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Abstract
The bionic polarization sensor (PS)/MEMS inertial measurement unit (MIMU) integrated system can provide reliable attitude and heading information for unmanned vehicles in the case of GNSS rejection. However, the existing measurement methods have poor adaptability to inclining, sheltering, and other harsh environments, and do not make full use of the complementary characteristics of the gyroscopes, accelerometers, and PS, which seriously affects the system performance. Therefore, this paper proposes an attitude and heading measurement method based on an adaptive complementary Kalman filter (ACKF), which corrects the gyroscopes according to the gravity measured by the accelerometers to improve the attitude accuracy and fuses the IMU heading and tilt-compensated polarization heading by Kalman optimal estimation. On this basis, the maximum correlation entropy of the measured gravity and the theoretical gravity is used to construct an adaptive factor to realize the adaptive complementary of the gyroscopes and the accelerometers. Finally, the effectiveness of the method is verified by the outdoor rotation test without occlusion and the vehicle test with occlusion. Compared with the traditional Kalman filter, the pitch, roll, and heading RMSE of the vehicle test are reduced by 89.3%, 93.2% and, 9.6% respectively, which verifies the great advantages.
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1. Introduction
The optical navigation system based on polarization sensor can provide robust navigation information for autonomous unmanned systems in scenarios of magnetic interference or the Global Navigation Satellite System (GNSS) rejection [1]. At present, the fully-autonomous navigation methods include inertial navigation [2], visual navigation [3,4], and atmospheric polarization navigation [5–7], etc. These navigation technologies are applied in different fields due to their different characteristics. The atmospheric polarization navigation system uses the stable spatiotemporal scattering mode of the natural skylight in the atmosphere to carry out the heading measurement [8–11], which has the advantages of no accumulated error, high autonomy, and the ability to resist electromagnetic interference and radio deception. Therefore, it has gradually become a research hotspot.
By studying the perception mechanism of animal organs in the natural environment to form navigation information [12,13], scholars have transformed the polarized skylight signal into the heading information and developed bionic polarization sensors based on point source [9,10] and polarization image [14–16], respectively. The robust bionic polarization heading measurement technology in complex environments such as desert [17], underwater [18], and urban [19,20] have also been studied. In addition, some scholars have also implemented all-silicon microdisplay through efficient hot carrier electronics, providing a new approach for polarization imaging [21]. However, there are many defects in the way of obtaining navigation information through a single sensor. For example, inertial navigation system (INS) errors will accumulate over time. And the polarimetric compass has poor environmental adaptability and is vulnerable to clouds, trees or buildings, resulting in a significant decline in measurement accuracy. Although Li [11] proposed a heading measurement method based on an improved Berry model to improve the polarization heading accuracy under multiple scattering effects, this method still does not combine the advantages of multiple sensors to achieve simultaneous measurement of attitude and heading.
The combination of INS and polarization compass enables better performance by simultaneously utilizing the non-cumulative-error characteristics of polarization heading and the short-term stability of INS, which can provide robust navigation information for fields such as aeronautic and aerospace [22]. Therefore, scholars have done a lot of researches on the integrated navigation technology aided by polarization compass. Xian [23] added the heading output by the point-source PS as a new constraint node to the visual inertial odometer, improving the overall position and attitude accuracy. However, point-source PS is vulnerable to carrier motion and occlusion interference. He [24] built an integrated navigation system based on MIMU/image-based PS/GNSS and used Kalman filtering to achieve navigation information fusion, with integrated heading accuracy better than 1$^\circ$. However, the impact of environmental interference is also not considered. Du [25,26] uses the Rayleigh scattering model proposed by [27] to solve the heading of the point source PS and use it to assist the strap-down inertial navigation system (SINS) in initial alignment. However, this method requires high openness of the environment. Hu [28] used polarized light to detect the spectral response of polymer gratings, improving the sensitivity of sensors to pressure and temperature, but it has not yet been applied in the field of polarization heading measurement. Xie [29] studied the multi-source fusion navigation system based on graph optimization and used the polarization sensor to constrain the heading, so as to reduce the heading drift under the GNSS rejection environment. Zhao [30] studied the seamless integrated navigation method for the image-based polarization sensor and IMU integrated system, used the long short-term memory (LSTM) neural network to learn the characteristics of polarization heading error, and used the LSTM network to output headings when the system is under the bad environments, with the error of heading measurement less than 0.53$^\circ$. However, this method does not consider the influence of inaccurate carrier attitude calculation on polarization heading measurement. Cai [31] proposed a fast transfer alignment method of polarized skylight-aided SINS applied in polar regions and carried out semi-physical simulation verification. Dou [32] studied the integrated navigation system based on SINS/ polarization sensor/odometer and improved the navigation performance using an adaptive filtering method. Hu [33] proposed a point-source polarization sensor/IMU attitude measurement method based on attitude feedback, which reduces the heading error under the condition of changing horizontal angles. Shen [34] proposed a new robust filtering method to realize the navigation information fusion of MIMU/GPS/polarization compass and improve the carrier positioning accuracy. Yang [1] studied the polarization-aided inertial navigation system under discontinuous observation, and the heading error under complex scenes is less than 1.1894$^\circ$. Yang [35] studied the INS/star sensor night attitude measurement system based on polarized skylight assistance, which provides a reference for navigation in the complex night environment. Although the above researches improve the navigation performance of the polarization-aided inertial system through multi-sensor fusion, the impact of inaccurate attitude calculation on the performance of the integrated system is not considered.
