An Intelligent Adaptive Kalman Filter for Integrated Navigation Systems

This paper presents an intelligent adaptive Kalman filter based on deep neural network and fuzzy logic for integrated navigation systems. The highlight is that the process noise covariance is obtained by analyzing inertial sensors’ output sequence rather than based on auxiliary measurements. Such novel method realizes the decoupling estimation of process noise covariance and measurement noise covariance, and the sequence-to-sequence process of sensors analyzer gains high real-time performance. The model of sensors analyzer is constructed and an incremental subset training method is proposed. The effects of noise covariance error are analyzed. A numerical experiment is conducted, which shows promising accuracy and robustness improvement of the proposed method.


I. INTRODUCTION
As explained in [1], there are mainly two categories of positioning techniques in navigation, namely, dead reckoning and direct positioning. Dead reckoning does not measure the position directly, but by accumulating increment to update the position recursively. Direct positioning obtains position directly via satellite, radio, astronomy, environmental features, etc. Their most representative examples are strap-down inertial navigation system (SINS) and global positioning system (GPS) respectively. The integration between SINS and auxiliary navigation system, such as GPS, is a the-state-of-the-art technique for many application scenarios [2]- [5].
Kalman filters and its variants are the most basic and widely used data fusion algorithms in integrated navigation systems. Under Gaussian and linear approximation, Kalman filter is an optimal estimation algorithm with the prerequisites of accurate system modeling and prior knowledge of noise. Unfortunately, inaccuracy in system modeling and the uncertainty of working environments are inevitable, which will reduce the precision and robustness of conventional Kalman filters. Therefrom, two main approaches of adaptive Kalman filters are developed, namely, innovation-based adaptive estimation (IAE) and multiple-model-based adaptive estimation (MMAE) [6], [7]. An IAE technique is developed to estimate The associate editor coordinating the review of this manuscript and approving it for publication was Liang Hu . process noise covariance Q and/or measurement noise covariance R by extracting noise information contained in measurement sequence during the process of filtering. Due to the complexity of integrated navigation systems and the limitation of auxiliary measurements, the IAE method is only effective in the estimation of R. In an MMAE technique, a bank of Kalman filters runs in parallel with different parameters, and the weights of each sub-filter are dynamically adjusted according to new measurements. After proper analysis, the sub-filter with the most accurate parameters prevails. However, the MMAE method needs more resources and does not certainly cover the true system model. There is also a limitation that both IAE and MMAE techniques need a sliding window over measurement sequence during the process of parameter estimation, which will lead to inaccuracy and delay.
Subsequently, adaptive filters in integrated navigation systems introduce more intelligent elements. Fuzzy logic techniques are broadly used as a selector in adaptive filters of integrated navigation systems, and such a selector dynamically adjusts the noise covariance matrix or chooses the optimal sensors [8]- [11]. Artificial neural networks (ANN) are extensively used owing to their capabilities of self-learning, storage, parallelism, and strong non-linearity. For SINS/GPS integrated navigation systems, SINS loses its calibration source during GPS outages. In order to avoid the system instability caused by GPS outages, many studies (such as [12]- [17]) utilized ANN as the calibration source for SINS.
These methods usually employ simple neural networks to fit missing data and require real-time training of neural networks. Therefore, they are inevitably computationally expensive and lack mathematical rigor.
Almost all the methods mentioned above are difficult to estimate the process noise covariance Q. Some new algorithms are proposed to cope with the case where Q is unknown, and many of them are dependent on innovation sequences [18]- [23]. But these methods still require Q not to change too fast. When Q varies rapidly, the estimation results of the above adaptive filters are deteriorated.
Furthermore, most methods rely heavily on GPS measurement. The innovation sequence derived from the difference between GPS and system state estimation contains the information of both measurement noise covariance R and process noise covariance Q. Thus, the simultaneous parameter estimation involves the coupling of multiple noises. The estimation errors of Q and R will affect each other. This makes the estimation more complex and difficult. For the adaptive Kalman filtering of integrated navigation systems, the weak observability of Q and the coupling between Q and R are the two main constraints. If an effective independent estimation of Q can be achieved, these shortcomings can be overcome.
