The working environment of the robot mainly refers to the terrain environment of the road surface (Tian and Fang, 2009). The mine disaster environment makes the underground road surface environment more complex: one is the three-dimensional complexity of the road surface, that is, the geometric shape of the road surface, including the ups and downs of the ramp, the trenches, and the bumps and depressions of the road surface; the second is the complexity of the physical state of the road surface, including the soil properties of the road surface, the sinking characteristics, the friction characteristics, etc. Different terrain features have different effects on the motion performance of the robot, which is a great challenge for the driving stability of the robot (Lin et al., 2023). Therefore, it is necessary to have a comprehensive analysis and estimation of the terrain environment in advance.

When the robot is working underground, the rugged and complex road conditions require the robot arm to actively adjust the configuration and maintain driving stability. Therefore, the structure and center of gravity of the robot will often change. Because the ground interference factors can be directly reflected in the contact force between the robot and the ground, measured by the force sensors, this paper proposes a stability criterion based on the contact force. In addition to the appropriate stability criterion, there must be a control system with strong function and high efficiency; the control system plays a pivotal role in motion control tasks, enabling the robot to promptly react to feedback from environmental information and thereby achieve autonomous control (Chen et al., 2024a). In an environment that cannot be fully predicted, the two combine with each other to deal with the current capsizing event in a way that quickly responds to the integrated task.

To analyze the stability of a robot, the friction coefficient is normally assumed to be sufficient to prevent the robot from slipping (Song et al., 2023). All contacts between the robot and contact surfaces are assumed to be point contacts and line contacts. Hence, all contacts do not lie on a single edge of the support polygon. According to the dynamic energy stability model, if the torque generated by the contact between the robot and the ground on any side of the support polygon is positive, that is, Mj>0 (j=1∼n), then the robot has dynamic stability. From Newton's law, the following equation must be satisfied about every jth edge of the support polygon: 1Min,j=Mg,j+Mm,j+Ms,j, where Min,j is the moment due to the inertial force, moment Mg,j is the moment due to the gravitational force, moment Mm,j is the moment due to the manipulation (external) forces and moments, and Ms,j is the moment due to the foot contact forces and moments. All of the moments are calculated about the jth edge of the support polygon. From Eq. (1), the following can be written: 2Ms,j=-Mg,j+Mm,j-Min,j. The right-hand side terms of Eq. (2) sum up to the net moment, acting about the jth edge of the support polygon, caused by the net force and moments acting on the system. Replacing the right-hand side of Eq. (2) with MN,j gives the following: 3Ms,j=-MN,j. Hence, for a robot to be considered to have a dynamically stable stance, the net moment about the jth edge of the support polygon caused by all the foot forces must be negative, and its magnitude must be the same as the Ms,j. Otherwise, the robot will rotate about that edge and tumble. For the detection robot for underground rescue studied in this paper, the net torque on the bottom support edge line is generated by the contact force of other touch points, so the stability of the robot can be expressed in the following ways.

The corners of the convex polygon on the bottom of the robot are marked sequentially in the clockwise direction in the case of overlooking, as shown in Fig. 4, which are recorded as pi(i=1∼n), forming the bottom edge of the polygon on the bottom of the robot, that is, the supporting edge line, recorded as 4ai=pi+1-pi(i=1∼n-1),5an=p1-pn, At time t, the robot's bottom polygon will be affected by the ground contact force. At this time, the overturning stability margin of the robot's side line ai can be expressed by the sum of the contact torques on the side line: 6Mai=∑F⋅R=∑f⋅R×1+u2(i=1∼n), where ∑f⋅R is the sum of the moments of the normal contact force on the side line ai at all contact points, and u represents the ground friction coefficient.

When the contact is a line contact, the moment of the contact force on the contact line to the side line ai can be regarded as the sum of the product of the contact force of multiple points and the position vector perpendicular to the side line ai. Considering the global stability of the robot, take 7Ki=min⁡(Mai)(i=1∼n). The overturning stability index based on the contact force stability criterion (the following referred to as the overturning stability index) is obtained by normalizing the weight of the robot: 8ECFSC=min⁡(Ki)mg-1(i=1∼n). When ECFSC>0, the robot meets the dynamic stability conditions in the standardized dynamic energy stability model and has dynamic stability; when ECFSC=0, the robot is in a critical overturning state; and when ECFSC<0, the robot is in a state of instability.

