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Abstract
In recent years, integrated polarized light/inertial heading measurement systems have been widely used to obtain autonomous heading measurements of small unmanned combat platforms in the case of satellite navigation rejection. However, existing polarized light/inertial heading measurement systems have certain limitations. For example, they can only measure the heading angle in environments where continuous observations can be obtained. When encountering a complex environment with trees and/or tall buildings, the measured heading angle will contain sharp noise which greatly affects its accuracy. In particular, when encountering an underpass, it will lead to the complete loss of lock of the polarized light compass signal. Therefore, for the problem of sharp noise arising from a complex environment, a robust Cubature Kalman filter (CKF) data-fusion algorithm is proposed and verified by experiments. It is proved that the robust CKF algorithm has a certain ability to filter out the effects of poor measurements. After application of the robust CKF algorithm, the Root Mean Square Error (RMSE) of the heading angle reaches 0.3612°. This method solves the problem of low precision and poor stability of the polarized light/inertial system when high buildings and/or trees are contained in a complex environment. Next, in view of the problem that the polarized light compass signal is completely lost due to passing through an underground passage, a random forest regression (RFR) neural network model is established and introduced into the combined system. Simulated and outdoor experiments are carried out to verify the designed model using data obtained with a vehicle. The RMSE of the heading angle obtained in the experiment is 1.1894°, which solves the problem that the polarized light/inertial system cannot utilize discontinuous observations when attempting to detect the carrier heading angle.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Due to the characteristics of small size, low power consumption and flexible operation, micro unmanned combat platforms, such as unmanned aerial vehicles (UAVs) and ground mobile robots, are used to undertake tasks in the complex environments in battlefields. The mission and working environment of a micro unmanned combat platform determine its special requirements when it attempts to obtain heading angle information, which poses a huge challenge in some environments. In modern war, a micro unmanned combat platform needs to work in conditions where the satellite signal it receives is either weak or completely absent, meaning it may be left in an unfamiliar area and/or a complex electromagnetic environment for a long time. The application of traditional satellite navigation, radio navigation or geomagnetic navigation is limited. In order to ensure the stable operation of a micro unmanned combat platform, it is necessary to develop a navigation system with strong autonomy that breaks through the key technical problems faced when the micro unmanned platform attempts to make heading measurements in the case of satellite navigation rejection.
Inertial systems can provide heading information completely and autonomously. They have the properties of autonomous navigation, strong anti-interference ability and good short-term stability. However, the accumulated errors of inertial systems will be magnified when the unmanned mission of a micro unmanned combat platform is long. Polarized light navigation is a kind of navigation method based on natural characteristics, which are not subject to interference in a large area, moreover, navigation errors do not accumulate with time. However, polarized light is easily affected by weather and light intensity, which will lead to the failure of polarized light navigation if the light is weak or completely blocked. Therefore, it is of great significance to develop an autonomous heading angle measurement method which can overcome the problems associated with complex environments and discontinuous observations.
At present, research on heading angle measurement methods using polarized light and inertial systems under the conditions of discontinuous observations or a complete loss of polarized light information has not been reported in the published literature. It is widely known that integrated navigation systems containing the global positioning system (GPS) and inertial navigation system (INS) can complement each other. Therefore, an integrated GPS/INS navigation system is the main research method used worldwide. In this paper, the research status of an integrated GPS/INS navigation model under the condition of discontinuous observations is reviewed to provide technical reference for polarized light/inertial integrated systems. Aimed at the problem of a loss of GPS lock, many scholars and experts have combined machine-learning algorithms such as artificial neural networks (ANNs) with traditional data-fusion methods. ANNs have been developed since the 1940s, and significant breakthroughs were made in the 1980s [1]. ANNs have strong learning, adaptability and flexibility capabilities, and when learning from a training set, an ANN can ascertain the complex logical relationship(s) in the data. In recent years, many scholars have used neural networks to simulate the nonlinear relationships between the errors of inertial navigation systems and their output [2–7]. Along these lines, Kalman-filter-based neural networks have become a research hotspot.
