A Secondary Particle Filter Photometric Data Inversion Method of Space Object Characteristics

Abstract

A secondary particle filter (SPF) inversion method for geostationary space object characteristics based on ground photometric data is presented. The method combines the estimation results of the standard particle filter (PF) algorithm and the resampling algorithm of the particle generation process. SPF first generates N particles according to the standard PF process, and performs the standard PF without resampling. Particle weight is an important indicator to determine the closeness of particles to the real state. With the progress of PF, the weight of particles closer to the real state will gradually increase. SPF takes the particle weight value as an important basis to judge the closeness of particles to the real state. By setting a threshold, the particles closest to the real state are screened out and roughened. The SPF method in this paper uses a particle filter twice and it is a new particle filter method. The first particle filter identifies particles near the real state. Before the second particle filter, it is equivalent to the actual state distribution of the system is known, so that the distribution of initial particles can be set more efficiently and effectively, and the number of particles close to the real state of the system can be increased. Experiment results show that the estimation error and the RMSE of the inversion error of SPF are less than PF, which not only shows that the inversion result based on SPF is better than the inversion result based on PF, but also proves the effectiveness of the inversion method based on SPF.

1. Introduction

Space domain awareness (SDA) is the premise and foundation of space control, which is of great significance to space security. As one of the means of SDA, the ground-based optical monitoring system can obtain the shape, size, position, attitude and other space object characteristics (SOC), which is very helpful to support SDA tasks [1,2]. However, for Geostationary Earth Orbit (GEO) space objects, due to the imaging resolution limitation of a telescope, only a few light spots with pixel brightness changes can be displayed in the view field of telescope [3]. Fortunately, since the photometric data of space object is a function of its SOC, the SOC can be reversed from photometric data of the space object through various Kalman filters and other estimation techniques.

At present, SOC inversion methods can be roughly divided into two categories: based on photometric curve features [3,4,5,6,7] and based on model construction [8,9,10,11,12,13]. If there is no prior knowledge of the space object, some SOC can be directly identified through the direct analysis of the photometric data characteristics. For example, we can inverse the platform type of the space object based on the overall characteristics of the photometric curve [3], the angular velocity and typical components of the space object based on the extreme value characteristics of the photometric data [5,6], or the spin period and working state of the space object based on the periodic characteristics of the photometric data [7]. The inversion method based on model construction improves the inversion accuracy by establishing a more complex model. This method quantitatively estimates the attitude, angular velocity, surface material and other SOC through the established shape model or photometric data model of the space object, such as the inversion methods based on two-panel model [8,9,10,11], closed facet model [12,13] and nonlinear filtering [14,15,16,17,18,19,20,21,22,23,24,25].

(1)

Based on two-panel model. Although the shape and motion of space objects are complex, most SOC can be ignored or integrated. One panel of the two-panel model represents the satellite solar panel, and the other represents the satellite body. The method requires that the surface of the satellite solar panel is approximate to the plane that has mirror and Lambert reflection characteristics. The satellite body is a complex three-dimensional shape with Lambert reflection characteristics, and has different observation attitude at each phase angle. The reflectivity, area and working state of GEO space object can be quickly inversed by this method.

(2)

Based on closed facet model. This method needs to establish a closed facet model, and requires each facet to have its own surface area, normal direction and surface scattering rule. This method can simultaneously identify the shape, rotation period, magnetic pole direction and scattering parameters of space objects through photometric data. It has good performance for the characteristic inversion of convex space objects; however, the identification results may not be unique for objects with other shapes.

(3)

Based on nonlinear filtering. The principle of this method is to estimate the characteristic parameters of the space object to be inversed as the unknown state parameters of the system. The precise phase angle data of the space object can update the orbit information near real-time, and the photometric data of the space object can provide its shape and attitude information; therefore, this method needs to fuse the phase angle data and photometric data. The shape, size, material, attitude and other SOC of space objects can be inversed from sparse photometric data, and the changes of these SOC can be detected.

Unscented Kalman filter (UKF) [24] and particle filter (PF) [25] are the main nonlinear filtering methods currently used for SOC inversion. As to particle filter, particle degradation is a prominent problem, efforts must be made to obtain better efficiency and effectiveness in state estimation [26]. Luca Martino used interacting parallel particle filters, each filter represents different model, according to the model’s posterior probability, the number of each filter’s particles can be adapted automatically [27]. Iman Askari used implicit particle filter to mitigate particle degeneracy according to the identification of particles in high-probability regions [28]. Reza Jalil Mozhdehi applied K-means clustering to assess the likelihood of the particles after the iterations, improving the utilization of prior distribution information [29].

