Research on the Model of a Navigation and Positioning Algorithm for Agricultural Machinery Based on the IABC-BP Network

Abstract

Improving the positioning accuracy and stability of a single BDS/INS sensor system in agricultural machinery is important for expanding the application scenarios of agricultural machinery. This paper proposes a navigation and positioning model based on an improved bee-colony-algorithm-optimized BP network (the IABC-BP model). The main aspect of this work involves introducing adaptive coefficients and speed adjustment coefficients that obey Gaussian distribution to ensure the balance between the rate of convergence, group flexibility, and searchability in the search process. The implicit adaptive layer formula of the BP network is proposed, and the BDS/INS navigation and positioning model for agricultural machinery is established using the IABC algorithm and the Kalman filter. Simulation tests and analyses of real-world application scenarios were conducted on the model, and the results showed that, compared with the original model, the performance of the model improved by 90.65%, 84.11%, and 25.96%, indicating that the proposed model has high accuracy and effectiveness. In the information fusion and compensation correction mode, the algorithm processes errors such as longitude and latitude within the target range and can achieve reliable navigation and positioning accuracy in a short period. At the same time, the model has good stability and generalization ability, and can be applied to other navigation scenarios in the future to expand its application scope.

1. Introduction

Global Navigation Satellite Systems (GNSSs) are one of the core technologies for civil navigation and positioning. As the application environment becomes more complex and profound, navigation and positioning are being continuously optimized for accuracy, stability, and reliability. Specifically, when GNSS signals are affected by occlusion or electromagnetic interference, this can easily cause delays or the short-term loss of positioning signals during transmission and reception. Being in the rejection environment for a short period rapidly increases errors in the system parameters over time, making it difficult to guarantee positioning accuracy and output stability. At this point, the limitations of using a single navigation and positioning method become apparent. To avoid this problem, scholars have introduced an additional navigation and positioning method, “GNSS+”, to build a combined navigation and positioning system (the connected navigation and positioning method). The more mature ones are GNSS and IMU, INS, DR, UWB, and vision [1,2,3,4] combined, which can provide complementary advantages and guarantee accuracy and stability. For example, one study [5] combined RTK-GPS, IMU, and a laser scanner to design a navigation and positioning system based on an automatically tracked tractor with data processing using a Kalman filter, and the experiments showed that the lateral error was less than 0.05 m in a straight line. The authors of [6] propose a combined navigation system based on SINS/GPS/TAN to address the problem of GPS/TAN signal failure in a denial environment, and a federal Kalman filter fuses the navigation information. The results show that the operating trajectory is close to the ideal course, and the positioning accuracy meets the requirements. In [7], a combined SINS/GPS/ADS/DVL navigation system based on a navigation switching strategy is proposed; it uses a federal filtering method to improve navigation accuracy while avoiding navigation serial data processing errors, and the reliability of the system is verified. In [8], a combined vision-assisted INS/odometer navigation and positioning method is proposed to calibrate the INS/odometer via image matching in a rejection environment, and a dual-rate Kalman filter is used to achieve the data fusion of vision, the inertial navigation system, and odometer. Its performance is verified by simulation and field tests. The authors of [9] proposed a navigation system for agricultural robots based on GPS + monocular vision + a LiDAR sensor. The analysis shows that such schemes apply more mathematical models and model them differently. According to their systems, they usually design different navigation and positioning algorithms, such as Kalman filtering methods (KF, EKF, and UKF), which are essential for undertaking combined navigation and positioning.

Meanwhile, with the extension of artificial intelligence and intelligent optimization algorithms to navigation and positioning and information fusion technologies, navigation and positioning systems’ intelligence, flexibility, and adaptability have been improved. Commonly used methods include neural network algorithms [10,11], genetic algorithms [12,13], particle swarm optimization algorithms [14,15], and artificial bee colony algorithms [16,17], as well as the mutual fusion of each algorithm. For example, in [18], for INS/GNSS systems, a combination of a trace-free Kalman filter (UKF) and external input (NARX) are proposed for navigation and positioning. In [19], an EKF fusion algorithm based on a MEMS multi-sensor IMU system is designed to realize the fusion processing of multi-sensor measurement data. Meanwhile, ref. [20] proposes an optimization method based on an adaptive Kalman filter, which maintains good navigation accuracy when GPS is interrupted. Moreover, ref. [21] proposed a greedy sample gossip distributed filter fusion algorithm, introducing information-weighted fusion and sample insatiable gossip averaging protocol. The authors of [22] presented an improved weighted particle-swarm-optimized Kalman filtering algorithm for agricultural machinery navigation. The study verified the feasibility and superiority of the modified weighted particle-swarm-optimized particle filtering through simulation experiments, which improved the positioning accuracy of the combined agricultural machinery navigation system. In [23], an improved AUV combined navigation method with a genetic neural network is proposed for the problem of EKF filtering divergence, and its performance is verified. In [24], an improved combined navigation positioning algorithm based on a multilayer perceptron neural network (MLPNN) was proposed to train the neural network when the GPS signal is valid and use the neural network to correct the navigation error of INS when it is out of the lock, in order to achieve the continuity of combined navigation.

It is worth noting that, because BP neural networks have strong self-learning and self-adaptive capabilities, and their fault tolerance and generalization capabilities are also excellent, they can also be optimized with the introduction of other algorithms and have a certain degree of open-source capability. Therefore, scholars have used this method to predict and correct the output of navigation and positioning [25,26,27]. Typically, a multi-sensor information fusion algorithm based on a GA (genetic algorithm)-BP network is proposed in the literature [28] for a combined RTK-GPS and IMU system, enabling the system to improve its accuracy to some extent. The authors of [29] proposed an ABCBP neural-network-assisted Kalman filtering fusion algorithm for BDS/INS navigation signal fusion and error accumulation after the loss of the lock. Meanwhile, ref. [30] proposed an improved UKF navigation and positioning algorithm based on particle-swarm-optimized BP networks to avoid the problem of UKF performance divergence caused by unknown system noise or inaccurate statistical features. In [31], a GA-BP neural-network-based UWB/IMU localization fusion method was proposed, and the average localization error of this method was around 9.6 cm. The above analysis shows that the application of such schemes in combined navigation and positioning usually involves the idea of optimization. However, the BP network has a slow convergence speed and quickly falls into local extremes, making it difficult to guarantee the accuracy and reliability of navigation and positioning in the denial environment, and there are relatively few practical applications.

