Innovation Adaptive UKF Train Location Method Based on Kinematic Constraints

Abstract

To address the issue of reduced positioning accuracy caused by satellite signal interruptions when trains pass through long tunnels, a novel train positioning method based on an innovative adaptive unscented Kalman filter (UKF) under kinematic constraints is proposed. This method aims to improve the accuracy of the location of trains during operation. By considering the dynamic characteristics of the train, a dynamic kinematic-constrained inertial navigation system (INS)/odometer (ODO) combination positioning system is established. This system utilizes kinematic constraints to correct the accumulated errors of the INS. Additionally, the algorithm incorporates real-time estimation of the measurement noise covariance using innovation sequences. The updated adaptive estimation algorithm is applied within the UKF framework for nonlinear filtering, forming the innovative adaptive UKF algorithm. At each time step, the difference between the ODO sensor data and the INS output is used as the measurement input for the innovative adaptive UKF algorithm, enabling global estimation. This process ultimately yields the actual positioning result for the train. Simulation results demonstrate that the innovative adaptive UKF train positioning method, incorporating kinematic constraints, effectively mitigates the impact of satellite signal interruptions. Compared with the traditional INS/ODO positioning method, the innovative adaptive UKF method reduces position errors by 34.35% and speed errors by 36.33%. Overall, this method enhances navigation accuracy, minimizes train positioning errors, and meets the requirements of modern train positioning systems.

1. Introduction

Train positioning technology is crucial for enhancing traffic safety and optimizing transportation efficiency. Currently, train positioning primarily depends on trackside equipment, which no longer meets the demands of the new generation of train control systems [1]. Advancements in high-speed rail signaling technology have driven the development of multi-sensor combination positioning methods, addressing the limitations of single-sensor systems and opening new possibilities for achieving high-precision train positioning [2,3].

In the field of land mobile navigation, sensor fusion technologies have been widely adopted, particularly in automobiles and autonomous aerial vehicles. Liu et al. employed an integrated global navigation satellite system (GNSS)/inertial navigation system (INS) in autonomous vehicles to update the measurement noise variance in real-time and introduce attenuation factors that increase the weight of the current measurement. This effectively suppresses white noise and mitigates environmental interference, leading to more accurate vehicle navigation [4,5]. Zhang et al. proposed an adaptive Kalman filtering method to address the issue of weak GNSS signals in the GNSS/INS integrated navigation system for unmanned aerial vehicles (UAVs) [6]. This method dynamically adjusts the GNSS measurement noise covariance using an accuracy classification model trained via supervised machine learning, effectively addressing the localization issue for UAVs when GNSS signals are disturbed. This approach provides a promising pathway for solving the challenge of determining the precise location of trains. The complex and dynamic operating environments of trains make it difficult to achieve accurate positioning based solely on satellite signals [7,8]. As a result, achieving precise train positioning often necessitates the integration of additional sensors into a multi-sensor fusion positioning system [9,10]. Numerous scholars have extensively researched train positioning through sensor fusion and its associated risk assessment technologies using GNSS [11,12,13,14]. However, challenges emerge when trains pass through tunnels, where GNSS signals are weak or interrupted. In these situations, the lack of GNSS corrections amplifies INS errors, leading to substantial drift and severely affecting the accuracy of the multi-sensor fusion positioning system [15]. Currently, there are two primary solutions to mitigate the rapid divergence of INS errors following satellite signal interference or interruption. One approach is to integrate additional sensors to improve train positioning accuracy [16,17]. To enhance positioning accuracy, Li et al. integrated a framework combining a strapdown inertial navigation system (SINS), odometer (ODO), and laser Doppler velocimetry. Additionally, they proposed the Schmidt EKF information fusion algorithm, which effectively reduces positioning errors [18]. In onboard railway positioning systems, some scholars combine SINS, ODO, and digital track maps. Track features are extracted during the offline stage and stored in the map to improve positioning accuracy [19,20]. During train operation, these stored track features are detected and matched to correct ODO errors, thereby improving positioning accuracy [21].

