© Z. Geng and T. Wan, Published by EDP Sciences, 2025

1 Introduction

Industrial robot arms are a key component in the field of industrial robotics, defined as automated devices that simulate human arm movements. These robotic arms are renowned for their high versatility, flexible mobility, excellent obstacle avoidance performance, and convenient maneuverability. They have been widely used in various industrial applications, including material handling, welding operations, painting processes, and cutting operations [1]. Due to the expanded production scale and diversified application environments, the operating environment of industrial robot arms has expanded from open spaces to confined spaces filled with obstacles. This technology has been proven to liberate human operators from dangerous and monotonous tasks, thereby improving the efficiency of the production process [2]. In this environment, academic attention has shifted towards path planning. The development of obstacle avoidance planning technology is of particular significance. The performance of this technology is directly related to the manipulation effect of the robot arm. Meanwhile, traditional static planning algorithms are difficult to address existing problems, and there is currently relatively little research on dynamic obstacle avoidance planning for robotic arms [3]. There are various interference factors in dynamic environments, such as the vibration of the robotic arm, external impacts, and uncertainty of obstacle positions. These interference may cause the robotic arm to deviate from the path planned for obstacle avoidance. Active Disturbance Rejection Control (ADRC) can compensate for these interference, enhancing the anti-interference ability of the robotic arm. The robotic arm must be able to track the path set by the obstacle avoidance plan with the highest accuracy. ADRC has been extensively applied in permanent magnet linear synchronous motor drive systems due to the strong anti-interference ability and fast response speed. However, some parameter adjustments have added difficulties to its application.

To address the issues discussed, Yuan C et al. proposed a sliding mode-based self-correcting parameter self-impedance control. Sliding Mode (SM) was combined Extended State Observer (ESO) to achieve parameter self-tuning. The proposed control method reduced jitter effects, exhibited excellent anti-interference performance, and provided fast dynamic response [4]. The European Control Association robot manipulator has been widely used in modern industry. The end effector needs to move precisely along a given trajectory. Ben M. S. et al. built a robust feedback control method to deal with this issue. The controller exhibited superiority in satisfying stability and target value requirements [5]. Nuo E et al. proposed a distributed control method with a proportional-integral-derivative scheme structure that is easy to implement. The controller used an internal model approach to suppress disturbances [6]. Switching systems often involve models that are difficult to determine. Flores Padilla M. et al. developed an adaptive controller for nonlinear discrete-time switching systems. The experimental results showed that the proposed controller exhibited closed-loop performance on discrete-time switching and disturbed systems [7].

Based on geometric feedforward controller, Xie H et al. proposed a path tracking framework based on composite interference suppression, which combined cascaded active interference suppression controller and online estimator. Especially, model-based cascaded ADRC consisted of an outer loop controller and an inner loop controller. In addition, to improve transient response, a geometric feedforward controller based on parameter self-learning was built. The effectiveness was evaluated, proving its superiority over traditional pure tracking algorithms [8]. The impact of pressure shock disturbance generated by the supply pump on the tracking performance of hydraulic robotic arms has been proven to be an important issue. To address this challenge, Yao S et al. proposed an impact disturbance compensation controller, which was developed to handle uncertainty and pressure impact disturbances in hydraulic systems. The experimental results indicated that the controller provided a theoretical guarantee for the asymptotic tracking performance of hydraulic manipulators under uncertainty and mixed disturbances. In addition, the maximum tracking error and average variance of Proportional-Integral-Derivative (PID) were reduced by 68.7% [9]. The mathematical model and optimal modulation scheme of Dual-Active Hybrid Full-Bridge (H-FDAB) Direct Current − Direct Current (DC-DC) Converter have not been fully explored. Luo S et al. proposed a DC-DC converter return power optimization strategy based on ADRC. The experiment ultimately proved the effectiveness and practicality of the designed control strategy. This method effectively solved the challenges of inherent converter efficiency reduction and output voltage fluctuations in the converter. Under traditional Proportional-Integral (PI) control, this converter generated a large amount of return power [10]. To avoid unnecessary movement and save energy, Jing A et al. built a correction adaptive ADRC. This new algorithm could be used for pitch angle control of underwater gliders. The ADRC could compensate for control inputs [11].

In summary, researchers have conducted some research on the control technology of industrial robotic arms, including ADRC, sliding mode self-tuning parameters, robust feedback control methods, composite interference suppression, and high-precision tracking control. However, the application of dynamic obstacle avoidance planning, parameter adaptive control, anti-interference ability, system stability, and optimization is not deep enough. Therefore, a control method for industrial robotic arms based on nonlinear ADRC technology is proposed, which innovatively adopts an improved ESO and integrates new functions to achieve efficient nonlinear control, providing a technical basis for control improvement of industrial robotic arms.

The article structure is as follows. The first part focuses on the industrial robotic arm control method based on nonlinear ADRC technology. The second part specifically verifies the proposed algorithm. The third part draws conclusions and shortcomings of this design and the directions that need to be further explored in the future.

2 Methods and materials

A control method for industrial robotic arms based on nonlinear ADRC technology is built. Firstly, the nonlinear ADRC technology is adopted and the ESO is improved. Secondly, the study applies the proposed improved nonlinear ADRC technology to industrial robotic arms and analyzes the control effectiveness.

2.1 Nonlinear ADRC technology based on improved SMESO

With the increasingly complex application environment and expanding industrial production of robotic arms, traditional control methods have been unable to meet the growing demand for industrial applications. Exploring the control method has become a popular development trend. ADRC technology was proposed in 1998. Its core idea is to consider the internal and external uncertainty factors as total disturbances, and estimate and compensate for them through ESO [12,13]. This method does not rely on the precise mathematical model, making ADRC perform well in controlling complex systems with nonlinearity, time-varying, and unknown disturbances. ADRC technology estimates the internal uncertainty and external disturbances by introducing ESO, and then takes a state feedback controller for accurate adjustment [14]. This technology not only significantly improves control accuracy, but also enhances the efficiency of the entire production process by reducing response time. Figure 1 displays the ADRC structure.

In Figure 1, r(t) is the system reference input signal, and ṙ(t) is the differential signal of the reference input; y(t) is the output of the controlled object, and y₀ (t) is the output feedback signal processed by the First order filter; w(t) is an external disturbance; Z₁, Z₂, Z₃ are the state observation values output by Nonlinear Extended State Observer, corresponding to different states of the system; Nonlinear combination is used to perform nonlinear processing on the error signal to obtain the intermediate control variable u₀; 1/b and b are control gain related parameters, and after calculation, the final control quantity u is output to act on the Controlled object, achieving control and disturbance suppression of the system. ADRC has three parts. The tracking differentiator can smooth the input signal, extract the differential signal, and arrange the transition process to avoid overshoot, solving the noise sensitivity in traditional differentiators and providing smooth input signals and their derivatives [15]. ESO can estimate state variables and total disturbances (including internal and external disturbances) of the system, treating the unknown dynamics and external disturbances as “extended states” and conducting online estimation through observers [16]. The Nonlinear State Error Feedback (NLSEF) Control Law can generate control signals through nonlinear combinations based on state errors and disturbance estimates, utilizing nonlinear characteristics to improve the dynamic response and disturbance rejection capability. Figure 2 displays the Simulink simulation diagram of the tracking differentiator.

In Figure 2, the tracking differentiator is a signal preprocessing algorithm used to extract smooth signals and their derivatives from noisy signals. The tracking differentiator algorithm is shown in equation (1).

$$\{\begin{array}{l}fh=fhan\left(v_1(k)-v(k),v_2(k),r_0,h_0\right)\hfill \\ v_1(k+1)=v_1(k)+h{v}_2(k)\hfill \\ v_2(k+1)=v_2(k)+hfh\hfill \end{array}$$(1)

In equation (1), h represents the sampling period. r₀ signifies the speed factor. fhan(v₁, v₂, r₀, h₀) represents the fastest control function. h₀ represents the filtering factor. v₁(k) represents the tracking signal at time k, v₂(k) represents the derivative of the tracking signal at time k, and fh represents the output of the fastest control function. Nonlinear function is an important core component of ADRC, as shown in equation (2).

$$\text{fal}(e,\alpha ,\delta )=\{\begin{array}{l}|e{|}^{\alpha }\text{sign}(e)|e|>\delta \hfill \\ \frac{e}{{\delta }^{1-\alpha }}\left|e\right|\le \delta \hfill \end{array}.$$(2)

In equation (2), δ represents the filtering factor. α represents the non-linear factor. e represents the tracking error. Mechanical systems generally exhibit strong nonlinearity, and traditional linear control methods are difficult to accurately describe these characteristics, resulting in model mismatch and decreased control accuracy. Therefore, a novel nonlinear function is selected for the study, as shown in equation (3).

$$nf(e,\alpha ,\delta )=\{\begin{array}{l}|e{|}^{\alpha }sign(e)|e|>\delta \hfill \\ m_{1}e+m_2{e}^2+m_3\tan e\left|e\right|\le \delta \hfill \end{array}.$$(3)

In equation (3), m₁, m₂, and m₃ represent joint mass parameters. The traditional ESO calculation is presented in equation (4).

$$\{\begin{array}{l}{e}_1=z_1-y\hfill \\ {{\displaystyle \dot{z}}}_1=z_2-{\beta }_1{e}_1\hfill \\ {{\displaystyle \dot{z}}}_2=z_3-{\beta }_{2}nf\left(e_1,{\alpha }_0,{\delta }_1\right)+b_{0}u\hfill \\ {{\displaystyle \dot{z}}}_3=-{\beta }_{3}nf\left(e_1,{\alpha }_1,{\delta }_1\right)\hfill \end{array}.$$(4)

In equation (4), z₁ represents the tracking signal of output y. β₁, β₂, and β₃ represent the gain coefficients. e₁ represents the tracking error. z₂ represents the differential signal of y. b₀ represents the estimated compensation coefficient. z₃ represents the observed total disturbance. α₀ and α₁ represent nonlinear factors. δ₁ represents the filtering factor.

The study adopts a Sliding Mode ESO (SMESO), which can better achieve good evaluation performance with lower efficiency and handle uncertainty and external disturbances. Firstly, the sliding surface is shown in equation (5).

$$\{\begin{array}{l}s={\displaystyle \dot{{\displaystyle \tilde{e}}}}+\Lambda {\displaystyle \tilde{e}}\\ e=q-z_1\end{array}.$$(5)

In equation (5), s represents the sliding surface. Λ ∈ R^3×3 represents a diagonal positive definite matrix, and $s={\left[\begin{array}{ccc}\hfill s_1\hfill & \hfill s_2\hfill & \hfill s_3\hfill \end{array}\right]}^T$. In summary, the SMESO design is shown in equation (6).

$$\{\begin{array}{l}e=q-z_1,s={\displaystyle \dot{{\displaystyle \tilde{e}}}}+\Lambda {\displaystyle \tilde{e}}\hfill \\ {{\displaystyle \dot{z}}}_1=z_2-k_{1}s-{\gamma }_{1}fal\left(s,{\alpha }_1,\delta \right)\hfill \\ {{\displaystyle \dot{z}}}_2={{\displaystyle \dot{z}}}_3-k_{2}s-{\gamma }_{2}fal\left(s,{\alpha }_1,\delta \right)+b_0{u}^{*}\hfill \\ {{\displaystyle \dot{z}}}_3=-k_{3}s-{\gamma }_{3}fal\left(s,{\alpha }_1,\delta \right)\hfill \end{array}.$$(6)

In equation (6), k > 0 and γ ∈ R^3×3 represent diagonal positive definite matrices. Figure 3 displays the improved nonlinear ADRC.

In Figure 3, v₁ is the smoothed signal output by the tracking differentiator, v₂ is the derivative of the smoothed signal, e₁ is the error between v₁ and the observed value z₁, e₂ is the error between v₂ and the observed value z₂, u₀ is the initial control quantity output by the nonlinear feedback, and u is the final control quantity after disturbance compensation. The tracking differentiator of the improved nonlinear ADRC is calculated in equation (7).

$$\{\begin{array}{l}{e}_1=v_1-z_1\hfill \\ e_2=v_2-z_2\hfill \\ u_0={\beta }_{1}nf\left(e_1,{\alpha }_0,{\delta }_2\right)+{\beta }_{2}nf\left(e_2,{\alpha }_1,{\delta }_2\right)\hfill \end{array}.$$(7)

In equation (7), δ₂ represents the filtering factor. β₁ signifies the gain coefficient of the error. β₂ signifies the gain coefficient of error differentiation. α₀ and α₁ represent nonlinear factors.

In ESO, after observing system disturbances, disturbance compensation is required to unify the internal and external disturbances as the total disturbance, and improve the control performance. The specific function is shown in equation (8).

$$u=u_0-\frac{z_3}{b_0}.$$(8)

In equation (8), $\frac{z_3}{b_0}$ signifies the total disturbance compensation. u₀ signifies the control signal before disturbance compensation. u signifies the control quantity of the controlled robotic arm.

Based on the Lyapunov stability theory, stability analysis was conducted on the improved nonlinear ADRC system. The tracking errors of the system is defined as e₁ = v₁ − z₁ and e₂ = v₂ − z₂, where v₁ and v₂ are the desired states and, and z₁ and z₂ are estimated values of ESO. The Lyapunov function $V(t)=\frac{1}{2}{e}_1^2+\frac{1}{2}{e}_2^2+\frac{1}{2{\beta }_3}{z}_3^2$ is constructed, wherez₃ is the estimated value of ESO for the total disturbance and β₃ is the positive definite gain coefficient. The Liapunov function is derived by substituting the system state equation (Eq. (6)) and the control law (Eq. (7)). Combined with the strict monotonicity of the nonlinear function nf(e), the result ${\displaystyle \dot{V}}(t)=-e_1{\beta }_1\text{nf}\left(e_1\right)-e_2{\beta }_2\text{nf}\left(e_2\right)-\frac{1}{{\beta }_3}{z}_3\left({\beta }_3\text{nf}\left(e_1\right)+d\right)$ is obtained. d is the total system perturbation. Since β₁, β₂, and β₃ >0 and nf(e) is a negative definite function, ${\displaystyle \dot{V}}(t)\le 0$ is constant, indicating that the system is stable in the Liapunovian sense. Further analysis shows that when t→ ∞ and V(t) → 0, i.e., the tracking error e₁, e₂ and the perturbation estimation z₃ converge to zero, the stability of the improved ADRC method is verified.

