Basic Kinematics

1. Basic Kinematics NatteeNiparnan

2. Recall • Robot Programming • Introduction to Control • PID • Motion Planning

3. Kinematics • Physical representation of manipulator • Description of robotics entity • Forward kinematics • Inverse kinematics

4. Motivation • Robot acts in the real world • We must know how robot interact with the real world • Where it is? • Where is its arm, upper arm, forearm, hand, etc. • How to reach to some position • Today, In static setting

5. What you will learn today • How to know the position of the robotic body? • Is that hard?

6. Today’s Protagonist • Manipulator

7. Robot Component • Links • Joints • End Effector

8. Joint

9. Link Something that connects joins

10. End Effector The last part of the robot

11. The Question: reprise • Where is my End Effecter? • Where are my Joints? • Where are my Links? • Demo • Looks ahead  motion planning

12. Description of Entity

13. Entities • Points • Orientation • Frame

14. Example We need coordinate Described by a vector P • Where is the end effecter? EE here P link joint link

15. Position Vector • P = What is the meaning of the value of a and b? Distance? From what?

16. The Origin We write P as Means that the value of x is related to the frame O • With respect to the origin yo P link link O xo

17. The Origin • With respect to the origin yo P a O b xo

18. Relative Description • The vector is related (referenced) to the specific frame • For now, let us assume that we know where the reference point is

19. Concrete Example • Object is a set of points, w.r.t. to some fixed point on the object

20. Another Example • Where is the end effecter? • Position is not enough • We need orientation link link X link link

21. Orientation • Rotation • Can we just simply use the angle?

22. Orientation by axes • Attach axes • Axis is a unit vector yE xE link link

23. Axes also described relatively yE d xE b a c

24. Angle <-> Axes equivalence • b = sin(θ) • a = cos(θ) • d = sin(θ+90o)=cos(θ) • c = cos(θ+90o)=-sin(θ) yE d xE b a c θ

25. Angle <-> Axes equivalence • We will soon knows that • Angle  axes is simple • Axes  angle is simple

26. Rotation Matrix • We write the orientation as a matrix • Rotation matrix is a matrix of column vectors that describe the axes

27. Recall the Dot Product A B |C| = A∙B/|B| C

28. Rotation Matrix • We write the orientation as a matrix yE d xE b a c

29. Frame Description • End Effecter can be described by Frame • Position and orientation  Frame • Let us call the “End Effecter Frame” as “Frame E” ( {E} ) • Describe the other frame related to the origin

30. Origin as a frame • Origin itself, is also a frame • x = (1,0)T • y = (0,1)T • P = (0,0)T • Hence, the description actually describe a frame relative to another frame

31. Extend to 3D : Position

32. Extend to 3D : Orientation

33. Extend to 3D : Frame Frame B is described by

34. Transformation

35. Mapping from Frame to Frame • If we know P relative to {B} and the frame {B} • What is P relative to {A}?

36. Translation Mapping

37. Rotational Mapping

38. Rotational Mapping

39. Rotational Mapping See how B cancelled out

40. A note on rotational matrix • As we have seen • What is RT ? • Hence, RT=R-1 Transpose is equal to the inverse

41. General Mapping

42. Mapping Example 5 10

43. Mapping Example R

44. Mapping Example given 7 3

45. Mapping Example given

46. Homogeneous Transform • Mapping using Matrix Multiplication • Instead of • We write Transformation matrix

47. Homogeneous Transform T Row of 0 and 1

48. Homogeneous Transform as a Frame Descriptor • Descriptor = (PBORG, RB) • Transform can also be regarded as a descriptor of a frame • is a description of frame {B} w.r.t to {A}

49. Operator on Points • T is an operator that performs “mapping” from one frame to another frame • Using matrix multiplication • There are also many other operators • Also matrix multiplication

50. Translational Operator • Translate point P1 by Q • What is P2 ?