ATTITUDE REPRESENTATION



Attitude cannot be represented by vector in 3-dimensional space, like
position or angular velocity, even though attitude is a “3-dimensional”
quantity.
Attitude is always specified as a rotation relative to a base, or reference
frame, just as vector position is specified as a displacement from a
reference point. However there is often confusion in the direction:

Rotation of the body frame to align with the reference frame

Rotation of the reference frame to align with the body frame
Rotations are described by various means

Direction Cosines Matrix (DCM)

Euler Angles

Euler Axis/Angle

Quaternion

Rodriquez parameters, Gibbs vector, etc.
P. Axelrad, D. Lawrence ASEN3200 Spring 2006
DIRECTION COSINES MATRIX

The DCM transforms a vector representation from one coordinate frame to
another, or rotates vectors from one attitude to another.
 r B

 r2 A
  R 2  r1  A
1
The DCM can be formed by dot products of unit vectors of two frames
T B
A

 T B  r  A or
A
 iB  iA

  jB  i A

 k B  iA
iB  j A
iB  k A 

jB  k A  or

k B  k A 
 i2  i1

1
jB  j A
 R 2  i2  j1

i2  k1
kB  jA
A
1 T
Note that if we set A=1 and B=2, T    R 
B
2
 
j2  i1
j2  j1
j2  k1
k2  i1 

k 2  j1 

k2  k1 
The nine elements are not independent because the DCM must be orthonormal
T B T A  T A T B
A
P. Axelrad, D. Lawrence ASEN3200 Spring 2006
B
B
A
=I
EULER ANGLES
Euler Angles are a particular sequence of three rotations about particular
reference frame axes. Both the sequence and the axes must be
specified to clearly define the attitude (rotation) of interest.

The same angle values used in a different sequence, or about different
axes, results in a different attitude

Example: Yaw-Pitch-Roll Euler angle sequence rotating the reference
frame (call it frame 1) into the body frame:
1) - Yaw the reference frame about its k-axis with angle y to
produce the 2-frame
2) - Pitch about the new j-axis with angle  to produce the 3-frame
3) - Roll about the new i-axis with angle  to produce the body
frame B
The resulting rotation matrix rotating 1-frame vectors v into their
corresponding body frame position is given by
 vB 1 =  R B  v1 1
1
where
P. Axelrad, D. Lawrence ASEN3200 Spring 2006
 R B   R( )B  R( )3  R(y )2
1
3
2
1
EULER ANGLE EXAMPLE
Yaw,Pitch,Roll (k,j,i) Sequence
i3
Reference Frame is Frame 1
pitch
i1
j2
j1
yaw
i2
j2,j3
i2
k1,k2
k2
i3,iB
Rotate about k1
(angle y
k3
Rotate about j2
(angle 
roll
Rotate about i3 j3
(angle 
Body frame is Frame B
jB
k3
kB
DCMs FOR GENERAL EULER ROTATIONS
1

1
 R    2  0

0
0
c
s
i
1

j
2
k
0 

 s 

c 
 c

1
 R    2   0

  s
c

1
 R    2   s

 0
P. Axelrad, D. Lawrence ASEN3200 Spring 2006
2
i

0
1
0
 s
c
0
s 

0

c 
1
j
0
 j
0

1 
k
i
1

2
k
Transformation Matrix for Euler Yaw,Pitch,Roll (k,j,i)
c cy


1
 R321 y , ,   B  c sy  s s cy

 s sy  c s cy
P. Axelrad, D. Lawrence ASEN3200 Spring 2006
c sy
c cy  s s sy
s cy  c s sy
s 

 s c 

c c 
EULER’S THEOREM (EULER AXIS/ANGLE REP.)


Any rigid body rotation can be expressed by a single rotation about a
fixed axis.
The rotation matrix [R] is given in terms of a unit vector along the “Euler
axis” e (a unit vector), and the angle, 
 R  n,  2  cos I + 1-cos  e1 e1  sin  [[e]1 ]
1
T
cos  
1
2
tr R  1
1
e
2
 R23  R32 

1 
 R31  R13 
e1 
2sin  

 R12  R21 

Shuster, M., "Survey of Attitude Representations," Journal of Astronautical Sciences, Vol. 41, No. 4, Oct.-Dec. 1993. pp. 439-517.
P. Axelrad, D. Lawrence ASEN3200 Spring 2006
NOTATION
r b   r B b   r1b1  r2b2  r3b3
T
Vector Dot Product
B
Vector Cross Product
 r  b   [[b ]B ] r B
B
Cross Product Matrix for vector
 0
[[b ]B ]   b3

 b2
c = cos()
s = sin()
P. Axelrad, D. Lawrence ASEN3200 Spring 2006
b3
0
b1
b2 
b1 

