Online Estimation of the Mounting Angle and the Lever Arm for a Low-Cost Embedded Integrated Navigation Module

Abstract

Multi-source fusion constitutes a research focus in the navigation domain. This article focuses on the online estimation of the mounting angles between the body frame and vehicle frame within low-cost embedded vehicle navigation modules and the lever arm between the global satellite navigation system (GNSS) antenna/odometer and the inertial measurement unit (IMU). An online mounting angle error estimation algorithm, using odometers and IMU speeds, has been developed to estimate the angle errors while vehicles are in motion. At the same time, an online estimation algorithm model for the GNSS antenna lever arm and odometer lever arm was constructed. These two types of lever arms are used as the estimated states, and then Kalman filters are used to estimate them. The algorithm can simultaneously estimate the IMU mounting angle error, GNSS antenna arm, and odometer arm online. The experimental outcomes demonstrate that the lever arm estimation algorithm presented herein is effective for tactical and MEMS-level inertial navigation, with an estimation error of less than 2 cm. Meanwhile, the proposed online estimation of the mounting angle algorithm has an accuracy comparable to that of the post-processing algorithm. After making up the mounting angle and lever arm, we found that the position and speed precision of the multi-source fusion navigation systems were significantly improved. The results indicate that the proposed online estimation of mounting angle error and lever arm algorithm are effective and may promote the practical and widespread application of integrated navigation systems in vehicles. It solves the shortcomings of traditional methods, including the cumbersome and inaccurate manual measurement of the lever arm. It provides a technical solution for developing a more accurate and convenient low-cost vehicle navigation module.

1. Introduction

Integrating the global satellite navigation system (GNSS) and inertial navigation system (INS) can complement each other’s defects and offer more precise and stable navigation information, thus providing high-precision positioning and navigation solutions for land vehicles [1,2]. However, when vehicles travel through complex urban environments, the GNSS may experience a decrease in navigation accuracy due to satellite signal interference or interruption. Especially when vehicles are traveling between high-rise buildings, under trees, or in tunnels, the navigation state often drifts based on low-cost inertial measurement units (IMUs) [3]. Multi-source information fusion can enhance the GNSS/INS integrated navigation accuracy as the signal state is poor. The commonly used external added sensor for land vehicle navigation is the odometer, which possesses the benefits of high cost-effectiveness and easy deployment and can supply vehicle forward speed information [4,5,6]. The GNSS signal is inaccessible, so odometer-assisted inertial navigation can dramatically enhance the positioning precision. While using odometry, two parameters significantly affect the navigation accuracy. The first is the mounting angle, which is the angle between the IMU body frame (b-frame) and vehicle frame (v-frame) [7]. When installing navigation equipment, angle deviation is inevitable. Although the deviation angle is small, it can significantly impact positioning accuracy. The second is the lever arm, which is the relative position between the IMU’s center and the odometer’s reference point [8,9]. If the lever arm is unknown, when the car turns, there will be significant differences in the estimation of vehicle speed between the IMU and the odometer due to the non-overlap between the IMU and the odometer, thus making it challenging to perform high-precision data fusion. In addition, the GNSS lever arm considerably influences navigation precision. Therefore, the present research focuses on accurately estimating the IMU mounting angles, odometer lever arm, and GNSS lever arm from the data processing perspective. The schematic diagram of the mounting angle and lever arm is shown in Figure 1.

1.1. Related Works

The mounting angles reflect the deviation between the b-frame and v-frame. After compensating for this error, a unified coordinate frame can be achieved, enhancing the precision of integrated navigation positioning. Reference [10] proposes to estimate roll and pitch angle errors using the forward or backward acceleration information of vehicles on a horizontal road and to estimate heading angle errors using acceleration information on a horizontal plane.

Coincidentally, reference [11] proposed a similar method for estimating mounting angle error using acceleration information. This method requires the carrier to collide, but the calculated accuracy is limited. A mounting angle error causes a velocity error between the vehicle and the inertial navigation system. Based on this relationship, references [12,13] proposed using velocity error information to estimate mounting angle error. Similar methods can also be seen in references [14,15,16]. Unlike methods based on velocity and acceleration as observation values, displacement-based methods use trajectory estimation to obtain position information and use it as measurement values. Reference [17] is a typical example of using two known coordinate positions as reference points to estimate the mounting angle. Still, this method has relatively low estimation accuracy for high-precision inertial navigation. Reference [18] proposes a method for accurately estimating IMU pitch and heading mounting angles using displacement increments based on the similarity principle between trajectory estimation and GNSS trajectory. From the above research, the mounting angle error can be estimated based on the integrated navigation system’s acceleration, velocity, and displacement information output. While all the above methods are post-processing, they cannot realize real-time online calculations for mounting angle errors.

Incorporating the lever arm state vector in Kalman filtering can effectively estimate the antenna lever arm online and rapidly estimate the gyroscope biases. Reference [19] regards the GNSS antenna lever arm as an estimated state and uses the Kalman filter to calculate it. For the arms between inertial devices and auxiliary devices, such as odometers or non-holonomic constraints [20], reference [12] proposes using the weighted recursive least squares method to evaluate the lever arms. Because there is space inside the device, the inertial device cannot be seen as a “point device”, and the error caused by the internal space is named as the inner arm error. In high-precision inertial navigation, this part cannot be ignored.

For this reason, the authors of [21] analyzed the generating mechanism of this error, derived its dynamic error model, and proposed an estimation method. A lever arm compensation procedure according to reinforcement learning is presented by the authors of [22]. The outcomes demonstrate that compared with the procedure, the parameter error of the presented strategy is decreased by 0.31%, and the navigation precision is enhanced.

