Adaptive neural network tracking control for unmanned electric shovel intelligent excavation system

Abstract

This study proposes an adaptive control strategy for unmanned mining shovel digging trajectory tracking based on radial basis function neural network (RBFNN) and a class of unmanned mining shovel time-varying systems with model uncertainty and external disturbances. A new set of Lagrangian dynamics differential equations is reconstructed by utilizing the kinematic model of the electric shovel and considering external disturbances along with modeling uncertainties. This approach lays the groundwork for subsequent adaptive controllers. The proposed controller is designed to regulate the position errors of the unmanned mining electric shovel system, which is characterized by a complex structure, high load, large size, and strong coupling. It takes the deviation values and their derivatives of the lifting and pushing movements as inputs and adjusts the output torque to converge the bucket position to the desired trajectory. The controller utilizes the RBFNN in the control law to compensate for uncertainties in this type of system with large disturbances and inertia. This compensation helps eliminate the impact of external disturbances and modeling uncertainties on the unmanned mining electric shovel’s ability to follow the excavation trajectory. The consistent boundedness of the closed-loop system’s ultimate limits is proven through Lyapunov stability theory. Finally, the effectiveness of the proposed solution is validated through simulation experiments.

Keywords

adaptive control / RBFNN / unmanned electric shovel / trajectory tracking

1 Introduction

The mining mechanical front shovel excavator, also known as a mining electric shovel, is a large-scale engineering equipment that combines excavation and loading in open-pit mining. Its performance directly determines the efficiency and safety of the entire mining operation. At present, mining electric shovels rely entirely on manual operation, and the challenging working environment contributes to issues such as low operational efficiency, elevated labor costs, and increased risks to personal safety. The rapid advancement of computer technology, sensing techniques, and artificial intelligence results in the paradigm shift of the traditional mining electric shovels toward intelligent and unmanned systems. This transition is considered an imperative choice to enhance operational efficiency, reduce operational energy consumption, and mitigate the occurrence of human-induced safety incidents [1–11]. In pursuit of achieving unmanned autonomous operations for mining electric shovels, a sequence of foundational research and exploration endeavors is necessitated to drive the transformation process toward unmanned capabilities [12]. Numerous researchers have conducted studies on excavators, with the primary focus being the intelligent design domain. In the realm of excavator intelligence, research efforts are predominantly concentrated on aspects such as trajectory planning and tracking control.

In the realm of trajectory planning, Zhang et al. [13,14] have proposed a multi-objective trajectory optimization framework. This framework is built on the pseudospectral method to establish a multi-objective optimization model. It enables the determination of optimal excavation trajectories and control variables for autonomous mining scenarios. Wang et al. [15] proposed a model-based trajectory planning method to address the limitations of traditional trajectory planning methods, which are constrained by curve types and cannot adequately or clearly obtain optimal trajectories. This approach considers time-related data as design variables for the optimal control problem and utilizes optimization solvers to obtain the optimal excavation trajectory. Compared with conventional trajectory planning methods, their proposed method demonstrates superior excavation performance and exhibits high flexibility under various working conditions. Fan et al. [16] proposed a multi-objective trajectory optimization method for hydraulic excavators to minimize energy consumption, execution time, and excavation volume. They employed a decomposed hybrid constrained multi-objective evolutionary algorithm to address the interdependencies between optimization objectives. Their method effectively mitigates adverse effects during the optimization process, enhances efficiency, and yields high-quality excavation trajectories. Fan et al. [17] investigated a novel electro-hydraulic composite cable shovel’s working trajectory planning. This investigation involved modeling the excavation process using shovel bucket kinematics and dynamics methods, encompassing dynamic loads on the shovel bucket and excavation resistance. A trajectory planning method utilizing segmental cubic polynomials to describe crowd and hoist speeds was implemented to minimize energy consumption per unit mass, thereby achieving the most energy-efficient trajectory for the specified excavation conditions. Zhang et al. [18] and Fu et al. [19] analyzed the physical and geometric constraints involved in operational scenarios. They devised a multi-objective optimization model incorporating excavation and loading processes. Their proposed multistage multi-objective collaborative design optimization strategy aims to enhance mining efficiency and minimize energy consumption. Unlike conventional optimization methods, their approach optimizes operational performance by simultaneously considering excavation and loading processes. Zhang et al. [20] utilized a sequential quadratic programming algorithm to address challenges such as local optima and low efficiency encountered by certain optimization algorithms in hydraulic excavator robot trajectory planning. This method demonstrates high computational efficiency, smooth optimal excavation trajectories, and gradual node convergence to target points, thereby minimizing impact during operation and saving working time. This outcome leads to improved stability and efficiency in the autonomous operation of excavators. Feng et al. [21] responded to the need for online planning of intelligent bucket excavation trajectories by introducing a multirobot system excavation trajectory planning method based on material surface perception. This method optimizes energy consumption and excavation time per unit mass as objective functions, thereby incorporating motor performance and geometric dimensions of the mining rope shovel as constraints. The proposed method demonstrates lifting and pushing forces consistent with real-world conditions. Thus, it can adapt to diverse excavation scenarios.

In the realm of tracking control, numerous researchers extensively employ various control algorithms. Dao et al. [22] proposed an extended state observer (ESO) to address tracking errors in excavator systems. They combined Lyapunov functions with an inversion method to develop a controller aimed at reducing contour error, tangential error, and directional error. This controller can handle unmeasurable velocities, aggregated disturbances, and uncertainties. The method is designed based on accurate mathematical models but does not consider the influence of the system itself. Hoang et al. [23] proposed a controller design method combining feedback linearization based on dynamic models with sliding mode control to address the issue of intensified vibration caused by the elastic properties of the ground. This method reduces vibration when the system operates on an elastic foundation while enhancing the precise tracking performance of the link mechanism. However, this approach is sensitive to vibration perception and exhibits rapid output force changes during control. Thus, it may not ensure smooth operation, particularly for heavy-duty equipment like unmanned electric shovels. Feng et al. [24] proposed a fuzzy adaptive sliding mode controller to enhance trajectory tracking accuracy and robustness in heavy-duty electro-hydraulic position systems. This controller addresses the chattering phenomenon caused by nonlinear switching terms using fuzzy switching techniques. It also provides 25 fuzzy rules for adjusting the sliding mode switching controller, thereby enabling the rapid and smooth tracking of the reference trajectory. However, the method solely relies on trigonometric functions as inputs and overlooks additional factors present in practical excavation scenarios, thereby underscoring the need for trajectories that can accurately represent real-world conditions. Egli et al. [25] presented a data-driven automatic arm control method for hydraulic excavators based on reinforcement learning. This approach involves directly applying control strategies trained in simulation to physical excavators, thereby achieving accurate and stable position tracking. The control strategy of this method involves outputting actuator commands and applying them directly to the machine without arbitrary filtering or modification, thereby enabling rapid control. However, this method requires a large number of samples for simulation and exhibits poor robustness. Thus, it is unsuitable for various real-world environments encountered in mining operations. Fan and Li [26] introduced an adaptive control system for electro-hydraulic shovels to tackle the challenge of unmeasurable full-state variables. This system comprises a terminal sliding mode controller and a novel neural network controller. The control law relies solely on the position signal of the system, which exhibits excellent adaptability to aggregated uncertainties, high computational efficiency, and good performance. Qin et al. [27] focused on the hydraulic arm system of a specific mining hydraulic excavator by examining its impact on control design. They introduced an adaptive robust impedance controller based on an ESO and the backstepping method. This controller demonstrates excellent position-tracking performance and minimized velocity fluctuations. Compared with systems lacking flow regeneration valves, the hydraulic arm system of a specific mining hydraulic excavator system reduces energy consumption by 27.32%. Lee et al. [28] introduced a precision motion control framework for industrial hydraulic excavator robots through data-driven model inversion techniques. They employed multiple neural networks to approximate the entire dynamics of the excavator, encompassing input delays and dead zones. Their framework achieves precise tracking performance even in the presence of severe soil interactions by constructing a modularly structured, physically inspired data-driven model. Although this method combines accurate mathematical models and neural networks for excavator control, not only external disturbances but also internal perturbations must be considered. Sandzimier and Asada [29] developed a data-driven automatic excavator statistical control method to address the highly complex and nonlinear interaction forces between the excavator bucket and the soil. This method considers the desired bucket fill factor and its associated variance, thereby enabling the excavator to collect the required volume of soil with high confidence.

