Shearer-Positioning Method Based on Non-Holonomic Constraints

Abstract

In the traditional shearer-positioning method, an odometer is used to assist the forward velocity correction of the inertial navigation system, but it cannot restrain the system’s error divergence. For this reason, this paper proposes a shearer positioning method based on non-holonomic constraints. In this method, an inertial measurement unit and odometer are installed in the middle of the shearer body and on the traction gear, respectively, the shearer attitude, speed, and position information are calculated through the inertial measurement-unit mechanization, and the shearer’s instantaneous velocity is calculated through the output of the odometer. The mechanization and error transfer process of the inertial navigation system are used to establish a Kalman filtering-state equation. The Kalman filtering observation equation is established through the difference between the projected velocity of the inertial navigation system at the joint and the output velocity of the odometer as the observation vector, and the non-holonomic constraint is introduced. Finally, the error feedback is derived from the results processed by the Kalman filtering algorithm, and the output of the inertial navigation system is corrected to obtain the optimal estimation of the shearer’s attitude, speed and position. The experiment shows that compared with the traditional inertial navigation and odometer combined positioning method, the degree of divergence in the positioning results over time is significantly reduced after adding the non-holonomic constraint. The positioning method has good tracking ability for the trajectory of the shearer. The error of the positioning results in the forward direction is reduced by 66%, the lateral direction is reduced by 62%, and the vertical direction is reduced by 67%.

1. Introduction

Efficient, green, and automatic mining is the development goal of China’s coal industry, and the precise determination of the shearer position is an important technology in the development of automatic mining in the fully mechanized long-wall face. It not only can it provide the accurate position of the shearer for memory cutting and face straightening, but it can also provide a reference for the linkage of three machines in the face and automatic mining. Therefore, it is necessary to develop a high-precision positioning method for shearers in the fully mechanized long-wall face.

At present, the shearer positioning method mostly relies on ab inertial navigation system [1,2] (INS for short) based on the inertial measurement unit (IMU), which is composed of a three-axis gyroscope and a three-axis accelerometer.

However, due to the influence of sensor error and environmental factors, positioning errors accumulate over time [3]. Therefore, it is necessary to introduce auxiliary information to suppress the error accumulation of the inertial navigation system. The authors of [4] combine the INS and the accurate geographical model of the fully mechanized long-wall face, and use matching map technology to restrict the error of the INS in the positioning process. However, since it is too difficult to establish the accurate geographical model of the fully mechanized long-wall face [5], the feasibility of this method remains to be determined. A different study [6] used the method of combining the odometer (ODO) and the inertial navigation system (INS) to correct the positioning results of the INS by collecting the speed or displacement information of the odometer. However, the odometer can adjust only the forward speed of the INS in the positioning process, and cannot correct its lateral and vertical speed. On the basis of establishing the ultrasonic positioning system, the authors of [7] used a Kalman filter (KF) algorithm to filter the combined positioning results of the inertial navigation system and the ultrasonic sensor. However, the underground non-line-of-sight (NLOS) environment reduces the measurement accuracy of the ultrasonic sensor and affects the accuracy of the ultrasonic positioning system [8]. The authors of [9,10] constructed the INS/UWB combined-positioning scheme, which can efficiently suppress the divergence of the inertial navigation system, but it is necessary to add a large number of additional UWB anchor nodes, which increases the system’s cost and complexity. In [11], a federated filter is used to form a sub filter with the inertial navigation system, the visual camera, and the odometer, respectively, in order to address the fact that single sensors with poor precision increase the positioning error of the whole positioning system in the traditional Kalman filter [12]. However, the large amount of dust, weak light, and strong vibration in the working face affect the positioning accuracy of the visual camera [13]. The authors of [14,15,16] established a Kalman-filter model with the distance between two INS kits as the constraint condition, which can effectively restrain the error divergence of a single kit of INS. However, the installation conditions of the two kits of the INS are strict, which affects the positioning accuracy of the model.

