Control of a Path Following Cable Trench Caterpillar Robot Based on a Self-Coupling PD Algorithm

Abstract

Underground cable trench inspection robots work in narrow, variable friction coefficient, and complex road environments. The running trajectory easily deviates from the desired path and leads to a collision, or even the destruction of the robot or cable. Addressing this problem, a path-following control method for the dual-tracked chassis robot based on a self-coupling PID (SCPID) control algorithm was developed. The caterpillar robot dynamics were modelled and both the unknown dynamics and external bounded disturbances were defined as sum disturbances, thus mapping the nonlinear system into a linearly disturbed system, then the self-coupling PD (SCPD) controller was designed. The system proved to be a robust stability control system and only one parameter, the velocity factor, needed to be tuned to achieve parameter calibration. Meanwhile, to solve the problem that the error-based speed factor is not universal and to improve the adaptive ability of the SCPD controller, an iterative method was used for adaptive tuning. The simulation results showed that the SCPID can achieve better control. The field test results showed that the SCPD’s maximum offset angle was 56.7% and 10.3% smaller than incremental PID and sliding mode control (SMC), respectively. The inspection time of the SCPD was 20% faster than other methods in the same environment.

1. Introduction

The underground cable trench is an important part of a city’s power supply network, and with the development of the power system, there have been more and more applications. Unfortunately, harmful gases such as methane gas and hydrogen sulfide can easily accumulate in the cable trench and cause harm to inspectors. At the same time, discharges caused by aging of the cable insulation can easily ignite these flammable gases, causing explosions and fires. More seriously, the accumulation of hazardous gases and the aging of insulation materials over time will also gradually increase the probability of dangerous accidents, so regular inspections of underground cable trenches to eliminate these safety hazards are essential. The underground cable trench space is narrow. Both sides are covered with cable brackets, resulting in limited access for the robot, and the inspection process needs to strictly follow the path of the task to avoid damage to the cable or the robot due to overshooting. Therefore, the development of intelligent inspection robots is of great significance in eliminating potential safety hazards and reducing the labor intensity of the staff.

The underground cable trench is narrow and covered with cable supports on both sides, resulting in limited options for the robot’s path. During the inspection process, the robot needs to strictly follow the path to perform the task to avoid damaging the cables or itself by overshooting. In addition, the road surface is mostly sand and gravel, garbage, and other obstacles, and the friction changes drastically, which requires the inspection robot to have a certain level of obstacle-crossing ability [1]. Among the common mobile chassis structures for inspection robots, caterpillar-tracked [2] has simple control and strong obstacle-crossing ability, which is suitable for underground cable trench environments. The path-following problem of a caterpillar-tracked inspection robot in an underground cable trench environment becomes the key to completing intelligent inspection safely and efficiently.

The path-following problem of inspection robots has received more attention in recent years. Positional PID [3,4] can get good results in controlling the vehicle speed and rotational speed. However, the PID parameters vary greatly for different controlled plants. Even if the PID controller is well tuned, if there is a sudden change in working conditions or the existence of external disturbances, it is necessary to re-tune the PID control parameters, and the PID controller has three parameters to tune. The incremental PID [5,6] method has the advantages of no overshooting as well as fast response and smoothness, but the control output is related to the past state and the calculation is large, which affects the real-time performance. The fuzzy controller [7] has strong robustness and self-adaptive ability without relying on the mathematical model. However, the fuzzy rules are difficult to establish and require a high level of expert knowledge and experience. The LQR algorithm [8] makes it easy to adjust the parameters and has better autonomous optimization ability in the path-following control. However, it relies on the high accuracy of the model, and the anti-disturbance robustness is insufficient. Sliding mode control [9,10,11,12,13,14] has strong robustness to the nonlinear system of caterpillar-tracked chassis systems on rough roads. However, the sliding mode control suffers from a high-frequency jitter vibration phenomenon, which can easily cause system oscillations. Although some methods can be used to effectively avoid the jitter vibration phenomenon, more parameters are involved and the complexity is greatly increased. Su et al. [15] developed an intelligent inspection robot for substations that can support real-time deep-learning models. Villasenor et al. [16] proposed a germinal center optimization algorithm to search for neuronally optimal control parameters for caterpillar-tracked chassis systems. Rios et al. [17] used a recursive higher-order neural network algorithm based on extended Kalman filtering and applied the inverse optimal control to a tracked chassis system. Cheng et al. [18] used a reinforcement learning algorithm to interact with the environment of a set path to achieve path-following in a non-autonomous wheeled mobile robot (NWMR). Saha et al. [19] controlled the robot to accomplish path following based on a deep neural network (DNN.) However, it is impossible to physically interpret the internal structure in neural networks (NN), and the parameters can only be tuned by trial-and-error methods, when the parameters of the controlled plant are mismatched, it will also make the control effect decrease obviously. In addition, NN has the problems of requiring numerous parameters, being sensitive to initial weights, and overfitting.

