Kinematics of Robot Manipulator

1. Introduction to ROBOTICS Kinematics of Robot Manipulator

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. 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? • Perform 4A tasks in 4D environments 4A: Automation, Augmentation, Assistance, Autonomous 4D: Dangerous, Dirty, Dull, Difficult

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. Robotic Manipulators • a robotic manipulator is a kinematic chain • i.e. an assembly of pairs of rigid bodies that can move respect to one another via a mechanical constraint • the rigid bodies are called links • the mechanical constraints are called joints

6. A150 Robotic Arm link 3 link 2

7. Joints • most manipulator joints are one of two types • revolute (or rotary) • like a hinge • allows relative rotation about a fixed axis between two links • axis of rotation is the z axis by convention • prismatic (or linear) • like a piston • allows relative translation along a fixed axis between two links • axis of translation is the z axis by convention • our convention: joint i connects link i – 1 to link i • when joint i is actuated, link i moves

8. Joint Variables • revolute and prismatic joints are one degree of freedom (DOF) joints; thus, they can be described using a single numeric value called a joint variable • qi : joint variable for joint i • revolute • qi = qi: angle of rotation of link i relative to link i – 1 • prismatic • qi = di : displacement of link i relative to link i – 1

9. Revolute Joint Variable link i qi link i – 1 • revolute • qi = qi: angle of rotation of link i relative to link i – 1

10. Prismatic Joint Variable link i – 1 link i di • prismatic • qi = di : displacement of link i relative to link i – 1

11. Common Manipulator Arrangments • most industrial manipulators have six or fewer joints • the first three joints are the arm • the remaining joints are the wrist • it is common to describe such manipulators using the joints of the arm • R: revolute joint • P: prismatic joint

12. Articulated Manipulator z0 z1 z2 q2 q3 shoulder forearm elbow q1 waist • RRR (first three joints are all revolute) • joint axes • z0 : waist • z1 : shoulder (perpendicular to z0) • z2 : elbow (parallel to z1)

13. Spherical Manipulator z0 z1 d3 q2 shoulder z2 q1 waist • RRP • Stanford arm • http://infolab.stanford.edu/pub/voy/museum/pictures/display/robots/IMG_2404ArmFrontPeekingOut.JPG

14. SCARA Manipulator z1 z2 z0 q2 d3 q1 • RRP • Selective Compliant Articulated Robot for Assembly • http://www.robots.epson.com/products/g-series.htm

15. Manipulators • Robot Configuration: Cartesian: PPP Cylindrical: RPP Spherical: RRP Hand coordinate: n: normal vector; s: sliding vector; a: approach vector, normal to the tool mounting plate SCARA: RRP (Selective Compliance Assembly Robot Arm) Articulated: RRR

16. 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

17. 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 how accurately a specified point can be reached how accurately the same position can be reached if the motion is repeated many times Which one is more important?

18. What is Kinematics • Forward kinematics Given joint variables End-effector position and orientation, -Formula?

19. What is Kinematics • Inverse kinematics End effector position and orientation Joint variables -Formula?

20. Example 1

21. Example II a2 q2 a1 q1 given the joint variables and dimensions of the links what is the position and orientation of the end effector?

22. Forward Kinematics (x, y) ? a2 q2 y0 a1 q1 x0 • choose the base coordinate frame of the robot • we want (x, y) to be expressed in this frame

23. Forward Kinematics (x, y) ? a2 q2 y0 a1 q1 ( a1cosq1 , a1 sin q1 ) x0 notice that link 1 moves in a circle centered on the base frame origin

24. Forward Kinematics (x, y) ? y1 a2 q2 y0 q1 a1 x1 q1 ( a1cosq1 , a1 sin q1 ) x0 choose a coordinate frame with origin located on joint 2 with the same orientation as the base frame

25. Forward Kinematics (x, y) ? y1 a2 ( a2cos(q1 + q2), a2sin(q1 + q2) ) q2 y0 q1 a1 x1 q1 ( a1cosq1 , a1 sin q1 ) x0 notice that link 2 moves in a circle centered on frame 1

26. Forward Kinematics (a1cosq1 + a2cos(q1 + q2), a1sinq1 + a2sin(q1 + q2) ) y1 a2 ( a2cos(q1 + q2), a2sin(q1 + q2) ) q2 y0 q1 a1 x1 q1 ( a1cosq1 , a1 sin q1 ) x0 because the base frame and frame 1 have the same orientation, we can sum the coordinates to find the position of the end effector in the base frame

27. Forward Kinematics y2 x2 a2 q2 y0 q1 a1 q1 x0 • we also want the orientation of frame 2 with respect to the base frame • x2 and y2 expressed in termsof x0 and y0

28. Forward Kinematics x2 = (cos(q1 + q2), sin(q1 + q2) ) y2 x2 y2 = (-sin(q1 + q2), cos(q1 + q2) ) a2 q2 y0 q1 a1 q1 x0

29. Inverse Kinematics y2 x2 (x, y) a2 q2 ? y0 a1 q1 ? x0 given the position (and possiblythe orientation) of the endeffector, and the dimensionsof the links, what are the jointvariables?

30. Inverse Kinematics (x, y) a2 y0 a1 x0 harder than forward kinematics because there is often more than one possible solution

31. Inverse Kinematics (x, y) a2 b q2 ? y0 a1 x0 law of cosines

32. Inverse Kinematics and we have the trigonometric identity therefore, We could take the inverse cosine, but this gives only one of the two solutions.

33. Inverse Kinematics Instead, use the two trigonometric identities: to obtain which yields both solutions for q2 . In many programming languages you would use the four quadrant inverse tangent function atan2 c2 = (x*x + y*y – a1*a1 – a2*a2) / (2*a1*a2); s2 = sqrt(1 – c2*c2); theta21 = atan2(s2, c2); theta22 = atan2(-s2, c2);

34. Inverse Kinematics

35. Preliminary • Robot Reference Frames • World frame • Joint frame • Tool frame T P W R

36. Points and Vectors point : a location in space vector : magnitude (length) and direction between two points

37. Coordinate Frames choosing a frame (a point and two perpendicular vectors of unit length) allows us to assign coordinates

38. Coordinate Frames the coordinates change depending on the choice of frame

39. O, O’ Preliminary • Coordinate Transformation • Reference coordinate frame OXYZ • Body-attached frame O’uvw Point represented in OXYZ: Point represented in O’uvw: Two frames coincide ==>

40. Mutually perpendicular Unit vectors Preliminary Properties: Dot Product Let and be arbitrary vectors in and be the angle from to , then Properties of orthonormal coordinate frame

41. Preliminary • Coordinate Transformation • Rotation only How to relate the coordinate in these two frames?

42. Preliminary • Basic Rotation • , , and represent the projections of onto OX, OY, OZ axes, respectively • Since

43. Preliminary • Basic Rotation Matrix • Rotation about x-axis with

44. Preliminary • Is it True? • Rotation about x axis with

45. Basic Rotation Matrices • Rotation about x-axis with • Rotation about y-axis with • Rotation about z-axis with

46. Preliminary • Basic Rotation Matrix • Obtain the coordinate of from the coordinate of Dot products are commutative! <== 3X3 identity matrix

47. Properties of Rotation Matrices RT = R-1 the columns of R are mutually orthogonal each column of R is a unit vector det R = 1 (the determinant is equal to 1)

48. Example • 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.

49. Example • 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.

50. 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