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

Another approach introduced in vandyke et al 2004

This preview shows page 60 - 61 out of 153 pages.

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)
Image of page 60

Subscribe to view the full document.

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.
Image of page 61
You've reached the end of this preview.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern