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Matrix Fisher–Gaussian Distribution on SO(3)×R for Attitude Estimation with a Gyro Bias

In this paper, we propose a new probability density function, referred to as the matrix Fisher–Gaussian (MFG) distribution, on the product of the special orthogonal group and the Euclidean space. MFG is constructed by conditioning a multivariate Gaussian distribution on the embedding Euclidean space, where the correlation between attitudes and linear components is formulated at the tangent space of the mean attitude. The desirable feature is that it can globally represent large uncertainties in the attitude of a rigid body correlated with any variable in the Euclidean space, thereby eliminating singularities and complexities inherent to local coordinates. Several stochastic properties and an approximate maximum likelihood estimation (MLE) are derived for MFG, and it is further utilized for unscented attitude estimation with a gyro bias. It is illustrated that the proposed attitude estimation scheme with MFG exhibits more accurate estimates than the multiplicative extended Kalman filter for a challenging case of large initial estimation errors.


I. INTRODUCTION
One of the most popular algorithms for attitude estimation is the Multiplicative Extended Kalman Filter (MEKF) [1], [2]. This algorithm exhibits excellent performances, and it has been successfully applied in several NASA spacecraft missions [3] while verifying both robustness and computational efficiency. However, MEKF has inherent limitations for attitude estimation, as it relies on Gaussian distributions propagated over linearized dynamics. In contrast, attitudes evolve on the three-dimensional orthogonal group, namely SO(3) that cannot be globally identified with any Euclidean space. Moreover, the wrapping nature of the compact manifold SO(3) makes the linear variance no longer proper statistics describing the dispersion of rotational data [4].
To address these issues, the matrix Fisher distribution defined on SO(3) and the Bingham distribution on the three sphere have been utilized to design new attitude estimators directly on the nonlinear compact manifold [5], [6], [7]. Both of these distributions are analogous to the Gaussian distribution on R 3 , and they are equivalent with each other [8]. By incorporating the geometry of SO(3), these filters inherently avoid the linearization and wrapping issues in MEKF. However, a key limitation of these filters is their incapability of estimating gyroscope biases which are known to change slowly over time [9]. The difficulty is that the matrix Fisher distribution or the Bingham distribution is formulated for attitude uncertainties, and as such, neither of them represent the correlation between attitudes on SO(3) and biases on R n . Recently, the Bingham distribution has been extended for the angular-linear correlation with unit quaternions [10]. However, the correlation term is formulated via a series of rotations that parameterize the four-dimensional special orthogonal group, and it requires a numerical optimization for maximum likelihood estimation (MLE).
Here we propose a new probability distribution, referred to as the Matrix Fisher-Gaussian (MFG) distribution, on the product manifold SO(3) ×R n . We adopt the matrix Fisher distribution and the Gaussian distribution, respectively for attitudes and linear components. The correlation between them is defined in the similar fashion as formulating linear correlations in Euclidean space, while providing intuitive geometric interpretations. Several statistical properties of MFG are presented, and it is shown that a maximum likelihood estimation can be approximated in a closed form by discarding some information in the linear components.
Furthermore, an unscented attitude estimator on SO(3) ×R n is developed with MFG to estimate the attitude and gyroscope bias concurrently. The unscented filter starts with selecting sigma points from MFG and noise distributions, propagating them through the gyroscope kinematic model, and recovering a new MFG from the propagated sigma points with MLE. After measurements are available, the propagated MFG is updated by Bayes' rule, and the corresponding moments are calculated to construct the updated MFG. It is shown by a numerical example that the proposed unscented filter exhibits more accurate estimates over the well-established MEKF for a challenging case of large initial estimation errors.
Beyond attitude and gyro bias estimation, MFG can be applied to a variety of estimation problems where the coupling between attitudes and translations is critical, such as GPS/IMU integration, and simultaneous estimation of position and attitude in robotics. As MFG is constructed directly on the product manifold, it may represent large uncertainties without singularities inherent to local parameterizations, which is the unique contribution of this paper.