Inspired by the self-coupling PID (SCPID) method [20,21], a self-coupling PD (SCPD) method with an adaptive speed factor was proposed for the path-following problem of inspection robots in an underground cable trench. The contributions of this paper can be summarized as follows:

• All complex factors of the caterpillar robot, including known or unknown dynamics and their uncertainty as well as external bounded disturbances, were defined as a total disturbance. The time-vary complex nonlinear system of the caterpillar chassis was matched to a linear disturbance system;
• The SCPD controller not only retains the advantages of the simple structure of the traditional PID controller but also the parameter tuning only needs to adjust the adaptive speed factor. The adaptive speed factor could further accelerate the convergence of parameter tuning. The computing cost of this method is pretty low;
• The SCPD control systems are critically damped, not only to ensure the stability of the control system but also good dynamic response characteristics.

The combination of the SCPD algorithm and caterpillar chassis was applied to the underground cable trench inspection robot, which can effectively reduce overshooting while ensuring inspection speed, avoiding damage to the cables or the robot, and providing a strong guarantee for the safe and efficient implementation of intelligent inspections.

2. Robot Model

The complex environment of the underground cable trench is shown in Figure 1 and the size was 100 m × 1.2 m × 1.5 m. The inspection robot should have a certain degree of obstacle-crossing ability and be able to cope with significant changes in friction. Therefore, the robot chosen was the double-track chassis structure. This structure has good obstacle-crossing and grasping ability, is easy to control, has low power consumption, and is small in size. The shape of the inspection robot is shown in Figure 2, and the actual parameters are shown in

Table 1. Main technical parameters of dual-motor caterpillar chassis.

Model	Parameters
Track Type	Rubber track
Mass/kg	20
Length × width × height/mm	500 × 300 × 120
Speed/ms−1	0.1~1
Track width/mm	80
Track gauge/mm	300
Track ground length/mm	440
Diameter of driving wheel pitch circle/mm	145

.

2.1. Mathematical Model of Caterpillar Chassis

There is no mechanical steering mechanism between the two active wheels of the dual-motor crawler chassis system, which are independently driven by the electric motors on both sides, respectively. Therefore, to realize the system steering of the dual-motor tracked chassis, the system mainly controls the rotational speed and torque of the motors on both sides through the differential steering system of the two wheels. Therefore, a dynamic model of the dual-motor tracked chassis system with 3 degrees freedom was established according to the above principle, which was the longitudinal motion around the x-axis, lateral motion around the y-axis, and transverse motion around the z-axis, respectively. The modelling and simplification of the dual-motor tracked chassis system were as follows:

• The origin of the kinetic coordinate system coincides with the center of mass.
• Neglecting the suspension effect, the robot only moves parallel to the ground.
• Neglecting the influence of the steering system, the mechanical characteristics of the two tracks are the same.

The simplified dual-tracked chassis dynamics model is shown in Figure 3.

