A Walking Trajectory Tracking Control Based on Uncertainties Estimation for a Drilling Robot for Rockburst Prevention

Abstract

A walking trajectory tracking control approach for a walking electrohydraulic control system is developed to reduce the walking trajectory tracking deviation and enhance robustness. The model uncertainties are estimated by a designed state observer. A saturation function is used to attenuate sliding mode chattering in the designed sliding mode controller. Additionally, a walking trajectory tracking control strategy is proposed to improve the walking trajectory tracking performance in terms of response time, tracking precision, and robustness, including walking longitudinal and lateral trajectory tracking controllers. Finally, simulation and experimental results are employed to verify the trajectory tracking performance and observability of the model uncertainties. The results testify that the proposed approach is better than other comparative methods, and the longitudinal and lateral trajectory tracking average absolute errors are controlled in 10.23 mm and 22.34 mm, respectively, thereby improving the walking trajectory tracking performance of the walking electrohydraulic control system for the coal mine drilling robot for rockburst prevention.

1. Introduction

The borehole pressure relief method has become an effective way to prevent coal mine rockburst in China, which can reduce the impact of dynamic disasters for deep coal mining [1,2,3]. During this period, moving the drilling robot to the target position is the precondition for realizing an effective borehole [4,5]. Furthermore, the accurate walking trajectory tracking control directly determines whether the drilling robot can move to the target position. In the moving process of the drilling robot, the walking electrohydraulic control system (WECS) enables the drilling rig to move, which is commonly driven by two hydraulic motors. The efficient operation of two hydraulic motors directly affects the walking position and deviation of the drilling rig. Nevertheless, the walking trajectory tracking control accuracy and stability of the walking system are not easy to obtain because of the nonlinear characteristics. Simultaneously, different walking conditions such as the coal mine roadway surface are uneven and external disturbances, leading to walking trajectory deviations from the expected path and even causing collision accidents.

To overcome the trajectory tracking control challenge, a variety of studies to improve the control performance have been conducted for the trajectory tracking control, e.g., speed control to track reference trajectory [6], speed predictive control [7], active disturbance rejection control [8,9], sliding mode control for path tracking [10,11], and adaptive control [12,13]. From these studies, the accurate trajectory tracking control behavior cannot be easily guaranteed in the presence of uncertainties or external disturbances, especially for the coal mine drilling robot for rockburst prevention (CMDR). Furthermore, sliding mode control (SMC) is selected as an effective control method for achieving expected tracking control performance [14,15], and the chattering phenomenon of the SMC can be reduced by the saturation function [16,17]. A sliding mode controller based on a high-gain observer was designed to improve the control performance [18]. A sliding mode observer-based controller is designed to ensure highly accurate tracking control characteristics [19]. A sliding mode controller is designed to improve the control precision and attenuate the disturbances, including model uncertainties and external disturbances [20]. Nevertheless, the relevant application of the SMC method in the WECS for the CMDR is still scarce. Moreover, the estimation of model uncertainties and disturbances are lacked for the trajectory tracking control of the WECS. Accordingly, a walking trajectory tracking control scheme urgently needs to be proposed to enhance the system performance of the WECS.

To reduce the adverse effects of model uncertainties and disturbances, some observer-based control methods have been applied to reduce the tracking deviation, e.g., a fuzzy observer-based composite nonlinear controller for trajectory tracking is presented to improve the tracking performance [21]. An extended state observer-based sliding mode controller is developed to estimate the lumped uncertainty and achieve robust tracking performance [22]. An unscented Kalman filter is used to estimate detection error, and the proposed trajectory tracking control strategy can effectively achieve target trajectory scheduling with a limited adjustment period [23]. A robust adaptive neural network trained with the integral sliding mode is presented to enhance the path-tracking performance, which has the capacity for state estimation and auxiliary control inputs [24]. A nonlinear disturbance observer is designed to estimate the system states, and a sliding mode controller is designed to improve the control efficiency and response characteristics [25]. An adaptive robust controller and a state feedback controller are proposed to effectively handle parametric uncertainties and uncertain nonlinearities, which can reveal the connection between achievable performances and control parameters [26]. Consequently, the model uncertainties and disturbances need to be estimated in the trajectory tracking control to eliminate tracking deviations in the regulation motion. However, there is still no effective way to handle the multiple uncertainties and disturbances in the WECS. Moreover, the mixture effects of model uncertainties and external disturbances in the WECS may bring more challenges for walking trajectory tracking control.

To exploit an adapted controller for the electrohydraulic system, sliding mode control and its extended methods are known as efficient nonlinear control algorithms for an engineering system. The observer-based tracking controller is designed to improve system response characteristics and has been applied in many systems [27,28,29]. Although many previous studies have solved the tracking control problem, the system characteristics also suffer in various working conditions because of parameter perturbations. In designing a controller for the WECS, when the walking trajectory loop experiences external disturbances, such as roadway excitation load and parameter mismatches, a state observer and control strategy are crucial to guarantee accurate trajectory tracking control and robustness. In the field of electrohydraulic control, a disturbance observer-based control is developed to estimate the uncertainties and guarantee the response performance [30,31]. Furthermore, the state estimator and SMC-based method are designed to enhance the system response characteristics [32,33]. Therefore, a feasible controller with an accurate trajectory tracking control needs to be applied in the walking motion of the CMDR. However, the accurate trajectory tracking of WECS is to cover some indeterminate parameters. Furthermore, the interference between these parameters may affect the controller regulating process.