In fact, for the SINS/GPS integrated navigation system, the variation of Q can be intuitively reflected in the output sequence of the inertial measurement unit (IMU). For example, engineering experiences show that the performance of IMU will be degraded by rapid attitude variations, severe vibrations, physical impacts, etc. Because the IMU is a sensor that is free from external interference, it is feasible to establish a knowledge base of the correspondence between motion patterns and sensor performance through statistics. Based on this, Q can be obtained by identifying these patterns and querying the knowledge base. Deep learning technology has developed quickly, and many specialized neural computing chips have appeared. Deep neural network (DNN) deployment is possible even in embedded applications. The complex structure makes DNNs have strong information refining capability, and well-defined networks have strong generalization capability. This inspired us to use DNN to identify motion patterns.
In this paper, a flexible deep neural network base on recursive neural networks is constructed and an incremental subset (IS) training method is proposed. It is a real-time sequence-to-sequence network. The DNN monitors the IMU output sequence, identifies certain motion patterns, and obtains the Q during the filtering process of the integrated navigation system. And the establishment of the pattern-to-Q knowledge base is simplified by using fuzzy logic. The auxiliary measurement, such as GPS, is easily affected by external disturbances, so it is difficult to use a similar method to establish the corresponding relationship between its output and measurement noise covariance. This paper uses the IAE technique to estimate R. The effect of misestimating process noise covariance is analyzed and a forgetting factor is introduced to reduce the influence of the misestimation. A numerical experiment is conducted, and the results show promising accuracy and robustness improvement of the approach. This paper innovatively uses artificial intelligence to estimate the noise of inertial sensors. The effectiveness of this method benefits from the independence and repeatability of inertial sensors. The entire estimation process is independent of the filter, so the coupling with auxiliary measurement is avoided. This method overcomes the two difficulties faced by conventional adaptive filters. The IAE technique is used to estimate the measurement noise parameters, and finally, the simultaneous estimation of process noise and measurement noise is achieved.
The rest of this paper consists of four parts. Firstly, the fuzzy-DNN adaptive Kalman filter algorithm is proposed and the SINS/GPS integrated navigation model is described. Then, the structure and the training method of the neural network are proposed. Next, the effect of estimation bias is analyzed. The last part is the numerical results and discussion.

II. FUZZY-DNN ADAPTIVE KALMAN FILTER AND INTEGRATED NAVIGATION SYSTEM MODELING A. FUZZY-DNN ADAPTIVE KALMAN FILTER ALGORITHM
Considering a general discrete-time Gaussian linear system: where x k ∈ R n denotes state vector; z k ∈ R m denotes measurement vector; k/k−1 ∈ R n×n denotes state transtion matrix; H k ∈ R m×n denotes measurement matrix. w k ∈ R n is process noise vector; v k ∈ R m is measurement noise vector and they meet the following restrictions: Under the linear Gaussian approximation, Kalman filters can obtain the optimal estimation of the state. But with insufficient prior knowledge, a Kalman filter may reduce the accuracy of the estimation or even produce a deviation. For example, if R or Q is smaller than the actual value, the uncertainty range of the estimated state will converge prematurely; if R or Q is bigger than the actual value, the filter will slow down the convergence process. The mismatch of noise parameters will lead to errors and even system instability.
