ME451
Kinematics and Dynamics
of Machine Systems
Review of Matrix Algebra – 2.2
September 13, 2011
© Dan Negrut, 2011
ME451, UW-Madison
Dan Negrut
University of Wisconsin-Madison
Before we get started…
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Due next week:
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Problems: 2.2.5, 2.2.8. 2.2.10 out of Haug’s book (http://sbel.wisc.edu/Courses/ME451/2010/bookHaugPointers.htm)
Due on Th:
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In class, if pen and paper version is submitted
At 23:59 PM if electronic form submitted
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Last time:
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Moving to full electronic submission starting after September
See Forum posting for some ideas on how to go about typing equations in your document in Windows
Covered Geometric Vectors & operations with them
Justified the need for Reference Frames (using a vector basis)
Introduced algebraic representation of a vector & related operations
Rotation Matrix (for switching from one RF to another RF)
Today:



Dealing with the kinematics of a body: rotation + translation
Quick review of matrix/vector algebra
Discuss concept of “generalized coordinates”
2
The Rotation Matrix A

Very important observation ! the matrix A is orthonormal:
3
Important Relation

Expressing a given vector in one reference frame
(local) in a different reference frame (global)
Also called a change of base.
4
Example 1
[Deals with the rotation of a body wrt a Global Reference Frame (GRF)]
B

Y

x’
y’
2222
5555O’
O
%
%
%
%
θ
X

Express the geometric vector
in the local reference frame
O’x’y’.
Express the same geometric
vector in the global reference
frame OXY
Do the same for the geometric
vector
L
E
5
Example 2
[Deals with the rotation of a body wrt a Global Reference Frame (GRF)]
G

θ
Y

L
2
22
2
O’
5
5
5
5
O
%
%
%
%
y’
X

Express the geometric vector
in the local reference frame O’x’y’.
Express the same geometric
vector in the global reference
frame OXY
Do the same for the geometric
vector
x’
P
6
The Kinematics of a Rigid Body:
Handling both Translation + Rotation

What we just discussed dealt with the case when you are interested in
finding the representing the location of a point P when you change the
reference frame, but yet the new and old reference frames share the same
origin

What if they don’t share the same origin (see
picture at right)? How would you represent the
position of the point P in this new reference
frame?
7
More on Body Kinematics

A lot of ME451 is based on the ability to look at the location of one
point P in two different reference frames: a local reference frame
(LRF) and a global reference frame (GRF)

Local reference frame is typically fixed (rigidly attached) to a body that is
moving in space

Global reference frame is the “world” reference frame: it’s not moving,
and serve as the universal reference frame
P
² In t he LRF, t he posit ion of point P is described by s0 (somet imes t he not at ion
used is ¹sP )
² In t he GRF, t he posit ion of point P is described by r P (see next slide)
8
ME451 Important Slide
Y
P
y’
s'
x’
P
f
rP
O’
r
O
X
9
Example

The location of point O’ in the OXY global RF is [x,y]T. The orientation of the
bar is described by the angle f1. Find the location of C and D expressed in the
global reference frame as functions of x, y, and f1.
C
2
y1′
1
2222
O′555
5
%
%
φ%
%
Y
1
O
X
2
x1′
D
10
Matrix Review
11
Recall Notation Conventions

A bold upper case letter denotes matrices


A bold lower case letter denotes a vector


Example: A, B, etc.
Example: v, s, etc.
A letter in italics format denotes a scalar quantity

Example:
a, b1
12
Matrix Review

What is a matrix? A tableau of numbers ordered by row and column.
éa
ê 11 a12
êa
a 22
21
ê
A= ê
¼
ê¼
êa
êë m 1 am 2

¼
¼
¼
¼
a1n ù
ú
a 2n ú
ú=
ú
¼ ú
am n ú
ú
û
éa a ¼
2
êë 1
éa T ù
ê 1ú
êa T ú
ê ú
ù
an ú= ê 2 ú
û êL ú
ê Tú
êa m ú
ë û
Matrix addition:
A  [aij ],
1  i  m,
1 j  n
B  [bij ],
1  i  m,
1 j  n
C  A  B  [cij ],

cij  aij  bij
Addition is commutative
A+ B= B+ A
13
Matrix Multiplication

Recall dimension constraints on matrices so that they can be multiplied:

# of columns of first matrix is equal to # of rows of second matrix
A = [aij ],
AÎ ¡
C = [cij ],
CÎ ¡
m *n
n *p
D = A·C = [dij ],
DÎ ¡
m *p
n
dij =
å
aikckj
k= 1

Matrix multiplication is not commutative

Distributivity of matrix multiplication with respect to matrix addition:
14
Matrix-Vector Multiplication
A column-wise perspective on matrix-vector multiplication (part of your HW)

