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1、More details and examples on robot arms and kinematics,Denavit-Hartenberg Notation,More details and examples on r,INTRODUCTION,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
2、 the hand be located at a particular point),INTRODUCTIONForward Kinematic,Direct Kinematics,Direct Kinematics,Direct Kinematics with no matrices,Where is my hand?,Direct Kinematics:HERE!,Direct Kinematics with no matr,Direct Kinematics,Position of tip in (x,y) coordinates,Direct KinematicsPosition o
3、f t,Direct Kinematics Algorithm,1) 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) Det
4、ermine joint angle i around Zi10+11) Write link transformation and concatenate,Often sufficient for 2D,Direct Kinematics Algorithm1),Kinematic Problems for Manipulation,Reliably position the tip - go from one position to another position Dont hit anything, avoid obstacles Make smooth motions at reas
5、onable speeds and at reasonable accelerations Adjust to changing conditions - i.e. when something is picked up respond to the change in weight,Kinematic Problems for Manipul,ROBOTS AS MECHANISMs,ROBOTS AS MECHANISMs,Robot Kinematics: ROBOTS AS MECHANISM,Fig. 2.1 A one-degree-of-freedom closed-loop f
6、our-bar mechanism,Multiple 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 M,Chapter 2Robot Kinematics: Position Analysis,Fig. 2.3 Representation of a point in space,A point P in space : 3
7、coordinates relative to a reference frame,Representation of a Point in Space,Chapter 2Robot Kinematics: Po,Chapter 2Robot Kinematics: Position Analysis,Fig. 2.4 Representation of a vector in space,A Vector P in space : 3 coordinates of its tail and of its head,Representation of a Vector in Space,Cha
8、pter 2Robot Kinematics: Po,Chapter 2Robot Kinematics: Position Analysis,Fig. 2.5 Representation of a frame at the origin of the reference frame,Each Unit Vector is mutually perpendicular. : normal, orientation, approach vector,Representation of a Frame at the Origin of a Fixed-Reference Frame,Chapte
9、r 2Robot Kinematics: Po,Chapter 2Robot Kinematics: Position Analysis,Fig. 2.6 Representation of a frame in a frame,Each Unit Vector is mutually perpendicular. : normal, orientation, approach vector,Representation of a Frame in a Fixed Reference Frame,The same as last slide,Chapter 2Robot Kinematics:
10、 Po,Chapter 2Robot Kinematics: Position Analysis,Fig. 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 Body,Chapter 2Robot Kinematics: Po,Chapter 2Robot Kinematics: Position Analys
11、is,A transformation matrices must be in square form. It is much easier to calculate the inverse of square matrices. To multiply two matrices, their dimensions must match.,HOMOGENEOUS TRANSFORMATION MATRICES,Chapter 2Robot Kinematics: Po,Representation of Transformations of rigid objects in 3D space,
12、Representation of Transformat,Chapter 2Robot Kinematics: Position Analysis,Fig. 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 o
13、f a Pure Translation,identity,Same value a,Chapter 2Robot Kinematics: Po,Chapter 2Robot Kinematics: Position Analysis,Fig. 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.
14、2.11 Coordinates of a point relative to the reference frame and rotating frame as viewed from the x-axis.,Representation of a Pure Rotation about an Axis,Projections as seen from x axis,x,y,z n, o, a,Chapter 2Robot Kinematics: Po,Fig. 2.13 Effects of three successive transformations,A number of succ
15、essive translations and rotations.,Representation of Combined Transformations,Order is important,x,y,z n, o, a,ni,oi,ai,T1,T2,T3,Fig. 2.13 Effects of three suc,Fig. 2.14 Changing the order of transformations will change the final result,Order of Transformations is important,x,y,z n, o, a,translation
16、,Fig. 2.14 Changing the order o,Chapter 2Robot Kinematics: Position Analysis,Fig. 2.15 Transformations relative to the current frames.,Example 2.8,Transformations Relative to the Rotating Frame,translation,rotation,Chapter 2Robot Kinematics: Po,MATRICES FORFORWARD AND INVERSE KINEMATICS OF ROBOTS,Fo
17、r positionFor orientation,MATRICES FORFORWARD AND INVER,Chapter 2Robot Kinematics: Position Analysis,Fig. 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