For SINS/GPS integrated navigation systems, the φ-angle error model of SINS is applied to construct (1) and z k is the difference between SINS and GPS. Thus Q is the noise covariance of IMU and R is the noise covariance of GPS. The Q of IMU is related to gyroscope drifts and accelerometer biases, which is difficult to be estimated precisely during the conventional filtering process. R is related to GPS accuracy, which is less coupled with the integrated navigation system and thus can be effectively estimated using the IAE technique. The estimation of R utilizes the covariance-matching technique developed in Sage-Husa adaptive Kalman filter [24]. From this point of view, this paper introduces a fuzzy-DNN VOLUME 8, 2020 Algorithm 1 Fuzzy-DNN Adaptive Kalman Filter inputs: measurements z k , IMU measurements (a, ω) outputs: state estimationx k initialization:x 0 , P 0 , Q 0 , R 0 , s (forgetting factor) 1: k ← 0 2: while Navigation do 3: if R k is undetermined then 10: adaptive Kalman filter for the situation where both Q and R are uncertain. The detailed procedure is as algorithm (1). Note that the s in step 8 is the forgetting factor which is used to limit the weight of historical information in the adaptive filter. The meanings of some notations in algorithm(1) are as follows: a denotes acceleration; ω denotes angular velocity; M pattern denotes motion pattern.
The innovation of the algorithm is to use DNN and fuzzy logic to estimate Q, which is hard for conventional adaptive methods, by analyzing the IMU's output sequence directly. The IMU error model is complex, the accuracy of IMU differs due to changes in working environments. That is to say, Q is related to the working status of IMU. [25] describes the most significant parameters affecting IMU performance: temperature, vibration, mechanical abuse, etc. In this paper, we assume that the temperature effect is weakened by temperature compensation and mechanical abuse such as impact is avoided in our application. Therefore, the function of fuzzy-DNN is simplified as the mapping between vibrations and Q. Due to the fact that there is a deviation of system modeling and the misestimation of Q, the usefulness of the filter may be nullified, and in order to limit the influence, a forgetting factor s is used to limit the memory of the filter by exponentially fading past data [26]. The estimation of R is conducted using the covariance matching technique, which is a conventional adaptive method.

B. SINS/GPS INTEGRATED NAVIGATION SYSTEM ERROR MODEL
The working of SINS involves conversion between multiple coordinate frames including inertial frame (i), earth frame (e), navigation frame (n), and body frame (b). The mutual conversion between different frames is realized by a rotation matrix C. The subscript behind C indicates the source frame, and the superscript indicates the target frame. The φ-angle error model is used in this paper, which is defined as the angle error between the calculated n -frame and the true n-frame during the calculation of SINS. The n-frame uses three directions of east-north-up (E-N-U) to define the positive direction in this paper.
For SINS/GPS integrated navigation system, an error model of SINS is usually established and the optimal estimation of the error state is obtained through a filter. Finally, the error state is used to correct SINS to realize the closed-loop of the integrated navigation system. The error state x contains attitude error (φ), velocity error (δv), position error ([δL, δλ, δh] T ), and sensor's biases ([ε g , ε a ] T ). The φ-angle error model of SINS is as follows: where ω denotes angular velocity; f denotes specific force; g denotes gravitational acceleration; R m denotes main curvature radius of meridian and R n denotes main curvature radius of prime unitary; τ denotes time constant of sensors' constant bias; w denotes white noise of sensors.
As the auxiliary measurement of SINS/GPS integrated navigation system, GPS is error-stable and outputs longitude, latitude, and height for navigation. Thus the measurement model consists of three states.
Equations (6-13) constitute the error model of SINS/GPS integration navigation system.

III. STRUCTURE OF THE FUZZY-DNN AND THE INCREMENTAL SUBSET TRAINING METHOD A. STRUCTURE OF THE FUZZY-DNN
The task of the DNN is to assist the filter in determining the validity of the IMU output. It monitors the output sequence of IMU and gives a real-time decision on what state the sensors are in. Therefore, it's a sequence-to-sequence pattern recognition process with high real-time requirements.