II. REVIEW OF MATRIX FISHER DISTRIBUTION
The attitude of a rigid body evolves on the special orthogonal group, defined as The rotation matrix R ∈ SO(3) is considered as the linear transformation of the coordinates of a vector from the bodyfixed frame to the inertial frame. Its Lie algebra, namely so(3) is composed of 3 × 3 skew-symmetric matrices. The hat map ∧ : R 3 → so(3) is defined such thatxy = x × y for any x, y ∈ R 3 , and the inverse of the hat map is denoted by the vee map ∨ : so(3) → R 3 . The i-th standard orthonormal basis for R n is denoted by e i ∈ R n . The ndimensional sphere is defined as S n = {x ∈ R n+1 | x = 1}. The set of circular shifts of (1, 2, 3) is defined as I = {(1, 2, 3), (2, 3, 1), (3, 1, 2)}. The vectorization of a matrix by its columns is denoted by vec(·), and the operator e trace(·) is abbreviated as etr(·). The Kronecker product is denoted by ⊗.
The matrix Fisher distribution on SO(3) is defined by a single matrix parameter F ∈ R 3×3 , and the corresponding density is given by with respect to the uniform distribution on SO (3), where c(F ) is a normalizing constant.
The following proper singular value decomposition of F is critical in analyzing a matrix Fisher distribution.
Definition 1: Let the singular value decomposition of F ∈ R 3×3 be given by where the rotation matrices U, V ∈ SO(3), and the diagonal matrix S ∈ R 3×3 are defined as The above proper singular value decomposition of F is used to interpret the mean attitude, principal axes, and dispersion along these axes of a matrix Fisher distribution [5].
Specifically, the normalizing constant of a matrix Fisher distribution depends only on the proper singular values, i.e. c(F ) = c(S). The first moment of R is given by for i ∈ {1, 2, 3}. Given a set of rotation matrices sampled from a matrix Fisher distribution, MLE of U , D and V are given by the proper singular value decomposition of the sample mean of R, i.e.
and MLE of S can be obtained by solving (4) from D.
Calculations of the normalizing constant and its derivatives up to the second order can be found in [11]. It should be noted that the arithmetic mean E[R] is not generally a rotation matrix. Instead, the best estimate of R ∈ SO(3) for a matrix Fisher distribution is usually chosen as the mean attitude that maximizes the density function (1), which is M U V T ∈ SO(3). The matrices U and V also determine the principal axes of a matrix Fisher distribution in the following sense. For θ i ∈ S 1 , let R(θ i ) ∈ SO(3) be a rotation of M about the axis U e i resolved in the inertial frame, or equivalently about V e i resolved in the frame spanned by the columns of M , by an angle θ i , i.e.
Then the density at R(θ i ) is given by where (i, j, k) ∈ I. The above equation indicates the rotations from M about the principal axes specified by U or V are uncorrelated from each other. More importantly, dispersion along the i-th principal axis is indicated by s j +s k : the larger s j +s k is, the more concentrated the distribution is along the rotations about the i-th principal axis; if s j + s k = 0, the distribution is uniform along the same rotations. More detailed descriptions for the properties of the matrix Fisher distribution are available in [5]. Finally, a matrix Fisher distribution can be obtained by conditioning a 9-variate Gaussian distribution onto SO(3) [12]. More specifically, if x R ∈ R 9 is distributed as for M ∈ O(3) and a symmetric, positive-definite K ∈ R 3×3 , then R vec −1 (x R ) T follows the matrix Fisher distribution with parameter F = M K, when conditioned on RR T = I 3×3 and det(R) = 1.

A. Definition
Now we propose a new density function on SO(3) ×R n , referred to as the matrix Fisher-Gaussian distribution (MFG).
Definition 2: The random variables (R, x) ∈ SO(3) × R n follow the matrix Fisher-Gaussian distribution with parameters µ ∈ R n , Σ ∈ R n×n , U, V ∈ SO(3), S = diag(s 1 , s 2 , s 3 ) ∈ R 3×3 and P ∈ R n×3 , if it has the following density function: where µ c ∈ R n is given by with for Q = U T RV , and Σ c ∈ R n×n is defined as Also, F = U SV T ∈ R 3×3 , and c(S) is the normalizing constant of the corresponding matrix Fisher distribution. This distribution is denoted as MG(µ, Σ, P, U, S, V ). It is apparent from (9) that the marginal distribution of R is a matrix Fisher distribution with the parameter F , and the distribution of x conditioned on R is a Gaussian distribution N (µ c , Σ c ). The angular-linear correlation between SO(3) and R n in a MFG is determined with P . Also, it is straightforward to see a MFG is well defined if and only if Σ c is positive-definite, and s 1 ≥ s 2 ≥ |s 3 | [5].

