A Cooperative Control Method for Excavation Support Robot with Desired Position/Posture

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An improved control method of desired position/posture coordinated adjustment by real-time posture data and sliding position mode control algorithm is proposed. A mathematical model of the non-longitudinal adjustment hydraulic system is presented, the coupling relationship between the stroke and posture of the lateral Adjustment Hydraulic Cylinder (AHC) considering the kinematic characteristics of the Excavation Support Robot (ESR) is deduced, and the controller is designed for optimization analysis. The simulation model desired position/posture adjustment control system is determined. The proposed control algorithm is verified by experiments, and the position deviation accuracy and posture adjustment coordination are greatly improved.

Abstract

Desired deviation of the position/posture of Excavation Support Robot (ESR) have a critical impact on the efficiency of underground mining. In this paper, a control method of desired position/posture coordinated adjustment by real-time posture data and sliding position mode control algorithm is proposed. The action stroke of the non-longitudinal Adjustment Hydraulic Cylinder (AHC) is controlled by adaptively adjusting the opening and time of the Electro-Hydraulic Servo (EHS) valve, thereby adjusting the position offset and posture coordination of the ESR. The mathematical model of hydraulic system is deduced. Considering the kinematic characteristics of the ESR and the coupling relationship between the stroke and posture of the lateral AHC, a coupled kinematics model of the robot and the support body is derived. The overall structure of the posture adjustment control system and the calculation method of the stroke adjustment amount of the AHC are presented, and the controller is further optimized. The simulation analysis model and test verification system of the desired posture adjustment control system are determined. These results show that the control method is effective and significant compared with the manual control system, which can meet the functional and precision requirements of position deviation and posture adjustment.

1. Introduction

An advanced hydraulic unit is the Excavation Support Robot (ESR). A scenario simulation diagram of the actual operation of the ESR is illustrated in Figure 1. The Adjustment Hydraulic Cylinder (AHC) is composed of a driving system as the ESR power actuator, and the excavation is alternately advanced. In particular, the roof supporting the excavation roadway is irreplaceable. Considering the complex construction road environment and the operation factors of on-site personnel, ESR is prone to position deviation and posture offset during actual operation. The accumulated consequences of position/posture deviation can easily lead to accidents such as side collision and overturning, which seriously affects mining efficiency and excavation safety. Therefore, whether the position/posture deviation of ESR can be adjusted automatically and quickly to achieve the expected collaborative value has become a key technical problem to be solved urgently.

Within the aforementioned context, the deviation of the position and the quick adjustment of the posture are the technical bottlenecks in the field of engineering control. Scholars have achieved fruitful results in the field of coordinated adjustment of position/posture [1,2]. The ESR posture adjustment control algorithm and the integral sliding mode control controller are designed to correct and adjust the position/posture deviation generated by the ESR during the execution of the task, which makes the deviation correction error converge with a negative exponent [3,4]. A memory network neural immune control algorithm for rapid posture adjustment of manned exploration device is proposed [5]. Based on the PID fuzzy algorithm with the optimal deviation path, the overall control scheme has a simple structure and a small amount of calculation, which can realize the optimal control of the position/posture action coordination [6,7,8]. A basic posture adjustment algorithm for singular robust path planning is expounded, so as to realize the rapid coordinated adjustment of the basic posture of space satellites [9,10]. A robust model-free adaptive control algorithm and strategy based on adaptive backward technology is discussed, which is oriented to the position/posture adjustment problem of quadrotor aircraft [11,12]. An adaptive sliding mode control method based on the neural network minimum parameter learning method is presented, which improves the convergence speed and better meets the requirements of real-time control of the magnetic levitation system [13]. A new coupled sliding mode control is investigated, which plays an important role in pendulum stability control of an inverted pendulum system [14]. A sliding mode control method for uncertain systems with nonlinear disturbance observer is proposed, which has better integral performance than the basic sliding mode control [15]. Considering the robust sliding surface configuration with the optimal coefficient ratio, a sliding surface algorithm based on the optimal coefficient ratio and an integral sliding mode robust tracking control method are proposed. Even in the presence of design parameter disturbances, the expected sliding surface obtained by this method is robust and optimal [16]. Several optimization algorithms in system optimization are mentioned, which make the control system have higher convergence accuracy and faster convergence speed [17,18,19]. Considering the problem of ESR posture adjustment in the process of excavation, an optimized plane linear correction curve model and a double closed-loop posture control strategy are expounded to correct the working posture of the single hydraulic support in the fully mechanized mining face [20,21,22]. Faced with complex driving conditions, a motion control algorithm for driving deviation correction is proposed, which can realize online real-time deviation correction position adjustment of control parameters and real-time position control of driving deviation [23,24,25]. Considering the inclined staggered posture of the fully mechanized mining hydraulic support, a theoretical model of inclination deviation adjustment is established, and an independent inclination deviation adjustment control method is proposed. In the initial support stage, the balance cylinder and cylindrical cylinder are used to adjust the posture of the top beam and the position of the deflection base, which can achieve precise support position/posture coordinated control [26,27,28]. To sum up, the current research mainly focuses on the fully mechanized mining face robot, vehicle-mounted equipment, and single hydraulic support. The research on ESR position/posture adjustment control method and collaborative strategy in fully mechanized mining and excavation is still in the preliminary exploration stage and needs to be revealed in depth. The execution of ESR action mainly relies on the centralized liquid supply of the hydraulic servo system and the manual control and adjustment of the hydraulic valve. Restricted by factors such as complex roadway environment and subjective judgment of personnel, ESR position deviation/posture bias is a thorny problem that always exists.