In 2006, Noureldin and his team [8] first used INS positions as input, where INS positions and GPS positions were given as output. In 2015, Adusumilli et al. [9] proposed a hybrid method of GPS and INS data fusion using random forest regression (RFR) and a principal component analysis. In 2017, Jaradat et al. [10] proposed an autoregressive neural network fusion architecture for the integration of low-cost GPS and inertial measurement unit (IMU) data. In 2018, Havyarimana et al. [11] proposed a hybrid method based on a fuzzy inference system (FIS) and a sparse random Gaussian (SRG) model to solve the problem of vehicle position predictions under the condition of losing GPS lock.
Similarly, a series of research has been carried out in China, such as using neural networks and the Kalman filter algorithm to solve the problem of losing GPS lock. For example, in 2013, Chen et al. [12] proposed the INS error compensation model, which combined the strong tracking Kalman filter (STKF) with the wavelet neural network (WNN) algorithm. In 2017, Jiang et al. [13] proposed a New Multi Fading Cubature Kalman Filter, called NMF-CKF, to control the influence of model errors and navigation system uncertainties. In 2017, Yao et al. [14] compared a hybrid fusion algorithm containing a Kalman filter and an improved multilayer perceptron network. When the GPS signal was unavailable, the fusion algorithm was able to predict and estimate pseudo GPS positions. In the same year, Li et al. [15] proposed a centralized learning method, which was similar to the strapdown inertial navigation system (SINS)/GPS position model constructed using neural networks. In 2018, Zhang et al. [16] proposed a dual model solution, which combined Multi Decay Factor Cubature Kalman Filter (MDF-CKF) and Random Forests (RFs) to model and compensate their GPS/INS integrated navigation system.
In a polarized light/inertial integrated system, when the polarized light signal is lost, the whole integrated system will be in the pure inertial system mode, and the error accumulation of the inertial system will not be corrected, which seriously affects the measurement accuracy of the heading angle determined by the integrated system. Although the heading angle measurement of polarized light/inertial integrated system can be seen in the published literature, it has not been studied under the discontinuous observation environment; fortunately, the basic ideas of these studies provide suitable reference for the paper [17,18].
In this paper, we report on the first successful implementation of combining CKF with H∞ robust filtering, which is used to weaken the sharp noise in complex environment. Then aimed at the problems faced by polarized light/inertial systems under the conditions of discontinuous observations, a discontinuous observations model containing an RFR neural network is designed here. Before the polarized light compass signal is loss of lock, the neural network always trains the heading angle information of the inertial system. Once the polarized light compass signal becomes loss of lock, the neural network will not train any more, but it will still provide measurements to the combined system, which allows the heading angle to be calculated by the inertial system in real time and improves the measurement accuracy of the heading angle. The rest of the paper is organized as follows: In Section 2, an overview of the integrated navigation system is introduced. The robust CKF data fusion algorithm and discontinuous observations model are also presented. Section 3 and Section 4 describes the experimental process as well as the results and discussions. Finally, conclusions are drawn in Section 5.
2. Methods
2.1 Combined heading measurement system
The experimental test platform of the combined system is shown in Fig. 1. It includes three independently developed channels, each containing a polarized light compass, an inertial system (gyro model: STIM202, accelerometer model: 1521L) and a Novartis high-precision positioning and orientation system (SPAN-LCI, with an accuracy of 0.008°). The polarized light compasses and inertial devices were used for the orientation research, and the SPAN-LCI provided the attitude reference. A polarized light compass was also installed on the top of a vehicle, which allowed it collect observations of the sky with a large field-of-view. In order to prevent the inertial system from being exposed to the sun for a long time, an inertial system was also installed on the interior base of the vehicle. The polarized light compass and the inertial system were parallel to the front axle of the vehicle to ensure that their coordinate systems coincided with the carrier. The inertial system provided the pitch angle and roll angle to the polarized compass in real time, and then the heading angle was determined after compass compensation. The high-precision heading angle output from the SPAN-LCI was used as the benchmark for comparison. The data output rate of the polarized compass was 2 Hz, while the data output rate of the inertial system and SPAN-LCI was 40 Hz.
In view of the unsynchronized sampling time and inconsistent sampling frequency of the polarized light/inertial system, conflicts can arise in the navigation data solved by each subsystem in the time domain, which limits an accurate description of the motion state of the carrier at certain times being determined, and thus limiting the effective fusion of the different kinds of information. In this paper, the accurate time of processing a group of data in real time by a polarized light compass was calculated using MATLAB, where the accuracy reached the millisecond level. Once calculated, the inertial system sent the collected heading angle information via a Bluetooth module in real time, which was collected by the polarized light compass and fused in real time to output the heading angle after Kalman filtering. But when the sky was blocked by buildings and/or trees, which will lead to a loss of polarization information and hence seriously affect the accuracy of the determined carrier heading angle. Therefore, the robust CKF data fusion algorithm should be established.