In the field of deep learning, the basic structure of complex neural networks can be roughly divided into five categories. They are multilayer perceptrons (MLP), convolutional neural networks (CNN), recurrent neural networks (RNN), graph neural networks (GNN), and residual networks (ResNet). Their structures are designed for specific application scenarios. Generally speaking, MLP has a simple structure and serves as a glue between networks. CNN reduces the number of network parameters to be optimized by extracting features of input data through multiple convolution layers which makes it an attractive deep learning structure. RNN transfers the state cyclically in its own network, so it is compatible with a wide range of time series structure inputs. GNN focuses on the topological structure in the data by modeling a set of objects and their relationships. It is a method of processing graph domain information. ResNet is proposed to solve the problem of network degradation when there are too many hidden layers in deep neural networks. It has advantages in the training of extremely deep networks.
Considering the characteristics of these basic neural networks, GNN and ResNet are excluded because they are not suitable for our application. CNNs are most suitable for classification scenarios, but it is also necessary to consider real-time performance. A CNN requires block data as input, so if the input data is serialized, then its output has a considerable delay for accumulating enough data. Hence, CNN does not meet the requirements of our application. RNN is capable to remember and recognize patterns in sequences, and it is easy to realize the structure of one-to-one correspondence between input and output. But simple RNN cannot solve the problem of exponential explosion or vanishing gradient with recursion, and it is difficult to capture long-term correlation. Gated recurrent unit (GRU) networks solve these problems by using an update gate and reset gate to decide what information should be passed to the output. The structure makes GRU be able to keep information from long ago without washing it through time and remove information which is irrelevant to the prediction [27].
In summary, GRU networks are used as the core of the fuzzy-DNN to identify motion patterns implicit in the sensor output sequence. At the same time, MLPs are used to standardize the output of GRU networks and integrate information in multi-sensor scenarios. The structure of the fuzzy-DNN is shown as Fig.(1). Each neural network subblock corresponds to a sensor, and finally, the outputs of these subblocks are synthesized through a full-connected network.
The other part of the fuzzy-DNN is the fuzzy logic unit. It is a huge project to establish correspondence between all motion patterns and noise covariance. This work can be simplified by using fuzzy logic. The details of the approach are to establish several key correspondences, and the others are approximated by the membership function of fuzzy logic. According to Remark 1, the negative effect of vibration on inertial sensors is measured by the power to maintain the vibration. Therefore, the neural network can use a scalar to label a vibration environment. The installation environment of inertial sensors has specific vibration characteristics. Under several common vibration conditions, the noise parameters of inertial sensors can be obtained by testing. Take this Algorithm 2 Incremental Subset Training Method inputs: data set A, DNN parameters θ, loss function L(A, θ), gradient g = ∇ θ L(A, θ) outputs: optimized DNN model θ o 1: Split A: A 1 ⊆ A 2 · · · ⊆ A k · · · ⊆ A 2: Set loss function target L 0 3: η ← 0.001, where η is the learning rate 4: β 1 ← 0.9, β 2 ← 0.999, ← 10 −8 5: while A k = A do 6: 11: as the reference database of fuzzy logic. The quantification of vibration environments by neural network makes it possible to measure the relationship between two different vibration environments by using memberships. According to the reference database and the membership, the noise parameters of inertial sensors in other vibration environments can be obtained.

B. THE INCREMENTAL SUBSET TRAINING METHOD
Backpropagation algorithms are usually used to efficiently train artificial neural networks through a gradient descent approach that exploits the chain rule. Firstly, a loss function is defined to describe the difference between the network output and the actual value. Then, backpropagation is used to adjust the parameters of neurons by the gradient descent optimization algorithm. The training of the neural network is completed until the value of the loss function drops to a steady-state. The gradient descent optimization algorithm adopted in this paper is called Adam optimizer [28]. The training of GRU networks is much harder than other networks such as CNNs. The time-dependent structure makes GRU networks difficult to train in parallel and the training speed is also limited by high memory bandwidth requirements.