The second approach involves utilizing track or train dynamic kinematic constraints to mitigate INS drift errors and achieve precise train positioning in tunnels [22]. By integrating GNSS and INS with motion constraints and considering the dynamic characteristics of track geometry measurement vehicles, navigation accuracy is improved. Liu et al. developed a high-precision laser integrated with the INS/GNSS positioning method, which significantly enhances system accuracy during GNSS signal interruptions by leveraging motion constraints [15]. Zhu et al. introduced the application of installation angle adjustments to improve the motion constraint model, integrating the motion characteristics of trains with inertial measurement unit (IMU) and ODO sensors. This model uses the GNSS system to estimate the installation angle of the IMU under favorable GNSS signal conditions [23]. By combining ODO data with dynamic kinematic constraints, inertial navigation errors are minimized, enhancing navigation accuracy in tunnels. Zhang et al., considering cost-effectiveness and practicality, employed train kinematic constraints to effectively reduce accumulated errors during INS operation, thereby improving train positioning accuracy [24]. Zhou et al. evaluated the spatial relative accuracy for measuring shortwave trajectory irregularities. This evaluation used specific indicators and calculations to assess relative accuracy, aiming to establish an accuracy threshold that meets the measurement constraints [25].

The multi-sensor information fusion algorithm is the core of combinational positioning and is crucial for ensuring system performance. Two widely used methods are the extended Kalman filter (EKF) and unscented Kalman filter (UKF) [26]. Based on the Kalman filter, the extended Kalman filter (EKF) addresses nonlinear system problems through linearization techniques. At each iteration, the EKF approximates the nonlinear state transition and observation models using a first-order Taylor series expansion around the current estimate, achieving local linearization. The recursive Kalman filter equations are then applied to this linearized model to predict and update the state, providing optimal estimation of the system state and its uncertainty [27]. The UKF algorithm utilizes calculated sigma points to perform the Unscented Transformation (UT), approximating the posterior probability density function of the system state [28]. Compared to the EKF algorithm, the UKF offers advantages such as simpler estimation of the posterior mean and variance of the state vector, better convergence performance, and significantly improved filtering accuracy when handling nonlinear systems [29]. An algorithm combining support vector regression (SVR) and adaptive UKF is proposed to predict and handle abnormal measurements using SVR. It adjusts the measurement noise covariance matrix in real-time, enhancing the robustness and accuracy of the UKF in environments with outliers [30]. Based on the covariance matching principle, Meng et al. designed an adaptive UKF capable of estimating the system’s noise variance online [31]. Lin et al. combined particle filtering with the UKF to propose an enhanced hybrid filtering method, enabling accurate estimation in highly nonlinear systems [32]. The performance advantage can be optimized by flexibly adjusting the distribution of sigma points and the number of particles to meet the requirements of different complexity levels in application scenarios. Gu et al. proposed an improved adaptive UKF algorithm that dynamically estimates and corrects the statistical characteristics of system noise in real-time. Additionally, an adaptive fading factor mechanism is introduced to effectively enhance the estimation accuracy and operational reliability of the system [33]. Hu et al. proposed a robust UKF algorithm based on the innovation orthogonality principle, designing a single measurement fault scale factor to implement a unified adjustment strategy for the measurement noise covariance matrix when handling system failures [34]. The results of simulation experiments fully demonstrate that the filtering algorithm is highly robust and effective when confronted with significant deviations in sensor measurements. Li et al. proposed an adaptive federated filtering method by introducing adaptive filter factors to develop information allocation strategies within federated filters [35]. In the application of this method, the precision single-point positioning system, assisted by an INS, provides a more accurate initial value estimation, thereby improving the overall performance of the positioning system. However, differences in sampling frequencies and error characteristics among various sensors can affect the timeliness of information fusion, which can, in turn, reduce positioning accuracy. Recent studies show that current train positioning methods primarily rely on multi-sensor fusion techniques, integrating GNSS with other sensors to enhance accuracy through algorithmic improvements. As railway infrastructures, such as the Sichuan-Tibet Railway, become more complex—with extensive mountainous terrain and numerous long tunnels—interruptions in satellite signals within these tunnels significantly degrade positioning accuracy.