There are also many studies on improving the Kalman filter to enhance the estimation effect of the lever arm. Reference [23] proposes a misalignment and lever arm estimation procedure in line with the error propagation formula. A dual-cascaded Kalman filter is crafted to fuse all data information based on environmental perception. An initial alignment procedure according to the Kalman filter is presented in reference [24]. To attenuate the consequences of IMU bias, lever arms, and mounting angles, researchers propose a closed-loop design using a linear state-space pattern to estimate and make up for the parameter errors simultaneously. A rapid regulation approach for lever arms is put forward in reference [25]. Researchers propose a backtracking regulation procedure using a reduced-order Kalman filter to reduce the regulation time and enhance precision. A 21-dimensional Kalman filter for regulating the parameters of the lever arm is presented by the authors of [26]. In reference [27], an information fusion system for land vehicles is presented according to decentralized system architecture. It adaptively fuses information from motion assist restrictions and odometer while compensating the lever arm.

1.2. An Overview of the Presented Method

This paper proposes an algorithm for the multi-source fusion vehicle navigation module to simultaneously estimate the online mounting angles, GNSS lever arm, and odometer lever arm. The algorithm implementation includes three stages, as demonstrated in Figure 2. In step 1, GNSS RTK/INS integrated navigation is performed to estimate the GNSS antenna arm. The estimated antenna arm is employed to acquire high precision position, velocity, and attitude information, and the speed and attitude information is transferred to step 2. Step 2 estimates the mounting angles by the velocity connection between the v-frame and navigation frame (n-frame). Then, the two mounting angles are transmitted to step 3. At last, in step 3, the odometer lever arm is estimated by extending it to the error state vector.

The mounting angle error in previous integrated navigation systems was obtained through post-processing, and their GNSS and odometer lever arm were not simultaneously estimated online. This situation is not conducive to the large-scale and widespread application of integrated navigation systems. Therefore, it is necessary to estimate them online to develop a low-cost, high-precision, multi-source integrated navigation system that can be applied on a large scale. We propose an online mounting angle error estimation algorithm and online estimation algorithms for GNSS antenna and odometer lever arms for low-cost embedded vehicle-mounted navigation systems to address this situation. The mounting angle error is estimated using the odometer and IMU velocities, and the lever arms are calculated using a Kalman filter. The effect of the presented algorithms is verified through real-time vehicle-mounted tests. This study aims to advance the positioning precision of integrated navigation modules in complex scenarios by estimating mounting angles and lever arms to promote their widespread application in vehicle navigation. The principal contribution of this study is twofold: firstly, proposing a low-cost embedded vehicle-mounted online mounting angle estimation algorithm to address the mounting angle errors from the b-frame to the v-frame, and secondly, proposing online estimation algorithms for GNSS antenna lever arms and odometer lever arms to address the need for compensating for lever arms between the GNSS/odometer and the IMU.

The configuration of the work is broken down as follows. Section 2 introduces the models of online mounting angle error estimation and lever arm online estimation algorithms and then explains the online estimation process. Section 3 describes the test route scenario and experimental platform. The corresponding results of the estimation are verified and analyzed in Section 4. At last, the discussion and conclusion of the results are provided in Section 5 and Section 6, respectively.

2. Methods

2.1. Online Estimation of IMU Mounting Angles

The mounting angles can be obtained for vehicle navigation using the velocity relationship between the b-frame and v-frame. Considering the lever arm between the center of the IMU and the touch point of the right rear wheel, the following relationship exists between the IMU’s center and the right rear wheel’s velocity of the non-steering wheel of the car:

$$v_{odo}^v=C_b^v{C}_n^b{v}^n+C_b^v({\mathsf{\omega }}_{nb}^b\times )l_{IMU-odo}^b$$

(1)

According to the matrix transformation, the preceding equation can be rephrased in the subsequent form:

$$C_v^b{v}_{odo}^v=C_n^b{v}^n+({\mathsf{\omega }}_{nb}^b\times )l_{IMU-odo}^b$$

(2)

The above equation shows that the second term on the right side of the equal mark is the velocity produced by the vehicle turning under the b-frame. This item should be ignored to reduce the influence of ${\mathsf{\omega }}_{nb}^b$ and $l_{GNSS-odo}^b$ on the estimation of mounting angles [28]. That is,

$$C_v^b{v}_{odo}^v=C_n^b{v}^n$$

(3)

Therefore, when calculating the mounting angle, the automobile should be traveling in a straight line. Figure 1 describes the correlation between the v-system and the b-system. In the v-system, the forward direction of the automobile corresponds to the forward direction of the x-axis, the lower part of the automobile corresponds to the forward direction of the z-axis, and the right side of the car corresponds to the positive direction of the y-axis. The directions of the three axes satisfy the right-hand rule. The mounting angle causes a certain deviation between the b-frame and the v-frame, expressed by Euler angles as $\Delta \alpha$, $\Delta \beta$, and $\Delta \gamma$. Then, the transition matrix $C_b^v$ from the b-system to the v-system could be written as [29,30]:

$$C_b^v=\left[\begin{array}{ccc}\cos \Delta \beta \cos \Delta \gamma & -\cos \Delta \alpha \sin \Delta \gamma +\sin \Delta \alpha \sin \Delta \beta \cos \Delta \gamma & \sin \Delta \alpha \sin \Delta \gamma +\cos \Delta \alpha \sin \Delta \beta \cos \Delta \gamma \\ \cos \Delta \beta \sin \Delta \gamma & \cos \Delta \alpha \cos \Delta \gamma +\sin \Delta \alpha \sin \Delta \beta \sin \Delta \gamma & -\sin \Delta \alpha \cos \Delta \gamma +\cos \Delta \alpha \sin \Delta \beta \sin \Delta \gamma \\ -\sin \Delta \beta & \sin \Delta \alpha \cos \Delta \beta & \cos \Delta \alpha \cos \Delta \beta \end{array}\right]$$

