1. Introduction
Attitude determination is a benchmark and important problem in astronautic engineering for the state estimation and control of spacecraft and robotic systems [
1,
2,
3,
4]. As a widely used attitude estimator, the Kalman filter (KF) is an optimal, numerically efficient, and widely used tool that infers based on all available information, i.e., the dynamic model of the system, sensor data, and the probabilities of both sensor signals and the algorithm’s numerical behavior [
5,
6]. Generally, the state estimation system is usually modeled as vector state space model in Euclidean space and solved with a Kalman filter for vector state variables. In recent years, building navigation and control systems on matrix Lie groups have been actively studied and the properties of the group system provide practical convenience for the definition and simplification of navigation and control system models [
7,
8]. For engineering applications, the Euclidean system models are usually corrupted by unknown noises or unpredictable disturbances; this is also true for the attitude estimation problems defined on matrix Lie groups and the performance of employed filtering methods is prone to be influenced by inaccurate covariance parameters [
6,
9,
10]. Therefore, for attitude estimation systems on matrix Lie groups, this work aims to study the Kalman filtering problem with unknown noise statistics parameters.
For attitude representation, the widely used quaternions can present a uniform approximation of attitude representation without gimbal-locks [
1,
9]. The quaternion-based algorithms do not need to compute trigonometric functions and allow for the writing of attitude changes as matrix–vector products. The multiplicative extended Kalman filter is an ad hoc modification of the extended Kalman Filter that accounts for the constraint of a unit quaternion, but its convergence property is ensured only at equilibrium points and the antipodal quaternions should be identified [
11]. Recently, there has been significant research interest in the estimation and control of matrix Lie group systems, including state observers and filters for attitude estimation on SO(3) and SE(3) [
12,
13]. Building attitude kinematics on matrix Lie groups can usually preserve the geometrical property of group systems and the resulting attitude representation does not suffer from the problems of singularities and unwinding [
11]. Nevertheless, these studies only generalize the extension to Euclidean Kalman filter theory for attitude estimation on matrix Lie groups and the symmetry properties of attitude estimation models are not sufficiently exploited.
Accounting for the invariance property of Lie groups’ attitude kinematics will lead to well-posed problems and boost the performance of attitude estimation algorithms [
14]. According to the theory of the invariant Kalman filter (IKF), using the mapping between matrix Lie groups and Lie algebra, attitude estimation systems on matrix Lie groups can be equivalently mapped into a Euclidean vector space and, therefore, the classical Kalman-type estimation methods can be applied to solve the corresponding problems [
15,
16]. The implementation steps of the IKF are derived based on the invariance and log-linearity of the linear invariant error, which contributes to better filtering convergence on a much bigger set of state trajectories [
16]. The necessary probability theory for uncertainty definition on matrix Lie groups has been studied in [
17,
18] and the concept of a
concentrated Gaussian distribution has been employed to describe the noisy and random variables on matrix Lie groups [
19,
20]. Note that the IKF actually obeys the classical Kalman theory and so its estimation performance also heavily depends on the parameters of the noise covariance matrices being accurate; however, in aerospace and satellite engineering, accurate noise statistics parameters are usually not available due to the presence of unpredictable noises and disturbances, which are sure to have a negative influence on the estimation performance of the invariant Kalman filter [
21,
22].
Motivated by above discussion, it is meaningful to further improve the filtering performance of the invariant Kalman filter for the matrix Lie group attitude estimation problem in the presence of unpredictable noises and disturbances. Note that, for conventional Euclidean space filtering problems, some adaptive techniques have already been developed [
23,
24] and the most common way is to directly scale or estimate the unknown noise parameters, i.e., the process and observation noise covariance [
25]. Recently, variational Bayesian (VB) methods have been applied to the joint estimation of system state and unknown noise parameters [
26,
27]. A novel adaptive Kalman filter utilizes the VB technique to obtain an approximate inference for inaccurate process and measurement noise covariance [
28]. Note that, to the best of our knowledge, for matrix Lie group systems there remains no investigation on adaptive methods for invariant Kalman filtering.