B. Motivation and Geometric Interpretation
In [13], a distribution on S 1 × R 1 is constructed by conditioning a tri-variate Gaussian distribution. This results in two parameters quantifying the correlation between the one-dimensional circular variable θ and the one-dimensional linear variable x. This is because the distribution of θ is conditioned from two dimensional Euclidean space, and the two correlation parameters with x are inherited after conditioning. However, as any vector on S 1 has unit-length, variations along the radial direction, or equivalently the direction normal to the tangent space at the mean angle, have minimal effects on the correlations.
The MFG in this paper is constructed similarly by conditioning a Gaussian distribution in the embedding space R 9 ×R n onto SO(3)×R n . However, the attitude-linear correlation is treated more carefully to capture only the first-order correlation without over-parameterization. More specifically, the tangent space of SO(3) embedded in R 9 at a specific R is the vector space spanned by {vec(Rb i )} i∈{1,2,3} , where {b i } i∈{1,2,3} is any set of basis on R 3 [14]. Here we show that MFG can be obtained by constraining the correlation terms of the Gaussian distribution on R 9 × R n to be nonzero only in the tangent space at the mean attitude with a specific choice of the basis.
Then suppose (x R , x) ∈ R 9+n follows the Gaussian distribution with the mean (µ R , µ) ∈ R 9+n and the covariance matrix Proof: From the properties of multivariate Gaussian distributions, the joint density of (x R , x) can be written as where c is a normalizing constant, As discussed at (8), the first exponential term in (13) reduces to etr(F R T ). Next, the second part of Σ c is . Let t i ∈ R 9 be equally split into a collection of three parts (t i1 , t i2 , t i3 ). Then the i, j-th entry ofΣ −1 R can be written as thus Σ c has the same expression as in (12). Furthermore, we have Thus µ c also has the same expression as in (10). In short, (13) reduces to (9) after conditioning.
In the above theorem, µ R and Σ R are the mean and covariance of x R before conditioning, {t 1 , t 2 , t 3 } is a basis of the tangent space of SO(3) at the mean attitude M in R 9 , and T is a transformation matrix in R 9 that rotates the correlation P expressed with respect to {t 1 , t 2 , t 3 } to P R expressed with respect to the natural basis of R 9 . By the definition of P R , all the correlations except those in the tangent space at M are constrained to be zeros. As such, the correlation is defined by 3n elements in P , instead of 9n elements in P R , so that the correlation between SO(3) and R n is specified by 3n parameters only. We particularly choose the principal axes of the matrix Fisher part, namely the columns of V to construct a basis of the tangent space. Consequently, the correlation expressed by P is defined with respect to the principal directions of the matrix Fisher distribution.
The correlation term of MFG has the following geometric interpretation. Suppose P ij > 0. This implies if x i becomes larger, the distribution of R will rotate about the axis U e j resolved in the inertial frame, or equivalently the axis V e j resolved in the frame spanned by the columns of M . See Fig. 1 for an illustration.

C. Moments
Next, we present selected moments of MFG.
where where E[ν R ν T R ] ∈ R 3×3 is a diagonal matrix with the i-th diagonal element given by for (i, j, k) ∈ I. Proof: Equation (14) follows immediately from the fact that the marginal distribution of R is a matrix Fisher distribution with parameter F . Next, for (16), we have: Also, for (17), we can integrate xx T directly and get The remaining (15) [11].

