文档介绍：Lecture 2
MECH572 Introduction To Robotics
Dept. Of Mechanical Engineering
Review
Overview of fields of Robotics
Concept of Vector Space and Linear Transformation
Ax = b linear system of equation
m×n n m Column Space(range), Null space
Properties: A(αx+y) = Ax + Ay
Useful Linear Transformation in 3-dimensional space
Projection
Reflection
Rotation
Important - Understand physical meaning
Review
Linear Trans.
Projection
(P)
Reflection
(R )
Rotation
(Q )
Definition
Properties
Det
(singular)
p
n
P'
n
p
P"
e
p
P'
Review
Linear Trans.
Projection
(P)
Reflection
(R )
Rotation
(Q )
Geometric interpretation
Matrix Representation
x
y
z
x
y
z
x
y
z
x
z
y
x
y
z
x
y
z
Review
Rotation Matrix
Alternative form
Canonical form – Euler Angles
A rotation sequence along different axes.
Roll: e is X Axis
Pitch: e is Y Axis
Yaw: e is Z Axis
Review
Example – Rotation about x axis
x = 1*x' + 0*y' + 0*z'
y = 0*x' + cos*y' - sin*z'
z = 0*x' + sin*y' + cos*z'
p = Q p'
Q maps p' into p
x
y
z
x'
y'
O
z'
Mathematical Background
Concept of Linear Invariants
Cartesian position of any 3x3 matrix A:
The vector of A has the following property:
The trace of A is defined as the eigenvalues of As
Symmetric
Skew-symmetric
Mathematical Background
Concept of Linear Invariants
For 3x3 matrix A defined in a certain coordinate frame:
Properties
vect(A) = 0 if A is symmetric
tr(A) = 0 if A is skew-symmetric
Vector
Trace
Mathematical Background
Linear Invariant of a rotation matrix
Recall:
Define
Group into 4-D vector
Linear Invariant
Points on the surface of 4-D sphere
Mathematical Background
Linear Invariant of a rotation matrix (cont'd)
Sign Problem
When = , Q is not uniquely defined