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Space: Science & Technology / 2022 / Article

Research Article | Open Access

Volume 2022 |Article ID 9796015 | https://doi.org/10.34133/2022/9796015

Yang Mo, Yaonan Wang, Hong Yang, Badong Chen, Hui Li, Zhihong Jiang, "Generalized Maximum Correntropy Kalman Filter for Target Tracking in TianGong-2 Space Laboratory", Space: Science & Technology, vol. 2022, Article ID 9796015, 15 pages, 2022. https://doi.org/10.34133/2022/9796015

Generalized Maximum Correntropy Kalman Filter for Target Tracking in TianGong-2 Space Laboratory

Received31 Aug 2021
Accepted16 Mar 2022
Published06 Apr 2022

Abstract

Target tracking plays an important role in the construction, operation, and maintenance of the space station by the robot, which puts forward high requirements on the accuracy of target tracking. However, the special space environment may cause complex non-Gaussian noise in target tracking data. And the performance of traditional Kalman Filter will deteriorate seriously when the error signals are non-Gaussian, which may lead to mission failure. In the paper, a novel Kalman Filter algorithm with Generalized Maximum Correntropy Criterion (GMCKF) is proposed to improve the tracking accuracy with non-Gaussian noise. The GMCKF algorithm, which replaces the default Gaussian kernel with the generalized Gaussian density function as kernel, can adapt to multi-type non-Gaussian noises and evaluate the noise accurately. A parameter automatic selection algorithm is proposed to determine the shape parameter of GMCKF algorithm, which helps the GMCKF algorithm achieve better performance for complex non-Gaussian noise. The performance of the proposed algorithm has been evaluated by simulations and the ground experiments. Then, the algorithm has been applied in the maintenance experiments in TianGong-2 space laboratory of China. The results validated the feasibility of the proposed method with the target tracking precision improved significantly in complex non-Gaussian environment.

1. Introduction

The space station is a bridgehead for human space exploration missions [1]. In the process of its construction, operation, and maintenance, there are a variety of tasks to be carried out [2]. However, the space environment has severe conditions such as microgravity, high vacuum, strong radiation, and large temperature differences, which seriously threaten the health and life safety of astronauts [3]. Since space robots are not restricted by human physiological conditions, they can carry out space exploration tasks with high quality for a long time [4, 5], which has become an important trend for space exploration [6, 7].

During space station missions, robots need to perform precise operations on tools or task objects, which relies on accurate tracking of the target [8, 9]. However, the light intensity in the shadow area and the light area changes dramatically due to the special space environment, and the thermal insulation coating of the space device will also form stray light, which seriously interferes with the target characteristic information [10]; high-energy particles in space will impact the process of image acquisition and transmission and cause complex interference. These comprehensive factors lead to the existence of complex non-Gaussian noise such as heavy tail noise and light tail noise in target tracking data, which brings challenges to accurate target tracking [11]. It is necessary to carry out research on filtering algorithms for complex non-Gaussian noises [12].

In order to solve the target tracking problem in Gaussian noise, R.E. Kalman proposed the Kalman Filter (KF) algorithm based on Bayesian theorem and recursive iteration [13], which has been widely used in engineering applications because of its outstanding performance. Even the trajectory of manned spacecraft in Apollo moon mission is estimated by KF algorithm [14]. However, Kalman Filter and their extensions, which are based on minimum mean square error (MMSE) criterion, may be sensitive to large outliers under non-Gaussian noise. Posterior estimate of system state may exist large error and result in deterioration of the robustness.

