1. The City College of New York 1 Dr. Jizhong Xiao Department of Electrical Engineering City College of New York jxiao@ccny.cuny.edu Kinematics of Robot Manipulator Introduction to ROBOTICS

2. The City College of New York 2 Outline Review Robot Manipulators –Robot Configuration –Robot Specification Number of Axes, DOF Precision, Repeatability Kinematics –Preliminary World frame, joint frame, end-effector frame Rotation Matrix, composite rotation matrix Homogeneous Matrix –Direct kinematics Denavit-Hartenberg Representation Examples –Inverse kinematics

3. The City College of New York 3 Review What is a robot? –By general agreement a robot is: A programmable machine that imitates the actions or appearance of an intelligent creature–usually a human. –To qualify as a robot, a machine must be able to: 1) Sensing and perception: get information from its surroundings 2) Carry out different tasks: Locomotion or manipulation, do something physical–such as move or manipulate objects 3) Re-programmable: can do different things 4) Function autonomously and/or interact with human beings Why use robots? 4A: Automation, Augmentation, Assistance, Autonomous 4D: Dangerous, Dirty, Dull, Difficult –Perform 4A tasks in 4D environments

4. The City College of New York 4 Manipulators Robot arms, industrial robot –Rigid bodies (links) connected by joints –Joints: revolute or prismatic –Drive: electric or hydraulic –End-effector (tool) mounted on a flange or plate secured to the wrist joint of robot

5. The City College of New York 5 Manipulators Robot Configuration: Cartesian: PPPCylindrical: RPPSpherical: RRP SCARA: RRP (Selective Compliance Assembly Robot Arm) Articulated: RRR Hand coordinate: n: normal vector; s: sliding vector; a: approach vector, normal to the tool mounting plate

6. The City College of New York 6 Manipulators Motion Control Methods –Point to point control a sequence of discrete points spot welding, pick-and-place, loading & unloading –Continuous path control follow a prescribed path, controlled-path motion Spray painting, Arc welding, Gluing

7. The City College of New York 7 Manipulators Robot Specifications –Number of Axes Major axes, (1-3) => Position the wrist Minor axes, (4-6) => Orient the tool Redundant, (7-n) => reaching around obstacles, avoiding undesirable configuration –Degree of Freedom (DOF) –Workspace –Payload (load capacity) –Precision v.s. Repeatability Which one is more important? how accurately a specified point can be reached how accurately the same position can be reached if the motion is repeated many times

8. The City College of New York 8 What is Kinematics Forward kinematics Given joint variables End-effector position and orientation, -Formula?

9. The City College of New York 9 What is Kinematics Inverse kinematics End effector position and orientation Joint variables -Formula?

10. The City College of New York 10 Example 1

11. The City College of New York 11 Preliminary Robot Reference Frames –World frame –Joint frame –Tool frame W R P T

12. The City College of New York 12 Preliminary Coordinate Transformation –Reference coordinate frame OXYZ –Body-attached frame O’uvw O, O’ Point represented in OXYZ: Point represented in O’uvw: Two frames coincide ==>

13. The City College of New York 13 Preliminary Mutually perpendicularUnit vectors Properties of orthonormal coordinate frame Properties: Dot Product Let and be arbitrary vectors in and be the angle from to, then

14. The City College of New York 14 Preliminary Coordinate Transformation –Rotation only How to relate the coordinate in these two frames?

15. The City College of New York 15 Preliminary Basic Rotation –,, and represent the projections of onto OX, OY, OZ axes, respectively –Since

16. The City College of New York 16 Preliminary Basic Rotation Matrix –Rotation about x-axis with

17. The City College of New York 17 Preliminary Is it True? –Rotation about x axis with

18. The City College of New York 18 Basic Rotation Matrices –Rotation about x-axis with –Rotation about y-axis with –Rotation about z-axis with

19. The City College of New York 19 Preliminary Basic Rotation Matrix –Obtain the coordinate of from the coordinate of <== 3X3 identity matrix Dot products are commutative!

20. The City College of New York 20 Example 2 A point is attached to a rotating frame, the frame rotates 60 degree about the OZ axis of the reference frame. Find the coordinates of the point relative to the reference frame after the rotation.

