Download 1 of 2

Agenda
 1. Tools
 2. Matrices
 3. Least squares
 4. Propagation of variances
 5. Geometry
5. Math
1
1. Tools
Excel
Matlab
Mathcad
Labview
5. Math
1. Tools
2
Excel
Spreadsheet
Readily available
Solver functions
5. Math
1. Tools
3
Matlab
Matrix based
Powerful analytical tool
Handles transforms well
Easy to program
5. Math
1. Tools
4
Mathcad
Mathematical tool
Evolving into handling transfer functions
Has special programming language
Documentation closer to real math
5. Math
1. Tools
5
Labview
Powerful analysis tool
Uses graphical language to translate
concepts into C-code and then execute
5. Math
1. Tools
6
2. Matrices (1 of 2)
Addition
Subtraction
Multiplication
Vector, dot product, & outer product
Transpose
Determinant of a 2x2 matrix
Cofactor and adjoint matrices
Determinant
Inverse matrix
5. Math
2. Matrices
7
Matrices (2 of 2)
Orthogonal matrix
Hermetian matrix
Unitary matrix
5. Math
2. Matrices
8
Addition (1 of 2)
C=A+B
1 -1 0
A= -2 1 -3
2 0 2
1
B= 0
-1
-1 -1
4 2
0 1
2
C= -2
1
-2 -1
5 -1
0 3
cIJ = aIJ + bIJ
5. Math
2. Matrices
9
Addition (2 of 2)
1
-2
2
A
-1
1
0
+
0
-3
2
1
0
-1
B
-1
4
0
-1
2
1
=
2
-2
1
C
-2
5
0
-1
-1
3
1. Highlight area for answer
2. Type "="
3. Highlight area of first matrix
4. Type "+"
5. Highlight area for second matrix
6. Type CTL+SHIFT+ENTER
Matrix addition using Excel
5. Math
2. Matrices
10
Subtraction (1 of 2)
C=A-B
1 -1 0
A= -2 1 -3
2 0 2
1
B= 0
-1
-1 -1
4 2
0 1
0 0 1
C= -2 -3 -5
3 0 1
cIJ = aIJ - bIJ
5. Math
2. Matrices
11
Subtraction (2 of 2)
1
-2
2
A
-1
1
0
0
-3
2
1
0
-1
B
-1
4
0
-1
2
1
=
0
-2
3
C
0
-3
0
1
-5
1
1. Highlight area for answer
2. Type "="
3. Highlight area of first matrix
4. Type "-"
5. Highlight area for second matrix
6. Type CTL+SHIFT+ENTER
Matrix subtraction using Excel
5. Math
2. Matrices
12
Multiplication (1 of 2)
C=A*B
1 -1 0
A= -2 1 -3
2 0 2
1
B= 0
-1
-1 -1
4 2
0 1
C=
1
1
0
-5 -3
6 1
-2 0
cIJ = aI1 * b1J + aI2 * b2J + aI3 * b3J
5. Math
2. Matrices
13
Multiplication (2 of 2)
1
-2
2
A
-1
1
0
*
0
-3
2
1
0
-1
B
-1
4
0
=
-1
2
1
1
1
0
C
-5
6
-2
-3
1
0
1. Highlight area for answer
2. Type "= MMULT(", or use INSERT FUNCTION
3. Highlight area of first matrix
4. Type ","
5. Highlight area for second matrix
6. Type CTL+SHIFT+ENTER
Matrix multiplication using Excel
5. Math
2. Matrices
14
Transpose (1 of 3)
B=AT
1 -1 0
A= -2 1 -3
2 0 2
1
B= -1
0
-2
1
-3
2
0
2
bIJ = aJI
5. Math