Researchers have conducted a series of studies to improve the Kalman Filter’s performance under non-Gaussian noise. To describe non-Gaussian noise, glint noise model [15], Gaussian mixture noise model [16], and colored heavy-tail noise models (e.g., Student’s T distribution [17], Gamma distribution [18], and Rayleigh distribution [19]) were proposed. A series of algorithms based on Huber function are proposed. Chang and Liu proposed M-estimator-based Robust Kalman Filter [20], which used Huber estimator in the recast linear regression to enhance the robustness of the Kalman Filter. Robust derivative unscented Kalman Filter (RDUKF) is proposed [21], which is sensitive to the outliers and has better performance under non-Gaussian noise. Chang used the Mahalanobis distance as outlier judging criterion and proposed a Robust Kalman Filter [22]. By rescaling the covariance of the observation, the effect of the outliers and the heavy-tailed noise was effectively resisted. A series of algorithms based on Student’s t distribution have also been proposed. Roth considered the filtering problem with heavy-tailed process and measurement noise and proposed Student’s t filter, which performs well under model mismatch [17]. Huang et al. used cubature rule (CR) to solve Student’s t integral and designed robust Student’s t-based stochastic cubature filter (RSTCKF) [23]. Based on Gaussian mixture distributions, Robust Kalman Filter (RKF) [24] is proposed to deal with the maneuvering target tracking problems. Based on the Monte Carlo analysis and the mix of multifilters, some other filters have been proposed to process non-Gaussian noise. In [25], the EKF algorithm is used to propose the prior value, and the PF algorithm is utilized to process the non-Gaussian noise. However, it may take much time to get good performance.

The Maximum Correntropy Criterion (MCC) [26] has good effect in evaluating non-Gaussian noise. There are many filters based on MCC that have been proposed [2730], which can obtain the error’s higher-order moments and filter the outliers effectively. They have made significant progress in solving non-Gaussian noise. The default kernel of MCC is Gaussian kernel. However, due to the limit of the Gaussian function, it cannot change the shape of correntropy freely to deal with various kinds of non-Gaussian noise. Therefore, the filter like the Maximum Correntropy Kalman Filter (MCKF) could not achieve the good performance under some complex non-Gaussian noise. The paper’s research emphasis is to propose a new kind of filter to realize better performance under the non-Gaussian noise with various forms that may encounter in the process of target tracking.

The Generalized Maximum Correntropy Criterion (GMCC) [31] utilizing the Generalized Gaussian Density (GGD) function replaces the traditional Gaussian density function as a kernel function has a more flexible form, and it would be a better selection. In this paper, a novel Kalman Filter named Generalized Maximum Correntropy Kalman Filter (GMCKF), which uses GMCC as evaluation criteria, was proposed to deal with optimal filtering problem under various types of non-Gaussian noise. And a kind of parameter automatic selection method was used to choose better shape parameter for GMCKF. The performance of the proposed algorithm is verified through simulations. Then, the algorithm is applied in the target tracking experiments both on the ground experiments and the data from the maintenance experiments in TianGong-2 space laboratory of China. Experiment results demonstrate that the tracking accuracy has been improved significantly.

The paper remaining is organized as follows: Section 2 briefly introduces the Generalized Correntropy and analyzes the difference between Gaussian kernel and Generalized Gaussian Density. The GMCKF algorithm is derived in Section 3. Section 4 presents how to select the shape parameter automatically. Section 5 studies the influence of shape parameter and verifies the validity of the proposed algorithm. The GMCKF algorithm is used to track the target in Section 6. The conclusions are presented in Section 7.

2. Generalized Correntropy

Suppose two random variables obey joint distribution function . Then, the generalized similarity measure (correntropy) between and can be defined as follows:

In (1), stands for the expectation operator, and is a shift-invariant Mercer kernel.

The default kernel function of correntropy is Gaussian kernel in general. The kernel’s expression is as follows: where is the error, is the standard deviation called kernel bandwidth, and is the normalization constant.

Due to the limit of Gaussian kernel, we can only adjust the kernel function by changing the bandwidth, which limits its application scope. As the expansion of the Gaussian function, GGD with zero-mean is given by [32], which is defined as follows: in which can be written as: where is the error, is the shape parameter which describes the rate of decay, is the scale parameter which models the dispersion of the distribution, is standard deviation called the kernel bandwidth, denotes the Gamma function, and is the normalization constant.

Remark 1. As shown in Figure 1, the GGD function is equivalent to the Gaussian distribution with , and is equivalent to Laplace distribution with , converges to uniform distribution with . Especially, is equal to when , and the GDD function is exactly the same as the Gaussian function.

Compared to Gaussian kernel, GGD function adds a parameter to change the shape of its Probability Distribution Function (PDF), so it can adjust to more noise distributions to get better performance.

The GGD function can be used as the kernel function of correntropy, that is:

In most practical applications, we cannot get the joint distribution , only a limited amount of data are available. We can use samples to estimate the generalized correntropy: where ; is the sample of .

