Rigid Body Attitude Estimation- An Overview and Comparative Stud.pdf

# By normalizing these weights to have n i 1 a i 1 it

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is the number of measurements. By normalizing these weights to have N i = 1 a i = 1, it is straightforward to show that L ( R ) = 1 - N X i = 1 a i ˆ W T i R ˆ V i = 1 - tr( RB T ) , (3.2) where tr denotes the trace operator and matrix B is defined as B = N X i = 1 a i ˆ W i ˆ V T i . (3.3) Equation (3.2) reduces the problem to finding the appropriate matrix R that maximizes the term tr( RB T ). It should be noted that Wahba’s problem addresses the attitude determi- nation in a closed-form reconstruction manner. For generalizations of this problem readers are referred to [Shuster, 2006] and [Psiaki, 2010]. Earlier solutions to the Wahba’s least squares problem included a method using po- lar decomposition of the matrix B proposed in [Farrell and Stuelpnagel, 1966], and other algorithms in [Wessner, 1966], [Velman, 1966], and [Brock, 1966]. Introduction of the Q- method [Keat, 1977], along with these algorithms divided the e ff orts of finding the optimal matrix R opt into two classes of solutions where the first class directly computes matrix R opt and the second tries to find the optimal quaternion associated with the orientation matrix. Structural di ff erences in numerous proposed algorithms belonging to each class result in di ff erent computational costs and execution times.

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C hapter 3. S tatic A ttitude D etermination 20 3.2 TRIAD The earliest attitude reconstruction method, known as TRIAD [Lerner, 1978], was designed to work with only two non-collinear unit reference vectors ˆ V 1 , ˆ V 2 in inertial frame and their corresponding unit observation vectors ˆ W 1 , ˆ W 2 in body frame to construct new orthonormal reference with bases (ˆ r 1 , ˆ r 2 , ˆ r 3 ) and observation vectors ( ˆ b 1 , ˆ b 2 , ˆ b 3 ): ˆ r 1 = ˆ V 1 , ˆ r 2 = ( ˆ V 1 × ˆ V 2 ) / | ˆ V 1 × ˆ V 2 | , ˆ r 3 = ( ˆ V 1 × ( ˆ V 1 × ˆ V 2 )) / | ˆ V 1 × ˆ V 2 | (3.4) ˆ b 1 = ˆ W 1 , ˆ b 2 = ( ˆ W 1 × ˆ W 2 ) / | ˆ W 1 × ˆ W 2 | , ˆ b 3 = ( ˆ W 1 × ( ˆ W 1 × ˆ W 2 )) / | ˆ W 1 × ˆ W 2 | (3.5) from which the attitude matrix can be simply found by R = 3 X i = 1 ˆ b i ˆ r T i . (3.6) Although this method seems to be very simple, in practice it su ff ers from the fact that parts of measurements are discarded. Therefore, the optimal attitude reconstruction methods were given more attention since they do not eliminate any parts of the observed vectors. 3.3 SVD and FOAM A descendant of the method proposed in [Farrell and Stuelpnagel, 1966], Singular Value Decomposition (SVD) method is a point-by-point algorithm to determine the optimal atti- tude matrix in the Wahba problem framework [Markley, 1988]. In this approach, similar to the other deterministic techniques, only sensor measurements are used and information about the system model is disregarded. The method consists of a direct “singular value” decomposition [Golub and Loan, 1983] of the matrix B that gives B = US V T , (3.7) where U and V are orthogonal matrices and S is a singular value diagonal matrix of the form S = diag( s 1 , s 2 , s 3 ) , (3.8)
C hapter 3. S tatic A ttitude D etermination 21 with the singular values s i , i = 1 , 2 , 3, obeying the inequalities s 1 s 2 s 3 0. Proper orthogonal matrices of U + and V + along with the diagonal matrix S 0 are defined as U + = U
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