(4)

Since the mounting angle is slight, the above equation can be equated to

$$C_b^v=\left[\begin{array}{ccc}1& -\Delta \gamma & \Delta \beta \\ \Delta \gamma & 1& -\Delta \alpha \\ -\Delta \beta & \Delta \alpha & 1\end{array}\right]=I_{3\times 3}+(\mathsf{\phi }\times ),\mathsf{\phi }=\left[\begin{array}{c}\Delta \alpha \\ \Delta \beta \\ \Delta \gamma \end{array}\right],$$

(5)

where $I_{3\times 3}$ is the unit matrix and $\mathsf{\phi }\times$ indicates the antisymmetric matrix. Transposing the above equation, we have

$$C_v^b={(C_b^v)}^{\mathrm{T}}=I_{3\times 3}-(\mathsf{\phi }\times )$$

(6)

The odometer speed $v_{odo}^v$ is the quantity corrected using the odometer scale factor, which can be stated in the v-frame as

$$v_{odo}^v={\left[\begin{array}{ccc}{v}_{odo,F}^v& 0& 0\end{array}\right]}^T$$

(7)

Substituting Equations (6) and (7) into Equation (3) yields that

$$(I_{3\times 3}-\mathsf{\phi }\times )v_{odo}^v=v_{odo}^v-(\mathsf{\phi }\times )v_{odo}^v=C_n^b{v}^n$$

(8)

Integrating the above equation yields that

$$\begin{array}{cc}\hfill v_{odo}^v-C_n^b{v}^n& =(\mathsf{\phi }\times )v_{odo}^v\hfill \\ \hfill & =(-v_{odo}^v\times )\mathsf{\phi }=\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& v_{odo,F}^v\\ 0& -v_{odo,F}^v& 0\end{array}\right]·\left[\begin{array}{c}\Delta \alpha \\ \Delta \beta \\ \Delta \gamma \end{array}\right]\hfill \end{array}$$

(9)

In the above equation, $C_n^b$ and $v^n$ can be expressed in the following form:

$$C_n^b=\left[\begin{array}{ccc}{C}_{11}& C_{12}& C_{13}\\ C_{21}& C_{22}& C_{23}\\ C_{31}& C_{32}& C_{33}\end{array}\right],v^n={\left[\begin{array}{ccc}{v}_1& v_2& v_3\end{array}\right]}^T$$

(10)

where the attitude matrix $C_n^b$ and the velocity $v^n$ can be represented by the convergence results of the GNSS RTK/INS integrated navigation. $C_{xy}$ represents components of attitude matrixes, x = 1–3, y = 1–3. $v_1$, $v_2$, and $v_3$ are velocities in the north, east, and down directions of the n-frame. After multiplying the $C_n^b$ and $v^n$ matrices, $C_n^b{v}^n$ can be expressed as follows:

$$C_n^b{v}^n=\left[\begin{array}{c}{C}_{11}{v}_1+C_{12}{v}_2+C_{13}{v}_3\\ C_{21}{v}_1+C_{22}{v}_2+C_{23}{v}_3\\ C_{31}{v}_1+C_{32}{v}_2+C_{33}{v}_3\end{array}\right]$$

(11)

Subtracting $v_{odo}^v$ from $C_n^b{v}^n$ yields

$$v_{odo}^v-C_n^b{v}^n=\left[\begin{array}{c}{v}_{odo,F}^v-\left(C_{11}{v}_1+C_{12}{v}_2+C_{13}{v}_3\right)\\ -\left(C_{21}{v}_1+C_{22}{v}_2+C_{23}{v}_3\right)\\ -\left(C_{31}{v}_1+C_{32}{v}_2+C_{33}{v}_3\right)\end{array}\right]$$

(12)

Combining Equations (9) and (12) yields the following equation:

$$\left[\begin{array}{c}{v}_{odo,F}^v-\left(C_{11}{v}_1+C_{12}{v}_2+C_{13}{v}_3\right)\\ -\left(C_{21}{v}_1+C_{22}{v}_2+C_{23}{v}_3\right)\\ -\left(C_{31}{v}_1+C_{32}{v}_2+C_{33}{v}_3\right)\end{array}\right]=\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& v_{odo,F}^v\\ 0& -v_{odo,F}^v& 0\end{array}\right]·\left[\begin{array}{c}\Delta \alpha \\ \Delta \beta \\ \Delta \gamma \end{array}\right]$$

(13)

Therefore, the pitch and heading of the mounting angles are

$$\{\begin{array}{c}\Delta \beta ={\displaystyle \frac{C_{21}{v}_1+C_{22}{v}_2+C_{23}{v}_3}{v_{odo,F}^v}}\\ \Delta \gamma ={\displaystyle \frac{-\left(C_{31}{v}_1+C_{32}{v}_2+C_{33}{v}_3\right)}{v_{odo,F}^v}}\end{array}$$

(14)

This paper presents a practical online mounting angle estimation algorithm, which has the advantages of being computationally small and without external assistance and is suitable for mass-market in-vehicle navigation and positioning applications. The online estimation of the mounting angles uses the velocity information and attitude information output from the integrated GNSS RTK/INS navigation to replace the high-precision external reference used in the post-processing assessment of the mounting angles. They are then calculated based on the correlation between the odometer velocity information and the speed information output from the integrated GNSS/INS navigation. The proposed online assessment of the IMU mounting angle algorithm is implemented according to Algorithm 1.

