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
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010
001
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
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c o ss i n0
s i nc o s0
001
z
y
x
z
y
x
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x
y
z
x'
y'
O
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z'
Mathematical Background
? Concept of Linear Invariants
Cartesian decomposition 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
Mathematical Background
? Linear Invariant of a Rotation Matrix - Example
e
x
y
z
x
y
z
Q
Mathematical Background
? Linear Invariant of a rotation matrix –Example
Solution
Use
Mathematical Background
? Linear Invariant of a rotation matrix –Example (cont'd)
2.3.7 Solution cont'd
Mathematical background
? Euler-Rodrigues Parameters – Quadratic Invariant
Use half angle,Problem at ? = ? solved
Pay attention to the definition of
Mathematical Background
? Coordinate Transformation – Common Origin
Assume,
Mathematical Background
? Coordinate Transformation – Example
Solution
X = 0*x + 0*y – 1* z
y = -1*x + 0*y + 0*z
Z = 0*x + 1*y + 0*z
Fig,2.3
Mathematical Background
? Coordinate Transformation – with origin shift
From geometry,
Mathematical Background
? Homogeneous Coordinates
- None homogeneous when there is a origin shift,
- Making the expression homogeneous by introducing 4-dimensional
homogeneous coordinates
Define
Then
Where
Concatenation,
Mathematical Background
? Coordinate Transformation - Example
,
XA
YA
ZA
OA
b X
y
Z
OB
u
v w
B
A
Mathematical Background
? Coordinate Transformation – Example (cont'd)
,
Mathematical Background
? Similarity Transformation
Assume two basis in the same vector space
Vector v can be expressed in two ways,
Transformation between two basis,
Matrix representation
which leads to
Mathematical Background
? Similarity Transformation (cont'd)
Let L be a linear transformation defined in the same vector space
Compare with
or (similarity Transformation)
Mathematical Background
? Similarity Transformation (cont'd)
Properties,
Under similarity transformation
a) Eigenvalues do not change (Characteristic polynomial identical)
b) Eigenvectors follow
c) For any integer k
d) Trace unchanged
Mathematical Background
? Similarity Transformation - Example
Mathematical Background
? Similarity Transformation – Example (cont'd)
Solution,
If we use
Then
which is a similarity transformation
Does not obey similarity
transformation rule
Mathematical Background
? Invariance Concepts
Definition,Scalar function
Vector function
Matrix function
p – position vector
If Invariant then in two different frames A and B
Scalar quantities associated with matrix do not change
(e.g.,trace,eigenvalues)
Mathematical Background
? Invariance Concepts (cont'd)
Definition,Matrix Moments
Mathematical Background
? Invariance Concepts (cont'd)
Properties,
Cayley-Hamilton Theorem
Take trace
Assignment #1
? 2.6,2.12,2.13,2.18,2.22,2.27
? Due in two weeks