2.2 Robust CKF data fusion algorithm
2.2.1 Matrix diagonalization transformation
In the standard CKF algorithm, Cholesky decomposition is used to decompose the covariance matrix, which requires that the matrix to be decomposed has positive definiteness. When the covariance matrix loses its positive definiteness, the filtering will be interrupted immediately, which poses a great challenge to the stability of the algorithm. Most filtering interruptions occur after filtering several times. The possible reason for the interruptions is that the covariance matrix caused by model mismatch and calculation error is not positive definite. However, when decomposing the covariance matrix, the method based on matrix diagonalization does not need the positive definiteness of the matrix.
If A is a real symmetric matrix of order n, there must be an orthogonal matrix V of order n so that VTAV = V-1AV = D, and where A = VTDV. Here D is a diagonal matrix where the n eigenvalues of A are diagonal elements [19,20].
In the Kalman filtering process, the covariance matrix P is a real symmetric matrix, i.e. P = VTDV, where D is a diagonal matrix which contains the n eigenvalues of P as diagonal elements, and V is an orthogonal matrix composed of two orthogonal eigenvectors corresponding to n eigenvalues.
We say that , thus is easy to know that S is a real symmetric matrix; so, P = SST. In the process of CKF calculation, when the square root matrix of the covariance matrix P needs to be obtained, the eigenvalue decomposition of P can be carried out to obtain its eigenvalue matrix D and eigenvector V. The square root matrix S of covariance matrix P can be obtained by Eq. (1). The square root matrix obtained by this method is the theoretical square root matrix. This method retains the original feature space information of the covariance matrix, making the covariance transfer more accurate, so it can effectively improve the accuracy of the filter estimation. The stability of the iterative calculation of the covariance matrix will be better guaranteed by using singular matrix diagonalization instead of square root decomposition.
2.2.2 Robust CKF combined with H∞ filtering
Different from the idea of Kalman filtering based on minimizing the variance of the estimated error, H∞ filtering is based on the minimax robust control criterion, i.e. minimizing the maximum possible error [21]. Specifically, it refers to obtaining the best estimated value of the state quantity under the condition that the prior information, such as the initial state of filtering, system noise and observation noise, is unknown. The cost function is defined as:
Equation (2) can be rewritten as:
We also assume the premise that the system’s process noise and measurement noise are finite energy signals, i.e. ${\sum\nolimits_{k = 0}^N {\left\|{{{\mathbf W}_k}} \right\|} ^2} = \infty$ and ${\sum\nolimits_{k = 0}^N {\left\|{{{\mathbf V}_k}} \right\|} ^2} = \infty$. Next, we find a filter to minimize the adverse effect of external interference on the state’s estimation results for a given minimum real number when the statistical characteristics of the system’s noise, the observation noise and the model estimation are inaccurate, that is:
Then, H∞ filtering is applied to CKF to transform the recurrence equation of the covariance matrix of the estimated state.
If the constraint parameter $\gamma$ at any time takes the minimum value satisfying condition ${{\mathbf P}_k} > 0$, then the optimal H∞ filter can be obtained. When the constraint parameter $\gamma$ tends to infinity, the H∞ filter degenerates into a Kalman filter. When $\gamma$ is a minimum, the robustness of the filter is the best, but the variance of the estimate is not necessarily the smallest. When $\gamma$ tends to infinity, the minimum variance estimation can be obtained, but the robustness is poor. Therefore, the balance between the measurement accuracy and the system’s robustness needs to be adjusted to meet the measurement needs by appropriately selecting the value of $\gamma$[22].