18、 known, one can calculate where the robot is at any instant. .,FORWARD AND INVERSE KINEMATICS OF ROBOTS,Chapter 2Robot Kinematics: Po,Chapter 2Robot Kinematics: Position Analysis,Forward Kinematics and Inverse Kinematics equation for position analysis : (a) Cartesian (gantry, rectangular) coordinate
19、s. (b) Cylindrical coordinates. (c) Spherical coordinates. (d) Articulated (anthropomorphic, or all-revolute) coordinates.,Forward and Inverse Kinematics Equations for Position,Chapter 2Robot Kinematics: Po,Chapter 2Robot Kinematics: Position Analysis,IBM 7565 robot All actuator is linear. A gantry
20、robot is a Cartesian robot.,Fig. 2.18 Cartesian Coordinates.,Forward and Inverse Kinematics Equations for Position (a) Cartesian (Gantry, Rectangular) Coordinates,Chapter 2Robot Kinematics: Po,Chapter 2Robot Kinematics: Position Analysis,2 Linear translations and 1 rotation translation of r along th
21、e x-axis rotation of about the z-axis translation of l along the z-axis,Fig. 2.19 Cylindrical Coordinates.,Forward and Inverse Kinematics Equations for Position:Cylindrical Coordinates,cosine,sine,Chapter 2Robot Kinematics: Po,Chapter 2Robot Kinematics: Position Analysis,2 Linear translations and 1
22、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.,Forward and Inverse Kinematics Equations for Position (c) Spherical Coordinates,Chapter 2Robot Kinematics: Po,Chapter 2Robot Kinematics: Position Analysis,3 rotations
23、 - Denavit-Hartenberg representation,Fig. 2.21 Articulated Coordinates.,Forward and Inverse Kinematics Equations for Position (d) Articulated Coordinates,Chapter 2Robot Kinematics: Po,Chapter 2Robot Kinematics: Position Analysis, Roll, Pitch, Yaw (RPY) angles Euler angles Articulated joints,Forward
24、and Inverse Kinematics Equations for Orientation,Chapter 2Robot Kinematics: Po,Chapter 2Robot Kinematics: Position Analysis,Roll: 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),F
25、ig. 2.22 RPY rotations about the current axes.,Forward and Inverse Kinematics Equations for Orientation (a) Roll, Pitch, Yaw(RPY) Angles,Chapter 2Robot Kinematics: Po,Chapter 2Robot Kinematics: Position Analysis,Fig. 2.24 Euler rotations about the current axes.,Rotation of about -axis (z-axis of the
26、 moving frame) followed byRotation of about -axis (y-axis of the moving frame) followed byRotation of about -axis (z-axis of the moving frame).,Forward and Inverse Kinematics Equations for Orientation (b) Euler Angles,Chapter 2Robot Kinematics: Po,Chapter 2Robot Kinematics: Position Analysis,Assumpt
27、ion : 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 possible,Denavit-Hartenberg Representation,Forward and Inverse Kinematics Equations for Orientation,Roll, Pitch, Yaw(RPY) Angles,Chapter 2R
28、obot Kinematics: Po,Forward and Inverse Transformations for robot arms,Forward and Inverse Transforma,Fig. 2.16 The Universe, robot, 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
29、the transposed matrix by its own minor (adjoint matrix). Divide the converted matrix by the determinant.,INVERSE OF TRANSFORMATION MATRICES,Fig. 2.16 The Universe, robot,Identity Transformations,Identity Transformations,We often need to calculate INVERSE MATRICESIt is good to reduce the number of su
30、ch operationsWe need to do these calculations fast,We often need to calculate INV,How to find an Inverse Matrix B of matrix A?,How to find an Inverse Matrix,Inverse Homogeneous Transformation,Inverse Homogeneous Transforma,Homogeneous Coordinates,Homogeneous coordinates: embed 3D vectors into 4D by
31、adding a “1”More generally, the transformation matrix T has the form:,a11 a12 a13 b1a21 a22 a23 b2a31 a32 a33 b3c1 c2 c3 sf,It is presented in more detail on the WWW!,Homogeneous CoordinatesHomogen,For various types of robots we have different maneuvering spaces,For various types of robots we,For va
32、rious types of robots we calculate different forward and inverse transformations,For various types of robots we,For various types of robots we solve different forward and inverse kinematic problems,For various types of robots we,Forward and Inverse Kinematics: Single Link Example,Forward and Inverse
33、 Kinematics,Forward and Inverse Kinematics: Single Link Example,easy,Forward and Inverse Kinematics,Robot-Arm-Kinematics=DH-intro：机器人手臂运动学=-DH-intro课件,Denavit Hartenberg idea,Denavit Hartenberg idea,Denavit-Hartenberg Representation :,Fig. 2.25 A D-H representation of a general-purpose joint-link co
