1. Introduction
Hydraulic support is an important safety support equipment in the automatic coal mining working face, which is of great significance to the safety of coal miners and the normal operation of coal mining equipment. In recent years, with the continuous development of intelligent mining technology [
1,
2], the roboticized hydraulic support technology has been paid more and more attention by scholars and coal mining managers. In particular, the precise estimation of the support attitude of hydraulic support directly determines the intelligent ability of hydraulic support. Therefore, it is urgent to study a new attitude sensing method to obtain accurate support attitude of hydraulic support [
3], especially in the special application scenarios with a large demand for the number of sensors and complex environment.
The existing studies on the support attitude sensing measurement of hydraulic support are mainly focused on the mechanism and kinematics of hydraulic support and trying to achieve the accurate estimation of support attitude by kinematic analysis of its mechanism [
4]. In [
5,
6], Polish scholars analyzed the structural composition and empirical formula of support strength of hydraulic support, and developed a new type of support mechanism for vertical displacement of hydraulic support, but which can only approximately estimate the yaw angle of hydraulic support by experience. In [
7,
8], through the analysis of the four-linkage mechanism and parameters of hydraulic support, the kinematics equation is established, and then the support attitude can only be roughly estimated due to the neglected clearance between the hinge joint and the pin shaft. In [
9], a laser radar detection method is used for measuring the relative attitude angle of hydraulic support based on the position of inspection robot, but the estimation result is disturbed greatly due to the influence of robot motion vibration. In [
10,
11], a mechanical-electrical-hydraulic coordination technique based on the dynamic response of the load is exploited to adjust the attitude of hydraulic support, but it is easily affected by the pressure of the pump station. In [
12], a virtual adjustment method of support attitude under the propulsive state of hydraulic support groups is proposed and the estimation accuracy is highly dependent on the modeling precision of the virtual model, and different hydraulic supports need different virtual models, which is difficult to establish and has poor universality. In [
13], an approach for monitoring the posture of hydraulic support with fiber Bragg grating tilt sensor based on gravity is presented, but it is not applied to measure yaw angle of the canopy. However, in automatic coal mining working face with the complicated geological structures, the above methods are characterized by complex solutions, difficult modelling, large estimation disturbance and lack of universality with estimating attitude or relative attitude indirectly based on kinematic parameters of mechanism, or can only estimate a single attitude angle. In this paper, we try to explore a new approach for estimating directly the support attitude of hydraulic support with comprehensive consideration of estimation accuracy, measurement method, universality and sensor size.
Inertial measurement unit (IMU) can be independent of the existing mechanism of hydraulic support to estimate support attitude directly, which is not affected by the mechanism of hydraulic support itself. In recent years, the development of microelectromechanical system technology has accelerated leaps and bounds the development process of IMU, which makes the IMUs become low in cost, low in power consumption and small in size. Additionally, it is widely applied in robotics [
14,
15], navigation [
16,
17] and virtual reality [
18]. Scientific research is not limited to sensor applications [
19], but also includes the performance of IMUs [
20]. The performance improvement and calibration analysis of IMU provide precise attitude estimation technologies for its application [
21], such as the method of improving estimation performance based of sine rotation vector; the acceleration and magnetic field measurements are transformed into the differences between the Euler angles of the measured attitude and the predicted attitude to correct the predicted attitude [
22]. However, the measurement accuracy and performance of these low-cost MEMS inertial measurement units are easily limited by the complex working environment [
23,
24], especially the automatic coal mining working face with complex magnetic interference produced by coal structure and high-power electromechanical equipment. Therefore, it is necessary to use a filtering algorithm to improve the estimation accuracy.