At the same time, researchers have also conducted a series of optimizations and improvements based on the BP model to enhance its stability and accuracy, achieving better results than the original BP model. The authors of [32] used artificial neural networks to simulate errors that occur in integrated navigation systems without satellite navigation system signals and proposed a new method to improve the coordinate and velocity accuracy of integrated navigation systems without receiving global navigation satellite system signals. Meanwhile, [33] proposes an extreme learning machine as a mechanism for learning stored digital elevation information to assist drones in traversing terrain without the need for GPS. This algorithm meets the needs of online implementation by supporting multi-resolution terrain access, thereby generating high-precision real-time paths within the allotted timeframe. The authors of [34] proposed a uniform sensor fusion optimization method, which can not only estimate the calibration constant of sensors in real time but also automatically suppresses degraded sensors while maintaining the overall accuracy of the estimation. The weights of the sensors are adaptively adjusted based on RMSE, involving the weighted average of all sensors. Elsewhere, ref. [35] utilized the Harris Hawks optimization algorithm (HHO) to optimize the random weights and thresholds of BPNN to obtain the optimal global solution that enhances UWB indoor positioning accuracy and NLOS resistance. This method has higher calibration accuracy and stability than BPNN, making it a feasible choice for scenarios that require high positioning accuracy.

Positioning information is diverse, scalable, and complex for combined systems. Therefore, navigation and positioning algorithms are designed with specific adaptability based on the basic requirements of robustness, parallel processing capabilities, and additional indicators of information samples such as operational speed and accuracy. Therefore, based on the above research analysis, this paper designs a Kalman system filter to improve the robustness of the information fusion of a BDS/INS navigation and positioning system for agricultural machinery, avoiding the problem of the gradual decrease in positioning accuracy and stability under a single INS system when the signal is out of the lock. On this basis, we propose a navigation and positioning algorithm based on an improved swarm-algorithm-optimized BP network (referred to as the IABC-BP algorithm). The main contributions of this paper are as follows:

• An improved swarm algorithm is proposed to ensure the balance between convergence speed, swarm flexibility, and searchability. We introduce an adaptive judgment factor ξ and a convergence speed adjustment factor α to improve the search formula, which improves the convergence speed and prevents the algorithm from falling into a local optimum; meanwhile, the adaptive judgment factor ξ is used to optimize the selection probability to improve the global search ability and effectiveness of the colony.
• This study proposes an adaptive implicit layer node selection formula by introducing a judgment coefficient σ to avoid the BP network’s overfitting problem and improve the network’s performance and generalization ability.
• Using the weights and thresholds of the BP network as the optimization objectives and the mean square error of the sample correction value and the actual value as the objective function values of the improved ABC algorithm, the enhanced swarm algorithm is reasonably integrated with the BP network. The model can be trained when the BDS signal is valid, and the INS navigation deviation can be corrected when it is out of the lock, which prevents filtering divergence and improves the robustness, reliability, and continuity of the system time.

This study is organized as follows: Section 2 introduces the experimental materials and methods, mainly introducing the BDS/INS model principle and related improvement work for experimental selection. Section 3 mainly introduces the results of the experiment, and analyzes and demonstrates the experimental results. Section 4 discusses the experimental results of this article and provides prospects for future research directions. Section 5 summarizes and elaborates on the work and achievements of the entire article.

2. Materials and Methods

2.1. Problem Description

2.1.1. BDS/INS Information Fusion Model

The BDS/INS integrated navigation method used in this study is in the loose coupling mode, and the model is in the northeast sky coordinate system. The main principle is that the velocity and position differences between BDS and INS are filtered to obtain the optimal estimates of the state variables. The estimated values are then used to correct the INS system using feedback to obtain the optimal desired output. The preliminary BDS/INS information fusion model is shown in Figure 1.

• State equation and state vector

$$\begin{array}{c}{\dot{X}}_t={\Phi }_{t,t-1}{X}_t+W_{t,t-1}\hfill \\ X=[{\alpha }_E,{\alpha }_N,{\alpha }_U,\delta V_E,\delta V_N,\delta V_U,\delta l,\delta b,\delta h,\Delta a_x,\Delta a_y,\Delta a_z,\Delta {\omega }_x,\Delta {\omega }_y,\Delta {\omega }_z,{\theta }_a,{\theta }_e,{\theta }_r]\hfill \end{array}$$

(1)

where $X_t$ and $X_{t-1}$ denote the 18th-order state vectors at times t and t − 1, respectively. α indicates the heading error. $\delta V_E$, $\delta V_N$, $\delta V_U$ represents the velocity error, and the subscripts ${\alpha }_E$, ${\alpha }_N$${\alpha }_U$ denote the eastward, northward, and skyward directions, respectively. δl, δb, and δh represent the errors in longitude, latitude, and altitude, respectively. ∆a and ∆ω are the acceleration and angular velocity information in 3 directions, respectively. ${\theta }_a$ denotes the azimuth angle, ${\theta }_e$ denotes the pitch angle, and ${\theta }_r$ denotes the roll angle. ${\Phi }_{t,t-1}$ denotes the system state transfer matrix and $W_{t,t-1}$ represents the system noise matrix.

2.

Observation equation and observation vector

The position and velocity information measured by INS and BDS receivers is used as the observation information:

$$Z_t=\left[\begin{array}{cc}{P}_{GNSS}& a_{INS}\\ V_{GNSS}& {\omega }_{INS}\end{array}\right]=H_t\times X_t+V_t$$

(2)

$$Z=[\delta V_E,\delta V_N,\delta V_U,\delta l,\delta b,\delta h]$$

(3)

where $V_t$ denotes the noise direction, $H_t$ denotes the observation matrix, O is the zero matrices, and letter I is the unit matrix.

$$H=\left[\begin{array}{l}\begin{array}{cccc}0& I& 0& 0\end{array}\\ \begin{array}{cccc}0& 0& I& 0\end{array}\end{array}\right]$$

(4)

2.1.2. IABC-BP Navigation and Positioning Algorithm Model

When there is a significant deviation between the Kalman filter and the working state of the farm machine, it will lead to the failure of the information fusion filtering. Additionally, considering the situation of signal masking and the loss of the lock, the artificial bee colony algorithm is introduced to optimize the BP neural network and is combined with the KF-BDS/INS model, which has the function of filtering correction and, at the same time, can compensate and correct the BDS data when the signal is weak; this is crucial for the improvement of positioning accuracy.