This topic is a more in-depth discussion based on the technical accumulation of ESR position/posture monitoring research results. Based on the ESR position/posture test data, ESR position deviation correction and posture offset adjustment are guided. Furthermore, an ESR desired position/posture coordinated adjustment control method based on real-time position/posture data and sliding mode control algorithm is proposed. The action stroke of the non-longitudinal hydraulic cylinder is controlled and coordinated according to the expected position/posture adjustment value, and the real-time deviation position and posture are quickly adjusted to ensure that the ESR reaches the optimal state of the expected position/posture coordination.

The subject is organized as follows: the multiple models of ESR, including structure, hydraulic servo system and kinematic models in Section 2, are presented. Section 3 focuses on the description of the proposed controlled structure of the position/posture adjustment system, and proposes a coordinated adjustment strategy for ESR position deviation, posture bias, and optimized controller design. In Section 4, the numerical analysis of the coordinated position/posture adjustment is presented., The field experiments are verified in Section 5, where the results are discussed and the corresponding conclusions are drawn, thus validating the cooperative control method proposed in this study.

2. ESR System Modeling and Description

2.1. Structural Model of ESR

The ESR motion process has six degrees of freedom, namely, lower left support, push left support, raise left support, lower right support, push right support, and raise right support. The ESR motion process is schematically shown in Figure 2. The detailed mechanical structure of ESR is illustrated in Figure 3. Herein, the upper lateral AHC is represented by U1–U3, the lower lateral AHC is indicated with U4–U5, and the cylindrical AHC is marked as C1–C8. The geometrical dimensions of the core structure of the ESR are shown in Figure 4, where, the length of a single ERS is indicated as L, the original value of the cylindrical AHC length is La0, and its extension maximum is La1. The original value of the lateral AHC length is Lb0 and its extension maximum is Lb1. The minimum value corresponding to the lateral AHC length of the labels U1–U5 is Lb0, and the maximum value is Lb0 + Lb1. The minimum value of the cylindrical AHC length of the labels C1–C8 is La0, and the maximum value is La0 + La1.

2.2. Hydraulic System Model with AHC for ESR

Considering the workload and other on-site environmental factors, a digital AHC cannot be used effectively for excavation operations. An ESR with a cylindrical AHC working stroke is regulated by controlling the opening degree of the Electro-Hydraulic Servo (EHS) valve and its opening time. According to the mechanical structure and movement form of the ESR, the two cylindrical AHCs on one side are connected to each other. The lateral cylindrical AHC works alone. A model of the hydraulic system with AHC for ESR is illustrated in Figure 5. As mentioned above, the lateral AHCs belong to the single-stage hydraulic cylinder. The EHS reversing valve is identified as an ideal spool valve structure, which makes the return oil pressure of the spool valve zero; the oil supply pressure of the whole system is stable, and is not affected by the oil in the spool valve cavity. The hydraulic system with the lateral AHC for ESR is indicated in Figure 6.

In Figure 6, three sets of balance equations are listed, such as the EHS diverter valve flow equation, the lateral AHC flow continuity equation, and the force balance equation with external load.

(1) The EHS diverter valve flow equation [29] is expressed as

$$Q_L=K_q{x}_v-K_c{p}_L$$

(1)

where $K_q$ is expressed as the flow gain of the EHS valve (directional valve) $({\mathrm{m}}^{-3}\cdot {\mathrm{s}}^{-1}/\mathrm{m})$, $x_v$ is the stroke (displacement) of the EHS valve spool (mm), $K_c$ indicates the flow pressure coefficient of the EHS valve $\left({\mathrm{m}}^3/\left(\mathrm{s}/\mathrm{Pa}\right)\right)$, and $p_L$ represents the lateral AHC pressure (Pa).

(2) The fluid continuity equations [30] for lateral AHC

The fluid flowing into the lateral AHC rodless cavity of the ESR is $Q_i$, which is expressed as:

$$Q_i=A_p\frac{d{x}_p}{dt}+\frac{V_i}{{\beta }_e}\frac{d{p}_i}{dt}+C_{pi}(p_i-p_0)+C_{pe}{p}_i$$

(2)

where $A_p$ represents the effective area of the AHC rod-end piston $({\mathrm{m}}^2)$, $x_p$ is the travel stroke (displacement) of the AHC piston $(\mathrm{m})$, $p_i$ is the oil pressure in the AHC rodless cavity $\left(\mathrm{Pa}\right)$, $p_0$ is the oil pressure of the AHC rod cavity $\left(\mathrm{Pa}\right)$, $C_{pi}$ is the leakage coefficient in the laterally adjusted EHS cylinder, $C_{pe}$ is the external leakage coefficient of the laterally adjusted EHS cylinder, and $V_i$ is the volume in the rodless cavity of the AHC ${(\mathrm{m}}^3)$; herein, $V_i=V_{i0}+A_p{x}_p$, ${\beta }_e$ is the elastic modulus of the effective volume $\left(\mathrm{Pa}\right)$.

The flow rate in the rod cavity of the lateral AHC of ESR is $Q_0$, which is expressed as:

$$Q_i=A_p^{\prime }\frac{d{x}_p}{dt}-\frac{V_0}{{\beta }_e}\frac{d{p}_0}{dt}+C_{pi}(p_i-p_0)-C_{pe}{p}_i$$

(3)

where $A_p^{\prime }$ is the effective area of the AHC rod end piston $({\mathrm{m}}^2)$, $A_p^{\prime }=A_p-\pi r^2$, and $V_0$ is the volume inside the AHC rodless cavity ${(\mathrm{m}}^3)$; herein, $V_0=V_0^{\prime }+A_p^{\prime }{x}_p$.

