Comment on “Highly Efficient Sigma Point Filter for Spacecraft Attitude and Rate Estimation”

Abstract

In light of the intuition that a better symmetrical structure can further increase the numerical accuracy, the paper by Fan and Zeng (2009) developed a new sigma point construction strategy for the unscented Kalman filter (UKF), namely, geometric simplex sigma points (GSSP). This comment presents a different perspective from the standpoint of the numerical integration. In this respect, the GSSP constitutes an integration formula of degree 2 with equal weights. Then, we demonstrate that the GSSP can be derived through the orthogonal transformation from the basic points set of degree 2. Moreover, the method presented in this comment can be used to construct more accurate sigma points set for certain dynamic problems.

---

With the intuition that a better symmetry property provides a better numerical behavior [1], addressed the construction strategies to make the best symmetric structure in simplex sigma point set and derived the so-called geometric simplex sigma points (GSSP) for Euclidean geometric space. As compared with the previously exiting simplex sigma points set, the GSSP has a symmetric structure and a lower computational expense, is numerically more accurate, and can be used in a variety of 3-dimensional modeled dynamic problems.

In this comment we will show that the GSSP can also be derived from the integration rule of degree 2. Embedding the Gaussian assumption in the Bayesian filter we can reach the idea that the functional recursion of the Bayesian filter reduces to an algebraic recursion operating only on conditional means and covariances which share the same structure of Gaussian weighted integrals whose integrands are all of the form nonlinear function × Gaussian density. The multidimensional integrals are usually intractable for systems involving nonlinearity, so the recursive estimation problem boils down to how to compute the integrals using approximate methods. There are many well-known numerical integration methods such as Gauss-Hermite quadrature, cubature rules, fully symmetric integration rule, and central-difference-based methods that can be used to handle such integrals [2–4]. The unscented transformation (UT) used in the traditional unscented Kalman filter (UKF) can be interpreted as either fully symmetric integration rule or cubature rule of degree 3. The simplex UT can also be interpreted as a numerical integration formula of degree 2 [3]. Next we will focus on the numerical integration formula of degree 2 in order to derive the GSSP.

Before getting involved in further details, we first introduce some definitions when constructing the exact monomials rule as follows [3, 4].

Definition 1. Consider the monomials of the form∏^𝑑_𝑖=1𝑥^𝛼𝑖_𝑖,where the powers𝛼_𝑖are nonnegative integers and∑^𝑑_𝑖=1𝛼_𝑖≤𝑝, a rule said to have precision𝑝if it can integrate such monomials accurately and it is not exact for monomials of degree𝑝+1.
The numerical integration formulas are conducted by approximating the integrals with the weighted sum of an elaborately chosen set of points as follows [5–7]:_𝑅𝑛𝑔(𝑥)⋅𝑊(𝑥)≈_𝑘𝛼_𝑘𝑔𝜒_𝑘, (1) where 𝑅^𝑛 is a region in an 𝑛-dimensional, real, Euclidean space, 𝑥=(𝑥₁,𝑥₂,𝑥₃,…,𝑥_𝑛) is the state variable, 𝛼_𝑘 are constants, and 𝜒_𝑘are points in the space. The integral weight is a Gaussian distribution as discussed above. Since an arbitrary Gaussian distribution with mean𝜇and covarianceΣcan always be transformed into the unit Gaussian distribution as (see prove in [4])𝑔(𝑥)⋅𝑁𝑥;𝜇,Σ𝑑𝑥=𝑔𝐴𝜉+𝜇⋅𝑁(𝜉;0,𝐼)𝑑𝜉, (2) where 𝐴𝐴^𝑇=Σ and𝐼is the identity matrix, we can start by considering the multidimensional unit Gaussian integral. Based on Definition 1, we can construct a rule, which is exact up to degree 2 by determining the weighted points set𝜒_𝑘 such that it is exact for selections𝑔_𝑖(𝜉)=1,𝑔_𝑖(𝜉)=𝜉_𝑖,𝑔_𝑖,𝑗(𝜉)=𝜉_𝑖𝜉_𝑗,𝑖≠𝑗,and𝑔_𝑖,𝑖(𝜉)=𝜉²_𝑖. The true values of the integrals are𝐼₀=()𝐼1⋅𝑁𝜉;0,𝐼𝑑𝜉=1,₁=𝜉_𝑖()𝐼⋅𝑁𝜉;0,𝐼𝑑𝜉=0,𝑖=1,2,…𝑛,₂=𝜉²_𝑖()𝐼⋅𝑁𝜉;0,𝐼𝑑𝜉=1,𝑖=1,2,…𝑛,_1×1=𝜉_𝑖𝜉_𝑗⋅𝑁(𝜉;0,𝐼)𝑑𝜉=0,𝑖≠𝑗=1,2,…𝑛. (3)(4) In [5] Stroud had proved that𝑛+1is the minimum number of points for equally weighted degree 2 formulas. Let us define𝑛+1equally weighted points𝜒=𝜒₁,𝜒₂,…,𝜒_𝑛+1, (5) where𝜒_𝑘=(𝜒_𝑘,1,𝜒_𝑘,2,…,𝜒_𝑘,𝑛)^𝑇,𝑘=1,2,…,𝑛+1. In order to calculate (4) accurately using these points through (1), we can get the following equations:1𝑛+1^𝑛+1_𝑘=1𝜒_𝑘1=0,𝑛+1^𝑛+1_𝑘=1𝜒_𝑘⋅𝜒^𝑇_𝑘=𝐼_𝑛, (6) where𝐼_𝑛is the 𝑛-dimensional identity matrix. Any equally weighted points set that fulfills (6) can approximate the unit Gaussian integral accurately up to degree 2. References [6, 7] have presented a basic points set that fulfills such conditions with the form as𝜒_{𝑘,2𝑟−1}=√2cos2𝑟𝑘𝜋,𝜒𝑛+1_𝑘,2𝑟=√2sin2𝑟𝑘𝜋,𝑛+1 (7)𝑟=1,2,…,[𝑛/2],and if𝑛is odd,𝜒_𝑘,𝑛=(−1)^𝑘. [𝑛/2] is the greatest integer not exceeding𝑛/2. When𝑛=3, the basic points set is𝑆₁=𝜒₁∣𝜒₂∣𝜒₃∣𝜒₄=0−√20√2√20−√20−11−11. (8) Next we will give a theorem through which we can get the GSSP from the basic points set of degree 2.