Motivated by the previous analysis, a walking trajectory tracking control method for the WECS is presented to improve performance. The main contributions can be stated as:

(1)

A coupled model is presented to analyze walking characteristics for the WECS, in which combing a walking motion model and an electrohydraulic control model to decompose the walking trajectory tracking control law, thereby improving the tracking control efficiency of the WECS.

(2)

Model uncertainties and external disturbances are observed by a state observer for the WECS based on an improved radial basis function. Additionally, a proper selection of the saturation function could reduce the chattering phenomenon and enhance the tracking capability of the WECS.

(3)

Compared with other existing control approaches, the presented control scheme for the WECS has a better performance in both walking longitudinal and lateral trajectory tracking control, which reduces the walking trajectory tracking average absolute error.

The remainder of this study is organized as follows. The configuration description and mathematical model of the WECS including the walking motion model and electrohydraulic control model are shown in Section 2. The state observer, walking longitudinal, and lateral trajectory tracking controller are designed in Section 3, and the walking trajectory tracking control strategy is proposed. In Section 4, simulations and experiments are conducted and compared with other existing methods. Conclusions are drawn in Section 5.

2. Walking Electrohydraulic Control System Analysis

2.1. Walking Process Description and Analysis

The WECS of the CMDR makes the tracked chassis move to achieve linear and steering walking, which is determined by the differential control of the left and right hydraulic motors. For the WECS, the linear and steering walking are controlled by two hydraulic motors that are installed on the left and right sides of the tracked chassis. When the drilling robot travels in a straight line, it is necessary to ensure that the left and right hydraulic motors output the same speed. And the steering walking relies on the differential speed control. These left and right hydraulic motors are coordinated and controlled by the electrohydraulic control system to achieve stable and accurate speed regulation, and the schematic diagram of a WECS of the CMDR is shown in Figure 1.

For the WECS, the straight and steering walking trajectory is regulated by the speed control of the left and right hydraulic motors, using the speed sensors as feedback input, the adjustment values are calculated and transmitted to the electrohydraulic proportional directional valves by the controller, generating different walking conditions of the coal mine robot for rockburst prevention. As a result, a speed sensor information continuous acquisition and electrohydraulic control system with proportional directional valves based on a WECS is proposed, as shown in Figure 2. The left and right hydraulic motors are separately controlled by proportional directional valves. Additionally, the controller and the host computer are utilized to obtain the speed signal, the attitude signal of the CMDR, the output electrohydraulic proportional directional valve control signal, and control parameters debugging.

2.2. Walking Model Analysis

Referring to Figure 1, the straight and steering walking trajectory is regulated by the speed control of the left and right hydraulic motors. Taking the forward direction of the tracked chassis as the x direction, the yaw motion direction of the tracked chassis as y direction, and the geometric center of the tracked chassis as the point of o, OXY is the inertial coordinate system fixed to the coal roadway and oxy is the implicated coordinate system fixed to the tracked chassis. Furthermore, the walking model of the tracked chassis can be described in Figure 3.

Furthermore, the walking function is achieved by adjusting the speed difference between the left and right hydraulic motors. Assuming that the state of the tracked chassis is (x₀, y₀, φ₀), and the kinematic model can be described as:

$$\left[\begin{array}{c}{\dot{x}}_0\\ {\dot{y}}_0\\ \dot{\phi }\end{array}\right]=\left[\begin{array}{ccc}\sin \phi & \cos \phi & 0\\ -\cos \phi & \sin \phi & 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{c}{v}_x\\ v_y\\ w_z\end{array}\right]$$

(1)

where φ is the heading angle of the tracked chassis, w_z is the yaw angle velocity of the tracked chassis.

Based on the kinematic model of Equation (1), combined with the model establishment process in the study [34], the Equation (1) can be rewritten as:

$$\left[\begin{array}{c}{\dot{x}}_0\\ {\dot{y}}_0\\ \dot{\phi }\end{array}\right]=\left[\begin{array}{ccc}\sin \phi & \cos \phi & 0\\ -\cos \phi & \sin \phi & 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{c}\frac{y_{\mathrm{L}}^{\prime }{v}_{\mathrm{jH}}-y_{\mathrm{H}}^{\prime }{v}_{\mathrm{jL}}}{y_{\mathrm{L}}^{\prime }-y_{\mathrm{H}}^{\prime }}{y}_{\mathrm{o}}^{\prime }\\ \frac{v_{\mathrm{jL}}-v_{\mathrm{jH}}}{y_{\mathrm{L}}^{\prime }-y_{\mathrm{H}}^{\prime }}{x}_0^{\prime }\\ -\frac{v_{\mathrm{jL}}-v_{\mathrm{jH}}}{y_{\mathrm{L}}^{\prime }-y_{\mathrm{H}}^{\prime }}\end{array}\right]$$

(2)

$$\left\{\begin{array}{cc}{y}_{\mathrm{L}}^{\prime }=\frac{v_x-v_{\mathrm{jL}}}{w_{\mathrm{z}}};\hfill & y_{\mathrm{H}}^{\prime }=\frac{v_x-v_{\mathrm{jH}}}{w_{\mathrm{z}}}\\ y_0^{\prime }=\frac{v_x}{w_{\mathrm{z}}};\hfill & x_{\mathrm{L}}^{\prime }=x_{\mathrm{H}}^{\prime }=x_0^{\prime }=-\frac{v_y}{w_{\mathrm{z}}}\end{array}\right.$$