The IABC-BP fusion algorithm model explored in this paper is shown in Figure 2. It comprises two main parts: the information fusion mode and the compensation correction mode. (1) The information fusion mode fuses the information received from multiple sensors and trains the model. (2) The compensation correction mode is used to correct and compensate the navigation and positioning system when the signal is weak.

The specific principle is as follows:

(1)

When the BDS signal is sent and received typically, switch S1 is connected and switch S2 is connected, and the information fusion mode is turned on. The position and velocity position differences of BDS and INS are used as the input of the Kalman filter, and the filtered information is output to train the IABC-BP neural network model. The secondary filtering and deep fusion of all the information are performed, and the optimal estimated value is finally obtained as navigation and positioning correction information. This model is commonly used.

(2)

When the BeiDou satellite signal fails briefly, switch S2 connects, switch S1 disconnects, and the compensation correction mode opens. At this time, the INS’s velocity and position information are used as the network input, and the IABC-BP model is used to predict the corrected velocity and position information. Then, it is output as the current information, and the acceleration and attitude information is solved via INS to ensure the accuracy and stability of combined navigation and positioning.

2.2. BP Neural Network Principle and Improvement Work

2.2.1. Principle of the BP Neural Network

A BP neural network is a feed-forward neural network based on error backpropagation, including two stages of forwarding propagation and backpropagation. It contains three layers of architecture: the input layer, implicit layer, and output layer.

Suppose the network has m nodes in the input layer, q nodes in the hidden layer, and n nodes in the output layer. The weights between the input and hidden layers are $w_{ji}$, and the weights between the hidden and output layers are $w_{kj}$. This principle is shown in Figure 3.

• Forward Propagation

The samples are propagated in the order of input–implicit–output (forward). At this point, the input $i{n}_i$ of the i-th neural node (i = 1, 2, ⋯, q) in the implicit layer is:

$$i{n}_i={\displaystyle \sum _{j=1}^q{w}_{ij}{x}_j+{\theta }_i}$$

(5)

The output $ou{t}_i$ of the i-th neural node in the hidden layer is:

$$ou{t}_i=f(i{n}_i)=f({\displaystyle \sum _{j=1}^q{w}_{ij}{x}_j+{\theta }_i})$$

(6)

The output $y_k$ of the k-th neural node in the output layer is:

$$y_k=f({\displaystyle \sum _{i=1}^n{w}_{ik}ou{t}_i+a_k})=f({\displaystyle \sum _{i=1}^n{w}_{ik}f({\displaystyle \sum _{j=1}^q{w}_{ij}{x}_j+{\theta }_i})+a_k})$$

(7)

where $x_j$ denotes the input of the j-th node of the input layer (j = 1, …, m). $w_{ij}$ denotes the weight of the j-th input layer node to the i-th node of the hidden layer. ${\theta }_i$ denotes the threshold of the i-th node of the hidden layer. f(x) refers to the incentive function of the hidden layer. $w_{ik}$ denotes the weight of the i-th node of the hidden layer to the k-th node of the output layer (i = 1, …, q). $a_k$ represents the threshold of the k-th node of the output layer (k = 1, …, n).

2.

Back Propagation

Suppose the total number of training samples is P, and the square error function is adopted, then the error criterion function of the P-th sample is:

$$E_p=\frac{1}{2}{\displaystyle \sum _{k=1}^L{(T_k-y_k)}^2}$$

(8)

The global error criterion function is:

$$E=\frac{1}{2}{\displaystyle \sum _{m=1}^P{\displaystyle \sum _{k=1}^L{(T_k^m-y_k^m)}^2}}$$

(9)

In this case, the squared error function used is:

$$f(x)=\tanh (x)=\frac{1-e^{-2x}}{1+e^{2x}}$$

(10)

2.2.2. BP Neural Network Improvement Work

In the process of establishing a BP neural network, the random setting of connection weights can lead to errors in the prediction results, and gradient descent training has the drawbacks of slow speeds and local minima, making it difficult to achieve global optimization in the training of the neural network. Therefore, it is necessary to optimize the network model to improve the accuracy and stability of training and operations. Thus, the selection of the number of hidden layer nodes in the network structure is highly important, as it not only has a significant impact on the performance of the neural network model but is also the direct cause of “overfitting” during training. The number of hidden layer nodes is not only related to the number of nodes in the input/output layer, but also to the complexity of the problem to be solved, the type of transformation function, and the characteristics of the sample data. Therefore, this study proposes an adaptive formula for selecting the number of hidden layer nodes to avoid overfitting during training as much as possible and to ensure sufficiently high network performance and generalization abilities.

$$H=\{\begin{array}{l}fix(\sqrt{I+O}+b_i),\sigma \le {10}^{-5}\\ 0.75\times I,\sigma >{10}^{-5}\end{array}$$

(11)

where fix( ) denotes rounding to the zero direction, I is the number of input layer nodes, H is the number of hidden layer nodes, and O is the number of output layer nodes. σ denotes the judgment coefficient, which is mainly related to the number of input and output samples. Hidden layer nodes are generally set to 75% of the number of input layer nodes.

$$\sigma =\frac{sum{(P_{In}-P_{out})}^2}{length(P)}$$

(12)

where $P_{In}$ is the number of input samples, and $P_{out}$ is the number of output samples.

2.3. Immune Ant Colony B Spline Interpolation Algorithm

2.3.1. Bee Colony Algorithm

The ABC algorithm can be divided into three types of bees: the leading bee, the following bee, and the scouting bee. The main steps of the algorithm are:

Step 1: The initial nectar source $x_{i,k}$ is randomly generated in the search space (−1, 1), as shown in Equation (13):

$$x_{i,k}=x_{k,\min }+rand(-1,1)(x_{k,\max }-x_{k,\min })$$

(13)

where $x_{k,\min }$ and $x_{k,\max }$ denote the lower and upper limits of the search space, respectively,

Step 2: According to Equation (14), the bees record the best nectar source so far and search for a new source within the current nectar neighborhood:

$$v_{i,k}=x_{i,k}+\phi (x_{i,k}-x_{j,k})$$

(14)

where j denotes a randomly selected nectar source not equal to i among N nectar sources. φ is an actual random number uniformly distributed within [−1, 1], determining the degree of perturbation. When a new nectar source $v_{i,k}$ is generated, its fitness value can be calculated using Equation (15):

$$fi{t}_i=\{\begin{array}{l}\frac{1}{1+f_i},f_i\ge 0\\ 1+abs(f_i),f_i<0\end{array}$$

(15)

where $f_i$ denotes the value of the function of the solution. When the fitness value of the new nectar source $v_i$ is better than $x_i$, the greedy selection strategy is used to replace the original nectar source with the new one; otherwise, $x_i$ is retained.