Although the traditional positioning technology of the combination of the inertial navigation system with the odometer (INS/ODO for short) directly uses the odometer output to correct the forward speed calculated by the INS, the ability to restrict the error divergence of the INS is very limited [17]. The shearer satisfies the characteristics of non-holonomic constraints (NHC) in the moving process, under the condition that if the shearer does not jump and side-slip, the lateral and vertical speeds at the connection between the traction gear and the crawler are zero. Combined with this characteristic, aiming at the current problems in the combined positioning technology of traditional inertial navigation system and odometer, this paper puts forward a shearer-positioning algorithm based on non-holonomic constraints, which can improve the positioning results of the combined-positioning technology in the traditional INS and odometer [18]. Furthermore, this paper verifies the proposed algorithm’s effectiveness and feasibility through positioning experiments.

2. Positioning Algorithm of Shearer

2.1. Calculation Process

Figure 1 shows the execution flow of the positioning algorithm of the shearer proposed in this paper: firstly, the information on shearer’s attitude, speed, and position are obtained by the output of the inertial measurement unit installed in the center of the shearer’s body through mechanical arrangement. The odometer output installed on the traction gear of the shearer is used to calculate the instantaneous speed of the shearer. Next, the mechanical arrangement results of the INS, the output of the odometer, and the non-holonomic constraints are fused through the Kalman-filter algorithm. The results obtained after the Kalman filter are used as the error feedback to modify the mechanical arrangement results of the INS to obtain the optimal estimation of the attitude, speed, and position information of the shearer. These three kinds of information reflect three different positions, and a unified coordinate system is required for a unified solution.

2.2. Definition of Coordinate System

Due to the continuous turning and moving of the shearer, and the inevitable alignment error when the inertial measurement unit is installed [19,20], a single coordinate system cannot directly describe the position of the shearer, nor is it conducive to position calculation [21]. Thus, the coordinate systems used in the process of solving the position information in this paper include a coordinate system of shearer body (M system), an inertial-measurement-unit coordinate system (B system), and a reference coordinate system (N system), as shown in Figure 2. They are respectively defined as follows:

Coordinate system of shearer body (M system): The coordinate origin is selected at the contact point between the shearer traction gear and the bottom plate. The forward direction of the shearer is the x-axis direction, the y-axis is perpendicular to the z-axis and points to the hydraulic support, and the z-axis is vertical and downward.

Inertial measurement unit coordinate system (B system): The origin of the B system coincides with the center of the inertial unit. The three coordinate axes are parallel to the three sensitive axes of the inertial measurement unit and follow the right-hand rule.

Reference coordinate system (N system): This system is equivalent to the local horizontal coordinate system in this paper. The origin of the N system is located in the measured center of the inertial measurement unit. The x-axis direction is north, the y-axis direction is east, and the z-axis direction is perpendicular to the plane formed by the other two axes and points upward.

In these three coordinate systems, the direction of M system and B system change during the working of shearer, while the direction of N system is always the same for reference. In addition, for the purpose of introducing non-holonomic constraints and ease of calculation, the origin of M system does not coincide with the other two.

2.3. Kalman-Filter-State Equation

To improve the filtering effect of Kalman filter, the INS mainly adopts indirect Kalman filter for combined-positioning solution, that is, the error of INS is used as the state variable, which can solve the problem of nonlinear change of positioning system under continuous time conditions. Partial error in INS is treated as the state vector of the whole positioning system, which consists of INS and odometer, namely:

$$X={\left[\begin{array}{cc}\begin{array}{ccc}\varphi & {\delta }_{v^{\mathrm{n}}}& {\delta }_{r^{\mathrm{n}}}\end{array}& \begin{array}{ccc}{b}_{\mathrm{g}}& b_{\mathrm{a}}& \begin{array}{cc}{s}_{\mathrm{g}}& s_{\mathrm{a}}\end{array}\end{array}\end{array}\right]}^{\mathrm{T}}$$

(1)

where $\varphi$ is the vector representing the three-dimensional error of shearer attitude, ${\delta }_{v^{\mathrm{n}}}$ represents the three-dimensional error of shearer velocity, ${\delta }_{r^{\mathrm{n}}}$ is the position-error vector in three directions, $b_g$ is the zero-bias vector of the three-axis gyroscope, $b_a$ is zero-bias vector of the three-axis accelerometer, $s_g$ is the three-axis gyroscope’s error vector of scale factor, and $s_a$ is the three-axis accelerometer’s error vector of scale factor [22].