In order to accelerate the training process, we present an incremental subset (IS) training method. The core idea of this method is to let the neural network quickly learn the outline of the data set, and then refine it in details. The procedure is shown as algorithm (2). The following is the detailed description of the method in the form of set theory. Set A might be thought of as the universe set or the set of all training data. A k , k ∈ N + might be thought of as the subset of A or the set of partial training data, and they obey the following formula: In incremental form, A k can be expressed as: where k−1 = A k \A k−1 represents set increment. With (15), recursively expanding (14), the incremental representation of A can be obtained: It is hard and not robust to train the GRU network directly using a huge volume of training sets. Pre-training the GRU network with a simple training subset is effective for accelerating further training process. Then the pre-trained network is further trained on more complex data sets.
Step by step, the retraining of the model is performed on a new data set A k until the last data set increment. The selection of subsets has some notes: the initial subset should be as simple as possible so that the network has an easy start for training; avoiding too much data increments per step can make each training step stable and fast. This method also helps to increase the scalability of the model. For example, adding a new situation is equivalent to a data increment k . The core idea of this method is to incrementally learn new knowledge on the basis of the original knowledge.

IV. ANALYSIS OF Q ESTIMATION BIAS AND EXTERNAL DISTURBANCE DYNAMICS A. ANALYSIS OF Q ESTIMATION BIAS
In fuzzy-DNN adaptive Kalman filter, the noise parameter Q is estimated in real-time by a fuzzy-DNN, so inevitably there is inaccuracy. The influence of this bias on the filter is analyzed as follows. Note that the estimation of Q k obtained by the fuzzy-DNN is a diagonal matrix.
Theorem 1: For Kalman filters, if the biased estimation of process noise covariance matrix Q * k can be expressed by a coefficient λ k , scilicet Q * k = λ k Q k . Then the effect of misestimation is equivalent to adding a coefficient 1/λ k to the measurement noise covariance matrix R k+1 , scilicet R * k+1 = 1/λ k · R k+1 .
Proof: Suppose the k-time estimated value Q * is expressed as: where Q k is the true value, and Q k is the estimation bias; λ k is the bias coefficient. The one step state prediction of DNN-based real-time adaptive Kalman filter is as follows: x The one-step state prediction covariance matrix can be obtained by the following formula: Substituting (17) into (19) gives: Multiply at both sides of 1/λ k−1 , thus: Take 1/λ · P * as a whole, and refer to as P, therefore: The Kalman gain can be obtained as follows: The state estimation is derived from the following formula: The state estimation covariance matrix is obtained by: Multiply at both sides of 1/λ k−1 , thus: Similar to (22), take 1/λ · P * as a whole, thus: Since the effect of the forgetting factor s is to reduce the filter inertia by attenuating the role of historical information, with careful selection it does not influence the filtering analysis. So we ignore it for the time being. Comparing Eqs. (18,22,23,24,27) with Kalman filter algorithms, the difference in form is 1/λ k−1 ·R k in (23) while it is R k in Kalman filter. Therefore, Q * k = λ k Q k ⇔ R * k+1 = 1/λ k · R k+1 . The theorem indicates that in k-time the misestimation of Q k indirectly has a reversed effect on R k+1 , e.g. misestimating Q as a larger value is equivalent to misestimating R as a smaller value. That is to say the greater the process noise misestimated, the smaller the measurement noise mistaken by the filter and vice versa. The actual state prediction covariance matrix P * k = λ k−1 P k . According to Kalman filter algorithms, P k is the suboptimal state estimation covariance matrix under condition Q k−1 , 1/λ k−1 · R k . Thus P * k is the suboptimal estimation covariance matrix under condition λ k−1 Q k−1 , R k . Therefore, the misestimation of Q k−1 has a co-directional influence on P * k . Scilicet, the greater the process noise misestimated, the greater the state estimation covariance matrix obtained by the filter and vice versa. In general, the misestimation of Q influences the credibility of the measurements for the filter and causes the bias of the state estimation covariance.