In summary, extensive research has been conducted by numerous scholars on various types of excavator trajectory planning and control tracking methods. The performance of equipment designed following these studies has undergone comprehensive improvement. However, the aforementioned methods are all based on precise mathematical models, which are studied in conjunction with specific objectives. Thus, the effects of inaccuracies in dynamic models and uncertainties during operation are overlooked. As the primary excavation equipment in open-pit mines, unmanned electric shovels feature a complex structure comprising numerous components that are closely interlinked. Consequently, an accurate mathematical model necessitates integration with the specific dimensional specifications of each component. However, various factors influence the components during actual production, resulting in differences between the dimensions and weights of the finished product and the ideal state. Moreover, the forces acting between components during operation cause deformation, further complicating the acquisition of an accurate dynamic model for electric shovels. In response to this situation, we addressed in this study the errors and uncertainties in the model by introducing model error terms into the dynamic model and designing corresponding controllers. As a crucial component of unmanned electric shovels, excavator trajectory tracking control requires consideration of the impact of various uncertain factors on tracking performance. Therefore, on the basis of prior research, we devised an adaptive neural network tracking control for an unmanned electric shovel intelligent excavation system. This system is designed to handle trajectory tracking control for unmanned electric shovel systems, accounting for disturbances and uncertainties arising from their inherent variability and environmental factors.

The key contributions of this study can be outlined as follows: (1) Unlike the approaches by Dao et al. [22], Lee et al. [28], and Sandzimier and Asada [29], our research investigates tracking control under nonprecise mathematical models. We incorporated some of the inherent disturbances of unmanned electric shovels into the controller design, thereby integrating disturbance terms into the design process. (2) In contrast to the methods proposed by Hoang et al. [23] and Qin et al. [27], our approach compensates for disturbance terms using neural networks, resulting in smooth controller outputs and stabilized motor operations. This outcome minimizes abrupt changes in the output force, thereby promoting the stable operation of unmanned electric shovels. (3) Finally, experimental validation conducted with optimal excavation trajectories planned in real mining environments confirms the practical feasibility of our proposed method.

The remaining sections of this paper are organized as follows: Section 2 provides an introduction to the components of the unmanned electric shovel system, with the focus being the excavation system. It analyzes the system and establishes the Lagrange dynamics differential equations. Section 3 reestablishes the Lagrange dynamics differential equations based on the kinematic model of the electric shovel and by considering external disturbances and modeled uncertainties. This foundation is used as the basis for introducing uncertainty to formulate adaptive control laws. The radial basis function neural network (RBFNN) is utilized to fit and compensate for disturbance terms by constructing an RBFNN adaptive controller along with adaptive laws. Section 4 constructs Lyapunov functions for the control laws and adaptive laws, and the stability of the controller based on Lyapunov’s theorem is analyzed. In Section 5, a simulation model is built in Simulink by using the optimal trajectory as input for tracking simulation. The path-tracking performance is evaluated based on the system states. The simulation results are analyzed to demonstrate that the proposed controller meets the control requirements. Section 6 concludes the study and summarizes the overall process of the research.

2 Unmanned electric shovel

2.1 System components

As a key component of intelligent mining operations, the unmanned mining electric shovel plays a crucial role in enhancing safety, thereby reducing costs, improving efficiency, and conserving energy in mining production. This equipment primarily consists of a motion system, an excavation system, and a main control room. The motion system comprises a crawler chassis and a motion power system, whereas the excavation system consists of an arm structure, lifting wheels, hoisting rope, bucket arm, bucket, transmission mechanism, and excavation power system. The main control room mainly includes environmental perception, intelligent decision-making, and execution. It is composed of industrial control computers (IPCs), an electrical control room, programmable logic controllers (PLCs), laser radar, and a positioning and orientation system, as shown in Fig.1.

Fig.1 WK-12 mining electric shovel.

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During the operation of the unmanned electric shovel, the system first utilizes laser radar, a positioning and orientation system, and various sensors (such as inclinometers, position sensors, and speed sensors) to conduct a comprehensive survey of the electric shovel and its surrounding environment. This process acquires detailed material information and precise positional data, providing a foundation for intelligent decision-making. Then, as the core of the system, the IPC integrates its attitude information with environmental data to perform optimization calculations on the excavation path. This process generates an optimal excavation trajectory that is efficient and precise. Then, the trajectory information is transmitted to the PLC. Finally, the PLC issues commands through the electrical control room to adjust the operation of the drive motors, thereby enabling the bucket to follow the predetermined excavation trajectory. The structure diagram of the unmanned electric shovel system is shown in Fig.2. The unmanned electric shovel ensures efficient and precise performance in complex working environments through this series of coordinated operations.

Fig.2 Structure diagram of the unmanned electric shovel system.

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2.2 Simplified kinematic models

The electric shovel is characterized by its large volume, heavy weight, and complex system. Thus, it faces challenges in mitigating disturbances generated during the excavation process. The impediments posed by unmanned mining electric shovels to excavation are necessary for smooth operations in mining activities. Hence, this study simplifies the structure of the electric shovel and establishes a two-dimensional coordinate system with the excavation trajectory starting point as the origin to develop a kinematic model.

The dynamic equations of a robot can typically be derived using either Newton–Euler formulas or Lagrange dynamics. The Lagrange dynamics formula can concisely and clearly express its structural characteristics for the working device of an unmanned electric shovel, which has only two degrees of freedom and involves complex mechanical calculations. Therefore, the Lagrange dynamic differential equations were established in this study by considering external disturbances and modeling uncertainties. The structure of the electric shovel is shown in Fig.3.

Fig.3 Schematic of the electric shovel structure.

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The mining electric shovel is a strongly coupled, high-inertia system with multiple inputs and outputs. The angle between the hoisting and thrusting motions varies with movement, and these two motions mutually influence each other. During the excavation process, the hoisting motion and thrusting motion hinder each other. The thrusting motion not only needs to counteract the resistance caused by the hoisting motion but also must overcome the radial cutting forces imposed by the minerals. Moreover, the primary motion of bucket hoisting involves lifting the bucket and materials. In this process, consideration must be given to the influence of materials, thrusting motion, and tangential cutting forces. Given the mutual coupling and complex structure of both motions, involving the collaborative operation of multiple components is unfavorable for subsequent controller design. Therefore, accounting for the influences of different components on the bucket tooth tip and decoupling the system are essential in establishing the kinematic equations. The thrusting structure utilizes the thrusting gear to control the extension of the dipper handle. This structure can be conceptualized as a linear module rotating around the thrusting gear, as illustrated in Fig.4.

Fig.4 Thrusting structure.

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The dynamic model in this study is constructed based on the structure shown in Fig.4. Tab.1 shows a detailed explanation of all variable symbols in the figure.

Tab.1 Explanation of symbols for the thrusting structure

Symbol	Detailed explanation
r	Telescopic length of the bucket arm
θ	Angle of the bucket arm rotation
Lba	Length of the bucket arm
Lb	Length of the bucket
mba	Mass of the bucket arm
mb	Mass of the bucket
ω	Angle between the hoisting rope and the bucket arm
α	Angle between the centerline of the boom and the vertical line

The Lagrange equation for the thrusting structure can be expressed as

(1)$\frac{\text{d}}{\text{d}t}\frac{\mathrm{\partial }\varsigma }{\mathrm{\partial }\dot{\mathit{q}}}-\frac{\mathrm{\partial }\varsigma }{\mathrm{\partial }\mathit{q}}=\mathit{\tau },$

where $\varsigma =E-E_{\text{p}}$ represents the Lagrange function, E denotes the system’s kinetic energy, $E_{\text{P}}$ represents the system’s potential energy, $\mathit{q}=[\begin{array}{cc}{q}_1& q_2\end{array}]$ represents the generalized coordinates, and $\dot{\mathit{q}}=[\begin{array}{cc}{\dot{q}}_1& {\dot{q}}_2\end{array}]$ represents the generalized velocities.