In the actual underground operation process, the robot also needs to consider the influence of external effects, such as impact, obstacle interference, and so on. The generated force and torque can be expressed as the center of gravity of the robot. It is assumed that the size of the external force can be seen as fo, the size of the external torque can be seen as mo, and the equivalent force along the overturning line can be expressed as 9fi∗=fi+l^i×a^ia^iTmoli, where li is a straight line perpendicular to the center of gravity and the toppling line ai, which can be expressed by the following equation: 10li=1-a^ia^iTpi+1. These effects can be reflected in the contact force between the robot and the ground. From this point of view, CFSC is a dynamic stability criterion, which is suitable for all the working conditions of the robot studied in this paper.

The real-time stability control strategy can continuously monitor the stability of the system according to the stability criterion and respond if necessary to enhance the stability of the system. Since the force sensor data can be measured intuitively, the overturning stability index can be directly integrated into the current control strategy as a stability measure, combined with the BP neural network to control the change of the robot's leading arm configuration, as shown in Fig. 5.

The system uses a BP neural network to coordinate and control the motor of the robot's leading arm and change its configuration and motion parameters, thereby changing the overturning stability index and ensuring that the overturning stability index is within the controllable range. The attitude information and motion parameters of the robot can be measured by the sensors. After organization and coordination, the necessary A/D (analog–digital) conversion is carried out and transmitted to the BP neural network controller. By coordinating the information provided by the sensing system and the calculated motion stability information, the controller obtains the expected trajectory of the robot's leading arm and drives the motor to work. At the same time, the system has a real-time feedback adjustment function. The backstepping sliding mode controller realizes the tracking control of the output angular position of the arm and ensures the robustness of the system.

According to the data information measured by the sensors, the overturning stability index of the robot can be obtained, and then a BP neural network controller is used to obtain the angular displacement of the robot's leading arm. The input variable is the overturning stability index; the ratio of training samples, verification samples, and test samples is 14:3:3; the number of neurons in the hidden layer is 10; the training algorithm is the inversion propagation algorithm (Ranjan et al., 2018); and the output is the swing angle of the swing arm. The sample data are a total of 658 groups. The training performance diagram is shown in Fig. 6, and the regression fitting diagram is shown in Fig. 7.

From Fig. 6, when the neural network training is iterated 37 times, the fitting relationship between input and output reaches the optimal solution. It can be seen from Fig. 7 that the R values of all data samples are higher than 0.99, indicating that the fitting relationship between input and output has reached a better level in theory, and stability control can be carried out.

To analyze and calculate the overturning stability index of the robot, it is necessary to study the contact force of the robot when driving on the unstructured road surface. Although the underground environment is complex, based on the size of the robot, it can be divided into two categories: obstacles that the robot can pass through and obstacles that the robot cannot pass through. When the robot encounters obstacles that can be passed, the body may tilt at a small angle. During these instances, the forces acting on the robot resemble those experienced on a gentle slope, allowing for substitution with such a slope for analysis. When encountering obstacles that it cannot pass, the robot will choose to detour. This article will not study such obstacles. Therefore, the following slope with both longitudinal and transverse components is used to simulate most of the terrain conditions that the robot can pass through underground (Mascarich et al., 2020). Because the track width is smaller than the wheelbase, its influence on the calculation is ignored in the study (Erden and Leblebiciolu, 2007). The normal force of each contact point of the mobile robot is shown in Fig. 8.

Force balance: 11N1+N2+N3=Gcos⁡φcos⁡θ. Moment equilibrium: 12N1c1+N2c-Gcos⁡φsin⁡θh-Gcos⁡φcos⁡θc1=0,13N2(b1+a)+N3(b2+a)-Gsin⁡φh-Gcos⁡φcos⁡θb=0. From Eqs. (11) to (13), we can find

14N1=Gcos⁡φsin⁡θhc1+Gcos⁡φcos⁡θ-Gcos⁡φsin⁡θhac+ac1c+Gsin⁡φhc1c+Gcos⁡φcos⁡θbc1c+Gcos⁡φsin⁡θhb2cb1c1+c2b2+ac1+cac1,N2=Gcos⁡φsin⁡θha+ac1+Gsin⁡φhc1+Gcos⁡φcos⁡θbc1+Gcos⁡φsin⁡θhb2b1c1+c2b2+ac1+ca,N3=Gcos⁡φsin⁡θhac2+ac1c2+Gsin⁡φhc1c2+Gcos⁡φcos⁡θbc1c2+Gcos⁡φsin⁡θhb2c2b1c1+c2b2+ac1+cac1-Gcos⁡φsin⁡θhc1,

where a is the distance between the center of gravity and the front axle, b is the distance between the center of gravity and the rear axle, c is the length between the two tracks, c1 is the distance between the center of gravity and the side line of the robot, c2 is the distance between the center of gravity and the right-side line of the robot, and h is the height of the center of mass.