Some previous researches have proposed several stability criteria of Kalman filters. However, it is very difficult to use these criteria for high-dimensional and time-varying systems. In addition, the working time of the filter in the real system is limited, and the absolute stability, in theory, will not appear. For the Kalman filter of integrated navigation systems, some states are weakly observable or even unobservable in practice. For example, the constant biases of the gyroscope and accelerometer are weakly observable states that are difficult to accurately estimate. But the error of these states can affect the accuracy and stability of the filter. Thus, in order to analyze the bound of filtering error, we first assume that the filter of the integrated navigation system is stable. In this way, the state estimation process of the Kalman filter can be equivalent to the optimal estimation of two vectors of the same dimension.
Assumption 1: Under the condition that noise parameters are accurately known, the Kalman filter of the integrated navigation system is stable.
Assumption 2: Larger errors in noise parameters lead to larger filter errors.
Theorem 2: With assumptions 1 and 2, the proposed filter has a certain bound of filtering error.
Proof: With assumption 1, the Kalman filtering process of the system state sequence and the measurement sequence can be equivalent to the optimal estimation of two vectors with full-rank covariances. These two vectors are represented here as m 1 ∈ R n and m 2 ∈ R n . They obey Gaussian distribution and the covariances are represented as 1 ∈ R n×n and 2 ∈ R n×n , where rank( 1 ) = rank( 2 ) = n. According to Bayes estimation, the optimal estimation of two vectors obeying Gaussian distribution is obtained by the following formulas.
where m denotes the optimal estimation; denotes the covariance of m. In Kalman filters, the K is called Kalman gain.
With assumption 2, there must be a scalar coefficient λ limiting the error boundary of process noise parameter estimation, namelyˆ 1 .ˆ where denotes within the bound. The upper bound of the filtering error can be determined by λ · 1 . Note that λ represents the degree of deviation from the accurate value. Thus, λ = 1 indicates that the deviation is the lowest. Let * 1 = λ · 1 . According to theorem 1, the bound of the estimated measurement noise covariance is * 2 = 1/λ · 2 . Thus, Therefore, m * and * in Eqs.(33, 34) are the bound of the filtering results. Thus, the error is also bounded. The theorem 2 can be obtained.

B. EXTERNAL DISTURBANCE DYNAMICS
SINS is fixedly mounted on the carrier, so it will be inevitably affected by the external environment which acts on the carrier during the process of navigation, such as jitters, turbulence, shocks, bumps, etc. Along with these external disturbance sources, vibration is a commonly generated long-term phenomenon and it has a significant impact on the performance of SINS. The detailed analysis of vibration effects on IMU is described as follows.
Firstly, the sensitive element of the IMU is approximated as a mass-spring-damper system, thus its position x can be described by the following differential equation.
where m denotes mass; c denotes damping coefficient; k denotes spring constants. Define two new parameters: ω = √ k/m, ζ = c/(2 √ km), then the formula can be rewritten as follows:ẍ The solution of the equation is When there is no external force, the system cannot maintain any motion state due to the effect of the damper. For IMU, the external disturbance provides an external force F maintaining a vibration state. In this case, the dynamic equation of the system is as follows: Now consider a special situation: the force F maintains the vibration at frequency f . Equation (38) is transformed as follows: According to (37), when ζ = 0, ω = 2π f , that is, k/(m + m ) = 2πf , (c + c ) = 0, the system maintains a vibration at frequency f . Therefore, where A denotes amplitude of the vibration. Substituting (41) into (40) gives: The power of force F is derived from following formula: The average power can be obtained by integrating and averaging.
Remark 1: The average power to maintain a simple harmonic vibration caused by external disturbance is P = 2cA(πf ) 2 . This means that high frequency and severe vibration have a greater negative impact on IMU performance.

A. TRAINING OF FUZZY-DNN
The fuzzy-DNN monitors sensor's output sequence and gives the identification results of motion patterns. As mentioned earlier in this paper, we focus on the correspondence between vibrations and IMU performance. The training of the DNN involves massive labeled fragments of sensor outputs. Considering the complexity of vibrations, we use the average maintaining power based on (44) to label the vibrations. The training data of the DNN is recorded from an actual inertial sensor (ADIS16488A from Analog Devices). And multiple stochastic simple harmonic vibrations are superimposed to simulate real vibrations (including random numbers, random amplitudes, random phases, and random frequencies). For convenience, the sensor output and the label are normalized to between 0 and 1.