3. Kalman Filter under GMCC

3.1. Derivation of the Algorithm

Kalman Filter can be used as an effective tool to estimate the linear systems’ states. However, under non-Gaussian noises, the performance of Kalman Filter may decrease significantly, especially for heavy-tailed impulse noise or light-tailed noise. To deal with the non-Gaussian noise, a new algorithm based on GMCC and the framework of Kalman Filter was derived in the paper.

Consider the linear and time-invariant system as follows: where is the discrete computing time, is the state vector, is the state transition matrix, is the measurement vector, is the measurement matrix, and are supposed to be time invariant. is the process noise with zero mean and nominal covariance matrix , is the non-Gaussian measurement noise with zero mean and nominal covariance matrix . In the paper, , , and are assumed to be mutually uncorrelated.

The prior estimate of can be derived from

Based on (7) and (8), we can construct model like where is the identity matrix; denotes

We can get the expectation

The variable can be get based on Cholesky decomposition of . The prior covariance matrix can be calculated as

Left multiplying both sides of (10) by , we get where

Since , the residual error of is white.

Based on (6), the cost function can be got as follows: where denotes the -th item of and where denotes the -th item of , denotes the -th row of , and denotes the dimension of .

According to the GMCC, when is maximum, can reach the optimal estimate.

The optimal estimate of can be derived by derivation method.

Using (18) and (21), we can get the following equation:

Let

Then, (22) can be represented as:

We can get the expression of . where with

Because can be presented as where and .

Using (16), (26), and (29), we can get:

With variables substitution as follows:

We can get

In a similar way, we can get

Combining (25), (32), and (33), we can get where

The optimal solution of is actually the solution of the equation with

Equation (38) can be solved by a fixed-point iterative algorithm. where represents the solution of the fixed-point at iteration of time .

Compare the value of the adjacent fixed-point iteration step; if the calculated value satisfies (41), set ; otherwise, set and continue the iteration;

Finally, update the posterior covariance matrix by (42).

The proposed Generalized Maximum Correntropy Kalman Filter consists of (9), (13), (15), (16), (19), (23), (27), (28), (34)–(37), and (42) and the implementation pseudocode for one step is shown in Algorithm 1.

Input: shape parameter ; kernel bandwidth ; threshold for fixed-point iterative algorithm ; initial states estimate; initial covariance matrix ; computing time .
Output: state estimate at next step ; posterior covariance matrix at next step
Predict:
1. Calculate the prior estimate state
2. Calculate the prior estimate and covariance matrix
Update:
3. Calculate using Cholesky decomposition
4.
5.
6. Iteration initialization
,
7. do
8. 
9. 
10. 
11. 
12. 
13. 
14. 
15. 
16. 
17. while
Obtain result:
18.
19.

Unlike traditional Kalman Filters, GMCKF updates the posterior states estimate using fixed-point algorithm under the evaluation criteria of GMCC. Because the initial value of fixed-point iteration algorithm is derived from prior estimation, its convergence rate will be very fast.

Both the shape parameter and the scale parameter have a significant impact on the performance of the GMCKF. Because can be calculated by and the kernel bandwidth using (4), GMCKF actually only adds a parameter compared with MCKF.

3.2. Computational Complexity

To run the GMCKF algorithm in real time with limited onboard computer resources, the complexity of the algorithm should be evaluated. In this subsection, we analyze the computational complexity based on the floating-point operations of the proposed algorithm. Table 1 gives the computational complexity of some basic equations. The other operation in the table means division, matrix inversion, Cholesky decomposition and exponentiation, and so on.


EquationAddition and multiplicationOther operations

(9)
(13)
(15)
(16)
(20)2n
(23)
(27)
(28)2nmm
(34)4nm
(35)
(36)
(37)
(42)

Due to the fixed-point iteration number is relatively small in general, the computational complexity of the GMCKF is controllable.

The average fixed-pointed iteration number in the algorithm is set to . According to Table 1, the computational complexity of GMCKF can be calculated as:

As a comparison, the computational complexity of Kalman Filter can be calculated as:

4. Parameter Automatic Selection

In contrast to the MCKF algorithm, GMCKF algorithm replaces Gaussian kernel with GGD function which adds a new shape parameter to get better performance under various non-Gaussian noises. This section will describe how to choose a proper parameter .