(3)

where $o^{\prime }\left(x_{\mathrm{o}}^{\prime },y_{\mathrm{o}}^{\prime }\right)$ is the instantaneous turning center of tracked vehicles in the inertial coordinate system; $o_{\mathrm{L}}\left(x_{\mathrm{L}}^{\prime },y_{\mathrm{L}}^{\prime }\right)$ is the instantaneous turning center of the low-speed side track in the inertial coordinate system; $o_{\mathrm{H}}\left(x_{\mathrm{H}}^{\prime },y_{\mathrm{H}}^{\prime }\right)$ is the instantaneous turning center of the high-speed side track in the inertial coordinate system; $v_{\mathrm{jL}},v_{\mathrm{jH}}$ are the winding speeds of the low-speed side and high-speed side tracks, respectively.

Based on Figure 3b, the dynamic model for the tracked chassis can be described as:

$$\left\{\begin{array}{l}\begin{array}{l}{F}_{C\mathrm{y}}-\left(F_{\mathrm{yL}}+F_{\mathrm{yR}}\right)-m{\ddot{x}}_o=0\\ F_{\mathrm{xR}}-F_{\mathrm{xL}}-\left(F_{\mathrm{rL}}+F_{\mathrm{rR}}\right)-F_{C\mathrm{x}}-\delta m{\ddot{y}}_o=0\\ \begin{array}{l}\left(-F_{\mathrm{xR}}+F_{\mathrm{xL}}+F_{\mathrm{rL}}+F_{\mathrm{rR}}\right)\frac{B}{2}+\left(F_{C\mathrm{y}}{x}_C+F_{C\mathrm{x}}{y}_C\right)\\ -M_C-J\ddot{\phi }=0\end{array}\end{array}\\ M_C=M_{\mathrm{CL}}+M_{\mathrm{CR}}\end{array}\right.$$

(4)

where m is the mass of the drilling robot; δ is the coefficient of the rotational mass for the drilling robot; M_C is total steering resistance moment; M_CL, M_CR are the left and right tracks’ steering resistance moment, respectively; J is inertia moment; F_C (F_Cx, F_Cy) is the centrifugal force; B is the track width between the left and right tracks.

Assuming that the steering speed is constant and the expected longitudinal speed and the structural parameters are given, the driven force can be described as:

$$\left\{\begin{array}{l}{F}_{\mathrm{xR}}=F_{\mathrm{rR}}-\frac{1}{B}\left(F_{C\mathrm{y}}{x}_C+F_{C\mathrm{x}}{y}_C-M_C\right)+\frac{1}{2}{F}_{C\mathrm{x}}\\ F_{\mathrm{xL}}=-F_{\mathrm{rL}}-\frac{1}{B}\left(F_{C\mathrm{y}}{x}_C+F_{C\mathrm{x}}{y}_C-M_C\right)-\frac{1}{2}{F}_{C\mathrm{x}}\\ {\lambda }_C=\frac{L{F}_{C\mathrm{y}}}{2\mu mg}+x_C\end{array}\right.$$

(5)

where μ is the resistance coefficient; L is the longitudinal length of the tracked chassis; g is the gravitational acceleration.

2.3. Mathematical Model of the Valve-Controlled Motor

For the WECS, the left and right hydraulic motors have the same motion function to support the adjustment of the walking of CMDR. As the walking trajectory tracking control condition for enhancing controllability, the designed control strategy must generate dynamic response performance [35,36]. The straight and steering walking of the WECS changes with the speed difference of the left and right hydraulic motors, which are controlled by the electrohydraulic proportional directional valves, i.e., the valve-controlled motor speed control system. Therefore, the mathematical model of the hydraulic motor can be described as:

$$\left\{\begin{array}{l}{q}_{\mathrm{L}}=K_{\mathrm{q}}{x}_{\mathrm{mv}}-K_{\mathrm{c}}{p}_{\mathrm{L}}\\ q_{\mathrm{L}}=D_{\mathrm{m}}{\dot{\theta }}_{\mathrm{m}}+\frac{V_{\mathrm{t}}}{4{\beta }_{\mathrm{e}}}{\dot{p}}_{\mathrm{L}}+C_{\mathrm{m}}{p}_{\mathrm{L}}\\ D_{\mathrm{m}}{p}_{\mathrm{L}}=J_{\mathrm{m}}{\ddot{\theta }}_{\mathrm{m}}+B_{\mathrm{m}}{\dot{\theta }}_{\mathrm{m}}+G_{\mathrm{m}}{\theta }_{\mathrm{m}}+T_{\mathrm{m}}\end{array}\right.$$

(6)

where p_L, q_L represent the load pressure and flow, respectively; K_q, K_c represent the flow and flow–pressure gain coefficient, respectively; D_m, θ_m represent the displacement volume and angular displacement of the hydraulic motor, V_t represents the total volume of the hydraulic motor; β_e represents the bulk modulus of hydraulic oil; C_m represents the total leakage coefficient; J_m, B_m, G_m, T_m represent the inertia moment, viscous damping coefficient, torsion spring stiffness, and load torque of the hydraulic motor, respectively.

