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

Diag1 1 det u 39 v v diag1 1 det v 310 s diag s 1 s 2

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[diag(1 , 1 , det U )] , (3.9) V + = V [diag(1 , 1 , det V )] , (3.10) S 0 = diag( s 1 , s 2 , s 3 (det U )(det V )) , (3.11) where det denotes the determinant of a matrix and (det U )(det V ) = ± 1. Then, the matrix B is decomposed into the following form B = U + S 0 V T + , (3.12) and the optimal matrix R opt , which minimizes the cost function (3.1), is found to be R opt = U + V T + = U [diag(1 , 1 , (det U )(det V ))] V T . (3.13) Another version of this method, known as Fast Optimal Attitude Matrix (FOAM) [Markley, 1993], uses the properties of the matrix B to rewrite the optimal rotation matrix (3.13) as R opt = [( κ + k B k 2 ) B + λ adj B T - BB T B ] /ξ, (3.14) where adj denotes the adjoint matrix and k B k 2 = s 2 1 + s 2 2 + s 2 3 , (3.15) and the scalar coe ffi cients κ , λ , and ξ are defined as κ = s 2 s 3 + s 3 s 2 + s 1 s 2 , λ = s 1 + s 2 + s 3 , ξ = ( s 2 + s 3 )( s 3 + s 1 )( s 1 + s 2 ) . (3.16) Since the values of these coe ffi cients depend on the SVD, FOAM takes advantage of an iterative computation strategy to avoid finding s 1 , s 2 , s 3 and instead, directly computing the
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C hapter 3. S tatic A ttitude D etermination 22 three scalar coe ffi cients. The coe ffi cients κ and ξ can be expressed in terms of λ and B as κ = 1 2 ( λ 2 - k B k 2 ) , ξ = κλ - det B . (3.17) Using (3.14) and the fact that λ = tr( R opt B T ), λ can be found by solving the following equation ( λ 2 - k B k 2 ) 2 - 8 λ det B - 4 k adj B k 2 = 0 . (3.18) Once this equation is recursively solved to find λ , all the other scalar coe ffi cients can be computed. These will determine the optimal rotation matrix from (3.14). In comparison to other methods. the FOAM algorithm is significantly higher in speed and is shown to be the most robust algorithm among the other deterministic attitude esti- mation methods. It also does not have problems in dealing with the special case of a 180 degrees rotation [Markley and Mortari, 2000], [Markley and Mortari, 1999]. The SVD and FOAM do not adopt quaternion parameterization and work entirely with a rotation ma- trix. This enables them to work without the requirement of computing eigenvalues and eigenvectors and save some computational time. 3.4 Q-Method and the QUEST Since the four component quaternion representation and the rotation matrix are related to each other by simple relations, it can be shown that a search for an optimal matrix R opt in Wahba’s problem leads to the computation of an optimal quaternion corresponding to that rotation matrix [Keat, 1977]. The method, known in literature as the Q-method, simplifies the previous optimization techniques by using the 4 × 1 quaternion vector instead of 3 × 3 rotation matrix. Given the observation pairs of ( ˆ V i , ˆ W i ) and the positive coe ffi cients a i , let us define the following 3 × 3 matrix, 3 × 1 vector z and scalar σ as S : = B + B T = N X i = 1 a i ˆ W i ˆ V T i + ˆ V i ˆ W T i , (3.19)
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C hapter 3. S tatic A ttitude D etermination 23 Z : = N X i = 1 a i ˆ W i × ˆ V i , (3.20) σ : = tr( B ) = N X i = 1 a i ˆ V T i ˆ W i . (3.21) Defining the 4 × 4 symmetric matrix K as K = S - σ I 3 Z Z T σ , (3.22) results in (3.1) to be written into the quadratic quaternion function 1 - L ( R ) = g ( Q ) = Q T KQ . (3.23) It is then clear that the minimization of L ( R ) is equivalent to finding the maximum value of
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