Set $Q_L=\frac{Q_i+Q_0}{2}$, $p_L=p_i-p_0$, combining Equations (2) and (3), leading to

$$Q_L=(A_p-\pi r^2)\frac{d{x}_p}{dt}+(C_{pi}+\frac{C_{pe}}{2})p_L+0.5{\beta }_e\left[\left(V_{i0}\frac{d{p}_i}{dt}-V_0^{\prime }\frac{d{p}_0}{dt}\right)+\left(A_p{x}_p\frac{d{p}_i}{dt}+A_p^{\prime }{x}_p\frac{d{p}_0}{dt}\right)\right]$$

(4)

Then, set to equalize the compression flow in the hydraulic cylinder, that is, $V_{i0}=V_0^{\prime }=V_0$. Assuming that $V_t=\frac{V_{i0}}{2}=\frac{V_0^{\prime }}{2}$ holds, with this in mind, $A_p{x}_p\ll V_0$,$A_p^{\prime }{x}_p\ll V_0$, with $\frac{d{x}_p}{dt}\approx 0$,$\frac{d{p}_0}{dt}\approx 0$. Equation (4) is simplified to:

$$Q_L=(A_p-\pi r^2)\frac{d{x}_p}{dt}+\frac{V_t}{4{\beta }_e}\frac{d{p}_L}{dt}+(C_{pi}+\frac{C_{pe}}{2})p_L$$

(5)

Further sorting out Equation (5) leads to:

$$Q_L=A_{pe}\frac{d{x}_p}{dt}+\frac{V_t}{4{\beta }_e}\frac{d{p}_L}{dt}+C_{pt}{p}_L$$

(6)

where $A_{pe}$ indicates the equivalent area of the AHC piston, $A_{pe}=A_p-\pi r^2$, $x_p$ represents the (displacement) stroke of the AHC piston, and $C_{pt}$ denotes the total leakage coefficient.

(3) The expression of force balance equation with external load is rewritten as

$$A_{pe}{p}_L=m_t\frac{d^2{x}_p}{dt}+c_t\frac{d{x}_p}{dt}+k_t{x}_p$$

(7)

where $m_t$ represents the effective total mass of the ESR $\left(\mathrm{kg}\right)$, $c_t$ is the viscous damping coefficient of the piston in a transverse AHC with an external load $(\mathrm{N}\cdot \mathrm{m}/\mathrm{s})$, and $k_t$ denotes stiffness coefficient with an external load $\left(\mathrm{N}/\mathrm{m}\right)$.

Equations (1), (6), and (7) are expanded by the Laplace Transform, leading to:

$$\left\{\begin{array}{l}{Q}_L(s)=K_q{x}_v(s)-K_c{p}_L(s)\\ Q_L(s)=A_{pe}s{x}_p(s)+\frac{V_t}{4{\beta }_e}s{p}_L(s)+C_{pt}{p}_L(s)\\ p_L(s)=\frac{(m_t{s}^2+c_{t}s+k_t)}{A_{pe}}{x}_p(s)\end{array}\right.$$

(8)

Notice that the spool displacement is the input and the piston displacement of the lateral AHC is the output. With this in mind, the transfer function of the system is expressed as:

$$G(s)=\frac{K_p}{\frac{m_t{V}_t}{4A_{pe}{\beta }_e}{s}^3+\left(\frac{c_t{V}_t}{4A_{pe}{\beta }_e}+\frac{(K_c+c_{pt})m_t}{A_{pe}}\right)s^2+\left(A_{pe}+\frac{(K_c+c_{pt})c_t}{A_{pe}}+\frac{k_t{V}_t}{4A_{pe}{\beta }_e}\right)s+\frac{(K_c+c_{pt})k_t}{A_{pe}}}$$

(9)

Assuming that $K_{cpt}=K_c+c_{pt}$ holds gives the simplified equation:

$$G(s)=\frac{K_p}{\frac{m_t{V}_t}{4A_{pe}{\beta }_e}{s}^3+\left(\frac{c_t{V}_t}{4A_{pe}{\beta }_e}+\frac{K_{cpt}{m}_t}{A_{pe}}\right)s^2+\left(A_{pe}+\frac{K_{cpt}{c}_t}{A_{pe}}+\frac{k_t{V}_t}{4A_{pe}{\beta }_e}\right)s+\frac{K_{cpt}{k}_t}{A_{pe}}}$$

(10)

Considering that the viscous damping coefficient $c_t$ of the lateral AHC piston under the external load is negligible, then $\frac{c_t{K}_{cpt}}{A_{pe}^2}\ll 1$ is satisfied. Simplified Equation (10) reads as

$$G(s)=\frac{K_p/A_{pe}}{\frac{m_t{V}_t}{4{\beta }_e{A}_{pe}^2}{s}^3+\left(\frac{c_t{V}_t}{4\beta A_{pe}^2{}_e}+\frac{K_{cpt}{m}_t}{A_{pe}^2}\right)s^2+\left(1+\frac{k_t{V}_t}{4{\beta }_e{A}_{pe}^2}\right)s+\frac{K_{cpt}{k}_t}{A_{pe}^2}}$$

(11)

Let $\sqrt{\frac{m_t{V}_t}{4{\beta }_e{A}_{pe}^2}}=\frac{1}{{\omega }_t},\frac{K_{cpt}}{A_{pe}}\sqrt{\frac{m_t{\beta }_e}{V_t}}={\xi }_t,\frac{4A_p^2{\beta }_e}{V_t}=K_h$ hold, and the simplified Formula (11) is expressed as

$$G(s)=\frac{K_p/A_{pe}}{\frac{1}{{\omega }_t^2}{s}^3+\frac{2{\xi }_t}{{\omega }_t}{s}^2+\left(1+\frac{k_t}{K_h}\right)s+\frac{K_{cpt}{k}_t}{A_{pe}^2}}$$

(12)

Set the assumption that $\frac{K_{cpt}\sqrt{m_t{k}_t}}{(1+\frac{k_t}{K_h})A_{pe}^2}\ll 1$ holds, further simplifying Equation (12), and is written as:

$$G(s)=\frac{K_p/A_{pe}}{\left[\left(1+\frac{k_t}{K_h}\right)s+\frac{K_{cpt}{k}_t}{A_{pe}^2}\right]\left(\frac{1}{{\omega }_{tn}^2}{s}^2+\frac{2{\xi }_{tn}}{{\omega }_{tn}}s+1\right)}$$

(13)

where ${\omega }_{tn}$ represents the system integrated natural frequency, and ${\omega }_{tn}={\omega }_t\sqrt{1+\frac{k_t}{K_h}}$ holds, ${\xi }_{tn}$ is the system integrated damping ratio, and ${\xi }_{tn}=\frac{1}{2{\omega }_{tn}}\left(\frac{4K_{cpt}{\beta }_e}{(1+\frac{k_t}{K_h})V_t}+\frac{c_t}{m_t}\right)$ holds.