In terms of multi-sensor fusion methods, Carbon [36] utilizes an extended Kalman filter (EKF) to fuse pressure and temperature sensors for internal combustion engine fault diagnosis. However, EKF has poor filtering accuracy and is prone to divergence for systems with strong nonlinearity. Aung [37] uses unscented Kalman filter (UKF) for satellite battery health management and utilizes Sigma points to estimate the covariance of battery life. Haghparast [38] utilizes the cubature Kalman filter (CKF) to identify the fuel sway parameters of the satellite and improve the accuracy of attitude feedback control. Although UKF and CKF approximate the posterior mean and variance of nonlinear functions through certain sampling strategies, their filtering accuracy suffers from certain losses. Vitorino [39] utilizes particle filtering (PF) to generate a large number of random particles to approximate the posterior distribution of the state, thereby achieving estimation of the internal charging state and temperature of lithium batteries. Compared to UKF and CKF, Vitorino has higher accuracy, but the algorithm efficiency is lower. Moreover, Wang [40] proposed a factor graph based combined positioning method that integrates inertial measurement units and Doppler velocity loggers to achieve underwater collaborative positioning of multiple unmanned systems. However, the system still cannot provide accurate attitude and heading information. Therefore, the above fusion methods have no corresponding adaptive mechanism for the fusion weight of sensors, and do not make full use of the complementary characteristics of different sensors to reduce the filtering accuracy.
To sum up, the image-based polarization sensor can obtain more robust navigation information than the point-source sensor by detecting the whole sky polarization pattern. However, there are still few researches on the image-based polarization sensor /MIMU integrated navigation system in the complex environment such as tilt and occlusion, and the existing methods have certain limitations. Therefore, this paper proposes the attitude and heading measurement method based on the adaptive complementary Kalman filter for PS/MIMU integrated system. Firstly, the adaptive complementary based on the maximum correlation entropy factor is constructed to correct the gyroscope and improve the attitude accuracy according to the non-cumulative error characteristic of the gravity measured by the accelerometers. Then, the polarization heading solving model is constructed in the inclined state, and the attitude information is used to compensate for the polarization sensor. Finally, the MIMU heading and the polarization heading are fused to improve the robustness of the integrated system. The main contributions of this paper are as follows:
1) A new attitude and heading measurement method based on the adaptive complementary Kalman filter is proposed, which can eliminate the accumulated error of the inertial navigation system through polarization sensors and accelerometers in complex environment.
2) The adaptive complementary factor is constructed with the maximum correlation entropy according to the relevance of the measured gravity and the local theoretical gravity, and further improves the attitude and heading accuracy.
The rest of this article is organized as follows. The integrated PS/MIMU attitude and heading measurement method is introduced in detail In Section 2. The Outdooro rotation test and the vehicle test of the unmanned platform are carried out, and the results are analyzed In Section 3. Finally, Section 4 summarizes the work of this paper.
2. Attitude and heading measurement based on the adaptive complementary Kalman filter
The coordinate system used in this study is defined as follows:
1) The geocentric inertial frame (i-frame): The origin is the center of the earth and ${O_i}{X_i}$ points to the vernal equinox.
2) The navigation frame ($n$-frame) is the reference coordinate system. The “East-North-Zenith” geographic coordinate system is selected as the axis direction.
3) The body frame ($b$-frame): The origin is defined as the center of the carrier and is firmly connected with the carrier. And the “Right-Front-Up” is selected as the coordinate reference direction.
4) The camera frame ($c$-frame): The origin is the center of the camera plane. ${{O}_{c}}{{Z}_{c}}$ is perpendicular to the camera plane. This article sets $c$-frame to coincide with $b$-frame.
5) The horizontal reference frame ($l$-frame): ${{O}_{l}}{{X}_{l}}{{Y}_{l}}$ is parallel to the horizontal plane of the navigation coordinate system and ${{O}_{l}}{{Z}_{l}}$ points to the zenith. ${{O}_{l}}{{X}_{l}}$ will coincide with the ${{O}_{c}}{{X}_{c}}$ axis after rotating the pitch angle and roll angle around the corresponding axis.
6) The incident light frame ($w$-frame): $P$ is the celestial observation point, the axis points to the observation direction, and $P{{X}_{w}}$ is on the observation meridian plane and perpendicular to the observation direction.