When the premise of the hoisting cable mass is not considered, the system’s kinetic energy E is composed of two parts: the bucket arm kinetic energy $E_{\text{ba}}$ and the bucket kinetic energy $E_{\text{b}}$. According to the parallel axis theorem, the following can be observed:

(2)$\{\begin{array}{rl}& E_{\text{ba}}={\displaystyle \frac{1}{2}}{m}_{\text{ba}}{\dot{r}}^2+{\displaystyle \frac{1}{2}}{m}_{\text{ba}}{\dot{\theta }}^2{r}^2-{\displaystyle \frac{1}{2}}{m}_{\text{ba}}{L}_{\text{ba}}{\dot{\theta }}^{2}r+{\displaystyle \frac{1}{6}}{m}_{\text{ba}}{L}_{\text{ba}}^2{\dot{\theta }}^2,\\ & E_{\text{b}}={\displaystyle \frac{1}{2}}{m}_{\text{b}}{\dot{r}}^2+{\displaystyle \frac{1}{2}}{m}_{\text{b}}{\dot{\theta }}^2{r}^2-{\displaystyle \frac{1}{2}}{m}_{\text{b}}{L}_{\text{b}}{\dot{\theta }}^{2}r+{\displaystyle \frac{1}{6}}{m}_{\text{b}}{L}_{\text{b}}^2{\dot{\theta }}^2,\end{array}$

where r is the telescopic length of the bucket arm, θ is the angle of the bucket arm rotation, $L_{\text{ba}}$ is the length of the bucket arm, $L_{\text{b}}$ is the length of the bucket, $m_{\text{ba}}$ is the mass of the bucket arm, and $m_{\text{b}}$ is the mass of the bucket.

The total potential energy of the bucket arm and the bucket can be expressed as follows by taking the x-axis of the $\left\{O\right\}$ coordinate system as the potential energy zero point:

(3)$\begin{array}{rl}{E}_{\text{p}}=& m_{\text{ba}}g\cdot \left[H-\cos \theta \cdot \left(r-\frac{L_{\text{ba}}}{2}\right)\right]\\ & +m_{\text{b}}g\cdot \left[H-\cos \theta \cdot \left(r+\frac{L_{\text{b}}}{2}\right)\right],\end{array}$

where the gravitational constant is g = 9.8 m/s².

The generalized forces for the relative generalized coordinates θ and r can be obtained as follows: based on the established generalized $\left\{O_1\right\}$ coordinates and by utilizing the principle of virtual work,

(4)$\{\begin{array}{l}{\tau }_1=F_{\text{r}}\cdot r\cdot \sin \omega ,\\ {\tau }_2=F_{\text{h}}-F_{\text{r}}\cdot \cos \omega ,\end{array}$

where $\omega =\arcsin \left[L_{\text{bm}}\sin \left(\alpha -\theta \right)/\phantom{L_{\text{bm}}\sin \left(\alpha -\theta \right)\sqrt{L_{\text{bm}}^2+r^2-2L_{\text{bm}}r\cos (\alpha -\theta )}}\sqrt{L_{\text{bm}}^2+r^2-2L_{\text{bm}}r\cos (\alpha -\theta )}\right]$ represents the angle between the hoisting rope and the bucket arm, and α = 135° represents the angle between the centerline of the boom and the vertical line. $F_{\text{r}}$ is the lift output, $F_{\text{h}}$ is the push output, and $L_{\text{bm}}$ is the distance between the lifting wheel and the pushing mechanical device.

The hoisting cable transforms the torque from the motor into the lifting force of the bucket through the pulley. The wrapping range between the cable and the pulley varies as the cable wraps around the pulley and the bucket is in different positions. Consequently, different positions slightly impact the angle between the hoisting rope and the bucket arm, as illustrated in Fig.5.

Fig.5 Angle between thrusting and hoisting movements.

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The Lagrange dynamic differential equation can be expressed as

(5)$\{\begin{array}{l}\begin{array}{c}\left[m_{\text{ba}}\left(r^2-L_{\text{ba}}r+\frac{1}{3}{L}_{\text{ba}}^2\right)+m_{\text{b}}\left(r^2+L_{\text{b}}r+\frac{1}{3}{L}_{\text{b}}^2\right)\right]\frac{\ddot{\theta }}{r\cdot \sin \omega }+\left[2\left(m_{\text{ba}}+m_{\text{b}}\right)r-\left(m_{\text{ba}}{L}_{\text{ba}}-m_{\text{b}}{L}_{\text{b}}\right)\right]\frac{\dot{\theta }\cdot \dot{r}}{r\cdot \sin \omega }\\ +\left[m_{\text{ba}}g\left(r-\frac{L_{\text{ba}}}{2}\right)\sin \theta +m_{\text{b}}g\left(r+\frac{L_{\text{b}}}{2}\right)\sin \theta \right]\frac{1}{r\cdot \sin \omega }=F_{\text{r}},\end{array}\\ \begin{array}{c}\left[m_{\text{ba}}\left(r^2-L_{\text{ba}}r+\frac{1}{3}{L}_{\text{ba}}^2\right)+m_{\text{b}}\left(r^2+L_{\text{b}}r+\frac{1}{3}{L}_{\text{b}}^2\right)\right]\frac{\ddot{\theta }}{r\tan \omega }+\left(m_{\text{ba}}+m_{\text{b}}\right)\ddot{r}\\ -\left[\left(m_{\text{ba}}+m_{\text{b}}\right)\cdot r-\frac{1}{2}\left(m_{\text{ba}}{L}_{\text{ba}}-m_{\text{b}}{L}_{\text{b}}\right)\right]{\dot{\theta }}^2+\left[2\left(m_{\text{ba}}+m_{\text{b}}\right)r-\left(m_{\text{ba}}{L}_{\text{ba}}-m_{\text{b}}{L}_{\text{b}}\right)\right]\frac{\dot{\theta }\cdot \dot{r}}{r\tan \omega }\\ -\left(m_{\text{ba}}+m_{\text{b}}\right)g\cos \theta +\left[m_{\text{ba}}g\left(r-\frac{L_{\text{ba}}}{2}\right)\sin \theta +m_{\text{b}}g\left(r+\frac{L_{\text{b}}}{2}\right)\sin \theta \right]\frac{1}{r\tan \omega }=F_{\text{h}}.\end{array}\end{array}$

Let $\mathit{q}=[\begin{array}{cc}\theta & r\end{array}{]}^{\mathrm{T}}$. According to the general form of the Lagrange equation, the above differential equation is reconstructed. The terms related to $\ddot{\mathit{q}}$ as $\mathit{M}\left(\mathit{q}\right)$ are denoted, and the remaining terms based on whether they are related to $\dot{\mathit{q}}$ into $\mathit{C}\left(\mathit{q},\dot{\mathit{q}}\right)$ and $\mathit{G}\left(\mathit{q}\right)$ are classified. The Lagrange dynamic differential equation becomes

(6)$\mathit{M}\left(\mathit{q}\right)\ddot{\mathit{q}}+\mathit{C}\left(\mathit{q},\dot{\mathit{q}}\right)\dot{\mathit{q}}+\mathit{G}\left(\mathit{q}\right)=\mathit{\tau },$

where $\mathit{M}\left(\mathit{q}\right)\in {\mathbb{R}}^{2\times 2}$ is a positive definite inertia matrix, $\mathit{C}\left(\mathit{q},\dot{\mathit{q}}\right)\in {\mathbb{R}}^{2\times 2}$ is the matrix of centrifugal and Coriolis force coefficients, $\mathit{G}\left(\mathit{q}\right)\in {\mathbb{R}}^{2\times 1}$ represents the gravitational term, and $\tau \in {\mathbb{R}}^{2\times 1}$ is the applied generalized force on the system. The detailed expressions for each element in the equation are as follows:

(7)$\{\begin{array}{l}{M}_{11}=\left[m_{\text{ba}}\left(r^2-L_{\text{ba}}r+{\displaystyle \frac{1}{3}}{L}_{\text{ba}}^2\right)+m_{\text{b}}\left(r^2+L_{\text{b}}r+{\displaystyle \frac{1}{3}}{L}_{\text{b}}^2\right)\right]{\displaystyle \frac{1}{r\cdot \sin \omega }},\\ M_{12}=0,\\ M_{21}=\left[m_{\text{ba}}\left(r^2-L_{\text{ba}}r+{\displaystyle \frac{1}{3}}{L}_{\text{ba}}^2\right)+m_{\text{b}}\left(r^2+L_{\text{b}}r+{\displaystyle \frac{1}{3}}{L}_{\text{b}}^2\right)\right]{\displaystyle \frac{1}{r\tan \omega }},\\ M_{22}=m_{\text{ba}}+m_{\text{b}},\\ C_{11}=0,\\ C_{12}=\left[2\left(m_{\text{ba}}+m_{\text{b}}\right)r-\left(m_{\text{ba}}{L}_{\text{ba}}-m_{\text{b}}{L}_{\text{b}}\right)\right]{\displaystyle \frac{\dot{\theta }}{r\cdot \sin \omega }},\\ C_{21}=-\left[\left(m_{\text{ba}}+m_{\text{b}}\right)\cdot r-{\displaystyle \frac{1}{2}}\left(m_{\text{ba}}{L}_{\text{ba}}-m_{\text{b}}{L}_{\text{b}}\right)\right]\dot{\theta },\\ C_{22}=\left[2\left(m_{\text{ba}}+m_{\text{b}}\right)r-\left(m_{\text{ba}}{L}_{\text{ba}}-m_{\text{b}}{L}_{\text{b}}\right)\right]{\displaystyle \frac{\dot{\theta }}{r\tan \omega }},\\ G_1=\left[m_{\text{ba}}g\left(r-{\displaystyle \frac{L_{\text{ba}}}{2}}\right)\sin \theta +m_{\text{b}}g\left(r+{\displaystyle \frac{L_{\text{b}}}{2}}\right)\sin \theta \right]{\displaystyle \frac{1}{r\cdot \sin \omega }},\\ G_2=-\left(m_{\text{ba}}+m_{\text{b}}\right)g\cos \theta +\left[m_{\text{ba}}g\left(r-{\displaystyle \frac{L_{\text{ba}}}{2}}\right)\sin \theta +m_{\text{b}}g\left(r+{\displaystyle \frac{L_{\text{b}}}{2}}\right)\sin \theta \right]{\displaystyle \frac{1}{r\tan \omega }}.\end{array}$

The dynamic model of the unmanned electric shovel exhibits the following dynamic characteristics.

Property 1 The positive definite inertia matrix $\mathit{M}\left(\mathit{q}\right)$ is bounded for $\mathrm{\forall }\mathit{q}\in {\mathbb{R}}^n$. This observation means that finite positive constants m₁ and m₂ exist such that $m_1{\Vert \mathit{x}\Vert }^2⩽{\mathit{x}}^{\mathrm{T}}\mathit{M}\left(\mathit{q}\right)\mathit{x}⩽m_2{\Vert \mathit{x}\Vert }^2$.

Property 2 In practical processes, $\mathit{G}\left(\mathit{q}\right)\in {\mathbb{R}}^n$ and $\mathit{C}\left(\mathit{q},\dot{\mathit{q}}\right)\in {\mathbb{R}}^n$ are always bounded for $\mathrm{\forall }\mathit{q}\in {\mathbb{R}}^n$.

3 Controller design

The development and deployment phases of unmanned electric shovels face unique challenges. During the development phase, the actual dimensions and weights of components often differ from the design specifications. These discrepancies accumulate during assembly, thereby leading to significant deviations from the ideal state that can severely impact equipment performance. In the deployment phase, various factors can cause deviations in the mathematical model. For example, the WK-12 model has a maximum excavation capacity of 12 m³ per load; in coal mining, each load can exceed 2 × 10⁴ kg. During material excavation, the bucket must bear the weight of the excavated material. The heavy load can cause deformation of critical load-bearing components, such as the hoisting rope, bucket arm, and boom, and these deformations are difficult to predict accurately. Additionally, prolonged operation leads to severe wear of the bucket teeth, thereby necessitating the replacement of the teeth or even the entire bucket.

These factors introduce uncertainties into the mathematical model that must be considered during the control process. Deformation of components can cause deviations in the dynamic model, thereby preventing the controller from accurately calculating the required drive force output. This scenario can result in significant trajectory tracking errors, potentially leading to equipment damage and reduced safety. An adaptive control system based on RBFNN is proposed to address these issues. This system approximates the uncertainties in the system and compensates for them in the control law, thereby mitigating their impact on system performance.

3.1 RBFNN

Given the powerful approximation capability of RBFNN, this study employs artificial neural networks to identify this nonlinear function. The closed-loop system is globally stable, and the controller is designed based on the nominal model of the mining electric shovel. RBFNN exhibits strong self-learning ability, which allows it to approximate accurately the various complex nonlinear functions and uncertainties f(x) with arbitrary precision. The approximation error ε of RBFNN is considered; thus, f(x) can be represented by the following [30]:

(8)$\{\begin{array}{c}f\left(\mathit{x}\right)=W^{\mathrm{T}}\mathit{h}\left(\mathit{x}\right)+\epsilon ,\\ h_j\left(\mathit{x}\right)=\exp \left(-\frac{{\Vert \mathit{x}-{\mathit{c}}_j\Vert }^2}{2b_j^2}\right),j=1,2,\dots ,m,\end{array}$

where $\mathit{W}={\left[{\omega }_1,\dots ,{\omega }_m\right]}^{\mathrm{T}}$ represents the weight vector of RBFNN, $\mathit{h}\left(\mathit{x}\right)={\left[h_1,h_2,\dots ,h_m\right]}^{\mathrm{T}}$ is the activation function of the hidden layer chosen as the Gaussian function, $\mathit{x}=\left[x_1,x_2,\dots ,x_n\right]$ is the input signal, m is the number of neurons in the hidden layer, ${\mathit{c}}_j=\left[c_{j1},\dots ,c_{jm}\right]$ is the center vector, b_j is the width of the jth radial basis function, and h_j is the Gaussian basis function of the jth neural network.

3.2 Adaptive controller design

Adaptive control is a closed-loop control method that automatically adjusts control parameters based on system dynamic characteristics. It holds significant importance in modern control systems, particularly when dealing with systems characterized by dynamic changes and uncertainties. This study chooses the adaptive control strategy mainly because of its outstanding performance in nonlinear systems and uncertain environments. This strategy can adjust control parameters in real time to cope with dynamic changes in the system and adapt to variations in system parameters and external disturbances. Adaptive control exhibits excellent robustness and control performance, thereby maintaining system stability and performance under a wide range of operating conditions. When selecting control rates and parameters, we primarily considered system stability and response speed. We used the Lyapunov function analysis to ensure that the selected parameters guarantee the asymptotic stability of the system. Numerous studies have demonstrated that adaptive control performs excellently in handling complex hydraulic excavator systems (Fan and Li [26]; Qin et al. [27]). These studies validate the effectiveness of adaptive control in such applications, thereby providing theoretical support for our design choice.

The control law is designed based on the Lyapunov function. We ensured through the Lyapunov function analysis that the selected parameters guarantee the system’s asymptotic stability. This approach demonstrates the stability and response speed of the system under various operating conditions while ensuring computational complexity and real-time requirements in practical applications. The dynamical equation governing the controlled system pertains to a mining electric shovel, which is represented by Eq. (6). In the actual motion control process, control rates are designed based on the given desired joint position motion information and the expected target values. ${\mathit{q}}_{\mathrm{d}}$ represents the desired angle and position, and q represents the actual angle and position. Corresponding velocities, accelerations, and respective deviations are obtained by differentiating q and q_d.

(9)$\{\begin{array}{c}\mathit{e}=\mathit{q}-{\mathit{q}}_{\mathrm{d}}\to 0,\\ \dot{\mathit{e}}=\dot{\mathit{q}}-{\dot{\mathit{q}}}_{\mathrm{d}}\to 0,\\ \ddot{\mathit{e}}=\ddot{\mathit{q}}-{\ddot{\mathit{q}}}_{\mathrm{d}}\to 0.\end{array}$

Designing the mining electric shovel controller based on the computed torque method results in

(10)$\mathit{\tau }=\mathit{M}\left(\mathit{q}\right)\left({\ddot{\mathit{q}}}_{\mathrm{d}}-k_{\text{d}}\dot{\mathit{e}}-k_{\text{p}}\mathit{e}\right)+\mathit{C}\left(\mathit{q},\dot{\mathit{q}}\right)\dot{\mathit{q}}+\mathit{G}\left(\mathit{q}\right).$

The expression for the closed-loop system is

(11)$\ddot{\mathit{e}}+k_{\text{d}}\dot{\mathit{e}}+k_{\text{p}}\mathit{e}=0.$

As previously mentioned, various disturbance factors exist in real world unmanned electric shovels. As a result, the dynamic expressions in the above equations cannot fully capture the actual behavior of the mining electric shovel. Given the uncertainty in the mining electric shovel system, $\mathit{M}\left(\mathit{q}\right)$, $\mathit{C}\left(\mathit{q},\dot{\mathit{q}}\right)$, and $\mathit{G}\left(\mathit{q}\right)$ can be expressed as