The Kalman filter (KF) and particle filter (PF) are the most studied and widely used filtering algorithms. The Kalman filter for linear filtering and the extended Kalman filter (EKF) with Jacobian matrix and the unscented Kalman filter (UKF) for nonlinear filtering are frequently used to process inertial information, including object tracking [
25] and rotation estimation [
26]. In order to reduce yaw drift of attitude rotation estimation in indoor scenarios, an improved EKF combining INS mechanization algorithm and zero velocity update methodology is proposed to improve the accuracy of yaw estimation [
27], but the Jacobian matrix is used to update the state transition matrix and the observation matrix in the linearization steps of system equation in attitude determination, which leads to poor stability and large estimation bias. Owing to the system, equations are updated based on the sigma-point approximation generated by unscented transformation in UKF [
28,
29], which has been demonstrated to have more attitude solution accuracy and strong robustness than EKF in estimating attitude with highly nonlinear kinematics models. In [
30,
31], a quaternion-based MEMS sensors fusion algorithm and heading estimation approach based on rotation matrix and principal component analysis are proposed to improve the heading estimation accuracy of indoor pedestrians, which shows that quaternion is easier to implement than other methods in sensor fusion technology. However, the particle filter is a sequential importance sampling filtering method based on Bayesian estimation [
32] and its idea is based on the Monte Carlo method to represent probability with particle set, which can be used in any state space models [
33,
34]. In [
35], a method of filtering and predicting time-varying signals under model uncertainty is proposed to realize dynamic estimation of the data. In [
36], a particle metropolis Hastings algorithm driven by multiple parallel particle filters is proposed to make inference on dynamic and static variables. In any case, although the particle filter has good estimation performance in nonlinear filtering, the mathematical process of particle filters is more complex, and the algorithm implementation and calculation are difficult. Moreover, the resampling, particle degradation and calculation amount in the calculation process lead to the complexity of the implementation process and poor real-time performance. Therefore, the operation time is much longer than that of UKF.
Through the above research on attitude estimation of nonlinear systems, and considering the estimation accuracy, calculation speed and real-time performance of algorithm, UKF is proved to be a powerful technique for estimating attitude angle and a superior alternative to the EKF and PF in various nonlinear system filtering [
37,
38]. Unfortunately, the key parameter of UKF selected by experience, process noise covariance Q, has a great influence on the estimation accuracy in the complex working environment [
39], so the research on tuning of parameter Q is very meaningful [
40,
41]. Therefore, the gradient descent algorithm [
42] with the advantages of fast global convergence, faster operation speed and simple realization could be used to tune the process noise covariance Q to improve the estimation performance of UKF.
In this paper, a support attitude sensing system with a character of intrinsic safety is designed to measure the support attitude of hydraulic support in the special application scenarios with a large demand for the number of sensors and complex environment. The designed support attitude sensing system is a nine-axis low-cost MEMS measurement unit composed of a gyroscope, magnetometer and accelerometer. The proposed gradient descent algorithm-optimized quaternion-based unscented Kalman filter makes full use of the characteristics of complementation of the magnetometers without long-term drift, accelerometers and gyroscopes which are not affected by magnetic disturbance to improve the support attitude estimation accuracy of hydraulic support.
The remaining parts of this paper are organized as follows. In
Section 2, the support attitude sensing system designed in our laboratory is described.
Section 3 introduces the proposed optimized quaternion-based unscented Kalman filter based on the complementary characteristics of MEMS sensors in detail. In
Section 4, an experiment is conducted and analyzed to validate the estimation performance of the proposed system and approach. Then, industrial applications and tests are carried out in
Section 5.
Section 6 summarizes our conclusions and future work.
3. The Optimized Quaternion-Based Unscented Kalman Filter
In this section, we propose an unscented Kalman filter based on quaternion for estimating the support attitude of hydraulic support. In order to improve the performance of the support attitude estimation algorithm, the gradient descent algorithm is used for tuning the key parameter of the unscented Kalman filter, i.e., process noise covariance Q. The whole procedure of accurate estimation of support attitude using the gradient descent algorithm-optimized quaternion-based unscented Kalman filter (GD-UKF) is also presented.
3.1. Gradient Descent Algorithm
Gradient descent algorithm with the advantages of simple implementation is a common method to solve unconstrained optimization problems. The basic idea of the gradient descent is to select an appropriate initial value , update the value of iteratively, and minimize the objective function until it converges. The input of gradient descent is the objective function , the gradient function and the calculation accuracy ; the output is the minimum point of . The process of the gradient descent can be summarized in detail as follows:
Step 1.1: Initial key parameters and k.
Step 1.2: Calculate the objective function .
Step 1.3: Calculate the gradient . If , stop iteration and let . Otherwise, let and then find for equation .
Step 1.4: Set the variable and update the function . If the expression or , stop iteration and let . Otherwise, let k = k + 1 and go back to Step 1.3 to find another better minimum.