Step 3: The following bees select a nectar source to attach to and follow according to the information (nectar amount and location) shared by the leader bee, which is calculated according to Equation (16) in a probabilistic selection manner:

$$P_i=\frac{fi{t}_i}{{\displaystyle \sum _{i=1}^{N}fi{t}_i}}$$

(16)

Due to the roulette strategy utilized, each nectar source generates a random number between [−1, 1], and, if this source is calculated to be larger than its corresponding random number, the following bee attaches to this source and transforms into the leading bee.

Step 4: During the search process, if the nectar source $x_i$ reaches a limit after an iterative search and no better nectar source is found, the nectar source will be abandoned, and the corresponding lead bee will be transformed into a scout bee. At this point, a new nectar source will be randomly generated in the search space instead of $x_i$.

2.3.2. Improvement of the Bee Colony Algorithm

According to the above analysis and the actual application process, it is known that the traditional ABC algorithm has poor local optimization abilities, leading to a slower rate of convergence of the algorithm. Therefore, in this study, the search strategy of bee colonies and the probability of honey source selection are improved to enhance the performance of the algorithm.

• Improvement of the Search Formula

In this paper, to improve the convergence speed of the swarm, the optimal global solution $bes{t}_k$ is used. The disadvantage of this approach is that it destroys the flexibility of the swarm and the searchability of the algorithm, and, to a certain extent, it leads the algorithm into premature maturity while the risk of local optima still exists. Therefore, to ensure the balance between the convergence speed, the flexibility of the swarm, and the searchability, the adaptive judgment coefficient ξ and the speed adjustment coefficient α obeying a Gaussian distribution $(N,D^2)$ are introduced to improve the search process.

$$\xi =\frac{1}{D\cdot \sqrt{2\pi }}\cdot \exp (-\frac{{(Maxcycle-N)}^2}{2D^2})-C$$

(17)

$$v_{{}_{i,k}}^{new}=\{\begin{array}{l}{\alpha }_1{x}_{i,k}+{\alpha }_1\phi (x_{i,k}-x_{j,k})+{\alpha }_2\phi (bes{t}_k-x_{j,k}),\zeta <\phi \text{'}\\ x_{i,k}+\phi (x_{i,k}-x_{j,k}),otherwise\end{array}$$

(18)

where C is an arbitrary constant. $\phi \text{'}$ denotes a randomly perturbed actual number uniformly distributed in [0, 1.5]. ${\alpha }_1$ denotes the approach speed of the new nectar source to the original nectar source and the neighboring nectar sources, and ${\alpha }_2$ denotes the approach speed of the new nectar source to the optimal nectar source as shown in Equation (19). iter denotes the current number of iterations, and Maxcycle denotes the maximum number of iterations.

$$\{\begin{array}{l}{\alpha }_1={\alpha }_{\max }-({\alpha }_{\max }-{\alpha }_{\min })(2-e^{iter\times \ln 2/Maxcycle})\\ {\alpha }_2={\alpha }_{\min }+({\alpha }_{\max }-{\alpha }_{\min })(2-e^{iter\times \ln 2/Maxcycle})\end{array}$$

(19)

By analyzing Equation (18), it is clear that the presence of ξ causes the algorithm to make judgments at convergence, increasing the fault tolerance of the algorithm. In particular, since the initial iterations iter is small, the value of ${\alpha }_1$ is small, and the value of ${\alpha }_2$ is large, the following bees have a higher probability of selecting a nectar source with high adaptation, and the selected nectar source approaches the global optimal nectar source faster, which improves the convergence rate. In the later stages of the algorithm, as the number of iterations increases, the ${\alpha }_1$ value gradually becomes larger and the ${\alpha }_2$ value becomes smaller, so that the probability of the following bees selecting a less adapted nectar source increases, making the algorithm increase its search capability in the later stages, reducing the influence of the optimal solution on the search and increasing the flexibility of the population.

2.

Improvement of Selection Probability

As shown by Equation (16) above, the higher the degree of adaptation, the higher the probability of the nectar source being selected, leading the algorithm to focus rapidly on the nectar source with a high degree of adaptation in the optimization process, which destroys the global search ability and generalization ability of the population and reduces the search effectiveness of the algorithm. Thus, the adaptive judgment factor ξ is still used to improve the probability calculation formula.

$$P_i=\{\begin{array}{l}\frac{fi{t}_i}{{\displaystyle \sum _{i=1}^{N}fi{t}_i}},\phi \ge \zeta \\ \frac{1}{fi{t}_i{\displaystyle \sum _{i=1}^N\frac{1}{fi{t}_i}}},\phi <\zeta \end{array}$$

(20)

2.3.3. Improved Bee Colony Algorithm to Optimize the BP Neural Network

The improvement of the bee colony algorithm enables it to transcend the limitations of the local optimum and provide the optimal initial weights for the BP neural network, which makes it possible to guarantee the localization accuracy and stability of the single-sensor system when the signal is out of the lock. At this time, the weights and thresholds of the BP network are used as the objectives of the algorithm optimization, and the mean square error MSE of the predicted and actual values of the BP network samples is used as the objective function value of the improved ABC algorithm (IABC for short). The flow chart is shown in Figure 4, and the specific implementation steps are described as follows:

Step 1: Determine the number of nodes according to the input and output requirements, build a BP neural network structure, and take the weights $w_{ij}$ connecting the input layer to the hidden layer and the weights $w_{ik}$ connecting the hidden layer to the output layer as the optimization objectives.

Step 2: Initialization of the swarm algorithm. Initialize the swarm size N, the number of leading bees $N_e$, the number of following bees, and the number of solutions $N_s$. The interrelationship between the ABC algorithm and the BP network is shown in Equation (21), where Limit denotes the upper iteration threshold, D is the number of dimensions, and round means rounding:

$$\{\begin{array}{l}N=N_e+N_0=2N_s=I\times H+H+H\times O+O\\ N_e=N_0\\ Limit=round(0.6\times D\times N)\end{array}$$

(21)

Step 3: The leading bees search for a new nectar source near the nectar source using Equation (22) and calculate its objective function value (mean square error of the i-th solution of the BP neural network) for the new nectar source using the BP algorithm, as shown in Equation (23). Then, the fitness $fi{t}_i$ of the new nectar source is calculated using Equation (15), and the old and new nectar sources are selected according to the greedy principle:

$$v_{{}_{i,k}}^{new}=\{\begin{array}{l}{\alpha }_1{x}_{i,k}+{\alpha }_1\phi (x_{i,k}-x_{j,k})+{\alpha }_2\phi (bes{t}_k-x_{j,k}),\zeta <\phi \text{'}\\ MSE+\phi (x_{i,k}-x_{j,k}),otherwise\end{array}$$

(22)

$$F_{MSE}=\frac{1}{N}({\displaystyle \sum _{i=1}^N{(s_i-c_i)}^2}$$

(23)

where $s_i$ and $c_i$ denote the predicted output value and the actual validation value of the i-th training sample of the BP network, respectively.