By analyzing the disturbance of INS, the differential equations for errors in attitude, velocity, and position in continuous time can be obtained:

$$\dot{\varphi }=-{\omega }_{\mathrm{in}}^n\times \varphi +{\delta }_{{\omega }_{\mathrm{in}}^{\mathrm{n}}}-C_{\mathrm{b}}^{\mathrm{n}}{\delta }_{{\omega }_{\mathrm{ib}}^{\mathrm{n}}}$$

(2)

$$\dot{{\delta }_{v^{\mathrm{n}}}}=C_{\mathrm{b}}^{\mathrm{n}}{\delta }_{f^{\mathrm{b}}}+C_{\mathrm{b}}^{\mathrm{n}}{f}^{\mathrm{b}}\times \varphi -\left(2{\omega }_{\mathrm{ie}}^{\mathrm{n}}+{\omega }_{\mathrm{en}}^{\mathrm{n}}\right)\times {\delta }_{v^{\mathrm{n}}}+v^{\mathrm{n}}\times \left(2{\delta }_{{\omega }_{\mathrm{ie}}^{\mathrm{n}}}+{\delta }_{{\omega }_{\mathrm{en}}^{\mathrm{n}}}\right)+{\delta }_{g_{\mathrm{l}}^{\mathrm{n}}}$$

(3)

$$\dot{{\delta }_{r^{\mathrm{n}}}}=-{\omega }_{en}^n\times {\delta }_{r^{\mathrm{n}}}+{\delta }_{v^{\mathrm{n}}}$$

(4)

where $C_{\mathrm{b}}^{\mathrm{n}}$ is the attitude matrix from the B system to the N system, ${\omega }_{\mathrm{in}}^n$ is the rotation speed of the N system, ${\delta }_{{\omega }_{\mathrm{in}}^{\mathrm{n}}}$ is measurement error of rotation-angle velocity of Earth, ${\delta }_{{\omega }_{\mathrm{ib}}^{\mathrm{n}}}$ is measurement error of angular velocity measured by three-axis gyroscope, ${\delta }_{f^{\mathrm{b}}}$ is the measurement error of the accelerometer, $f^{\mathrm{b}}$ is the output value of the three-axis accelerometer, ${\omega }_{\mathrm{ie}}^{\mathrm{n}}$ is projection of rotation speed of Earth in the N system, ${\omega }_{\mathrm{en}}^{\mathrm{n}}$ is projection of rotational angular velocity of the N system relative to the earth under the N system, ${\delta }_{{\omega }_{\mathrm{ie}}^{\mathrm{n}}}$ is measurement error of ${\omega }_{\mathrm{ie}}^{\mathrm{n}}$, ${\delta }_{{\omega }_{\mathrm{en}}^{\mathrm{n}}}$ is measurement error of ${\omega }_{\mathrm{en}}^{\mathrm{n}}$, and ${\delta }_{g_{\mathrm{l}}^{\mathrm{n}}}$ is measurement error of gravity vector [23]. A first-order Gauss Markov process is used to describe the zero-bias and scale-factor error propagation equation of three-axis gyroscope and three-axis accelerometer, and its differential equation is

$$\{\begin{array}{c}\begin{array}{c}\dot{b_g}\left(t\right)=-\frac{1}{T_{bg}}{b}_g\left(t\right)+w_{b_g}\left(t\right)\\ \dot{b_a}\left(t\right)=-\frac{1}{T_{ba}}{b}_a\left(t\right)+w_{b_a}\left(t\right)\end{array}\\ \begin{array}{c}\dot{s_g}\left(t\right)=-\frac{1}{T_{sg}}{s}_g\left(t\right)+w_{s_g}\left(t\right)\\ \dot{s_a}\left(t\right)=-\frac{1}{T_{sa}}{s}_a\left(t\right)+w_{s_a}\left(t\right)\end{array}\end{array}$$

(5)

where, in the Gauss Markov process described above, $T_{bg}$, $T_{ba}$, $T_{sg}$, and $T_{sa}$ are the correlation time, $w_{bg}$, $w_{ba}$, $w_{sg}$, and $w_{sa}$ are the driving white noise [24].