(12)$\{\begin{array}{l}\mathit{M}\left(\mathit{q}\right)={\mathit{M}}_0\left(\mathit{q}\right)-\mathrm{\Delta }\mathit{M}\left(\mathit{q}\right),\\ \mathit{C}\left(\mathit{q},\dot{\mathit{q}}\right)={\mathit{C}}_0\left(\mathit{q},\dot{\mathit{q}}\right)-\mathrm{\Delta }\mathit{C}\left(\mathit{q},\dot{\mathit{q}}\right),\\ \mathit{G}\left(\mathit{q}\right)={\mathit{G}}_0\left(\mathit{q}\right)-\mathrm{\Delta }\mathit{G}\left(\mathit{q}\right),\end{array}$

where ${\mathit{M}}_0\left(\mathit{q}\right)$, ${\mathit{C}}_0\left(\mathit{q},\dot{\mathit{q}}\right)$, and ${\mathit{G}}_0\left(\mathit{q}\right)$ represent the corresponding nominal models; $\mathrm{\Delta }\mathit{M}\left(\mathit{q}\right)$, $\mathrm{\Delta }\mathit{C}\left(\mathit{q},\dot{\mathit{q}}\right)$, and $\mathrm{\Delta }\mathit{G}\left(\mathit{q}\right)$ represent the respective uncertainty terms; $\mathit{d}$ is the disturbance term.

The actual dynamic model of the mining electric shovel is given by

(13)$\begin{array}{rl}& \left({\mathit{M}}_0\left(\mathit{q}\right)-\mathrm{\Delta }\mathit{M}\left(\mathit{q}\right)\right)\ddot{\mathit{q}}+\left({\mathit{C}}_0\left(\mathit{q},\dot{\mathit{q}}\right)-\mathrm{\Delta }\mathit{C}\left(\mathit{q},\dot{\mathit{q}}\right)\right)\dot{\mathit{q}}\\ & +{\mathit{G}}_0\left(\mathit{q}\right)-\mathrm{\Delta }\mathit{G}\left(\mathit{q}\right)=\mathit{\tau }+\mathit{d}.\end{array}$

The adaptive controller is obtained according to Eq. (10) and takes the following form:

(14)$\begin{array}{rl}\mathit{\tau }=& {\mathit{M}}_0\left(\mathit{q}\right)\left({\ddot{\mathit{q}}}_{\mathrm{d}}-k_{\text{d}}\dot{\mathit{e}}-k_{\text{p}}\mathit{e}\right)+{\mathit{C}}_0\left(\mathit{q},\dot{\mathit{q}}\right)\dot{\mathit{q}}+{\mathit{G}}_0\left(\mathit{q}\right)\\ & -\left(\mathrm{\Delta }\mathit{M}\left(\mathit{q}\right)\ddot{\mathit{q}}+\mathrm{\Delta }\mathit{C}\left(\mathit{q},\dot{\mathit{q}}\right)\dot{\mathit{q}}+\mathrm{\Delta }\mathit{G}\left(\mathit{q}\right)+\mathit{d}\right)\\ =& {\mathit{M}}_0\left(\mathit{q}\right)\left({\ddot{\mathit{q}}}_{\mathrm{d}}-k_{\text{d}}\dot{\mathit{e}}-k_{\text{p}}\mathit{e}\right)+{\mathit{C}}_0\left(\mathit{q},\dot{\mathit{q}}\right)\dot{\mathit{q}}+{\mathit{G}}_0\left(\mathit{q}\right)-f(\cdot ),\end{array}$

where the modeling error term is $f(\cdot )=\mathrm{\Delta }\mathit{M}\left(\mathit{q}\right)\ddot{\mathit{q}}+\mathrm{\Delta }\mathit{C}\left(\mathit{q},\dot{\mathit{q}}\right)\dot{\mathit{q}}+\mathrm{\Delta }\mathit{G}\left(\mathit{q}\right)+\mathit{d}$. Suppose that $\ddot{\mathit{q}}$ is a function of $\mathit{\tau }$, $\mathit{q}$, and $\dot{\mathit{q}}$; ${\mathit{D}}_0$, $\mathrm{\Delta }\mathit{D}$, and $\mathrm{\Delta }\mathit{G}$ are functions of $\mathit{q}$; $\mathrm{\Delta }\mathit{C}$ is a function of $\mathit{q}$ and $\dot{\mathit{q}}$; and $\mathit{d}$ is an external disturbance. In this case, the function f(∙) is a nonlinear function of variables $\mathit{q}$, $\dot{\mathit{q}}$, $\ddot{\mathit{q}}$, ${\mathit{q}}_{\mathrm{d}}$, ${\dot{\mathit{q}}}_{\mathrm{d}}$, and ${\ddot{\mathit{q}}}_{\mathrm{d}}$. Given that $\mathit{e}=\mathit{q}-{\mathit{q}}_{\mathrm{d}}$, $\dot{\mathit{e}}=\dot{\mathit{q}}-{\dot{\mathit{q}}}_{\mathrm{d}}$, and $\ddot{\mathit{e}}=\ddot{\mathit{q}}-{\ddot{\mathit{q}}}_{\mathrm{d}}$, let $\mathit{x}=[\begin{array}{ccc}\mathit{e}& \dot{\mathit{e}}& \ddot{\mathit{e}}\end{array}{]}^{\mathrm{T}}$. Moreover, the function f(∙) can be concluded as a nonlinear function of the state variables $\mathit{x}$, ${\mathit{q}}_{\mathrm{d}}$, ${\dot{\mathit{q}}}_{\mathrm{d}}$, and ${\ddot{\mathit{q}}}_{\mathrm{d}}$. The tracking error vector $\mathit{x}$ belongs to a compact set, where the tracking error $\mathit{x}$ falls within the region $\mathrm{\wp }\left({\mathit{M}}_x\right)=\left\{\mathit{x},\Vert \mathit{x}\Vert ⩽{\mathit{M}}_x\right\}$, which is a sphere with a radius ${\mathit{M}}_x$ that is sufficiently large. Given that ${\mathit{q}}_{\mathrm{d}}$, ${\dot{\mathit{q}}}_{\mathrm{d}}$, and ${\ddot{\mathit{q}}}_{\mathrm{d}}$ are known as a priori, the nonlinear function f(∙) can be denoted as f(x). In practice, the state variable x is a priori unknown; thus, the nonlinear function f(x) is also a priori unknown. Therefore, the modified controller mentioned above cannot be implemented. However, the controller suggests that improved tracking performance for unmanned mining electric shovels can be achieved by using a well-estimated function $\hat{f}\left(\mathit{x}\right)$ for the unknown nonlinear function f(x).

3.3 Adaptive neural controller design

This study addresses the unmodeled dynamics and external disturbances by designing corresponding compensators to mitigate their impact on the system.

The output of the RBFNN can be rewritten as

(15)${\hat{f}}^{\ast }\left(\mathit{x}\right)={\left({\mathit{W}}^{\ast }\right)}^{\mathrm{T}}\mathit{h}\left(\mathit{x}\right)+\epsilon ,$

where ${\hat{f}}^{\ast }\left(\mathit{x}\right)$ represents the neural network output, ${\mathit{W}}^{\ast }$ is the optimal weight, and $\epsilon$ is the minimum approximation error.

The estimated output of the RBFNN is as follows:

(16)$\hat{f}\left(\mathit{x}\right)={\hat{\mathit{W}}}^{\mathrm{T}}\mathit{h}\left(\mathit{x}\right),$

where ${\hat{\mathit{W}}}^{\mathrm{T}}=\left[{\hat{\omega }}_1,\dots ,{\hat{\omega }}_m\right]$ represents the estimated weight values of ${\mathit{W}}^{\ast }$. The optimal weight matrix of the RBFNN is bounded; thus,

(17)$\text{tr}\left\{{\left({\mathit{W}}^{\ast }\right)}^{\mathrm{T}}{\mathit{W}}^{\ast }\right\}⩽{\mathit{W}}_M,$

where ${\mathit{W}}_M$ is a positive constant.

In this section, an adaptive neural network controller is designed to control an unmanned mining electric shovel with unknown dynamic characteristics. The control diagram is depicted in Fig.6.