To address these challenges, this paper proposes an innovative adaptive UKF-based train positioning method that integrates kinematic constraints. By leveraging the dynamic characteristics of train operations, a kinematically constrained INS/ODO positioning system is developed. This system employs dynamic kinematic constraints to correct the accumulated errors in INS data. At the same time, real-time estimation of measurement noise covariance is combined with innovative sequences, improving the adaptive estimation algorithm within the UKF framework for nonlinear filtering. This results in the development of an advanced adaptive UKF algorithm. In the proposed method, the difference between the outputs of the ODO sensor and INS serves as the measurement data, which is vital for the adaptive UKF algorithm to achieve precise global estimation and provide accurate train positioning results.

2. INS/ODO Train Combination Positioning System

2.1. Structure of Train Combination Positioning System

The train positioning system discussed in this article primarily integrates data from an INS and an ODO. By combining these sensor inputs, the system performs data collection, preprocessing, fusion, transmission, and display of position information. This process enables real-time monitoring of the train’s dynamics and ensures accurate positioning. Figure 1 illustrates the INS/ODO combined train positioning system, highlighting its integration with dynamic kinematic constraints.

The kinematically constrained INS/ODO train positioning system employs kinematic constraints to mitigate the accumulated errors inherent in the INS. It integrates the differential data between ODO velocity measurements and INS outputs as part of the measurement inputs for an innovative adaptive UKF algorithm. This approach estimates global errors and corrects INS deviations by incorporating feedback related to zero bias and specific force factors. Additionally, refined gyroscope and accelerometer data from the INS are used to enhance updates on train angular velocity, acceleration, and system error estimates. These components work synergistically to provide a cohesive and accurate navigation solution for the train.

2.2. Train Combination Positioning System Model

The mathematical model of the INS/ODO combined train positioning system consists of two components: the state model and the measurement model.

2.2.1. The State Model of the System

The system state variables are categorized into two distinct types: INS errors and ODO errors. The state vector of the system is expressed as follows:

$$x={\left[\delta v_E,\delta v_N,\delta v_U,{\epsilon }_X,{\epsilon }_Y,{\epsilon }_Z,\delta f_x,\delta f_y,\delta f_z,\delta \psi ,\delta K_D,\delta \theta \right]}^T$$

(1)

Among these, $(\delta v_E,\delta v_N,\delta v_U)$ represent the velocity errors of the train in the east, north, and vertical directions; $({\epsilon }_X,{\epsilon }_Y,{\epsilon }_Z)$ and $(\delta f_x,\delta f_y,\delta f_z)$ are the gyroscope and acceleration errors of the INS; and $(\delta \psi ,\delta K_D,\delta \theta )$ represent the ODO errors.

By combining the state equations of the INS and ODO, it can be simplified as follows:

$$\dot{x}\left(t\right)=F(x\left(t\right),t)+G\left(t\right)w\left(t\right)$$

(2)

where $\delta \dot{x}$ denotes the differential of the system error state vector; $\delta x$ denotes the system error state vector; $F(·)$ denotes a nonlinear state transition function; $G\left(t\right)$ denotes the noise array; and $w\left(t\right)$ denotes the process noise of the system.

2.2.2. System Measurement Model

In the integrated positioning system, the velocity outputs from the INS and ODO are compared by subtracting them to derive the difference, serving as the measurement information for the positioning system. The true value and error of the INS velocity can be expressed as follows:

$${\tilde{v}}^n=v^n+\delta v^n$$

(3)

where ${\tilde{v}}^n$ denotes the carrier velocity calculated by INS, $v^n$ denotes the true value of the train carrier velocity, and $\delta v^n$ denotes the velocity error. Then, we subtract the carrier velocity ${\tilde{v}}_D^n$ calculated by the ODO from ${\tilde{v}}^n$; It can be obtained as follows:

$$\Delta v={\tilde{v}}^n-{\tilde{v}}_D^n=\left(v^n+\delta v^n\right)-\left(v^n+\phi \times v^n\right)-C_b^n{\tilde{v}}_D^b=\delta v^n-\phi \times v^n-C_b^n{\tilde{v}}_D^b$$

(4)

Among these, $\phi$ denotes the attitude error vector; and $C_b^n$ denotes a $3\times 3$ transformation matrix. The velocity error $\Delta v$ can be represented along the three axes of the navigation coordinate system as follows:

$$\Delta v=\left[\begin{array}{c}\Delta v_E^n\hfill \\ \Delta v_N^n\hfill \\ \Delta v_U^n\hfill \end{array}\right]$$