D. Maximum Likelihood Estimation
Here we consider a maximum likelihood estimation problem to construct MFG from its samples. Given a set of samples (R i , x i ) Ns i=1 , the log-likelihood function of the parameters, after omitting some constants, is given by whereĒ[·] represents the sample mean of a random variable. It is challenging to obtain a closed form solution, as the log-likelihood function must be maximized jointly for the matrix Fisher part and the Gaussian part. From the construction process of MFG, this is comparable to the MLE of a (9 + n)-variate Gaussian distribution with prescribed linear constraints on the covariance matrix, which is already known as a complicated problem [15].
Instead of jointly maximizing the likelihood, we exploit the fact that the marginal distribution for R is a matrix Fisher distribution, and the conditional distribution for x is Gaussian. More specifically, the log-likelihood for the marginal distribution corresponds to the first two terms of the right hand side of (22), and the marginal MLE is solved by the proper singular value decomposition ofĒ[R] to obtain U , S and V , as discussed in Section II.
Next, the log-likelihood of the conditional density for x|R is the last two terms of the right hand side of (22), and the corresponding conditional MLE is addressed as follows.
Theorem 3: Let U, V ∈ SO(3) and S ∈ R 3×3 be the solution of the marginal MLE for R.
. . , N s . Also define the following sample covariance matrices: Then the solution of the conditional MLE for x and the correlation, namely P , µ, and Σ, is given by (28) Proof: Take the derivatives of (22) with respect to µ and P to obtain By setting the derivatives zero, the MLE of µ and P can be obtained as in (27) and (26). Next, take the derivative of (22) with respect to Σ −1 c to have Setting the derivative to zero and substitute the MLE of µ and P into the above equation, (28) can be obtained. The given marginal-conditional MLE is only an approximation to the joint MLE, because the information of U , S and V in x i is simply discarded. Intuitively, the correlation between x and vec(R T ) indicated by the samples is not necessarily constrained in the tangent space at U V T calculated fromĒ[R], but this is required by the MFG as suggested in Theorem 1. By analyzing the Fisher information matrix, if R is highly concentrated, x is widely dispersed, or the correlation between them is weak, the information of U , S and V is mostly contained in R i , which implies the marginalconditional MLE is close to the joint MLE.

IV. UNSCENTED ATTITUDE ESTIMATION WITH MATRIX FISHER-GAUSSIAN DISTRIBUTION
In this section, we design an unscented attitude estimator to concurrently estimate attitudes and gyroscope biases based on the proposed MFG. This is achieved by selecting a set of samples, referred to as sigma points, that characterize a MFG.

A. Sigma Points for MFG
In order to select sigma points from MFG, we first give a canonical form of MFG where the mean values are centered and all the correlations are eliminated. which finishes the proof. In the above canonical form, the Gaussian part is decoupled from the matrix Fisher part. As such, the sigma points of the canonical MFG is the union of the sigma points for the Gaussian distribution and those for the matrix Fisher distribution [5]. These can be transformed back to the original MFG according to the above theorem as follows.
Definition 3: Consider MG(µ, Σ, P, U, S, V ). Define the 7 + 2n sigma points for its canonical distribution as where 0 ≤ θ i ≤ π is chosen according to for i, j, k ∈ I. The weights for the first three pairs of sigma points are given by The weights for the next 2n sigma points, namely from the 7-th through (6 + 2n)-th sigma points are w G 2n , and the weight for the last one is The sigma points for MG(µ, Σ, P, U, S, V ) are defined as (R, x) i .
In others words, each pair of sigma points are designed to capture the dispersion along each principal axis of MFG. Parameters w M , w G and w 0 are used to adjust the weights, respectively for the attitude, the linear components and the sigma point at identity. This selection of sigma points are justified by the following theorem stating that the above sigma points recover the original MFG after the MLE described in the prior section.
Theorem 5: The marginal-conditional MLE of the sigma points given in Definition 3 are (µ, Σ, P, U, S, V ).
Proof: The marginal MLE for U , S and V are the same as the MLE of matrix Fisher distribution [11]. Sincē According to Theorem 3, these prove the MLE of P , µ and Σ are the same as the original MFG.

