes.Forward and Inverse Kinematics Equations for Orientation (a)Roll,Pitch,Yaw(RPY)AnglesChapter 2Robot Kinematics:Positio
19、n 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-axis of the moving frame).aoaForward and Inverse Kinematics Equations for Orientation (b)Eul
20、er AnglesChapter 2Robot Kinematics:Position Analysis)()(,noazyxcartHRRPYPPPTT)()(,EulerTTrsphHR Assumption:Robot is made of a Cartesian and an RPY set of joints.Assumption:Robot is made of a Spherical Coordinate and an Euler angle.Another Combination can be possibleDenavit-Hartenberg RepresentationR
21、oll,Pitch,Yaw(RPY)AnglesFig.2.16 The Universe,robot,hand,part,and end effecter frames.Steps of calculation of an Inverse matrix:1.Calculate the determinant of the matrix.2.Transpose the matrix.3.Replace each element of the transposed matrix by its own minor(adjoint matrix).4.Divide the converted mat
22、rix by the determinant.INVERSE OF TRANSFORMATION MATRICES1.We often need to calculate INVERSE MATRICESHomogeneous Coordinates Homogeneous coordinates:embed 3D vectors into 4D by adding a“1”More generally,the transformation matrix T has the form:FactorScalingTrans.Perspect.Vector Trans.MatrixRot.Ta11
23、 a12 a13 b1a21 a22 a23 b2a31 a32 a33 b3c1 c2 c3 sfIt is presented in more detail on the WWW!easy 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,regardless of its sequence or complexity.Tran
24、sformations 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 ROBOTChapter 2Robot Kinematics:Position Analysis :A rotation angle between two links,about the z-axis(revol
25、ute).d:The distance()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.Symbol Terminologies:Links are in 3D,any shape associated with Zi alwaysOnly rotationOnly t
26、ranslationOnly offsetOnly offsetOnly rotationAxis alignmentChapter 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”angle between
27、 two successive z-axes(Joint twist)(revolute)Only and d are joint variables.Symbol Terminologies:Example with three Revolute Jointsi (i-1)a(i-1)di i 0 0 0 0 0 1 0 a0 0 1 2-90 a1 d2 2 Z0X0Y0Z1X2Y1Z2X1Y2d2a0a1Apply firstApply lastAlpha applied firstOrder of multiplication of matrices is inverse of ord
28、er of applying themHere we show order of matricesDenavit-Hartenberg Representation of Joint-Link-Joint TransformationAlpha is applied firstEXAMPLE:Denavit-Hartenberg Representation of Joint-Link-Joint Transformation for Final matrix from previous slidesubstitutesubstituteNumeric or symbolic matrices
29、1000coscossincossinsinsinsincoscoscossin0sincosi1)(i1)(i1)(ii1)(iii1)(i1)(i1)(ii1)(ii1)(iiidda:Just like the Homogeneous Matrix,the Denavit-Hartenberg Matrix is a transformation matrix Using a series of D-H Matrix multiplications and the D-H Parameter table,the final result is a transformation matri
30、x from some frame to your initial frame.Z(i-1)X(i-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 ln=0dn=0Type 4 LinkOrigins coincideStart point:Assign joint number n to the first shown joint.Assign a local reference frame for each and every joint before or after
31、these joints.Y-axis is not used in D-H representation.1?All joints are represented by a z-axis.(right-hand rule for rotational joint,linear movement for prismatic joint)2.The common normal is one line mutually perpendicular to any two skew lines.3.Parallel z-axes joints make a infinite number of com
32、mon normal.4.Intersecting z-axes of two successive joints make no common normal between them(Length is 0.).DENAVIT-HARTENBERG REPRESENTATION Procedures for assigning a local reference frame to each joint:Chapter 2Robot Kinematics:Position Analysis :A rotation about the z-axis.:The distance on the z-
33、axis.:The length of each common normal(Joint offset).:The angle between two successive z-axes(Joint twist)Only and d are joint variables.Symbol Terminologies:Chapter 2Robot Kinematics:Position Analysis(I)Rotate about the zn-axis an able of n+1.(Coplanar)(II)Translate along zn-axis a distance of dn+1 to make xn and xn+1 colinear.(III)Translate along the xn-axis a distance of an+1 to bring the origins of xn+1 together.(IV)Rotate zn-axis about xn+1 axis an angle of n+1 to align zn-axis with zn+1-axis.DENAVIT-HARTENBERG REPRESENTATION The necessary motions to transform to the next.