The dynamics equation of the dual-motor tracked chassis system is:

$$\{\begin{array}{l}m{\dot{v}}_x=(F_L+F_R)\cos \theta +(F_{x1}+F_{x2})\cos \theta -(F_{y1}+F_{y2})\sin \theta -(R_L+R_R)\cos \theta \hfill \\ m{\dot{v}}_y=(F_L+F_R)\sin \theta +(F_{x1}+F_{x2})\sin \theta -(F_{y1}+F_{y2})\cos \theta -(R_L+R_R)\sin \theta \hfill \\ I\dot{\omega }=(F_R+F_{x2}-F_{x1}+R_L-R_R)B/2-M_{\mu }\hfill \\ v_x=V\cos \theta \hfill \\ v_y=V\sin \theta \hfill \\ \max (F_L,F_R)\le \phi mg/2\hfill \\ M_{\mu }=\mu mgL/4\hfill \\ \mu ={\mu }_{\max }/(0.925+0.15R/B)\hfill \\ R_R=R_L=mgf/2\hfill \end{array}$$

(1)

In the above equation, $m$ is the mass of the dual-motor tracked chassis system, and $v_x$,$v_y$ is the velocity component of the dual-motor tracked chassis system in the X, Y direction. $F_L$, $F_R$ is the driving force of the left and right tracks of the dual-motor tracked chassis system; $F_{x1}$, $F_{x2}$ is the force in the X direction on the left and right tracks of the dual-motor tracked chassis system; $F_{y1}$, $F_{y2}$ is the force in the Y direction on the left and right tracks of the dual-motor tracked chassis system; $R_L$, $R_R$ is the resistance to moving on the left and right tracks of the dual-motor tracked chassis system; $I$ is the inertia of the dual-motor tracked chassis system; $\omega$ is the angular speed of the dual-motor tracked chassis system; $B$ is the track distance of the two-motor tracked chassis system; $M_{\mu }$ is the steering resistance moment; $V$ is the moving speed of the two-motor tracked chassis system; $\phi$ is the coefficient of adhesion between the track and the ground; $g$ is the gravity acceleration; $L$ is the length of the track; and $f$ is the coefficient of rolling friction of the track.

From the chassis unified force decomposition and velocity direction of the tracked robot in Figure 3, while combining with the system, (1) can be obtained. The control system of the tracked robot is shown in Equations (2) and (3):

$$\{\begin{array}{c}\ddot{x}=w_x+u_x\\ \begin{array}{l}\ddot{y}=w_y+u_y\hfill \\ \ddot{\theta }=w_{\theta }+T_{\delta }\hfill \end{array}\end{array}$$

(2)

$$\{\begin{array}{l}{u}_x=\left(F_L+F_R\right)\cos \theta /m\\ u_y=\left(F_L+F_R\right)\sin \theta /m\\ T_{\delta }=0.5B{F}_R/I\end{array}$$

(3)

where $w_x=[(F_{x1}+F_{x2})\cos \theta -(F_{y1}+F_{y2})\sin \theta -(R_L+R_R)\cos \theta ]/m$ is disturbance in the x direction, $w_y=[(F_{x1}+F_{x2})\sin \theta -(F_{y1}+F_{y2})\cos \theta -(R_L+R_R)\sin \theta ]/m$ is a disturbance in the y direction, and $w_{\theta }=[(F_{x2}-F_{x1}+R_L-R_R)\frac{B}{2}-M_{\mu }]/I$ is a disturbance in the angle.

The decoupling from Equation (3) gives the relationship between $u_x$, $u_y$, and $F_R$, $F_L$ as shown in Equation (4):

$$\{\begin{array}{l}{F}_R=2I{T}_{\delta }/B\\ F_L=m\sqrt{u_x^2+u_y^2}-2I{T}_{\delta }/B\end{array}$$

(4)

where ${\theta }_d=\arctan (\frac{u_y}{u_x})$ is the angle controller.

The relationship between $F_R$ and $F_L$ with the left and right wheel motor speeds. ${\omega }_1$ and ${\omega }_2$ is shown in Equation (5). The path-following requires control of the left and right motors to make the robot reach the desired position. According to the relationship among the control inputs, the left and right track driving forces, and angular velocity, the control strategy diagram of the robot based on SCPD can be shown in Figure 4.

$$\{\begin{array}{l}{F}_R=1946P_e\pi \eta i/{\omega }_{2}r\\ F_L=1946P_e\pi \eta i/{\omega }_{1}r\end{array}$$

(5)

2.2. Robot Overall Control System Structure

After establishing the track dynamics model and performing force decomposition, the motors were taken as the main control object and the controller controls the driving force of the robot’s left and right wheels to realize trajectory tracking. The control of the caterpillar track inspection robot was essentially performed by controlling the caterpillar driven by the single-side motors. The closed-loop speed regulation system was mainly composed of an embedded controller, a DC motor, and a speed encoder; the overall flowchart is shown in Figure 5. The speed encoder is in charge of converting and encoding the motor speed into corresponding square wave pulse signals for the embedded controller to process. The path control is given back to the host computer through the LiDAR and depth camera, and the desired path is changed to form a closed-loop control system.