2.1 System model construction
To take advantage of the short-term stability of MIMU and the non-cumulative error characteristics of polarization sensor, we first construct the PS/MIMU integrated system model. First, we introduce the strapdown inertial attitude calculation method. The attitude differential equation in the navigation frame (n-frame) is
where $\mathbf {C}_{b}^{n}$ represents the attitude matrix of the body frame ($b$-frame) relative to the $n$-frame. ${\omega }_{nb}^{b}$ is the angular velocity of the $b$-frame relative to the $n$-frame. Due to the influence of MIMU noise, the direction cosine matrix updating equation of the carrier can be obtained after ignoring the rotation velocity of the earth: where $k$ represents the time ${{t}_{k}}$, $\mathbf {C}_{b(k)}^{b(k-1)}$ represents the attitude changing of b-frame from ${{t}_{k-1}}$ to ${{t}_{k}}$, $\mathbf {C}_{b(k-1)}^{n(k-1)}$ and $\mathbf {C}_{b(k)}^{n(k)}$ represents the strap-down attitude matrix at time ${{t}_{k-1}}$ and ${{t}_{k}}$, respectively. According to the coning error compensation algorithm and Rodrigues formula, $\mathbf {C}_{b(k)}^{b(k-1)}$ can be solved:Then we can construct the state equation according to the attitude error propagation equation. The state vector of the attitude and heading measurement system can be expressed as
The heading angle calculated by the bionic polarized light sensor is selected as the observation value. So, the observation equation can be expressed as:
${\mathbf{H}} = \left [ {\begin {array}{*{20}{c}} 0 & 0 & 1 & {{0_{1 \times 6}}} Where \end {array}} \right ]$, and ${\mathbf{V}} = [{w_{polar}}]$ represents the Gaussian noise of polarization heading.
2.2 Polarization heading measurement in the inclined state
The polarization mode is a special distribution mode formed in the sky by the polarized light generated by the sunlight scattered by small particles, which has a significant distribution rule. Under sunny weather conditions, scattering particles are mainly composed of atmospheric molecules, and their size is far smaller than the wavelength of light. Therefore, the single-order Rayleigh scattering model can be used to describe the atmospheric scattering process under sunny weather, that is, the direction of the E vector of scattering light is perpendicular to the scattering surface. The process of detecting sky polarization mode by polarization sensor is shown in Fig. 1.
Where $O$ represents the position of the observer, and $S$ represents the direction of the sun on the celestial sphere, expressed by zenith angle ${\gamma _s}$ and azimuth angle ${\alpha _s}$. $\overrightarrow {OP}$ represents the observation direction corresponding to a pixel point. In the camera frame, ${\gamma ^c}$ and ${\alpha ^c}$ are the zenith angle and azimuth angle of $\overrightarrow {O{P^c}}$, so $\overrightarrow {O{P^c}}$ can be expressed as Eq. (11). The transformation relationship from 2D pixel points to 3D observation vectors can be obtained from the internal parameters and distortion parameters of the fish-eye camera calibrated by Zhang’s calibration method [41].
The angle of polarization (AOP) and degree of polarization (DOP) of the polarization unit corresponding to $\overrightarrow {O{P^c}}$ is $\psi = 0.5\arctan \left ( {{s_2},{s_1}} \right )$ and $d = \sqrt {s_1^2 + s_2^2} /{s_0}$, according to [11]. However, the attitude angle of the polarization sensor will affect the sky area observed by the sensor, and then affect the measured AOP. Therefore, it is necessary to find the corresponding relationship between the pixel and the incident light direction, so as to realize the inclination error compensation of the bionic polarization sensor.
Figure 1 shows the AOP measurement process in the inclined state from the local meridian through the observation celestial sphere. ${\gamma ^l}$ and ${\alpha ^l}$ is the zenith angle and azimuth of the observation point under the horizontal reference frame. And the expression of $\overrightarrow {O{P^l}}$ is
whereThe detection process of polarized light is shown in Fig. 2 . After natural light is scattered by the atmosphere, it forms a regular polarized light. Firstly, the polarized light converges through the lens and passes through an array of four-channel polarizers. Then it is incident onto a photosensitive sensor based on complementary metal oxide semiconductors (CMOS) to achieve the conversion of light signals to charge signals. After the transmission electrode gate is opened, the charge signal is transmitted to the floating diffusion layer, and then converted into a voltage signal through the amplification effect of the metal oxide semiconductor field-effect transistor (MOSFET) [42]. Finally, the analog signal is converted into digital signal and output the polarization image to obtain the final polarization angle and polarization degree information:
Fig. 2. Schematic diagram of the process of detecting polarized light for bionic polarization sensor.