As mentioned previously, the poles of the system are expressed as $s=-\frac{K_{cpt}{k}_t{K}_h}{(K_{cpt}+k_t)A_{pe}^2},$ $s=-{\omega }_{tn}({\xi }_{tn}\pm \sqrt{{\xi }^2{}_{tn}-1})$. If the spool motion stroke $x_v$ of the EHS reversing valve is a step signal, the piston displacement $x_p$ of the lateral AHC reads

$$x_p(s)=\frac{K_p/A_{pe}}{\left[\left(1+\frac{k_t}{K_h}\right)s^2+\frac{K_{cpt}{k}_t}{A_{pe}^2}s\right]\left(\frac{1}{{\omega }_{tn}^2}{s}^3+\frac{2{\xi }_{tn}}{{\omega }_{tn}}{s}^2+s\right)}$$

(14)

Take Equation (14) as the inverse Laplace transform, leading to:

$$x_p(t)=-\left[\frac{C_1}{C_2}+\frac{C_1{C}_3{e}^{-{\omega }_{tn}{\xi }_{tn}t}\left(\cos C_5-\frac{\sin C_5(C_2{\omega }_{tn}{\xi }^2{}_{tn}-C_3{\xi }_{tn}-C_2{\omega }_{tn})}{C_6\sqrt{{\xi }_{tn}-1}}\right)}{C_4}+\frac{C_1{C}_2^2}{C_3}{e}^{\frac{C_3}{C_2}t}\right]$$

(15)

where $C_1=K_p/A_{pe}$, $C_2=1+\frac{k_t}{K_h}$, $C_3=\frac{K_{cpt}{k}_t}{A_{pe}^2}$, $C_4=-\left[{(1+\frac{k_t}{K_h})}^2{\omega }_{tn}^2+2{\xi }_{tn}{\omega }_{tn}(1+\frac{k_t}{K_h})\frac{K_{cpt}{k}_t}{A_{pe}^2}-\frac{K_{cpt}^2{k}_t^2}{A_{pe}^4}\right]$,$C_5=\left({\omega }_{tn}\sqrt{{\xi }_{tn}^2-1}\right)t$, $C_6=-\left[2{\omega }_{tn}{\xi }_{tn}(1+\frac{k_t}{K_h})-\frac{K_{cpt}{k}_t}{A_{pe}^2}\right]$.

Assuming that the current $I$ denotes the input parameter, the spool displacement xv of the EHS reversing valve is the output parameter, and the transfer function of the EHS reversing valve is expressed as:

$$G(s)=\frac{x_v}{I}=\frac{K_{tv}/K_p}{\frac{1}{{\omega }_{tv}^2}{s}^2+2\frac{{\xi }_{tv}}{{\omega }_{tv}}s+1}$$

(16)

where $I$ represents the current input parameter, $K_{tv}$ denotes the flow gain coefficient of the EHS reversing valve, ${\xi }_{tv}$ is the damping ratio of EHS directional valve, and ${\omega }_{tv}$ is the natural frequency of EHS directional valve.

2.3. Kinematic Model with AHC for ESR

The ESR proposed in this study is based on the extension and retraction of the lateral AHC to adjust position/posture. The coupled motion model of the ESR and the support body is illustrated in Figure 7, where the scroll angles of ESR are set as ${\gamma }_L$ and ${\gamma }_R$, and the declination angles are set as ${\alpha }_L$ and ${\alpha }_R$. The distances from the position key points to the defined edges are, respectively, identified as $d_1$ to $d_8$. The specified space width is $B$ and the height is $H$. The expected distances between the ESR and the two edge sidewalls are identified as $d_{Le}$ and $d_{Re}$. The overall longitudinal length of the ESR is $L$, and the lateral width is $b$. The distances from the support point of the core position to its center point are respectively calibrated $m$ and $n$. Taking the geometric center points ($x_{L0},y_{L0},Z_{L0}$) and ($x_{R0},y_{R0},Z_{R0}$) of the unilateral ESR as the origin of the body coordinate system, the ESR body coordinate systems ($O_L-X_L{Y}_L{Z}_L$) and ($O_R-X_R{Y}_R{Z}_R$) are both feasible in the kinematic model. The geometric center of the roadway section is set as the origin, the direction of the roadway is the $X_0$ axis direction of the coordinate system, and the direction perpendicular to the side of the roadway is the axis direction of the coordinate system. The right-hand rule determines the direction of the $Z_0$ axis coordinate system, which follows the roadway absolute coordinate system ($O_0-X_0{Y}_0{Z}_0$). In the ESR ontology coordinate system, the left support is taken as an example to illustrate, and its position key point coordinates are marked ($m,B/2,H/2$), ($-n,B/2,H/2$), ($m,B/2,-H/2$), and ($-n,B/2,-H/2$).