3. Design of the SCPD Controller

The general engineering problems can be attributed to controlling variables such as position, velocity, and acceleration. In this paper, the problem in controlling the path-following caterpillar robot was to control the position and angle. From the control structure diagram of the robot chassis system, it can be concluded that three controllers must be designed for position x-direction, position y-direction, and angle $\theta$. The robot in the underground cable trench needs good control accuracy, anti-disturbance robustness, and real-time requirements, so the design of the controllers should satisfy these indicators. The SCPID control [22,23] algorithm defines the complex unknown dynamics and external disturbance as the total disturbance, which converts the nonlinear uncertain system into a linear uncertain system with good robustness. Furthermore, the SCPID couples the proportional gain, integral gain, and differential gain into one parameter, simplifying the parameter tuning and enhancing the real-time capability. Therefore, the SCPD controller in this paper was designed based on the SCPID method.

3.1. Design of the x-Direction Controller

The purpose of designing the displacement x direction controller $u_x$ was to decouple the driving force of the left and right wheels of the robot in the following ways.

Let the desired value of the system is $x_d$, and the actual value is $x$, according to this can be obtained tracking error $e_{11}=x_d-x$, differential error $e_{12}={\dot{e}}_{11}={\dot{x}}_d-x_1$, where $x_1=\dot{x}$, combined with the system (2) can be obtained controlled error system as follows:

$$\{\begin{array}{l}{\dot{e}}_{11}=e_{12}\\ {\dot{e}}_{12}={\widehat{w}}_x-u_x\end{array}$$

(6)

where ${\widehat{w}}_x={\ddot{x}}_d-w_x$.

On the SCPD controller, the proportional control input $u_{1\mathrm{p}}$ and differential control input $u_{1\mathrm{d}}$ are defined as follows:

$$\{\begin{array}{l}{u}_{1\mathrm{p}}=z_{1\mathrm{c}}^2{e}_{11}\\ u_{1\mathrm{d}}=2z_{1\mathrm{c}}{e}_{12}\end{array}$$

(7)

where $0<z_{1c}<\infty$ is the iterative speed factor.

According to Equation (7), the x-displacement SCPD controller can be formed:

$$u_x=u_{1\mathrm{p}}+u_{1\mathrm{d}}$$

(8)

3.2. Design of the y-Direction Controller

The purpose of designing the displacement y direction controller $u_y$ is to decouple to get the driving force of the left and right wheels of the robot in the following way.

Let the desired value of the system be $y_d$, and the actual value is $y$. According to this, the tracking error can be obtained $e_{21}=y_d-y$, and differential error $e_{22}={\dot{e}}_{22}={\dot{y}}_d-y_1$, where $y_1=\dot{y}$, combined with the system (2) can be obtained controlled error system as follows:

$$\{\begin{array}{l}{\dot{e}}_{21}=e_{22}\\ {\dot{e}}_{22}={\widehat{w}}_y-u_y\end{array}$$

(9)

where ${\widehat{w}}_y={\ddot{y}}_d-w_y$.

On the SCPD controller, the proportional control input $u_{2\mathrm{p}}$ and differential control input $u_{2\mathrm{d}}$ are defined as follows:

$$\{\begin{array}{l}{u}_{2\mathrm{p}}=z_{2\mathrm{c}}^2{e}_{21}\\ u_{2\mathrm{d}}=2z_{2\mathrm{c}}{e}_{22}\end{array}$$

(10)

where $0<z_{2c}<\infty$ is the iterative speed factor.

According to Equation (10), the y-displacement SCPD controller can be formed:

$$u_y=u_{2\mathrm{p}}+u_{2\mathrm{d}}$$

(11)

3.3. Design of the Angle Direction Controller

The purpose of designing the angle controller $T_{\delta }$ was to decouple the drive forces of the left and right wheels of the robot in the following way.