0 
where
 b B
 b1 
 b2 
 
b3 
QUATERNION REPRESENTATION OF ATTITUDE






Only one redundant element
requiring use of a constraint | q | = 1
Only ambiguity is a sign
Can be combined easily to produce
successive rotations
DCM computation given by multiply
& add of quaternion elements (no
trig functions)
Propagation requires integration of
only 4 kinematic equations
Widely used because of simplicity of
operations and small dimension,
together with lack of representation
singularity
 q1 
 
 q2 
q 
 q3 
 
 q4 
Shuster, M., "Survey of Attitude Representations," Journal of the Astronautical Sciences, Vol. 41, No. 4, Oct.-Dec. 1993. pp. 439-517.
P. Axelrad, D. Lawrence ASEN3200 Spring 2006
QUATERNION REPRESENTATION
Given Euler Axis e and angle 
 q1 
 
 q1 
 q2 
 


q   q2   sin e, q4  cos , q   
2
2
 q3 
 
 
 q3 
 q4 
q  q4  1 (q must be constrained to unit length)
2
2
P. Axelrad, D. Lawrence ASEN3200 Spring 2006
Quaternion versus Rotation Matrix (DCM_

R(q , q4 )  q4  q
2
2

I  2 q q T  2q4 [[q ]]
 q12  q2 2  q32  q4 2

R (q )   2  q1q2  q4 q3 

 2  q3 q1  q4 q2 

2  q1q2  q4 q3 
q1  q2  q3  q4
2
2
2
2  q3 q2  q4 q1 
2  q1q3  q4 q2 
2


2  q2 q3  q4 q1  

2
2
2
2
q1  q2  q3  q4 
1
1  trR ,
2
1
1
1
q1 
R

R
,
q

R

R
,
q

 23 32  2
 31 13  3
 R12  R21 
4q4
4q4
4q4
trR  4q4  1, q4  
2
P. Axelrad, D. Lawrence ASEN3200 Spring 2006
Quaternion Composition (Successive Rotations)
RCA  RCB RBA
 q3 C   q2 C   q1 B
A
A
B
 q3 C   q1 B  q2 C
A
B
 q4

 q3
 q   
 q2

  q1
P. Axelrad, D. Lawrence ASEN3200 Spring 2006
A
(Note the swapped order)
q3
q2
q4
q1
q1
q4
 q2
 q3
q1 

q2 

q3 

q4 
Kinematics
Relationship between angular velocity and attitude representations
d  R (t ) 
dt
=  R(t ) [[  t ]]
 q4 (t )

d  q (t )  1  q3 (t )
= 
dt
2  q2 (t )

  q1 (t )
P. Axelrad, D. Lawrence ASEN3200 Spring 2006
q3 (t )
q4 (t )
q1 (t )
 q2 (t )
q2 (t ) 
  1 (t ) 
q1 (t )  

 2 (t ) 
q4 (t )  

 3 (t ) 
q3 (t ) 
SMALL ANGLE APPROXIMATIONS

For a small angles ,
sin( ~  , cos( ~ 1

The rotation DCM for a sequence of three small Euler angles is:
 1
 R   I  [[ ]]   3
  2
P. Axelrad, D. Lawrence ASEN3200 Spring 2006
3
1
1
 2 
1 
1 
ATTITUDE DETERMINATION PROBLEM






Use standard attitude sensors such as a star tracker or sun sensor
Sensor axes are calibrated with respect to body-fixed reference frame
(B)
Direction to reference object (sun or star) is found in an inertial frame (I)
using star catalog, ephemeris prediction, etc.
Direction to reference object is also measured by the on-board sensors
and expressed in the (B) frame.
Now have one or more unit vectors to objects expressed in both (I) and
in (B). Note that a minimum of 2 “independent” objects is required to
determine 3-D attitude
Calculate the attitude DCM
P. Axelrad, D. Lawrence ASEN3200 Spring 2006
ATTITUDE DETERMINATION PROBLEM

Given measurements of two unit vectors (pointing to two objects) in a
body frame and a reference frame
v1 A , v1 B , v2 A , v2 B

How can the DCM representing attitude be determined? T must
simultaneously satisfy
v1 B  T B v1 A
A

and
v2 B  T B v2 A
A
Deterministic method - TRIAD

Use two of the measured vectors to define a set of three orthogonal
unit vectors in the two frames.

Create a matrix equation from the three vector equations and use
this to solve for the attitude DCM
P. Axelrad, D. Lawrence ASEN3200 Spring 2006
DETERMINISTIC ATTITUDE DETERMINATION
Given unit vectors
Construct
v1 A , v1 B , v2 A , v2 B
v1 A  v2 A
r1   v1  A , r2 
, r3  r1  r2
v1 A  v2 A
v1 B  v2 B
s1   v1 B , s2 
, s3  s1  s2
v1 B  v2 B
M R   r1 r2 r3  M s   s1 s2 s3 
T B  M s M RT
A
P. Axelrad, D. Lawrence ASEN3200 Spring 2006
Transformation DCM estimate
(note rotation DCM  R  A is the
B
transpose of this)
Attitude Representations and Attitude Determination
REFERENCES

Shuster, M., "Survey of Attitude Representations," Journal of the
Astronautical Sciences, Vol. 41, No. 4, Oct.-Dec. 1993. pp. 439-517.

Shuster, M. D. and Oh, S. D., "Three-Axis Attitude Determination from
Vector Observations," Journal of Guidance and Control, Vol. 4, No. 1,
Jan.-Feb. 1981, pp. 70-77.

Wertz, J. R., ed. Spacecraft Attitude Determination and Control, Kluwer
Academic Publishers, Dordrecht, Netherlands, 1978.
P. Axelrad, D. Lawrence ASEN3200 Spring 2006