(5)

Then, the obtained measurement error is as follows:

$$\Delta v=\left[\begin{array}{c}\delta v_E^n\hfill \\ \delta v_N^n\hfill \\ \delta v_U^n\hfill \end{array}\right]-\left[\begin{array}{ccc}0& v_U^n& -v_N^n\\ -v_U^n& 0& v_E^n\\ v_N^n& -v_E^n& 0\end{array}\right]{\left[{\phi }_E,{\phi }_N,{\phi }_U\right]}^T-C_b^n{v}_D{\left[\delta \psi ,\delta K_D,\delta \theta \right]}^T$$

(6)

Among these, $({\phi }_E,{\phi }_N,{\phi }_U)$ represent the attitude error information of the train in the east, north, and vertical directions. The measurements in the INS/ODO cooperative navigation system include pseudorange, differential pseudorange, and relative distance measurements. Relative distance measurements are obtained by calculating the difference between the INS equivalent relative distance and the ODO distance measurement. The measurement equation for the INS/ODO combined positioning system is expressed as follows:

$$z\left(t\right)=H\left(x\right(t),t)+V\left(t\right)$$

(7)

where $z\left(t\right)$ represents the measurement vector, $V\left(t\right)$ represents the noise matrix, and $H(·)$ represents a nonlinear measurement function.

2.3. Dynamic Kinematic Constraints of Trains

In traditional combined positioning solutions, it is generally assumed that the lateral and vertical motions of the train—specifically, the velocities along the lateral (Y-axis) and vertical (Z-axis) axes of the car body—are negligible. This assumption is based on the premise that the train remains stable on its tracks. As a result, dynamic kinematic constraints suggest that velocities along both the Y-axis and Z-axis in the car body coordinate system are consistently zero. This simplification leverages the physical characteristics of trains to improve model accuracy. However, prior methodologies often set a fixed variance for these dynamic kinematic constraints—commonly around 0.1 m/s or 0.2 m/s [24]—which may not adequately reflect the actual movements of the train in complex environments. Track curvature, in particular, can significantly influence both the train’s attitude and displacement during turns and gradient transitions. By incorporating track curvature constraints, the angular velocity, acceleration, and displacement estimates derived from INS can be corrected to reduce attitude errors. Track geometry often exhibits non-linear characteristics, especially at complex intersections or sharp curves. Using a UKF algorithm allows the integration of these non-linear constraints into the state estimation process, ultimately enhancing the accuracy of the overall system.

To further enhance the precision of the solution, this paper proposes a method for dynamically adjusting kinematic constraint variances based on INS outputs. Specifically, train motion stability is evaluated by calculating the noise variance from INS output data over one-second intervals. When the calculated noise variance falls below a predetermined threshold, it indicates relative stability in the train’s movement. This allows for dynamic adjustment of the kinematic constraint variances according to the noise levels from the INS sensors. This approach enables greater “trust” in dynamic kinematic constraints during steady states, where stricter variances can be applied to improve positional accuracy. Conversely, under unstable conditions, where confidence in these constraints diminishes, increased variances prevent erroneous results caused by overly restrictive limitations. By adaptively adjusting the kinematic constraint variances, the integrated INS/ODO positioning system can respond more flexibly to varying operational conditions. It applies stringent kinematic parameters when the train is in a stable state while adjusting variances based on noise levels during periods of dynamic changes. This assumption implies that the train’s velocities along the y-axis and z-axis are both zero within the coordinate system. The specific process can be described as follows:

$$\left\{\begin{array}{c}\delta v_{y0}^b=0\hfill \\ \delta v_{z0}^b=0\hfill \end{array}\right.$$

(8)

Among these, $\delta v_{y0}^b$ and $\delta v_{z0}^b$ represent the train’s velocity in the coordinate system’s y direction and z direction, respectively.

