MECH572A
Introduction To Robotics
Lecture 5
Dept,Of Mechanical Engineering
Midterm Exam
? Date & Time,19:00 - 21:00,Oct 25,2004
? Open Book
? Chapters 2 & 3 of the text book
Note,Regular lecture will take place 18:00 –18:45 on Oct 25
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
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
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)
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)
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)
Rotating pair –
Revolute (R)
Sliding pair –
Prismatic (P)
Robot Kinematics Overview
? Robot Manipulators
Kinematic Chains, Link + Joint
Rigid bodies Kinematic Pairs
? Basic Topologies of Kinematic Chain
Open Chain Tree Necklace
Robot Kinematics Overview
? Basic Problems in Robotic Kinematics
Direct Kinematics
Inverse Kinematics
.,,
X
Y
Z
O
Base
End Effector
?1
?2
?i
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θ
px,,py,pz
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Joint Variables
Cartesian
Variables
θθfx ?? ???
Linear relationship between
Cartesian rate of EE and joint rates
? x
Direct
Inverse
)(θfx? (Joint)
(Cartesian)
Denavit-Hartenberg Notation
? Purpose
– To uniquely define architecture of robot manipulator (Kinematic
chains)
? Assumptions
– Links, 0,1,…,n - n+1 links
– Pairs,1,2,…,n - n pairs
– Frame Fi (Oi - Xi -Yi -Zi) is attached to (i-1)st frame
(NOT ith frame)
Denavite-Hartenberg Notation
? Definition of Axes
– Zi - Axes of the pair (Rotational/translational)
Zi Zi
Denavite-Hartenberg Notation
? Definition of Axes (cont'd)
– Xi - Common perpendicular to Zi+1 and Zi directed from
Zi+1 to Zi (Follow right hand rule if intersect)
– Yi = Zi ? Xi
(d)
Zi-1 Z
i X
i undefined
DH Notation
? Joint Parameters & Joint Variables
– ai - Distance between Zi and Zi+1
– bi - Z-coordinate of Zi and Xi+1 intersection point (absolute
value = distance between Xi and Xi+1 )
– ?i - Angle between Zi and Zi+1 along +Xi+1 (R.H.R)
– ?i - Angle between Xi and Xi+1 along +Zi (R.H.R)
– Joint Variables
?i - R joint
bi - P joint
DH Notation
? Summary
Oi-1
Oi
Oi+1
Zi-1
Zi
Zi+1
i-1
i
i+1
Xi-1
Xi X
i+1
Revolute joints
bi-1
bi
?i
ai-1
ai
?i ?i-1
?i-1
DH Notation
? Summary – Prismatic joint
Xi+1
i - 1
i
Xi
Zi
bi
?i
bi – Variable
?i - Constant
DH Notation
? Example - PUMA
DH Notation
? Example - PUMA
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
DH Notation
? Example - Stanford Arm
DH Notation
? Example - Stanford Arm
X1 Y1
Z1
X2
Z2
X3
Z3
X4
X5
X6
Z4
Z5
Z6
X7
Z7
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
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
DH Notation
? Relative Orientation and Position Analysis
– Orientation
Xi'
Yi'
Zi'
Xi+1
Yi+1
Zi+1
Xi
Xi'
Yi'
Zi Zi'
Yi
? i about Zi
?i about Xi'
Rotation Decomposition (a) & (b)
(a)
(b)
DH Notation
? Relative Orientation and Position Analysis
– Orientation (cont'd)
(a) (Xi,Yi,Zi) (Xi',Yi',Zi')
(b) (Xi',Yi',Zi') (Xi+1,Yi+1,Zi+1)
DH Notation
? Relative Orientation and Position Analysis
– Orientation (cont'd)
DH Notation
? Relative Orientation and Position Analysis
– Position
To find the position vector ai in Fi frame (position vector connecting
Oi to Oi+1
DH Notation
? Relative Orientation and Position Analysis
– Position
– Observation,Changing
Constant
DH Notation
? Relative Orientation and Position Analysis
– Summary
Orientation
Position
Direct Kinematics
? 6-R Serial Manipulator
Problem description,
Known ?1 … ?n,find Q and p in the base frame
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 = [Qi]i
Direct Kinematics
? 6-R Serial Manipulator
2,Position
3,Homogeneous form (position + orientation)
Direct Kinematics
? Some useful properties of Qi