2. Matrices
15
Transpose (2 of 3)
1
-2
2
A
-1
1
0
0
-3
2
A-transpose
1
-2
2
-1
1
0
0
-3
2
1. Highlight area for answer
2. Type "= TRANSPOSE(", or use INSERT FUNCTION
3. Highlight area of matrix
4. Type CTL+SHIFT+ENTER
Matrix transpose using Excel
5. Math
2. Matrices
16
Transpose (3 of 3)
(AB)T = BT AT
1 -1 0
A= -2 1 -3
2 0 2
1 -2
AT = -1 1
0 -3
5. Math
2
0
2
1
B= 0
-1
1
BT = -1
-1
-1 -1
4 2
0 1
0
4
2
2. Matrices
-1
0
1
1
(AB)T = -5
-3
BTAT =
1
-5
-3
1 0
6 -2
1 0
1 0
6 -2
1 0
17
Vector, dot & outer products (1 of 2)
A vector v is an N x 1 matrix
Dot product = inner product = vT x v = a
scalar
Outer product = v x vT = N x N matrix
5. Math
2. Matrices
18
Vector, dot & outer products (2 of 2)
v
1
2
3
1
v'
2
3
inner
v'*v
14
1
2
3
outer
v*v'
2
3
4
6
6
9
Matrix inner and outer products using Excel
5. Math
2. Matrices
19
Determinant of a 2x2 matrix
B
=
1 -1
-2 1
= -1
2x2 determinant = b11 * b22 - bI2 * b21
5. Math
2. Matrices
20
Cofactor and adjoint matrices
1 -1 0
A= -2 1 -3
2 0 2
1 -3
-2 -3
0 2
2 2
B = cofactor = - -1
0
0
2
1
2
0
2
-1 0 - 1
0 -3
-2
C=BT = adjoint=
5. Math
0
-3
-2
2
-
1
0
1 -1
2 0
2 -2 -2
= 2 2 -2
3 3 -1
1 -1
-2 1
2 2 3
-2 2 3
-2 -2 -1
2. Matrices
21
Determinant
determinant of A =
1 -1
0
1 -1 0
-2 1 -3
2 0 2
2
-2
-2
=4
=4
The determinant of A = dot product of any row in A times
the corresponding column in the adjoint matrix =
dot product of any row (or column) in A times
the corresponding row (or column) in the cofactor matrix
5. Math
2. Matrices
22
Inverse matrix (1 of 3)
B = A-1 =adjoint(A)/determinant(A) =
1 -1 0
-2 1 -3
2 0 2
0.5 0.5 0.75
-0.5 0.5 0.75
-0.5 -0.5 -0.25
0.5 0.5 0.75
-0.5 0.5 0.75
-0.5 -0.5 -0.25
1 0 0
= 0 1 0
0 0 1
Inverse
5. Math
2. Matrices
23
Inverse matrix (2 of 3)
1
-2
2
A
-1
1
0
0
-3
2
inv(A)
0.5 0.5 0.75
-0.5 0.5 0.75
-0.5 -0.5 -0.25
1. Highlight area for answer
2. Type "= MINVERSE(", or use INSERT FUNCTION
3. Highlight area of matrix
4. Type CTL+SHIFT+ENTER
Matrix inverse using Excel
5. Math
2. Matrices
24
Inverse matrix (3 of 3)
(AB)-1 = B-1 A-1
1 -1 0
A= -2 1 -3
2 0 2
1
B= 0
-1
0.5 0.5 0.75
A-1 = -0.5 0.5 0.75
-0.5 -0.5 -0.25
2
B-1 = -1
2
0.5 1
0 -1
0.5 2
-1 -1
0.25 0.75 1.625
4 2 (AB)-1 = 0
0 -0.5
0 1
-0.25 0.35 1.375
0.25 0.75 1.625
0 -0.5
B-1A-1 = 0
-0.25 0.35 1.375
Inverse of a product
5. Math
2. Matrices
25
Orthogonal matrix
An orthogonal matrix is a matrix whose
inverse is equal to its transpose.