Furthermore, in order to calculate the solar vector ${\mathbf{e}}$ using the Rayleigh scattering model, it is necessary to calculate the expression of the polarization vector in the $w$-frame:
By substituting Eq. (18) into Eq. (17), the polarization vector ${{\mathbf{e}}^l}$ can be compensated for the inclination error in the $l$-frame. Further, according to the single Rayleigh scattering model, the vector $E$ of the scattered light is perpendicular to the scattering plane, and in the same way, it is perpendicular to the solar vector $\overrightarrow {O{S^l}}$, which is denoted as ${\left ( {{e^l}} \right )^T}\overrightarrow {O{S^l}} = {\left ( {{e^l}} \right )^T}{s^l} = 0$. Therefore, the direction of the solar vector can be estimated by two non-collinear $E$. Define $E = \left [ {{\mathbf{e}}_1^l \cdots {\mathbf{e}}_n^l} \right ]$, in which $n$ represents the number of valid pixels, so that ${E^T}{s^l} = 0$. And solving the solar vector in the $l$-frame can be expressed as an optimization problem as follows
The characteristic vector ${\mathbf{\lambda }}$ corresponding to the minimum eigenvalue of the matrix $E{E^T}$ is the best estimate of the solar vector, which can be obtained by singular value decomposition (SVD). Then, the solar meridian direction in the $l$-frame is $\alpha _{sun}^l = \arctan \left ( {{{{\mathbf{\lambda }}(2)} \mathord {\left / {\vphantom {{{\mathbf{\lambda }}(2)} {{\mathbf{\lambda }}(1)}}} \right. } {{\mathbf{\lambda }}(1)}}} \right )$.
On the other hand, according to the celestial ephemeris, the sun’s zenith angle ${{\gamma }_{S}}$ and azimuth angle ${{\alpha }_{S}}$ in the navigation coordinate system can be determined from local time and position. The heading can be calculated from the azimuth angle in the $n-$frame and the measured values $\alpha _{sun}^{l}$ in the $l-$frame
There is 180$^\circ$ ambiguity of the heading angle, which can be determined by the integrated navigation system.
2.3 Adaptive complementary Kalman filter
Although the traditional Kalman filtering method can use the polarization heading to suppress the heading drift of MIMU. However, the temperature drift of the gyroscope will be serious when it runs for a long time, resulting in the drift of the attitude calculated by the gyroscope alone. The attitude error of the system will further affect the output value of the polarization heading and reduce the heading accuracy of the integrated system. Therefore, the absolute gravity vector measured by the accelerometers and the angular rate of the gyroscope are used to realize adaptive complementarity.
Set the three-axis output of the gyroscopes and the accelerometers as $\mathrm { }\!\!\theta \!\!\text { }={{[{{\theta }_{x}},{{\theta }_{y}},{{\theta }_{z}}]}^{T}}$ and $\mathbf {a}={{[{{a}_{x}},{{a}_{y}},{{a}_{z}}]}^{T}}$. The rotation matrix of the carrier in the current attitude is ${[0,0, - 1]^T}$. First, use the rotation matrix to project the gravity vector under the $n$-frame to the $b$-frame:
Then carry out cross vector product operation between the gravity vector ${{\mathbf{g}}^b}$ under $b$-frame and the unit measured value of the accelerometer output value:
Then, we substitute the result of cross vector product $Eg$ into the PI regulator to obtain the drift compensation of gyroscope $\Delta {\mathbf{\theta }}$:
where $\Delta t$ represents the sampling interval of IMU. $ki$ is usually taken as 1. $kp$ is determined by the deviation between the resultant force ${\left | {\mathbf{a}} \right |}$ of the accelerometers and the theoretical gravity value ${\left | {{{\mathbf{g}}_{local}}} \right |}$. The traditional complementary filtering method uses the piecewise function to take the value of $kp$, which is difficult to select the appropriate interval and weight. According to the maximum correlation entropy criterion [40], the correlation between the theoretical local gravity $g_{local}$ and the measured gravity ${\left | {\mathbf{a}} \right |}$ in the process of carrier movement can be expressed by the correlation entropy based on the Gaussian kernel. Therefore, this paper proposes an adaptive complementary filtering method based on the maximum correlation entropy factor $kp$, which is expressed asThus, the corrected gyroscope measurement value ${\mathbf{\tilde \theta = \theta - }}\Delta {\mathbf{\theta }}$ can be obtained. ${\mathbf{\tilde \theta }}$ is substituted into (4) to realize the coning error compensation and attitude update method based on accelerometer correction and reduce the cumulative error of the gyroscope. Then, the time update and measurement update of the system state Eq. (7) and measurement Eq. (21) are realized by the Kalman optimal estimation. Set $\mathbf {\hat {X}}$ as the optimal estimation of the state variable $\mathbf {X}$, and the specific steps at the $k$-th moment are as follows:
Step1: State one-step prediction
Step2: Mean square error matrix of state one-step prediction
Step3: Observation discrimination According to the difference between the measured heading value $headin{{g}_{polar}}$ of the polarization sensor and the IMU heading update value $headin{{g}_{IMU}}$, the observations are screened. When $abs\left ( headin{{g}_{polar}}-headin{{g}_{IMU}} \right )<\varepsilon$, ${{Z}_{k}}$ is considered to be effective, Kalman filtering is performed to gradually complete the status update and measurement update; If $abs\left ( headin{{g}_{polar}}-headin{{g}_{IMU}} \right )>\varepsilon$, ${{Z}_{k}}$ is considered invalid, and skip the measurement update step of Kalman filter, and directly use the IMU attitude update result based on complementary filter as the output value of the system.