Therefore, for aerospace, satellite, and robotics engineering, this work aims to investigate the benchmark matrix Lie group attitude estimation problem. Note that, in practice, the noise parameter of onboard observation sensors usually depends on the utilized sensing technique and can be determined offline, but it is impractical to accurately evaluate the unpredictable disturbances in the kinematics/dynamics model [
23,
24]. In this work, the matrix Lie group attitude estimation problem with inaccurate process noise covariance is investigated and a variational Bayesian iteration-based adaptive invariant Kalman filter (VBIKF) is proposed. In the VBIKF, the a priori error covariance is not propagated by the conventional steps but directly calibrated in an iterative manner based on the posterior sequences. The main advantage is that the statistics parameter of the system process noise is no longer required and so the IKF’s hard dependency on accurate process noise statistics can be reduced significantly. The mathematical foundation for the new VBIKF is presented and its superior performance in adaptability and simplicity is further demonstrated by numerical simulations.
3. Variational Iteration-Based Invariant Kalman Filter for Attitude Estimation
In the invariant Kalman filter, given the prior error covariance matrix
, the predicted probability density function (PDF)
and the likelihood probability density function
are assumed to be a
concentrated Gaussian distribution, i.e.,
where the prior estimate
is obtained with (12) and the prior error covariance
is propagated through the calculus (13) with
required. Obviously, if the true value of the process noise covariance
cannot be accurately determined in advance, the
is sure to be misled by an incorrect
, which will in turn degrade the estimation results of the IKF.
3.1. Distribution Definition for the Prior Error Covariance
To deal with the trouble caused by an inaccurate , in this work we choose to infer the rotation group state together with the prior error covariance . The basic idea is that, before calculating the required PDF , the probability density function of the prior error covariance should first be calculated based on all the historical observation sequences, i.e., by calculating the probability density function .
As to the definition of the conjugate prior distribution PDF
for
, in this work we define it as the inverse Winshart distribution, which has been commonly employed as the conjugate for the prior covariance matrix of Gaussian distributions with known mean [
28]. The inverse Winshart probability density function of a symmetric positive definite
random matrix
is formulated as:
where
is the inverse Winshart distribution;
denotes the degrees of freedom (dof) for the inverse Winshart distribution;
denotes the inverse scale matrix, which should be a
symmetric and positive definite matrix;
is the matrix determinant operation; and
denotes the
d-variate gamma function [
28]. For covariance matrix
with an inverse Winshart distribution, i.e.,
, the expectation of the matrix inverse
is:
Therefore, to infer the prior error covariance
, the following inverse Winshart distribution is employed in this work to describe the distribution of
.
where
are respectively the degree of freedom parameter and the inverse scale matrix parameter for the PDF
that needs to be determined.
3.2. Variational Bayesian Approximations of Posterior PDF
To infer the rotation group state
together with the prior error covariance
, the joint posterior probability density function
should be calculated. In this work, we use the variational Bayesian method to obtain an approximation factored from [
33], i.e.,
where
denotes the approximate posterior probability density function
; and
and
can be determined by minimizing the Kullback–Leibler divergence (KLD) [
28,
33] between
and
, i.e.,
where
denotes the KLD between
and
.
The optimal solution to (23) can be obtained based on the following equations:
where
denotes the mathematical expectation only for the variable
and
denotes the mathematical expectation only for the variable
; and
are the constants with respect to
and
, respectively. Note that fixed-point iterations are needed in order to solve above two coupled equations; the approximation to the posterior PDF
is updated as
at the
i+1th iteration using the approximate PDF
, while the approximate
is updated as
at the
i+1th iteration using the
[
33].