1
0
0
5. Math
0
0
cos  sin 
-sin  cos 
1
0
0
0
0
1 0 0
cos  -sin  = 0 1 0
sin  cos 
0 0 1
2. Matrices
26
Hermetian matrix (1 of 3)
A Hermetian matrix is a matrix that is
equal to its own Hermetian transpose
• A = AH
The Hermetian transpose of A is the
complex conjugate transpose of A
• AH = AT
Hermetian matrix
5. Math
2. Matrices
27
Hermetian matrix (2 of 3)
A=
1
1-I
1+I 3
2
-i
2
i
0
AT =
1
1-I
2
1+I
3
+i
2
-i
0
1
1-I
1+I 3
2
-i
2
i
0
AT =
=A
Example
5. Math
2. Matrices
28
Hermetian matrix (3 of 3)
1
1+i
2
H
1-i
3
i
2
-i
0
H'
1
1+i
1-i
3
2
-i
2
i
0
conj(H')
1
1-i
2
1+i
3
-i
2
i
0
1. Use COMPLEX to enter complex numbers
2. Use IMCONJUGATE to convert cell-by-cell
Note: Cell operations; not matrix
Hermetian matrix using Excel
5. Math
2. Matrices
29
Unitary matrix
A matrix is unitary if its inverse equals its
Hermetian transpose
• U-1 = UH
DFT and inverse DFT are unitary matrices
5. Math
2. Matrices
30
3. Least squares
Example 1
Example 2
5. Math
3. Least squares
31
Example 1 (1 of 9)
x + 2y + 3z = 14
-2x +
+ z= 1
2x + y
= 4
1 2 3
A = -2 0 1
2 1 0
-1 3 2
A-1 = -1/3 2 -6 -7
-2 3 4
x
y
z
= A-1 b =
14
b= 1
4
1
2
3
Solve 3 equations and 3 unknowns
5. Math
3. Least squares
32
Example 1 (2 of 9)
x + 2y + 3z = 14
-2x +
+ z= 1
2x + y
= 4
3x + y - z = 2
x
y
z
=
x + 2y + 3z = 13
-2x +
+ z= 1
2x + y
= 4
3x + y - z = 3
x
y
z
= ?
1
2
3
What happens if we have 4 equations and 3 unknowns
5. Math
3. Least squares
33
Example 1 (3 of 9)
e1
e2
e3
e4
= x + 2y + 3z - 13
= -2x +
+ z- 1
= 2x + y
- 4
= 3x + y - z - 3
Minimize J = (e12 + e22 + e32 + e42)
Minimize the sum of squares
5. Math
3. Least squares
34
Example 1 (4 of 9)
x
y
x
0.46 3.37 1.91
coefficients
1.0 2.0 3.0
-2.0 0.0 1.0
2.0 1.0 0.0
3.0 1.0 -1.0
sum of squares
b
e
e^2
13.0
1.0
4.0
3.0
-0.1
0.0
0.3
-0.2
0.0
0.0
0.1
0.0
0.11
1. Set up matrix as shown
2. Select Solver
3. Select the cell containing the sum of squares
4. Select "minimize"
5. Set "by changing cells" to the unknowns -- x, y, z
6. Select solve
Solve using Solver in Excel
5. Math
3. Least squares
35
Example 1 (5 of 9)
e1
e2
e3
e4
A=
ATA s = AT b
= x + 2y + 3z - 13
= -2x +
+ z- 1
= 2x + y
- 4
= 3x + y - z - 3
1
-2
2
3
2
0
1
1
3
1
0
1
13
b=
1
4
3
s = [ATA]-1 AT b =
x
y
z
0.46
= 3.37
1.91
Solve using matrices
5. Math
3. Least squares
36
Example 1 (6 of 9)
A=
a1x
a2x
a3x
a4x
AT
a1y
a2y
a3y
a4y
a1z
a2z
a3z
a4z
b = b1
b2
b3
b4
a1x a2x a3x a4x
A= a a a a
1y 2y 3y 4y
a1z a2z a3z a4z
=
a1x
a2x
a3x
a4x
AT = a1x a2x a3x a4x
a1y a2y a3y a4y
a1z a2z a3z a4z
a1y
a2y
a3y
a4y
a1z
a2z
a3z
a4z
 akx akx  aky akx  akz akx
 akx aky  aky aky  akz aky
 akx akz  aky akz  akz akz
Express matrix solution in more general terms
5. Math
3. Least squares
37
Example 1 (7 of 9)
AT b =
 akxbk
 akxbk
 akzbk
Express matrix solution in more general terms (cont)
5. Math
3. Least squares
38
Example 1 (8 of 9)
J = [a1xx + a1yy + a1zz - b1]2 +
[a2xx + a2yy + a2zz - b2]2 +
[a3xx + a3yy + a3zz - b3]2 +
[a4xx + a4yy + a4zz - b4]2
J/ x = 2[a1xa1xx + a1ya1xy + a1za1xz - a1xb1] +
[a2xa2xx + a2ya2xy + a2za2xz - a2xb2] +
[a3xa3xx + a3ya3xy + a3za3xz - a3xb3] +