The implementation of the online estimation algorithm for mounting angles can be separated into four steps:

(1) Receive GNSS and sensor data and determine whether the present GNSS RTK is a fixed solution and whether the inertial navigation has completed the initial alignment at the same time. If the judgment condition is false, then continue to dynamic alignment; if it is true, the initial inertial navigation alignment is successful, and then the GNSS RTK/INS combination navigation is performed for attitude convergence. At the same time, we determine the traveling state of the vehicle. If the car is traveling straight, save the GNSS RTK/INS integrated navigation speed value and odometer output speed value; if it is not traveling straight, return to continue the following received data and execute the integrated navigation operation.

(2) Calculate the scale factor K of the odometer in the straight-ahead state from the velocity of the carrier in the b-frame and the velocity of the odometer output and then use K to obtain the odometer speed in the v-frame.

(3) Using the velocity correlation between the v-frame and the b-frame, the pitch and heading angles are then obtained according to the small angle theory.

Save the heading and pitch angles, remove the greatest and least values of both, and then average the remaining data to obtain the pitch and heading angles for the final estimated mounting angles.

Algorithm 1 Online Estimation of IMU Mounting Angles

Input: Specific force increment, angular velocity increment, RTK, odometer velocity

Calculation of $v^n$ and $v_{odo}^v$ 1. Judge whether the vehicle is going straight according to the gyro output, then save the GNSS RTK/INS integrated navigation $v^n$ and odometer velocity $v_{odo}^v$.

Calculation of $C_n^b$ 2. Compute the attitude matrix using GNSS RTK/INS integrated navigation.

Calculation of the odometer scale factor and $v_{odo}^v$ 3. Compute the odometer scale factor K according to odometer velocity and vehicle velocity in the b-frame, then compute $v_{odo}^v$ by the scale factor.

Calculation of $\Delta \beta$ and $\Delta \gamma$ 4. Compute the pitch and heading mounting angles with the Formula (10).

Output: $\Delta \beta$, $\Delta \gamma$, $C_n^b$, $v^n$, $v_{odo}^v$

2.2. Online Estimation of Lever Arm

In the INS, the inconsistency between the position coordinates is computed by the GNSS and the coordinates are computed by the inertial navigation mechanization, as shown in Figure 1. it is necessary to make up for the lever arm influence when fusing the two data sets to improve the integrated navigation positioning precision [21,31]. The position between the central points of the GNSS antenna and the IMU satisfies the following relationship:

$$r_{GNSS}^n=r_{IMU}^n+C_b^n{l}_{GNSS}^b$$

(15)

According to Equation (15), the observation vector of the GNSS lever arm can be expressed as the distinction between the INS location and the GNSS location observation.

$$\delta z_{GNSS\_lever}=r_{IMU}^n-r_{GNSS}^n=\delta r-C_b^n{l}_{GNSS}^b$$

(16)

2.3. Online Estimation of Lever Arm between Odometer and IMU

In the procedure of integrated odometer-assisted inertial navigation, the positions of the two do not overlap, resulting in inconsistencies between the measured and true speeds when the carrier vehicle turns. The velocity error is more extensive, especially when the distance between the two is farther. Therefore, in the process of odometer-assisted inertial navigation, the lever arm that the IMU connects to the odometer is evaluated using the extended Kalman filter and compensated, which can enhance the positioning precision of systems [32,33,34].

Assuming the spatial location correlation between the IMU mounting position and the odometer shown in Figure 1, it can be expressed as

$$l_{odo}^b={\left[\begin{array}{ccc}{l}_x& l_y& l_z\end{array}\right]}^T$$

(17)

The velocity correlation between the inertial navigation and the odometer is

$${\widehat{v}}_{odo}^v=C_b^v{C}_n^b{v}_{IMU}^n+C_b^v({\mathsf{\omega }}_{nb}^b\times )l_{odo}^b$$

(18)

The second expression positioned to the right side of the formula represents the lever arm influence. In line with the above equation, the observed vector of the odometer lever arm can be expressed as the distinction between the INS and the odometer speed in the v-frame

$$\begin{array}{cc}\hfill \delta z_{odo\_lever}& =v_{IMU}^v-v_{odo}^v\hfill \\ \hfill & =C_b^v{C}_n^b\delta v^n-C_b^v{C}_n^b(v^n\times )\varphi -C_b^v({\mathsf{\omega }}_{nb}^b\times )l_{odo}^b\hfill \end{array}$$

(19)

