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## MECH572A Introduction To Robotics

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### Robotic Kinematics Overview. Basic Concepts. Robot Kinematics - Study robot motion without resorting to ... Direct Kinematics. Some useful properties of Qi ... – PowerPoint PPT presentation

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Title: MECH572A Introduction To Robotics

1
MECH572AIntroduction To Robotics
• Lecture 5

2
Midterm Exam
• Date Time 1900 - 2100 ,Oct 25, 2004
• Open Book
• Chapters 2 3 of the text book
• Note Regular lecture will take place 1800
1845 on Oct 25

3
Review
• New concepts
• Twist of rigid body
• Wrench (static analysis)
• Instantaneous Screw of rigid-body motion
• Define by direction one point
• Similarity between Velocity and Force/Moment
Analysis
• Screw-like force and moment property Wrench axis

4
Review
• Acceleration Analysis
• Fixed reference frame
• Moving Reference frame
• Corilios term in the expression
• Basics in Rigid Body Dynamics
• Mass properties - Mass 1st 2nd Moment
Parallel Axes Theorem
• Principle
Axes/Moments (Eigenvectors/values)
• Equation of Motion Newton-Euler Equations

Acceleration tensor
5
Robotic Kinematics Overview
• Basic Concepts
• Robot Kinematics - Study robot motion without
resorting to force and mass properties. Dealing
with position, velocity and acceleration
• Kinematic Chain - A set of rigid bodies connected
by kinematic pairs
• Kinematic Pairs
• Upper Pair - Line/point contact (gear,
cam-follower)
• Lower Pair - Surface contact (revolute, prismatic)

6
Robotic Kinematics Overview
• Basic Concepts (cont'd)
• Typical Lower Kinematic Pairs
• Revolute (R) - 1 Dof (Rotation)
• Prismatic (P) - 1 Dof (Translation)
• Cylindrical (C) - 2 Dof (Rotation
Translation)
• Helical (H) - 1 Dof (Coupled
Rotation/Translation)
• Planar (E) - 2 Dof (Translation
in 2 directions)
• Spherical (S) - 3 Dof (Rotation in
3 directions)

7
Robotic Kinematics Overview
• Basic Concepts (cont'd)
• Two Basic Types of Kinematic Pairs - R P
• All six lower pairs can be produced from
Revolute (R) and Prismatic (P)

Sliding pair Prismatic (P)
Rotating pair Revolute (R)
8
Robot Kinematics Overview
• Robot Manipulators
• Rigid bodies
Kinematic Pairs
• Basic Topologies of Kinematic Chain

Necklace
Open Chain
Tree
9
Robot Kinematics Overview
• Basic Problems in Robotic Kinematics
• Direct Kinematics
• Inverse Kinematics

px, , py,pz
?????
Joint Variables
Cartesian Variables
Direct
? x
(Joint)
(Cartesian)
Linear relationship between Cartesian rate of EE
and joint rates
Inverse
10
Denavit-Hartenberg Notation
• Purpose
• To uniquely define architecture of robot
manipulator (Kinematic chains)
• Assumptions
• Pairs 1, 2, , n - n pairs
• Frame Fi (Oi - Xi -Yi -Zi) is attached to
(i-1)st frame
(NOT ith frame)

11
Denavite-Hartenberg Notation
• Definition of Axes
• Zi - Axes of the pair (Rotational/translational)

Zi
Zi
12
Denavite-Hartenberg Notation
• Definition of Axes (cont'd)
• Xi - Common perpendicular to Zi1 and Zi
directed from Zi1 to Zi (Follow right hand
rule if intersect)
• Yi Zi ? Xi

Zi-1
Zi
Xi undefined
(d)
13
DH Notation
• Joint Parameters Joint Variables
• ai - Distance between Zi and Zi1
• bi - Z-coordinate of Zi and Xi1 intersection
point (absolute value distance between Xi and
Xi1 )
• ?i - Angle between Zi and Zi1 along Xi1
(R.H.R)
• ?i - Angle between Xi and Xi1 along Zi
(R.H.R)
• Joint Variables
• ?i - R joint
• bi - P joint

14
DH Notation
• Summary

15
DH Notation
• Summary Prismatic joint

Xi1
bi Variable ?i - Constant
16
DH Notation
• Example - PUMA

17
DH Notation
• Example - PUMA

18
DH Notation
• Example PUMA
• DH Parameters of PUMA
Robot

i ai bi ?i ?i
1 0 b1 90 ?1
2 a2 0 0 ?2
3 a3 b3 90 ?3
4 0 b4 90 ?4
5 0 0 90 ?5
6 0 b6 ? ?6
19
DH Notation
• Example - Stanford Arm

20
DH Notation
• Example - Stanford Arm

21
DH Notation
• Example - Stanford Arm (cont'd)
• DH Parameters of Stanford
Arm

i ai bi ?i ?i
1 0 b1 90 ?1
2 0 b2 90 ?2
3 0 b3 (var) 90 90
4 0 0 90 ?4
5 0 b5 0 ?5
6 0 b6 0 ?6
22
DH Notation
• Summary

ith pair R joint P joint Number of parameters/variable
Joint Parameters (Constant) ai, bi, ?i ai, ?i, ?i 3
Joint Variable (Changing) ?i bi 1
If there are n joint, there will be 3n joint
parameters and n joint variables
23
DH Notation
• Relative Orientation and Position Analysis
• Orientation

(a)
(b)
Rotation Decomposition (a) (b)
24
DH Notation
• Relative Orientation and Position Analysis
• Orientation (cont'd)
• (Xi, Yi, Zi) (Xi', Yi', Zi')
• (Xi', Yi', Zi') (Xi1, Yi1, Zi1)

25
DH Notation
• Relative Orientation and Position Analysis
• Orientation (cont'd)

26
DH Notation
• Relative Orientation and Position Analysis
• Position
• To find the position vector ai in Fi frame
(position vector connecting Oi to Oi1

27
DH Notation
• Relative Orientation and Position Analysis
• Position
• Observation

Changing
Constant
28
DH Notation
• Relative Orientation and Position Analysis
• Summary
• Orientation
• Position

29
Direct Kinematics
• 6-R Serial Manipulator
• Problem description
• Known ?1 ?n, find Q and p in the
base frame

30
Direct Kinematics
• 6-R Serial Manipulator
• 1. Orientation
• With DH Parameter defined, Q1, Q6 can
be calculated.

Similarity transformation to individual frame
Abbreviated notation Qi Qii
31
Direct Kinematics
• 6-R Serial Manipulator
• 2. Position
• 3. Homogeneous form (position
orientation)

32
Direct Kinematics
• Some useful properties of Qi