1 / 57

Kinematics – Frame Assignment using Denavit-Hartenberg Convention

Kinematics – Frame Assignment using Denavit-Hartenberg Convention. Professor Nicola Ferrier ME Room 2246, 265-8793 ferrier@engr.wisc.edu. Coordinate Transformations. End-effector. Z. Base. Supply. Table. Goal. Y. X. Coordinate Transformations. End-effector. Base. Supply. Goal.

samantha-horton
Download Presentation

Kinematics – Frame Assignment using Denavit-Hartenberg Convention

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. Kinematics – Frame Assignment using Denavit-Hartenberg Convention Professor Nicola Ferrier ME Room 2246, 265-8793 ferrier@engr.wisc.edu

  2. Coordinate Transformations End-effector Z Base Supply Table Goal Y X

  3. Coordinate Transformations End-effector Base Supply Goal Table

  4. Coordinate Transformations Robot forward kinematic model

  5. Manipulator Forward Kinematics • Motion is composition of elementary motions for each link End-effector Base

  6. Relative Pose between 2 links i i-1

  7. Relative Pose between 2 links • Frames can be chosen arbitrarily • Denavit-Hartenberg convention is used to assign frames – described in §3.2.2 of Spong, Hutchinson, Vidyasagar Text • Iterative process (start at base, assign frames for each link from base to end-effector)

  8. DH Frame assignment • Frame {i} moves with link i when joint i is actuated • Zi axis is along joint axis i+1 • Zi is axis of actuation for joint i+1 Zi Link i-1 Link i+1 Link i Zi-1

  9. DH convention: Assign Z axes • Use actuation as a guide • Prismatic – joint slides along zi • Revolute – joint rotates around zi • Establish base frame {0}: • Nearly arbitrary • Start at base and assign frames 1,…,N • Pick x-axis and origin • y-axis chosen to form a right hand system

  10. Robot Base • Often base is “given” or some fixed point on the work-table is used. • z0 is along joint axis 1 • Original: • any point on z0 for origin • Modified DH: • {0} is defined to be completely co-incident with the reference system {1}, when the variable joint parameter, d1 or q1 , is zero.

  11. DH convention: Assign X axes • Start at base and assign frames 1,…,N • Pick x-axis and origin • y-axis chosen to form a right hand system • Consider 3 cases for zi-1 and zi: • Not-coplanar • Parallel • Intersect

  12. DH convention: x axis • zi-1 and zi are not-coplanar • Common normal to axes is the “link” axis • Intersection with zi is origin Usually, xi points from frame i-1 to i zi-1 Xi zi

  13. DH convention: x axis • zi and zi-1 are parallel • Infinitely many common normals • Pick one to be the “link” axis • Choose normal that passes through origin of frame {i-1} pointing toward zi • Origin is intersection of xi with zi Xi zi-1 zi

  14. DH convention: x axis zi If joint axes zi-1 and zi intersect, xi is normal to the plane containing the axes xi = (zi-1  zi ) zi-1 link i Xi

  15. DH convention: Origin non-coplanar Z Origin of frame {i} is placed at intersection of joint axis and link axis zi xi

  16. DH convention: y axis • Yi is chosen to make a right hand frame Zi xi points from frame i-1 to i Yi xi

  17. DH convention: Origin parallel Z • zi and zi-1 are parallel • Origin is intersection of xi with zi zi-1 zi xi

  18. DH convention: x axis - parallel Z • zi and zi-1 are parallel • Origin is intersection of xi with zi • Yi is chosen to make a right hand frame yi zi-1 zi xi

  19. DH convention: origin If joint axes intersect, the origin of frame {i} is usually placed at intersection of the joint axes zi zi-1 link i xi

  20. DH convention: y axis Yi is chosen to make a right hand frame zi zi-1 yi link i xi

  21. End-Effector Frame • Frame to which the gripper is attached • Sometimes {n} is used • denoted by {e} (or {n+1} in many texts) • Often simple translation along Xn axis Z4 Ze Xe

  22. End-Effector Frame • Frame to which the gripper is attached – • denoted by {e} (or {n+1} in many texts) • Often simple translation along Xn axis • Often: • Origin between grippers • Z points outward (approach) • Y points along pinch direction (sliding) • X points normal Z4 ye xe ze