We assume the relationship between the input control signal and proportional directional valve spool displacement is given as:

$$x_{\mathrm{mv}}=k_{\mathrm{mv}}{u}_{\mathrm{mv}}$$

(7)

where k_mv is the gain coefficient between the valve spool displacement x_mv and control input signal u_mv for the hydraulic motor.

Simultaneously, with Equations (6) and (7), the transfer function between the x_mv and θ_m can be obtained as:

$$\frac{{\theta }_{\mathrm{m}}}{x_{\mathrm{mv}}}=\frac{K_{\mathrm{q}}/D_{\mathrm{m}}}{\left[\frac{V_{\mathrm{t}}{J}_{\mathrm{m}}}{4{\beta }_{\mathrm{e}}{D}_m^2} s^2+\left(\frac{V_{\mathrm{t}}}{4{\beta }_{\mathrm{e}}{D}_m^2}{B}_{\mathrm{m}}+\frac{J_{\mathrm{m}}{K}_{\mathrm{ce}}}{D_m^2}\right)s+1\right]}=\frac{K_{\mathrm{q}}/D_{\mathrm{m}}}{\left(\frac{s^2}{w_h^2}+\frac{2{\zeta }_h}{w_h}s+1\right)}$$

(8)

$$w_h=\sqrt{\frac{4{\beta }_{\mathrm{e}}{D}_{\mathrm{m}}^2}{ V_{\mathrm{t}}{J}_{\mathrm{m}}^2}}$$

(9)

$${\zeta }_h=\frac{K_{\mathrm{ce}}}{D_{\mathrm{m}}}\sqrt{\frac{{\beta }_{\mathrm{e}}{J}_{\mathrm{m}}}{V_{\mathrm{t}}}}+\frac{B_{\mathrm{m}}}{4D_{\mathrm{m}}}\sqrt{\frac{V_{\mathrm{t}}}{{\beta }_{\mathrm{e}}{J}_{\mathrm{m}}}}$$

(10)

$$K_{\mathrm{ce}}=K_{\mathrm{c}}+C_{\mathrm{m}}$$

(11)

Simultaneously, with Equations (6) and (8), the transfer function between the θ_m and T_L can be obtained as:

$$\frac{{\theta }_{\mathrm{m}}}{T_{\mathrm{m}}}=\frac{-\frac{K_{\mathrm{ce}}}{D_{\mathrm{m}}}\left(1+\frac{V_{\mathrm{t}}}{4{\beta }_{\mathrm{e}}{K}_{\mathrm{ce}}}s\right)}{\left(\frac{s^2}{w_h^2}+\frac{2{\zeta }_h}{w_h}s+1\right)}$$

(12)

Simultaneously, with Equations (8) and (12), the transfer function of the valve-controlled motor speed control system can be obtained, as shown in Figure 4.

The walking of the drilling robot is driven by the left and right hydraulic motors, which are transmitted through a mechanical gear reducer mechanism for proportional reduction. The deceleration transmission process can be described as:

$${\dot{\theta }}_{\mathrm{s}}=\frac{{\dot{\theta }}_{\mathrm{m}}}{{\mathrm{n}}_{\mathrm{m}}}$$

(13)

where ${\dot{\theta }}_{\mathrm{s}}$ represents the output speed acting on the track after being output by the reducer; $n_{\mathrm{m}}$ represents the reduction ratio of the reducer.

And then, the linear speed acting on the ground of the track can be described as:

$$\left\{\begin{array}{l}{v}_{\mathrm{jL}}={\dot{\theta }}_{\mathrm{s}}{R}_{\mathrm{sL}}\\ v_{\mathrm{jH}}={\dot{\theta }}_{\mathrm{s}}{R}_{\mathrm{sH}}\end{array}\right.$$

(14)

Based on Figure 4 and Equation (14), the mathematical model is constituted for the speed control of the valve-controlled hydraulic motor.

3. Design of Walking Trajectory Tracking Control Strategy

3.1. Design of the State Observer Based on Improved RBF Neural Network

In the walking process of the tracked chassis for the CMDR, the driving resistance force of F_rR, F_rL, lateral resistance force of F_yR, F_yL, the centrifugal force of F_C (F_Cx, F_Cy), and external disturbances are not easy to acquire because of the roadway being uneven and model uncertainties. Accordingly, a state observer is designed to estimate these uncertainties based on an improved radial basis function (RBF) neural network [37], which can be used to approximate the uncertainty items. The state observer based on an improved RBF neural network can be described as:

$$F_{f1}\left(x\right)=W_1^{\mathrm{T}} S{n}_1\left(x\right)+{\epsilon }_1$$

(15)

where F_f1(x) is the uncertainty item; x is the input values of the RBF neural network; ε₁ is the approximate error; W₁ = [w₁₁, w₁₂, w₁₃, …, w_1n]^T is the connection weights among network nodes; Sn₁(x) = [sn₁₁, sn₁₂, sn₁₃, …, sn_1n]^T is the radial basis function vector.