Assuming that the train is traveling at a constant velocity at a given moment, we can infer the velocity of the train in the direction of travel as follows:

$$v_x^b(k+1)=v_x^b\left(k\right)$$

(9)

Here, $v_x^b\left(k\right)$ can be obtained through the output data of INS and ODO sensors, as follows:

$$v_x^b\left(k\right)=\sqrt{{\left(v_{I/D}^{bx}\right)}^2+{\left(v_{I/D}^{by}\right)}^2+{\left(v_{I/D}^{bz}\right)}^2}$$

(10)

Among these, $v_{I/D}^{bx}$, $v_{I/D}^{by}$, and $v_{I/D}^{bz}$ denote the train velocities in the coordinate system; we can obtain the following:

$$v_{I/D}^b=C_n^b{v}_{I/D}^n$$

(11)

Therefore, the error measurement of train velocity under dynamic kinematic constraints is as follows:

$$z_v\left(t\right)=C_b^v{C}_n^b{v}_x^b\left(t\right)+n_v\left(t\right)$$

(12)

The above equation represents the measurement model of the UKF algorithm, where $n_v$ represents the noise matrix of the dynamic kinematics constraint, which is calculated as follows:

$$n_v=\left[\begin{array}{cc}v{x}_{std}& 0\\ 0& v{y}_{std}\end{array}\right]$$

(13)

where $v{x}_{std},v{y}_{std}$ represent the variance of the accelerometer on the X and Y axes within 1 s, respectively.

3. Innovative Adaptive UKF Information Fusion Algorithm

Compared to the EKF and particle filter (PF), the UKF offers significant advantages in the INS/ODO combined positioning system for trains. These advantages are mainly reflected in UKF’s ability to handle nonlinear problems, higher computational efficiency, and robustness in high-noise environments [28]. UKF avoids errors caused by linearization through unscented transformation, allowing it to better capture the system’s nonlinear dynamics. Additionally, UKF does not require the computation of a Jacobian matrix, simplifying its implementation and enabling it to handle the complex track and environmental changes more effectively. In contrast, EKF is constrained by the limitations of the linearization process, while PF, despite its theoretical capability to handle nonlinearities, suffers from high computational complexity, making it less suitable for real-time applications [29]. As a result, UKF is well-suited for such positioning systems. To mitigate the accumulation of errors in INS calculations, dynamic kinematic constraints are applied based on the train’s operational characteristics. These constraints help compensate for errors by aligning the calculations with the actual motion dynamics of the train. At the same time, the difference between the ODO sensor and INS output data serves as the measurement input for the innovative adaptive UKF algorithm, which performs global state estimation to determine the train’s position at the current moment. Subsequently, the innovative adaptive UKF algorithm is used to predict the train’s location for the next time step. The final navigation results are then output and fed back into the INS system to correct any errors that have accumulated during the previous calculation cycle. Figure 2 illustrates the process of the INS/ODO train positioning information fusion algorithm, incorporating dynamic kinematic constraints.

As shown in Equation (3), the state equation of the INS/ODO combined positioning model for trains exhibits nonlinear characteristics. To more accurately estimate the system’s state, the UKF algorithm is employed to address the state estimation problem during the information fusion process.

The state model (3) and measurement model (8) of the train’s combined positioning system are discretized and can be expressed as follows:

$$\left\{\begin{array}{c}{x}_k=f(x_{k-1},k-1)+G_{k-1}{w}_{k-1}\hfill \\ z_k=h\left(x_k,k\right)+v_k\hfill \end{array}\right.$$

(14)

Among these, $f(·)$ denotes a nonlinear state transition function; $G_{k-1}$ denotes the noise array; $h(·)$ denotes a nonlinear measurement function; $w_{k-1}$ denotes the system noise vector, $v_k$ denotes the measurement noise vector.

The specific process of the UKF algorithm is as follows [28]:

(i) Calculate the (2n + 1)-th sigma point as follows:

$$\left\{\begin{array}{c}{\eta }_0={\widehat{x}}_{k-1}\hfill \\ {\eta }_i={\widehat{x}}_{k-1}+\left(\alpha \sqrt{n{\widehat{p}}_i}\right)e_i\hfill & i=1,2,\cdots ,n\hfill \\ {\eta }_i={\widehat{x}}_{k-1}-\left(\alpha \sqrt{n{\widehat{p}}_{i-n}}\right)e_{i-n}\hfill & i=n+1,n+2,\cdots ,2n\hfill \end{array}\right.$$

(15)

In the formula, $\alpha \sqrt{n{\widehat{p}}_i}$ denotes the i column of the square root of the covariance matrix. $\alpha =\sqrt{n+\lambda }$, $\lambda ={\gamma }^2(n+k)-n$, $\gamma$ takes 0∼1, and $k=3-n$.