Step4: Filter gain calculation
Step5: State estimation
Step6: Mean square error matrix of state estimation
where $\varepsilon$ takes the empirical value of 2$^\circ$, and the initial value covariance matrix $\mathbf {P}_0$ of the Kalman filter prediction update process is set as the identity matrix. $\mathbf {Q}_0$ and $\mathbf {R}_0$ are defined as covariance matrices of system noise and observation noise respectively, and their values are related to the device errors of IMU and polarization sensor [43]. The flow chart of the attitude and heading measurement method based on adaptive complementary Kalman filter for PS/MIMU integrated system is shown in Fig. 3.3. Experiments and results
The attitude and heading measurement method based on the adaptive complementary Kalman filter for the polarization-aided inertial integrated system is tested and verified. In the experiment, the polarization sensor is composed of a polarization camera and a fish-eye lens to detect all-sky polarization mode and form an integrated attitude measurement system with the MEMS inertial measurement unit. The navigation system composed of self-made fiber-optic inertial navigation system (FINS) and GNSS is used as the verification benchmark for the heading and attitude, and the heading accuracy and attitude accuracy can reach 0.02$^{\circ }/secL$ and 0.01$^{\circ }$ respectively. First, we conducted an outdoor rotation test to verify the dynamic performance of the system in an open and non-interference environment. Then, the vehicle tests with the occlusion interference to verify the attitude and heading measurement effect of the system in practical application. The equipments and specific parameters are shown in Table 1.
Table 1. Instrument specifications.
3.1 Outdoor dynamic rotation test
The equipments used for the outdoor dynamic rotation test are shown in Fig. 4. The test was conducted on the afternoon of December 15, 2022, near the Aerospace Museum of Harbin University of Technology (longitude 126.6258, latitude 45.7278, altitude 148.74m). The polarization sensor, MIMU, and FINS are jointly fixed on the aluminum alloy bracket to avoid relative displacement. Then the aluminum alloy support is fixed on the electric turntable. After the system is powered up, it is preheated and initially aligned first, and then rotated periodically through the remote control. During this process, polarized skylight images, raw data of the MIMU gyroscope and accelerometer, and the pose information of the FINS are collected. Finally, MIMU pure inertial navigation solution algorithm, traditional Kalman filter (KF) without complementary filter, and adaptive complementary Kalman filter (ACKF) are used for attitude and heading measurement respectively.
Pitch and roll angle results for the outdoor rotation tests are shown in Fig. 5. When attitude updating is performed directly using the MIMU, the measured attitude drifts continuously due to accumulated errors of the gyroscope. When the carrier rotates in one direction, the pitch and roll angle also drift in the fixed direction. However, when the direction of rotation changes, the direction of IMU zero bias error also changes, resulting in changes in the divergence direction of pitch angle and roll angle, showing a trend of constant fluctuation. Moreover, the traditional KF method also does not make full use of the complementary characteristics of gyroscope and accelerometer, and the attitude solution results show a drift trend. However, the proposed ACKF method can effectively reduce the attitude accumulation error of MIMU. Pitch angle and roll angle errors calculated by different methods are shown in Fig. 6. The root mean square error (RMSE) values of heading results are shown in Table 2, which are calculated as follows:
$T$ is the total number of sampling points during the data collection time. ${at{t_m}}$ is the pitch, roll, or heading angle measured with different methods. ${at{t_{FINS}}}$ is the pitch, roll, and heading reference values output by FINS. The pitch and roll angle RMSE of ACKF are reduced by 91.4${\% }$ and 98.3${\% }$ relative to KF, respectively. This is because complementary filtering can simultaneously utilize the short-term stability of the gyroscopes and the no-cumulative-error characteristics of the gravity vector to reduce the random error of the accelerometers. In addition, since we have improved the traditional complementary filter by the maximum correlation entropy factor, the probability density distribution between the local theoretical gravity and the measured gravity has been further improved. Therefore, the accuracy of attitude calculation can be greatly improved. The pitch and roll angle RMSE of ACKF are similarly reduced by 91.4${\% }$ and 98.3${\% }$ relative to MIMU, which is almost equivalent to that of KF. This is because KF does not have corresponding feedback channels for the east and north misalignment angles, and the cumulative errors of pitch and roll angles cannot be effectively suppressed. However, KF can integrate the heading output from the polarization sensor, effectively suppressing heading divergence, which will be further analyzed in the following text.
Then the MIMU and polarization sensor are further fused to improve the heading accuracy. The results of the heading measurement for the rotation test are shown in Fig. 5. The accumulated heading error of the MIMU increases gradually. Although there is no cumulative error in the polarization heading, the random error is large due to the influence of the complex environment. However, the fusion-based method can combine the advantages of the two sensors and achieves higher heading accuracy. The heading errors calculated by different methods are shown in Fig. 6, and the RMSE values are shown in Table 2. Since ACKF uses adaptive factors to describe the relativity of gravity to achieve more accurate attitude compensation of the polarization sensor. Therefore, the heading accuracy has been further improved. The heading angle RMSE of ACKF is reduced by 9.5${\% }$ relative to KF, which shows the good application potential of the proposed method. Subsequently, the practical application ability of this method will be further verified.