Let the desired value of the system be ${\theta }_d$, where the actual value is $\theta$. According to this, the tracking error can be obtained $e_{31}={\theta }_d-\theta$, and differential error $e_{32}={\dot{e}}_{31}={\dot{\theta }}_d-{\theta }_1$, where ${\theta }_1=\dot{\theta }$, combined with the system (2) can be obtained controlled error system as follows:

$$\{\begin{array}{l}{\dot{e}}_{31}=e_{32}\\ {\dot{e}}_{32}={\widehat{w}}_{\theta }-T_{\delta }\end{array}$$

(12)

where ${\widehat{w}}_{\theta }={\ddot{\theta }}_d-w_{\theta }$.

On the SCPD controller, the proportional control input $u_{3\mathrm{p}}$ and differential control input $u_{3\mathrm{d}}$ are defined as follows:

$$\{\begin{array}{l}{u}_{3\mathrm{p}}=z_{3\mathrm{c}}^2{e}_{31}\\ u_{3\mathrm{d}}=2z_{3\mathrm{c}}{e}_{32}\end{array}$$

(13)

where $0<z_{3c}<\infty$ is the iterative speed factor.

According to Equation (13) that can be formed y-displacement SCPD controller:

$$T_{\delta }=u_{3\mathrm{p}}+u_{3\mathrm{d}}$$

(14)

3.4. Stability Analysis

To verify that the controllers defined by Equations (8), (11), and (14) can ensure robustness, the stability theorem and its theoretical analysis are as follows. The stability is proven based on the Laplace transform.

Theorem 1.

The x-direction displacement control system (6) based on the SCPD control strategy can guarantee robust stability if $\left|{\widehat{w}}_x\right|\le {\epsilon }_1<\infty$ and $0<z_{1\mathrm{c}}<\infty$, the steady-state errors are then bounded by $\left|e_{11}(\infty )\right|<{\epsilon }_1/z_{1\mathrm{c}}^2$.

Proof. 

Substituting Equations (7) and (8) into the controlled error system (6), the SCPD control system for the x direction displacement can be obtained as:

$$\{\begin{array}{l}{\dot{e}}_{11}=e_{12}\\ {\dot{e}}_{12}=w_x-z_{1\mathrm{c}}^2{e}_{11}-2z_{1\mathrm{c}}{e}_{12}\end{array}$$

(15)

As the SCPD control system (15) is an error-dynamic system under the excitation of the composite total disturbance ${\widehat{w}}_x$, and thus is a causal system. A Laplace transform of the system (15) is taken and the detailed transformation as shown in Appendix A:

$$E_{11}(s)=\frac{1}{{(s+z_{1\mathrm{c}})}^2}{W}_x(s)$$

(16)

From Equation (16), the transfer function of the system is obtained as:

$$H_{11}(s)=\frac{E_{11}(s)}{W_x(s)}=\frac{1}{{(s+z_{1\mathrm{c}})}^2}$$

(17)

when $0<z_{1c}<\infty$, there are two real poles $s_1=s_2=-z_{1c}<0$ on the real axis in the left half-plane of the complex frequency domain $H_{11}(s)$. It is a typical critical system, and thus the system (17) or (15) is stable. Since $z_{1c}$ is independent of the dynamic model of the system, the system (17) or (15) is robustly stable.

The unit impulse response of the system is obtained from Equation (17) as:

$$h_{11}(t)=t{\mathrm{e}}^{-z_{1\mathrm{c}}t}$$

(18)

The time domain form of the controlled error system can be obtained from Equation (18) as:

$$e_{11}(t)=h_{11}(t)\ast w_x(t)={\displaystyle {\int }_0^t{h}_{11}(\tau )}{w}_x(t-\tau )d\tau$$

(19)

where “∗” refers to convolutional integral operation.

