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

The function g q it is also easy to show that the

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the function g ( Q ). It is also easy to show that the optimal quaternion that maximizes (3.23) is the eigenvector associated with the largest positive eigenvalue of the matrix K . In other words, KQ opt = λ max Q opt . (3.24) Substituting (3.24) into (3.23) and applying the quaternion norm constraint gives the fol- lowing expression for the optimized loss function L ( R opt ) = 1 - λ max . (3.25) Finding the largest eigenvalue of the matrix K and its corresponding optimal quaternion have been the target of many works, including two versions of ESOQ algorithm (EStima- tion of Optimal Quaternion) [Mortari, 1997], [Mortari, 2000], [Markley and Mortari, 2000], and most importantly, Shuster’s algorithm QUEST (QUaternion ESTimation) presented in [Shuster and Oh, 1981]. The latter is a popular algorithm for finding the optimal quater- nion Q opt and since it does not require the minimization of a cost function, it has been a fast attitude determination technique for real-time applications. The QUEST relies on applying the Cayley-Hamilton theorem on matrix S , which yields S 3 = tr( S ) S 2 - tr(adj S ) S + det( S ) I 3 . (3.26)
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C hapter 3. S tatic A ttitude D etermination 24 This characteristic equation is used to find an optimal vector Y opt defined as Y opt = X /γ, (3.27) where X = ( α I + ( λ max - 1 2 tr S ) S + S 2 ) Z , (3.28) and γ = ( λ max + 1 2 tr S ) α - det S , (3.29) with α = λ 2 max - ( 1 2 tr S ) 2 + tr(adj S ) . (3.30) By relating the definition of vector Y opt to the optimal quaternion, one can find Q opt as follows Q opt = 1 p γ 2 + | X | 2 γ X . (3.31) Although the definition of Y opt is similar to the definition of a Gibbs vector [Shuster, 1993], the author avoided explicitly using this vector in the definition of unit quaternion since the Gibbs vector becomes infinite when the rotation angle passes ± π . It can be seen from (3.28- 3.30) that in this method, the computation of the optimal quaternion requires the value of λ max to be known. This value is provided by solving the following forth-order characteristic equation λ 4 - ( a + b ) λ 2 - c λ + ( ab + c 2 tr S - d ) , (3.32) where a = ( 1 2 tr S ) 2 - tr(adj S ) , b = ( 1 2 tr( S )) 2 + Z T Z , c = det S + Z T S Z , d = Z T S 2 Z . (3.33) Numerical algorithms, such as Newton-Raphson method, can be applied on the char- acteristic (3.32) with λ = 1 as a starting point. Based on (3.25) and assuming that λ max is close to unity, its value is easily computed by a very few number of iterations (generally
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C hapter 3. S tatic A ttitude D etermination 25 just a single iteration) with desirable accuracy. However, because of the problems associ- ated with the reliability of these numerical methods, it is commonly believed that QUEST is less robust than the other point-to-point methods. The ESOQ (Estimator of the Optimal Quaternion) [Mortari, 1997] is another approach that has the same structure as QUEST for obtaining λ max , but a di ff erent method for finding Q opt . It is based on defining the matrix H = K - λ max I , (3.34) where in light of (3.24), it is clear that the optimal quaternion is orthogonal to all the columns of H . Now, by representing K in terms of its eigenvectors and eigenvalues (among which one is λ max ) and using the orthonormality of eigenvectors, the following equation can
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