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1、More details and examples on robot arms and kinematics,Denavit-Hartenberg Notation,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 the hand be

2、 located at a particular point),Direct Kinematics,Direct Kinematics with no matrices,Where is my hand?,Direct Kinematics: HERE!,Direct Kinematics,Position of tip in (x,y) coordinates,,Direct Kinematics Algorithm,1) Draw sketch 2) Number links. Base=0, Last link = n 3) Identify and number r

3、obot joints 4) Draw axis Zi for joint i 5) Determine joint length ai-1 between Zi-1 and Zi 6) Draw axis Xi-1 7) Determine joint twist i-1 measured around Xi-1 8) Determine the joint offset di 9) Determine joint angle i around Zi 10+11) Write link transformation and concatenate,Often sufficient for 2

4、D,Kinematic 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 and at reasonable accelerations Adjust to changing conditions - i.e. when something is picked up respond to the chang

5、e in weight,ROBOTS AS MECHANISMs,Robot Kinematics: ROBOTS AS MECHANISM,Fig. 2.1 A one-degree-of-freedom closed-loop four-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,Chapter 2Robot Kinematic

6、s: Position Analysis,Fig. 2.3 Representation of a point in space,A point P in space : 3 coordinates relative to a reference frame,Representation of a Point in Space,Chapter 2Robot Kinematics: Position Analysis,Fig. 2.4 Representation of a vector in space,A Vector P in space : 3 coordinates of it

7、s tail and of its head,Representation of a Vector in Space,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 Ori

8、gin of a Fixed-Reference Frame,,,,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 2R

9、obot 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: Position Analysis,A transformation matrices must be in squa

10、re form. It is much easier to calculate the inverse of square matrices. To multiply two matrices, their dimensions must match.,HOMOGENEOUS TRANSFORMATION MATRICES,Representation of Transformations of rigid objects in 3D space,Chapter 2Robot Kinematics: Position Analysis,Fig. 2.9 Representation of

11、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,identity,Same value a,Chapter 2Robot Kinematics: Position Analysis,Fig. 2.10 Coor

12、dinates 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 frame as viewed from the x-axis.,Representation of a Pure R

13、otation about an Axis,Projections as seen from x axis,x,y,z n, o, a,Fig. 2.13 Effects of three successive transformations,A number of successive translations and rotations.,Representation of Combined Transformations,Order is important,x,y,z n, o, a,,,,ni,oi,,ai,T1,T2,T3,Fig. 2.14 Changing the order

14、of transformations will change the final result,Order of Transformations is important,x,y,z n, o, a,,translation,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,MA

15、TRICES FORFORWARD AND INVERSE KINEMATICS OF ROBOTS,For position For orientation,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 al

16、l robot joint variables are known, one can calculate where the robot is at any instant. .,FORWARD AND INVERSE KINEMATICS OF ROBOTS,Chapter 2Robot Kinematics: Position Analysis,,Forward Kinematics and Inverse Kinematics equation for position analysis : (a) Cartesian (gantry, rectangular) coordinat

17、es. (b) Cylindrical coordinates. (c) Spherical coordinates. (d) Articulated (anthropomorphic, or all-revolute) coordinates.,Forward and Inverse Kinematics Equations for Position,Chapter 2Robot Kinematics: Position Analysis,IBM 7565 robot All actuator is linear. A gantry robot is a Cartesian rob

18、ot.,Fig. 2.18 Cartesian Coordinates.,Forward and Inverse Kinematics Equations for Position (a) Cartesian (Gantry, Rectangular) Coordinates,Chapter 2Robot Kinematics: Position Analysis,2 Linear translations and 1 rotation translation of r along the x-axis rotation of about the z-axis translation o

19、f l along the z-axis,Fig. 2.19 Cylindrical Coordinates.,Forward and Inverse Kinematics Equations for Position: Cylindrical Coordinates,cosine,sine,Chapter 2Robot Kinematics: Position Analysis,2 Linear translations and 1 rotation translation of r along the z-axis rotation of about the y-axis rotat

20、ion of along the z-axis,Fig. 2.20 Spherical Coordinates.,Forward and Inverse Kinematics Equations for Position (c) Spherical Coordinates,Chapter 2Robot Kinematics: Position Analysis,3 rotations - Denavit-Hartenberg representation,Fig. 2.21 Articulated Coordinates.,Forward and Inverse Kinematics Equ

21、ations for Position (d) Articulated Coordinates,Chapter 2Robot Kinematics: Position Analysis, Roll, Pitch, Yaw (RPY) angles Euler angles Articulated joints,Forward and Inverse Kinematics Equations for Orientation,Chapter 2Robot Kinematics: Position Analysis,Roll: Rotation of about -axis (z-axis of

