Aerospace and Electronic Systems Magazine July 2018 - 63
Mehrjardi et al.
trix, measured vector in the body frame maps to the observation
in the reference frame. Two main attitude determination methods
from at least two vector measurements are discussed.
Attitude determination algorithm solves the following equation
for A,
bi = Ari , i = 1, , N
(3)
where bi's are a set of observation unit vectors in the body frame,
ri's are the ones in the reference frame, and N is the number of vector sensors. Since measured vectors are inevitably contaminated
by some noises, therefore, an exact answer for the rotation matrix
A does not exist.
TRIAD ALGORITHM
The TRIaxial Attitude Determination (TRIAD) algorithm [23]-
[25] is the most common attitude determination method because
of its simplicity. This method delivers a deterministic and nonoptimal solution based on two nonparallel unit vector measurement
pairs. Suppose that there are two observed unit vectors, b1 and b2,
in the spacecraft body frame that can be observed by star tracker
or the Sun sensor. Each of these unit vectors contains two different scalar information of the attitude. Moreover, obtaining two
measured unit vectors r1 and r2 in some reference frame, such as
inertial frame, is necessary. The rotation matrix is the matrix that
rotates vectors from the reference frame to the spacecraft body
frame. The exact solution of rotation matrix satisfies the following
two equations [26]:
Ar1 = b1
Ar2 = b2
the attitude matrix A is
3
A w1; w2 ; w3
v1 , v2 , v3 T w1v1T w2v2T w3v3T wi viT
i 1
The rotation matrix A represents the transformation between
the reference frame and body frame and transfer vi to the wi by
wi = Avi , i = 1, 2,3
Q -METHOD
In the TRIAD method, the combination of sensor observations is
not optimal, and it is a major disadvantage of the TRIAD method
[27]. An optimal solution of attitude determination is to minimize
the cost function,
A = arg min
A
JULY 2018
1 N −2
σ i bi − Ari
2 i =1
2
(9)
where σi is the weight assigned to the ith pair based on the confidence in each sensor. The cost function is the weighted sum
squared of the difference between body observations and transformed vectors. The cost function based on parameterized attitude
matrix, A, in terms of quaternion vector, (q), is
(4)
Based on the TRIAD idea, when there is an orthogonal righthanded triad of vectors [v1, v2, v3] in the reference frame, and a corresponding triad [w1, w2, w3] in the spacecraft body frame:
b
w1 = 1
b1
w1 × b2
w2 =
w1 × b2
w3 = w1 × w2
(8)
The negative point of the TRIAD method is that at least two
measurement pairs are required. If there are many measurement
unit vectors, the TRIAD algorithm is able to be repeated for every
two pairs. Then, the final attitude matrix can integrate all determinations by some means.
A = arg min
r
v1 = 1
r1
v1 × r2
v2 =
v1 × r2
v3 = v1 × v2
(7)
(5)
A
1 N −2
σ i bi − A(q)ri
2 i =1
2
(10)
To minimize this cost function, deriving a unit length quaternion is desired. There is an explanation of the relation between the
Euler axis of rotation and quaternion in Wertz [6]. It is shown that
every rotation about three axes can be explained as a single rotation about one vector by an angle, ϕ. If the Euler axis is
eˆ e1 e2
e3 T
(11)
quaternion vector components are given as
, q2 e
, q3 e3
sin , q4 cos
q1 e
1 sin
2 sin
2
2
2
2
(6)
(12)
The required quaternion vector is the normalized eigenvector
of matrix, k, corresponding to the largest eigenvalue. Formation
of the matrix, k, starts by the construction of σ, B, S, and z as follows [28]:
IEEE A&E SYSTEMS MAGAZINE
63
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