2.4. State Space Model

Considering both the GNSS lever arm error and odometer lever arm error, the state space model of the GNSS lever arm and odometer lever arm estimation for integrated GNSS/inertial navigation/odometer navigation can be obtained [35]. Let $X$ be the state and $Z$ be the observation, and the state space model of GNSS and odometer lever arm evaluation is obtained:

$$\begin{array}{l}X=\mathsf{\Phi }\mathbf{X}+GW\\ Z=HX+V\end{array}\},$$

(20)

where $\mathsf{\Phi }$ is responsible for state transitions, $G$ stands for the system noise driver matrix, $W$ stands for the system state noise, $H$ stands for the system design matrix, and $V$ stands for the observation noise. $W$ and $V$ are Gaussian white noise. The error state vector of the extended Kalman filter comprises the position errors, the velocity errors, the attitude errors, the gyro bias, the acceleration bias, the GNSS antenna arm, and the odometer arm and is written as

$$X={\left[\begin{array}{ccccccc}\delta r^n& \delta v^n& \varphi & b_g& b_a& {\mathrm{lever}}_{GNSS}& {\mathrm{lever}}_{ODO}\end{array}\right]}^{\mathrm{T}}.$$

(21)

where $\delta r^n=\left[\begin{array}{ccc}\delta r_N& \delta r_E& \delta r_D\end{array}\right]$ and $\delta v^n=\left[\begin{array}{ccc}\delta v_N& \delta v_E& \delta v_D\end{array}\right]$ are the position and velocity errors. $\varphi =\left[\begin{array}{ccc}{\varphi }_{roll}& {\varphi }_{pitch}& {\varphi }_{yaw}\end{array}\right]$ is the attitude error, $b_g=\left[\begin{array}{ccc}{b}_{g\_x}& b_{g\_y}& b_{g\_z}\end{array}\right]$ is the gyroscope bias, $b_a=\left[\begin{array}{ccc}{b}_{a\_x}& b_{a\_y}& b_{a\_z}\end{array}\right]$ is the acceleration bias, ${\mathrm{lever}}_{GNSS}=\left[\begin{array}{ccc}{l}_{g\_x}& l_{g\_y}& l_{g\_z}\end{array}\right]$ is the GNSS lever arm, and ${\mathrm{lever}}_{ODO}=\left[\begin{array}{ccc}{l}_{odo\_x}& l_{odo\_y}& l_{odo\_z}\end{array}\right]$ is the odometer lever arm.

The observation matrix of the GNSS lever arm is

$$H_{GNSS\_lever}=\left[\begin{array}{ccccccc}{I}_{3\times 3}& 0_{3\times 3}& 0_{3\times 3}& 0_{3\times 3}& 0_{3\times 3}& -C_b^n& 0_{3\times 3}\end{array}\right]$$

(22)

The observation matrix of the odometer lever arm is

$$H_{odo\_lever}=\left[\begin{array}{ccccccc}{0}_{3\times 3}& C_b^v{C}_n^b& -C_b^v{C}_n^b(v^n\times )& 0_{3\times 3}& 0_{3\times 3}& 0_{3\times 3}& -C_b^v({\mathsf{\omega }}_{nb}^b\times )\end{array}\right]$$

(23)

The state transfer matrix is

$$\mathsf{\Phi }=\left[\begin{array}{ccccccc}{F}_1& F_2& 0& 0& 0& 0& 0\\ F_3& F_4& F_5& 0& F_6& 0& 0\\ 0& 0& F_7& F_8& 0& 0& 0\\ 0& 0& 0& F_9& 0& 0& 0\\ 0& 0& 0& 0& F_{10}& 0& 0\\ 0& 0& 0& 0& 0& F_{11}& 0\\ 0& 0& 0& 0& 0& 0& F_{12}\end{array}\right],$$

(24)

where

$F_1={\left[I_{3\times 3}+(-{\mathsf{\omega }}_{en}^n\times )\Delta t\right]}_{3\times 3}$,

$F_2={\left[I_{3\times 3}\Delta t\right]}_{3\times 3}$,

$F_3=diag{\left(\begin{array}{ccc}{\displaystyle \frac{-g_l^n\Delta t}{\sqrt{R}+h}}& {\displaystyle \frac{-g_l^n\Delta t}{\sqrt{R}+h}}& {\displaystyle \frac{2g_l^n\Delta t}{\sqrt{R}+h}}\end{array}\right)}_{3\times 3}$,

$F_4={\left[I_{3\times 3}+[(-2{\mathsf{\omega }}_{ie}^n-{\mathsf{\omega }}_{en}^n)\times ]\Delta t\right]}_{3\times 3}$,

$F_5={\left[\left(f^n\times \right)\Delta t\right]}_{3\times 3}$,

$F_6={\left[C_b^n\Delta t\right]}_{3\times 3}$,

$F_7={\left[I_{3\times 3}-\left[({\mathsf{\omega }}_{ie}^n+{\mathsf{\omega }}_{en}^n)\times \right]\Delta t\right]}_{3\times 3}$,

$F_8={\left[-C_b^n\Delta t\right]}_{3\times 3}$,

$F_9=diag{\left(\begin{array}{ccc}{e}^{-\Delta t/T_{gb}}& e^{-\Delta t/T_{gb}}& e^{-\Delta t/T_{gb}}\end{array}\right)}_{3\times 3}$,

$F_{10}=diag{\left(\begin{array}{ccc}{e}^{-\Delta t/T_{ab}}& e^{-\Delta t/T_{ab}}& e^{-\Delta t/T_{ab}}\end{array}\right)}_{3\times 3}$,

$F_{11}=I_{3\times 3}$,

$F_{12}=I_{3\times 3}$,

$f^n$ is the applied force, $R=R_M{R}_N$, $R_M$ and $R_N$ are the radius of the meridian circle and the prime circle. $T_{gb}$ and $T_{ab}$ are the gyro and accelerometer zero bias correlation times.

3. Test Route Scene and Experiment Platform

The Fozuling area in Wuhan, Hubei Province, was selected as the testing site, with an open, unobstructed testing area and good GNSS signal observation conditions. Figure 3 shows the test trajectory, which includes a straight-line trajectory and rich turning movements to enhance the detectability of the lever arm.