Therefore, if the longitudinal slope, roll angle, and center of gravity of the terrain are known, the normal contact force can be obtained. At the same time, when the configuration of the robot arm changes, the normal contact force can be determined by changing the values of a, b, and c. The overturning stability index of the robot in this state can be obtained by substituting the obtained contact force into Eq. (14).

During the underground operation of the rescue detection robot, due to the influence of terrain, ground force imbalance, and other factors, the robot system will present front and back pitching, left and right tilting, and their combination states (Roan et al., 2010). The robot's stability is directly reflected in its attitude changes during movement, particularly in terms of roll and pitch angles (Chen et al., 2024b). The centroid of the robot system is taken as the origin of the coordinate, and the rectangular coordinate system o-xyz is established. The positive direction of the robot's forward direction is taken as the x axis, the direction perpendicular to the forward direction to the right is taken as the y axis, and the direction perpendicular to the robot's body plane is taken as the z axis. For the robot, the technical parameters of the whole vehicle can be approximately regarded as unchanged when the terrain changes. The changes are the longitudinal inclination angle, lateral inclination angle, center of gravity position, and the direction and size of the normal contact force with the ground. The change of terrain is equivalent to the forward and backward pitch, left and right tilt, and their combination of the robot system (Wong et al., 2019). Assuming forward and backward pitch, the slope of left and right tilt is less than 30°, the random terrain changes around the xaxis and the y axis are defined as 15φ(t)=(δu(t)-0.5)⋅π/6, 16θ(t)=(δv(t)-0.5)⋅π/6, where δu(t) and δv(t) are random numbers that change with time t between 0 and 1, and the sizes are not necessarily the same. At time t, the mobile robot system first tilts around the x axis angle φ and then tilts around the y axis angle θ, which is brought into Eq. (14). The normal contact force between the robot and the ground can be obtained. The obtained force is brought into Eq. (6), and the overturning stability index at time t can be obtained.

To verify the reliability and efficiency of CFSC, it is compared with the ZMP, FASM, and NESM methods, which are commonly used stability analysis methods for mobile robots. The mass of the detection robot for underground rescue is 24 kg, the friction coefficient between the track and the ground is u1=2.9, and the friction coefficient between the front guide wheel and the ground is u2=0.8 (Haikal, 2023).

Using t to represent the time of each stage of the robot, the robot in the simulation scene has gone through four stages: when 0<t≤10 s, the longitudinal slope of the ground gradually changed from 0 to 25°; when 10<t≤20 s, the leading mechanical arm is gradually elongated by 30 mm from the initial state; when 20<t≤30 s, the arm is gradually deflected to the left by 26 degrees; when 30<t≤40 s, the lateral tilt angle of the terrain to the left gradually changed from 0 to 25°. The curve of the stability margin of the four methods with time is shown in Fig. 9. At t, the calculation time of the four methods is shown in Table 2.

Comparing the stability margin trends of the four stages of the four methods, the following results can be obtained. In the first stage, as the longitudinal slope of the ground increases, the stability margin of the robot decreases and the stability becomes worse. In the second stage, after the robot arm is elongated, the stability margin increases slightly and the stability is improved. In the third stage, when the deflection angle of the arm is changed, the stability margin of the robot is obviously improved, which indicates that the deflection of the arm improves the stability of the robot. In the fourth stage, the stability margin decreases, indicating that the change of slope angle reduces the stability of the robot. And CFSC and the other three methods have the same instability point and change trend, which verifies the accuracy of CFSC. By comparing the calculation time of the four methods at time t with MATLAB, it is found that CFSC is shortened by half, which shows that the method is simple and efficient.

When the robot is driving on the simulated road surface, it will be affected by the inertia force and the control force. The two forces have different effects on the overturning stability of the robot: the robot's overturning due to the inertia force often occurs in the start-up and braking stages on the large slope pitch slope; the control force is mainly generated by the change of the arm configuration, which affects the stability of the body when the arm is deflected.