Step 4: The following bees calculate the probability according to the Equation (16), find a new nectar source, and calculate the value of the fitness function to determine whether it is better than the previous source.

Step 5: It is determined whether the scout bees appear or not. If some nectar source remains unchanged after the trail cycles, abandon the source. When the limit is reached, the corresponding leading bees become scout bees, and new solutions are randomly generated according to Equation (15).

Step 6: When the number of updates is less than Maxcycle, return to Step 4 and restart the optimization process until the number of updates exceeds Maxcycle. Then, the training is finished.

Step 7: Apply the obtained optimal weights and thresholds to the BP neural network model, optimize the BP neural network, and output the predicted values. At this time, the output of the neural network is shown as Equation (24):

$$y_k=f({\displaystyle \sum _{i=1}^{n}fi{t}_i\times w_{ik}f({\displaystyle \sum _{j=1}^{q}fi{t}_i\times w_{ij}{x}_j+{\theta }_i})+a_k})$$

(24)

3. Results

3.1. Test Preparation and Protocol Design

The BDS/INS navigation and positioning system of the agricultural machine used in the experiment is shown in Figure 5a, and the INS is embedded in the package box and connected to the industrial control computer. The test location is the engineering building and exhibition hall area of Qingdao Agricultural University (120.40° E, 36.32° N), and the movement route is shown in Figure 5b, with the black curve indicating a straight path and the red angle indicating a turn. The test process shown in Figure 6a shows the base station that was erected, and its operation conditions can be controlled according to the experimental needs. Figure 6b shows the test process, and the agricultural system operates normally.

In this paper, the simulation software used is MATLAB R2020a, and the model of the experimental agricultural machinery is Weituo TY300 tractor, produced in Shandong Province, China, provided by Weifang Zhongte Agricultural Equipment Co., Ltd. In addition, the experimental device also includes: BDS receiver, INS (model: Y61 six axis attitude angle sensor), control execution device, and front wheel angle sensor.

For the purposes of illustration, the locations of this experiment are all latitude and longitude coordinates, with an initial position of (120.40° E, 36.32° N), an initial heading of 65.8° (north of east), an initial velocity of 0 m/s, a period of T = 310 s, and a sampling interval of 1 s. The experiments in this study are validated in three main aspects:

The feasibility, flexibility, and other performance metrics of the improved bee colony algorithm are assessed via test functions;

The performance, accuracy, and practical effect of the model before and after combining the Kalman filter are verified via the IABC-BP algorithm;

The performance of the combined BDS/INS navigation system after applying the IABC-BP fusion algorithm is verified and the comparison is illustrated.

The parameters of the swarm algorithm are described as follows: Maxcycle = 1000, the number of swarms N is related to the number of nodes in the BP network, ${\alpha }_{\max }$ = 1.1, and ${\alpha }_{\min }$ = 0.1. The parameters of the BP network are described as follows: the number of training step epochs = 1000, the learning rate $l_r$ = 0.01, the minimum error of the training target $ms{e}_{goal}$ = 0.000001, and the input I and output node O are determined according to the actual number of samples are determined.

3.2. Improved Algorithm Performance Testing

The primary purpose of this experiment is to verify the feasibility and superiority of the improved bee colony algorithm proposed in this paper. Three test functions are selected for this experiment to verify this improved algorithm and compare it with the traditional ABC algorithm in order to demonstrate its superior performance.

The test functions are shown in

Table 1. Expression and scope of test function.

Function	Expression	Scope
Sphere	$f_1(x)={\displaystyle \sum _{i=1}^D{x}_i^2}$	[−100, 100]
Ackley	$f_2(x)=-20\exp (0.2\sqrt{\frac{1}{D}{\displaystyle \sum _{i=1}^D{x}_i^2}})+\exp (\frac{1}{D}{\displaystyle \sum _{i=1}^D\cos (2\pi x_i))+20+e}$	[−32, 32]
Rastrigin	$f_3(x)={\displaystyle \sum _{i=1}^D[x_i^2-10\cos (2\pi x_i)+10]}$	[−5.12, 5.12]

, where the swarm size N is set to 100. Since the convergence speed of the improved algorithm is accelerated, dimension D is set to 5, 50, and 100 as independent variables to avoid algorithm discontinuity for the purposes of comparison and illustration. The test results are shown in

Table 2. Test results of different algorithms according to test function (D = 5).

Function	Algorithms	Range	Best	Worst	Mean	Mean Square Error
Sphere	ABC	[−1, 1]	3.92 × 10 -23	1.16 × 10 -22	7.73 × 10 -23	1.46 × 10 -45
	IABC	[0, 1]	1.79 × 10 -45	3.30 × 10 -45	2.54 × 10 -45	1.07 × 10 -45
		[−1, 1]	3.10 × 10 -171	4.21 × 10 -171	3.66 × 10 -171	0
Ackley	ABC	[−1, 1]	3.79 × 10 -12	1.32 × 10 -11	8.84 × 10 -12	2.20 × 10 -23
	IABC	[0, 1]	8.88 × 10 -16	4.44 × 10 -15	2.66 × 10 -15	2.51 × 10 -15
		[−1, 1]	8.88 × 10 -16	4.44 × 10 -15	2.66 × 10 -15	2.51 × 10 -15
Rosenbrock	ABC	[−1, 1]	7.71 × 10 -01	8.24 × 10 -01	7.98 × 10 -01	7.04 × 10 -04
	IABC	[0, 1]	8.40 × 10 -01	1.19 × 10 +00	1.02 × 10 +00	2.49 × 10 -01
		[−1, 1]	1.00	2.39	1.69	9.82 × 10 -01

,

Table 3. Test results of different algorithms according to test function (D = 50).