21. The City College of New York 21 Example 3 A point is the coordinate w.r.t. the reference coordinate system, find the corresponding point w.r.t. the rotated OU-V-W coordinate system if it has been rotated 60 degree about OZ axis.

22. The City College of New York 22 Composite Rotation Matrix A sequence of finite rotations –matrix multiplications do not commute –rules: if rotating coordinate O-U-V-W is rotating about principal axis of OXYZ frame, then Pre-multiply the previous (resultant) rotation matrix with an appropriate basic rotation matrix if rotating coordinate OUVW is rotating about its own principal axes, then post-multiply the previous (resultant) rotation matrix with an appropriate basic rotation matrix

23. The City College of New York 23 Example 4 Find the rotation matrix for the following operations: Post-multiply if rotate about the OUVW axes Pre-multiply if rotate about the OXYZ axes

24. The City College of New York 24 Coordinate Transformations position vector of P in {B} is transformed to position vector of P in {A} description of {B} as seen from an observer in {A} Rotation of {B} with respect to {A} Translation of the origin of {B} with respect to origin of {A}

25. The City College of New York 25 Coordinate Transformations Two Special Cases 1. Translation only –Axes of {B} and {A} are parallel 2. Rotation only –Origins of {B} and {A} are coincident

26. The City College of New York 26 Homogeneous Representation Coordinate transformation from {B} to {A} Homogeneous transformation matrix Position vector Rotation matrix Scaling

27. The City College of New York 27 Homogeneous Transformation Special cases 1. Translation 2. Rotation

28. The City College of New York 28 Example 5 Translation along Z-axis with h: O, O’ h

29. The City College of New York 29 Example 6 Rotation about the X-axis by

30. The City College of New York 30 Homogeneous Transformation Composite Homogeneous Transformation Matrix Rules: –Transformation (rotation/translation) w.r.t (X,Y,Z) (OLD FRAME), using pre- multiplication –Transformation (rotation/translation) w.r.t (U,V,W) (NEW FRAME), using post- multiplication

31. The City College of New York 31 Example 7 Find the homogeneous transformation matrix (T) for the following operations:

32. The City College of New York 32 Homogeneous Representation A frame in space (Geometric Interpretation) Principal axis n w.r.t. the reference coordinate system (X’) (y’) (z’)

33. The City College of New York 33 Homogeneous Transformation Translation

34. The City College of New York 34 Homogeneous Transformation Composite Homogeneous Transformation Matrix ? Transformation matrix for adjacent coordinate frames Chain product of successive coordinate transformation matrices

35. The City College of New York 35 Example 8 For the figure shown below, find the 4x4 homogeneous transformation matrices and for i=1, 2, 3, 4, 5 Can you find the answer by observation based on the geometric interpretation of homogeneous transformation matrix?

36. The City College of New York 36 Orientation Representation Rotation matrix representation needs 9 elements to completely describe the orientation of a rotating rigid body. Any easy way? Euler Angles Representation

37. The City College of New York 37 Orientation Representation Euler Angles Representation (,, ) –Many different types –Description of Euler angle representations Euler Angle I Euler Angle II Roll-Pitch-Yaw Sequence about OZ axis about OZ axis about OX axis of about OU axis about OV axis about OY axis Rotations about OW axis about OW axis about OZ axis

38. The City College of New York 38 x y z u'u' v'v'   v " w"w" w'=w'= =u" v'"  u'" w'"= Euler Angle I, Animated

39. The City College of New York 39 Orientation Representation Euler Angle I

40. The City College of New York 40 Euler Angle I Resultant eulerian rotation matrix:

41. The City College of New York 41 Euler Angle II, Animated x y z u'u' v'v'   =v" w"w" w'=w'= u" v"'  u"' w"'= Note the opposite (clockwise) sense of the third rotation, .

42. The City College of New York 42 Orientation Representation Matrix with Euler Angle II Quiz: How to get this matrix ?

43. The City College of New York 43 Orientation Representation Description of Roll Pitch Yaw X Y Z Quiz: How to get rotation matrix ?

44. The City College of New York 44 Thank you! Homework 1 is posted on the web. Next class: kinematics II