[a4xa4xx + a4ya4xy + a4za4xz - a4xb4]
2[ akx akx x  aky akx y  akz akxz -  akxbk ]
=0
Minimize by calculus
5. Math
3. Least squares
39
Example 1 (9 of 9)
 akx akx x  aky akx y  akz akxz -  akxbk = 0
 akx aky x  aky aky y  akz akyz -  akybk = 0
 akx akz x  aky akz y  akz akzz -  akzbz = 0
 akx akx  aky akx  akz akx
 akx aky  aky aky  akz aky
 akx akz  aky akz  akz akz
x
y
z
-
 akxbk
 akybk
 akzbk
=0
Minimize by calculus (continued)
5. Math
3. Least squares
40
Example 2 (1 of 3)
x=
1.1000
1.9000
2.9000
4.0000
5.0000
6.0000
y=
2.2000
3.0000
4.1000
5.0000
6.1000
6.9000
Fit a curve to the following data
5. Math
3. Least squares
41
Example 2 (2 of 3)
Fit z = a + b xi + c xi2
A = [[1;1;1;1;1;1], x, x.*x] =
b=y
a
b
c
=
(ATA)-1
AT
b =
1.0000 1.1000 1.2100
1.0000 1.9000 3.6100
1.0000 2.9000 8.4100
1.0000 4.0000 16.0000
1.0000 5.0000 25.0000
1.0000 6.0000 36.0000
1.0126
1.0949
-0.0184
Fit curve z to data
5. Math
3. Least squares
42
Example 2 (3 of 3)
error = a + b x + c x2 - y =
7
6.5
6
5.5
5
-0.0052
0.0266
-0.0668
0.0980
-0.0726
0.0200
4.5
4
3.5
3
2.5
2
1
2
3
4
5
6
Error in curve fit
5. Math
3. Least squares
43
4. Propagation of variance
Combining variance
Multiple dimensions
Example -- propagation of position
Example -- angular rotation
5. Math
4. Propagation of variables
44
Combining variances
Variances from multiple error sources can
be combined by adding variances
 Example
xorig = standard deviation in original position = 1 m
vorig = standard deviation in original velocity = 0.5 m/s
T = time between samples = 2 sec
xcurrent = error in current position
= square root of [(xorig)2 + (vorig * T)2] = sqrt(2)
5. Math
4. Propagation of variables
45
Multiple dimensions
When multiple dimensions are included,
covariance matrices can be added
P1 = covariance of error source 1
P2 = covariance of error source 2
P = resulting covariance = P1 + P2
When an error source goes through a
linear transformation, resulting covariance
is expressed as follows
T = linear transformation
TT = transform of linear transformation
Porig = covariance of original error source
P = T * P * TT
5. Math
4. Propagation of variables
46
Example -- propagation of position
xorig = standard deviation in original position = 2 m
vorig = standard deviation in original velocity = 0.5
m/s
T = time between samples = 4 sec
xcurrent = error in current position
xcurrent = xorig + T * vorig
vcurrent = vorig
T= 1 4
0 1
Porig =
Pcurrent = T * P orig * TT =
5. Math
1
0
4
1
22 0
0 0.52
4
0
0
0.25
4. Propagation of variables
1
4
0
1
= 16 4
4 0.25
47
Example -- angular rotation
Xoriginal = original coordinates
Xcurrent = current coordinates
T = transformation corresponding to angular rotation
y
y’
T = cos -sin 
where  = atan(0.75)
sin  cos 
Porig =
5. Math
x
1.64 -0.48
-0.48 1.36
Pcurrent = T * P orig * TT =

x’
0.8 -0.6
0.6 0.8
1.64 -0.48
-0.48 1.36
5. Statistics
0.8 0.6
-0.6 0.8
= 2
0
0
1
48
5. Geometry
Unit vectors
Angle between two lines
Perpendicular to a plane
Pointing
5. Math
5. Geometry
49
Unit vectors
A unit vector is a vector of length 1.