The mass of the robot is 24 kg, and the acceleration is 0.32 m s-2 when the driving motor outputs at full power. The inertia force of the robot at this time is 7.68 N calculated by Newton's second law of motion. Due to the slow speed of the robot, it can maintain a consistent speed even when the passable terrain changes, and the resulting inertial force is smaller than that when the motor is started. To prove the effectiveness of stability control under extreme conditions, the inertial force is set to 10 N. On the pitch terrain, it is assumed that 10 N inertial force is applied when the robot goes uphill (taking the forward direction of the robot as positive), and -10 N inertial force is applied when it goes downhill to represent the robot's acceleration uphill and braking downhill processes. The elongation range of the arm is controlled to be 0–30 mm. After calculation, the maximum pitch angle of the robot is 27° when the driving motor is at full power. Therefore, the range of road slope is 0–25°, the sampling time is 21 s, and the number of sampling data points is 21. The overturning stability index of the robot under the pitch road can be calculated by Eqs. (6)–(15), as shown in Fig. 10.

At the same time, the overturning stability index of the robot with the mechanical arm elongation of 30 mm, the mechanical arm elongation of 15 mm, and the initial state decreases in turn, indicating that on the pitch road surface, when the inertial force is 10 N, the dynamic stability of the robot gradually increases with the increase in the length of the mechanical arm.

Figure 11 simulates the stability control process of the robot while driving on different terrains and considers the influence of the control force. When the overturning stability index is less than 0.05, the deflection of the arm is controlled to prevent roll, and the change trend of the overturning stability index with time is obtained.

It can be seen from Fig. 11 that the stability control of the arm can improve the overturning stability index of the robot on the inclined terrain and improve the stability of the robot to prevent overturning, and different deflection angles of the arm have different effects on the stability under different lateral slopes. On the simulated pitch–tilt combined road surface, an inertial force of 10 N is applied, and the arm control system is added. The change of overturning stability index with time is shown in Fig. 11b.

It can be seen from Fig. 11b that on the pitch–tilt combined road surface, the arm control system can improve the overturning stability index of the robot, that is, improve the dynamic stability of the robot and the anti-overturning ability at critical overturning.

Because the size of the inertial force is also affected by the road conditions when the robot is working, the size of the inertial force is random. To study the influence of different inertial forces on the stability of the robot with different arm elongations, the 25° longitudinal slope is taken. The surface of the overturning stability index with the inertial force and the arm elongation is obtained, as shown in Fig. 12a.

It can be seen from Fig. 12a that for the 25° positive slope, with the increase in the elongation of the arm, the overturning stability index increases linearly, indicating that the dynamic stability is improved. For the same arm configuration, when the direction of inertia force is consistent with the forward direction of the robot, with the increase in inertia force, the overturning stability index increases linearly and the dynamic stability increases. When the direction of the inertial force is opposite to the direction of the robot, with the increase in the inertial force, the overturning stability index decreases linearly and the dynamic stability decreases.

From the analysis results of Fig. 11a, the influence of the deflection angle of the arm on the stability under different lateral slopes changes always. Therefore, it is necessary to analyze the influence of different deflection angles on the stability of the robot under different lateral slopes, and obtain the overturning stability index with the lateral slope β and the deflection angle of the arm α, as shown in Fig. 12b.

It can be seen from Fig. 12b that for the transverse slope, as the slope becomes steeper, the overturning stability index of the robot decreases linearly, indicating that its dynamic stability gradually decreases. On the same slope, considering the control force, when the deflection angle of the arm in the inclined direction is less than 5° or greater than 15°, the overturning stability index of the robot is improved, and the dynamic stability of the robot is improved. When the deflection angle of the arm is 5–10°, the overturning stability index of the robot does not change significantly.

According to the specific design size of the underground rescue detection robot, the robot model and the simulated road surface are established in the simulation software Adams. The motion pair and drive are added to the robot, and the contact between the robot and the simulated road surface is added. The body and the bearing mechanism are set to be metal materials, and the wheels and tracks are set to be rubber materials.

The purpose of this simulation is to verify the effectiveness of the stability criterion of the robot. By comparing the change trend of the centroid displacement, velocity, acceleration, and overturning stability index when the robot is driving on the simulated road, the effectiveness of the overturning stability index is proved. Compared with the rugged and random road surface, the data of centroid displacement and contact force have obvious regularity when the robot is driving on the washboard road. The contrast effect of the obtained data is more obvious after visual processing, which is conducive to the proof of effectiveness. In Fig. 13, the simulated road surface of the robot is a washboard road, which is composed of a strip cuboid with a length of 250 mm, a width of 50 mm, and a height of 10 mm. The spacing between adjacent cuboids is 200 mm. The washboard road is closer to the underground rough and bumpy road surface, and the simulation results are of practical significance.

The contact point between the front guide wheel and the ground of the robot simulation model is taken as a point contact, and two point contacts are taken from the contact part of the two tracks of the robot model and the ground. The simulation time is 15 s, and the number of sampling points is 30. The relationship between the size of the five normal contact forces and the displacement of the center of mass along the Z axis with time can be obtained, as shown in Fig. 14.

