Particle Filtering for Attitude   Estimation Using a Minimal  Local - Error Representation:   A Revisit  Lubin Chang  Department of navigation engineering, Naval University of Engineering ,   China  F or spacecraft attitude representation ,   t he quaternion is preferred o ver   other representations  due to its   bilinear nature of the   kinematics   and the   singularity - free property . However, when  applied in the attitude   estimation   filters, its unity   norm   constraint   can be easily destroyed by  the   quaternions   averag ing operation. Al though the   brute - force quaternion normalization   can  be   implement ed, this strategy is   crude   and may be not sufficient for some filters which  necessitate   the state covariance calculations [2 - 4]. In [1] a   quaternion - based   p article   f iltering  (Q PF )   for spacecra ft attitude estimation has been   proposed based on   the local/global attitude  representation .   For   the   local/global   representation ,   the   unconstrained   three - dimensional  representation , say   modi fi ed Rodrigues parameter (MRP) , is used   for the local - error   space  w hile the quaternion   is used   for the global representation   and   propagation.   With this   ingenious  structure,   the unconstrained local - error representation   is   used directly   as the state   in the  developed QPF, which can   maintain   the   unity norm of   the quaternion   i n a nature way .  Therefore, there is no need to perform the   brute - force quaternion normalization   any more.  Since the MRP and quaternion based p article s are used   simultaneous ly in the filtering cycle of  the QPF, they   should   be transformed to each other, whic h   necessitate s a   fiducial   attitude  quaternion (see   Eq. (1 8 )   in [1]). In the QPF, the   estimated quaternion   that is given by  1  ˆ   ˆ   ˆ k   k   k  -   +   +  +   é   ù = W   ë   û q   q  w   (1)  is used as the   fiducial   quaternion. The explicit   definitions   and   expression s of the terms  involved i n   Eq.   (1) can be found in [1] and they are not presented for   brevity . It is shown that
Eq.   (1)   inherit s a similar manner as the extended Kalman filter (EKF) for state propagation [5].  It is well known that the p article   f iltering   (PF) bears the   superiority   in terms of accuracy and  robustness over other filtering algorithms with no exception of the EKF due to large amount  of   particles   being used to   make probabilistic inference .   In this respect,   the quaternion  propagation in   Eq.   (1) has not made full use of th e   superiority   of the PF. With this  consideration, the   estimated quaternion   by the following q uaternions   a veraging   may be more  preferred.  (   )   (   )  1   1   1  1  N   i   i  k   k   k  i  w -  +   +   +  =  =   å % q   q   (2)  where   (   ) 1  i  k   + q   are the quaternion based   particles   which can be obtai ned through   Eq. (1 6 )   in [1]  and   (   ) 1  i  k w   +   are the corresponding weights. U nfortunate ly, as has been pointed out that   the norm  constraint   of the   quaternion estimate given   by the   averag ing operation in   Eq. ( 2 )   can be easily  destroyed. Actual ly, this is just the motivation of the   local/global representation   structure  being used, that is,   avoid ing the quaternions   averag ing operation is the   filtering   recursion .  If the   norm   constraint   preserv ing quaternion can be derived using the quaternion base d  particles , it can be used to   substitute   the   estimated quaternion   in   Eq.   (1) as the   fiducial   attitude  quaternion for the QPF. F ortunate ly,   either the minimum mean - square   error   (MMSE) [6, 7]   or  maximum a posteriori   (MAP) [6]   approach   can be used to derive   a   normalized quaternion  estimate   without any   brute - force quaternion normalization.   Since the   MAP   approach   usually  yields noisier estimates than the   MMSE   approach , the MMSE   approach   is more preferred. In  this note, the QPF is revised and modified by making   use of the MMSE approach to derive  the   fiducial   attitude quaternion based on the   aforementioned   discussion.