According to the opinion mentioned in the book [33], the kernel filter can get better performance when the function of the kernel is similar to the PDF of noise. Based on the above theory, this paper uses the GDD to estimate the distribution of noise to get the parameter of GMCKF.

There are many methods to estimate the parameters of a GGD distribution [34]. In this paper, we choose the moment-matching estimators (MMEs) to estimate the parameter because it is simple and accurate enough for parameters estimation. Mallat [35] first proposed it to simply match the data set’s moments with the assumed distribution to obtained the statistical description of GGD distributions.

The r-order absolute central moment of a GGD distribution can be calculated as: where is the average value of data , is the GGD distribution, and and are the shape parameter and scale parameter of GGD distribution, respectively.

According to the kurtosis of the GGD distribution [36], the shape parameter can be estimated. A similar ratio is shown in (46), and it is equal to the reciprocal of the square root of kurtosis of the GGD distribution.

For independent identically distributed random samples , which is waiting for processing, the estimate of is where denotes the inverse function of ; , , and are the average, sample standard deviation, and fourth-order central moment of the samples, respectively.

To estimate the shape parameter , the inverse function needs to be solved. Because the inverse function has no explicit forms, cubic Hermite interpolation method was used in practical applications.

Due to the significant influence of outlier for parameter estimation, we need to process the data to remove some of the outliers. Comparing with the mean , the farthest 10 percent of data are removed to enhance the estimation precision. In addition, when , a small increase in will cause a sharp increase in . Therefore, the upper bound of is set to 10 to avoid very large gradient. If , we set , and its influence will be discussed in Section 5.

In practical application, the historical data of states (more than 1000 points) are collected. Then, the MMEs method will be used to estimate the shape parameter of GMCKF. Because the percentage of outliers has significant influence on the calculation, the scale parameters or kernel bandwidth are usually set manually or optimized by trial-and-error methods, just like what has been done in the MCKF algorithm.

5. Simulations for Performance Evaluation

The influence of the additional shape parameter and the validity of the parameter automatic selection algorithm for GMCKF have been studied in the simulations before the experiments for target tracking. Because the influence of the kernel bandwidth and the stop threshold of fixed-point iteration have been fully validated in the MCKF algorithms [27], they will not be covered here.

5.1. Simulation Setting

Analyzing the process that vector rotates around the origin of coordinates at the speed of , the system state equation is shown as follows: where the sampling period , angular velocity , and and are uncorrelated process error of state vectors and at instant with zero mean, respectively.

Set its observation equation as: where is observation noise at instant which is uncorrelated with process noise.

To validate the filter’s performance in various non-Gaussian noises, the observation noise model can be set as where is a binary independent and identically distributes process which obeys the Bernoulli distribution with parameter :

In (51), denotes a noise process with smaller variance which occupies the majority, and denotes a noise process with larger variance to describe the large outliers that may occur during the process.

Based on the observation noise model proposed in (51), the filtering performance of the state vectors and in the Kalman Filter (KF), MCKF, Measurement-Specific Correntropy Filter (MSCF) [37], and a novel robust Student’s t-based Kalman Filter (NRSKF) [38] and the proposed GMCKF algorithm are tested in the simulations.

In the following simulations, c is set at 0.1, and obeys the Gaussian distribution with zero mean and variance 100. is set to obey several different noise distributions to verify the filtering effect under different noise conditions. We consider three distributions: (a): Gaussian distribution with zero mean and variance 0.01(b): Laplace distribution with zero mean and variance 1(c): Uniform distribution over

The kernel bandwidth of MCKF in the simulations is set at the value in which the MCKF algorithm can get better performance. The kernel bandwidth of the GMCKF is exactly the same as the of MCKF. The iteration end condition for the fixed-point iteration of both MCKF and GMCKF is set at . The kernel size of the MSCF is set as , and the parameters of the NRSTF are set as . All the filters are coded with MATLAB.

5.2. Computational Complexity Comparison Experiment

According to the above simulation setting, we test the real running time for different filters when the observation noise obeys the Gaussian distribution. The test computer is installed with Windows 10 and equipped with Intel Core i5-1135G7 2.42GHz ×8. The version of Matlab is R2021a. All the tests are computed as the average over 100 independent Monte Carlo simulations, each simulation contains 1000 steps.