In the calculation process of PF, the generation mode of such particles is blind, and the distribution of particles is unknown. The precision of PF depends on the closeness of particles to the real state to a certain extent. The closer the particles are, the higher the precision is. Therefore, another way to improve the precision of PF is to directly increase the number of particles to make particles closer to the real state.

However, for high-dimensional systems or systems with large uncertainties, this method requires a large number of particles to significantly improve the precision of PF, and may also introduce more outliers to the inversion results. How to generate initial particles and increase the number of particles more effectively has become an urgent problem to be solved, which can improve the precision of PF, and reduce the occurrence of outliers. The secondary particle filter (SPF) photometric data inversion method proposed in this paper can solve this problem well.

2. The SPF Method

We have a good foundation for this part. In the early stage, we established the satellite apparent magnitude model based on the Cook–Torrance bidirectional reflectance distribution function (BRDF) model. Besides, we have also established rotational kinematics, rotational dynamics and attitude inversion model of space object. Relevant work has been elaborated in the literature [30].

2.1. Standard PF Method

PF is a recursive Bayesian filter algorithm based on Monte Carlo methods, which represents the probability distribution of state through randomly selected particles. It is not constrained by the nonlinear conditions of the filter system, and is very suitable for state estimation in strong nonlinear systems.

Suppose that N independent particles $x^{(i)}$, $i=1,\dots ,N$ with the same distribution are extracted from the distribution function $p(x)\approx (1/N){\displaystyle {\sum }_{i=1}^N\delta (x-x^{(i)})}$, then the approximation of an expected value can be calculated by Equation (1) as follows:

$${\displaystyle \int f(x)}p(x)dx\approx \frac{1}{N}{\displaystyle \sum _{i=1}^{N}f(x^{(i)})}.$$

(1)

The ideal Monte Carlo sampling method assumes that the particles are directly sampled randomly from $p(x)$, and there are enough particles to represent the particle mode. When the number of particles tends to infinity, the above integral can approach the real state. Each particle corresponds to a weight $w^{(i)}\propto p(x^{(i)})$, which represents the probability of each particle state and ${\displaystyle \sum _{i=1}^N{w}^{(i)}=1}$.

However, it is difficult for PF to extract particles from $p(x)$ by using Monte Carlo method. It is relatively easy to extract particles from an important density function $q(x)$; therefore, the integral of Equation (1) is updated by Equation (2) as follows:

$${\displaystyle \int f(x)}p(x)dx={\displaystyle \int f(x)w(x)q(x)dx}.$$

(2)

where $w(x)=\frac{p(x)}{q(x)}$ is the importance weight.

In the sequential importance sampling PF, the posterior probability density function can be approximately expressed by Equation (3) as follows:

$$p(x_{0:k}|y_{1:k})\approx {\displaystyle \sum _{i=1}^N{w}_k^{(i)}\delta (x_{0:k}-x_{0:k}^i)}.$$

(3)

In the formula, $\delta (·)$ represents Dirac function, and the value is 1 when the independent variable is 0, otherwise the value is 0. When the number of particles is large, the above discrete weighted formula can be used to approximate the true posterior probability density function, and the particle set ${\left\{x_{0:k}^{(i)}\right\}}_{i=1}^N$ can be obtained from the important density function $q(x_{0:k}^{}|y_{1:k})$, and the weight of each particle is shown in Equation (4).

$$w_k^{(i)}\propto \frac{p(x_{0:k}^i|y_{1:k})}{q(x_{0:k}^i|y_{1:k})}.$$

(4)

According to Bayesian theorem, the posterior probability density function is shown in Equation (5).

$$p(x_{0:k}|y_{1:k})=\frac{p(y_k|x_{0:k},y_{1:k-1})p(x_{0:k}|y_{1:k-1})}{p(y_k|y_{1:k-1})} .$$

(5)

By the Markov property [31], it is further simplified by Equation (6) as follows:

$$\begin{gathered} p(x_{0:k}|y_{1:k})=\frac{p(y_k|x_k)p(x_k|x_{k-1})}{p(y_k|y_{1:k-1})}p(x_{0:k-1}|y_{1:k-1}) \\ \hspace{1em}\hspace{1em}\hspace{1em}\propto p(y_k|x_k)p(x_k|x_{k-1})p(x_{0:k-1}|y_{1:k-1}). \end{gathered}$$

(6)

The importance density function can be decomposed into Equation (7).