The joint PDF
can be factored based on the conditional independence properties as follows:
Note that the PDF
is actually the distribution of the noisy observation
conditioned on the true rotation state
and, according to (7) and (14), we have:
Besides,
is actually the prior distribution of the rotation group state
; then, according to the equivalence of
and
in the probability distribution, we have:
Then, using (24) and (25) in (28), we have:
where
is the iterative approximation of PDF
at the
i+1th iteration;
represents the mathematical expectation at the
ith iteration; and
is given by:
where
is the correction term at the
i+1th iteration and will be calculated later. Note that, since the PDF of
is considered to be an inverse Wishart distribution, the updated PDF
in (29) with the updated
and
can be written as:
Then, according to (20), the
in (30) can be calculated as:
3.3. The Variational Bayesian Iteration-Based Invariant Kalman Filter
Define the propagated
at the
i+1th iteration as:
then the propagated PDF
at the
i+1th iteration can be written as:
Then, using (27) and (36)~(38) in (30) yields:
According to the above equations, the
can be updated as a Gaussian PDF with the mean being
and the covariance being
, i.e.,
and the propagated PDF
at the
i+1th iteration can be updated as the
concentrated Gaussian distribution with the mean being
and the covariance being
, i.e.,
where the mean
and the covariance
at the
i+1th iteration are calculated as:
Additionally, according to the definition of
invariant error , the correction term
that is required in (31) can be calculated as follows:
Then, for
N iterations, the variational Bayesian approximation of posterior PDFs is:
Therefore, the filtering steps of proposed approach include the prediction step (12), the initialization of the
and
for inverse Wishart distribution, and the variational Bayesian iteration steps of (31)~(35), (37), and (43)~(48). The details of the implementation of the proposed method for attitude estimation is presented in Algorithm 1.
Algorithm 1. The filtering steps of one time instant in the proposed approach to attitude estimation. |
Inputs: , , , , , , d = 3, , |
Time update: |
1: |
2: |
Measurement update: |
3: Initialization: , , , , |
for i from 0 to N−1 |
update given : |
4: , , |
update given : |
5: , |
6: |
7: , |
8: |
end for |
9: , , , , |
Outputs: , , , |
3.4. Parameter Selection for the Proposed Approach to Attitude Estimation on SO(3)
For the attitude estimation problem, the matrix Lie group system (6) and (7) for rotation
is converted to the Euclidean space system (11) and (14) for error
; therefore, according to the equivalence of
and
in the probability distribution, the propagation of the filtering parameters for rotation
mimics that of the classical Kalman filter for error
. Note that the projected Euclidean space system (3) and (7) for error
is actually a linear time-invariant system, for which the optimal Kalman filtering parameters, including the prior error covariance
and the Kalman gain, would converge to constants [
14,
22,
24]. Similarly, for attitude estimation the parameters
and the Kalman gain of the invariant Kalman filter will also converge to their optimal values, which has been validated in [
14] and can be used to help simplify the parameter selection for the proposed approach. Therefore, to initialize the prior error covariance
, the covariance
of the last time instant can be used as the initial estimate
before the iteration so that the negative influence caused by the inaccurate covariance
can be considerably reduced.
In the conventional invariant Kalman filter, the prior error estimate
is actually based on all the historical observations
before time instant
k. In this work, the covariance parameter
is also inferred using the historical observations
until time instant
k. As to the inverse Wishart distribution of
, at time instant
k the dof parameter is set as
and the inverse scale matrix parameter is set as
because the following conditions should be satisfied according to (19) and (20):
The number of iterations N is also a crucial parameter for the proposed approach and, generally, it should be set to a value larger than d to guarantee the convergence of variational Bayesian iterations; nevertheless, a value that is too large is sure to increase the computational cost of the algorithm’s implementation and so a balance between precision and cost should be considered according to the particular application.
Note that, for invariant Kalman filtering without an accurate process noise covariance, it is impossible to directly obtain the optimal estimates. In the related work [
21,
22,
23,
24,
25,
26,
27], some suboptimal approximations and assumptions were used to asymptotically approach the optimal results. In this sense, the proposed variational iteration-based invariant Kalman filter is actually a suboptimal approach for the following reasons:
- (1)
In the parameter setting shown in Algorithm 1, the initialization of the parameter
at the
kth time instant is based on the estimate
of the last time instant, i.e.,
; the advantage of this setting is that usage of an inaccurate
can be avoided, but the validity of the parameter
actually assumes that the filtering remains around its steady state. A similar usage can be found in [
21,
23,
24,
25].
- (2)
The variational Bayesian iteration method is based on fixed-point iterations that are only guaranteed to converge to a local optimum [
28], and iteratively updating steps are employed to reduce the negative influence caused by an inaccurate covariance parameter. A similar usage can be found in [
26,
27].