The numerical algorithms for roll and declination angles are written as

$$\left\{\begin{array}{l}{\alpha }_L=\arccos \frac{(m+n)}{\sqrt{{(m+n)}^2+{(d_{i+1}-d_i)}^2}}(i=1,3)\\ {\alpha }_R=\arccos \frac{(m+n)}{\sqrt{{(m+n)}^2+{(d_{i+1}-d_i)}^2}}(i=5,7)\end{array}\right.$$

(17)

$$\left\{\begin{array}{l}{\gamma }_L=\arccos \frac{H\cos {\gamma }_L}{\sqrt{H^2{\cos }^2{\gamma }_L+{(d_3-d_1)}^2{\cos }^2{\gamma }_L}}=\arccos \frac{H}{\sqrt{H^2+{(d_{i+2}-d_i)}^2}}(i=1,2)\\ {\gamma }_R=\arccos \frac{H\cos {\gamma }_L}{\sqrt{H^2{\cos }^2{\gamma }_L+{(d_3-d_1)}^2{\cos }^2{\gamma }_L}}=\arccos \frac{H}{\sqrt{H^2+{(d_{i+2}-d_i)}^2}}(i=5,6)\end{array}\right.$$

(18)

The main view based on the kinematics model can be derived, which is expressed by the distance $x_L$ from the coordinate origin $O_L$ to the left edge of the roadway side

$$x_L=\left(\frac{m}{(m+n)}\left(d_1+d_3\right)+\frac{b}{2}\right)\cos {\alpha }_L$$

(19)

Then, the distance $x_R$ from the coordinate origin $O_R$ to the right edge of the roadway is expressed as

$$x_R=\left(\frac{m}{(m+n)}\left(d_3+d_4\right)+\frac{b}{2}\right)\cos {\alpha }_R$$

(20)

The distance from the core position key point to the space edge is written as

$$x_i=(b+d_i)\cos \alpha (i=1-8)$$

(21)

When solving for the distance to the left of the edge, set $\alpha ={\alpha }_L$ to hold. When solving the distance to the right side of the edge, set $\alpha ={\alpha }_R$ to hold. The deflection angle ${\delta }_i$ of the lateral AHC is expressed as:

$${\delta }_i=\arccos (\frac{B-x_i-x_{i+4}}{H_i}) (i=1-4)$$

(22)

The distance difference $d_x$ between the two sides edge along the X-axis is written as:

$$d_x=H_2\sin {\delta }_2+\left(\frac{L}{2}-n\right)(\cos {\alpha }_L-\cos {\alpha }_R)-\frac{b}{2}(\sin {\alpha }_L-\sin {\alpha }_R)$$

(23)

The top view in the kinematics model shows that the distance $y_L$ from the coordinate origin $O_L$ to the left side of the edge denotes:

$$y_L=\frac{(b+d_2+d_4)\cos {\gamma }_L}{2}$$

(24)

Then, the distance $y_R$ from the coordinate origin $O_R$ to the right-side edge of denotes:

$$y_R=\frac{(b+d_6+d_8)\cos {\gamma }_R}{2}$$

(25)

The distance from the core key position point to space edge side is written as follows:

$$y_i=(d_i+b)\cos \gamma (i=1-8)$$

(26)

Herein, when solving for the distance to the left of the edge, let $\gamma ={\gamma }_L$ hold. When solving for the distance to the right of the edge, setting $\gamma ={\gamma }_R$ holds.

3. Cooperative Control Method

3.1. Overall Controlled Structure Model of the ESR System

When the ESR system is running, the dynamic posture monitoring system is subject to position information. The desired position of the ESR and the real-time position captured by monitoring are identified as inputs to the control system. The desired posture control system brings the ESR to the desired position/posture due on the coordinated adjustment between the ESR position and the lateral AHC stroke. The general configuration of the desired position/posture adjustment control system for is illustrated in Figure 8.

Figure 8 describes the three processes of ESR: position/posture detection, position/posture discrimination and position/posture adjustment. The sliding mode controller controls the EHS valve to adjust multiple lateral AHC strokes according to the set desired value, which dynamically adjusts the ESR even in the presence of deviations.

3.2. Position/Posture Detailed Description for ESR

Considering the space environment conditions in which the ESR is located and its working process constraints, the position/posture deviation of different states is caused. The position/posture morphological motion trend sets of ESR are represented in Figure 9.

As shown in Figure 9, in the pre-set presentable state, the declination and scroll angles of the ESRs on both sides are both 0°, and the distance from the edge sidewall is within the expected limit. Under the influence of the space environment and ESR motion, the ESRs on both sides of the edge are prone to different degrees of deviation such as declination and lateral incline.

3.3. Desired Position/Posture Coordination Strategy

The ESR moves along a preset path for a certain displacement, which requires timely adjustment of the positional deviation and support posture and is specifically expressed as the adjustment phase in which one side of the ESR is raised to a supported fixed posture and the other side of the ESR is lowered to the initial supported position. The desired position/posture coordination adjustment process of the ESR is illustrated in Figure 10.

As shown in Figure 10, once the initial position and support posture are fixed and maintained in a steady state, the ESR enters the left and right position/real-time posture adjustment process after completing a certain stroke, and then judges the monitoring data such as the unilateral declination angle, scroll angle, and edge wall distance value of the ESR in turn. The stroke adjustment amounts $\Delta L_1,\Delta L_2,\Delta L_3,\Delta R_1,\Delta R_2$, and $\Delta R_3$ of the lateral AHC are set based on the desired position/posture. Coordinate the ESR declination angle, scroll angle, and distance to the edge walls on both sides one by one until the most expected position/posture is adjusted.

