Title: MECH572A Introduction To Robotics
1MECH572AIntroduction To Robotics
2Midterm 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
3Review
- 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
4Review
- 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
5Robotic 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)
6Robotic 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)
7Robotic 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)
8Robot Kinematics Overview
- Robot Manipulators
- Kinematic Chains Link Joint
- Rigid bodies
Kinematic Pairs - Basic Topologies of Kinematic Chain
Necklace
Open Chain
Tree
9Robot 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
10Denavit-Hartenberg Notation
- Purpose
- To uniquely define architecture of robot
manipulator (Kinematic chains) - Assumptions
- Links 0, 1, , n - n1 links
- Pairs 1, 2, , n - n pairs
- Frame Fi (Oi - Xi -Yi -Zi) is attached to
(i-1)st frame
(NOT ith frame)
11Denavite-Hartenberg Notation
- Definition of Axes
- Zi - Axes of the pair (Rotational/translational)
Zi
Zi
12Denavite-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)
13DH 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
14DH Notation
15DH Notation
Xi1
bi Variable ?i - Constant
16DH Notation
17DH Notation
18DH 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
19DH Notation
20DH Notation
21DH 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
22DH Notation
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
23DH Notation
- Relative Orientation and Position Analysis
- Orientation
? i about Zi
(a)
?i about Xi'
(b)
Rotation Decomposition (a) (b)
24DH Notation
- Relative Orientation and Position Analysis
- Orientation (cont'd)
- (Xi, Yi, Zi) (Xi', Yi', Zi')
- (Xi', Yi', Zi') (Xi1, Yi1, Zi1)
25DH Notation
- Relative Orientation and Position Analysis
- Orientation (cont'd)
-
26DH Notation
- Relative Orientation and Position Analysis
- Position
- To find the position vector ai in Fi frame
(position vector connecting Oi to Oi1
27DH Notation
- Relative Orientation and Position Analysis
- Position
- Observation
Changing
Constant
28DH Notation
- Relative Orientation and Position Analysis
- Summary
-
- Orientation
- Position
29Direct Kinematics
- 6-R Serial Manipulator
- Problem description
- Known ?1 ?n, find Q and p in the
base frame
30Direct 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
31Direct Kinematics
- 6-R Serial Manipulator
- 2. Position
- 3. Homogeneous form (position
orientation) -
32Direct Kinematics
- Some useful properties of Qi