Function	Algorithms	Range	Best	Worst	Mean	Mean Square Error
Sphere	ABC	[−1, 1]	7.78 × 10 +03	7.80 × 10 +03	7.79 × 10 +03	3.83 × 10 +01
	IABC	[0, 1]	5.44 × 10 -04	7.04 × 10 -04	6.24 × 10 -04	1.14 × 10 -04
		[−1, 1]	3.88 × 10 -82	4.36 × 10 -82	4.12 × 10 -82	3.42 × 10 -83
Ackley	ABC	[−1, 1]	1.61 × 10 +01	1.74 × 10 +01	1.68 × 10 +01	4.79 × 10 -01
	IABC	[0, 1]	3.54 × 10 -03	3.60 × 10 -03	3.57 × 10 -03	4.68 × 10 -05
		[−1, 1]	8.88 × 10 -16	4.44 × 10 -15	2.66 × 10 -15	2.51 × 10 -15
Rosenbrock	ABC	[−1, 1]	3.50 × 10 +03	4.02 × 10 +03	3.76 × 10 +03	6.68 × 10 +04
	IABC	[0, 1]	4.76 × 10 +01	4.81 × 10 +01	4.78 × 10 +01	3.07 × 10 -01
		[−1, 1]	4.85 × 10 +01	4.85 × 10 +01	4.85 × 10 +01	8.20 × 10 -03

and

Table 4. Test results of different algorithms according to test function (D = 100).

Function	Algorithms	Range	Best	Worst	Mean	Mean Square Error
Sphere	ABC	[−1, 1]	1.92 × 10 +05	2.35 × 10 +05	2.13 × 10 +05	4.48 × 10 +08
	IABC	[0, 1]	2.29 × 10 +00	2.39 × 10 +00	2.34 × 10 +00	6.92 × 10 -02
		[−1, 1]	7.52 × 10 -70	1.01 × 10 -69	8.81 × 10 -70	1.83 × 10 -70
Ackley	ABC	[−1, 1]	2.09 × 10 +01	2.10 × 10 +01	2.09 × 10 +01	2.47 × 10 -03
	IABC	[0, 1]	4.38 × 10 -01	4.56 × 10 -01	4.47 × 10 -01	1.23 × 10 -02
		[−1, 1]	8.88 × 10 -16	4.44 × 10 -15	2.66 × 10 -15	2.51 × 10 -15
Rosenbrock	ABC	[−1, 1]	3.02 × 10 +04	3.37 × 10 +04	3.19 × 10 +04	3.11 × 10 +06
	IABC	[0, 1]	9.93 × 10 +01	9.96 × 10 +01	9.94 × 10 +01	2.35 × 10 -01
		[−1, 1]	9.81 × 10 +01	9.87 × 10 +01	9.84 × 10 +01	4.09 × 10 -01

.

In this experiment, the effect of the randomness factor is taken into account. Therefore, under the same test conditions, the function optimization experiments of both traditional and modified swarm algorithms are conducted 20 times, and data such as the optimal value and mean squared error are recorded when downloading different dimensions. The optimal value reflects the quality of understanding, and the mean square deviation reflects the robustness and stability of the algorithm. Analyzing the data in Table 2 and Table 4, we can see that:

(1)

The optimal values of both algorithms show a decreasing trend with a decrease in dimensionality. When the dimensionality is 100 (D = 100), the optimization index values of the algorithm proposed in this study are better than those of the ABC algorithm.

(2)

When the dimension of the function is reduced to 5 dimensions, the advantage of this algorithm is more pronounced. The accuracy of the optimal value, the worst value, the average value, and the mean squared error in different dimensions (especially in high dimensions) is higher than that of the ABC algorithm, which indicates the high quality of the solution of the IABC algorithm and the improved robustness and stability of the algorithm.

(3)

The comparison of the IABC algorithm in the two search intervals [−1, 1] and [0, 1] shows that the optimal solution and the variance are significantly smaller in the interval [−1, 1], which indicates that the IABC algorithm achieves better results in the overall search capability and in jumping out of the local optimum.

For the three functions listed in Table 1, the ABC algorithm is compared with the IABC algorithm in 50 dimensions (interval [0, 1]); this method reflects the performance of the algorithm more intuitively, and the comparison results are shown in Figure 7a–c. The comparison clearly shows that the improved algorithm has substantially improved convergence speed, accuracy, and global optimization capabilities. A comprehensive analysis shows that the IABC algorithm achieves better results and can be used for subsequent optimization work.

3.3. BP Neural Network Performance Analysis

This section focuses on the performance verification of the improved BP network and IABC-BP network algorithms, and the experimental data of the motion path trajectory in Section 3.1 are selected. This experiment compares the BP network, Kalman filter–BP neural network (KF-BP), the IABC-BP network, and Kalman filter–IABC-BP network (KF-IABC-BP). The performance of the four methods was evaluated by filtering and correcting the velocity information values, and the comparison of the error curves of the velocity correction values of the four methods are shown in Figure 8 and Figure 9.

Considering the need for the convenient and intuitive analysis of the model’s performance in the experiment, the mean absolute error MAE, mean square error MSE, root mean square error RMSE, and correlation coefficient R are used as the judging indexes, and the related data statistics are shown in

Table 5. Comparison of speed filter performance.

Algorithm	MAE	MSE	RMSE	R
BP	0.15205	0.08412	0.2900	0.8815
KF-BP	0.0843	0.0291	0.1705	0.9580
IABC-BP	0.0185	0.0013	0.0366	0.9983
KF-IABC-BP	0.0143	0.0007	0.0271	0.9988

.

Based on the analysis of the experimental results, it can be concluded that:

(1)

All four models can filter the speed. However, there are certain differences in the indicators achieved by different models.

(2)

The BP neural network model is relatively poor and has the largest error. The MAE, MSE, and RMSE values of the BP model are the largest, which indicates that the model is less accurate and does not reflect the actual situation of the error. In particular, the smallest R-value indicates that the traditional BP model has certain defects, which are mainly shown by the increase in the error with time. The performance of the BP model is improved by introducing the Kalman filter based on the BP model.

(3)

Both the IABC-BP and KF-IABC-BP models exhibit better experimental results, and the error values are less than 0.037 regarding the differences between various evaluation indexes.

(4)

Compared with other models, KF-IABCBP has the best performance, The value of the correlation coefficient R reached 0.9988, which is better than that of the other models; the relationship between the variables is closer, and the error will stabilize with time.