Unit vectors are frequently used to denote
vectors that have the same direction, such
as those parallel to a chosen axis of a
coordinate system
5. Math
5. Geometry
50
Angle between two lines (1 of 10)
The dot product is the result of multiplying
the length of a vector A times the length of the
component of vector B that is parallel to A
A • B = |A| |B| cos , where  is the angle
between the vectors
Dot product
5. Math
5. Geometry
51
Angle between two lines (2 of 10)
To find the angle between two lines,
• Establish a vector A and a vector B
along each line
• Solve for  = arccos[A • B /( |A| |B| )]
• 0  
Solving for  using dot product
5. Math
5. Geometry
52
Angle between two lines (3 of 10)
A = [1 2], B = [2 1]
|A| = SQRT(12 + 22) = SQRT(5)
|B| = SQRT(22 + 12) = SQRT(5)
A • B = [1 2] • [2 1]T = [2 • 1 + 2 • 1] = 4
4 = SQRT(5) • SQRT(5) cos 
cos  = 4/5
y
A

B
x
Example using dot product
5. Math
5. Geometry
53
Angle between two lines (4 of 10)
A
1
2
B
2
1
A'
1
2
B'
2
1
A'*A B'*B A'*B
5
5
4
2.24 2.24
1. Use dot product to compute square of hypotenuse
angle(radians)
0.64
angle(degrees) = angle (radians)*180/pi
36.9
1. Use ACOS to compute angle in radians
2. Use 180/pi to convert angle to degrees
3. Use PI function to compute pi
Note: PI must be followed by "(" if typed
Using Excel to compute values
5. Math
5. Geometry
54
Angle between two lines (5 of 10)
The cross product is the result of
multiplying the length of a vector A times
the length of the component of vector B
that is perpendicular to A
A x B = |A| |B|sin , where  is the angle
between the vectors
The vector A x B is perpendicular to the
plane containing A and B
Cross product
5. Math
5. Geometry
55
Angle between two lines (6 of 10)
To find the angle between two lines,
• Establish a vector A and a vector B
along each line
• Solve for  = arcsin[A x B /( |A| |B| )]
• - /2    /2
Solving for  using cross product
5. Math
5. Geometry
56
Angle between two lines (7 of 10)
A=
=
i
j
k
Ax Ay Az
Bx By Bz
i
1
2
j
2
1
k
0
0
= -3k
Example using cross product
5. Math
5. Geometry
57
Angle between two lines (8 of 10)
A = [1 2], B = [2 1]
|A| = SQRT(12 + 22) = SQRT(5)
|B| = SQRT(22 + 12) = SQRT(5)
A x B = -3 k
-3 = SQRT(5) • SQRT(5) sin 
sin  = -3/5
Example using cross product (continued)
5. Math
5. Geometry
58
Angle between two lines (9 of 10)
 = atan2(sin  , cos  )
Combining dot product and cross product
5. Math
5. Geometry
59
Angle between two lines (10 of 10)
x
-0.6
y
-0.8
ATAN
53.13
ATAN2
-127
1. Use ATAN2 for four quadrant arctangent
