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Introduction to Robotics Lecture II

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  1. Introduction to RoboticsLecture II Alfred Bruckstein Yaniv Altshuler

  2. Denavit-Hartenberg • Specialized description of articulated figures • Each joint has only one degree of freedom • rotate around its z-axis • translate along its z-axis

  3. Denavit-Hartenberg • One degree of freedom : very compact notation • Only fourparameters to describe a relation between two links : • link length • link twist • link offset • link rotation

  4. Denavit-Hartenberg • Link length ai • The perpendicular distance between the axes of jointi and jointi+1

  5. Denavit-Hartenberg • Link twist αi • The angle between the axes of jointi and jointi+1 • Angle around xi-axis

  6. Denavit-Hartenberg • Link offset di • The distance between the origins of the coordinate frames attached to jointi and jointi+1 • Measured along the axis of jointi

  7. Denavit-Hartenberg • Link rotation (joint angle) φi • The angle between the link lenghts αi-1 and αi • Angle around zi-axis

  8. Denavit-Hartenberg • How to compute the parameters to describe an articulated figure : • Compute the link vector ai and the link length • Attach coordinate frames to the joint axes • Compute the link twist αi

  9. Denavit-Hartenberg • Compute the link offset di • Compute the joint angle φi • Compute the transformation (i-1)Ti which transforms entities from linki to linki-1

  10. Denavit-Hartenberg • Let’s do it step by step • Compute the link vector ai and the link length • Attach coordinate frames to the joint axes • Compute the link twist αi • Compute the link offset di • Compute the joint angle φi • Compute the transformation (i-1)Ti which transforms entities from linki to linki-1

  11. Denavit-Hartenberg The link length ai is the shortest distance between the joint axes jointi and jointi+1. Let the joint axes be given by the expression : Where pi is a point on axis of jointi and ui is one of its direction vectors (analogous for jointi+1).

  12. Denavit-Hartenberg

  13. Denavit-Hartenberg • There are three methods to compute the link vector ai and the link length

  14. Denavit-Hartenberg • Method 1 : The Pseudo-naive approach The shortest distance aiis the length of the vector connecting the two axes, and perpendicular to both of them. Which can be expressed :

  15. Denavit-Hartenberg Let’s find the points oi and oai where this distance exists.

  16. We can go some distance s from pi along axisi, and then the distance ai along the unit vector and finally some distance t along axisi+1 to arrive at point pi+1. Denavit-Hartenberg

  17. Denavit-Hartenberg Multiplying respectively by ui and ui+1, we obtain the two following equations:

  18. Denavit-Hartenberg Solution :

  19. Denavit-Hartenberg Finally, using and we obtain :

  20. Denavit-Hartenberg • Method 2 : The Geometric approach The vector ui x ui+1 gives the perpendicular vector to both axes. Let’s find out where it is located on the joint axes. We can go some distance s from point pi along the axisi, and then go some distance k along ui x ui+1. Finally go some distance t along the axisi+1 to arrive at point pi+1.

  21. Denavit-Hartenberg We obtain the equation : There are three unknowns.

  22. Denavit-Hartenberg Let’s first eliminate the unknown k from the equation : by multiplying by ui:

  23. Denavit-Hartenberg Let’s first eliminate the unknown k from the equation : by multiplying by ui+1:

  24. Denavit-Hartenberg Now we shall eliminate the s and t from the equation : by multiplying by ui x ui+1:

  25. Denavit-Hartenberg We have obtained a system of three equations in the unknowns s, t, k :

  26. Denavit-Hartenberg From , it can be seen that the shortest distance between jointi and jointi+1 is given by the vector : Where

  27. Denavit-Hartenberg From and , we can compute s and t :

  28. Denavit-Hartenberg Finally, using and we obtain :

  29. Denavit-Hartenberg • Method 3 : The Analytic approach The distance between two arbitrary points located on the joint axes jointi and jointi+1 is :

  30. Denavit-Hartenberg The link length of linki, ai, is the minimum distance between the joint axes :

  31. Denavit-Hartenberg A necessary condition is :

  32. Denavit-Hartenberg Which is equivalent to their numerators being equal to 0 :

  33. Denavit-Hartenberg Rewriting this system yields :

  34. Denavit-Hartenberg Whose solution are :

  35. Denavit-Hartenberg Finally, using and we obtain :

  36. Denavit-Hartenberg oi and oaiare the closest points on the axes of jointi and jointi+1. We deduce that the link vector aiand the link length ai :

  37. Denavit-Hartenberg The link vector ai:

  38. Denavit-Hartenberg Calculating the scalar products and, both equal to 0, proves that the vector ai is perpendicular to both axes of jointi and jointi+1

  39. Denavit-Hartenberg • Three methods • How do we actually compute ai and ||ai||2 ?

  40. Denavit-Hartenberg The link vector ai is perpendicular to both of the axes of jointi and jointi+1. The unit vector : is parallel to the link vector ai.

  41. Denavit-Hartenberg Given two points pi and pi+1on the axes of jointi and jointi+1, the link length can be computed as : And the link vector :

  42. Denavit-Hartenberg • Special cases : • The joint axes intersect • The shortest distance ai is equal to zero • The link vector is the null vector

  43. Denavit-Hartenberg • The joint axes are parallel • There is no unique shortest distance oi can be chosen arbitrarily, so we should chose values that offset the most of Denavit-Hartenberg parameters

  44. Denavit-Hartenberg • The first joint • There is no link preceding it • We use a base link : link0 • Its link frame should coincide with the link frame of link1 • Most of the Denavit-Hartenberg parameters will be equal to zero

  45. Denavit-Hartenberg • The last joint • There is no link succeding it • We use arbitrary values so that most of Denavit-Hartenberg parameters are equal to zero

  46. Denavit-Hartenberg • Compute the link vector ai and the link lenght • Attach coordinate frames to the joint axes • Compute the link twist αi • Compute the link offset di • Compute the joint angle φi • Compute the transformation (i-1)Ti which transforms entities from linki to linki-1

  47. Denavit-Hartenberg • Identify the joint axes • Identify the common perpendiculars of successive joint axes • Attach coordinate frames to each joint axes

  48. Denavit-Hartenberg Identifying the joint axes

  49. Remember, is the point where the shortest distance to jointi+1 exists Denavit-Hartenberg Identifying the common perpendiculars

  50. the origin Denavit-Hartenberg Attaching the frames