CKF uses a set of volume points with equal weights to calculate the posterior’s probability density. The state estimation accuracy of the model accurate system can reach the third order. However, when there is a large system model error or a sudden change in the system’s state, a large estimation error will still be generated [23,24]. In practical applications, a single CKF algorithm can sometimes find it difficult to meet the measurement requirements. However, by combining CKF with H∞ robust filtering, in the framework of the CKF algorithm, the robust characteristics of H∞ filtering are added to the covariance matrix updating calculations. In addition, matrix diagonalization transformation is used to replace Cholesky decomposition in the standard CKF to increase the calculation stability of the filtering algorithm. The specific process is as follows:
- (1) Calculated the volume point: where m = 2n, n is the dimension of state vector. ${s_k} = {{\mathbf V}^T}\sqrt {\mathbf D} {\mathbf V}$, D and V are the eigenvalues of decomposed ${{\mathbf P}_k}$ used to get the eigenvalue matrix and eigenvector, ${{\mathbf P}_0} = {s_k}s_k^T$. ${\xi _i}$ is the volume point, i.e. ${\xi _i} = \sqrt n {[1 ]_i}$, and ${[1 ]_i}$ represents the i-th column of the point set generated by changing the element symbols and arranging the elements of the n-dimensional unit vector ${\boldsymbol e} = {[{1,0, \cdots ,0} ]^T}$.
Volume point $\delta \chi _k^i$ consists of two parts: an error quaternion vector and gyro drift.
$$\delta \chi _k^i = \left[ \begin{array}{l} \delta \chi_{{\boldsymbol q}v,k}^i\\ \delta \chi_{b,k}^i \end{array} \right]. $$The quaternion and the partial volume point of gyro drift are, respectively:
$$\left\{ \begin{array}{l} \chi_{q,k}^i = \delta \chi_k^i \otimes {{\hat{{\boldsymbol q}}}_k},\chi_k^{i + n} = \delta \chi_k^{i + n} \otimes {{\hat{{\boldsymbol q}}}_k}\\ \chi_{b,k}^i = {{\hat{b}}_k} + \delta \chi_{b,k}^i,\chi_{b,k}^{i + n} = {{\hat{b}}_k} - \delta \chi_{b,k}^i \end{array} \right., $$where i = 1, 2, ……, n. - (2) Calculate the volume points transferred by the equation of state:$$\chi _{{\boldsymbol q},k + 1|k}^i = \left[ \begin{array}{l} \cos \left( {\frac{1}{2}\left\|{\hat{\omega }_k^i} \right\|\Delta t} \right){{\mathbf I}_{3 \times 3}} - [{\hat{\psi }_k^i} ]\textrm{ }\hat{\psi }_k^i\\ - {({\hat{\psi }_k^i} )^T}\textrm{ }\cos \left( {\frac{1}{2}\left\|{\hat{\omega }_k^i} \right\|\Delta t} \right){{\mathbf I}_{3 \times 3}}\textrm{ } \end{array} \right] \otimes \chi _{{\boldsymbol q},k}^i. $$where $\hat{\omega }_k^i = {\omega _k} - \chi _{b,k}^i$.
- (3) Calculate the state prediction value at time k+1:$$\hat{{\boldsymbol q}}_{k + 1|k}^i = \arg \max \chi _{{\boldsymbol q},k + 1|k}^iM{({\chi_{{\boldsymbol q},k + 1|k}^i} )^T}, $$where $M{({\chi_{{\boldsymbol q},k + 1|k}^i} )^T} = \frac{1}{m}{\sum\nolimits_{i = 1}^m {\chi _{{\boldsymbol q},k + 1|k}^i({\chi_{{\boldsymbol q},k + 1|k}^i} )} ^T}$.