Fig.6 Adaptive neural controller design.

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The expression for the adaptive neural network controller is

(18)$\mathit{\tau }={\mathit{M}}_0\left(\mathit{q}\right)\left({\ddot{\mathit{q}}}_{\mathrm{d}}-k_{\text{d}}\dot{\mathit{e}}-k_{\text{p}}\mathit{e}\right)+{\mathit{C}}_0\left(\mathit{q},\dot{\mathit{q}}\right)\dot{\mathit{q}}+{\mathit{G}}_0\left(\mathit{q}\right)-\hat{f}(\cdot ).$

After substituting the above expression into the dynamic system of the unmanned electric shovel and rearranging, we obtain

(19)$\begin{array}{rl}& \mathit{M}\left(\mathit{q}\right)\ddot{\mathit{q}}+\mathit{C}\left(\mathit{q},\dot{\mathit{q}}\right)\dot{\mathit{q}}+\mathit{G}\left(\mathit{q}\right)\\ =& {\mathit{M}}_0\left(\mathit{q}\right)\left({\ddot{\mathit{q}}}_{\mathrm{d}}-k_{\text{d}}\dot{\mathit{e}}-k_{\text{p}}\mathit{e}\right)+C_0\left(\mathit{q},\dot{\mathit{q}}\right)\dot{\mathit{q}}+{\mathit{G}}_0\left(\mathit{q}\right)-f(\cdot )+\mathit{d}.\end{array}$

We obtain the following by combining the above equation with Eq. (15) and simplifying the combined equations:

(20)$\begin{array}{rl}& \mathrm{\Delta }\mathit{M}\left(\mathit{q}\right)\ddot{\mathit{q}}+\mathrm{\Delta }\mathit{C}\left(\mathit{q},\dot{\mathit{q}}\right)\dot{\mathit{q}}+\mathrm{\Delta }\mathit{G}\left(\mathit{q}\right)+\mathit{d}\\ =& {\mathit{M}}_0\left(\mathit{q}\right)\ddot{\mathit{q}}-{\mathit{M}}_0\left(\mathit{q}\right)\left({\ddot{\mathit{q}}}_{\mathrm{d}}-k_{\text{d}}\dot{\mathit{e}}-k_{\text{p}}\mathit{e}\right)+\hat{f}(\cdot )\\ =& {\mathit{M}}_0\left(\mathit{q}\right)\left(\ddot{\mathit{e}}+k_{\text{d}}\dot{\mathit{e}}+k_{\text{p}}\mathit{e}\right)+\hat{f}(\cdot ).\end{array}$

The following can be derived from the above equation:

(21)$\begin{array}{rl}& \ddot{\mathit{e}}+k_{\text{d}}\dot{\mathit{e}}+k_{\text{p}}\mathit{e}+{\mathit{M}}_0^{-1}\left(\mathit{q}\right)\hat{f}(\cdot )\\ =& {\mathit{M}}_0^{-1}\left(\mathrm{\Delta }\mathit{M}\left(\mathit{q}\right)\ddot{\mathit{q}}+\mathrm{\Delta }\mathit{C}\left(\mathit{q},\dot{\mathit{q}}\right)\dot{\mathit{q}}+\mathrm{\Delta }\mathit{G}\left(\mathit{q}\right)+\mathit{d}\right).\end{array}$

After rearranging, we obtain

(22)$\ddot{\mathit{e}}+k_{\text{d}}\dot{\mathit{e}}+k_{\text{p}}\mathit{e}={\mathit{M}}_0^{-1}\left(f(\cdot )-\hat{f}(\cdot )\right).$

Given the equation, the following is obtained:

(23)$\begin{array}{rl}f(\cdot )-\hat{f}(\cdot )=& f(\cdot )-{\hat{f}}^{\ast }(\cdot )+{\hat{f}}^{\ast }(\cdot )-\hat{f}(\cdot )\\ & =\eta +{\left({\mathit{W}}^{\ast }\right)}^{\text{T}}\mathit{h}-{\hat{\mathit{W}}}^{\mathrm{T}}\mathit{h}=\eta -{\tilde{\mathit{W}}}^{\mathrm{T}}\mathit{h},\end{array}$

where $\tilde{\mathit{W}}=\hat{\mathit{W}}-{\mathit{W}}^{\ast }$ and $\eta =f(\cdot )-{\hat{f}}^{\ast }(\cdot )$.

We obtain the following by simplifying:

(24)$\dot{\mathit{x}}=\mathit{A}\mathit{x}+\mathit{B}\left(\eta -{\tilde{\mathit{W}}}^{\mathrm{T}}\mathit{h}\right),$

where

(25)$\mathit{A}=\left(\begin{array}{cc}0& 1\\ -k_{\text{p}}& -k_{\text{d}}\end{array}\right),\mathit{B}=\left(\begin{array}{c}0\\ {\mathit{M}}_0^{-1}\left(\mathit{q}\right)\end{array}\right).$

The parameter update law for W is chosen as

(26)${\dot{\hat{\mathit{W}}}}^{\mathrm{T}}=\gamma {\mathit{B}}^{\mathrm{T}}\mathit{P}{\mathit{x}\mathit{h}}^{\mathrm{T}},$

where $\gamma \in \mathbb{N}$ represents the adaptation coefficient, which implies

(27)$\dot{\hat{\mathit{W}}}=\gamma {\mathit{h}\mathit{x}}^{\mathrm{T}}\mathit{P}\mathit{B}.$

4 Stability analysis

For the mining electric shovel system described above, suppose that the RBFNN weight updating law is chosen as in Eq. (27), and the control law is designed as in Eq. (18). In this case, achieving the asymptotic convergence of the system tracking error to zero is possible.

The following Lyapunov function is considered:

(28)$V=\frac{1}{2}{\mathit{x}}^{\mathrm{T}}\mathit{P}\mathit{x}+\frac{1}{2\gamma }{\Vert \tilde{\mathit{W}}\Vert }^2,$

where $\lambda >0$.

The matrix $\mathit{P}$ is symmetric positive definite and satisfies the following Lyapunov equation:

(29)$\mathit{P}\mathit{A}+{\mathit{A}}^{\mathrm{T}}\mathit{P}=-\mathit{Q},$

where $\mathit{Q}⩾0$ (positive semidefinite).

The following is yielded by taking the derivative of $V$:

(30)$\begin{array}{rl}\dot{V}& ={\displaystyle \frac{1}{2}}\left[{\mathit{x}}^{\mathrm{T}}\mathit{P}\dot{\mathit{x}}+{\dot{\mathit{x}}}^{\mathrm{T}}\mathit{P}\mathit{x}\right]+{\displaystyle \frac{1}{\gamma }}\text{tr}\left({\dot{\tilde{\mathit{W}}}}^{\mathrm{T}}\tilde{\mathit{W}}\right)\\ & ={\displaystyle \frac{1}{2}}\left[{\mathit{x}}^{\mathrm{T}}\mathit{P}\left(\mathit{A}\mathit{x}+\mathit{B}\left(\eta -{\tilde{\mathit{W}}}^{\mathrm{T}}\mathit{h}\right)\right)+\left({\mathit{x}}^{\mathrm{T}}{\mathit{A}}^{\mathrm{T}}+{\left(\eta -{\tilde{\mathit{W}}}^{\mathrm{T}}\mathit{h}\right)}^{\mathrm{T}}{\mathit{B}}^{\mathrm{T}}\right)\mathit{P}\mathit{x}\right]+{\displaystyle \frac{1}{\gamma }}\text{tr}\left({\dot{\tilde{\mathit{W}}}}^{\mathrm{T}}\tilde{\mathit{W}}\right)\\ & ={\displaystyle \frac{1}{2}}\left[{\mathit{x}}^{\mathrm{T}}\left(\mathit{P}\mathit{A}+{\mathit{A}}^{\mathrm{T}}\mathit{P}\right)\mathit{x}+\left({\mathit{x}}^{\mathrm{T}}\mathit{P}\mathit{B}\eta -{\mathit{x}}^{\mathrm{T}}\mathit{P}\mathit{B}{\tilde{\mathit{W}}}^{\mathrm{T}}\mathit{h}+{\eta }^{\mathrm{T}}{\mathit{B}}^{\mathrm{T}}\mathit{P}\mathit{x}-{\mathit{h}}^{\mathrm{T}}\tilde{\mathit{W}}{\mathit{B}}^{\mathrm{T}}\mathit{P}\mathit{x}\right)\right]+{\displaystyle \frac{1}{\gamma }}\text{tr}\left({\dot{\tilde{\mathit{W}}}}^{\mathrm{T}}\tilde{\mathit{W}}\right)\\ & =-{\displaystyle \frac{1}{2}}{\mathit{x}}^{\mathrm{T}}\mathit{Q}\mathit{x}+{\mathit{\eta }}^{\mathrm{T}}{\mathit{B}}^{\mathrm{T}}\mathit{P}\mathit{x}-{\mathit{h}}^{\mathrm{T}}\tilde{\mathit{W}}{\mathit{B}}^{\mathrm{T}}\mathit{P}\mathit{x}+{\displaystyle \frac{1}{\gamma }}\text{tr}\left({\dot{\tilde{\mathit{W}}}}^{\mathrm{T}}\tilde{\mathit{W}}\right),\\ & \end{array}$