Note: First argument is X and not Y as in FORTRAN
Using Excel to compute arctangents
5. Math
5. Geometry
60
Perpendicular to a plane
The cross product defines the direction
perpendicular to the plane defined by the
two vectors A and B
5. Math
5. Geometry
61
Pointing (1 of 14)
y0
B (2,3,2)
A (3,1,1)
camera
x0
 Change pointing of camera so that points A and
B are on the same level
Point camera as directed
5. Math
5. Geometry
62
Pointing (2 of 14)
y1
y0
B (2,3,2)
x1
A (3,1,1)

x0
z0 and z1 are positive out of page
Pan camera to point at A in the x0-y0 plane
5. Math
5. Geometry
63
Pointing (3 of 14)
T01
 = atan2(3,1) = 18.4o
cos  sin  0
-sin  cos  0
0
0
1
3
1
1
cos  sin  0
-sin  cos  0
0
0
1
2
3
2
=
=
3.16
0.00
1.00
2.85
2.22
2.00
Determine T01 as follows
5. Math
5. Geometry
64
Pointing (4 of 14)
y1
z1 is positive out of page
B (2.85,2.22,2)
camera
x1
A (3.16,0,1)
Redraw problem in x1-y1
5. Math
5. Geometry
65
Pointing (5 of 14)
z1
y1 is positive into page
B (2.85,2.22,2)
A (3.16,0,1)
camera
x1
View x1-z1 plane
5. Math
5. Geometry
66
Pointing (6 of 14)
z1
z2
y1 and y2 are positive into page
B (2.85,2.22,2)
x2
A (3.16,0,1)

x1
Elevate camera to point at A in x1-z1 plane
5. Math
5. Geometry
67
Pointing (7 of 14)
T12
 = atan2(1,3.16) = 17.5o
cos  0
0
1
-sin  0
sin 
0
cos 
3.16
0.00 =
1.00
3.16
0.00
0.00
cos  0
0
1
-sin  0
sin 
0
cos 
2.85
2.22 =
2.00
3.32
2.21
1.05
Determine T12 as follows
5. Math
5. Geometry
68
Pointing (8 of 14)
x2 is positive into page
z2
B (3.32,2.21,1.05)
y2
A (3.16,0,0)
View y2-z2 plane
5. Math
5. Geometry
69
Pointing (9 of 14)
z2
z3
y3
B (3.32,2.21,1.05)

y2
A (3.16,0,0)
x2 and x3 are positive into page
Roll camera so that A and B are on horizontal line
5. Math
5. Geometry
70
Pointing (10 of 14)
 = atan2(1.05.2.21) = 25.4o
T23
1
0
0
0 cos  sin 
0 -sin  cos 
3.32
2.21 =
1.05
3.32
2.45
0.00
Determine T23 as follows
5. Math
5. Geometry
71
Pointing (11 of 14)
x3 is positive into page
y3
z3
A (3.16,0,0)
B (3.32,2.45,0)
View y3-z3 plane
5. Math
5. Geometry
72
Pointing (12 of 14)
T01T T12T T23T
0
0
1
=
-0.12
-0.49
0.86
Express unit vector perpendicular to AB in x0-y0-z0 plane
5. Math
5. Geometry
73
Pointing (13 of 14)
A=
=
i
j
k
Ax Ay Az
Bx By Bz
i
3
2
j
1
3
k
1
2
= (- i - 4j +7k)/sqrt(66)
=
-0.12
-0.49
0.86
Compare perpendicular unit vector to cross product
5. Math
5. Geometry
74
Pointing (14 of 14)
T01, T12, T23, and any of their products
are examples of direction cosine matrices
The element in aij is the cosine between
axis i and axis j
Define direction cosine matrix
5. Math
5. Geometry
75