Direct Kinematics The Arm Equation

1
Direct Kinematics- The Arm Equation

• Link Coordinates and Kinematics Parameters
  (Contd)
• Another example A 5-axis articulated robot
  (Rhino XR-3)
• (refer to class notes for details)
• Exercise Determine the link coordinates and the
  kinematics parameters of
• the Motoman robot arm
• the SONY arm (LAB-2)

2
The Arm Equation

• For a n-axis robot arm, the arm equation is
• P0 T Pn
• where P0 is a point (vector) w.r.t. frame L0 ,
  ie. The base frame
• Pn is a point (vector) w.r.t. frame Ln,
  ie. The tool frame
• T is the transformation matrix.
• Question What is T?
• T is the composite transformation from one frame
  to another and is expressed in terms of the
  kinematics parameters.

3
Arm Equation

• Consider the transformation from one frame to
  another, say from frame Lk-1 to Lk.
• Idea is find the transformation that will bring
  Lk-1 to align with Lk .
• 4 fundamental motions
• First two motions
• Rotate Lk-1 about zk-1 to bring xk-1 parallel to
  xk , ie. by an angle of qk gt pure rotation
  Rot(qk , zk-1 )
• Translate Lk-1 along zk-1 to bring xk-1 align
  with xk , ie. by a distance of dk gt pure
  translation Trans(dk , zk-1 )
• gt Screw (dk , qk , zk-1 )

4
Arm Equation

• Second two motions
• Translate Lk-1 along xk-1 (xk ) to bring Lk-1 and
  Lk (the two origins) to coincide, ie. by a
  distance of ak gt pure translation Trans(ak ,
  xk-1 )
• Rotate Lk-1 about xk-1 (xk ) to bring zk-1 to
  align with zk , ie. by an angle of ak gt pure
  rotation Rot(ak , xk-1 )
• gt Screw (ak , ak , xk-1 )
• Let Tk-1k denote the transformation from frame
  Lk-1 to frame Lk,
• i.e. Pk-1 Tk-1k Pk

5
Arm Equation

• Question
• Tk-1k Screw (dk , qk , zk-1 ) Screw (ak , ak ,
  xk-1 ) A ?
• or
• Tk-1k Screw (ak , ak , xk-1 ) Screw (dk , qk ,
  zk-1 ) B ?
• Note Refer to class notes for the details to
  answer the above question.