It can be seen from the kinematic model in the previous chapter that if the declination angle of the ESR on the left and right sides is adjusted to the desired value, the stroke coordination amounts $\Delta L_1$ and $\Delta R_1$ of the lateral AHC between the ESRs are written as

$$\left\{\begin{array}{l}\Delta L_1\approx -\left((\frac{L}{2}-m)\sin {\alpha }_{Le}+(d_1+\frac{b}{2})\cos {\alpha }_L\right)+x_L\\ \Delta R_1\approx -\left((\frac{L}{2}-m)\sin {\alpha }_{Re}+(d_5+\frac{b}{2})\cos {\alpha }_R\right)+x_R\end{array}\right.$$

(27)

If the scroll angles of the ESRs on both sides are adjusted close to the desired expected values [31], the stroke coordination amounts $\Delta L_2$ and $\Delta R_2$ of the lateral AHC between the ESRs are expressed as

$$\left\{\begin{array}{l}\Delta L_2\approx -\left(\frac{H}{2}\sin {\gamma }_{Le}+(d_2+\frac{b}{2})\cos {\gamma }_L\right)+y_L\\ \Delta R_2\approx -\left(\frac{H}{2}\sin {\gamma }_{Re}+(d_6+\frac{b}{2})\cos {\gamma }_R\right)+y_R\end{array}\right.$$

(28)

Assuming that the distance between the ESR and the edge sidewall is adjusted to the most desired expected value, first determine the minimum distance between the left and right ESRs and the edge sidewall, let the left $d_{mL}=\min (d_1,d_2,d_3,d_4)$ and the right $d_{mR}=\min (d_5,d_6,d_7,d_8)$ hold true, and comparing $d_{mL}$ and $d_{mR}$ with the expected distance $d_{Le}$ and $d_{Re}$ of the edge sidewall, the amounts of stroke coordination $\Delta L_3$ and $\Delta R_3$ for the lateral AHC between the ESRs are identified as

$$\Delta L_3=\left\{\begin{array}{l}{d}_{mL}-d_{Le} (d_{mL}\ge d_{Le})\\ d_{Le}-d_{mL} (d_{mL}<d_{Le})\end{array}\right.$$

(29)

$$\Delta R_3=\left\{\begin{array}{l}{d}_{mR}-d_{Re} (d_{mR}\ge d_{Re})\\ d_{Re}-d_{mR} (d_{mR}<d_{Re})\end{array}\right.$$

(30)

To sum up, the ESRs on the left and right sides are pre-adjusted to achieve the most desired position/posture, and the stroke coordination amount of the lateral AHC is denoted as:

$$\left\{\begin{array}{l}\Delta L=\Delta L_1+\Delta L_2+\Delta L_3\\ \Delta R=\Delta R_1+\Delta R_2+\Delta R_3\end{array}\right.$$

(31)

where if the lateral AHC shrinks, the values $\Delta L_i$ and $\Delta R_i$ are negative, and when the lateral AHC extends, the values $\Delta L_i$ and $\Delta R_i$ are positive. The expected optimal length $E_i$ of the ESR lateral AHC corresponding to each desired position key point can be expressed as:

$$E_i=H_i+\Delta L+\Delta R (i=1-4)$$

(32)

The desired expected stroke value (displacement) for each lateral AHC is written as

$$e_i=E_i-L_{b0} (i=1-4)$$

(33)

3.4. Control Algorithm for Coordinated ESR Position/Posture

This subject adopts the sliding mode control algorithm design, and then seeks the posture adjustment controller to dominate the stroke of the lateral AHC in the ESR, which realizes the coordination of the ESR relative to the edge space support posture in stages.

For any nth order MIMO (Multiple Input Multiple Output) system [32],

$$y^{(n)}=f(y^{(n-1)},\cdots ,y,t)+\Delta f(y^{(n-1)},\cdots ,y,t)+b(y^{(n-1)},\cdots ,y,t)u+d(t)$$

(34)

where $y(y\in R^m)$ denotes the output vector of the ESR system, $u(u\in R^m)$ is the input vector of the ESR system, and $f(f\in R^m)$ and $b(b\in R^m)$ represents the state matrix of the position/posture morphology of the ESR system.

Convert expression (34) in the form of state equation and write it as:

$$\left\{\begin{array}{l}{x}_1=x_2\\ x_2=x_3\\ ⋮\\ x_n=f(X,t)+\Delta f(X,t)+b(X,t)u+d(t)\end{array}\right.$$

(35)

where $x_n=y^{(n-1)}$, $X={\left[x_1^T,x_2^T,\cdots ,x_n^T\right]}^T$.

By tracking the expected desired position/posture $X_d={\left[x_{1d}^T,x_{2d}^T,\cdots ,x_{nd}^T\right]}^T$ in real time, the error vector is expressed as:

$$E=X-X_d=\left[\begin{array}{cccc}{e}_1& e_2& \cdots & e_n\end{array}\right]$$

(36)

The designed sliding mode switching surface is expressed as:

$$s(X,t)=CE-W(t)$$

(37)

where $C=\left[C_1,C_2,\cdots ,C_n\right]$, $W(t)=C{p}_i(t)$,

herein, $p_i(t)=\left\{\begin{array}{l}{\displaystyle \sum _{k=0}^n\frac{1}{k_i}{e}_i{(0)}^{(k)}{t}^k+{\displaystyle \sum _{j=0}^n\left({\displaystyle \sum _{L=0}^n\frac{{\alpha }_{jL}}{T^{j-L+n+1}}}{e}_i{(0)}^{(L)}\right)t^{j+n+1}}}\\ 0 (t>T)\end{array}\right.(0\le t\le T)$

The sliding mode controller is expressed as

$$s(X,t)=CE-P(t)=C_n\left[f(X,t)+\Delta f(X,t)+b(X,t)u+d(t)-x_{1d}^{(n)}-p{(t)}^{(n)}\right]+{\displaystyle \sum _{k=1}^{n-1}{C}_k\left[e^{(k)}-p{(t)}^{(k)}\right]}$$

(38)

The desired position/posture coordination adjustment system controller in ESR is written as:

$$u(t)=-b{(X,t)}^{-1}\left[f(X,t)-x_{1d}^{(n)}-p{(t)}^{(n)}\right]+C_n^{-1}{\displaystyle \sum _{k=1}^{n-1}{C}_k\left[e^{(k)}-p{(t)}^{(k)}\right]}=-b{(X,t)}^{-1}\frac{C_n^{T}s}{‖C_n^{T}s‖}\left[f(X,t)+d(t)+k\right]$$

(39)

4. Simulation Calculation of Desired Expected Value for ESR

4.1. Simulink Controlled System Model

The desired position/posture adjustment control system for ESR proposed in this study includes a posture monitoring subsystem, a lateral AHC subsystem, and a position/posture adjustment controller. The Simulink model of the desired position/posture adjustment control system is revealed in Figure 11.