$$q(x_{0:k}|y_{1:k})=q(x_k|x_{0:k-1},y_{1:k})q(x_{0:k-1}|y_{1:k-1}).$$

(7)

Then, the importance weight formula is updated with Equation (8).

$$w_k^{(i)}\propto \frac{p(y_k|x_k^i)p(x_k^i|x_{k-1}^i)p(x_{0:k-1}^i|y_{1:k-1})}{q(x_k^i|x_{0:k-1}^i,y_{1:k})q(x_{0:k-1}^i|y_{1:k-1})}=w_{k-1}^{(i)}\frac{p(y_k|x_k^i)p(x_k^i|x_{k-1}^i)}{q(x_k^i|x_{0:k-1}^i,y_{1:k})}.$$

(8)

Through the above analysis, the standard PF includes four steps: prediction, update, resampling and regularization, and constitutes a loop. The PF algorithm process is as follows.

2.1.1. Prediction

The particles at $t_k$ and $t_{k+1}$, their weights are represented by $\left\{x_k^{(i)},w_k^{(i)}\right\}$ and $\left\{x_{k+1}^{(i)},w_{k+1}^{(i)}\right\}$, respectively. At $t_k$, the weight of particles remains unchanged when they are transferred according to the state equation. The particle column $\left\{x_{k+1}^{(i)},w_{k+1}^{(i)}\right\}$ at $t_{k+1}$ implies the predicted probability density function information, $w_{k+1}^{(i)}$ is the normalized particle weight, and $w_{k+1}^{(i)}\propto p(w_{k+1})$. The process noise and measurement noise are generally defaulted to zero mean white noise and are independent of the initial particle state. The state posterior density at $t_k$ is approximately expressed as follows:

$$p(x(t_k))\approx {\displaystyle \sum _{i=1}^N{w}^{(i)}\delta (x(t_k)-x^i(t_k))}.$$

(9)

2.1.2. Update

In the update phase, after the measured value of each particle is obtained, the particle weight is updated using the likelihood function according to Bayesian theorem, and $w_{k+1}^{(i)}=w_k^{(i)}p({\tilde{y}}_{k+1}|x_{k+1}^{(i)})$. If the measurement noise is additive, the likelihood function in the above formula is simplified as follows:

$$p({\tilde{y}}_{k+1}|x_{k+1}^{(i)})=p({\tilde{y}}_{k+1}-\mathrm{H}(x_{k+1}^{(i)})).$$

(10)

The normalized weight can be written as follows:

$$w_{k+1}^{(i)}=\frac{w_{k+1}^{(j)}}{{\displaystyle {\sum }_{j=1}^N{w}_{k+1}^{(j)}}}.$$

(11)

The estimated mean value obtained by PF is shown as follows:

$${\widehat{\mathrm{x}}}_{k+1}^{+}={\displaystyle \sum _{i=1}^N{w}_{k+1}^{(i)}{x}_{k+1}^i}.$$

(12)

The covariance is expressed as follows:

$$P_{k+1}={\displaystyle \sum _{i=1}^N{w}_{k+1}^{(i)}(x_{k+1}^i-}{\widehat{\mathrm{x}}}_{k+1}^{+}){(x_{k+1}^i-{\widehat{\mathrm{x}}}_{k+1}^{+})}^T.$$

(13)

2.1.3. Resample

In PF based on sequential importance sampling, particle degradation is inevitable. With the passage of time, the variance of particle weight increases continuously. Finally, only one particle weight cannot be ignored, and the weight of other particles is almost zero. Low weight particles consume a lot of time and have little meaning in the calculation process. In order to overcome this defect, the resampling method is introduced. The idea of resampling is to copy the high weight particles and discard the low weight particles by comparing the weights so as to finally get the children particles with the same weight. Resampling methods include systematic resampling, polynomial resampling and residual resampling. Because residual resampling has a small Monte Carlo variance and a small amount of computation, we chose residual resampling to resample the updated particles. The expectation of number of times of being resampled for each particle is $N·w_{k+1}^{(i)}$, the residual weight of each particle can be normalized as follows:

$$w_{k+1}^{(i)}=\frac{N{w}_{k+1}^{(i)}-⌊N{w}_{k+1}^{(i)}⌋}{N-{\displaystyle {\sum }_{j=1}^N⌊N{w}_{k+1}^{(j)}⌋}}.$$

(14)

where ⌊·⌋ is a rounding down symbol. After resampling, the particle column $\left\{x_{k+1}^{(i)},w_{k+1}^{(i)}\right\}$ updates to $\left\{x_{k+1}^{(i)},1/N\right\}$. More illustrative operations of residual resampling can be found in the literature [27].