After processing the image data, the relationship between the robot's overturning stability index value and time can be obtained, which is compared with the displacement of the center of mass along the z axis: when the robot's overturning stability index is small, the center of mass displacement is large, and the robot stability is poor. When the overturning stability index is large, the displacement of the center of mass is small, and the stability of the robot is good, which verifies the correctness of the theory and proves that CFSC can be used as the criterion for the stability of the robot.

It can be seen from the theoretical analysis results that the adaptive control system can improve the stability of the robot. To verify the results, two virtual simulation experiments are set up in this section. (1) First, the lateral slope angle is continuously improved to find the critical overturning condition of the robot. (2) Second, we control the arm to adjust the configuration and observe the change of the stability index.

The transverse slope is set as the virtual simulation road surface, and the slope is continuously improved, as shown in Fig. 15. With the increase in the slope, the initial state of the robot capsizes on the 25° transverse slope, indicating that 25° is the critical overturning angle of the robot's transverse slope. Compared with the theoretical analysis of 27°, the error is 8 %. The reason is as follows: in the theoretical analysis, it is assumed that the external force received by the robot is a fixed value. In the simulation experiment, the external force received by the robot during operation changes at any time, resulting in an error. Through simulation, the relationship between the normal contact force of the point contact of the robot's non-overturning edge line along the Z axis and time can be obtained, and the overturning stability index can be calculated, as shown in Fig. 16.

It can be seen from the data in Fig. 16 that when the robot reaches the critical overturning state, the overturning stability index gradually decreases to 0, which is consistent with the theoretical analysis.

In the critical overturning state, the mechanical arm is controlled to deflect forward along the X axis. Figure  17 shows the simulation experiment of adjusting the deflection angle of the arm on the 25° transverse slope. The results show that the robot gradually enters a stable state during the deflection process of the arm.

By analyzing the simulation process, the change of the normal contact force along the Z axis of the point contact of the robot's non-overturning edge line with time can be obtained, and the overturning stability index can be calculated, as shown in Fig. 18. It can be seen from Fig. 18 that under dynamic conditions, with the increase in the deflection angle of the arm, the overturning stability index of the robot increases first, then does not change significantly, and then increases again, which verifies the correctness of the theoretical analysis and proves the reliability of the adaptive control system.

The same stage performed in the simulation process is applied to the physical robot. The first stage of the robot runs on a gentle road surface for 5 s. The second stage of the robot climbs a 15° longitudinal slope for 10 s. The third stage is when the robot's leading arm extends on the longitudinal slope for 10 s. In the three stages, the data measured by the sensors are sent to the host computer every second, and the overturning stability index is calculated. Figure 20 shows the three stages of the robot, and Fig. 21 is the comparison of simulation and experimental results.

It can be seen from Fig. 21 that the overturning stability index of the robot in the first stage remains at a high value and the change range is small; the stability is good. When the robot moves up the longitudinal slope in the second stage, the center of gravity moves; the force measured by the sensors in contact with the plane becomes larger, and the force in contact with the slope decreases. At the same time, the overturning stability index of the robot decreases, and the stability of the system decreases. At this time, if the overturning stability index of the robot becomes 0, the robot will capsize. In the third stage, the control robot gradually extends the arm out. Currently, due to the forward movement of the center of gravity, the overturning stability index of the robot gradually increases and the stability is improved. The experimental results are consistent with the simulation results, which proves the accuracy of the criterion, indicating that the stability control on the longitudinal slope can improve the stability of the robot.

Figure 22 shows the process of the robot driving on a 15° transverse slope and the deflection of the arm to maintain stability. It can be divided into two stages: uphill and deflection of the arm, which lasts for 30 s. The overturning stability index of the robot is calculated by the sensor data. Figure 23 shows the overturning stability index curve with time and the simulation results.

The experimental results show that the deflection process of the arm on the side slope can be divided into three stages from 10 s onwards. In the first stage, when the deflection angle of the arm is in the range of 0–5°, the overturning stability index of the robot increases gradually with the increase in the deflection angle. In the second stage, when the deflection angle of the arm is in the range of 5–15°, the overturning stability index of the robot does not change significantly. In the third stage, when the deflection angle of the arm is greater than 15°, the overturning stability index of the robot increases gradually with the increase in the deflection angle of the arm. The overturning stability index of the three stages is consistent with the change trend of the simulation curve, and the error is within 5 %. The correctness of the theory is verified, and the effectiveness of the adaptive control system is proved.