MECH572-lecture3.ppt

MECH572 Introduction To Robotics
Lecture 3
Dept. Of Mechanical Engineering
Review
Rigid-body Rotation - Representations
Representation
Matrix
Linear Invariant
Quadratic Invariant
(Euler Parameters)
Natural Invariant
Definition
Number of Elements
9
4
4
4
Constraints
||e1|| = 1, ||e2|| = 1,
||e3|| = 1
e1•e2= 0, e2•e3= 0,
e3•e1= 0
||e|| = 1
Independent Elements
9 - 6 = 3
4 – 1 = 3
4 – 1 = 3
4 – 1 = 3
Review
Alternative form to represent a rotation – Euler Angles
A sequence of rotation:
Q = Q()Q()Q()
, ,  rotation angles about certain axes.
Coordinate Transformation
General form
Homogeneous form
Origin offset
Review
Similarity Transformations
- Transformation of matrix entries (compare with vector entries which uses linear transformation)


The concept of invariance
After transformation between frames, certain quantities are unchanged or frame invariant (inner product, trace, moments, etc)
Overview of Rigid-Body Mechanics
Purpose – Lay down foundations of ostatics (kinematics + statics) and dynamics of rigid bodies using matrix method
Scope
Linear and angular displacement, velocity, and acceleration analysis
Static analysis
Mass & Inertial properties
Equation of motion for single rigid body
Useful tools/concepts to be introduced
Screw theory
Twist
Wrench
Q
O
P
P'
A
p
a
p'
a'
P – Arbitrary
A - Reference
Rigid-Body Mechanics
Description of a Rigid-Body Motion
Rigid-Body motion preserves distance
Define
Left multiply
Take transpose
= 0
A'
Displacement of any point projected onto the rotation axis are same
Rigid-Body Mechanics
General Rigid-Body Motion
e
Rigid-Body Mechanics
General Rigid-Body Motion
Geometric Interpretation
A
B
A'
B'
O
2-D case
3-D case
e

Pitch p0
Any rigid body motion in 2-D can be regarded as a pure rotation around one point O
Any rigid body motion in 3-D can be regarded as a Screw-like motion along an axis, e
Rigid-Body Mechanics
Screw Motion of a Rigid-Body
A ri