Additionally, the RBF can be described as:

$$S{n}_{1j}\left(x\right)=\exp \left(\frac{{‖x-{\sigma }_{1j}‖}_2}{2b_{1j}^2}\right)$$

(16)

where σ_1j(x) = [σ_11j, σ_12j, σ_13j]^T and b_1j are the center and width of the jth node, respectively; ${\sigma }_{1j}\in R, i=1,2,3, j=1,2,\cdots ,n$.

RBF networks can approximate uncertainty items by adjusting the hidden layer neurons and connection weights, and an inappropriate number of hidden layer neurons and connection weights will adversely affect the learning speed and training accuracy of the RBF network. Consequently, dynamically regulating the hidden layer neurons based on the data sample deviation from the central sample is proposed. The minimum distance can be described as:

$$l_1=\arg \underset{j}{\min }{‖x^{1i}-{\sigma }_{1j}‖}_2$$

(17)

where x¹ⁱ is the ith sample.

Additionally, the redundant neuron nodes can merge based on the node center distance among neurons. The merged neuron node center can be described as:

$${\overline{\sigma }}_{1p}=\frac{{\sigma }_{1p}+{\sigma }_{1q}}{2}$$

(18)

where ${\sigma }_{1p}$ and ${\sigma }_{1q}$ are any two-neuron node centers; when ${‖{\sigma }_{1p}-{\sigma }_{1q}‖}_2<{\omega }_1$, two neurons merge into one neuron, and ${\omega }_1$ is the determination coefficient.

Assuming the W₁ is an unknown term, the ${\widehat{W}}_1$ is the estimation value of the W₁, and the estimation error can be described as:

$${\tilde{W}}_1=W_1-{\widehat{W}}_1$$

(19)

Consequently, the estimation value of the uncertainty items can be described as:

$${\widehat{F}}_{f1}\left(x\right)={\widehat{W}}_1^{\mathrm{T}} S{n}_1\left(x\right)$$

(20)

3.2. Design of the Walking Longitudinal Trajectory Tracking Controller

For the walking trajectory control of the WECS, the accuracy of the walking longitudinal trajectory is a prerequisite for tracking its forward displacement and ensuring accurate movement. In this process, the actual walking trajectory of the drilling robot is determined by the differential speed controlled by the left and right hydraulic motors. When the left and right tracks travel at the same speed, the drilling robot will move in a straight trajectory. Correspondingly, the ideal walking trajectory is determined by the ideal speed $v_{\mathrm{jL}},v_{\mathrm{jH}}$ on the left and right hydraulic motors. Considering that the functions of the left and right hydraulic motors are the same, we take the left hydraulic motor as an example to establish a mathematical model. To this end, an integral sliding surface is designed to guarantee the control effectiveness and accuracy for the WECS.

$$s_c=e_{c1}+k_{c1}{\displaystyle {\int }_0^t{e}_{c1}d\mathrm{t}}$$

(21)

where s_c is the integral sliding surface; k_c1 > 0 is the coefficient of the sliding mode control; e_c1 is the speed error of the controlled hydraulic motor.

$$e_{c1}=v_{\mathrm{L}}-v_{\mathrm{L}\exp }$$

(22)

where v_Lexp and v_L represent the expected and actual speed, respectively.

We calculate the derivative of Equation (20), and then:

$${\dot{s}}_c={\dot{e}}_{c1}+k_{c1}{e}_{c1}$$

(23)

In addition, the saturation function is added in the sliding mode reaching law, and then:

$${\dot{s}}_c=-{\epsilon }_{c1}sat\left(s_c\right)-k_{c2}{s}_c$$

(24)

where ε_c1 > 1, and k_c2 > 0.

If the Lyapunov function is employed as:

$$V_1=\frac{1}{2}{s}_c^2$$

(25)

then the derivative of the Equation (25) is described as:

$$\begin{array}{l}{\dot{V}}_1={\dot{s}}_c{s}_c=s_c\left({\dot{e}}_{c1}+k_{c1}{e}_{c1}\right)=s_c\left({\dot{v}}_{\mathrm{L}}-{\dot{v}}_{\mathrm{L}\exp }+k_{c1}{e}_{c1}\right)\\ =s_c\left({\dot{\theta }}_{\mathrm{s}}{R}_{\mathrm{sL}}-{\dot{v}}_{\mathrm{L}\exp }+k_{c1}{e}_{c1}\right)\\ =s_c\left(g_1(x_{\mathrm{mv}},t)u_{\mathrm{mv}}{R}_{\mathrm{sL}}-{\dot{v}}_{\mathrm{L}\exp }+k_{c1}{e}_{c1}\right)\end{array}$$

(26)

The walking longitudinal trajectory tracking controller based on sliding mode control can be designed as:

$$u_{\mathrm{mv}}=\frac{1}{g(x_{\mathrm{mv}},t)R_{\mathrm{sL}}}\left({\dot{v}}_{\mathrm{L}\exp }-k_{c1}{e}_{c1}-k_{c3}{s}_c\right)$$

(27)

where k_c3 > 0.

Combing Equations (27) and (26),

$$\begin{array}{l}{\dot{V}}_1=s_c\left(g_1(x_{\mathrm{mv}},t)u_{\mathrm{mv}}{R}_{\mathrm{sL}}-{\dot{v}}_{\mathrm{L}\exp }+k_{c1}{e}_{c1}\right)\\ =s_c\left({\dot{v}}_{\mathrm{L}\exp }-k_{c1}{e}_{c1}-k_{c3}{s}_c-{\dot{v}}_{\mathrm{L}\exp }+k_{c1}{e}_{c1}\right)\\ =-k_{c3}{s}_c^2\end{array}$$

(28)

where ${\dot{V}}_1<0$ indicates the system is stable.