The cyclic structure of GRU networks makes its training more difficult than other networks of the same size. In our practice, some optimizers are less effective, such as SGD, RMSprop, and Adadelta. In the comparison of training methods, two more suitable optimizers Adam and Adamax are used. The commonly used mean squared error (MSE) loss function, which describes the difference between two sequences, is used as the target function in training. As shown in Fig.(2a) and Fig.(2b), the proposed IS training method significantly speeds up the training process. According to IS training method, the training task is divided into two progressive tasks. Firstly, a subset of the whole training data set is used to train the DNN until the loss function reaches a preset target. Then, the universal data set is used to retrain the DNN until the training target is achieved. The core idea of this method is outlining and depicting, where outlining speeds up the training and depicting makes up the generalization. The effectiveness of the method is well demonstrated in the current experiment. Fig.(2c) shows an evaluation sample of the fuzzy-DNN, where the multi-frequency compound vibration is described as an intensity value in real-time. The evaluation of the DNN shows that the DNN can process about 33 seconds of data per second using a laptop graphic card (Nvidia GTX 1060MQ).
The correspondence between vibrations and covariances is obtained by the statistics of device tests, but it is not realistic to test every possible vibration. We divide the vibrations into four cases: none, slight, moderate, and severe, and then TABLE 1. The performance of SINS in different motion conditions. (ε denotes gyro constant bias. w denotes random walk. Subscripts g and a denote gyroscope and accelerometer respectively. The unit mg denotes milli earth gravity). obtain the covariance under each case as shown in table (1). And the others are calculated by fuzzy logic as shown in Fig.(2d).

B. INTEGRATED NAVIGATION SYSTEM SIMULATION
The simulation of the SINS/GPS integrated navigation system is to verify the adaptability of the proposed algorithm to the variations of sensor parameters. The performance variation of IMU and the error fluctuation of GPS are evaluated during the simulation. They correspond to the variations of Kalman filter's process noise parameters Q and measurement noise parameters R respectively. The flight trajectory of the unmanned aerial vehicle (UAV) is shown in Fig.(3). The flight trajectory is generated using actual aircraft parameters (based on F-16 fighter). In the three flight segments, the UAV is in three states: none vibration, slight vibration, and severe vibration. And in each segment, GPS produces error fluctuations.
The detailed error model of gyroscopes and accelerometers is very complicated. The simplified version is a combination of zero drift and random walk, which can be described as constant bias plus Gaussian white noise, and the variation of IMU performance caused by external disturbances can be described by the variation of the two parameters. In the numerical experiment, two situations which are defined as the i-type and the ii-type respectively are evaluated. The i-type denotes that only the noise parameters vary with the external disturbance, while the ii-type denotes that both noise parameters and constant biases vary with the external disturbance. Which type is suitable depends on the specific inertial sensors. The details are shown in table (1), and the values in parentheses belong to i-type sensors. For the i-type situation, appropriate modification of filter parameters Q can accurately describe the variation. However, conventional methods are hard to deal with the ii-type situation due to that the states of constant biases are hardly observable for Kalman filters. The proposed method in this paper can deal with this situation commendably. Benefit from that the fuzzy-DNN directly analyzes the IMU output and is very sensitive to the variations of its performance, the action that appropriately increasing the filter states covariance matrix P to temporarily reduce the reliability of the filter states can be promptly implemented. This action can reduce the contamination of the filter caused by parameter mismatch. Figs. (4,5) separately show the two cases, where 3σ denotes 99.7% boundary of Gaussian distribution. If the states fall outside the 3σ range, it indicates deviations in the state estimation of Kalman filters.