éa
ê 11 a12
êa
a 22
21
ê
Av = ê
¼
ê¼
êa
êë m 1 am 2

Example:
1
2
Av  
 1

0
¼
¼
¼
¼
ù
a1n ùé
úêv1 ú
ú
a2n úê
úêv2 ú=
ú
¼ úê
úêL ú
amn úê
vn ú
úê
ûë ú
û
éa a ¼
2
êë 1
év ù
ê 1ú
êv ú
ù
an úêê 2 ú
=
ûêL ú
ú
êv ú
êë n ú
û
15
å
a i vi
i= 1
0  1   1 
 4
2
0
7
 3
1
1
8
3 1 1  2   2 
     ·(1)   ·(2)   ·(1)   ·(1)   
0
1
 1
 3
0 1 1  1  1
   
 
 
 
 
1 1 2   1   0 
1 
 1
 2
1
4
2
A row-wise perspective on matrix-vector multiplication:

n
éa T ù
éa T v ù
ê 1ú
ê 1 ú
êa T ú
êa T v ú
ê 2ú
ê
ú
Av = ê úv = ê 2 ú
êL ú
êL ú
ê Tú
ê T ú
êa m ú
êa m v ú
ë û
ë
û
Orthogonal & Orthonormal Matrices

Definition (Q, orthogonal matrix): a square matrix Q is orthogonal if the
product QTQ is a diagonal matrix

Matrix Q is called orthonormal if it’s orthogonal and also QTQ=In
 Note that in many cases people fail to make a distinction between an
orthogonal and orthonormal matrix. We’ll observe this distinction

Note that if Q is an orthonormal matrix, then Q-1=QT
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Example, orthonormal matrix:
16
Exercise
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Prove that the orientation A matrix is orthonormal
2
A = 4
cosÁ ¡ sin Á
sin Á
cosÁ
3
5
17
Remark:
On the Columns of an Orthonormal Matrix

Assume Q is an orthonormal matrix


In other words, the columns of an orthonormal matrix have unit norm
and are mutually perpendicular to each other
18
Matrix Review [Cntd.]

Scaling of a matrix by a real number: scale each entry of the matrix
 ·A   ·[aij ]  [ ·aij ]

Example:
1
2
(1.5)·
 1

0

0   1.5
6
3
0 
3 1 1  3
4.5 1.5 1.5 


0 1 1  1.5 0
1.5 1.5
 

1 1 2   0
1.5 1.5 3 
4
2
Transpose of a matrix A dimension m£n: a matrix B=AT of
dimension n£m whose (i,j) entry is the (j,i) entry of original matrix A
1
2

 1

0
T
4 2 0
1
4
3 1 1
 
2
0 1 1


1 1 2 
0
2 1 0 
3 0 1

1 1 1

1 1 2 
19
Linear Independence of Vectors

Definition: linear independence of a set of m vectors, v1,…, vm :
v 1 ; ::::; v m 2 Rn

The vectors are linearly independent if the following condition holds
1v1  ....   m vm  0n


1    m  0
If a set of vectors are not linearly independent, they are called dependent

Example:

Note that v1-2v2-v3=0
1
v1   1
 
 2 
0
v 2  1 
 
 4 
1
v 3   3
 
 6 
20
Matrix Rank

Row rank of a matrix


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Column rank of a matrix

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Largest number of rows of the matrix that are linearly independent
A matrix is said to have full row rank if the rank of the matrix is equal to
the number of rows of that matrix
Largest number of columns of the matrix that are linearly independent
NOTE: for each matrix, the row rank and column rank are the same


This number is simply called the rank of the matrix
It follows that
21
Matrix Rank, Example

What is the row rank of the matrix J?
 2 1 1 0 
J   4 2 2 1 


 0 4 0 1 

What is the rank of J?
22
Matrix Review [Cntd.]

Symmetric matrix: a square matrix A for which A=AT
Skew-symmetric matrix: a square matrix B for which B=-BT
Examples:

Singular matrix: square matrix whose determinant is zero

Inverse of a square matrix A: a matrix of the same dimension, called A-1,
that satisfies the following:


 2 1 1
A1 0 3


 1 3 4 
 0 1 2 
B   1 0 4


 2 4 0 
23
Singular vs. Nonsingular Matrices

Let A be a square matrix of dimension n. The following are equivalent:
24
Other Useful Formulas
[Pretty straightforward to prove true]

If A and B are invertible, their product is invertible and

Also,

For any two matrices A and B that can be multiplied

For any three matrices A, B, and C that can be multiplied
25
Absolute (Cartesian) Generalized Coordinates
vs.
Relative Generalized Coordinates
26