Theorem 1. Assume that𝑛+1equally weighted points set as that in (5) constitutes an integration formula of degree 2. 𝐴is an 𝑛×𝑛 orthogonal matrix. Then,𝐴𝜒also constitutes an integration formula of degree 2.

Proof. By defining a matrix 𝜒𝑀=^1,1𝜒_2,1𝜒_3,1⋯𝜒_𝑛+1,1𝜒_1,2𝜒_2,2𝜒_3,2⋯𝜒_𝑛+1,2𝜒⋮⋮⋮⋱⋮_1,𝑛𝜒_2,𝑛𝜒_3,𝑛⋯𝜒_𝑛+1,𝑛, (9) we can rewrite (6) as 𝑀𝑀^𝑇=(𝑛+1)𝐼_𝑛, (10) where𝐼_𝑛is the 𝑛-dimensional identity matrix. 𝐴is an orthogonal matrix, so 𝐴𝑀(𝐴𝑀)^𝑇=𝐴𝑀𝑀^𝑇𝐴^𝑇=(𝑛+1)𝐼_𝑛𝐴𝐴^𝑇=(𝑛+1)𝐼_𝑛. (11) Hence, 𝐴𝜒also fulfills (6) which completes the proof.

For the 3-dimensional Euclidean space, there are many orthogonal matrixes. Here, we use the direction cosine matrix (DCM) which is widely used in the practical systems such as guidance and navigation [8]. The DCM that rotates an angle𝜙about𝑢=[001]^𝑇is𝐶𝜙=cos𝜙sin𝜙0−sin𝜙cos𝜙0001. (12) By a simple computation, it is obvious that𝑆₂=𝐶3𝜋4𝑆₁=11−1−1−111−1−11−11. (13) Since all the points share equal weight, 𝑆₂is virtually just the GSSP derived in [1].

Up to this point we have derived the GSSP through the numerical integration formulas method. Compared with the intuitionistic method in [1], our method is more principled in mathematical terms. Although Theorem 1 is proposed for integration formula of degree 2, it can be generalized for different degrees, that is, the orthogonal transformation on the numerical integration formula will not change its accurate degree. Reference [7] also presented the points set of degree 3, that is, 𝛾=𝛾₁,𝛾₂,…,𝛾_2𝑛, (14) where𝛾_𝑘=(𝛾_𝑘,1,𝛾_𝑘,2,…,𝛾_𝑘,𝑛)^𝑇,𝑘=1,2,…,2𝑛with𝛾_{𝑘,2𝑟−1}=√()2cos2𝑟−1𝑘𝜋𝑛,𝛾_𝑘,2𝑟=√()2sin2𝑟−1𝑘𝜋𝑛, (15)𝑟=1,2,…,[𝑛/2],and if𝑛is odd, 𝛾_𝑘,𝑛=(−1)^𝑘. [𝑛/2]is the greatest integer not exceeding𝑛/2. It can be proven that the points set (14) can be derived through orthogonal transformation on the cubature points set with the form [4]𝜆=√𝑛𝑒_𝑘−√𝑛𝑒_𝑘,𝑘=1,…,𝑛, (16) where𝑒_𝑖denotes a unit vector to the direction of coordinate axis𝑖.

As can be seen from (16) the distance of the cubature point from the mean is proportional to √𝑛. So, for high-dimensional problems, the cubature points set bears the nonlocal sampling problem [9–11]. For many kinds of nonlinearities (such as exponents or trigonometric functions), this can lead to significant difficulties. In contrast, the points set (14) does not bear such nonlocal sampling problem. Under this condition, the points set (14) is more accurate than the cubature points set (16). Therefore, to this respect, we can derive different sigma points set which may be more accurate for certain dynamic problems by the presented method.