As shown in Table 2, the filter KF costs the least time and the filter NRSTF costs the most time. The running time of filter GMCKF is longer than filter KF, but it is within the acceptable range. The running time of the filter and the average iteration number of GMCKF algorithm decrease with the value of increases.


FilterRunning time for 1000 time (ms)Average iteration numbers

KF2.3104/
MCKF77.96055.52
NRSTF409.9463/
MSCF24.5846/
GMCKF ()291.029125.967
GMCKF ()145.767812.915
GMCKF ()76.23845.53
GMCKF ()57.49053.223
GMCKF ()54.61662.286

5.3. Performance under Different Non-Gaussian Noise

The MSEs and Probability Density Function (PDF) of and are used to evaluate the performance of different algorithms. The simulated MSEs are computed as the average over 100 independent Monte Carlo simulations, each simulation contains 1000 steps.

Set the initial states with , and run 1000 steps with formulae (49)–(52). We can get the state data with three kinds of noises (data xi) to estimate the shape parameter . Table 3 presents the shape parameter obtained by parameter automatic selection algorithm under three different noises . As we can see, when , the shape parameter is close to 2, and when , the shape parameter is close to 1. This is consistent with the property of the GGD function mentioned in Remark 1. According to this property, when , the shape parameter is going to approach infinity. However, this paper set the upper bound of equal to 10 in the parameter automatic selection algorithm. Therefore, the shape parameter in the third noise condition was estimated as 10.


Type of noise process Value of auto-selected

1.93
1.016
10

Figure 2 and Table 4 present the performance of five kinds of filters when the in the model of observation noise obeys the Gaussian distribution . The kernel bandwidth and the stop condition for the fixed-point iteration of GMCKF algorithm are set to be the same as that in MCKF which makes MCKF get good performance. The shape parameter is set to around the value 1.93 which is selected by parameter automatic selection algorithm. All the values of are {0.5, 1, 2, 4, 6, 1.93}.


FilterMSE of MSE of

KF0.50380.4684
MCKF0.04670.0356
NRSTF0.04630.0329
MSCF0.16600.1125
GMCKF ()0.04670.0356

Figure 2 illustrates the PDF of the estimation state errors of and . As shown in the figure, the performance of GMCKF will deteriorate when the shape parameter is too small or too large. When the shape parameter is set to 2 or 1.93 (auto-selected value), the GMCKF can achieve almost the same performance as the MCKF and the NRSTF. The performance of MCKF, NRSTF, MSCF, and GMCKF can be better than the KF significantly, achieving a desirable PDF with smaller dispersion and a higher peak.

Table 4 shows the MSEs of and under the KF, MCKF, NRSTF, MSCF, and GMCKF with auto-selected . As we can see, the GMCKF achieves almost the same performance as MCKF and NRSTF, which is significantly better than KF.

Figure 3 and Table 5 present the performance of five kinds of filters when the . The parameter and of GMCKF algorithm are still set to be the same as that in MCKF. The shape parameter is set to the value 1.016 which is selected by parameter automatic selection algorithm. All values of are {0.5, 1, 2, 4, 6, 1.016}.


FilterMSE of MSE of

KF0.68250.5469
MCKF0.33750.2032
NRSTF0.31590.1983
MSCF0.52270.3481
GMCKF ()0.29230.1835

The probability densities function of and in Figure 3 shows that in this noise condition, the GMCKF can get nearly the best performance when , which outperforms than other filters, with smaller dispersion and a higher peak.

The MSEs of and calculated from KF, MCKF, NRSTF, MSCF, and GMCKF with are shown in Table 5. The data shows that GMCKF achieves up to , , , and improvement in performance than KF, MCKF, NRSTF, and MSCF, respectively.

Figure 4 and Table 6 present the performance of five kinds of filters when the . The parameter and of GMCKF algorithm are still set to be the same as that in MCKF. The shape parameter is set to different values {1, 2, 4, 6, 10, 100}, in which the value 10 is selected by parameter automatic selection algorithm.


FilterMSE of MSE of

KF0.47860.3783
MCKF0.12250.0703
NRSTF0.12590.0691
MSCF0.29490.2244
GMCKF ()0.10310.0589

As we can see in Figure 4, performance of GMCKF becomes better as the value of increases (that is, the PDF curves of and become higher and more compact). When shape parameter , the GMCKF achieves the better performance than MCKF. Especially, when , the performance of GMCKF is almost the best in all other filters. When value of is higher than the upper bound 10 set in Section 4, the performance would improve slightly.