3.3. Design of the Walking Lateral Trajectory Tracking Controller

In the actual walking movement of the drilling robot, it is easily affected by the roadway being uneven and external disturbances, leading to walking lateral errors. For the yaw angle of the drilling robot, an integral sliding surface is designed as:

$$s_a=e_{a1}+k_{a1}{\displaystyle {\int }_0^t{e}_{a1}d\mathrm{t}}$$

(29)

where s_a is the integral sliding surface; k_a1 > 0 is the coefficient; e_a1 is the error of the controlled hydraulic motor.

$$e_{a1}={\phi }_{\mathrm{t}}-{\phi }_{\exp }$$

(30)

where φ_exp and φ_t represent the expected and actual heading angles, respectively.

We calculate the derivative of Equation (29), and then:

$${\dot{s}}_a={\dot{e}}_{a1}+k_{a1}{e}_{a1}$$

(31)

The saturation function is added in the sliding mode reaching law, and then:

$${\dot{s}}_a=-{\epsilon }_{a1}sat\left(s_a\right)-k_{a2}{s}_a$$

(32)

where ε_a1 > 1, k_a2 > 0.

If the Lyapunov function is designed as:

$$V_2=\frac{1}{2}{s}_a^2$$

(33)

then the derivative of the Equation (33) is obtained:

$$\begin{array}{l}{\dot{V}}_2={\dot{s}}_a{s}_a=s_a\left({\dot{e}}_{a1}+k_{a1}{e}_{a1}\right)=s_a\left({\dot{\phi }}_{\mathrm{t}}-{\dot{\phi }}_{\exp }+k_{a1}{e}_{a1}\right)\\ =s_a\left(g_2(x_{\mathrm{mv}},t)u_{\mathrm{mv}}-{\dot{\phi }}_{\exp }+k_{a1}{e}_{a1}\right)\end{array}$$

(34)

The walking lateral trajectory tracking controller based on sliding mode control can be designed as:

$$u_{\mathrm{mv}}=\frac{1}{g_2(x_{\mathrm{mv}},t)}\left({\dot{\phi }}_{\exp }-k_{a1}{e}_{a1}-k_{a3}{s}_a\right)$$

(35)

where k_a3 > 0.

We combine Equations (35) and (34), and then

$$\begin{array}{cc}\hfill {\dot{V}}_2& =s_a\left(g_2(x_{\mathrm{mv}},t)u_{\mathrm{mv}}-{\dot{\phi }}_{\exp }+k_{a1}{e}_{a1}\right)\hfill \\ & =s_a\left({\dot{\phi }}_{\exp }-k_{a1}{e}_{a1}-k_{a3}{s}_a-{\dot{\phi }}_{\exp }+k_{a1}{e}_{a1}\right)\hfill \\ & =-k_{a3}{s}_a^2\hfill \end{array}$$

(36)

where ${\dot{V}}_2<0$ represents that the system is stable.

Based on Equations (29) and (32), the adaptive sliding mode control law can be designed, and Equation (35) can be rewritten as:

$$u_{\mathrm{mv}}=\frac{1}{g_2(x_{\mathrm{mv}},t)}\left({\widehat{F}}_{f1}\left(x\right)+{\dot{\phi }}_{\exp }-k_{a1}{e}_{a1}-k_{a3}{s}_a-{\epsilon }_{a1}sat\left(s_a\right)-k_{a2}{s}_a\right)$$

(37)

The adaptive estimation of the connection weights W₂ in the function of $F_{f1}\left(x\right)$ can be designed as:

$${\dot{\widehat{W}}}_2=\frac{1}{F_{f1}\left(x\right)}{s}_{a}S{n}_2\left(x\right)$$

(38)

If the Lyapunov function is employed as:

$$V_3=\frac{1}{2}{s}_a^2+\frac{1}{2}{F}_{f1}\left(x\right){\tilde{W}}_2^{\mathrm{T}}{\tilde{W}}_2$$

(39)

The derivative of Equation (39) is obtained:

$$\begin{array}{l}{\dot{V}}_3={\dot{s}}_a{s}_a=s_a\left({\dot{e}}_{a1}+k_{a1}{e}_{a1}\right)-F_{f1}\left(x\right){\tilde{W}}_2^{\mathrm{T}}{\dot{\widehat{W}}}_2\\ =s_a\left({\dot{\phi }}_{\mathrm{t}}-{\dot{\phi }}_{\exp }+k_{a1}{e}_{a1}\right)-F_{f1}\left(x\right){\tilde{W}}_2^{\mathrm{T}}\dot{\widehat{W}}\\ =s_a\left(g_2(x_{\mathrm{mv}},t)u_{\mathrm{mv}}-{\dot{\phi }}_{\exp }+k_{a1}{e}_{a1}\right)-{\tilde{W}}_2^{\mathrm{T}}{s}_{a}S{n}_2\left(x\right)\\ =s_a\left(-k_{a3}{s}_a-{\epsilon }_{a1}sat\left(s_a\right)-k_{a2}{s}_a+{\widehat{F}}_{f1}\left(x\right)-{\tilde{W}}_2^{\mathrm{T}}S{n}_2\left(x\right)\right)\\ =s_a\left(-k_{a3}{s}_a-{\epsilon }_{a1}sat\left(s_a\right)-k_{a2}{s}_a\right)\end{array}$$

(40)

where ${\dot{V}}_3<0$ represents that the system is stable.