The vehicle-mounted test platform used for algorithm testing is shown in Figure 4, including tactical-level and MEMS-level multi-source fusion navigation equipment NovAtel SPAN CPT-6 (dual frequency GNSS RTK/fiber IMU), MAP Space Time M39 (consumer grade MEMS IMU), self-developed GM916 (low-cost chip-level MEMS IMU), and self-developed GM917(low-cost chip-level MEMS IMU). The Chinese characters on the GM916 and GM917 chips mean Wuhan University. Throughout the entire testing process, the output of the NovAtel SPAN CPT-6 was used as a reference truth value, and data were collected simultaneously to validate the algorithm. In addition, the raw data of M39, GM916, and GM917 were also collected for algorithm validation. We have also installed an odometer on the right rear wheel to help MEMS-INS improve positioning accuracy.

Table 1. IMU performance specifications.

IMU Level	Tactical Level	MEMS
IMU Model	SPAN CPT6	M39	GM916	GM917
Bias (deg/h)	1	8	10	15
ARW (deg/sqrt(h))	0.0667	0.12	0.24	0.61
Bias (mGal)	50	20	25	25
VRW (m/s/sqrt(h))	0.03	0.09	0.042	0.08

lists the performance standards for different IMUs on the four devices.

4. Experimental Verification and Result Analysis

4.1. Impact of Mounting Angle Estimation Error

To authenticate the correctness of the presented online mounting angle error estimation algorithm, three methods were used to calculate the mounting angle error: post-processing with GINS software (version 1.0), post-processing with CPT high-precision data (using the attitude information and the velocity output from CPT to calculate the mounting angles), and online real-time estimation. Then, they were subjected to ten sets of experimental tests, and the outcomes of the experiments are exhibited in

Table 2. Estimation results for mounting angles.

Calibration Experiment	GINS Software Post-Processing		Post-Processing (CPT for Reference)		Online Estimation
	Pitch	Heading	Pitch	Heading	Pitch	Heading
1	1.108	−0.297	1.060	−0.430	1.171	−0.299
2	1.105	−0.291	1.088	−0.159	1.096	−0.459
3	1.109	−0.278	1.05	−0.879	1.129	−0.578
4	1.110	−0.257	1.123	−0.076	1.045	−0.526
5	1.111	−0.273	1.141	−0.532	1.117	−0.303
6	1.100	−0.283	1.136	−0.498	1.050	−0.563
7	1.095	−0.287	1.038	−0.542	1.129	−0.568
8	1.131	−0.294	1.080	−0.203	1.081	−0.377
9	1.098	−0.320	1.061	−0.569	1.055	−0.376
10	1.127	−0.308	1.137	−0.494	1.079	−0.372
Aver	1.109	−0.288	1.091	−0.538	1.095	−0.442

. The table shows that the mounting error angles of pitch and heading angles fluctuate less, and the estimation precision of the three methods is equivalent.

In the process of odometer-assisted inertial navigation, it is imperative to convert the speed information in the v-frame of the odometer to the b-frame, so mounting angles are required. To further verify the effect of the post-processing and online estimated mounting angles on the positioning accuracy, the GNSS was interrupted for 30 s, during which only the odometer-assisted inertial navigation was available, and the navigation positioning errors of the two mounting angle estimation methods were compared by testing. During the odometry-assisted inertial navigation, the odometry measurement update frequency was 10 Hz. The position, velocity, and attitude errors of the two methods are shown in Figure 5, from which it can be observed that the mounting angles estimated by the post-processing of the GINS software (version 1.0) have almost the same effect on the integrated navigation accuracy as the mounting angles assessed online. This also reflects the effectiveness and feasibility of the online estimation of the mounting angles strategy presented in this study.

In addition, the impact of the mounting angle errors calculated by online estimation methods and CPT data post-processing methods on navigation positioning accuracy was also compared, as shown in Figure 6. This comparison method also interrupts GNSS for 30 s, during which only the odometer assists in inertial navigation. The figure shows that the mounting angle error calculated by the two methods has almost the same impact on positioning accuracy.

4.2. Impacts of Lever Arm Estimation Error

To confirm the correctness of the lever arm estimation algorithm, we collected the data from SPAN CPT6, M39, and GM916, which were used to estimate their respective lever arm values. Meanwhile, the weighted recursive least squares method is used to evaluate the lever arm and contrast with the method presented in this study.

The outcomes of the GNSS antenna lever arm and the odometer lever arm of the SPAN CPT6, M39, and GM916 devices utilizing the approach outlined in this study and the weighted recursive least squares method are illustrated in Figure 7a, b, c, d, e, and f, respectively. It can be seen that both methods can estimate the lever arm correctly, but the proposed method converges faster. The SPAN CPT6 antenna arms converge to [−0.71, 0.13, −0.17], and the odometer arms converge to [−0.84, 0.77, 1.30]. The M39 antenna arm converges to [0.46, −0.06, −1.08], and the odometer arm converges to [0.87, 0.62, 0.31]. The GM916 antenna arm converges to [−0.40, 0.15, −0.05], and the odometer arm converges to [−0.52, 0.70, 1.18]. To verify the repeatability of the online estimation algorithm, we utilize three sets of data (A, B, and C) to estimate the lever arm values and perform longitudinal comparisons, as shown in

Table 3. Statistics for GNSS lever arm estimate (unit: meters).