In Figure 11, the expected position/posture value and the real-time position/posture data obtained by monitoring are used as the input of the ESR position/posture adjustment control system. Based on the real-time data analysis and judgment of position/posture, the lateral AHC stroke in ESR is coordinated by using the position and posture adjustment control process.

4.2. Numerical Results and Simulation Analysis

The lateral AHC parameter settings of the ESR position/posture adjustment control system with fifth order are shown in

Table 1. Parameters of lateral AHC for ESR.

Parameters Names	Values	Units
Flow gain $(k_q)$	4.15	$\left({\mathrm{m}}^2/\mathrm{s}\right)$
Pressure coefficient $(K_c)$	5.32	$({\mathrm{m}}^3/\mathrm{s}\cdot \mathrm{Pa})$
Effective cross-sectional area of AHC piston $(A_{pe})$	0.68	${(\mathrm{m}}^2)$
Leakage coefficient $(c_{tp})$	$0.78\times {10}^{-11}$	$({\mathrm{m}}^3/\mathrm{s}\cdot \mathrm{Pa})$
Effective volume modulus $({\beta }_e)$	$6.9\times {10}^8$	$\left(\mathrm{Pa}\right)$
Equivalent volume $(V_t)$	0.386	${(\mathrm{m}}^3)$
AHC length $(L_{a0})$	1.48	$\left(\mathrm{m}\right)$
Maximum stroke $(L_{a1})$	1.19	$\left(\mathrm{m}\right)$

. The parameters of the ESR position/posture adjustment controller are as follows: $C_1=100,C_2=30,C_3=30,C_4=10,C_3=1.0$.

Set the most desired position/posture parameters expected by ESR, involving left declination angle ${\alpha }_{Le}=0°$, left inclination angle ${\gamma }_{Le}=0°$, left distance $d_{Le}=1.0 \mathrm{m}$, right declination angle ${\alpha }_{Re}=0°$, right inclination angle ${\gamma }_{Re}=0°$, right distance $d_{Re}=1.0 \mathrm{m}$, and so on.

In the research of this subject, 20 groups of different initial position/posture support forms are randomly selected, and the key position/support posture is coordinated, calculated, and analyzed, and then the expected position/posture time-varying angles and distances of ESR is simulated as shown in Figure 12, Figure 13 and Figure 14.

Figure 12 analyzes the simulation calculation results of the ESR declination angle adjustment on both sides. With 20 groups of different initial positions/support posture states, based on the coordinated adjustment of the position/support posture control system, the ESR declination angle on both sides drops to about 0° within 10 s, the collaborative adjustment numerical error is within ±0.5°, and the minimum deviation between the simulation calculation results and the desired expected value is only within 0.2°.

Figure 13 shows the simulation results of the scroll angle adjustment of the ESRs on both sides. With 20 groups of different initial positions/support postures, based on the coordinated adjustment of the position/posture adjustment control system, the scroll angle of the ESRs on both sides can also be reduced to about 0° within 10 s, the numerical error of the coordinated adjustment remains within ±0.5°, and the minimum deviation between the simulation calculation results and the desired expected value is only within 0.10°.

Figure 14 demonstrates the simulation calculation results of the distance adjustment between the edge walls on both sides of the ESR. Based on the adjustment of the position/posture adjustment control system, the distance between the edge walls on both sides of the ESR reaches about 1.0 m within 10 s, and the adjustment value error is within ±0.03 m. The minimum deviation between the simulation calculation result and the desired expected value is controlled at not more than 0.01 m.

In summary, from the simulation calculation results in Figure 12, Figure 13 and Figure 14, aiming at the ESR key position/support posture adjustment control, the desired expected posture adjustment control method quickly adjusts the deviation/posture state in a short time, based on the collaborative adjustment proposed in this paper. The control method is effective, the deviation adjustment precision is high, and the support posture stability is reliable.

5. Experimental Verification and Analysis

In the actual space, the position/posture coordinated adjustment control system is tested, and the coordinated adjustment performance of the ESR key position/support posture is verified on site. Field testing and actual verification of ESR are illustrated in Figure 15.

In the field test and performance verification, the ESR on both sides of the space edge has different degrees of heading deflection and lateral tilt in the initial time after startup. The difference between the distance between the key position of ESR and the side edge wall of the roadway is not as optimistic. Set the desired expected position/posture of ESR, including left declination angle ${\alpha }_{Le}=0°$, left inclination angle ${\gamma }_{Le}=0°$, left distance $d_{Le}=1.0\mathrm{m}$, right declination angle ${\alpha }_{Re}=0°$, right inclination angle ${\gamma }_{Re}=0°$, and right distance $d_{Re}=1.0\mathrm{m}$. The ESR position/posture monitoring system is used to collect the position/posture state at the initial moment after each ESR movement. The experimental results of the coordinated ESR position/posture adjustment control system is indicated in

Table 2. Experimental data of declination angle of ESR.