2.1.4. Regularization

Copying high weight particles in the resampling process will produce a large number of identical particles, destroying the diversity of particles. Generally, particles need to be modified through regularization steps. Regularization is carried out to maintain the diversity of particles by adding a small noise disturbance to the resampled particles. Usually, an independent small amount is extracted from the Gaussian distribution and added to the same particles to increase the diversity.

The perturbed particles added in the regularization process obey Gaussian distribution $\mathbb{N}(0,h_{opt}^2{\sum }_{k+1})$. $h_{opt}$ is the adjustment parameter of Gaussian kernel and it can be calculated as follows:

$$h_{opt}={\left(\frac{4}{N(n_x+2)}\right)}^{\frac{1}{n_x+4}}.$$

(15)

where $n_x$ is the dimension of the state variable.

${\sum }_{k+1}$ is the covariance matrix related to state particles, and the calculation method of each quantity is shown in Equations (16) and (17).

$${\sum }_{k+1}=\frac{1}{N-1}{\displaystyle \sum _{i=1}^N{\tilde{\mathrm{x}}}_{k+1}^{(i)}{\tilde{\mathrm{x}}}_{k+1}^{(i)}{}^T}.$$

(16)

$${\tilde{\mathrm{x}}}_{k+1}^{(i)}=x_{k+1}^{(i)}-{\widehat{\mathrm{x}}}_{k+1}^{+}.$$

(17)

In the above formula, ${\widehat{\mathrm{x}}}_{k+1}^{+}$ is the mean value of the state estimation.

The above four steps essentially completely describe the process of standard PF. However, since the standard PF takes the prior density function $p(x_k^i|x_{k-1}^i)$ as an important density function $q(x_k^i|x_{k-1}^i,y_{1:k})$, it does not take into account the impact of the current measurement value. Particularly when the observation model is very accurate, the overlap area between the prior probability density function and the real posterior probability density function is very narrow, causing serious particle degradation, which ultimately leads to the filter failure when the number of particles is not high. Even if the number of particles is increased and a better result is obtained, the time consumed is uneconomical; therefore, PF needs to be improved in practical applications.

2.2. The SPF Method

According to the previous analysis, PF is the most accurate SOC inversion method. However, the particle distribution at the initial time in the calculation process of PF is unknown, and it is generally generated randomly, which is blind. The SPF inversion method proposed in this paper considers the tracking characteristics of PF, which can generate initial particles pertinently and increase the number of particles more effectively.

The SPF inversion method combines the estimation results of the standard PF algorithm and the resampling algorithm of the particle generation process. SPF first generates N particles according to the standard PF process, and performs the standard PF without resampling. Particle weight is an important indicator to determine the closeness of particles to the real state. With the progress of PF, the weight of particles closer to the real state will gradually increase. SPF takes the particle weight value as an important basis to judge the closeness of particles to the real state. By setting a threshold, the particles closest to the real state are screened out and roughened. The SPF method in this paper uses a particle filter twice and it is a new particle filter method. The first particle filter identifies particles near the real state. Before the second particle filter, it is equivalent to the actual state distribution of the system that is known, so that the distribution of initial particles can be set more efficiently and effectively, and the number of particles close to the real state of the system can be increased. The flow chart of SPF method is shown in Figure 1, and its specific steps can be described as follows.

• At the initial time, N particles are randomly generated according to the standard PF process;
• The particles are brought into the system dynamic equation to calculate the prior output value at the next time. According to the prior output of each particle and the actual output at the next time, the likelihood probability of each particle is calculated. The process executes for time 1~K;
• Arrange the weight of particles from the largest to the smallest, and select the first HW particles;
• Set the roughness coefficient RH which represents the degree of roughness (The larger the coefficient, the greater the degree of roughness) and the number of particles after roughening NSP, roughen HW particles with high weight, and generate NSP new particles;
• Take the new particle as the initial particle at the initial time, and implement the standard PF process as described in Section 2.1;
• Set SOC to be inverted as system state, and the ground based photometric data of the space object as system observation value. Then, the SOC is inverted according to the SPF process of 1–5.

