Quaternion estimator algorithm

The quaternion estimator algorithm (QUEST) is an algorithm designed to solve Wahba's problem, that consists of finding a rotation matrix between two coordinate systems from two sets of observations sampled in each system respectively. The key idea behind the algorithm is to find an expression of the loss function for the Wahba's problem as a quadratic form, using the Cayley–Hamilton theorem and the Newton–Raphson method to efficiently solve the eigenvalue problem and construct a numerically stable representation of the solution.

The algorithm was introduced by Malcolm D. Shuster in 1981, while working at Computer Sciences Corporation.[1] While being in principle less robust than other methods such as Davenport's q method or singular value decomposition, the algorithm is significantly faster and reliable in practical applications,[2][3] and it is used for attitude determination problem in fields such as robotics and avionics.[4][5][6]

Formulation of the problem

edit

Wahba's problem consists of finding a rotation matrix   that minimises the loss function

 

where   are the vector observations in the reference frame,   are the vector observations in the body frame,   is a rotation matrix between the two frames, and   are a set of weights such that  . It is possible to rewrite this as a maximisation problem of a gain function  

 

defined in such a way that the loss   attains a minimum when   is maximised. The gain   can in turn be rewritten as

 

where   is known as the attitude profile matrix.

In order to reduce the number of variables, the problem can be reformulated by parametrising the rotation as a unit quaternion   with vector part   and scalar part  , representing the rotation of angle   around an axis whose direction is described by the vector  , subject to the unity constraint  . It is now possible to express   in terms of the quaternion parametrisation as

 

where   is the skew-symmetric matrix

 .

Substituting   with the quaternion representation and simplifying the resulting expression, the gain function can be written as a quadratic form in  

 

where the   matrix

 

is defined from the quantities

 

This quadratic form can be optimised under the unity constraint by adding a Lagrange multiplier  , obtaining an unconstrained gain function

 

that attains a maximum when

 .

This implies that the optimal rotation is parametrised by the quaternion   that is the eigenvector associated to the largest eigenvalue   of  .[1][2]

Solution of the characteristic equation

edit

The optimal quaternion can be determined by solving the characteristic equation of   and constructing the eigenvector for the largest eigenvalue. From the definition of  , it is possible to rewrite

 

as a system of two equations

 

where   is the Rodrigues vector. Substituting   in the second equation with the first, it is possible to derive an expression of the characteristic equation

 .

Since  , it follows that   and therefore   for an optimal solution (when the loss   is small). This permits to construct the optimal quaternion   by replacing   in the Rodrigues vector  

 .

The   vector is however singular for  . An alternative expression of the solution that does not involve the Rodrigues vector can be constructed using the Cayley–Hamilton theorem. The characteristic equation of a   matrix   is

 

where

 

The Cayley–Hamilton theorem states that any square matrix over a commutative ring satisfies its own characteristic equation, therefore

 

allowing to write

 

where

 

and for   this provides a new construction of the optimal vector

 

that gives the conjugate quaternion representation of the optimal rotation as

 

where

 .

The value of   can be determined as a numerical solution of the characteristic equation. Replacing   inside the previously obtained characteristic equation

 .

gives

 

where

 

whose root can be efficiently approximated with the Newton–Raphson method, taking 1 as initial guess of the solution in order to converge to the highest eigenvalue (using the fact, shown above, that   when the quaternion is close to the optimal solution).[1][2]

See also

edit

References

edit
  1. ^ a b c Shuster and Oh (1981)
  2. ^ a b c Markley and Mortari (2000)
  3. ^ Crassidis (2007)
  4. ^ Psiaki (2000)
  5. ^ Wu et al. (2017)
  6. ^ Xiaoping et al. (2008)

Sources

edit
  • Crassidis, John L; Markley, F Landis; Cheng, Yang (2007). "Survey of nonlinear attitude estimation methods". Journal of Guidance, Control, and Dynamics. 30 (1): 12–28. Bibcode:2007JGCD...30...12C. doi:10.2514/1.22452.
  • Markley, F Landis; Mortari, Daniele (2000). "Quaternion attitude estimation using vector observations". The Journal of the Astronautical Sciences. 48 (2). Springer: 359–380. Bibcode:2000JAnSc..48..359M. doi:10.1007/BF03546284.
  • Psiaki, Mark L (2000). "Attitude-determination filtering via extended quaternion estimation". Journal of Guidance, Control, and Dynamics. 23 (2): 206–214. Bibcode:2000JGCD...23..206P. doi:10.2514/2.4540.
  • Shuster, M.D.; Oh, S.D. (1981). "Three-axis attitude determination from vector observations". Journal of Guidance and Control. 4 (1): 70–77. Bibcode:1981JGCD....4...70S. doi:10.2514/3.19717.
  • Wu, Jin; Zhou, Zebo; Gao, Bin; Li, Rui; Cheng, Yuhua; Fourati, Hassen (2017). "Fast linear quaternion attitude estimator using vector observations" (PDF). IEEE Transactions on Automation Science and Engineering. 15 (1). IEEE: 307–319. doi:10.1109/TASE.2017.2699221. S2CID 3455346.
  • Yun, Xiaoping; Bachmann, Eric R; McGhee, Robert B (2008). "A simplified quaternion-based algorithm for orientation estimation from earth gravity and magnetic field measurements". IEEE Transactions on Instrumentation and Measurement. 57 (3). IEEE: 638–650. doi:10.1109/TIM.2007.911646. hdl:10945/46081. S2CID 15571138.
edit