Notes_04_02
page 1 of 3
Two-Dimensional Coordinate Transformations
y2’
P
d
y1
c
x2’

O2
b
x1
a
ri P  ri   A i s i ' P
x 2  P x 2   __________ __________  x 2  ' P
     
 
y 2 
y 2   __________ __________  y 2 
__________  __________   __________



__________  __________   __________
__________  __________ 


__________  __________ 
__________  __________   __________



__________  __________   __________
__________  __________ 


__________  __________ 
C i
 S i 
C i 
A i   
 S i
Ai   f̂ i  ĝ i 
f̂ '  10
i
 
ĝ i '  
0

1
f̂ and ĝare unit vectors
f̂  A f̂ '
s i P  A i s i ' P
[A] matrices are orthonormal
[A] -1 = [A] T
● all columns are unit vectors
● all columns are mutually orthogonal
● all rows are unit vectors
● all rows are mutually orthogonal
● det( [A] ) = +1
i
i
i
ĝ i   A i ĝ i '
s i ' P  A i 1 s i P
Notes_04_02
page 2 of 3
Provide numerical values for the three coordinate transformations shown below.
y4’
x4’
P
Body 4
y7’
45 deg
x7’
y1
Body 7
15 deg
x1
Ground
rP
3
 
6
units = cm
s 4 ' P  
1.4142 cm
__________
1.4142 cm
__________ 
s 7 ' P  
~ -2.8 cm __ 
__________
~ +5 cm __ 
__________

r4 P  r4   A 4  s 4 ' P
r4   
3 cm
__________


4
cm
__________ 
A 4   

C(45°)
__________
S(45°)
__________
- S(45°) 
__________
C(45°) 
__________
4 = +45°
- S(15°) 
__________
C(15°) 
__________

7 = +15°
r7 P  r7  A 7  s 7 'P
7 cm
__________ 
r7    2 cm 
__________ 
C(15°)
 __________
A 7    S(15°)
 __________
 ij attitude angle for body j with respect to body i
A 
ij
ij   j  i
attitude matrix for body j with respect to body i
cos ij
A ij    sin 

A 47   
ij
 sin ij 
T
 A i  A j 

cos ij 
C(-30°)
__________
S(-30°)
 __________
- S(-30°)  47 = -30°
__________
C(-30°) 
__________
check using MATLAB
A 47   A 4 T A7 
Notes_04_02
% n0402.m - check Notes_04_02
% HJSIII, 14.02.03
clear
% constants
d2r = pi / 180;
% body
r4 = [
s4pP=[
phi4 =
A4 = [
4
3 4 ]';
1.4142 1.4142 ]';
45 * d2r;
cos(phi4) -sin(phi4) ;
sin(phi4)
cos(phi4) ];
s4P = r4 + A4 *s4pP
% body
r7 = [
s7pP=[
phi7 =
A7 = [
7
7 2 ]';
-2.8 5 ]';
15 * d2r;
cos(phi7) -sin(phi7) ;
sin(phi7)
cos(phi7) ];
s7P = r7 + A7 *s7pP
% relative attitude
A47 = A4' * A7
% bottom of n0402
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
>> n0402
s4P =
3.0000
6.0000
s7P =
3.0013
6.1049
A47 =
0.8660
-0.5000
0.5000
0.8660
page 3 of 3