To facilitate calculation, the state equation in continuous time needs to be discretized first [25]. The simplified form of Kalman-filter-state equation after discretization can be obtained from Equations (2) to (5).

$$X_{k+1,k}=F_{k+1,k}{X}_k+G_{k+1,k}{w}_k$$

(6)

where $X$ is the inertial navigation system error-state vector,$F$ is the inertial navigation system-error-transfer matrix, $G$ is the noise-driving matrix, $w$ is the system noise, and the subscript $k$ is the time mark. The establishment of the state equation can integrate the inertial navigation system error into Kalman filter as the state information, and provide the predicted value for Kalman filter.

2.4. Kalman-Filter-Observation Equation

This paper is based on the combined positioning of inertial navigation system and odometer, and introduces non-holonomic constraints as auxiliary information to suppress the divergence of INS error. For the shearer, in the cutting process, the inertial measurement unit must be affected by the installation angle and the shearer vibration; therefore. the non-holonomic constraint is not fully established at the installation of the inertial measurement unit, but for the connection between the shearer traction gear and the crawler, the traction gear is tightly stuck on the continuous crawler, and the non-holonomic constraint is established. Therefore, the speed obtained from the mechanical arrangement of the INS is projected to the joint of traction gear and crawler, and the non-holonomic constraint and the odometer-output speed are used to form a three-dimensional speed constraint on the shearer.

When the shearer meets the non-holonomic constraint, the speed $v_m$ of the connection between the traction gear and the crawler in the coordinate system of the shearer body is as follows:

$$v_m={\left[\begin{array}{ccc}{v}_t& 0& 0\end{array}\right]}^T$$

(7)

where $v_t$ is the real cutting speed of the shearer.

When projecting the INS mechanization result onto the M system, it is necessary to consider the alignment error caused by the installation of the inertial unit and the offset between the respective origin of the B system and the M system. Therefore, the velocity ${\widehat{v}}_m$ which is projected from B system to m system is

$$\begin{array}{c}{\widehat{v}}_m=C_b^m{\widehat{C}}_n^b{\widehat{v}}^n+C_b^m\left({\widehat{\omega }}_{nb}^n\times \right)l_{\mathrm{IMU}}^b\\ \approx C_b^m{C}_n^b\left[I+\left(\varphi \times \right)\right]\left(v^n+{\delta }_{v^{\mathrm{n}}}\right)+C_b^m\left({\omega }_{nb}^n\times \right)l_{\mathrm{IMU}}^b+C_b^m\left({\delta }_{{\omega }_{\mathrm{ib}}^{\mathrm{b}}}\times \right)l_{\mathrm{IMU}}^b\\ \approx v_m+C_b^m{C}_n^b{\delta }_{v^{\mathrm{n}}}-C_b^m{C}_n^b\left(v^n\times \right)\varphi -C_b^m\left(l_{\mathrm{IMU}}^b\times \right){\delta }_{{\omega }_{\mathrm{ib}}^{\mathrm{b}}}\end{array}$$

(8)

where $C_b^m$ represents the attitude matrix [26] from the B system to the shearer body, which is determined by the installation angle of the inertial measurement unit. ${\widehat{C}}_n^b$ represents the attitude matrix from the N system to the B system with alignment error, which is determined by the attitude angle calculated by the INS. ${\widehat{v}}^n$ is calculated velocity for INS. ${\widehat{\omega }}_{nb}^n$ is a calculation result by INS and projected from B system to N system, which express the rotational angular velocity relative to the N system, $l_{IMU}^b$ is the projection in B system of the position vector between the two origins from M system to B system, ${\delta }_{{\omega }_{ib}^b}$ is measurement error of the three-axis gyroscope, and $C_n^b$ is the attitude matrix from the N system to the B system.