22、 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, Pitch, Yaw(RPY) Angles,Chapter 2Robot Kinemat

23、ics: Position Analysis,Fig. 2.24 Euler rotations about the current axes.,Rotation of about -axis (z-axis of the moving frame) followed by Rotation of about -axis (y-axis of the moving frame) followed by Rotation of about -axis (z-axis of the moving frame).,Forward and Inverse Kinematics Equations

24、 for Orientation (b) Euler Angles,Chapter 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 an Euler angle.,,Another Combination can be possible,Denavit-Hartenberg Representation,Forward

25、and Inverse Kinematics Equations for Orientation,Roll, Pitch, Yaw(RPY) Angles,Forward and Inverse Transformations for robot arms,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 mat

26、rix. Replace each element of the transposed matrix by its own minor (adjoint matrix). Divide the converted matrix by the determinant.,INVERSE OF TRANSFORMATION MATRICES,Identity Transformations,We often need to calculate INVERSE MATRICES It is good to reduce the number of such operations We need t

27、o do these calculations fast,How to find an Inverse Matrix B of matrix A?,,Inverse Homogeneous Transformation,,Homogeneous Coordinates,Homogeneous coordinates: embed 3D vectors into 4D by adding a “1” More generally, the transformation matrix T has the form:,a11 a12 a13 b1 a21 a22 a23 b2 a31 a32 a33

28、 b3 c1 c2 c3 sf,It is presented in more detail on the WWW!,,,,,,,For various types of robots we have different maneuvering spaces,,,For various types of robots we calculate different forward and inverse transformations,,,For various types of robots we solve different forward and inverse kinematic p

29、roblems,Forward and Inverse Kinematics: Single Link Example,Forward and Inverse Kinematics: Single Link Example,,easy,,,Denavit 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 join

30、ts 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 REPRESENTATION OF FORWARD KINEMATIC EQUATIONS OF R

31、OBOT,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 normal (Joint offset). : The “twist” angle between two successive z-axes (Joint t

32、wist) (revolute) Only and d are joint variables.,DENAVIT-HARTENBERG REPRESENTATION Symbol Terminologies :,Links are in 3D, any shape, associated with Zi always,,,,,,Only rotation,Only translation,Only offset,Only offset,Only rotation,Axis alignment,DENAVIT-HARTENBERG REPRESENTATION for each link,4

33、link parameters,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 normal (Joint offset). : The “twist” angle between two successive z-ax

34、es (Joint twist) (revolute) Only and d are joint variables.,DENAVIT-HARTENBERG REPRESENTATION Symbol Terminologies :,Example with three Revolute Joints,Denavit-Hartenberg Link Parameter Table,The DH Parameter Table,Apply first,Apply last,Denavit-Hartenberg Representation of Joint-Link-Joint Transfo

35、rmation,Notation for Denavit-Hartenberg Representation of Joint-Link-Joint Transformation,Alpha applied first,Four Transformations from one Joint to the Next,Order of multiplication of matrices is inverse of order of applying them Here we show order of matrices,Joint-Link-Joint,Denavit-Hartenberg Re

36、presentation of Joint-Link-Joint Transformation,Alpha is applied first,,How to create a single matrix A n,EXAMPLE: Denavit-Hartenberg Representation of Joint-Link-Joint Transformation for Type 1 Link,Final matrix from previous slide,substitute,substitute,Numeric or symbolic matrices,The Denavit-Hart

37、enberg 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 to the next. Using a series of D-H Matrix multiplications and the D-H Parameter table, the final result is a transformat

38、ion matrix from some frame to your initial frame.,Put the transformation here for every link,In DENAVIT-HARTENBERG REPRESENTATION we must be able to find parameters for each link So we must know link types,,,Links between revolute joints,,,,,ln=0,Type 3 Link,Joint n+1,Joint n,dn=0,Link n,xn-1,xn,ln=

39、0 dn=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,Links between prismatic joints,,,,,Forward and Inverse Transformations on Matrices,,,Start point: Assign joint number n to the first shown joint. Assign a local reference frame for each and every joint bef

40、ore or after these joints. Y-axis is not used in D-H representation.,DENAVIT-HARTENBERG REPRESENTATION PROCEDURES, 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 s

41、kew 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:,Chapter 2Robot Kinematics: Posi

42、tion 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 Reminder:,Chapter 2Rob

43、ot 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 from one reference frame to the next.,