Table 2. Attitude and heading RMSE (($^\circ$) of the dynamic rotation test.
3.2 Vehicle test of the unmanned platform
To further verify the validity of the proposed method, the unmanned platform vehicle test was carried out on December 16, 2022, near the Aerospace Museum of Harbin University of Technology. The test equipment is shown in Fig. 7. During the test, the bionic polarization sensor, MIMU, FINS, GNSS receiver, GNSS antenna, and power supply module were fixed on the unmanned platform. The polarization sensor and MIMU form the attitude and heading measurement system, and the FINS and GNSS are combined as the reference frame. The motion trajectory near the aerospace museum is shown in Fig. 8, with the green marking as the starting point and the red marking as the ending point. In addition, because the polarized light is inevitably affected by streetlights, trees, and buildings in the actual moving process, the sky AOP image detected by the polarization sensor will be seriously affected.
The pitch and roll angle results of different methods are shown in Fig. 9 and the measurement errors are shown in Fig. 10. The RMSE values are shown in Table 3. During the actual movement of the carrier, the random error of the pitch and roll angle increases obviously due to the influence of vibration, turning, and road conditions. However, the attitude accuracy of the proposed method is still significantly improved compared with pure MIMU solution and KF. The pitch and roll angle RMSE of ACKF are reduced by 89.3${\% }$ and 93.2${\% }$ relative to KF respectively.
Table 3. Attitude and heading RMSE ($^\circ$) of the vehicle test.
The heading results of different methods are shown in Fig. 9 and the heading errors are shown in Fig. 10. The heading RMSE values are shown in Table 3. Due to the longer duration of the vehicle test, the drift of MIMU headings increases significantly, and the heading RMSE is 5.4721$^\circ$. And due to the interference of AOP image, the polarization heading error changes abruptly, as shown in Fig. 10. However, the proposed ACKF can measure the heading and attitude through the adaptive complementary Kalman filter, and can well suppress the accumulated error of MIMU and the random gross error of polarization sensor. Besides, ACKF uses adaptive factors to describe the relativity of gravity to achieve attitude compensation for the polarization sensor. Therefore, the heading angle RMSE of ACKF is reduced by 9.6${\% }$ compared with KF, which shows the great advantages in the actual unmanned system navigation application.
In addition, from the enlarged inset on the left side of the heading result in Fig. 9, it can be seen that when the heading state changes, the filtered results (such as KF and ACKF) require a longer time to stabilize compared to the heading output by MIMU and polarization sensors. This is because the filter needs to estimate the expected and variance of the current moment by combining the historical state and measurement, and the heading correction requires a certain number of steps to gradually feedback into the heading measurement results, resulting in a certain lag phenomenon in the output of the heading measurement results when the state changes. In addition, this phenomenon also exists in the measurement results of the roll angle, as shown at 325s in the second row of Fig. 9. The idea of strong tracking filtering can be combined, introducing time-varying suboptimal fading factors, and adjusting the covariance matrix and gain matrix of state prediction errors in real time to improve the tracking ability for sudden changes in states. Further related research will be conducted in the future.
4. Conclusion
Aiming at the problem that the polarization sensor/MIMU integrated attitude and heading measurement system is vulnerable to inclining, occlusion, and other adverse environments, this paper proposed an attitude and heading measurement method based on the adaptive complementary Kalman filter. Combined with the non-cumulative-error characteristic of the gravity vector measured by the accelerometer, the maximum correlation entropy of the measured gravity and the local theoretical gravity is used to construct an adaptive factor to realize the adaptive complementary of the gyroscopes and the accelerometers, and improve the attitude accuracy. On this basis, the IMU heading and tilt-compensated polarization heading are fused by the Kalman optimal estimation method to improve the robustness of heading measurement. Finally, the effectiveness of the proposed method was verified by the outdoor rotation test and the vehicle test of the unmanned platform. Compared with the traditional Kalman filter, the pitch, roll, and heading angle RMSE of the vehicle test were reduced by 89.3${\% }$,93.2${\% }$ and, 9.6${\% }$ respectively, which shows great practical application value.
In the future, we will further study the robust multi-sensor fusion algorithms to measure the attitude and heading in complex environments.
Funding
National Natural Science Foundation of China (52071121, 52271311); Postdoctoral Foundation of Heilongjiang Province Government (LBH-Z22161), Chinese Aeronautical Establishment (ASFC-2022Z073077005, ASFC-2022Z022077002).
Disclosures
The authors declare no conflicts of 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.