Table 6 shows the MSEs of and calculated from KF, MCKF, NRSTF, MSCF, and GMCKF with . The data shows that GMCKF achieves up to improvement in performance than KF, up to improvement in performance than MCKF, up to improvement in performance than NRSTF, and up to improvement in performance than MSCF.

It can be found through the above simulations that: (a)When the kernel shape is similar to the basic noise distribution (for ; for ; for ), the GMCKF can achieve better performance(b)The GMCKF can outperform than MCKF with proper parameter (same effect when , obviously advanced when and )(c)The GMCKF has better performance than NRSTF and MSCF(d)The shape parameter selected automatically by algorithm can almost reach better performance for GMCKF

In conclusion, with the additional shape parameter and the automatic selection algorithm, GMCKF can achieve good performance under various non-Gaussian noises.

6. Experiments for Target Tracking

The target may be floating in the space due to the microgravity environment. In this situation, it is important to track the target with high accuracy for space robot operation. However, the vision sensors like cameras in space are exposed to high-energy particles for a long time; there may be complex non-Gaussian noise in process of image acquisition and transmission. Uncontrolled lighting, covered by other floating objects, and complex background can also bring complex non-Gaussian noise to target tracking. To get more accurate target position, GMCKF algorithm is used to filter the non-Gaussian noise contained in the target tracking.

6.1. System Model

In the process of target tracking, the acceleration of the target is not constant due to the interference in the space. The Singer model [39] can be used to establish the system model.

According to the Singer model, the state equation of position tracking for target can be expressed as: with where is the time interval, is the process noise, and is the maneuvering frequency which represents the frequency of motion changes and is usually adjusted by experience.

When , the Singer model can be regarded as Constant Acceleration (CA) motion.

When , the Singer model can be regarded as Constant Velocity (CV) motion.

Therefore, the singer model can express the motion between CA and CV. In this way, it can describe the motion of target in space well.

The observation equation is with where is the observation noise.

6.2. Ground Experiments

Before the application in the space station, ground experiments have been done to validate the effect of GMCKF algorithm for target tracking.

6.2.1. Experimental system

To simulate the target tracking in the space environment, the experimental system was built as shown in Figure 5. The system includes a 6-DOF industrial robot (RS10N of Kawasaki), a pair of binocular cameras (MER-125-30UM of Imavision), a spherical target, and a laser tracker (AT-960-MR of Leica).

The positions of industrial robot base, binocular camera, and laser tracker are fixed. The repeatability of industrial robot is 0.04 mm. The horizontal reach and vertical reach of robot are 1450 mm and 2582 mm, respectively. The target is fixed in the end of the robot, and it is driven by industrial robot to simulate the motion in the space. The frame rate of binocular vision is 30fps, the focus length is 8 mm, and the resolution of the camera is 1292964. The position of target is derived from the algorithm based on Halcon, and the measuring accuracy is 1 mm.

To validate the performance of target tracking, we can get the precise position in the process of target motion by laser tracker (measuring accuracy is 15+6) and take it as the real motion state of target. Because there is not enough complex noise in laboratory environment, we deliberately add some non-Gaussian noise to the measure data from binocular camera to simulate the non-Gaussian observation noise generated by various factors in the space station. Then, we transition coordinate of the estimation position derived from algorithm and the measure position derived from laser tracker into the coordinate of robot base. The performance of the GMCKF algorithm can be evaluated in this way, and the detailed process is shown in Figure 6.

6.2.2. Experiment sets

The motion of target in space can be simulated by industrial robot. The robot end moves according to singer model. The maneuvering frequency in the experiments is set to . The initial position (unit: mm), velocity (unit: mm/s), and acceleration (unit: mm/s) of robot end are set as follows:

The covariance matrix is set to where is the sampling period and is the covariance of acceleration. Default parameters in equation (58) are set as follows: , , , P is the unit matrix.

The non-Gaussian observation noise added in the system is set as follows:

6.2.3. Experiment results

Before the GMCKF algorithm is applied, the robot moves following the experiment sets. And more than 1000 points of the position data in direction are collected to estimate the shape parameter of GMCKF.