B. Unscented Uncertainty Propagation
We consider the following gyroscope noise model which is most frequently used in attitude estimation literature [3]: where Ω m , Ω ∈ R 3 denote the measured angular velocity, and the true angular velocity, respectively, and x ∈ R 3 is the gyro bias. The vectors η v , η u ∈ R 3 are white noises with In other words, the measured angular velocity is perturbed by a white noise with the isotropic strength of σ v > 0, as well as a time-varying bias modeled as a Wiener process with the isotropic strength of σ u > 0.
For a specific sampling period ∆t, the discretized gyroscope kinematics model is given by where the subscript k denotes the value of any variable at the k-th discrete time step. We assume the attitude and the gyroscope bias (R k , x k ) follow matrix Fisher-Gaussian distribution with parameters (µ k , Σ k , P k , U k , S k , V k ) and n = 3. We wish to find the distribution of (R k+1 , x k+1 ) and match it to a new MFG. This is accomplished by the following steps: 1) Select sigma points and weights {(R, x, w) i1 } 1≤i1≤13 from MG(µ k , Σ k , P k , U k , S k , V k ).
and calculate weights as w i1,i2,i3 = w i1 w i2 w i3 . 5) From {(R, x, w) i1,i2,i3 } 1≤i1≤13,1≤i2,i3≤7 , recover the parameters at k + 1 according to the marginalconditional MLE described in Section III-D. In short, this unscented uncertainty propagation scheme is simply to select sigma points from the MFG and two Gaussian noises, to propagate them through the gyroscope kinematics equations, and to recover the propagated MFG parameters from the MLE [17].

C. Measurement Update
Next, we consider how to update the propagated MFG when measurements are available. In this paper, we only deal with the case when attitude is directly measured or some vectors associated with attitude, such as magnetic or gravity field, are measured.
First, suppose the attitude is measured by N a attitude sensors as Z i ∈ SO(3), whose error is disturbed by a matrix Fisher distribution with the parameters F i ∈ R 3×3 for i = 1, . . . , N a . That is, given the true attitude R t ∈ SO(3), the measurement error R T t Z i ∈ SO(3) follows the matrix Fisher distribution with the parameter F i .
Next, suppose there are also N v reference vectors a j ∈ S 2 expressed in the inertial reference frame, which are measured by direction sensors in the body-fixed frame as z j ∈ S 2 for j = 1, . . . , N v . Furthermore, given the true attitude R t , assume the noisy measurement z j follows Von Mises Fisher distribution defined on S 2 [4] with mean direction R T t B j a j ∈ S 2 and concentration parameter κ j > 0. The parameter B j ∈ SO(3) specifies the constant bias of the direction sensor, and κ j specifies the concentration of its random noise.
If the prior distribution of (R, x) before measurement is MFG with parameters (µ, Σ, P, U, S, V ), then by Bayes' rule and Theorem III.2 in [5], the posterior density conditioned on measurements is where F , µ c and Σ c are defined as in Definition 2. The above posterior distribution of (R, x) is no longer MFG. But, we project it into a MFG by moment matching as in the common assumed density filters. We compute the necessary moments from the above posterior density, and match them to a MFG through MLE as follows.
and let its proper singular value decomposition be F = U S (V ) T . Also, let where (49) can be calculated using the second order moments of Q [11] analogously to (19). Proof: By direct integration over (41). From these moments, a updated MFG can be matched through the MLE in Section III-D. The complete procedures of this proposed unscented filter based on MFG are summarized at Table I. V. NUMERICAL SIMULATIONS In this section, the proposed unscented filter based on MFG is compared with the well-established MEKF in a very challenging simulation scenario.

A. Simulation Parameters
We consider a rotational motion where the roll, pitch, and yaw angles (body-fixed 3-2-1 Euler) follow sinusoidal functions with the amplitude of π, π/2, and π, respectively, and the frequency of 0.35 Hz. This results in the average angular velocity of 6.17 rad/s. The gyro bias is modeled as a Wiener process starting from [0.2, 0.2, 0.2] T rad/s. The attitude is assumed to be measured by a full attitude sensor. The sampling frequency of the gyroscope is 50Hz, and that of the attitude sensor is 10Hz.
The gyroscope angle random walk noise is set as σ v = 10 deg / √ s, and the bias random walk noise is σ u = 500 deg /h/ √ s. The parameters are larger than most commercial gyroscopes. However, we use this challenging setting to signify the potential linearization and wrapping issues associated with the MEKF. In addition, the attitude measurement noise is specified by the error rotation vector in MEKF. More specifically, the attitude measurement is where δθ ∼ N (0, σ 2 a I 3×3 ) with σ a = 0.2 rad. For the unscented filter based on MFG, at the beginning of each simulation, we calculate the empirical momentĒ[exp(δθ)] using a large number of independent samples from the distribution of δθ, and match this moment to a matrix Fisher Parameter F 1 at (41).
The initial attitude is set as the true attitude rotated by 180 • about the body-fixed e 1 -axis. The initial confidence in this guess is set as δθ t=0 ∼ N (0, σ 2 t=0 I 3×3 ), where σ 2 t=0 = 1/200 rad 2 . In other words, this represents a challenging case where each estimator is falsely too confident about a wrong attitude. The initial bias and its variance are chosen as [0, 0, 0] T rad/s and (0.1 rad/s) 2 , respectively.