where ${\mathit{x}}^{\mathrm{T}}\mathit{P}\mathit{B}{\tilde{\mathit{W}}}^{\mathrm{T}}\mathit{h}={\mathit{h}}^{\mathrm{T}}\tilde{\mathit{W}}{\mathit{B}}^{\mathrm{T}}\mathit{P}\mathit{x}$ and ${\mathit{x}}^{\mathrm{T}}\mathit{P}\mathit{B}\eta ={\eta }^{\mathrm{T}}{\mathit{B}}^{\mathrm{T}}\mathit{P}\mathit{x}$.

Given that

(31)${\mathit{h}}^{\mathrm{T}}\tilde{\mathit{W}}{\mathit{B}}^{\mathrm{T}}\mathit{P}\mathit{x}=\text{tr}\left({\mathit{B}}^{\mathrm{T}}{\mathit{P}\mathit{x}\mathit{h}}^{\mathrm{T}}\tilde{\mathit{W}}\right),$

by rearranging, Eq. (30) yields

(32)$\dot{V}=-\frac{1}{2}{\mathit{x}}^{\mathrm{T}}\mathit{Q}\mathit{x}+\frac{1}{\gamma }\text{tr}\left(-\gamma {\mathit{B}}^{\mathrm{T}}{\mathit{P}\mathit{x}\mathit{h}}^{\mathrm{T}}\tilde{\mathit{W}}+\dot{\tilde{\mathit{W}}}\tilde{\mathit{W}}\right)+{\eta }^{\mathrm{T}}{\mathit{B}}^{\mathrm{T}}\mathit{P}\mathit{x}.$

We can deduce from the above that $\dot{\tilde{\mathit{W}}}=\dot{\hat{\mathit{W}}}$. By substituting $\dot{\tilde{\mathit{W}}}=\dot{\hat{\mathit{W}}}$ and $\dot{\hat{\mathit{W}}}=\gamma {\mathit{h}\mathit{x}}^{\mathrm{T}}\mathit{P}\mathit{B}$ into the above expression, we obtain

(33)$\dot{V}=-{\displaystyle \frac{1}{2}}{\mathit{x}}^{\mathrm{T}}\mathit{Q}\mathit{x}+{\eta }^{\mathrm{T}}{\mathit{B}}^{\mathrm{T}}\mathit{P}\mathit{x}.$

The following can be deduced from the previously known conditions:

(34)$\{\begin{array}{c}\Vert {\eta }^{\mathrm{T}}\Vert ⩽\Vert {\eta }_0\Vert ,\\ \Vert \mathit{B}\Vert =\Vert {\mathit{M}}_0^{-1}\left(\mathit{q}\right)\Vert ,\end{array}$

and

(35)$\begin{array}{rl}\dot{V}⩽& \frac{1}{2}{\lambda }_{min}\left(\mathit{Q}\right){\Vert \mathit{x}\Vert }^2+\Vert {\eta }_0\Vert \Vert {\mathit{M}}_0^{-1}\left(\mathit{q}\right)\Vert {\lambda }_{max}\left(\mathit{P}\right)\Vert \mathit{x}\Vert \\ & =-\frac{1}{2}\Vert \mathit{x}\Vert \left[{\lambda }_{min}\left(\mathit{Q}\right)\Vert \mathit{x}\Vert -2\Vert {\eta }_0\Vert \Vert {\mathit{M}}_0^{-1}\left(q\right)\Vert {\lambda }_{max}\left(\mathit{P}\right)\right],\end{array}$

where ${\lambda }_{max}\left(\mathit{P}\right)$ and ${\lambda }_{min}\left(\mathit{Q}\right)$ are the maximum eigenvalue of matrix $\mathit{P}$ and the minimum eigenvalue of matrix $\mathit{Q}$, respectively.

The following is required to ensure that $\dot{V}\le 0$,

(36)${\lambda }_{min}\left(\mathit{Q}\right)⩾\frac{2\Vert {\mathit{M}}_0^{-1}\left(\mathit{q}\right)\Vert {\lambda }_{max}\left(\mathit{P}\right)}{\Vert \mathit{x}\Vert }\Vert {\eta }_0\Vert ,$

i.e.,

(37)$\Vert \mathit{x}\Vert =\frac{2\Vert {\mathit{M}}_0^{-1}\left(\mathit{q}\right)\Vert {\lambda }_{max}\left(\mathit{P}\right)}{{\lambda }_{max}\left(\mathit{Q}\right)}\Vert {\eta }_0\Vert .$

We can conclude that under the assumption of bounded $\tilde{\omega }=\hat{\omega }-{\omega }^{\ast }$, increasing the eigenvalues of $\mathit{Q}$, decreasing the eigenvalues of $\mathit{P}$, or reducing ${\eta }_0$ can improve the convergence performance of x.

5 Simulation

Finally, experiments on optimal excavation trajectory tracking are conducted in this study by comparing the proposed control method with traditional adaptive control methods. The computer setup includes an Intel Core i7-8700 CPU paired with 16 GB RAM. All simulation experiments are conducted using MATLAB software. The effectiveness of the proposed controller in unmanned electric shovel systems is confirmed by the experimental results, thereby demonstrating its strong tracking performance under optimal excavation trajectories. This result supports the feasibility of applying the control method in practical mining site scenarios. In this study, a polynomial trajectory is employed as the input, and the trajectory function can be represented by [18,19]

(38)${\mathit{S}}_{\text{d}}=\left[\begin{array}{c}{X}_{\text{d}}\\ Y_{\text{d}}\end{array}\right]=\left[\begin{array}{c}{a}_6{t}^6+a_5{t}^5+a_4{t}^4+a_3{t}^3+a_2{t}^2+a_{1}t+a_0\\ b_6{t}^6+b_5{t}^5+b_4{t}^4+b_3{t}^3+b_2{t}^2+b_{1}t+b_0\end{array}\right].$

Let ${\mathit{A}}_i=[\begin{array}{cc}{a}_i& b_i\end{array}]$. As shown in Tab.2, the optimal excavation trajectory parameters based on the actual material surface are designed according to Refs. [18,19].

Tab.2 Polynomial coefficients

	Value of ai	Value of bi
${\mathit{A}}_6$	−1.828 × 10−5	1.944 × 10−5
${\mathit{A}}_5$	8.387 × 10−4	−6.657 × 10−4
${\mathit{A}}_4$	−1.331 × 10−2	6.204 × 10−3
${\mathit{A}}_3$	7.479 × 10−2	−7.917 × 10−3
${\mathit{A}}_2$	1.193 × 10−16	4.106 × 10−16
${\mathit{A}}_1$	4.387 × 10−15	−2.193 × 10−15
${\mathit{A}}_0$	3.478	−8.000

The trajectory is established in the $\left\{O_2\right\}$ coordinate system, and the unmanned mining electric shovel utilizes the push length and the angle of the dipper stick as position information. During the simulation, the trajectory signal must be converted into length and angle expected tracking commands.

(39)${\mathit{q}}_{\text{d}}=\left[\begin{array}{c}{\theta }_{\text{d}}\\ r_{\text{d}}\end{array}\right]=\left[\begin{array}{c}\arctan \left({\displaystyle \frac{Y_{\text{d}}}{X_{\text{d}}}}\right)+{\displaystyle \frac{\text{π}}{2}}\\ \sqrt{X_d^2+Y_d^2}\end{array}\right].$

This simulation experiment focuses on the WK-12 and utilizes the Simulink software toolbox for simulation. The control law is chosen as in Eq. (18), and the adaptive updating law is selected as in Eq. (27).