	SPAN CPT6			M39			GM916
	X	Y	Z	X	Y	Z	X	Y	Z
A	0.722	0.120	0.167	0.480	0.104	1.153	0.401	0.152	0.067
B	0.689	0.138	0.172	0.494	0.105	1.181	0.409	0.150	0.059
C	0.697	0.112	0.166	0.483	0.097	1.162	0.410	0.147	0.068

and

Table 4. Statistics for odometer lever arm estimate (unit: meters).

	SPAN CPT6			M39			GM916
	X	Y	Z	X	Y	Z	X	Y	Z
A	−0.833	0.765	1.312	0.892	0.666	0.258	−0.499	0.690	1.120
B	−0.856	0.779	1.279	0.881	0.667	0.275	−0.511	0.683	1.152
C	−0.852	0.769	1.323	0.873	0.650	0.268	−0.498	0.656	1.143

. In the three experiments of GNSS lever arm estimation, SPAN CPT6 converges to [0.69, 0.12, 0.16], M39 converges to [0.48, 0.1, 1.16], and GM916 converges to [0.4, 0.15, 0.06]. In the three experiments of odometer lever arm estimation, SPAN CPT6 converges to [−0.85, 0.77, 1.31], M39 converges to [0.88, 0.66, 0.26], and GM916 converges to [−0.49, 0.68, 1.14]. By comparing the results of three sets of experiments with the convergence results, it can be observed that the estimation error of GNSS and odometer lever arm in different experiments does not exceed 3 cm. This indicates that the lever arm estimation algorithm has good consistency and reliable results.

According to the definitions of the mounting angles and lever arm, after compensating for the mounting angles and lever arm, if the errors fluctuate around zero in the integrated navigation results, it indirectly indicates that the mounting angle and lever arm estimation is accurate. Taking the compensation of GM917 and GM916 as an example, the errors under the four compensation modes were compared to demonstrate the effectiveness of this paper’s estimation algorithms. The different compensation modes are listed in

Table 5. List of different compensation modes.

Compensation Mode	Mounting Angles	Odometer Lever Arm	GNSS Lever Arm
Mode #1	NO	NO	NO
Mode #2	YES	NO	NO
Mode #3	YES	YES	NO
Mode #4	YES	YES	YES

.

Figure 8 demonstrates the positioning error changes of two different modules, GM917 and GM916, under four compensation modes. In mode #1, where the mounting angles, odometer, and antenna lever arm are not compensated, the position error is not zero. The maximal position error of the two modules in the north direction reaches 2.1 m and 2.13 m. The maximal position error in the east direction reaches 2.03 m and 2.29 m. The same in the down direction reaches 1.61 m and 1.43 m. In mode #2, compensating only for the mounting angles can effectively reduce the location errors in the north, east, and down directions, indicating that the estimation of the mounting angle is effective. The maximal position error of the two modules in the north direction reaches 0.73 m and 0.71 m. The maximal position error in the east direction achieves 0.53 m and 0.52 m. The same in the down direction achieves 1.29 m and 1.38 m. After making up for the mounting angles and odometer lever arm in mode #3, the reduction in position error is not significant. That is because the odometer lever arm only affects the vehicle’s speed when turning. The maximal position error of the two modules in the north direction achieves 0.516 m and 0.51 m. The same in the east direction achieves 0.52 m and 0.51 m. The same in the down direction achieves 1.31 m and 1.32 m. In mode #4, the mounting angles, odometer lever arm, and GNSS lever arm are compensated at the same time, and the position error is significantly reduced, which reflects the significant impact of the GNSS lever arm on positioning accuracy and also proves the correctness of the arm estimation algorithm. In this case, after convergence, the maximal position error of the two modules in the north direction achieves 0.09 m and 0.08 m. The same in the east direction achieves 0.07 m and 0.09 m. The same in the down direction achieves 0.05 m and 0.04 m.

Figure 9 shows the velocity error changes of two different modules, GM917 and GM916, under four compensation modes. In mode #1, the speed error is relatively large, and after convergence, the maximal velocity error of the two modules in the north direction reaches 2.85 m/s and 2.9 m/s. The maximal velocity error in the east direction reaches 2.69 m/s and 3.53 m/s. The maximal velocity error in the down direction reaches 0.91 m/s and 1.57 m/s. After the mounting angle compensation in mode #2, The maximal velocity error of the two modules in the north direction reaches 0.57 m/s and 0.54 m/s. The maximal velocity error in the east direction reaches 0.51 m/s and 0.53 m/s. The maximal velocity error in the down direction reaches 0.92 m/s and 1.58 m/s. When the vehicle turns, it will generate lateral speed. After further compensating for the odometer lever arm in mode #3, the speed in the north and east directions significantly decreased during turning. The maximal velocity error of the two modules in the north direction reaches 0.18 m/s and 0.24 m/s. The maximal velocity error in the east direction reaches 0.23 m/s and 0.18 m/s. The maximal velocity error in the down direction reaches 0.91 m/s and 1.59 m/s. In mode #4, the mounting angles, odometer, and antenna lever arm are simultaneously compensated, and the speed error fluctuates around zero. After convergence, the maximal velocity error of the two modules in the north direction reaches 0.03 m/s and 0.04 m/s. The maximal velocity error in the east direction reaches 0.04 m/s and 0.05 m/s. The maximal velocity error in the down direction reaches 0.91 m/s and 1.59 m/s. However, compensating for the GNSS antenna lever arm and odometer lever arm has a negligible impact on the longitudinal speed, and the effect is not apparent from the figure. That is because after making up the mounting angle and lever arm, the longitudinal speed is greatly affected by the damping system, acceleration, deceleration, etc.

