More details and examples on robot arms and kinematics

1. More details and examples on robot arms and kinematics Denavit-Hartenberg Notation

2. INTRODUCTION Forward Kinematics: to determine where the robot’s 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)

3. Direct Kinematics

4. Where is my hand? Direct Kinematics with no matrices Direct Kinematics: HERE!

5. Direct Kinematics • Position of tip in (x,y) coordinates

6. Direct Kinematics Algorithm Often sufficient for 2D 1) Draw sketch 2) Number links. Base=0, Last link = n 3) Identify and number robot joints 4) Draw axis Zifor joint i 5) Determine joint length ai-1between Zi-1and 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

7. Kinematic Problems for Manipulation • Reliably position the tip - go from one position to another position • Don’t 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 change in weight

8. ROBOTS AS MECHANISMs

9. Robot Kinematics: ROBOTS AS MECHANISM Multiple type robot have multiple DOF. (3 Dimensional, open loop, chain mechanisms) Fig. 2.1 A one-degree-of-freedom closed-loop four-bar mechanism Fig. 2.2 (a) Closed-loop versus(b) open-loop mechanism

10. Representation of a Point in Space Chapter 2Robot Kinematics: Position Analysis A point P in space : 3 coordinates relative to a reference frame Fig. 2.3 Representation of a point in space

11. Representation of a Vector in Space Chapter 2Robot Kinematics: Position Analysis A Vector P in space : 3 coordinates of its tail and of its head Fig. 2.4 Representation of a vector in space

12. Representation of a Frame at the Origin of a Fixed-Reference Frame Chapter 2Robot Kinematics: Position Analysis Each Unit Vector is mutually perpendicular. : normal, orientation, approach vector Fig. 2.5 Representation of a frame at the origin of the reference frame

13. Representation of a Frame in a Fixed Reference Frame Chapter 2Robot Kinematics: Position Analysis Each Unit Vector is mutually perpendicular. : normal, orientation, approach vector The same as last slide Fig. 2.6 Representation of a frame in a frame

14. Representation of a Rigid Body Chapter 2Robot Kinematics: Position Analysis An object can be represented in space by attaching a frame to it and representing the frame in space. Fig. 2.8 Representation of an object in space

15. HOMOGENEOUS TRANSFORMATION MATRICES Chapter 2Robot Kinematics: Position Analysis • 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.

16. Representation of Transformations of rigid objects in 3D space

17. Representation of a Pure Translation Chapter 2Robot Kinematics: Position Analysis • 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. Same value a identity Fig. 2.9 Representation of an pure translation in space

18. Representation of a Pure Rotation about an Axis Chapter 2Robot Kinematics: Position Analysis x,y,z  n, o, a Assumption : The frame is at the origin of the reference frame and parallel to it. Projections as seen from x axis Fig. 2.10 Coordinates of a point in a rotating frame before and after rotation around axis x. Fig. 2.11 Coordinates of a point relative to the reference frame and rotating frame as viewed from the x-axis.

19. Representation of Combined Transformations A number of successive translations and rotations…. T1 ai Fig. 2.13 Effects of three successive transformations oi ni T2 T3 x,y,z  n, o, a Order is important

20. x,y,z n, o, a translation Order of Transformations is important Fig. 2.14Changing the order of transformations will change the final result

21. Transformations Relative to the Rotating Frame Chapter 2Robot Kinematics: Position Analysis Example 2.8 translation rotation Fig. 2.15 Transformations relative to the current frames.

22. MATRICES FORFORWARD AND INVERSE KINEMATICS OF ROBOTS • For position • For orientation

23. FORWARD AND INVERSE KINEMATICS OF ROBOTS Chapter 2Robot Kinematics: Position Analysis • 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. • . Fig. 2.17 The hand frame of the robot relative to the reference frame.

24. Forward and Inverse KinematicsEquations for Position Chapter 2Robot Kinematics: Position Analysis Forward Kinematics and Inverse Kinematics equation for position analysis : (a) Cartesian (gantry, rectangular) coordinates. (b) Cylindrical coordinates. (c) Spherical coordinates. (d) Articulated (anthropomorphic, or all-revolute) coordinates.

25. Forward and Inverse KinematicsEquations for Position (a) Cartesian (Gantry, Rectangular) Coordinates Chapter 2Robot Kinematics: Position Analysis IBM 7565 robot • All actuator is linear. • A gantry robot is a Cartesian robot. Fig. 2.18 Cartesian Coordinates.

26. Forward and Inverse KinematicsEquations for Position: Cylindrical 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 of l along the z-axis cosine sine Fig. 2.19 Cylindrical Coordinates.

27. Forward and Inverse KinematicsEquations for Position (c) Spherical Coordinates Chapter 2Robot Kinematics: Position Analysis 2 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.

28. Forward and Inverse KinematicsEquations for Position (d) Articulated Coordinates Chapter 2Robot Kinematics: Position Analysis 3 rotations -> Denavit-Hartenberg representation Fig. 2.21 Articulated Coordinates.

29. Forward and Inverse KinematicsEquations for Orientation Chapter 2Robot Kinematics: Position Analysis  Roll, Pitch, Yaw (RPY) angles  Euler angles  Articulated joints

30. Forward and Inverse KinematicsEquations for Orientation (a) Roll, Pitch, Yaw(RPY) Angles 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) Chapter 2Robot Kinematics: Position Analysis Fig. 2.22 RPY rotations about the current axes.

31. Forward and Inverse KinematicsEquations for Orientation (b) Euler Angles 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). Chapter 2Robot Kinematics: Position Analysis Fig. 2.24 Euler rotations about the current axes.

32. Forward and Inverse KinematicsEquations for Orientation Chapter 2Robot Kinematics: Position Analysis Roll, Pitch, Yaw(RPY) Angles • 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

33. Forward and Inverse Transformations for robot arms

34. INVERSE OF TRANSFORMATION MATRICES • 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. Fig. 2.16 The Universe, robot, hand, part, and end effecter frames.

35. Identity Transformations

36. We often need to calculate INVERSE MATRICES It is good to reduce the number of such operations We need to do these calculations fast

37. How to find an Inverse Matrix B of matrix A?

38. Inverse Homogeneous Transformation

39. 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 b3 c1 c2 c3sf It is presented in more detail on the WWW!

40. For various types of robots we have different maneuvering spaces

41. For various types of robots we calculate different forward and inverse transformations

42. For various types of robots we solve different forward and inverse kinematic problems

43. Forward and Inverse Kinematics: Single Link Example

44. Forward and Inverse Kinematics: Single Link Example easy

46. Denavit – Hartenbergidea

47. DENAVIT-HARTENBERG REPRESENTATION OF FORWARD KINEMATIC EQUATIONS OF ROBOT Fig. 2.25 A D-H representation of a general-purpose joint-link combination • Denavit-Hartenberg Representation : @ 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.

48. DENAVIT-HARTENBERG REPRESENTATION Symbol Terminologies : 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 twist) (revolute)  Only  anddare joint variables.

49.  associated with Zi always Links are in 3D, any shape

50. Only translation Only offset Only rotation Only rotation Axis alignment Only offset