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

# And be rewritten as a function of δ θ to give the

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and be rewritten as a function of δ θ to give the new loss function in the two following forms J ( δ θ ) = 1 - tr[( I 3 × 3 - S ( δ θ ) + 1 2 S ( δ θ ) 2 ) RB T ] = 1 - ˆ Q T K ˆ Q + 1 2 δ θ T F θθ δ θ . (3.66) From (3.66), the Fisher information matrix can be computed by taking the partial derivative of J ( δ θ ) twice with respect to δ θ : F θθ = 2 J ∂θ∂θ = tr[ RB T ] I 3 × 3 - RB T . (3.67) Here, assuming that a priori attitude estimate ˆ R and covariance matrix P - 1 θθ F θθ are known, the attitude profile matrix B can be computed as B = 1 2 tr( F θθ ) I 3 × 3 - F θθ ˆ R . (3.68)

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C hapter 3. S tatic A ttitude D etermination 33 Equation (3.68) can be used to find the a priori estimate B - of the profile matrix, pro- vided that a priori estimated quaternion ˆ Q - and covariance matrix P - θθ are available. The attitude matrix ˆ R can also be computed from the transformation of the a priori quaternion into the matrix form. The attitude profile matrix B - constructs an a priori Davenport’s K matrix denoted as K - . This matrix is added to the matrix K m constructed from the new measurements, and gives the modified Davenport matrix K + = K m + K - . A posteriori optimal attitude estimate is then computed by solving the familiar equation of K + ˆ Q + = λ ˆ Q + , (3.69) for ˆ Q + = q + 0 , ˆ q + ), using any appropriate solution method to the normal Wahba problem. Once this optimal estimate is found, the update to the auxiliary parameters vector δβ is computed as δ ˆ β + = δβ - - 2( F - ( ββ ) ) - 1 F - ( βθ ) Ψ ( ˆ Q - ) ˆ Q + , (3.70) with Ψ ( ˆ Q - ) = h ˆ q - 0 I 3 - S q - ) - ˆ q - i . (3.71) The remaining steps are to compute the updated covariance matrix and its corresponding propagation along with the propagated state vector to the next step. Details of this proce- dure can be found in [Christian and Lightsey, 2010]. The SOAR filter is a relatively computationally expensive method because it relies on the calculation of some inverse matrices during the recursion. However, since it is similar in structure to a Multiplicative Extended Kalman Filter (MEKF), it behaves in a similar way when the errors are small and linearization assumptions hold. 3.8 Simulations This section deals with the performance of the QUEST algorithm under di ff erent measure- ment conditions that vary from completely ideal conditions to more realistic cases.
C hapter 3. S tatic A ttitude D etermination 34 In the ideal case, all the measurements and filter inputs are assumed to be noise-free. This means that, for instance, for the case of magnetic field vector, the value of this vector in the surrounding environment is constant and known in the inertial frame and the value in the body frame is available without any added measurement noise. Also, it is assumed that there are no magnetic field generators (such as powerful electric motors) in the environ- ment. For the case of accelerometers, the ideal measurement is defined as the measurement of the Earth gravity vector the body frame.
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