(ii) Time update.

After determining the sigma point sampling, the time update equation can be changed to the following:

$$\left\{\begin{array}{c}{\widehat{x}}_{k,k-1}=\sum w_{i,n}{\eta }_{i,k,k-1}\hfill \\ P_{x,k,k-1}=\sum w_{i,c}\left({\eta }_{i,k,k-1}-{\widehat{x}}_{k,k-1}\right)\times \left({\eta }_{i,k,k-1}-{\widehat{x}}_{k,k-1}\right)+Q_k\hfill \end{array}\right.$$

(16)

In the formula, ${\eta }_{k,k-1}$ denotes the sigma point from time $k-1$ to k, ${\widehat{x}}_{k,k-1}$ denotes the predicted estimated value of the system state from time $k-1$ to k, and $P_{k,k-1}$ denotes the covariance matrix for predicting and estimating this state, quantifying the error range of the predicted estimation values.

(iii) Measurement update.

$$\left\{\begin{array}{c}{\widehat{z}}_k=\sum _{i=0}^{2n}{w}_{i,n}{z}_{i,k,k-1}\hfill \\ P_{zz}=\sum _{i=0}^{2n}{w}_{i,c}\left(z_{i,k,k-1}-\widehat{z}\right)\times {\left(z_{i,k,k-1}-\widehat{z}\right)}^T+R_k\hfill \end{array}\right.$$

(17)

In the formula, ${\widehat{z}}_{k,k-1}$ denotes the predicted measurement value from time $k-1$ to k, ${\widehat{z}}_k$ denotes the predicted measurement value at time k, and $P_{zz}$ denotes the covariance matrix of ${\widehat{z}}_k$.

For the nonlinear system of Equation (16), the UKF algorithm is updated with measurements, and the predicted value is as follows:

$${\widehat{z}}_k=\sum _{i=0}^{2n}{w}_i{\eta }_i^{*}$$

(18)

where ${\eta }_i^{*}$ denotes the i measurement point; the calculation update is as follows:

$$\begin{array}{cc}\hfill \delta {\widehat{z}}_k=& z_k-\sum _{i=0}^{2n}{w}_i{\eta }_i^{*}\hfill \\ \hfill =& \left(z_k-{\widehat{z}}_k\right)+\left({\widehat{z}}_k-\sum _{i=0}^{2n}{w}_i{\eta }_i^{*}\right)\hfill \\ \hfill =& v_k+\left(\sum _{i=0}^{2n}{w}_i{\widehat{z}}_k-\sum _{i=0}^{2n}{w}_i{\eta }_i^{*}\right)\hfill \\ \hfill =& v_k+\sum _{i=0}^{2n}{w}_i\left({\widehat{z}}_k-{\eta }_i^{*}\right)\hfill \end{array}$$

(19)

Real-time estimation of measurement noise variance based on innovation sequences. The innovation sequence is defined as follows:

$${\epsilon }_k=z_k-h_k{x}_{k,k-1}=h_k\left(x_k-x_{k,k-1}\right)+v_k$$

(20)

When the innovation sequence follows a zero-mean, independently and identically distributed Gaussian white noise model, the innovation covariance matrix is calculated as follows:

$$S_k=E\left[{\epsilon }_k{\epsilon }_k^T\right]$$

(21)

Therefore, adaptive factor ${\beta }_k$ is used to adjust the innovation covariance matrix $S_k$ in the Kalman gain, so that the performance of the algorithm can be optimized online.

$$K_k=P_{zz,k,k-1}{\left({\beta }_k{S}_k\right)}^{-1}={\displaystyle \frac{1}{{\beta }_k}}{P}_{zz,k,k-1}{S}_k^{-1}$$

(22)

Finally, the measurement equation is updated, and the updated state and covariance estimation are obtained as follows:

$${\widehat{x}}_k={\widehat{x}}_{k,k-1}+K_k{ϵ}_k$$

(23)

$$P_k=P_{x,k,k-1}-K_k{S}_k{K}_k^T$$

(24)

4. Verification and Analysis

To evaluate the effectiveness of the innovation adaptive UKF approach in train positioning under dynamic kinematic constraints, a train motion model was developed using MATLAB R2022b software, and a simulation experiment was conducted. Real data from the Longnan to Guangyuan section of the Lanzhou–Chongqing Railway were utilized to simulate the positioning performance of the INS/ODO combined positioning system in a tunnel environment where satellite signals are unavailable. The simulation emulated various stages of train operations in a continuous tunnel group, including acceleration, climbing, turning, deceleration, and waiting to avoid obstacles, with speed variations ranging from 60 to 200 km/h. During the process, both INS and ODO data were recorded. The specific sensor parameters are presented in

Table 1. Parameter settings for positioning sensors.