The speed ${\tilde{v}}_m$ provided by the odometer can be expressed as

$${\tilde{v}}_m=v_m+V$$

(9)

where $V$ is the observation noise.

To sum up, the observation vector of Kalman filter can be determined by Equations (7)–(9), and the expression is

$$Z={\tilde{v}}_m-{\widehat{v}}_m=-C_b^m{C}_n^b{\delta }_{v^{\mathrm{n}}}+C_b^m{C}_n^b\left(v^n\times \right)\varphi +C_b^m\left(l_{\mathrm{IMU}}^b\times \right){\delta }_{{\omega }_{\mathrm{ib}}^{\mathrm{b}}}+V$$

(10)

Furthermore, the observation equation is

$$Z_{k+1}=H_{k+1} X_{k+1,k}+V$$

(11)

In the formula, $H$ is the observation matrix. By establishing the observation equation, the non-holonomic constraints and the speed of odometer output can be integrated into Kalman filter as the observation information to form a three-dimensional speed constraint on the shearer.

2.5. Kalman-Filter Algorithm

Kalman-filter algorithm is a kind of algorithm that can fuse prediction information and observation information to produce the best estimation of the system state at the next time. The Kalman-filter algorithm is divided into two parts, prediction and update, which can be expressed by the following five iterative equations.

(1)

Prediction algorithm

$$X_{k+1,k}=F_{k+1,k}{X}_k$$

(12)

$$P_{k+1,k}=F_{k+1,k}{P}_k{F}_{k+1,k}{}^T+Q_k$$

(13)

(2)

Update algorithm

$$K_{k+1}=P_{k+1,k}{H}_k{}^T{\left(H_k{P}_{k+1,k}{H}_k{}^T+R_k\right)}^{-1}$$

(14)

$$X_{k+1}=X_{k+1,k}+K_{k+1}\left(Z_{k+1}-H_{k+1}{X}_{k+1,k}\right)$$

(15)

$$P_{k+1}=\left[I-K_k{H}_k\right]P_{k+1,k}{\left[I-K_k{H}_k\right]}^T+K_k{R}_k{K}_k{}^T$$

(16)

In the formula, $P$,$Q$, and $R$ form the covariance matrix; $P$ represents the prediction error, $Q$ represents the system noise in the state equation, and $R$ represents the observation noise. $K$ is the filter-gain matrix.

With this algorithm, optimal estimation of the shearer’s posture, speed, and position information can be calculated.

3. Experimental Validation

3.1. Scheme Design

To verify the feasibility and positioning accuracy of the algorithm, and to compare it with the traditional INS/ODO combined-positioning scheme, this paper reports the experiments on a laboratory-simulation platform. The experimental equipment used an inertial measurement unit featuring an ICM42605 gyroscope and an ADXL355 accelerometer.

Table 1. Performance parameters of IMU.

Gyroscope		Accelerometer
Dynamic range (°/s)	$\pm 500$	Dynamic range (g)	$\pm 2$
Zero bias (°/s)	$\pm 0.5$	Zero bias (g)	$\pm 0.025$
Random error (°/ $\sqrt{h}$ )	$0.24$	Random error ($\mathsf{\mu }\mathrm{m}/\mathrm{s}·\sqrt{h})$	$9$
Scale-factor error (%)	$\pm 0.82$	Scale-factor error ($\%)$	$\pm 1.46$

shows the relevant parameters of the components. The E6H-CWZ6C encoder was installed on the traction gear of the shearer as the observation source data of the odometer, and the encoder resolution was 3600 P/R. The inertial measurement unit and coder were installed according to the requirements described in Section 1. The data of the inertial measurement unit and coder were collected in real time over the course of the experiment. The data-collection frequency was 100 Hz, and the data were processed after the experiment.

