The traditional robust extended Kalman filter (EKF) usually fails to resist small and continuous outliers effectively when the abnormality occurs in GNSS data. To solve this issue, we propose a PSO-SVR aided robust EKF for GNSS/INS integrated navigation. First, this paper uses the particle swarm optimization (PSO) algorithm to optimize the SVR kernel function and penalty coefficient as well as establishes the PSO-SVR intelligent prediction model. Next, on the basis of robust EKF, a PSO-SVR model is employed to predict the navigation algorithm during abnormal periods of GNSS. Finally, this algorithm is verified by the vehicle field test, and the robustness of this algorithm to different outliers is further tested by artificially adding small and continuous outliers of different sizes. According to the experimental results, compared with the REKF algorithm, this PSO-SVR-REKF algorithm can resist outliers effectively whether the outlier of a single point is significant or not. A 10s-abnormality of GNSS data can bring out an improvement of accuracy by 64%. GNSS/INS integrated navigation, extended Kalman filter, support vector regression, particle
approach to solve the problem of poor accuracy of GNSS/INS integrated navigation systems in abnormal periods of GNSS [1]. The common machine learning algorithms to assist GNSS/INS integrated navigation include Support Vector Machine (SVM) [2, 3], Recurrent Neural Network (RNN) [4, 5], Residual Attention Network (RAN) [6], Long Short-Term
Controlled Recurrent Unit (GRU) [9], etc. The above algorithms can be divided into intelligent prediction based on neural network and regression prediction based on statistics, both of which are of shallow network layers to conduct time-series prediction easily. However, the prediction algorithm based on neural networks may be easily puzzled by local optimization, uncertain network topology, the “curse of dimensionality”, and other issues [10, 11]. SVR, based on the statistical learning theory, is a method proposed to study statistical regression and prediction problems that can help solve small samples, nonlinearity, high dimension, and local minimum issues. Hence, it is very suitable for predicting GNSS abnormal information, and it is verified to be more efficient than the neural-network prediction algorithm.
However, to construct the SVR model, the parameter optimization must be carried out since choosing training parameters by experience will affect the regression accuracy and generalization ability. Though being widely applied in the navigation system, the genetic algorithm-support vector regression (GA-SVR) model is of low-level efficiency, as it requires complex processes such as selection, crossover, and mutation during the operation process [12]. Particle Swarm Optimization (PSO) is a population-based parallel optimization technique with few adjustable parameters, fast convergence speed, simple and easy operation, which is suitable for processing real-time navigation data [13].
2020 Copyright for this paper by its authors.
optimized kernel function parameters and penalty coefficients, taking the angular rate and acceleration velocity of the inertial navigation system (INS), the velocity and attitude estimations by mechanization equations as the input, and the increment of GNSS/INS integrated navigation solution as the output;
The standard Kalman filtering algorithm generally assumes that the system and observation equations are linear. However, GNSS/INS tight coupled navigation system usually cannot satisfy this assumption. EKF algorithm can realize the linear approximation of nonlinear systems to further improve the solution accuracy. The nonlinear system is assumed by the following expression: = −1( −1) + , ~ (0, )
= ℎ ( ) + , ~ (0, )
The prediction residual is =
− ̂ , The GNSS/INS navigation system can be considered nearly linear but not absolutely linear, and EKF can effectively solve nonlinear problems to deliver better state estimation. In addition, as the first-order estimation is taken into the Taylor series expansion, the prediction residual is enough to describe dynamic characteristics even though failing to represent the real value.
The equivalent EKF gain matrix is constructed, a similar expression to the IGG Ⅲ weight function:
̂ (−) = −1( ̂ −1(+)) ̂ = ℎ ( ̂ (−))
(−) = −1 −1(+) −1 + −1 ̅
= (−) [ (−) + ]−1 ̂ (+) = ̂ (−) + ̅ (
− ̂ ) = ̂ (−) + ̅
(+) = [
− ̅ ( )] (−)
(
1 − 0 0, > 1 0 × [ 1 − ]2, 0 ≤ ≤ 1
Wherein 0 and 1 are threshold parameters with 0 as 2.5-3.5 and 0 as 3.5-4.5; , wherein and denote the dimensions of state and observation vectors respectively, and refers to the prediction residual, redundant observation component, and measurement standard deviation of the observation vector separately, and the iteration will be carried out after each update.
, (−) = , −1(+) , (−) =
− , (−)
Wherein the state prediction value , (−) at the t iteration is confirmed by the state filter value and its prediction residual after the T-1 iteration. After calculating the equivalent gain matrix according to
If the difference between , (+) and , −1(−) is the estimated value of standard EKF at the time k, and the posterior covariance matrix will be: , (+) = , −1(−) + ̃ ,
(+) = [ − ̃ , ] (−)
Wherein ̃ , means the final equivalent Kalman filter gain matrix at the end of an iteration.
3.1. Support vector regression( SVR )
SVR, one of machine learning methods, is commonly employed to study statistical regression and data prediction to solve the problems of small samples, nonlinearity, high dimensions, and local minimum. The basic concept of SVR is to map the dataset x to high-dimensional feature space for regression analysis and prediction by defining a kind of nonlinear transformation. Given the training samples ( xi , yi ), xi R, yi R, i = 1 n ， we will set the objective regression function: ( ) = ⟨ ⋅ ⟩ + with constraint condition To obtain the optimal parameters w and b , we establish the optimization function as follows: 1 2
Wherein C is the penalty factor to weigh the overall model and sample errors. The larger the value of C , the higher level of data fitting. refers to the loss function used in reflecting the generalization ability of models. The larger of , the lower level of data fitting. The above functions are transformed into the following optimal problem through the kernel method and corresponding duality theory.