The comparative experiment is carried out between IAE-based adaptive Kalman filter and fuzzy-DNN adaptive Kalman filter. Fig.(4) shows comparisons under the i-type case. Under the condition of no vibration or slight vibration, the accuracy of the two methods is not much different. The reason for this phenomenon is that the floating range of noise is small relative to the constant biases and the latter is still the main error of IMU. But there are still some mismatches of Q for IAE-based filter. The 3σ range is slightly smaller than the actual range, so some states are slightly exceeded. Under the condition of severe vibration, the fuzzy-DNN adaptive Kalman filter still works well. While the IAE-based adaptive Kalman filter produces worse mistakes and the estimation of R is almost ineffective. The reason for this phenomenon is the coupling between Q and R, and the mismatch of Q directly leads to the error of R estimation. Fig.(5) shows comparisons under the ii-type case. Some results are similar to those in Fig.(4), but with greater errors. The difference is that the IAE-based method produces a significant upward error under  the condition of slight vibrations. The variation of constant bias directly leads to the variation of filter state, which cannot be described by Q. At the same time, the altitude of the UAV changes less in the flight process, resulting in a lack of correction in altitude. These two factors lead to large errors in height. In comparison, the fuzzy-DNN adaptive Kalman filter still works well. Fig.(6) shows the comparison between fuzzy-DNN adaptive Kalman filter and fuzzy-DNN adaptive unscented Kalman filter. The advantage of nonlinear filters is that it can handle nonlinearity better, but the cost is greater computational complexity. The nonlinearity of SINS is mainly reflected in the angle error. The nonlinearity is not significant when the angle error is small, so there are only slight differences between the two filters. It can be observed from the comparison that the slight advantage of the fuzzy-DNN adaptive unscented Kalman filter is not worth several times the computational cost. In summary, the fuzzy-DNN adaptive Kalman filter is good enough for our application.

VI. CONCLUSION
A novel intelligent adaptive Kalman filter is proposed to deal with the variability of the sensors' performance in GPS/SINS integrated navigation systems. The proposed technique obtains process noise parameters in real-time by monitoring the output sequence of IMU, and the measurement noise parameters are obtained by an IAE-based algorithm. The advantage of this method is to realize the decoupling of complex objects and direct observation of deep states. And it can be easily grafted on conventional filters, so it has good practicability and flexibility. The numerical results of the integrated navigation system show that the proposed method achieved higher navigation accuracy and better robustness.
FEI YAN was born in Suqian, Jiangsu, China, in 1992. He received the B.S. degree in automation from the Nanjing University of Science and Technology (NUST), Nanjing, China, in 2014, where he is currently pursuing the Ph.D. degree in control theory and control engineering. His current research interests include intelligent adaptive filter, integrated navigaiton systems, and deep learning.
SHENG LI was born in Xuzhou, Jiangsu, China, in 1976. He received the Ph.D. degree in control theory and control engineering from the Nanjing University of Science and Technology (NUST), in 2005. He is currently an Associate Professor with the School of Automation, NUST. His research interests include nonlinear system control, process control, integrated navigation, and robot control.
ENZE ZHANG was born in Yangzhou, Jiangsu, China, in 1989. She received the B.S. degree in automation and the M.S. and Ph.D. degrees in control theory and control engineering from the Nanjing University of Science and Technology (NUST), in 2011 and 2017, respectively. She is currently a Lecturer with Yangzhou University, Yangzhou. Her research interests include multi-objective optimization and particle swarm algorithm.
QINGWEI CHEN was born in 1963. He received the B.S. degree in electrical engineering from Jiangsu University, Zhenjiang, China, in 1985, and the M.S. degree in automation and the Ph.D. degree in control theory and control engineering from the Nanjing University of Science and Technology (NUST), Nanjing, China, in 1988 and 2005, respectively. Since 2001, he has been a Professor with the School of Automation. His research interests include intelligent control and intelligent systems, networked control systems, and high-precision tracking control system for moving body. VOLUME 8, 2020