- (4) Calculate the covariance matrix of the state error at time k+1:$${{\mathbf P}_{k + 1/k}} = \left[ \begin{array}{l} {{\mathbf P}_{1,k + 1/k}}{{\mathbf P}_{2,k + 1/k}}\\ {{\mathbf P}_{3,k + 1/k}}{{\mathbf P}_{4,k + 1/k}} \end{array} \right] + {{\mathbf Q}_k}, $$where$${{\mathbf P}_{1,k + 1|k}} = \frac{1}{m}\sum\limits_{i = 1}^m {\delta \chi _{qv,k + 1|k}^i{{({\delta \chi_{qv,k + 1|k}^i} )}^T}}, $$$${{\mathbf P}_{2,k + 1|k}} = \frac{1}{m}\sum\limits_{i = 1}^m {\delta \chi _{qv,k + 1|k}^i{{[{\chi_{b,k + 1|k}^i - {{\hat{b}}_{k + 1|k}}} ]}^T}}, $$
- (5) Calculate the updated state volume points: where ${s_{k + 1|k}} = {{\mathbf V}^T}\sqrt {\mathbf D} {\mathbf V}$, ${{\mathbf P}_{k + 1|k}} = {s_{k + 1|k}}s_{k + 1|k}^T$ and
- (6) Calculate the volume point via the measurement equation:$$\left\{ \begin{array}{l} \eta_{{\boldsymbol q},k + 1|k}^i = \delta \eta_{{\boldsymbol q},k + 1|k}^i \otimes {{\hat{{\boldsymbol q}}}_{k + 1|k}}\\ \eta_{{\boldsymbol q},k + 1|k}^{i + n} = {({\delta \eta_{{\boldsymbol q},k + 1|k}^{i + n}} )^{ - 1}} \otimes {{\hat{{\boldsymbol q}}}_{k + 1|k}} \end{array} \right., $$where i = 1, 2, ……, n.
- (7) Calculate the predicted measurement value at time k+1:$${\hat{y}_{{\boldsymbol q},k + 1|k}} = \arg \max \eta _{{\boldsymbol q},k + 1|k}^iM{({\eta_{{\boldsymbol q},k + 1|k}^i} )^T}, $$where $M{({\eta_{{\boldsymbol q},k + 1|k}^i} )^T} = \frac{1}{m}\sum\nolimits_{i = 1}^m {\eta _{{\boldsymbol q},k + 1|k}^i} {({\eta_{{\boldsymbol q},k + 1|k}^i} )^T}$ and
- (8) Estimate the measurement error covariance and the one-step prediction covariance matrix at time k+1:$${\mathbf P}_{k + 1}^y = \frac{1}{m}\sum\limits_{i = 1}^m {\delta y_{{\boldsymbol q}v,k + |k}^i{{({\delta y_{{\boldsymbol q}v,k + |k}^i} )}^T}} + {{\mathbf R}_k}. $$$${\mathbf P}_{k + 1}^{xy} = \left[ \begin{array}{l} \frac{1}{m}\sum\limits_{i = 1}^m {\delta \chi_{{\boldsymbol q}v,k + 1|k}^i{{({\chi_{{\boldsymbol q}v,k + 1|k}^i} )}^T}} \\ \frac{1}{m}\sum\limits_{i = 1}^m {({\chi_{b,k + 1|k}^i - {{\hat{b}}_{k + 1|k}}} ){{({\delta y_{k + 1|k}^i} )}^T}} \end{array} \right]. $$
- (11) Equation (6) is used to update the covariance matrix of state error at time k+1.
In this method, the CKF and H∞ filter are combined, where the high accuracy of the CKF algorithm and the robustness of the H∞ filter are used to improve the stability of the numerical calculations. The polarized light and inertial fusion measurement system filter at the time of coincident data, using polarized light data to suppress the drift of the fused heading angle with time, and hence obtaining a stable and robust heading angle output. But when the polarized light signal is lost, the whole integrated system will be in the pure inertial system mode, and the error accumulation of the inertial system will not be corrected. Therefore, the model and algorithm design in discontinuous observation should be established.
2.3 Model and algorithm design in discontinuous observation
2.3.1 Filter structure design of combined heading measurement system in the case of losing the polarized light lock
The fused information obtained with the integrated system allows us to estimate the error of the inertial system optimally by introducing the measured signal of the polarized light, which is fed back to the inertial system for error correction. However, when the sky is completely blocked due to e.g. an underground passage, the signal of the polarized light compass will be lost. At this time, the combined system will work in the pure inertial system mode, which cannot be calibrated. If a method can be found to provide observations for the combined system when the polarized light signal lose lock, the combined system can continue to work normally.
Neural networks can learn from input data and output results that fit the nonlinear system model, and does not need to model the system accurately. The combination of neural networks in composite systems has become a research hotspot in recent times, but it is complex to get neural network algorithms to perform calculations, and the accuracy of the chosen algorithm depends on the training data. The neural network model designed in this paper is shown in Fig. 2. When the polarized light compass signal is present, the neural network always trains the heading angle information outputted by the inertial system. Once the polarized light compass signal is received, the neural network will no longer train, but instead provide measurements to the combined system, correct the heading angle calculated by the inertial system in real time, and improve the measurement accuracy of the combined system. Because the RFR algorithm can simultaneously deal with continuous and discrete attributes, and it possesses high efficiency, strong robustness and anti-noise properties, as well as preventing over fitting, this paper uses the RFR neural network model to realize an auxiliary system for the existing combined system.