In the simulation experiments, we chose control parameters $\mathit{Q}=850\cdot eye\left(4\right)$, $\alpha =3$, and $\gamma =25$, where increasing the value of $\mathit{Q}$ helps improve the accuracy of position tracking. In the initial experiments, we estimated the relevant parameters based on the types and magnitudes of disturbances that the unmanned electric shovel might encounter, and we repeatedly optimized them in subsequent experiments to enhance system performance.

In the RBFNN, the parameters of the Gaussian function are set as $c_i=$ [–5 –3.8889 –2.7778 –1.6667 –0.5556 0.5556 1.6667 2.7778 3.8889 5] and $b=3$, and the initial weights are set to 0.01. The selection of the Gaussian function parameters and initial weights is based on the system’s dynamic characteristics and the disturbances encountered during actual operation. Additionally, the neural network settings are based on the control process errors. For the unmanned electric shovel, the desired error in position tracking is zero. Thus, the RBFNN nodes are estimated to diffuse outward from zero as the center point. This approach ensures that the neural network can effectively learn and adapt to the system’s dynamic changes, thereby achieving precise control. This selection is meticulously adjusted and validated based on experimental results to ensure optimal performance and accuracy in handling input data by the network. According to the actual environment, the initial excavation position of WK-12 is the initial state, which is $[\begin{array}{cccc}\theta & \dot{\theta }& r& \dot{r}\end{array}{]}^{\mathrm{T}}=[\begin{array}{cccc}23.5& 0& 6.3553& 0\end{array}{]}^{\mathrm{T}}$. Some physical parameters of the unmanned electric shovel WK-12’s excavation system are shown in Tab.3.

Tab.3 Physical parameters of the bucket

Bucket parameter	Symbol	Value	Unit
Arm quality	mba	15700	kg
Arm extension length	Lba	10.767	m
Mass	mb	26700	kg
Length	Lb	2.368	m

Let $\mathrm{\Delta }\mathit{M}\left(\mathit{q}\right)=0.1\cdot \mathit{M}\left(\mathit{q}\right)$, $\mathrm{\Delta }\mathit{C}\left(\mathit{q},\dot{\mathit{q}}\right)=0.1\cdot \mathit{C}\left(\mathit{q},\dot{\mathit{q}}\right)$, and $\mathrm{\Delta }\mathit{G}\left(\mathit{q}\right)=0.1\cdot \mathit{G}\left(\mathit{q}\right)$. The external disturbance to the system is $d=d_1+d_2\Vert \mathit{e}\Vert +d_3\Vert \dot{\mathit{e}}\Vert$, where d₁ = 2, d₂ = 3, and d₃ = 6.

Fig.7 and Fig.8 depict bucket arm angle and extension length plots, respectively, demonstrating that the proposed method aligns the desired trajectory closely with the actual trajectory. Moreover, traditional adaptive control exhibits inferior tracking performance in elongation. Fig.9 and Fig.10 illustrate the convergence of tracking errors in angle and elongation. The proposed method gradually reduces the angular tracking error to zero, and the arm extension error remains within ±0.005 m. The proposed method achieves smoother error control and better convergence than the traditional adaptive control. For large-scale equipment such as unmanned electric shovels, where angular changes exceed 70° and elongation exceeds 2 m during operation, the aforementioned errors meet the control requirements.

Fig.7 Bucket arm angle.

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Fig.8 Bucket arm extension length.

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Fig.9 Bucket arm angle errors.

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Fig.10 Bucket arm extension length errors.

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Fig.11 and Fig.12 depict velocity tracking plots, illustrating that the proposed method exhibits smooth and slightly fluctuating velocity control. This characteristic aids in reducing vibrations during the operation of heavy-duty equipment. Furthermore, the control performance of the proposed method is superior to the performance of the traditional adaptive control, with the desired and actual velocities being closely aligned. By contrast, traditional adaptive control exhibits large velocity fluctuations, which can lead to fatigue damage in large-scale equipment. Fig.13 and Fig.14 represent velocity tracking error curves. Initially, some oscillations exist in the actual velocity; however, after 2 s, the velocity tracking error converges to zero with minimal oscillation. The method proposed in this study exhibits smaller error fluctuations and shorter convergence time than the traditional adaptive control. The plots indicate that the proposed method maintains angular velocity error within a range of 0–0.004 °/s and velocity error within ±0.015 m∙s⁻¹, thereby meeting production requirements.

Fig.11 Bucket arm angular velocity.

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Fig.12 Bucket arm extension velocity.

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Fig.13 Bucket arm angular velocity errors.

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Fig.14 Bucket Arm extension velocity errors.

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Fig.15 illustrates excavation trajectories, where the optimal excavation trajectory represents positional information on the XOY coordinate system. However, the operational apparatus of unmanned electric shovel systems primarily revolves around the bucket arm, thereby necessitating the conversion of the bucket arm’s angle and extension length into positional information on the XOY coordinate system. The transformed trajectory aligns closely with the optimal excavation trajectory, with an absolute error value being less than 0.005 m. Thus, the production standards are met. Therefore, the proposed method can meet control requirements, thereby enabling the unmanned electric shovel control to excavate materials along the desired trajectory. Fig.16 and Fig.17 display output curve plots, demonstrating that using the proposed method compensates for disturbances in the unmanned electric shovel system. This outcome results in smooth variations in the system output force, which helps suppress the vibrations during the excavation process. By contrast, traditional adaptive control exhibits significant fluctuations in the output force, leading to frequent vibrations that are detrimental to the operation and maintenance of heavy-duty equipment.

Fig.15 Excavation trajectory.

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Fig.16 Lift output.

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Fig.17 Push output.

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Fig.18 and Fig.19 depict uncertainty fitting graphs, illustrating the neural network’s capability to fit the uncertainties effectively. Fig.20 represents the uncertainty approximation error graph, showing significant differences between the initial weights and the optimal weights. These differences lead to considerable fluctuation in the fitting curve during the initial stages. However, the adaptive law tends to converge toward optimal values after the adaptive law adjustments to the neural network weights. Moreover, the fitting error gradually converges to zero after 2 s.

Fig.18 Uncertainties of lift.

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Fig.19 Uncertainties of push.

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Fig.20 Uncertainty approximation error.

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The simulation results indicate that the proposed controller ensures that the bucket position of the unmanned electric shovel overcomes disturbances and tracks the desired trajectory within a short time. Furthermore, the adaptive law $\hat{W}\left(t\right)$ converges to its true value within a short time under sustained excitation signals, thereby facilitating the convergence of the fitting curve. The position error and the velocity error converge to zero by integrating the compensation controller’s fitting results into the control output torque. This observation demonstrates that the control system achieves the control requirements within a short time.

6 Conclusions

This study proposes an adaptive control algorithm for trajectory tracking of unmanned electric shovels based on RBFNN. Initially, the study establishes the kinematic model of the electric shovel and simplifies it for decoupling. Subsequently, the model is reestablished based on the errors of the dynamic model and uncertainties during operation. An RBFNN compensation controller is introduced to fit and compensate for these factors, thereby addressing the inaccuracies in the dynamic model and the impact of uncertainties during operation. The controller’s weights are adjusted through an adaptive law to approach their true values. An RBFNN adaptive controller is established in conjunction with the compensation controller by using position and velocity errors during the excavation process as inputs and calculating adjustments for different trajectory velocities as outputs to achieve trajectory tracking. Lyapunov stability theory is employed to prove the consistent boundedness of the closed-loop system. Simulations using the optimal trajectory as input show the effectiveness of the proposed control method in terms of position, velocity, and stability. The results indicate that the proposed controller meets the requirements for error correction and overshoot avoidance, effectively mitigates frequent system oscillations, and exhibits good tracking performance. In future research, the impact of load variations will be considered for assessing shovel tracking performance. We also aim to optimize the trajectory tracking control of the unmanned electric shovel in future work by incorporating practical factors, such as digging resistance, bucket load, and material spillage, into the controller design to enhance the system’s practical performance.

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Acknowledgements

This work was supported by the Major Science and Technology Project of Shanxi Province, China (Grant No. 20191101014) and the National Natural Science Foundation of China (Grant No. 52075068).

Conflict of Interest

The authors declare no conflict of interest.

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