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

# Another approach introduced in vandyke et al 2004

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Another approach introduced in [VanDyke et al., 2004] takes advantage of the system’s rotational dynamics equations to estimate both the attitude and the angular velocity. The state vector used in this unscented filter is defined as x = δ q ω , (4.29) where δ q is the vector part of the error quaternion defined by δ Q = Q ˆ Q - 1 . (4.30) The state in step k is initialized by assuming that the error quaternion is zero ˆ x k = [ δ ˆ q T k ω T k ] T = [0 0 0 ω T k ] T . (4.31) As in (4.21), ˆ x k = ˆ x + k is used to compute the sigma points x ( i ) k = [ δ q ( i ) T k ω ( i ) T k ] T for i = 1 , ..., 2 N x + 1 . (4.32) The δ q ( i ) k error vector in each sigma point is then transformed to its associated quaternion using the unit norm constraint δ Q ( i ) k = [ δ q ( i ) T k q 1 - δ q ( i ) T k δ q ( i ) k ] T for i = 1 , ..., 2 N x + 1 . (4.33) Substituting these quaternions into (4.30) gives the four-element sigma point quaternions. The quaternion and angular velocity sigma points are then propagated using system dy- namic equation ˙ Q = 1 2 Q (0 , ω ) , ˙ ω = I - 1 b ( - ω × I b ω + u ) , (4.34) to give Q ( i ) k + 1 and ω ( i ) k + 1 . Using the quaternion-rotation matrix transformation, the measure- ments vector with two vectorial measurements is subsequently defined as y ( i ) k + 1 = R T ( Q ( i ) k + 1 ) v ( i ) 1 R T ( Q ( i ) k + 1 ) v ( i ) 2 ω ( i ) k + 1 (4.35)

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C hapter 4. D ynamic A ttitude F iltering and E stimation 53 where v 1 and v 2 are known vectors in the inertial reference frame. The method then pro- ceeds in the same manner as the UKF as discussed before. Once the updated state vector ˆ x k + 1 = [ δ ˆ q + k + 1 ω + k + 1 ] T is found, it is easy to find the new quaternion error estimate δ ˆ Q + k + 1 us- ing the same approach as (4.33), and then obtaining the estimated quaternion in step k + 1 using ˆ Q + k + 1 = δ ˆ Q + k + 1 ˆ Q k . (4.36) The authors show that their algorithm outperforms the EKF in the presence of noisy mea- surements and poor initial estimates [Sekhavat et al., 2007]. On the other hand, one disad- vantage is that as evident from (4.34), the rigid body’s inertia matrix must be exactly known in order to find the propagated angular velocity points. This may not be preferable in prac- tice where the lack of accurate knowledge of this matrix may result in poor performance of the filter in its propagation phase. The recently developed Particle Filter (PF) algorithm is a generalization of the UKF based on random sample (or particle ) representations of Probability Density Function (PDF) of the states [Gordon et al., 1993]. Instead of having only 2 N x + 1 particles, as in UKF, unlimited number of particles in PF allows reconstructing the states PDF within a sampling process. The advantage of this strategy is better filter performance when the sys- tem is strongly nonlinear or the measurements are contaminated with non-Gaussian noises.
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