3.4. The Proposed Walking Trajectory Tracking Control Strategy

To realize the precise walking trajectory tracking control of the CMDR, the control output is often executed under operating conditions to accommodate the nonlinear motion characteristics of WECS. Speed and attitude sensors and electrohydraulic proportional directional valves are used to obtain the feedback signals and output the expected controlling trajectory. Finally, the proposed walking trajectory tracking control scheme with the proposed method is shown in Figure 5. Furthermore, the proposed control strategy improves the walking trajectory tracking control accuracy and robustness, and the saturation function is used to suppress the chattering phenomenon.

4. Results and Discussion

4.1. Main Parameters

A comparison of the proportion integration differentiation (PID), SMC, and the proposed walking trajectory tracking control based on sliding mode control with the state observer (SMC-SO) was conducted for the purpose of verifying the control accuracy and effectiveness. A simulation model is established based on AMESim and MATLAB/Simulink. The main parameters of the WECS are listed in

Table 1. Main parameters of the WECS.

Parameters	Characteristics	Values
Kq	Flow gain	1.5 × 10−10 m3/(s∙MPa)
Kc	Flow–pressure gain	4.6 × 10−5 m3/(s∙MPa)
Jm	Inertia moment	1.28 Kg/m2
Dm	Motor displacement	1.6 × 10−4 m3/rad
Bm	Viscous damping coefficient	0.353 N·m·s/rad
Gm	Torsional stiffness	1.2 N·m/rad
Jm	Inertia moment	1.28 Kg/m2
ρ	Hydraulic oil density	900 Kg/m3
βe	bulk modulus	1.7 × 109 Pa

. The controller parameters of the PID are configured as k_P = 13, k_I = 8, k_D = 0.06; the controller parameters of the SMC-SO are configured as ε_c1 = 2, k_c1 = 0.6, k_c2 = 8, k_c3 = 8, ε_a1 = 2, k_a1 = 0.6, k_a2 = 8, k_a3 = 8.

4.2. Simulation Results and Analysis

4.2.1. Walking Longitudinal Trajectory Tracking and Analysis

Figure 6 illustrates a comparison of the PID, SMC, and SMC-SO methods in tracking step signal to analyze the walking longitudinal tracking performance. Figure 6a shows that the steady-state time of the PID, SMC, and SMC-SO is 2.07 s, 0.87 s, and 0.58 s, respectively. Figure 6b shows that the X-direction trajectory tracking average absolute errors of the PID, SMC, and SMC-SO are 12.66 mm, 8.84 mm, and 7.86 mm, respectively, which indicates that the SMC-SO has a better tracking performance, and the longitudinal trajectory tracking average absolute error of the SMC-SO is 37.91% and 11.08% smaller than the PID and SMC, respectively. Figure 6c shows that the Y-direction trajectory tracking average absolute errors of the PID, SMC, and SMC-SO are 42.83 mm, 26.39 mm, and 17.88 mm, respectively. It can be indicated that the SMC-SO has a better tracking performance in the Y-direction trajectory, and the trajectory tracking average absolute error of the SMC-SO is 58.25% and 32.24% smaller than the PID and SMC, respectively. Figure 6d shows that the heading angle average absolute errors of the PID, SMC, and SMC-SO are 0.26 rad, 0.17 rad, and 0.11 rad, respectively. It can be shown that the heading angle can be controlled at a smaller angle than the PID and SMC, and the heading angle average absolute error of the SMC-SO is 57.69% and 35.29% smaller than the PID and SMC, respectively. Consequently, the SMC-SO has a better walking longitudinal trajectory tracking performance than the PID and SMC.

4.2.2. Walking Lateral Trajectory Tracking Performance and Analysis

Figure 7 illustrates a comparison of the PID, SMC, and SMC-SO methods in tracking a given curve signal to analyze the walking lateral trajectory tracking performance. Figure 7a shows that the steady-state time of the PID, SMC, and SMC-SO is 0.58 s, 0.41 s, and 0.32 s, respectively. Figure 7b shows that the X-direction trajectory tracking average absolute errors of the PID, SMC, and SMC-SO are 18.36 mm, 11.65 mm, and 10.08 mm, respectively, which indicates that the SMC-SO has a better tracking performance, and the lateral trajectory tracking average absolute error of the SMC-SO is 36.54% and 13.47% smaller than the PID and SMC, respectively. Figure 7c shows that the Y-direction trajectory tracking average absolute errors of the PID, SMC, and SMC-SO are 45.94 mm, 26.66 mm, and 19.16 mm, respectively. It can be indicated that the SMC-SO has a better tracking performance in the Y-direction trajectory, and the trajectory tracking average absolute error of the SMC-SO is 58.29% and 28.13% smaller than the PID and SMC, respectively. Figure 7d shows that the heading angle average absolute errors of the PID, SMC, and SMC-SO are 0.24 rad, 0.17 rad, and 0.13 rad, respectively. It can be shown that the heading angle can be controlled at a smaller angle than the PID and SMC, and the heading angle average absolute error of the SMC-SO is 45.83% and 23.52% smaller than the PID and SMC, respectively. Based on these simulation results, the designed SMC-SO can improve the walking trajectory tracking control performance, which has a strong robustness to reject the disturbance.