When the objective function is a convex function, the solution of gradient descent is the global optimal solution.
3.2. Unscented Transformation
For unscented Kalman filter in a nonlinear system, unscented transformation (UT) is the core technique for propagating mean and covariance based on Cholesky decomposition, and can effectively approximate mean and covariance changes of the random variables when it undergoes a nonlinear transformation, including cross-correlation between state and measurement. The basic principle of UT can be considered in this way. We assume that the mean and covariance of a random variable X with n dimensions are and P, respectively, and mean and covariance P propagate through a nonlinear function . In order to calculate the statistics of the variable y, 2n + 1 sigma points and corresponding weights are formed and calculated as follows:
- (1)
Calculate 2n + 1 sigma points
where
,
is the
i-th column of the matrix of the square root.
- (2)
Calculate the corresponding weight of sigma point
where
is the scaling factor for reducing total prediction error;
is set to a small positive value to control the distribution of sigma points;
is the parameter to be selected to ensure a positive semidefinite and is usually set to 0; and
is a nonnegative weight coefficient and is set for 2 for Gaussian distribution in this paper. The subscript
m and
c represent covariance and mean, respectively.
3.3. Unscented Kalman Filter Design
Generally, the dynamic system of the unscented Kalman filter can be described in two system equations: firstly, state equation; and secondly, observation equation. Considering the discrete-time nonlinear dynamic model of support attitude estimation of hydraulic support, the model state variable is , which is a vector composed of four elements of quaternion; and the model observation variable is , which is a vector composed of attitude angles. The state variable is unknown and is constantly updated according to the gyroscope; the observation variable is known and measured by the accelerometer and magnetometer. The details of the mathematical model are described with the following steps.
- (1)
State equation
The state prediction is based on the previous optimal estimation, and the discrete-time nonlinear dynamic state equation of unscented Kalman filter is shown as:
where
and
are the prior estimation at time
k and the posterior estimation at time
k − 1, respectively;
is the process noise with covariance Q and simplified as independent Gaussian white noise.
is the state transition matrix and can be obtained as:
- (2)
Observation equation
The correction state is the essential step for refining measurement estimation. The observation equation is expressed as
where
is the independent Gaussian measurement noise with noise covariance R.
is the output function and can be calculated as follows:
3.4. Noise Covariance of Process and Observation
The observation noise covariance and process noise covariance are the two key parameters of unscented Kalman filter, which sometimes leads to large support attitude estimation errors. However, it is essential to optimize the noise covariance of the process and observation to minimize the support attitude estimation errors of hydraulic support.
(1) Process noise covariance
In the process of hydraulic support moving and canopy supporting in the automatic coal mining working face, the uneven floor could cause random acceleration of the hydraulic support movement. Meanwhile the coupling effect of roof rock and hydraulic support can produce random impact on the hydraulic support in the moving process and static state, and the left and right adjacent hydraulic support could also produce impact vibration on hydraulic support. Therefore, the coupling effect between hydraulic support and the surrounding environment could lead to random vibration and acceleration on the support attitude sensing system in the static and moving process of hydraulic support, which leads to difficulty in determining the process covariance Q.
We assume that
,
,
where
are the mean of
;
are the process deviations of the support attitude sensing system caused by the disturbance of coupling interaction between the hydraulic support and the surrounding environment.
,
and
are the variance of
. Thus, Equation (11) can be rewritten as:
The second term on the right of Equation (22) can be considered as the process noise
and the process noise covariance Q is:
However, it can be determined from Equation (23) that the optimal Q of UKF can be obtained by optimizing variance , and based on the optimization algorithm.
(2) Observation noise covariance
The observation noise covariance R is determined by the measurement process and related to the characteristics of the measuring instrument, which can be obtained through long-term probability statistics of sensor measurement data. In this paper, the observation noise covariance R is determined by the measurement deviation of accelerometer and magnetometer.
We assume that
,
and
are the observation deviation of accelerometer three-axis output
,
and
. Then, the first-order Taylor series expansion of trigonometric function is applied to Equation (8), and the deviation of the roll and pitch angles measured by the accelerometer can be derived as
Through Equation (24), the covariance of pitch angle and roll angle based on the accelerometer is as follows:
where
,
and
is the variance of accelerometer three-axis output
, and
.