When   N   weighted   quaternion based   particles   have been obtained through   Eq. (1 6 )   in [1], the  normalized quaternion   estimate   ca n be derived by solving the following maximization  problem  3 1   arg max   T  k   M -  +   Î  =  (   (  q   S  q   q   q   (3)  where   M   is   a   4   4  ́   matrix   given by  (   )   (   )   (   )  1   1   1  1  N   i   i   i   T  k   k   k  i  M   w   +   +   +  =  =   å   q   q   (4)  S imilarly to   the   well - known Davenport’s q method, th e average quaternion that solves the  maximization problem of Eq. ( 3 ) is the normalized eigenvector of   M   corresponding to the  maximum   eigenvalue.  With the   normalized quaternion   estimate   1 k  -  +  (  q   ,   Eq. (1 8 )   in [1] can   be   modified   as  (   )  (   )  (   )  (   )   (   )  1  1 1  1   1   1  4 k  i  k i   i  k   k   k i  q   +  - +   -  +   +   +  é   ù  ê   ú º   =   Ä  ê   ú ë   û  (  q   q   q  dr  d   d   (5)  Meanwhile,   the updated quaternion given by Eq. ( 23 )   in [1] should also be   modified   as  1   1   1  ˆ   ˆ k   k   k  +   +   -  +   +   + =   Ä   (  q   q   q  d   (6)  With the aforementioned modification, the QPF can not only maintain the virtue of preser ving  quaternion   norm   constraint   naturally , but also make full use of the advantage of the PF in  terms of accuracy and robustness.  Making use of the MMSE   approach   to derive a   normalized quaternion   estimate   has been  investigated in the PF in [6]. However, in   [6] the quaternion is used directly as the state and  the   local/global representation   structure has not been used. This is because that there is no  need to calculate the state covariance in the designed PF in [6]. In contrast, the   attitude - error  covariance   (for MRP based state) should be calculated to p erturb   the state p article s in [1],  which   necessitate s   the   unconstrained   three - dimensional   local - error   representation .   More  specifically,   the quaternion   is a four - element   parameterization   that is used to repre sent the
three   dimensional attitude , the rank of the   4   4  ́   covariance matrix corresponding to the  quaternion is virtually three. Therefore, carrying out square root decomposition   or   invers ion  on the quaternion covariance matrix, which   a re usually   necessary procedure s   in filtering  algorithms , can result in undesired numerical difficulty.   In this respect, the   3   3  ́   full rank  covariance matrix should be   determined for the attitude   (actually the attitude error)   based  stat e.  The   concept   of modification discussed in this paper has also been used to modify the  UnScented Quaternion   Estimator   ( USQUE ) in [8]. Actually, the   local/global representation  structure is just   populari zed by USQUE in [9, 10] and has been extended to othe r   nonlinear  sampling based filtering algorithm with the QPF in [1] as a representation. Therefore, the  concept   of modification discussed in this note and [8] is   promising to   modify the   local/global  representation   structure based filtering algorithm, especi ally the sampling based nonlinear  Kalman filtering algorithms which   necessitate   the state covariance calculations [11 - 13].  Acknowledgments  T his work was supported in part by the National Natural Science Foundation of China  (61 304241, 61374206 ) .  Referenc es  [1]   Cheng, Y.,   and   Crassidis, J. L., “Particle Filtering for Attitude   Estimation Using a  Minimal   Local - Error Representation,”   Journal of Guidance, Control, and Dynamics , Vol.  3 3 ,   No. 4, July – Aug. 20 10 , pp. 1 305 – 1 310 .  [2]   Shuster, M. D., “A Survey of Attitude Re presentations,”   Journal of the   Astronautical  Sciences , Vol. 41, No. 4, Oct. – Dec. 1993, pp. 439 – 517.
[3]   Markley,   F.   L.,   “Attitude   Error   Representations   for   Kalman   Filtering,”   Journal   of  Guidance, Control, and Dynamics , Vol. 63, No. 2, 2003,   pp. 311 – 317.  [4]   Crassi dis, J. L., Markley, F. L., and Cheng, Y., “A Survey of Nonlinear   Attitude  Estimation Methods,”   Journal of Guidance, Control, and   Dynamics , Vol. 30, No. 1,  Jan. – Feb. 2007, pp. 12 – 28.  [5]   Crassidis, J. L., and Junkins, J. L., Optimal Estimation of Dynamic   Syste ms, CRC Press,  Boca Raton, FL, 2004, pp. 2 85 – 29 2.  [6]   Oshman, Y., and Carmi, A., “Attitude Estimation from Vector   Observations Using a  Genetic - Algorithm - Embedded Quaternion   Particle Filter,”   Journal of Guidance, Control,  and Dynamics ,   Vol. 29, No. 4, July – Aug.   2006, pp. 879 – 891.  [7]   Markley, F. L., Cheng, Y., Crassidis, J. L., and Oshman, Y., “Averaging   Quaternions,”  Journal of Guidance, Control, and Dynamics , Vol. 30,   No. 4, July – Aug. 2007, pp.  1193 – 1196.  [8]   Chang, L. B., Hu, B. Q., and   Ch ang ,   G .   B. ,   “ Modified   UnScen ted Q U aternion   Estimator  based on Quaternion Averaging, ”   Journal of Guidance, Control, and Dynamics ,   V ol.3 7 ,  N o. 1,   Jan. – Feb.   2014,   pp.   305 - 309 .  [9]   Crassidis,   J.   L.,   and   Markley,   F.   L.,   “Unscented   Filtering   for   Spacecraft   Attitude  Estimation,”   Journal of Guid ance, Control, and Dynamics ,   Vol. 26, No. 4, July – Aug.  2003, pp. 536 – 542.  [10]  Crassidis,   J.   L.,   “Sigma - Point   Kalman   Filtering   for   Integrated   GPS   and   Inertial  Navigation,”   IEEE Transactions on Aerospace and Electronic   Systems , Vol. 42, No. 2,  2006, pp. 750 – 756.
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