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

Albeit this comes with high computational expense

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Albeit, this comes with high computational expense [Crassidis et al., 2007]. The idea has also been used for attitude estimation purposes. The authors in [Cheng and Crassidis, 2004] have presented a PF that utilizes Modified Rodrigues Parame- ters (MRPs) for its estimation of attitude and gyro bias. In [Liu et al., 2007], a separation of the nonlinear dynamics, like the orientation, and the linear dynamics, like the gyro bias, in the system has led to a less computationally expensive algorithm. Other PFs aimed for atti- tude estimation can be found in [Carmi and Oshman, 2009b], [Carmi and Oshman, 2009a].
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C hapter 4. D ynamic A ttitude F iltering and E stimation 54 4.2.2 Invariant Kalman Filters Invariant Extended Kalman Filter (IEKF) [Bonnabel et al., 2009b], is a newly-proposed filter for AHRS systems, based on the symmetry-preserving observers design approach [Bonnabel et al., 2008], [Bonnabel et al., 2009a]. The approach simply exploits the natural symmetries in the rigid body system dynamics and uses this property to design filters and observers that remain invariant by body-fixed rotations and linear translations. The method starts with proposing a pre-observer for the original system, with the cor- rection terms having the same invariant properties. This results in an error system whose trajectory does not depend on the original system trajectory and input. The choice of strat- egy to obtain the correction term gains can either be on a Kalman filtering base or an observer design trend. In [Bonnabel et al., 2009b], the authors use the following system dynamic model ˙ Q = 1 2 Q ( ω meas - ω b ) , ˙ v = g + 1 a s Q a meas Q - 1 , ˙ ω b = 0 , ˙ a s = 0 , (4.37) where a s > 0 is a constant scaling factor for the accelerometer reading, which measures a meas = a s a , with a being the specific acceleration. It will be shown in section (4.4) that the model remains invariant under a constant rotation of Q g and linear body-fixed translation V g . The task of the Invariant EKF is to use the IMU data along with GPS to estimate at- titude, velocity, gyro bias, and accelerometer unknown scaling factor. For this, a pre-
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C hapter 4. D ynamic A ttitude F iltering and E stimation 55 observer is given as follows: ˙ ˆ Q = 1 2 ˆ Q ( ω meas - ˆ ω b ) + ˆ Q ( K Q E ) , ˙ ˆ v = ge 3 + 1 ˆ a s ˆ Q a meas ˆ Q - 1 + ˆ Q ( K v E ) ˆ Q - 1 , ˙ ˆ ω b = K ω E , ˙ ˆ a s = ˆ a s K a E , (4.38) with K Q , K v , K ω , K a being the filter gains. The invariant output error is given by E = ˆ Q - 1 v - y v ) ˆ Q ˆ Q - 1 B ˆ Q - y B , (4.39) where y v is the GPS-obtained linear velocity of the rigid body, B is the Earth magnetic field in NED coordinates known in inertial frame, and y B is the noise-contaminated magnetome- ter reading in the body frame. The next step consists in defining the state error vector η : = μ ν β α = Q - 1 ˆ Q ˆ v - v ˆ ω b - ω b ˆ a s - a s .
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