When $\left|w_x\right|\le {\epsilon }_1<\infty$, there is:

$$\left|e_{11}(\infty )\right|\le {\epsilon }_1{\displaystyle {\int }_0^{\infty }\left|h_{11}(\tau )\right|}d\tau$$

(20)

From Equation (18), when $0<t<\infty$, $h_{11}(t)>0$, and $h_{11}(\infty )=0$, thus there is:

$${\displaystyle {\int }_0^{\infty }\left|h_{11}(\tau )\right|}d\tau ={\displaystyle {\int }_0^{\infty }{h}_{11}(\tau )}d\tau =H_{11}(0)=\frac{1}{z_{1\mathrm{c}}^2}$$

(21)

Substituting Equation (21) into Equation (20), the steady-state error is obtained as:

$$\left|e_{11}(\infty )\right|\le \frac{{\epsilon }_1}{z_{1\mathrm{c}}^2}$$

(22)

From Equation (22), when the composite total disturbance is bounded, the steady-state error is bounded, independent of the specific model of the controlled system, and thus the SCPD control system (6) has good stability and robustness against disturbances. □

Similarly, it can be proved that the y-direction displacement system (9) and the angle system (12) have good stability and robustness against disturbances.

3.5. Iterative Speed Factor

The speed factor is a control parameter, similar to k_p, k_d in PD control. The speed factor tightly couples the proportional gain and the differential gain, which makes these two functionally different gains show a coordinated control behavior with the same goal during the control process, and effectively improves the gain robustness and anti-disturbance robustness of the control system. Theoretically, the larger the speed factor $z_c$, the faster the dynamic response of the system and the stronger the ability to anti-disturbance. However, if the speed factor is too large, the control force will be too large, and the system will generate overshoot and oscillation. Therefore, in the initial stage of a dynamic response, due to the large tracking error, the speed factor is required to take a smaller value. After entering the steady-state, the speed factor is required to take a larger value to improve the ability of anti-disturbance, so an adaptive speed factor based on the rate of change of the error is designed for this purpose.

$$z_c=z_0\exp (-\beta \left|e_2\right|)$$

(23)

$z_0=20\alpha /(\alpha +1)t_r$ is the minimum speed factor, and $1<\alpha \le 10$, $\beta =1+0.1\alpha$, $z_{1c}=z_{2c}=z_0\exp (-\beta \left|e_2\right|)$, $z_{3c}=2z_{1c}$. $t_r$ is the transition time from the dynamic to the steady state, this paper took $t_r=1$.

The error-based adaptive speed factor is effective in avoiding overshooting phenomenon, but not universal, for different systems have to carry out a certain amount of tuning. To further improve the adaptive control capability of the designed controller in this paper, this paper also combines the parameter calibration with the steepest descent method [24]. The steepest descent method is simple and easy to realize, with small computational effort and fast convergence. Taking the x-direction controller as an example:

Let the performance index be: $J=0.5e_{11}^2(k+1)$, and discretize Equation (6) to obtain:

$$\{\begin{array}{l}{e}_{11}(k)=x_d(k)-x(k)\hfill \\ e_{12}(k)={\dot{e}}_{11}(k)=e_{11}(k)-e_{11}(k-1)\hfill \end{array}$$

(24)

$$u_x(k)=z_{1c}^2{e}_{11}(k)+2z_{1c}{e}_{12}(k)$$

(25)

Combined with the steepest descent method, the iterative speed factor is updated as:

$$\begin{array}{l}\Delta z_{1c}=-{\eta }_i\frac{\partial J}{\partial z_{1c}}=-{\eta }_i\frac{\partial J}{\partial e_{11}(k+1)}\frac{\partial e_{11}(k+1)}{\partial x(k+1)}\frac{\partial x(k+1)}{\partial u_x(k)}\frac{\partial u_x(k)}{\partial z_{1c}}\\ \begin{array}{cc}& \end{array}={\eta }_i{e}_{11}(k+1)\frac{\partial x(k+1)}{\partial u_x(k)}\left[2z_{1c}{e}_{11}(k)+2e_{12}(k)\right]\end{array}$$

(26)

As $e_{11}(k+1)$ and $\frac{\partial x(k+1)}{\partial u_x(k)}$ are related to future errors and future states, it makes Equation (26) difficult to compute. However, as long as the algorithm is convergent, the inequality $\left|e(k+1)\right|\le \left|e(k)\right|$ must be valid. Therefore, in this paper, $e(k)$ is used to replace $e(k+1)$. Also, the size of $\frac{\partial x(k+1)}{\partial u_x(k)}$ determines the convergence speed, and the sign determines the direction of convergence. Therefore, this paper uses the sign function instead, which is:

$$\{\begin{array}{l}{x}_{u_x}(k)=\frac{\partial x(k+1)}{\partial u_x(k)}\approx sign[\frac{x(k)-x(k-1)}{u_x(k-1)-u_x(k-2)}],u_x(k-1)\ne u_x(k-2)\hfill \\ x_{u_x}(k)=\frac{\partial x(k+1)}{\partial u_x(k)}\approx sign[x(k)-x(k-1)],\begin{array}{cc}& \end{array}{u}_x(k-1)=u_x(k-2)\hfill \end{array}$$