Parameter Entry	Initial Value
The constant error of the accelerometer	$0.05\mathrm{g}$
Random movement of the accelerometer	${10}^{-4}\mathrm{g}/\sqrt{\mathrm{s}}$
Zero bias characteristic of the gyroscope	$0.{05}^{\circ }/\mathrm{h}$
Random movements of gyroscopes	$0.{01}^{\circ }/\sqrt{\mathrm{h}}$
Sampling frequency of INS	100 Hz
Sampling frequency of ODO	10 Hz

. The train operation data from GPST 457250 to 457900 were analyzed over a total duration of 650 s, and a simulated train trajectory was constructed. Satellite signal interruptions were simulated in five segments, with each interruption lasting 60 s. The results showed that the performance of the INS/ODO train positioning system, constrained by the innovative adaptive UKF algorithm and dynamic kinematics, was superior in the absence of satellite signals. To evaluate the effectiveness of the innovative adaptive UKF algorithm, its accuracy was compared to that of the UKF algorithm during simulated satellite signal interruptions, as shown in

Table 2. Simulated satellite signal interruption.

Interrupt	Start Time (GPST)	End Time (GPST)	Interrupt Duration
#1	457250	457310	60 s
#2	457350	457410	60 s
#3	457450	457510	60 s
#4	457550	457610	60 s
#5	457650	457710	60 s

.

The longitude -latitude track of the train is shown in Figure 3, where the curve represents the actual railway line in this section. Using high-precision SPAN-FSAS-integrated navigation results as a reference, the errors of both the UKF algorithm and the innovative adaptive UKF algorithm were calculated through epoch-wise differencing. A comparative analysis of the results was then conducted.

Figure 4 illustrates the positional errors of the train in the north, east, and vertical directions, while Figure 5 depicts the velocity errors in the corresponding directions. In both figures, five satellite signal interruptions are highlighted in red. For instance, in Figure 4, these interruptions are labeled as interrupts no. 1 through no. 5.

The comparative analysis of Figure 4 and Figure 5 clearly demonstrates that during satellite signal interruptions, the error associated with the innovative adaptive UKF algorithm is significantly lower than that of the standard UKF algorithm. This highlights the superior performance of the proposed innovative adaptive UKF in information processing and data fusion. The algorithm shows an enhanced ability to integrate multiple data sources, effectively reducing positioning errors in scenarios where satellite signals are unstable or interrupted.

To further compare and analyze the positioning accuracy of the two algorithms, detailed statistics for various error cases were compiled, with the relevant data summarized in

Table 3. Comparison of position errors between two algorithms (m).

	Innovation Adaptive UKF			UKF
	North	East	Vertical	North	East	Vertical
#1	2.71	3.90	7.06	2.10	6.22	9.32
#2	3.12	2.24	6.56	5.62	0.68	7.74
#3	0.82	0.61	1.81	1.32	0.71	2.43
#4	0.40	0.32	0.22	0.83	0.32	0.37
#5	1.72	2.15	3.96	1.72	2.91	5.21
Mean	0.17	0.28	1.22	0.25	0.33	1.56
RMSE	1.41	1.34	2.15	1.13	2.00	2.76

and

Table 4. Comparison of velocity errors between two algorithms (m/s).