References
1. J. T. Yang, X. Y. Xu, X. Chen, et al., “Polarized light compass-aided inertial navigation under discontinuous observations environment,” Opt. Express 30(11), 19665–19683 (2022). [CrossRef]
2. L. C. Wang, K. Li, Y. P. Chen, et al., “Single-axis rotation/azimuth-motion insulation inertial navigation control system with fogs,” Opt. Express 25(25), 30956–30975 (2017). [CrossRef]
3. Y. Zhao, H. T. Yu, K. Zhang, et al., “Fpp-slam: indoor simultaneous localization and mapping based on fringe projection profilometry,” Opt. Express 31(4), 5853–5871 (2023). [CrossRef]
4. D. Wang, J. H. Wang, Y. H. Tian, et al., “Pal-slam: a feature-based slam system for a panoramic annular lens,” Opt. Express 30(2), 1099–1113 (2022). [CrossRef]
5. H. J. Zhao, W. J. Xu, Y. Zhang, et al., “Polarization patterns under different sky conditions and a navigation method based on the symmetry of the aop map of skyligh,” Opt. Express 26(22), 28589–28603 (2018). [CrossRef]
6. X. Wang, J. Gao, and N. W. Roberts, “Bio-inspired orientation using the polarization pattern in the sky based on artificial neural networks,” Opt. Express 27(10), 13681–13693 (2019). [CrossRef]
7. Z. H. Wan, K. C. Zhao, Y. H. Li, et al., “Measurement error model of the bio-inspired polarization imaging orientation sensor,” Opt. Express 30(1), 22–41 (2022). [CrossRef]
8. W. Sturzl and N. Carey, “A fisheye camera system for polarisation detection on uavs,” in 12th European Conference on Computer Vision (ECCV), vol. 7584 of Lecture Notes in Computer Science (BERLIN, 2012), pp. 431–440.
9. J. Yang, X. Liu, Q. Y. Zhang, et al., “Global autonomous positioning in gnss-challenged environments: A bioinspired strategy by polarization pattern,” IEEE Trans. Ind. Electron. 68(7), 6308–6317 (2021). [CrossRef]
10. J. Yang, T. Du, X. Liu, et al., “Method and implementation of a bioinspired polarization-based attitude and heading reference system by integration of polarization compass and inertial sensors,” IEEE Trans. Ind. Electron. 67(11), 9802–9812 (2020). [CrossRef]
11. G. M. Li, Y. Zhang, S. W. Fan, et al., “Robust heading measurement based on improved berry model for bionic polarization navigation,” IEEE Trans. Instrum. Meas. 72, 1–11 (2023). [CrossRef]
12. D. Száz and G. Horváth, “Success of sky-polarimetric viking navigation: revealing the chance viking sailors could reach greenland from norway,” R. Soc. Open Sci. 5(4), 172187 (2018). [CrossRef]
13. S. E. Temple, M. J. How, S. B. Powell, et al., “Thresholds of polarization vision in octopus,” J. Exp. Biol. 224(7), 240812 (2021). [CrossRef]
14. W. Sturzl, “A lightweight single-camera polarization compass with covariance estimation,” in 16th IEEE International Conference on Computer Vision (ICCV), (Ieee, NEW YORK, 2017), IEEE International Conference on Computer Vision, pp. 5363–5371.
15. G. L. Han, L. L. Zhang, X. F. He, et al., “A novel orientation method for polarized light compass under tilted conditions,” IEEE Sens. J. 20(18), 10554–10563 (2020). [CrossRef]
16. P. Hu, J. Yang, L. Guo, et al., “Solar-tracking methodology based on refraction-polarization in snell’s window for underwater navigation,” Chin. J. Aeronaut. 35(3), 380–389 (2022). [CrossRef]
17. J. Dupeyroux, J. R. Serres, and S. Viollet, “Antbot: A six-legged walking robot able to home like desert ants in outdoor environments,” Sci. Robot 4(27), 12 (2019). [CrossRef]
18. S. B. Powell, R. Garnett, J. Marshall, et al., “Bioinspired polarization vision enables underwater geolocalization,” Sci. Adv. 4(4), 8 (2018). [CrossRef]
19. Z. H. Wan, K. C. Zhao, and J. K. Chu, “Robust azimuth measurement method based on polarimetric imaging for bionic polarization navigation,” IEEE Trans. Instrum. Meas. 69(8), 5684–5692 (2020). [CrossRef]
20. Q. H. Li, Y. Hu, Q. Hao, et al., “Skylight polarization patterns under urban obscurations and a navigation method adapted to urban environments,” Opt. Express 29(25), 42090–42105 (2021). [CrossRef]
21. K. Wu, H. Zhang, Y. Chen, et al., “All silicon microdisplay using efficient hot-carrier electroluminescence in standard 0.18μm cmos technology,” IEEE Electron Device Lett. 42(4), 541–544 (2021). [CrossRef]
22. C. Marques, A. Leal-Júnior, and S. Kumar, “Multifunctional integration of optical fibers and nanomaterials for aircraft systems,” Materials 16(4), 1433 (2023). [CrossRef]
23. Z. W. Xian, X. F. He, J. X. Lian, et al., “A bionic autonomous navigation system by using polarization navigation sensor and stereo camera,” Auton Robot 41(5), 1107–1118 (2017). [CrossRef]
24. X. F. He, L. L. Zhang, C. Fan, et al., “A mimu/polarized camera/gnss integrated navigation algorithm for uav application,” in 2nd International Conference on DGON Inertial Sensors and Systems (ISS), (Ieee, NEW YORK, 2019).
