Accuracy evaluation of marginalized unscented Kalman filter

Firstly, the reason that there is an important gap between the filter error of MUKF and that of UKF for systems with differential state equations is analyzed and then the condition where the accuracy of MUKF is equivalent to that of UKF is given. In this paper, the following conditionally linear differential state equation is considered:

dx/dt = g(x) = d(x^nl) + G(x^nl)x^l

where x ∈ ℝⁿ denotes the n-dimensional state vector, x^l ∈ ℝⁿ_l is the conditionally linear states, x^nl ∈ ℝⁿ_nl is nonlinear states, and n = n_l + n_nl. g: ℝⁿ → ℝⁿ, d: ℝⁿ_nl → ℝⁿ, and G: ℝⁿ_nl → ℝ^n×n_l are known nonlinear functions.

For the conventional UKF, the following discrete-time state equation need to be known by analytical solution (if exists) or numerical integration:

However, since Eq. 2 is not conditionally linear in general if Eq. 2 is discretized from Eq. 1, the Euler method is required to propagate the states so that every small step of Euler method leads to a conditionally linear discrete-time. More specifically, the interval [t_k, t_k-1] is spilt into M ≥ 1 subintervals. The mean and covariance obtained by iterating UKF M times (multistep) are different from those directly obtained in 1-step propagation (Fig. 1). Since MUKF is a special case of multistep UKF when the numerical integration algorithm is the Euler method, the accuracy of MUKF is not equivalent to the conventional UKF. However, there is a special condition that can make the accuracy of MUKF equivalent to UKF. The condition is that the propagation of the states to be sampled and the propagation of the states not to be sampled are independent of each other. Therefore, the discretization of the 2 parts of states is independent of each other as well:

It can be seen that under this special condition, the discrete- time state equation is still conditionally linear so that there is no need to use multistep propagation any more. Therefore, MUKF will achieve the similar accuracy with UKF.

Fig. 1. Comparison of 1-step and multistep propagation.

Then, it is proven mathematically that for systems with measurement biases, the differences of the computed means and covariances between MUKF and UKF are the same order infinitesimal as the square of the scaling parameter. After considering the measurement biases, the state and measurement equations can be written as:

where x_k ∈ ℝⁿ is the state and b_k ∈ ℝⁿ_b is the measurement bias. For UKF, it can be derived that

For MUKF, it can be obtained that

Then,

Therefore, the 1-step predicted means obtained by UKF and MUKF have equivalent accuracy. Moreover, if sigma points are sampled only for nonlinear variables as MUKF does, the P^f_zz, P^f_yz, and P^f_xz computed by UKF and MUKF are in the equivalent accuracy as well.

Finally, a numerical simulation of MUKF and UKF on the systems with measurement biases is provided to verify the accuracy estimation. In this example, an orbit determination method based on geomagnetic field is introduced, which estimates the position and velocity through a 3-axis magnetometer. The state dynamics only considering Earth’s central force and the J₂ perturbation is adopted, considering other perturbation factors as system noises. The true orbit ephemerides are calculated by a high-precision numerical orbit simulator. The Adams-Bashforth-Moulton method is used for numerical integration. The perturbing forces of the true orbit include Earth’s nonspherical gravitational forces, atmospheric drag, and third- body attractions from the Sun and the Moon. To ensure the reliability of the results, 100 independent experiments are conducted based on the initial covariances, and the average errors and total computation time are considered as the final simulation results. The RMSE curves for each algorithm are illustrated in Figs. 3 to 5. It can be seen that the estimation error of MUKF is extremely close to that of UKF, which means that MUKF has the equivalent accuracy compared with UKF. In the meanwhile, he time cost of MUKF is remarkably less than UKF, and the ratio of the filtering time between MUKF and UKF (0.7032) is close to the ratio of the number of the sigma points between 2 algorithms (0.6842).

Fig. 3. Position estimation RMSE.

Fig. 4. Velocity estimation RMSE.

Fig. 5. Biases estimation RMSE.