  23. Link Parameters ai+1 Zi Z’i Zi-1 Zi+1 Link i ai ai+1 ai

  24. Joint Parameters i di+1 i+1 di i

  25. Original DH -1 Frame is placed at distal end of link xi screw motion zi-1 screw motion

  26. DH Frames and Parameters

  27. Robot Revolute Joint DH

  28. Prismatic Joint DH

  29. Link Transformations • Described by 4 parameters: • ai : twist • ai : link length • di : joint offset • qi : joint angle • Joint variable is di or qi • Build Table with values for each link:

  30. Link Transformations • Described by 4 parameters: • ai : twist • ai : link length • di : joint offset • qi : joint angle • Joint variable is di or qi • Link Transformation is zi-1 screw motion xiscrew motion

  31. A-matrices Ai = contains only one variable or Equation 3.10 in Spong, Hutchinson, Vidyasagar

  32. Original DH -1 ! Frame is placed at distal end of link zi-1 screw motion xi screw motion

  33. Modified DH zi yi xi Zi+1 ! Zi Zi+2 Frame is placed at proximal end of link xi-1 screw motion zi screw motion

  34. Modified DH – text figure

  35. DH Example: “academic manipulator” 3 revolute joints Shown in home position joint 1 R Link 2 Link 3 Link 1 joint 2 joint 3 L1 L2

  36. DH Example: “academic manipulator” Zi is axis of actuation for joint i+1 Z0 Z0 and Z1 are not co-planar Z1 and Z2 are parallel 1 3 2 Z1 Z2

  37. DH Example: “academic manipulator” Z0 and Z1 are not co-planar: x0 is the common normal Z0 1 x1 x2 x3 x0 3 2 Z3 Z1 Z2

  38. DH Example: “academic manipulator” Z0 and Z1 are not co-planar: x0 is the common normal Z0 1 x1 x2 x3 x0 3 2 Z3 Z1 Z2 Z1 and Z2 are parallel : x1 is selected as the common normal that lies along the center of the link

  39. DH Example: “academic manipulator” Z0 and Z1 are not co-planar: x0 is the common normal Z0 1 x1 x2 x3 x0 3 2 Z3 Z1 Z2 Z2 and Z3 are parallel : x2 is selected as the common normal that lies along the center of the link

  40. DH Example: “academic manipulator” Shown with joints in non-zero positions Z0 x3 z3 3 2 x2 x1 Z2 1 x0 Z1 Observe that frame i moves with link i

  41. DH Example: “academic manipulator” Link lengths given 1 = 90o(rotate by 90o around x0 to align Z0 and Z1) R Z0 L2 L1 x1 x2 x3 1 x0 Z3 Z1 Z2

  42. DH Example: “academic manipulator” Build table R Z0 L2 L1 1 x1 x2 x3 x0 1 3 2 Z3 Z1 Z2

  43. DH Example: “academic manipulator”

  44. DH Example: “academic manipulator”

  45. DH Example: “academic manipulator” z0 x3 z3 3 2 x2 x1 z2 1 x0 z1 x1 axis expressed wrt {0} y1 axis expressed wrt {0} z1 axis expressed wrt {0} Origin of {1} w.r.t. {0}

  46. DH Example: “academic manipulator” z0 x3 z3 3 2 x2 x1 z2 1 x0 z1 x2 axis expressed wrt {1} y2 axis expressed wrt {1} z2 axis expressed wrt {1} Origin of {2} w.r.t. {1}

  47. DH Example: “academic manipulator” z0 x3 z3 3 2 x2 x1 z2 1 x0 z1 x3 axis expressed wrt {2} y3 axis expressed wrt {2} z3 axis expressed wrt {2} Origin of {3} w.r.t. {2}

  48. DH Example: “academic manipulator” where

  49. DH Example: “academic manipulator” – alternate end-effector frame Zi is axis of actuation for joint i+1 Z0 Z0 and Z1 are not co-planar Z1 and Z2 are parallel 1 Pick this z3 3 2 Z1 Z2

  50. DH Example: “academic manipulator” – alternate end-effector frame Z0 y2 1 x1 x2 x0 1 Z3 3 2 Z1 Z2 Would need to rotate about y2 here!

More Related