25. T. Du, C. Z. Tian, J. Yang, et al., “An autonomous initial alignment and observability analysis for sins with bio-inspired polarized skylight sensors,” IEEE Sens. J. 20(14), 7941–7956 (2020). [CrossRef]
26. Q. F. Dou, T. Du, S. P. Wang, et al., “A novel polarized skylight navigation model for bionic navigation with marginalized unscented Kalman filter,” IEEE Sens. J. 22(5), 4472–4483 (2022). [CrossRef]
27. A. Bucholtz, “Rayleigh-scattering calculations for the terrestrial atmosphere,” Appl. Opt. 34(15), 2765 (1995). [CrossRef]
28. X. Hu, D. Saez-Rodriguez, C. Marques, et al., “Polarization effects in polymer fbgs: study and use for transverse force sensing,” Opt. Express 23(4), 4581–4590 (2015). [CrossRef]
29. J. Xie, X. F. He, J. Mao, et al., “A bio-inspired multi-sensor system for robust orientation and position estimation,” in IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), (2021), pp. 1570–1577.
30. D. H. Zhao, X. C. Liu, H. J. Zhao, et al., “Seamless integration of polarization compass and inertial navigation data with a self-learning multi-rate residual correction algorithm,” Measurement 170, 108694 (2021). [CrossRef]
31. J. Cai, J. H. Cheng, J. X. Liu, et al., “A polar rapid transfer alignment assisted by the improved polarized-light navigation,” IEEE Sens. J. 22(3), 2508–2517 (2022). [CrossRef]
32. Q. F. Dou, T. Du, Z. B. Qiu, et al., “An adaptive anti-disturbance navigation method for polarized skylight-based autonomous integrated navigation system,” Measurement 202, 111847 (2022). [CrossRef]
33. P. W. Hu, P. P. Huang, Z. B. Qiu, et al., “A 3d attitude estimation method based on attitude angular partial feedback for polarization-based integrated navigation system,” Sensors 22(3), 710 (2022). [CrossRef]
34. C. Shen, Y. F. Xiong, D. H. Zhao, et al., “Multi-rate strong tracking square-root cubature Kalman filter for mems-ins/gps/polarization compass integrated navigation system,” Mech. Syst. Signal Process. 163, 108146 (2022). [CrossRef]
35. Y. T. Yang, Y. Wang, X. Yu, et al., “Moonlit polarized skylight-aided ins/cns: An enhanced attitude determination method,” Control. Eng. Pract. 132, 105408 (2023). [CrossRef]
36. D. A. Carbot-Rojas, G. Besancon, R. F. Escobar-Jimenez, et al., “Ekf based sensor fault diagnosis for an internal combustion engine,” in 2019 23RD INTERNATIONAL CONFERENCE ON SYSTEM THEORY, CONTROL AND COMPUTING (ICSTCC), R. Precup, ed. (2019), International Conference on System Theory Control and Computing, pp. 43–48. 23rd International Conference on System Theory, Control and Computing (ICSTCC), Sinaia, ROMANIA, OCT 09-11, 2019.
37. H. Aung, J. J. Soon, S. T. Goh, et al., “Battery management system with state-of-charge and opportunistic state-of-health for a miniaturized satellite,” IEEE Trans. Aerosp. Electron. Syst. 56(4), 2978–2989 (2020). [CrossRef]
38. B. Haghparast, H. Salarieh, H. N. Pishkenari, et al., “A cubature Kalman filter for parameter identification and output-feedback attitude control of liquid-propellant satellites considering fuel sloshing effects,” Aerosp. Sci. Technol. 144, 108813 (2024). [CrossRef]
39. V. Biazi, A. C. Moreira, J. L. Pinto, et al., “A particle filter-based virtual sensor for estimating the state of charge and internal temperature of lithium-ion batteries: Implementation in a simulated study case,” J. Energy Storage 61, 106814 (2023). [CrossRef]
40. Q. Wang, S. Fan, Y. Zhang, et al., “A novel adaptive sliding observation-based cooperative positioning algorithm under factor graph framework for multiple uuvs,” Ieee Trans. on Ind. Informatics p. 11 (2022).
41. Z. Y. Zhang, “A flexible new technique for camera calibration,” IEEE Trans. Pattern Anal. Machine Intell. 22(11), 1330–1334 (2000). [CrossRef]
42. K. Xu, “Silicon electro-optic micro-modulator fabricated in standard cmos technology as components for all silicon monolithic integrated optoelectronic systems,” J. Micromech. Microeng. 31(5), 054001 (2021). [CrossRef]
43. Q. Sun, Z. Tang, J. P. Gao, et al., “Short-term ship motion attitude prediction based on lstm and gpr,” Appl. Ocean. Res. 118, 102927 (2022). [CrossRef]