- (3)
The precision and convergence performance of the proposed approach can be further improved by regulating the filtering process into a steady state; for example, using a larger for the first few time instants of the filtering process to initialize the of the proposed approach will contribute to better results.
4. Numerical Simulations
To further demonstrate the performance of the variational Bayesian iteration-based invariant Kaman filter, the attitude estimation system (6) and (7) was simulated with
The real attitude trajectories were generated with the true
and the number of total time steps was set to 10,000 s. Note that, according to (11), the control parameter
is independent of the error propagation process; in this work, it was set to a time-dependent random rotation matrix. To study the filtering adaptability to
of various qualities, the filtering of the proposed approach (denoted VBAIKF), the IKF [
14,
16], and the QeIKF [
23,
24] was conducted using an inaccurate
with
ranging from 1 to 10. To implement the VBIKF and the QeIKF, the filtering instants before
k = 8 were initialized using the IKF with an inaccurate
and, for this attitude model, at time step 8 the covariance parameters of the IKF gradually converge. The error variable in the Lie algebra of 5000 random simulations was employed to evaluate the root mean square error
during the filtering processes and the average root mean square error
, i.e.,
where
denotes the true estimate at the
k-th time instant of the
l-th simulation run and
is the corresponding estimate for the true
; and
denotes the Euclidean vector norm.
- (1)
Although for
(i.e.,
= 1) the precision of the VBIKF is not optimal, the
ARMSE value of the VBIKF (0.0356) is only slightly inferior to that of the IKF (0.0353) and better than that of the QeIKF (0.0359) as shown in
Figure 1 and
Table 1;
- (2)
For all cases of the biased with 1, the presented ARMSE and data clearly demonstrate that the proposed VBIKF not only shows better filtering precision than the QeIKF but its filtering stability is obviously superior to that of QeIKF;
- (3)
Note that, for the biased with different , although the ARMSE of the proposed VBIKF is still influenced to some extent (i.e., the higher ARMSE value 0.0376 for = 10), the negative influence caused by the inaccurate is significantly reduced compared with and smaller than the 0.0422 of the QeIKF and the 0.443 of the IKF;
- (4)
The respective errors of three elements of
with the corresponding 3
boundary for the VBIKF are presented in
Figure 8 using the scaled
with
= 1, 2, 4, 6, 8, and 10, which clearly shows that most of the time estimation errors would fall within the 3
boundary.
- (5)
As to the computational cost, the usage of extra fixed-point iterations introduces a longer running time than that of the conventional methods. For example, in this work the iteration number N was set to 8 and the average running time was about 6 times that of the conventional IKF. Obviously, an N that is too large is sure to increase the computational cost of the algorithm’s implementation and so a balance between precision and cost should be considered according to the particular application.
As to the
ARMSE result displayed in
Figure 1 and
Table 1, if the scale parameter
is close or equal to 1 (
= 1), the employed covariance
is close to its true
. Then, the
ARMSE value of the standard IKF is the lowest among the three methods and its performance is optimal in the least square sense, which is also certified by the data on
displayed in
Figure 2. However, when
becomes larger (for instance,
2), the accuracy of the employed
is biased by the scaling parameter
and the
ARMSE value of the IKF increases significantly, which means that the optimal estimation performance of the IKF is destroyed by the inaccurate
. We can conclude that the performance of the IKF is rather sensitive to the accuracy of
and, for cases of an inaccurate
, adaptive methods are necessary to improve the performance of attitude estimation.
Note that the
ARMSE result of the QeIKF shows some adaptability to the inaccurate
with
, which can also be verified by the data on
in
Figure 4,
Figure 5,
Figure 6 and
Figure 7; however, the estimation precision is still degraded by the inaccurate parameter and its effective range is limited. For the cases of
, the estimation precision is still degraded by the inaccurate parameter and its effective range is rather small and limited. Moreover, as the
data show in
Figure 5,
Figure 6 and
Figure 7, the process data on the QeIKF show some instability during the initializing stage; in fact, the critical issue with the QeIKF is that its filtering is not stable and rather prone to divergence [
28]. In conclusion, the filtering precision and stability of the available QeIKF could not effectively reduce the influence of the inaccurate
.