4.3. Experimental Results and Analysis

4.3.1. Experimental Platform of the WECS

Figure 8 shows the experimental platform for validating the designed walking trajectory tracking controller for the WECS. The designed walking trajectory tracking controller was employed utilizing PLC 1511C, including speed and attitude sensors information acquisition, electrohydraulic proportional directional valve signal output, proposed method calculating, and the sampling frequency of the speed tracking loop is 1 kHz. Additionally, the steering speed is a constant speed when conducting testing experiments, and the walking trajectory tracking performance tests were carried out to validate the practicability and effectiveness of the designed controller for the WECS.

4.3.2. Walking Longitudinal Trajectory Tracking Performance Verification

To verify the walking longitudinal trajectory tracking performance of the proposed SMC-SO method for the WECS, Figure 9 illustrates a comparison of the PID, SMC, and SMC-SO methods in tracking a given straight line signal. Figure 9a shows that the steady-state time of the PID, SMC, and SMC-SO for the walking speed is 2.11 s, 1.07 s, and 0.93 s, respectively. Figure 9b shows that the X-direction trajectory tracking average absolute errors of the PID, SMC, and SMC-SO are 16.37 mm, 12.14 mm, and 10.23 mm, respectively, which indicates that the SMC-SO has a better tracking performance. The longitudinal trajectory tracking average absolute error of the SMC-SO is 10.23 mm, which is 37.51%, 15.73% less than the PID and SMC. Figure 9c shows that the Y-direction trajectory tracking average absolute errors of the PID, SMC, and SMC-SO are 26.91 mm, 15.95 mm, and 11.59 mm, respectively. It can be indicated that the SMC-SO has a better tracking performance in the Y-direction trajectory, and the trajectory tracking average absolute error of the SMC-SO is 56.93% and 15.22% smaller than the PID and SMC, respectively. Figure 9d shows that the heading angle average absolute errors of the PID, SMC, and SMC-SO are 0.29 rad, 0.18 rad, and 0.14 rad, respectively. It can be shown that the heading angle can be controlled at a smaller angle than the PID and SMC, and the heading angle average absolute error of the SMC-SO is 51.72% and 22.22% smaller than the PID and SMC, respectively. Therefore, the proposed SMC-SO can further improve the walking longitudinal trajectory tracking performance.

4.3.3. Walking Lateral Trajectory Tracking Performance Verification

To verify the walking lateral trajectory tracking performance of the proposed SMC-SO method for the WECS, Figure 10 illustrates a comparison of the PID, SMC, and SMC-SO methods in tracking a given curve line signal. Figure 10a shows that the steady-state time of the PID, SMC, and SMC-SO is 0.77 s, 0.54 s, and 0.48 s, respectively. Figure 10b shows that the X-direction trajectory tracking average absolute errors of the PID, SMC, and SMC-SO are 27.31 mm, 16.27 mm, and 14.59 mm, respectively, which indicates that the SMC-SO has a better tracking performance, and the lateral trajectory tracking average absolute error of the SMC-SO is 46.57% and 10.33% smaller than the PID and SMC, respectively. Figure 10c shows that the Y-direction trajectory tracking average absolute errors of the PID, SMC, and SMC-SO are 46.42 mm, 28.97 mm, and 22.34 mm, respectively, which can indicate that the SMC-SO has a better tracking performance in the Y-direction trajectory. The lateral trajectory tracking average absolute error of the SMC-SO is 22.34 mm, which is 51.87% and 22.89% less than the PID and SMC. Figure 10d shows that the heading angle average absolute errors of the PID, SMC, and SMC-SO are 0.36 rad, 0.21 rad, and 0.19 rad, respectively. It can be shown that the heading angle can be controlled at a smaller angle than the PID and SMC, and the heading angle average absolute error of the SMC-SO is 47.22% and 9.52% smaller than the PID and SMC, respectively. Therefore, it can be indicated that the proposed SMC-SO has a better walking lateral trajectory tracking performance than the PID and SMC, which can effectively improve the walking trajectory tracking capability of the WECS.

5. Conclusions

A walking trajectory tracking control scheme for the WECS using the SMC-SO is proposed. Targeting the walking trajectory tracking deviation of the WECS for the CMDR, an SMC-SO method using a state observer, was proposed to realize system rapid response characteristics and disturbance rejection performance. Then, the improved RBF is employed to approximate the uncertainties. In addition, the saturation function is used to reduce the sliding mode chattering phenomenon. Furthermore, simulation and experimental results indicate that the proposed method has a better walking trajectory tracking performance and stronger robustness than the comparison methods. Meanwhile, the walking longitudinal and lateral trajectory performance of the SMC-SO is verified by the comparative simulation and experimental results under different operating conditions. In the future, the proposed method could also be applied to speed or position tracking control in other systems, such as the electrohydraulic rotary system of a geological exploration drilling rig, electrohydraulic drive system of the heavy mobile equipment, etc. Additionally, a more detailed chattering suppression method, disturbance rejection control approach, and more detailed trajectory tracking error analysis strategy could be explored in future research.