1 / 85

Introduction to Robotics Lecture II

Introduction to Robotics Lecture II. Alfred Bruckstein Yaniv Altshuler. 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. Denavit-Hartenberg.

jennifer-snyder
Télécharger la présentation

Introduction to Robotics Lecture II

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

More Related