Left/Right Edge Side	Number of Groups	Initial Data	Simulation Data
Left edge side ${\alpha }_L(°)$	1	3.62	0.42
	2	−2.43	−0.31
	3	2.94	0.42
	4	−4.57	−0.49
	5	2.62	0.42
Right edge side ${\alpha }_R(°)$	1	2.13	0.49
	2	1.64	0.42
	3	−2.36	−0.42
	4	−2.81	0.23
	5	1.92	0.32

,

Table 3. Experimental data of scroll angle of ESR.

Left/Right Edge Side	Number of Groups	Initial Data	Simulation Data
Left edge side ${\gamma }_L(°)$	1	3.83	0.33
	2	−3.16	−0.42
	3	2.37	0.23
	4	3.72	0.52
	5	−3.34	−0.49
Right edge side ${\gamma }_R(°)$	1	2.43	0.42
	2	1.82	−0.23
	3	−2.06	0.33
	4	−2.42	−0.52
	5	2.93	0.23

and

Table 4. Experimental data of distance.

Left/Right Edge Side	Number of Groups	Initial Data	Simulation Data
Left edge side $d_{mL}(\mathrm{m})$	1	0.696	0.986
	2	1.236	1.035
	3	0.787	0.982
	4	1.102	1.042
	5	0.993	1.031
Right edge side $d_{mR}(\mathrm{m})$	1	0.832	0.973
	2	0.879	0.998
	3	1.187	1.023
	4	1.236	1.039
	5	0.819	0.989

.

The experimental data of the ESR declination angle adjustment on both sides of the edge space are shown in Table 2. The ESR position/posture adjustment system cooperates with five groups of declination postures, and within 15 s, the ESR declination angle on both sides is adjusted to be close to 0°, and the declination angle adjustment result is controlled within ±0.5°. The minimum deviation of the experimental results from the desired expected value does not exceed 0.2°.

The experimental data of ESR scroll angle adjustment on both sides of the edge are shown in Table 3. The position/posture adjustment system cooperates with five sets of deviation postures. Within 15 s, the ESR scroll angle of the two side walls tends to 0°, and the scroll angle adjustment data is coordinated and controlled to be no more than ±0.5°. The experimental data is in line with the expected expectations, and the minimum deviation value is not higher than 0.2°.

The experimental data for adjusting the distance between the ESRs on both sides to the sidewall of the space edge are shown in Table 4. After the coordinated adjustment of the position/posture adjustment system, the distance between the ESR on both sides and the side wall of the space edge can achieve the desired expected value of 1.0 m within 15 s. By adjusting the position/posture field test, the distance error to the side wall of the space edge is limited to the expected value of ±0.03 m, and the minimum deviation between the experimental value and the desired expected value is controlled within the range of no more than 0.01 m.

The field test data in the above Figures and Tables demonstrates that the ESR position/posture adjustment control system has a fast response speed, and coordinately adjusts the deviation position and supporting posture of the ESR in a short period of time. Parameters such as the posture angles and the distances from the key position point to the side wall of the space edge in the verification process are adjusted by the system to the desired state and close to the minimum deviation area of the expected value. It is pointed out here that the maximum error of the posture angle is controlled to be no more than 0.5°, and the maximum error of the distance between the key position point and the edge of the space domain is no more than 0.03 m, which meets the actual collaborative adjustment accuracy requirements. Compared with the traditional manual adjustment method, the adjustment accuracy is significantly improved, the synergistic effect is credible, and the promotion and application prospects are promising.

6. Discussion of the Results

Compared with the simulation data, the desired co-adjustment time of the expected position deviation and the support posture in the experimental test is prolonged. The main reason is that the fluctuation effect of the support base of the ESR space environment has contributed to the additional resistance in the adjustment process, prolonged the coordination time, and led to an increase in the dispersion of the test data relative to the desired expected value.

The data parsed by the simulation calculation is almost close to the lower limit of the error range, whereas most of the experimental test data in the verification process reaches the upper limit of the error range. The main reason for the analysis is attributed to the combined effect of ESR’s hydraulic system leakage and experimentally verified field system response delay.

Regarding the comprehensive analysis of the simulation calculation results and test data, it is confirmed without hesitation that the desired expected position/posture coordinated adjustment control method based on the sliding mode control algorithm is effective and credible. The sliding mode controller has the characteristics of high reliability, stability, and adjustment accuracy.

Compared with the traditional manual control of the lateral AHC stroke to adjust the position deviation and support posture of the ESR, the method proposed in this paper has a more significant effect and higher adjustment accuracy, and realizes the lateral AHC posture coordination in the position deviation adjustment. The specific parameterization has important practical value and industrial promotion significance.

7. Conclusions

(1) The desired expected position/posture coordinated adjustment control method of ESR is effective, and the proposed ESR control system is adapted to the performance requirements of field applications.

(2) The minimum error of the declination angle in the simulation calculation results is less than 0.2°, and the error is limited to within ±0.5°. The minimum error of the roll angle in the results does not exceed 0.1°, and the error is controlled within ±0.5°. The minimum error of the simulated sidewall distance is not more than 0.01 m, and the error range is within ±0.03 m of the desired expected value.

(3) The minimum error of the declination angle in the experimental test results of the field verification link is controlled within 0.2°, and the error does not exceed ±0.5°. The minimum error of the scroll angle in the test results is less than 0.2°, and the error is limited to within ±0.5°. The minimum error of the margin in the test results is not more than 0.01 m, and its error range is within the desired expected value of ±0.03 m.

(4) The main reason for ESR position deviation/support posture inclination is the combined influence of the unaccounted instantaneous fluctuation of the excavation floor and the ESR transient push-pull effect in the space environment. The position stroke of each lateral adjustment hydraulic cylinder is adjusted to coordinate the deviation state of the ESR support posture, which verifies the effectiveness and feasibility of the proposed adjustment strategy and cooperative control process.