Table 7 summarizes the MSEs of target position , , and for KF, MCKF, NRSTF, MSCF, and GMCKF. The results confirm again that the proposed GMCKF can outperform than other filters significantly when the system is disturbed by various non-Gaussian noises.


FilterMSE of MSE of MSE of

KF7.06796.642421.107
MCKF0.66120.61231.1069
NRSTF0.64730.61071.0856
MSCF0.81840.78501.6573
GMCKF0.48260.46510.7969

6.3. Practical Application

The simulations and the ground experiments have validated the performance of the GMCKF algorithm. To confirm the algorithm effect in the real environment, the GMCKF algorithm was used to deal with the experimental data from the on-orbit robot maintenance experiments of Chinese TianGong-2 space laboratory.

The purpose of the experiment is to verify the robot’s ability to carry out maintenance operations autonomously in the harsh space conditions. Filtering out complex non-Gaussian noise effectively caused by special space environment would help the robot improve operation accuracy and complete the task successfully. The experimental setup of the on-orbit robot maintenance experiments is shown in Figure 7. A 6 degrees-of-freedom manipulator was installed in the side wall of the space laboratory. The manipulator was equipped with a dexterous hand in the end and a hand-eye camera to complete the tasks. Global binocular cameras (Lt425 of Lumenera) were installed in the top wall of the space laboratory to guide the manipulator. Various on-orbit robot maintenance experiments (such as equipment installation and removal and catching floating objects) have been done using this system.

Figure 7 shows the experiment of catching a ball in the space laboratory. The global cameras can identify and locate the red ball. The measurement data of the distance between the red ball and the coordination of binocular camera can be obtained to guide the manipulator to complete the catch mission. The experimental data have been transmitted back to the ground from the space laboratory. We can regard the data as the real-time measurement data in the true space environment and verify the algorithm using the practical application data.

The GMCKF algorithm was used to filter the measurement data and estimate the position of the target. Figure 8 shows the position data of the red ball during the experiments. The figure’s horizontal axis shows the time range for about 40s. The vertical axis of the figure shows the distance between the ball and the coordination of global cameras (). The blue lines are the raw measurement data from global cameras and the red dot curve is the estimation data from GMCKF algorithm.

As can be seen in Figure 8, the ball moves about 160 mm due to the air disturbance. During the moving process, there is impulse noise with a maximum amplitude value of 10 mm contained in the raw measurement data due to the complex space environment. Under the action of GMCKF algorithm, the estimation curve of the distance becomes smoother and the ball could be tracked more accurately. The GMCKF algorithm can help the robot to better locate the target and complete the tasks. This practical experiment validates the effectiveness of the algorithm again.

7. Conclusions

This paper presents a novel Kalman Filter algorithm, called Generalized Maximum Correntropy Kalman Filter, which is derived from the Generalized Maximum Correntropy Criteria instead of minimum mean square error criterion. In GMCKF, the prior estimation of the state vectors and covariance matrix is obtained in the same way as KF, and the posterior estimation is based on the fixed-point algorithm and the GMCC criterion. GMCKF is significantly better than KF, MCKF, NRSTF, and MSCF with proper parameters, especially when the system is disturbed by various non-Gaussian noises. The shape parameter of the GMCKF can be selected automatically according to the PDF of the noise to help GMCKF achieve better performance. The effectiveness and robustness of the algorithm under non-Gaussian noise are validated by simulations and experiments. With GMCKF algorithm, target can be tracked more accurately in complex application, which is verified by ground experiments and the practical application in the robot maintenance experiments on TianGong-2 space laboratory.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon reasonable request.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this article.

Authors’ Contributions

Yang Mo is responsible for overall coordination, manuscript writing, and data collection. Yaonan Wang provided supervisory support and contributed to the writing of the paper. Hong Yang contributed to the data acquisition and the writing of the paper. Badong Chen assisted in the simulation experiments and the revision of the paper. Hui Li and Zhihong Jiang are the supervisor of the investigation efforts and offered many valuable help and suggestions. Hui Li and Zhihong Jiang are the corresponding authors.

Acknowledgments

The authors would like to acknowledge the National Key Research and Development Program of China (2018YFB1305300), the National Natural Science Foundation of China (Grant Nos. 62103141, 61733001, 61873039, U1713215, U1913211, U2013602), and the China Postdoctoral Science Foundation of China (2021M690963) for their support and funding of this paper.

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