Finally, after compensating the mounting angle, odometer, and GNSS lever arm, the proposed method is compared with the weighted recursive least squares strategy for position, velocity, and attitude errors. As shown in Figure 10, Figure 11 and Figure 12, the proposed method has slightly better position errors in the north and east directions compared with weighted recursive least squares. After the convergence of the proposed method, the maximal position errors in three directions are 0.1 m, 0.08 m, and 0.05 m, and the maximal velocity errors are 0.03 m/s, 0.032 m/s, and 1.58 m/s. The maximal pitch, roll, and heading errors are 2.9 degrees, 3.7 degrees, and 0.38 degrees. However, the longitudinal velocity error is large and is mainly affected by the vehicle damping system and acceleration/deceleration. At the same time, the error of pitch and roll is also large, mainly because the mounting angle and lever arm have little influence on it, resulting in no obvious improvement effect. In addition, it can be seen from Figure 10, Figure 11 and Figure 12 that without compensation, the maximal position errors in three directions are 0.55 m, 0.52 m, and 1.32 m, and the maximal velocity errors are 0.16 m/s, 0.18 m/s, and 0.92 m/s. The maximal roll, pitch, and heading errors are 3.02 degrees, 3.75 degrees, and 2.46 degrees. By comparison, it can be concluded that the error is significantly reduced after compensation utilizing the method presented in this article.

Next, different mounting angles of the same IMU are tested, and their effects on position and speed are analyzed. The experiment was divided into three groups. In the first group, the mounting angles were estimated using the proposed method, and the position and speed of integrated navigation were calculated using the results. In the second group, after making up the mounting angles in the first step, the pitch and heading mounting angles are deviated 3 degrees from the correct direction, and then the values are used to compute the position and speed. The third group, after compensating the mounting angles in the first step, deviates the pitch and heading mounting angles from the correct direction by 5 degrees, and then used them to calculate the position and speed. The test results are shown in Figure 13 and Figure 14.

It can be seen from the figures that correctly compensating for the mounting angles, the maximal position errors in the north, east, and down directions are 0.09 m, 0.08 m, and 0.05 m, and the maximal velocity errors are 0.03 m/s, 0.032 m/s, and 0.87 m/s, as shown by the green line in the figure. Deviating from the correct direction by 3 degrees, the maximal position errors in three directions are 0.18 m, 0.23 m, and 0.39 m, and the maximal velocity errors are 0.12 m/s, 0.13 m/s, and 0.9 m/s, as shown by the blue line in the figure. Five degrees off the correct direction, the maximal position errors in three directions are 0.34 m, 0.31 m, and 0.44 m, and the maximal velocity errors are 0.2 m/s, 0.21 m/s, and 0.91 m/s, as shown by the red line in the figure. Therefore, it could be observed in the figure that different mounting angles of the same IMU greatly influence position and velocity errors, reflecting the effect of the method presented in this study.

5. Discussion

It is worth noting that compared with GINS post-processing and CPT high-precision reference post-processing methods, the proposed online mounting angle error estimation algorithm shows considerable accuracy in estimating pitch and yaw mounting angle errors. In addition, under the condition of odometer-assisted inertial navigation, GNSS is interrupted for 30 s to test the impact of mounting angle error on navigation positioning accuracy. After research, it was found that the impact of mounting angle errors estimated using online methods is almost the same as that of GINS post-processing methods. However, the online estimated method showed slightly better performance in heading errors. These findings suggest that the estimation method for online mounting angles is practical and feasible.

Regarding the lever arm estimation experiment, the article estimated the lever arms of three different levels of IMU devices: SPAN CPT6, M39, GM916, and GM917. It is worth noting that they all achieve rapid convergence. The SPAN CPT6 antenna arms converge to [−0.71, 0.13, −0.17], and the odometer arms converge to [−0.84, 0.77, 1.30]. The M39 antenna arm converges to [0.46, −0.06, −1.08], and the odometer arm converges to [0.87, 0.62, 0.31]. The GM916 antenna arm converges to [−0.40, 0.15, −0.05], and the odometer arm converges to [−0.52, 0.70, 1.18]. Taking GM916 and GM917 as examples, as demonstrated in Figure 8 and Figure 9, the position and speed errors of multi-source fusion navigation are significantly reduced after compensating for the lever arm. These outcomes indicate that the proposed estimation algorithm is effective in actual driving situations.

6. Conclusions

This study proposes a parameter estimation algorithm that can run on a low-cost GNSS/MEMS IMU multi-source fusion navigation module. The algorithm can simultaneously estimate the IMU mounting angle error, GNSS antenna arm, and odometer arm online. The experimental outcomes demonstrate that this study’s lever arm estimation algorithm is effective for tactical and MEMS-level inertial navigation, with an estimation error of less than 2 cm. Meanwhile, the proposed online estimation of the mounting angle algorithm has an accuracy comparable to that of the post-processing algorithm. After making up the mounting angle and the lever arm, the evaluation found that the position and speed precision of the multi-source fusion systems were significantly improved. Therefore, the proposed algorithm can potentially promote the practical and widespread application of vehicle multi-source fusion navigation modules, addressing the shortcomings of traditional methods, including the cumbersome and inaccurate manual measurement of the lever arm. In conclusion, this study provides valuable technical solutions for developing low-cost vehicle multi-source fusion navigation modules with better accuracy and convenience.