(27)

According to Equations (26) and (27), the adaptive parameter updating formula can be obtained as:

$$z_{1c}=z_{1c}+{\eta }_i{e}_{11}(k)x_{u_x}(k)\left[2z_{1c}{e}_{11}(k)+2e_{12}(k)\right]$$

(28)

where $0<{\eta }_i<1$ is the learning rate, this paper took ${\eta }_i=0.2$.

Similarly, it can be obtained:

$$z_{2c}=z_{2c}+{\eta }_i{e}_{21}(k)y_{u_y}(k)\left[2z_{2c}{e}_{21}(k)+2e_{22}(k)\right]$$

(29)

$$z_{3c}=z_{3c}+{\eta }_i{e}_{31}(k){\theta }_{T_{\delta }}(k)\left[2z_{3c}{e}_{31}(k)+2e_{32}(k)\right]$$

(30)

According to the above equation, when adjusting the controller parameters, the gain parameter only involves multiplication and addition operations, which is a small amount of calculation. And, the speed factor couples the parameters of proportional gain and differential gain together.

The error-based adaptive speed factor $1<\alpha \le 10$, while $\alpha$ is smaller, $z_0$ is larger, and $z_c$ is larger. The larger the speed factor, the greater the stability, the greater the disturbance resistance, and the higher the control accuracy, but the larger the overshoot will also be, which may cause the control force to oscillate. Therefore, two different adaptive speed factors, $\alpha =1$ adaptive speed factor 1 and $\alpha =10$ adaptive speed factor 2, were selected for comparison experiments with the iterative speed factor. The simulation was tested in Matlab/Simulink 2021. A sudden change signal was added to simulate the perturbation added to the robot when t = 0.3 s, and a square wave signal was added to simulate the perturbation added to the motor during 5–6 s. The simulation result of the path tracking is shown in Figure 6. From the path following the error in Figure 7, it can be seen that the error-based adaptive speed factor cannot eliminate the jitter phenomenon, while the iterative speed factor can eliminate it. From Figure 8, it can be seen that the control force of the iterative speed factor is smoother and the vibration amplitude is smaller.

4. Simulation of Path Following Control

4.1. Practical Disturbance of Underground Cable Trenches

The caterpillar robot in this paper was moved in the underground cable trench environment. Underground cable trench surfaces were characterized by uneven road surfaces, uneven friction coefficients, and random distribution of debris. These related random factors had different degrees of effect on the caterpillar robot. To get the effect of the disturbance, the caterpillar robot moved straight and explored the change of speed under a complex road environment, where the speed was 0.2 m/s.

As can be seen in Figure 9, in the actual cable trench environment, the minimum speed value of the crawler motor was 0.17 m/s and the maximum speed value was 0.23 m/s when it was perturbed. So, the influence of the disturbance factor on the speed of the caterpillar motor was ±0.3 m/s. It provided a basis for the subsequent simulation of the controller.

4.2. Simulation

The simulation was based in Matlab/Simulink. The response speed, overshoot, control accuracy, robustness, and anti-disturbance ability of incremental PID, SMC, and SCPD were tested based on the good operation of the simulation. The underground cable trench was generally a narrow channel, so in this paper, the slope signal was selected as the desired path to follow. According to the analysis in Section 4.1, the square wave with an amplitude of 0.3 was selected to simulate the disturbance during 5–6 s.

After tuning the parameters according to several attempts, the simulation results are shown in Figure 10. Figure 11 and Figure 12 show the details of overshoot and perturbation, respectively.

From Figure 11, it is shown that SCPD had the smallest overshoot of 0.15 and stabilized in 0.2 s, while SMC and incremental PID had overshoots of 0.22 and 0.51, which stabilized in 0.45 s and 0.7 s, respectively. It can be seen that the response time of SCPD was shorter and the robustness was higher.