B. Estimation Results and Discussions
Each numerical simulation is performed for sixty seconds, and it is repeated for sixty times with randomly generated sensor noises. The root mean squared errors of attitude and gyroscope bias averaged across sixty simulations are 8 12 Attitude RMSE (deg) shown at Fig. 2. It is shown that the attitude and gyroscope bias estimated from the proposed unscented filter based on MFG are more accurate than MEKF under this challenging simulation setting, which illustrates the effectiveness of the proposed unscented filter with MFG. The attitude and bias errors for one particular simulation are also presented at Fig. 3 for sixty seconds, and at Fig.4 for the first two seconds. First, the bias error for the proposed estimator is noticeably lower than MEKF throughout the simulation period. Next, for the attitude estimation error, the proposed estimator exhibits distinct behaviors over the initial transient response, as illustrated by Fig. 4 and 5. The accuracy benefit of the proposed unscented filter can be explained by the uncertainties of attitude estimates.

Accuracy comparison between MEFK and MFG Unscented
For the proposed unscented filter with MFG, the attitude estimate does not quickly shift toward the measured attitude for t ≤ 0.3, as there is more confidence in the initial estimate. However, due to the conflicts between the current belief and the measurements, the uncertainty in the estimated attitude grows rapidly. Later at t = 0.4, the attitude uncertainty becomes comparable to the uncertainties in attitude measurements, at which the posterior estimate becomes closer to the measured attitude with higher confidence. This explains the sudden decrease of the attitude estimator error at t = 0.4. Afterwards, as more measurements become available, the uncertainty in the attitude estimate decreases gradually.
On the other hand, the attitude uncertainty in MEKF is not affected by the large discrepancy between the prediction and the measurements, as the covariance update in the correction step is not relevant to the measurement in KF. Consequently, the attitude estimation error does not converge, until a large number of measurements become available to eliminate the impact of the false initial guess. Similarly, the uncertainty of the bias in the MFG unscented filter is larger than that in the MEKF, as shown in Fig. 5(b).
After the attitude and gyroscope bias converge, the performance of MFG unscented filter is almost identical to MEKF. This is because MEKF is developed for highly concentrated cases, and the proposed MFG behaves similarly as a Gaussian distribution for such cases.
VI. CONCLUSION In this paper, we have presented a new probability distribution, namely the matrix Fisher-Gaussian distribution on The attitude uncertainty is calculated as the square root of the first diagonal term of the attitude covariance matrix in the inertial frame, i.e. RΣR T for MEKF and U (tr(S)I 3×3 −S)U T for MFG unscented, where R is the attitude estimate and Σ is the covariance matrix for attitude in MEKF. The bias uncertainty for MEKF is given as the square root of the corresponding diagonal term in the covariance matrix, and for MFG unscented is given by the square root of the third diagonal term in cov(x, x) calculated using (15) and (17). SO(3) × R n by conditioning a (9 + n)-variate Gaussian distribution. MFG models the correlation between attitude and linear components in a similar fashion as the linear correlation in Euclidean space, and thus it provides an intuitive geometric interpretation. The unique feature is that large uncertainties in attitudes correlated with a linear variable of an arbitrary dimension are formulated globally on the manifold. We also design an unscented filter based on MFG for concurrent estimation of attitudes and gyroscope biases, and under some challenging simulation scenarios, it exhibits a higher accuracy over the well-known MEKF. The future work includes solving the stochastic differential equation with MFG for attitude estimation, and utilizing MFG for other estimation problems involving angular-linear correlations in robotics and controls.