34、mbination, Simple way of modeling robot links and joints for any robot configuration, regardless of its sequence or complexity., Transformations in any coordinates is possible., Any possible combinations of joints and links and all-revolute articulated robots can be represented.,DENAVIT-HARTENBERG R
35、EPRESENTATION OF FORWARD KINEMATIC EQUATIONS OF ROBOT,Denavit-Hartenberg Representa,Chapter 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 norma
36、l (Joint offset). : The “twist” angle between two successive z-axes (Joint twist) (revolute) Only and d are joint variables.,DENAVIT-HARTENBERG REPRESENTATION Symbol Terminologies :,Chapter 2Robot Kinematics: Po,Links are in 3D, any shape, associated with Zi always,Links are in 3D, any shape as,Only
37、 rotation,Only translation,Only offset,Only offset,Only rotation,Axis alignment,Only rotationOnly translationO,DENAVIT-HARTENBERG REPRESENTATION for each link,DENAVIT-HARTENBERG REPRESENTAT,4 link parameters,4 link parameters,Chapter 2Robot Kinematics: Position Analysis, : A rotation angle between t
38、wo 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
39、 Symbol Terminologies :,Chapter 2Robot Kinematics: Po,Example with three Revolute Joints,Denavit-Hartenberg Link Parameter Table,The DH Parameter Table,Apply first,Apply last,Example with three Revolute Jo,Denavit-Hartenberg Representation of Joint-Link-Joint Transformation,Denavit-Hartenberg Repres
40、entat,Notation for Denavit-Hartenberg Representation of Joint-Link-Joint Transformation,Alpha applied first,Notation for Denavit-Hartenber,Four Transformations from one Joint to the Next,Order of multiplication of matrices is inverse of order of applying themHere we show order of matrices,Joint-Link
41、-Joint,Four Transformations from one,Denavit-Hartenberg Representation of Joint-Link-Joint Transformation,Alpha is applied first,How to create a single matrix A n,Denavit-Hartenberg Representat,EXAMPLE: Denavit-Hartenberg Representation of Joint-Link-Joint Transformation for Type 1 Link,Final matrix
42、 from previous slide,substitute,substitute,Numeric or symbolic matrices,EXAMPLE: Denavit-Hartenberg Re,The Denavit-Hartenberg Matrix for another link type,Similarity to Homegeneous: Just like the Homogeneous Matrix, the Denavit-Hartenberg Matrix is a transformation matrix from one coordinate frame t
43、o 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.,Put the transformation here for every link,The Denavit-Hartenberg Matrix,In DENAVIT-HARTENBERG REPRESENTATION we must be able to fi
44、nd parameters for each linkSo we must know link types,In DENAVIT-HARTENBERG REPRESEN,Robot-Arm-Kinematics=DH-intro：机器人手臂运动学=-DH-intro课件,Links between revolute joints,Links between revolute joints,Robot-Arm-Kinematics=DH-intro：机器人手臂运动学=-DH-intro课件,ln=0,Type 3 Link,Joint n+1,Joint n,dn=0,Link n,xn-1,x
45、n,ln=0Type 3 LinkJoint n+1Joint,ln=0dn=0,Type 4 Link,Origins coincide,n-1,Joint n+1,Joint n,Part of dn-1,Link n,xn-1,yn-1,xn,n,ln=0Type 4 LinkOrigins coincid,Links between prismatic joints,Links between prismatic joints,Robot-Arm-Kinematics=DH-intro：机器人手臂运动学=-DH-intro课件,Robot-Arm-Kinematics=DH-intro
46、：机器人手臂运动学=-DH-intro课件,Forward and Inverse Transformations on Matrices,Forward and Inverse Transforma,Robot-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. Assign a local reference frame for each and every joint before or after these joints. Y
47、-axis is not used in D-H representation.,DENAVIT-HARTENBERG REPRESENTATION PROCEDURES,Start point: DENAVIT-HARTENBER, All joints are represented by a z-axis. (right-hand rule for rotational joint, linear movement for prismatic joint)The common normal is one line mutually perpendicular to any two ske
48、w lines. Parallel z-axes joints make a infinite number of common normal. 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:, All joints are represented b,Chap
49、ter 2Robot Kinematics: Position Analysis, : A rotation about the z-axis. d : The distance on the z-axis. a : The length of each common normal (Joint offset). : The angle between two successive z-axes (Joint twist) Only and d are joint variables.,DENAVIT-HARTENBERG REPRESENTATION Symbol Terminologies
50、 Reminder:,Chapter 2Robot Kinematics: Po,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