From Figure 12, the SCPD had the strongest anti-disturbance ability and recovered to the desired path very fast, the SMC control also had good anti-disturbance ability, while the incremental PID had poor anti-disturbance ability and could not recover to the desired path before the disturbance disappeared.

5. Experiment-Field Test

To prove the effectiveness of the SCPD algorithm proposed in this paper, the experiment was carried out in an underground cable trench environment, as shown in Figure 13. This experiment aimed to verify the response speed, control accuracy, and disturbance robustness of the three control algorithms. Since the robot had a host computer for localization, mapping, and path planning, the underground cable trench was mainly narrow and straight paths. To ensure the convenience of the experiment, the design strategy of this paper was to send a straight-line command to control the speed and accuracy of the dual-motor caterpillar robot. Meanwhile, an obstacle was added at 5-6 s to test the robustness and anti-disturbance ability of the control algorithms. This experiment used STM32f1 as the chassis controller and connects the motor driver to achieve the purpose of driving the motor. The two-wheel differential robot kinematics model was adopted to operate, and the effect of the control algorithm was assessed by observing the change in the angle of the robot. The angle remained consistent when the robot ran in a straight line.

The trajectories run by the different control algorithms at the underground cable trench site are shown in Figure 14. The purple square in the figure shows the dynamic obstacles to avoid collision between the robot and the cable, and the blue square shows the obstacles after the expansion coefficient. From Figure 15, the robot had the highest smoothness under SCPD control, which greatly reduced the total length of the path. To further verify the reliability of the algorithm, different starting and ending points were set in the cable trench environment with distances of 10 m, 20 m, and 30 m, respectively, and the results are shown in

Table 2. Comparison of field experiments with different control algorithms.

Algorithm	Detection Channel Length/(m)	Average Time/(s)	Average Path Length/(m)
Incremental PID	10	79.4	12.2
	20	164.3	24.6
	30	240.7	35.7
SCPD	10	61.2	11.4
	20	131.6	21.8
	30	197.8	32.7

. Compared with incremental PID, SCPD increased the robot’s moving speed by 21% and reduced the path by 6.5%.

The superiority of the SCPD algorithm can be seen in Figure 15. Although incremental PID, SMC, and SCPD are stable under regular operation, compared with incremental PID and SMC, SCPD was more stable at the beginning of the operation and had better control accuracy. When crossing an obstacle, the angle of the incremental PID varied the most, within an adjustment range of ±4.7 degrees; the angle of the SMC varied within an adjustment range of ±3.2 degrees; and the angle of the SCPD varied within an adjustment range of approximately −3 degrees to 1.5 degrees. So, the maximum offset angle of SCPD was 56.7% and 10.3% smaller than incremental PID and SMC, respectively. Furthermore, although the three algorithm’s variation in adjustment time was not significant, the SCPD was also the best one.

In the comprehensive evaluation, compared with incremental PID and SMC, SCPD had the advantages of fast response, low overshoot, strong robustness, and strong anti-disturbance ability. In addition, the parameter of SCPD was easier to tune than incremental PID and SMC control algorithms because it had only one parameter.

6. Conclusions

The underground cable trench inspection robot usually needs to operate in a narrow environment with various uncertainties such as uneven road surface, uneven coefficient of friction, and a random distribution of cables. However, both the cables and the robot are valuable and cannot afford to be lost due to collisions. Compared with other control methods, the performance of the SCPD control method in this paper, such as high accuracy, strong robustness, and fast response, can better solve this problem. In addition, to make the SCPD control more accurate and efficient as well as the parameter adjustment easier, the speed factor was updated and iterated in this paper. The experiment proves that the SCPD control algorithm with an optimized velocity factor is an effective and more suitable algorithm for inspection robots in an underground cable trench environment.

This method improves the development of cable trench inspection. First, the caterpillar chassis is more adaptable to the complex environment of an underground cable trench. Then, the SCPD control algorithm can achieve error-free path following. In addition, the iteration speed factor makes the control input smoother and the vibration amplitude smaller. In conclusion, this method makes it simple to tune the parameters and there are no out-of-control phenomena during rapid robot detection.