The experimental process simulated the process of the slanted shearer cutting and triangular coal cutting in the long-wall face. The cutting distance was 0.3 m and the total length of the whole process was 30 m. In order to facilitate the comparison and observation of the experimental results, a face-coordinate system (F system) was established, with the starting position of the shearer as the origin, the heading direction of the shearer as the x-axis, the advancing direction of the face as the y-axis, and the z-axis conforming to the right-hand rule. The positioning results were transformed from the N system to the F system. During the movement process of the shearer, a series of marker points were made on the moving track of the shearer artificially and irregularly, the coordinates of the marker points in the coordinate system of the long-wall face were measured, the marker time was recorded, and the marker points were numbered. In this way, a series of marker points for the shearer’s actual moving track were obtained. Tagged points were used as location-reference points, and the corresponding point coordinates of the trajectory calculated by the algorithm presented in this paper were compared and analyzed.

3.2. Experimental Results and Analysis

Figure 3 shows the movement trajectory of a shearer in the face’s coordinate system. It compares the trajectory of the shearer in the traditional INS/ODO scheme and the positioning method based on non-holonomic constraints which is proposed in this paper, and adds the coordinates of the marker points as a reference. From Figure 3, it can be seen that the traditional INS/ODO scheme has rapid error divergence because only the odometer can restrict the forward speed of the shearer. After adding the non-holonomic constraint, the positioning error does not diverge with time due to the increase in the motion-constraint information in the lateral and vertical direction of the shearer, which has good tracking performance for the actual track.

The absolute values of the positioning errors in the three directions of the two schemes were counted. The results are shown in Figure 4, and

Table 2. Maximum positioning error.

Axis	Maximum Positioning Error/m
	INS/ODO	NHC
x	0.88	0.30
y	0.08	0.03
z	0.12	0.04

records the maximum positioning error in all directions. Through comparison, it can be found that the max errors in all three axes decrease significantly after adding the non-holonomic constraint, and the error of the positioning results in the forward direction is reduced by 66%, from 0.88 to 0.30; the lateral direction is reduced by 62%, from 0.08 to 0.03; and the vertical direction is reduced by 67%, from 0.12 to 0.04.

In Table 2, the data show that although the algorithm based on the non-holonomic constraint can greatly improve the accuracy of the shearer positioning in three directions, the deviation of the positioning result on the x-axis is still much greater than that in the other two directions. The reason is that although there is an odometer to take concerted action with the INS to constrain the shearer’s forward speed, the odometer itself also has errors, such as the scale factor, which accumulate through the continuous iteration of the filter and are then superimposed in the direction of the shearer. The output of the odometer can be corrected by using a more accurate odometer or adding a new sensor, thereby reducing the impact of the accumulated errors of the odometer [27].

4. Conclusions

Aiming at the divergence problem of the positioning error in the automatic face-0positioning system in coal mines, this paper presented in-depth research around error sources, error-elimination algorithms, and positioning-algorithm experiments, and obtained the following conclusions:

• On the basis of the traditional INS/ODO combined-positioning method, according to the motion information in the working process of the shearer, this paper proposes a shearer-positioning method that combines the non-holonomic constraints and the output speed of the odometer to form a three-dimensional speed constraint on the shearer, which effectively suppresses the divergence of the INS mechanization result.
• An experimental system was set up to verify the validity of the positioning method proposed in this paper. The experimental results show that, compared with a traditional INS/ODO combined-positioning scheme without non-holonomic constraints, the error of the positioning results in the forward direction was reduced by 66%, while in the lateral direction, it was reduced by 62% and in the vertical direction, it was reduced by 67% when the non-holonomic constraint was added.
• Although the positioning method mentioned in this paper effectively improves the accuracy of shearer positioning, it was found that the positioning error of the shearer in the forward direction after the application of this method is still greater than the lateral and vertical errors, which are caused by the accumulation of errors such as the scale factor of the odometer. Further analysis and research are needed in future work.