(5)

With the RMSE value as the main evaluation index, the performance of the KF-IABC-BP model is improved by 90.65%, 84.11%, and 25.96% compared with the other models, which indicates that the Kalman-filter-based IABC-BP neural network algorithm model established in this paper has sufficient feasibility.

3.4. Performance Analysis under Different Modes

In this section, the performance of the BDS/INS system for agricultural machinery is verified after applying the IABC-BP navigation and positioning algorithm model. We ascertain the rationality, flexibility, and timeliness of information fusion, as well as demonstrating the working performance when the satellite signal is out of the lock or weak and the accuracy of the prediction correction of the data. The experimental protocols are:

(1)

To verify the performance of the BDS/INS system when applying the IABC-BP model for information fusion and filter correction when the signal is not out of lock, and all sensor data are collected normally.

(2)

To verify the performance of the IABC-BP model algorithm and the accuracy of the predicted output compensation system when setting the lock loss condition under the above conditions.

3.4.1. Performance Analysis of the System Information Fusion Model

When the signal is not out of the lock, the BDS receiver data are generally sent and received in the RTK positioning mode (sampling frequency 1 HZ), and the INS system data are collected normally (sampling frequency 10 HZ). At this time, model switch S1 is connected, S2 is not connected, and the information fusion mode is turned on. The comparison of longitude and latitude trajectories before and after filtering with the IABC-BP neural network fusion algorithm is shown in Figure 10, and the comparison of velocity and course is shown in Figure 11.

By comparing and analyzing the longitude and latitude trajectories, speed, and heading before and after comprehensive filtering, it can be seen that filtering speed and heading can eliminate coarse errors, improve accuracy, make speed changes smoother and more stable, and enable agricultural machinery to achieve a uniform acceleration movement.

The statistics of position, velocity, heading, and other related deviations are shown in

Table 6. Position and speed deviation statistics.

Evaluating Indicator		MAE	MSE	RMSE	R	ErrorMax
Position Error (m)	Latitude	6.13 × 10 -06	1.32 × 10 -10	1.15 × 10 -05	0.99988	6.17 × 10 -05
	Longitude	8.55 × 10 -06	1.90 × 10 -10	1.38 × 10 -05	0.99981	4.03 × 10 -05
	Altitude	9.90 × 10 -03	5.96 × 10 -04	2.44 × 10 -02	0.99998	1.29 × 10 -01
Velocity Error (m/s)		0.0137	0.0007	0.0257	0.99994	2.81 × 10 -01
Heading Error (°)		1.3477	6.5312	2.5556	0.99978	8.41 × 10 +00

, which can be analyzed as follows:

(1)

After the latitude and longitude data are processed by the model, the corresponding maximum deviation values are significantly smaller, but the data for altitude processing deviation are relatively large.

(2)

MAE, MSE, and other indicators show that the model exhibits good performance in data signal fusion. The velocity deviation also falls within a small range, indicating that the agricultural machine can maintain a uniform acceleration movement.

(3)

For heading processing, due to the small number of training samples, the maximum heading deviation is relatively large, with a value of 8°. In future research, the heading accuracy will continue to be improved.

Overall, the effectiveness and feasibility of the algorithm are relatively high and meet the requirements of high accuracy.

The processing results of the acceleration and attitude information collected using inertial sensors are shown in Figure 12, which can be analyzed as follows:

Within a running time of 3500 ms, the model can effectively process acceleration and altitude information. The model handles the X, Y, and Z directions well.

Although the attitude changes quickly during the experiment, with the direction adjustment angle being especially large, the model still shows a good processing effect.

The model is relatively well trained, and the data-filtering process is maintained within a specific error range, with small errors that meet the requirements of the model.

The corresponding statistics are given in

Table 7. Statistics related to acceleration and attitude deviation.

Evaluating Indicator		MAE	MSE	RMSE	R
Position Error (m)	X	1.3371 × 10 -03	4.5249 × 10 -06	2.1272 × 10 -03	0.99999
	Y	9.9046 × 10 -04	2.0930 × 10 -06	1.4467 × 10 -03	0.99998
	Z	2.0930 × 10 -06	2.1664 × 10 -05	4.6544 × 10 -03	0.99981
Attitude Error (°)	Azimuth	8.6624 × 10 -02	2.7088 × 10 -02	1.6459 × 10 -01	0.99989
	Elevation	5.7800 × 10 -03	7.1427 × 10 -05	8.4514 × 10 -03	0.99972
	Roll	6.6599 × 10 -03	1.1186 × 10 -04	1.0576 × 10 -02	0.99999

, which shows that the error indicators are relatively small and the correlation coefficient R > 0.90, meaning that the algorithm model accurately predicts the data and guarantees the subsequent correction of the compensated BDS information.

3.4.2. Performance Analysis of the System Compensation Correction Mode

In the agricultural navigation test, no signal loss lock occurred during the motion. However, to verify the performance of the IABC-BP model in the case of a satellite lockout, a satellite lockout duration of the 60 s was set manually, setting the following interference condition: turn off the base station during the test. The trajectory diagram and each part of the out-of-lock time are shown in Figure 13.

Based on the above setup, corresponding experiments were conducted to compare three navigation and positioning methods before and after the satellite loss of lock: the INS system alone, Kalman filter correction after the loss of the lock, and the predictive correction of data using the IABC-BP algorithm proposed in this paper. The experimental results are shown in Figure 14.

By analyzing the above experimental results, it can be concluded that:

(1)

After the satellite loss of lock, the position, velocity, and attitude values all undergo unknown changes. At this point, the drift from the loss of lock is not changed because there are no observations of the satellite signal to update the measurement equations of the Kalman filter, resulting in the model’s inability to estimate the information values correctly (blue line).

(2)

Since the IABC-BP model proposed in this paper is trained on the information data of BDS/INS when the signal is not out of the lock, the corresponding data are stored. Moreover, when the BDS signal is weak or out of the lock, the IABC-BP model proposed in this paper can be used instead of the BDS system to perform prediction corrections for the out-of-lock signal (green line).

(3)

The IABC-BP model predicts the observed quantities. Therefore, the observed quantities can be filtered together with the state equation of the INS system data, which can better compensate for the information data in a loss of lock, avoiding the generation of large errors and the complete dependence on historical data.

A comprehensive analysis shows that the compensation correction model of the IABC-BP model proposed in this paper can be applied to a longer period of satellite loss of lock and has a better effect and performance.