Similarly, through Equations (3) and (5) in the first-order Taylor series expansion of trigonometric function, the deviation of yaw angle measured by magnetometer can be derived as
where
,
and
are the deviation of magnetometer output
,
and
.
Through Equation (26), the covariance of yaw angle based on magnetometer is as follows:
where
,
and
are the variances of the magnetometer output
,
and
, respectively.
Finally, it can be determined from Equations (25) and (27) that the observation noise covariance R is:
3.5. Unscented Kalman Filter Based on Gradient Descent
The UKF with Kalman linear filtering framework is not the traditional method of linearizing nonlinear functions, which uses unscented transformation to propagate mean and covariance. Compared with the extended Kalman filter, the UKF without deriving Jacobian matrix can make the state variable approximate the probability density distribution of nonlinear function and is more robust and accurate. However, the process noise covariance Q affects the filtering performance to a certain extent, and gradient descent can be used to improve the unscented Kalman filter. The flow of gradient descent algorithm-optimized quaternion-based unscented Kalman filter (GD-UKF) can be elaborated as below:
Step 2.1: The key parameters of the initial setup are dimension (n = 4), initial value
, covariance initial value
of random variable. Through Equations (16) and (17), a set of sigma points and the corresponding weights are computed by:
Step 2.2: The predicted value and covariance matrix of system state variable calculated through the state Equation (18) based on the sigma points obtained in Step 2.1 are expressed as:
Step 2.3: Through using UT again for the predicted value in Step 2.2, the new sigma points are calculated as follows:
Step 2.4: The predicted value and covariance matrix of observation variable calculated through the observation in Equation (20) based on the new sigma points obtained in Step 2.3 are expressed as follows:
Step 2.5: The unscented Kalman filter gain matrix
K is calculated as:
Step 2.6: The state update and covariance update of the unscented Kalman filter system are calculated as:
Step 2.7: The process noise covariance Q is updated by calculating and updating process variance
of the support attitude sensing system based on gradient descent. And the objective function of gradient descent is
where
is the loss function;
is the root mean square error of estimation results;
x is the true state value of the support attitude sensing system. The flowchart of the proposed GD-UKF is presented in
Figure 4.
5. Industrial Experiment and Application
In this section, our support attitude sensing system based on the proposed GD-UKF is applied in the automatic coal mining working face to test practical performance, as shown in
Figure 9. The industrial experiment and application were tested at the 13230 coal mining working face of Gengcun Mine of Yima Coal Industrial Group Co., Ltd. (Yima, China). The intrinsically safe micro inertial sensor was installed on the canopy of hydraulic support to measure support attitude of hydraulic support. The attitude angle information was transmitted to the remote monitoring center by the network switch, and then the support attitude of hydraulic support was adjusted by the support attitude controller to satisfy the support requirements of coal mining.
The initial pitch angle of the hydraulic support canopy was 3.5°, then the angle was adjusted to −2.6° and remained stationary. In this process, the roll angle fluctuated and finally returned to the initial angle of 10° which was the inclination of the 13,230 coal mining working face, and the yaw angle was also changed from 6.9° to 3.9°. The industrial application results using the support attitude sensing system are shown in
Figure 10. The pitch and roll estimation errors of support sensing system base GD-UDF were less than 1°, which could meet the requirements of the automatic coal mining working face. In order to verify the feasibility and superiority of the proposed algorithm for magnetic disturbances filtering, an industrial experiment of yaw angle was carried out, and the test results were shown in
Figure 11. However, we could obviously find out that the yaw angle had a larger variation range than the other two angles and the yaw estimation error was relatively large, as shown in
Figure 11a, the measurement error of original data was about 5° and that of yaw angle based on GD-UKF was less than 2°. The reason is that the high-power equipment, such as a shearer in the automatic coal mining working face, can generate a certain complex external magnetic field interference, which has a great influence on the yaw estimation of hydraulic support and is difficult to eliminate.
The industrial application has verified that the support attitude estimation of hydraulic support with our support attitude sensing system using gradient descent algorithm-optimized quaternion-based unscented Kalman filter for attitude solution has high measurement accuracy, but the movement conditions of the automatic coal mining working face are complex, especially the magnetic disturbance. Therefore, the experiments for improving the yaw angle estimation algorithm need to be carried out under more complicated conditions.