The roughness method can be varied. In this paper, we use a roughness method similar to unscented transform in the calculation process. Suppose that a particle to be roughened is $\stackrel{⌢}{x}$, we can select 2n sigma points around the particles as follows:

$$\begin{array}{l}{\tilde{x}}_{}^{(i)}=\sqrt{i}·\stackrel{⌢}{x}\hspace{1em}\hspace{1em}i=1,\cdots ,n\\ {\tilde{x}}_{}^{(n+i)}=-\sqrt{i}·\stackrel{⌢}{x}\hspace{1em}i=1,\cdots ,n\end{array}$$

(18)

where, the value of n can be selected by the user according to the calculation cost.

The pseudo code of SPF is as shown in Algorithm 1.

Algorithm 1. Secondary_particle_filter $({\chi }_{t-1},u_t,z_t)$

${\overline{\chi }}_t={\chi }_t=\varnothing$

for n = 1 to N do

sample $x_t^{[n]}\sim p(x_t|u_t,x_{t-1}^{[n]})$

$w_t^{[n]}\sim p(z_t|x_{t-1}^{[n]})$

${\overline{\chi }}_t={\overline{\chi }}_t+\langle x_t^{[HW]},w_t^{[HW]}\rangle$

endfor

roughen HW particles to NSP particles

for m = 1 to NSP do

sample $x_t^{[m]}\sim p(x_t|u_t,x_{t-1}^{[m]})$

$w_t^{[m]}\sim p(z_t|x_{t-1}^{[m]})$

${\overline{\chi }}_t={\overline{\chi }}_t+\langle x_t^{[m]},w_t^{[m]}\rangle$

endfor

for m = 1 to NSP do

draw i with probability $\propto w_t^{[i]}$

add $x_t^{[i]}$ to ${\chi }_t$

endfor

return ${\chi }_t$

3. Results

The SPF inversion method is verified with scaling experiments with the scene of attitude inversion. The transmission process of light in scale experiment, the halogen lamp and parallel light tube used in the experiment can be seen in the literature [31]. The satellite model is shown in Figure 2. The size of the satellite model is 8 cm × 3.5 cm × 4 cm. The surface material is aluminized polyimide film and the monocrystalline silicon.

The experimental layout and the light source image in CCD are shown in Figure 3. The satellite model rotates on the turntable. The parameters in the experiment are shown in

Table 1. Parameters in the experiment.

Parameters in the Experiment
total pixel	1060 × 1040
effective pixel	1024 × 1024
cell size	6.45 μm × 6.45 μm
focal length of the camera	10 mm
viewing angle	23.7°
fixed exposure time	50 ms

.

Figure 4 shows the gray level changes under different rolling angles when the phase angle is 25° and the satellite rotates around the Y axis of the maximum inertia axis. The pitch angle and yaw angle are both 0°.

The simulation time is set to be from 22:00 to 23:00 on 25 November 2022, the observation station is set to Beijing and the satellite is set to Kompsat-2B. The phase angle change of the satellite during the observation period is calculated as the basis for the phase angle measurement of the scaling experiment. The initial attitude of the simulation is taken as the initial attitude of the scaling experiment. A part of the photometric measurement data is shown in Figure 5.

The number of particles is set to 5000. Figure 6 shows the comparison of the attitude inversion results of space object based on SPF and PF at different times. It can be seen that the estimation error of the SPF is less than that of PF, especially at the moment when the error of PF algorithm is large such as time 288, 290, 316, 318, etc., and the estimated value is closer to the real characteristic parameters of the space object, which shows that the SPF inversion method can effectively improve estimation accuracy. The improved accuracy of SPF compared with PF is shown in Figure 7.

Figure 8 shows the RMSE of the inversion error of SPF and PF filter after 30 experiments. During these simulations, the estimation in Equation (13) of the covariance matrix is all stable. It can be seen that the RMSE value of SPF is about 0.1 less than that of PF, which not only shows that the inversion result based on SPF is better than the inversion result based on PF, but also proves the effectiveness of the inversion method based on SPF.

4. Discussion

This paper provides a new inversion method based on SPF, which combines the estimation results of the standard PF algorithm and the resampling algorithm of the particle generation process. The first particle filter identifies particles near the real state. Before the second particle filter, it is equivalent to the actual state distribution of the system that is known, so that the distribution of initial particles can be set more efficiently and effectively, and the number of particles close to the real state of the system can be increased. Experiment results show that the estimation error and the RMSE of the inversion error of SPF is less than those of PF, which shows that the inversion result based on SPF is better than the inversion result based on PF. The SPF method can inverse SOC from photometric data and has strong universality, which can be integrated with other PF methods such as UPF. However, sometimes its results are difficult to converge, which is a problem that we need to further study and overcome in the future.