In addition, the statistical data related to the position, velocity, and attitude errors of the three methods were further processed and analyzed, and the excellent performance of the proposed method was further demonstrated through the experimental indicator data. The comparison results are shown in

Table 8. Deviation statistics of position and attitude.

Description	Position			Velocity (m/s)	Attitude (°)
	Latitude	Longitude	Altitude		Azimuth	Elevation	Roll
Before and after losing the lock	1.90 × 10 -03	3.68 × 10 -03	2.02 × 10 -01	1.50	1346.20	74.22	349.29
	5.87 × 10 -07	1.62 × 10 -06	6.51 × 10 -03	0.56	12,114.74	48.5539	864.25
	7.66 × 10 -04	1.27 × 10 -03	8.07 × 10 -02	0.75	110.07	6.97	29.40
KF filtering	1.69 × 10 -03	3.70 × 10 -03	1.73 × 10 +00	0.75	1447.76	122.39	330.92
	4.47 × 10 -07	1.63 × 10 -06	6.00 × 10 -01	0.19	25,897.81	161.92	1077.97
	6.69 × 10 -04	1.28 × 10 -03	7.75 × 10 -01	0.44	160.93	12.72	32.83
Predictive compensation	2.85 × 10 -04	3.70 × 10 -03	1.74 × 10 +00	0.75	315.10	106.17	203.47
	1.11 × 10 -08	1.63 × 10 -06	6.07 × 10 -01	0.19	13,301.76	124.47	378.58
	1.05 × 10 -04	1.28 × 10 -03	7.79 × 10 -01	0.44	115.33	11.16	19.45

.

Table 8 shows that:

(1)

When the BDS signal is out of the lock, both the navigation and positioning accuracy approaches are better than the single INS system, whether the Kalman filter is used for the correction output or the prediction compensation output using the Kalman-filter-based IABC-BP model in this paper.

(2)

For the compensation correction of position and velocity, the MAE, RMSE, and MSE values of the Kalman-filter-based IABC-BP algorithm model are the smallest, which indicates that the model proposed in this paper has higher accuracy and better robustness and stability.

(3)

At the same time, the compensation correction for the azimuth information is also minimal, and the pitch and roll angles are relatively small, which indicates that the farm machine can maintain a more stable motion when the model is corrected, and problems such as sideslip and cross-swing will not arise.

Overall, the model proposed in this paper exhibits relatively high accuracy and can effectively replace receiver information for the correction of outputs in case of lock loss.

4. Discussion

In this paper, the proposed model achieved significant results in improving the positioning accuracy and system stability of agricultural machinery. First, the BP neural network was improved through the process of selecting the number of nodes in the adaptive hidden layer. Then, the bee colony algorithm was improved via the search strategy of the bee colony and the probability of honey source selection, which improved the performance of the algorithm. Finally, an optimized BP network navigation and positioning model was constructed using the improved bee colony algorithm, continuously updating the network’s weights and thresholds to find the optimal solution, improving the accuracy and stability of the model.

However, further research and practice are needed in the following areas:

• The performance of different algorithms may vary in different application scenarios, and it is necessary to expand the application of the model in multiple scenarios. Therefore, it is necessary to explore the performance of the IABC-BP algorithm in different scenarios in the future, such as scenarios with different field sizes, terrains, and crops, in order to verify the robustness of the model and compare it with other intelligent algorithms for network operation timing.
• The actual running time and frequency of the proposed IABC-BP algorithm still need to be improved, requiring long-term and sustained experimental research in different environments and the further optimization of the model’s performance based on potential issues.
• The proposed IABC-BP algorithm can only achieve better performance by taking reliable hardware devices and computing power as its foundation, and by continuously adjusting the relevant parameters in combination with the constantly changing environment and actual circumstances. Therefore, future research should also explore issues such as the portability of the model to different hardware, as well as continuously expanding the application scope of the model.

5. Conclusions

This paper proposes a navigation and positioning algorithm model based on an improved bee-colony-algorithm-optimized BP network, which solves the problem of the reduced positioning accuracy and stability of BDS/INS systems in rejection environments. This study fully considers the rationality of information fusion between system sensors. Dividing the algorithm model into information fusion mode and the correction compensation mode can accurately locate the position, speed, and attitude information of agricultural machinery, while improving the stability and robustness of the system. At the same time, an experimental analysis was conducted to verify the performance of the model in practical applications.

The innovation of the method proposed in this paper lies in the fusion processing of longitude and latitude, height, speed, and other information in different sensors under the two modes of information fusion and correction compensation. The real-time analysis and fusion processing of the dynamic operation data of agricultural machinery were achieved, ensuring the accuracy of the predicted data. The proposed IABC-BP algorithm can better grasp the motion trajectory of agricultural machinery, effectively improving the operational performance of the model and the reliability of practical applications. The main conclusions are as follows:

• The optimization test result of the test function shows that the proposed IABC algorithm has higher accuracy than the traditional ABC algorithm in terms of the optimal value, mean, and mean square deviation in the search for the optimal value, which indicates the higher quality of the IABC algorithm’s solution and the better performance of the ABC algorithm in terms of robustness and stability.
• The IABC-BP algorithm is applied to the BDS/INS navigation system, the IABC-BP navigation and positioning algorithm are modeled based on the Kalman filter, and a comparison experiment is set up with the BP model, the KF-BP model, IABC-BP, and KF-IABC-BP in four models. The experimental results show that the KF-IABC-BP model improves performance by 90.65%, 84.11%, and 25.96% compared with the previous three models. The Kalman-filter-based IABC-BP neural network algorithm model established in this paper has sufficient feasibility.
• Practical experiments were conducted on two modes of the IABCBP navigation and positioning algorithm model, both of which can ensure the system’s navigation and positioning accuracy in a short time. The MSE values after fusion processing for longitude, latitude, altitude, velocity, and heading in the information fusion mode are 1.32 × 10 ^-10, 1.90 × 10 ^-10, 5.96 × 10 ^-04, 0.0007, and 6.5312, respectively, indicating that the algorithm model has high accuracy, effectiveness, and feasibility. Meanwhile, the algorithm model can accurately predict the data and has good filtering and fusion effects because the correlation coefficients R > 0.90. In the compensation correction mode, the predicted data from the trained IABC-BP model are output instead of BDS, and the MSE values of the predicted longitude, latitude, altitude, velocity, and attitude data are 1.11 × 10 ^-08, 1.63 × 10 ^-06, and 6.07 × 10 ^-01 and 13301.76, 124.47, and 378.58, respectively, and the accuracy is reduced compared with the first mode.