2.3.2 RFR algorithm
In 2001, the RFR algorithm was proposed by Leo Breiman and Adele Cutler, which is an integrated learning algorithm based on decision trees. It is widely used in the fields of economy, medicine, biology and so on. It has a series of prediction advantages, mainly including (1) good prediction effect, (2) training can be highly parallelized, which has advantages for large sample training speeds in the era of big data, (3) it can process high-dimensional features without feature filtering in advance, (4) it dose not easily over-fit data, (5) the model training speed is fast, which is desirable for parallel processing, and (6) it can rank the importance of features.
The algorithm flow involved in this paper is as follows:
- Step 1: The sample subset is randomly generated according to the concept of bagging.
- Step 2: Based on the idea of random subspace, f features are randomly extracted to the segmentation node, and a single regression decision tree is constructed.
- Step 3: Repeat steps 1 and 2 to construct T free growth regression decision trees without pruning, and finally form a forest.
- Step 4: Take the average of the predicted value of the T decision tree as the final result.
The implementation flow of the RFR model is shown in Fig. 3.
3. Results and discussion
3.1 Simulation of the combined system with discontinuous observations
3.1.1 Simulation verification of the robust CKF data-Fusion algorithm
In order to verify the feasibility and effectiveness of the proposed robust CKF data-fusion algorithm, a series of simulations were carried out with the built combined measurement system with the help of a developed measurement program.
The complementary filtering stores the SPAN-LCI data in a flash drive after the vehicle experiment. The data collected by the polarized light compasses and the inertial system are stored in a computer, and the robust CKF data-fusion algorithm is used for offline data-fusion processing. The heading angles collected by SPAN-LCI are used to evaluate the filtering effect. In order to verify the advantages and disadvantages of the robust CKF filtering algorithm in terms of data fusion, we comprehensively evaluated the performance of our proposed model. The standard CKF filtering algorithm and KF filtering algorithm were used for comparison in this paper. Two groups of experimental data were collected. The first group contained data collect by the vehicle, which had a running time of 35 s (group 1), while the second group were also collected by the experimental vehicle, but it had a running time of 70 s (group 2). The integrated heading angle and its error are shown in Figs. 4 and 5.
Fig. 4. Simulated heading angles of the first group using the polarized light/inertial combination system. (a) Calculated heading angles. (b) Calculated heading angles between 22-28 s. (c) Heading angle error comparison.
Fig. 5. Simulated heading angles of the second group using the polarized light/inertial combination system. (a) Calculated heading angles. (b) Calculated heading angles between 5-65 s. (c) Heading angle error comparison.
It can be seen from the comparison data in Fig. 4 and 5 that, due to the influence of weather, the heading angle errors in the presence of polarized light are large. However, the errors contained in the output data of the composite system are greatly reduced after the data were processed by the robust CKF filtering algorithm. Although the errors of the inertial system accumulate with time, the robust CKF filtering algorithm was able to effectively suppress the errors caused by the time drift of the inertial system. The comparison of the experimental data is also shown in Table 1.
Table 1. Simulated heading angles for groups 1 and 2.a
According to the experimental data in Table 1, the robust CKF filtering algorithm, the CKF filtering algorithm and the KF filtering algorithm have obvious filtering effects on fused heading angle data. However, the effect of the fused CKF data is more obvious because the filtering accuracy is higher, the optimal RMSE reached 1.1688°, the stability is better, and the nonlinear approximation characteristics are strong.
3.1.2 Simulation and verification of the RFR neural network model
In order to verify the validity of the RFR neural network model established in this paper, the data obtained during the vehicle heading measurement experiment were processed offline. We carried out a simulated experiment of two groups of polarized light compass signals, where both times the signal was lost. The time when the polarized light signal was lost was 130 s (group 3) and 100 s (group 4). In order to further verify the advantages and disadvantages of the designed RFR neural network model and evaluate its performance comprehensively, The long short-term memory (LSTM) and radial basis function (RBF) neural network prediction models were used as comparisons for experimental verification. The predicted heading angles from the three neural network models are shown in Figs. 6 and 7.