	Innovation Adaptive UKF			UKF
	North	East	Vertical	North	East	Vertical
#1	0.41	0.60	0.38	0.44	0.63	0.39
#2	0.32	0.23	0.48	0.35	0.25	0.56
#3	0.22	0.22	0.17	0.24	0.24	0.19
#4	0.16	0.16	0.18	0.21	0.18	0.23
#5	0.36	0.61	0.23	0.39	0.66	0.25
Mean	0.02	0.02	0.04	0.04	0.03	0.05
RMSE	0.07	0.08	0.12	0.11	0.13	0.14

. As shown in these tables, during the five-stage satellite signal interruption, the maximum positional errors in the north, east, and vertical directions under the innovative adaptive UKF algorithm were 3.1 m, 3.9 m, and 7.06 m, respectively. These results outperformed the UKF algorithm, which had maximum errors of 5.6 m, 6.2 m, and 9.32 m in the same direction. Additionally, during the five-stage satellite signal interruption, the maximum velocity errors for the north, east, and vertical directions under the innovative adaptive UKF algorithm were 0.41 m/s, 0.61 m/s, and 0.48 m/s, respectively. These also surpassed the UKF algorithm, which exhibited velocity errors of 0.44 m/s, 0.66 m/s, and 0.56 m/s.

Based on the findings presented above, it can be concluded that during satellite signal interruptions, the proposed innovative adaptive UKF algorithm significantly reduces positioning errors compared to the UKF algorithm. Analyzing data from the five segments with signal interruptions, it is evident that the innovative adaptive UKF algorithm consistently demonstrates smaller velocity, mean position error (Mean), and root mean square error (RMSE) values compared to the UKF algorithm. The innovative adaptive UKF algorithm achieved RMSE reductions of 32.02% in the north direction, 33.14% in the east direction, and 21.74% in the vertical direction. Similarly, velocity errors in the north, east, and vertical directions were reduced by 36.36%, 38.46%, and 14.39%, respectively. These results indicate that during satellite signal interruptions, the innovative adaptive UKF algorithm significantly enhances the global optimization of INS and ODO information, leading to improved train positioning accuracy.

To evaluate the impact of incorporating dynamic kinematic constraints on the accuracy of the INS/ODO train positioning system in tunnel environments, we conducted a comparative simulation analysis of two approaches. These approaches included the standard INS/ODO train combination positioning system and the dynamic kinematic constraint INS/ODO train combination positioning system. In both systems, the innovative adaptive UKF algorithm was used as the information fusion method. The comparative simulation results for these two positioning methods are presented in Figure 6 and Figure 7.

By analyzing the position error comparison data in Figure 6, it is evident that the incorporation of dynamic kinematic constraints significantly improved the positioning accuracy of the INS/ODO train combination positioning system. Specifically, with these constraints, the maximum positioning error was reduced to 6.38 m, and the average error improved by 34.35% compared to the system without dynamic kinematic constraints. Similarly, the velocity error comparison in Figure 7 demonstrates that the velocity error of the system with dynamic kinematic constraints exhibited relatively smooth variations, further indicating enhanced positioning accuracy. Compared to the INS/ODO system without dynamic constraints, the maximum velocity error in the constrained system was reduced to 0.38 m/s, with an average error reduction of 36.33%. These results clearly demonstrate that the innovative adaptive UKF train positioning system, when operating under dynamic kinematic constraints, maintains high navigation accuracy even in environments where satellite signals are unavailable, such as tunnels.

5. Conclusions

To address the challenge of reduced positioning accuracy caused by satellite signal interruptions in tunnels, we developed an integrated INS/ODO train positioning system. Leveraging the train’s motion characteristics, we proposed an innovative adaptive UKF train positioning method that incorporates dynamic kinematic constraints. This method enhances the fusion of data within the INS/ODO system using an innovative adaptive UKF algorithm. Experimental results demonstrated that the innovative adaptive UKF algorithm offers significant improvements in estimating train position errors compared to the UKF algorithm. When applied to the dynamic kinematic constraints INS/ODO train positioning system, the innovative adaptive UKF method achieved a 34.35% reduction in position errors and a 36.33% reduction in velocity errors compared to the system without constraints. This confirms that the proposed approach effectively improves positioning accuracy, particularly in environments such as tunnels, where satellite signals are unavailable. Looking ahead, further optimization of the train positioning method will be required, especially in scenarios like forced or unexpected stops within long tunnels. Enhancing the system with additional sensors and addressing the issue of error divergence in the INS system will be crucial. Techniques such as zero velocity correction and other strategies should be explored to further mitigate error divergence and enhance the reliability of train positioning systems.