( , ∗) = ∑
=1 ( − ∗ ) − ∑
=1( + ∗) −
1
2 =1, =1( − ∗) ( − ∗) ( , ) （16）
∑
(
( − ∗) = 0,0 ≤ ≤ , = 1,2, ⋯ , = 1,2, ⋯ , 0 ≤ ∗ ≤ = ∑ =1( − ∗) ， = −
∑ =1( − ∗) ( ⋅ ) + ， i = 1, 2, n, j = 1, 2, n
( ) = ∑ =1( − ∗) ( , ) + （17）
The PSO algorithm first initializes a group of particles in the feasible solution space, each of which represents a potential optimal solution to extremal optimization. The particle characteristics are indicated by three indexes: position, velocity, and fitness value. The fitness value can be calculated by the fitness function, which reflects the advantages and disadvantages of particles. The moving particle in the solution space updates its individual position by tracking the individual extremum Pbest and the group extremum Gbest; while the individual extremum Pbest refers to the optimal position of fitness value calculated from the position experienced by the individual, and the group extreme Gbest means the optimal position of fitness searched by all particles in the population. The fitness value will be calculated upon each update of the particle position, and the positions of individual extremum Pbest and group extremum Gbest are updated by comparing the fitness value of the new particle with those of individual extreme and group extremums.
Assume that a D-dimensional space for searching has a population X = (X1, X2, . . . Xn) which includes n number of particles, among which the i-th particle is represented as a D-dimensional vector Xi = [xi1, xi2, . . . xiD]T, representing the position of the i-th particle in the D-dimensional space as well as a candidate solution. The fitness value corresponding to the position Xi of each particle can be calculated by the objective function. The velocity of the i-th particle is Vi = [Vi1, Vi2, . . . ViD]T, its individual extremum is Pi = [Pi1, Pi2, . . . PiD]T , and the group extremum of the population is Pg =
T [Pg1, Pg2, . . . PgD] .
+1 = + 1 1( − ) + 2 2( − ) (18) +1 = + +1 (19)
Wherein refers to the inertia weight with =1, 2, ..., D; i = 1,2, . . . , n; k means the current number of iterations; indicates the velocity of particles; 1 and 2 are non-negative constants called acceleration factors; and r1 and r2 are random numbers distributed between [0,1].
no abnormality in GNSS data, we can enter the PSO-SVR training mode. The input of training samples is the position, speed, and attitude calculated by INS mechanization equations, as well as the acceleration and angular rate data output by INS. The output of training samples is the position increment of the navigation solution. We use the input and output of training samples to train the PSO
mode starts, in which the input is still the position, velocity, and attitude output by INS mechanization equations, as well as the acceleration and angular rate output by INS. Next, the input data will be put into the well-trained PSO-SVR model to predict the increment of navigation solution; finally, the error correction will be carried out in the position, velocity, and attitude calculated by INS to obtain the estimated values.
experiments below: (
1. Single point abnormality of GNSS
Figure 5(a) shows the performance comparison of REKF and EKF after adding the 20-meter outlier to GNSS. It’s obvious that the 20-meter outlier is a significant outlier and the robust EKF performs 5(a), the robust EKF will carry out suspicious section to weaken the effect of outliers on results (see Figure 5(b)). If the outlier is within the normal range, EKF will perform standard section to deliver the same results as obtained by standard EKF (see Figure 5(c)). Hence, robust EKF will improve the accuracy most significantly by eliminating the abnormal data without the assistance of PSO - SVR. However, when the outliers gradually reduce, the diagnosis ability of robust EKF will decrease accordingly, which largely affects the accuracy of results in the end. As can be referred to in Table 3, the robust EKF cannot resist the effect of outliers when the observation outlier equals 10 meters.
(a) 20-meter outlier (b) 15-meter outlier (c) 10-meter outlier Figure 5: Comparison of REKF and EKF (with different outliers)
meter, 15-meter, and 10-meter outliers to train and predict PSO-SVR. The input-out data of the first 150s are taken as PSO-SVR training samples to train the model and predict the position increment after the 150s as indicated in Figure 6. As can be seen, the abnormality occurs in the position increment of
abnormality. We replace the original REKF solution increment with prediction increment and restore it to the position data, the trajectory of which is suggested in Figure 7. Obviously, the PSO-SVR-aided robust EKF model can further resist the influence of outliers based on the robust solution results.
(a) 20-meter outlier (b) 15-meter outlier 2. Continuous abnormality of GNSS
meter outlier during 192-201s (lasting for 10s). As a result, the robust EKF performs elimination section to resist outliers to the greatest extent (see Figure 8). However, when the abnormality continues longer, affected by the observation covariance matrix of the Kalman filter, the abnormality diagnosis ability of
paper builds up a PSO-SVR-aided robust EKF algorithm along with its process. The main conclusions are as follows:
When the outliner of GNSS is equal to or above 10m, REKF will lose its robustness; when the abnormality lasts over 10s in GNSS (the outliner is 20m), the robustness of REKF will gradually decrease. In other words, the less significant the outliner becomes, the longer it will last, and the weaker the robustness of REKF will be.
2) Whether the abnormality is significant or not, the PSO-SVR-REKF model can resist the outliner to the largest extent. When the abnormality of GNSS lasts more than 10s, it can also resist the influence of outliers, which means, the PSO-SVR-REKF algorithm can be less affected by the significance and duration of abnormal data than REKF, and can resist outliers below 10 meters. When the abnormality of GNSS continues for 10s, the accuracy will be improved by 64%. 7. References