Fig. 6. Predicted heading angles for the third group found by the different neural network models. (a) Heading angle calculations. (b) Calculated heading angles between 15-125 s. (c) Heading angle error comparison.
Fig. 7. Predicted heading angles for the forth group found by the various neural network models. (a) Heading angle calculations. (b) Calculated heading angles between 20-50 s. (c) Heading angle error comparison.
If there is no neural network model, the polarized light/inertial integrated system will work in the pure inertial system mode during periods where the polarized light signal is lost. However, the heading angle calculation errors of the inertial system accumulate rapidly, which eventually leads to a divergence of navigation information. From the simulation we can see that the divergence of the heading angle information calculated by the inertial system is obvious. Therefore, the LSTM, RBF and RFR neural networks are used to train the heading angle outputted by the inertial system before the polarized light signal is lost.
According to the simulated results shown in Table 2, all three methods are able to restrain the divergence of the heading angle information to a certain extent when the polarized light signal is lost. Among them, the RFR neural network model proposed in this paper has a better prediction effect and higher prediction accuracy. The RMSE of the optimal heading angle prediction is 0.6578°, which solved the problem that the polarized light/inertial system is unable to detect the carrier heading angle under the condition of discontinuous observations.
Table 2. Comparison of predicted heading angle using different neural network models.a
3.2 Application of the combined system with discontinuous observations
3.2.1 Application of the composite system in complex environments
Due to the rapid development of urbanization and the advocacy of environmental protection, the sky can be blocked by buildings and/or trees in urban environments, which will lead to a loss of polarization information and hence seriously affect the accuracy of the determined carrier heading angle. When the sky is occluded by trees or buildings, only a small part of the middle area is available, and the heading angle calculated by the polarization information provided by this part of the available area will produce peak noise. Therefore, combined with the proposed robust CKF data-fusion filtering algorithm and the developed measurement program, more accurate heading angle information can be calculated.
In this experiment, the experimental site was a road section covered by some trees and tall buildings at the Taiyuan University of Science and Technology. The experiment was conducted on August 4, 2019. The weather was clear and the visibility was good, which were conducive to the proper extraction of the polarization data from the observations. The SPAN-LCI and integrated polarized light/inertial navigation system were mounted on the vehicle were debugged normally, which met the hardware conditions of the vehicle experiment. The acquisition of polarization information in the occluded environment is shown in Fig. 8, while the calculated heading angles and the heading angle errors are compared in Fig. 9. Figure 9(a) shows a comparison of the heading angle between the robust CKF filtering algorithm and CKF and KF, and Fig. 9(b) is a comparison of their errors. The experimental data are shown in Table 3.
Fig. 9. Comparison of the heading angle information measured in occluded test environment (reference, polarized light, inertial system, KF, CKF, robust CKF). (a) Heading angle calculations. (b) Comparison of the heading angle errors.
Table 3. Comparison of the experimental results. a
It can be seen from Fig. 9 that when the polarized light compass are occluded by trees or buildings, the heading angle accuracy decreased and there was random spike noise. The robust CKF suppressed the errors caused by the time drift of the inertial system, which allowed the vehicle to follow a stable heading angle. Although CKF and KF also have the characteristics of restraining drift, their output noise was large, whether it was peak noise from the polarized light or jitter noise from the inertial system, which is obviously reflected in the fusion results. Robust CKF was better than CKF and KF in weakening the fusion result following the measurement noise and system noise changes, which shows that the robust CKF algorithm has obvious advantages and comprehensive performance in terms of data fusion. From the experiment data shown in Table 3, it can be concluded that the RMSE of the heading angle achieved by robust CKF algorithm reached 0.3612°, which solves the problem of low accuracy and poor stability of the polarized light/inertial system in complex environments containing high buildings and/or trees.
3.2.2 Application of the composite system to discontinuous observations
Nowadays, more and more underpasses appear in urban construction. When a vehicle enters an underpass, the polarized light compass will lose the signal completely for a short time. Therefore, to overcome this problem, in the polarized light/inertial system, an algorithm based on neural networks was proposed to calculate the combined heading angle. Before the polarized light signal loses lock, the neural network is used to train the heading angle information outputted by the inertial system. When the polarized light signal lose lock, the neural network will not be trained any more. The combined system will then calibrate the inertial system in real time using the training values outputted by the neural network as the observation values.
