ECE 450 Introduction to Robotics

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ECE 450 Introduction to Robotics

• Section 50883
• Instructor Linda A. Gee
• 9/21/99
• Lecture 06

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Links, Joints and the Parameters to describe them

• A mechanical manipulator has a sequence of rigid
  bodies called links.
• The links are connected by revolute and prismatic
  joints
• Each joint-link pair defines 1 degree of freedom

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Robotic Link
Jointi
Jointi1
Linki1
ai
Link length
?i
ai
Link angle between joint axes
?i
common normal
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Links and Joints contd

• An N-degree of freedom manipulator suggests there
  are N joint-link pairs link 0 is attached to the
  supporting base where the inertial coordinate
  frame is defined for the dynamic system
• Link 0 is NOT considered part of the robot
• The last link, link N is attached with a tool

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More about Links and Joints

• Joints and links are numbered from the base
  forward
• For example, joint 1 refers to the connection
  point between the base and link 1
• Each link is connected to at most 2 other links

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More on Joints and Links

• The joint axis for jointi is established at the
  connection of two links
• The joint axis has two normals associated with it
  (i.e. one for each link)
• The relative position for two connected links
  linki-1 and linki is given by
• di distance measured along the joint axis
  between normals

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More with Joints and Links

• The joint angle ?i between normals is measured in
  the plane normal to the joint axis
• di distance between adjacent links
• ?i angle between adjacent links
• di and ?i determine the relative position of
  neighboring links

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More info on Joints and Links

• Link i is connected to at most two other links
  and has two joint axes that are defined at both
  ends of connections
• Parameters ai and ?i describe the length and the
  twist angle respectively of linki

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Four Important Parameters

• There are four parameters that are used to
  completely describe revolute or prismatic joints
• (ai, ?i) describe the structure of the link
• (di, ?i) describe the relative position of
  neighboring links

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Robotic Joint
Zi-1
Zi
Zi1
Linki
Jointi1
Jointi-1
Jointi
Linki1
ai-1
?i
di
common normal
ai
?i
common normal
di
Distance along joint axes between normals
?i
Joint angle between normals to joint axes
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Denavit-Hartenberg Representation

• Coordinate frames are established according to
  the following rules
• 1. Zi-1 axis lies along the axis of motion of the
  ith joint
• 2. Xi axis is normal to the Zi-1 axis and points
  away
• 3. Yi axis completes the right-hand-rule
• Depends on four parameters associated with each
  link which are used to describe any revolute or
  prismatic joints

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D-H Parameters

• The parameters to describe links
• ?i joint angle from xi-1 axis to the xi about
  the zi-1 axis
• di distance from the origin of (i-1)th
  coordinate frame to intersection of zi-1 axis
  with the xi axis along the zi-1 axis
• ai offset distance from the intersection of
  zi-1 axis with xi axis to the origin of the ith
  frame along the xi axis (shortest distance
  between zi-1 and zi)
• ?i offset angle from zi-1 axis to zi axis about
  xi (use the right hand rule)

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PUMA Robot
Fu, page 37
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Stanford Robot
Fu, page 38
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Establishing Link Coordinate Systems

• Obtain the link coordinates for the robot
• Develop the homogeneous transformation matrix
  that relates the ith coordinate frame to the
  (i-1)th coordinate frame

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Hand Coordinate System
Fu, page 43
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Link Coordinate Systems contd
i-1Ai Tz,d Tz,? Tx,a Tx,?
n normal vector of hand s sliding vector of
hand a approach vector of hand p position
vector of hand
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Kinematic Equations for Manipulators
0Ti specifies the transformation matrix for
the ith coordinate frame with respect to
the base coordinate frame
i
0Ti 0A1 1A2 . . . i-1Ai ? j-1Aj
for i 1, 2, n
j1
T 0A6 for the manipulator
B transformation manipulator related to the
reference coordinate frame H tool attached to
the last joint
refTtool B 0T6 H
B refA0
H 6Atool
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Position and Orientation for End Effector
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Transformation Matrices for End Effectors
Position
Orientation
1 0 0 px
0
n s a
or R ???
0 1 0 py
Tposition

0
0 0 1 pz
0
0 0 0 1
0 0 0 1