In this paper, three groups of experiments were carried out. The experimental site was a section containing an underground passage at the Taiyuan University of Science and Technology. The experiment data was August 5, 2019. The weather was helpful for measuring the polarization signal. In the three experiments, the time of polarized light signal loss were 5 s (group 5), 10 s (group 6) and 25 s (group 7), respectively. The RFR neural network model was used in the experiments, which was compared with the LSTM and RBF neural network models. Figures 10–12 show a comparison of the heading angles and their errors for the three groups as well as those between the proposed system and the pure inertial system when polarized the light compass signal lose lock.
Fig. 10. Predicted heading angles for the fifth group of neural network models. (a) Calculated heading angles. (b) Heading angle error comparison.
Fig. 11. Predicted heading angles for the sixth group of neural network models. (a) Calculated heading angles. (b) Heading angle error comparison.
Fig. 12. Predicted heading angles for the seventh group of neural network models. (a) Calculated heading angles. (b) Heading angle error comparison.
The experiment data are shown in Table 4. From the three groups of data, it can be concluded that the LSTM, RBF and RFR neural networks each have a certain effect on the predicted heading angles. However, the RMSE of the heading angles predicted by the RFR neural network model was the smallest, and it had the best optimal predicted heading angle accuracy of 1.1894°. Moreover, its ability to restrain divergence was better, which solves the problem that the polarized light/inertial system was unable to detect the heading angle of the carrier from discontinuous observations.
Table 4. Comparison of the experimental results under different neural network models.a
4. Conclusions
In this paper, we designed a system aimed at overcoming the problem that polarized light is easily affected by weather and light intensity. These problems reduce the accuracy of the measured heading angle determined by polarized light/inertial systems in complex environments containing trees and/or tall buildings, or in an underpass, which can partially or completely block the polarized light signal. On the basis of the research results of GPS/INS, an autonomous measurement method for measuring heading angles was developed using polarized light as the navigation information source which was combined with inertial navigation technology. The combined system was able to overcome the problems associated with complex environments and discontinuous observations. A robust CKF filtering algorithm and a combined heading angle calculation algorithm based on neural networks were proposed. Then, a series of simulation were performed to verify the built combined measurement system and the developed measurement program. On this basis, the combined measurement system was used to carry out vehicle experiments in a complex environment containing high buildings and trees, as well as discontinuous observations arising when the vehicle travelled through an underground passage.
In the complex environment containing obstructive tall buildings and trees, the robust CKF algorithm was able to effectively suppress the errors caused by the time drift of the inertial system. It was also able to weaken the peak noise coming from the polarized light and the jitter noise coming from the inertial system. It was better than CKF in weakening the fusion result following the measurement noise and the system noise change. With the robust CKF algorithm, the RMSE of the heading angle reached 0.3612°, which solved the problem of poor stability of the polarized light/inertial system in complex environments containing occlusion by high buildings and/or trees.
In the final experiment, where the vehicle passed through an underpass, a complete loss of the polarized light compass signal occurred. In this experiment, the RFR, LSTM and RBF neural network models were used to train the heading angle of the system in the early stage which were used to predict the heading angle after polarized light signal was lost. The RMSE of heading angles predicted by the RFR neural network model was the smallest, reaching 1.1894°. Moreover, the RFR neural network model had a more precise heading angle prediction and it was better able to restrain heading information divergence.
Overall, our proposed system solved the problem that normal polarized light/inertial systems face, in terms of being unable to detect the heading angle with discontinuous observations.
Funding
National Natural Science Foundation of China (61905172); Award Fund Project for outstanding doctors from Shanxi Province (20192068); Doctoral Research Start-up Fund Project (20192015); Scientific and Technological Innovation Programs of Higher Education Institutions in Shanxi (2021L295).
Acknowledgments
We thank International Science Editing (http://www. international science editing.com) for editing this manuscript. We acknowledge the financial support from the Natural Science Foundation of China (61905172), and the Scientific and Technological Innovation Programs of Higher Education Institutions in Shanxi (2021L295), and the Doctoral Research Start-up Fund Project (20192015), and the Award Fund Project for outstanding doctors from Shanxi Province(20192068).
Disclosures
The authors declare no competing financial interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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