Fig. 1 Structure diagram of ADRC.

Fig. 2 Simulink simulation diagram of tracking differentiator.

Fig. 3 Structure diagram of improved nonlinear ADRC.

2.2 Application of industrial robotic arm based on nonlinear ADRC technology

After improving the active disturbance rejection controller mentioned above, the study applies the controller to industrial robotic arms. As the core equipment of intelligent manufacturing, industrial robotic arms are widely used in welding, assembly, handling, and other scenarios [17,18]. However, the robotic arm system has problems such as nonlinearity, strong coupling, and external interference, and traditional control methods are difficult to satisfy high-precision control requirements. Nonlinear ADRC technology provides a new approach for solving the above problems by compensating for internal and external disturbances in real time through ESO [19].

The study first constructs a dynamic model for the robotic arm, as shown in equation (9).

$$\tau =M(q){\displaystyle \ddot{q}}+C(q,{\displaystyle \dot{q}}){\displaystyle \dot{q}}+G(q)+F_d.$$(9)

In equation (9), M(q) signifies the inertia matrix. $C(q,{\displaystyle \dot{q}}){\displaystyle \dot{q}}$ signifies the matrix of Coriolis force and centrifugal force. G(q) signifies the gravity term. F_d represents the external disturbance. q, ${\displaystyle \dot{q}}$, and ${\displaystyle \ddot{q}}$ represent the rotation angle, angular velocity, and angular acceleration vectors. τ signifies the generalized moment vector. The joint-link modeling is presented in Figure 4.

In Figure 4, the study adopts the parameter method, which is a standardized approach used to depict the relative position and attitude between connecting rods in the robotic arm. The length of the Connecting Rod (CR) is the distance from the current joint axis to the next joint axis along the previous joint axis. The torsion angle of the CR is the angle between the previous and the current joint axis. CR offset is the distance from the origin of the previous one to the origin of the current CR along the current joint axis. Joint angle is the rotation angle of the current joint axis [20,21].

According to the Lagrange equation, a dynamic equation is construed for the robotic arm system, as shown in equation (10).

$${\tau }_i=\frac{d}{\mathrm{d}t}\left(\frac{\partial L}{\partial {{\displaystyle \dot{q}}}_i}\right)-\frac{\partial L}{\partial q_i},i=\mathrm{1,2},\cdots ,n.$$(10)

In equation (10), i represents the joint. q_i signifies the generalized coordinates of the joint. ${{\displaystyle \dot{q}}}_i$ signifies the generalized velocity. τ_i represents the generalized moment. The transfer function between the motor output and input is shown in equation (11).

$$\frac{\mathrm{\Omega }(s)}{U_a(s)}=\frac{K_e}{\left(R_a+L_{a}s\right)(f+Js)+K_e{K}_a}.$$(11)

In equation (11), Ω(s) represents the output quantity. U_a(s) represents the input quantity. K_a represents the back electromotive force constant. L_a signifies the inductance of the armature circuit. R_a signifies the resistance of the armature circuit. f signifies the coefficient of viscous friction. K_e signifies the motor torque constant. J signifies the moment of inertia. The mathematical model of motor position is shown in equation (12).

$$\frac{d^2\theta }{d{t}^2}=\frac{K_e{K}_a}{J{R}_a}\omega -\frac{T_d}{J}+\frac{K_e}{J{R}_a}{u}_a.$$(12)

In equation (12), T_d represents the load torque with interference factors. ω signifies the angular velocity. u_a represents the armature circuit voltage. θ represents the angle at which the motor rotates.

The joints of industrial robotic arms generally use DC torque motors, as displayed in Figure 5.

In Figure 5, T_d (t) represents the load disturbance torque, i_a (t) represents the armature current, and T_e (t) represents the electromagnetic torque of the motor; Ra is the armature resistance, L_as is the armature inductance, K_e is the back electromotive force coefficient, f is the viscous friction coefficient, Js is the moment of inertia, and K_a is the feedback coefficient; G₁(s)=1/(Ra+La) s) is the transfer function of the armature circuit, G₂(s)=1/((f+J) s) is the mechanical transfer function of the motor, and Ea is the feedback signal of the back electromotive force. The overall structure principle of the DC torque motor from electrical signal input to torque, speed output, and feedback regulation is presented. It is the core component of the industrial robotic arm joint drive system. By increasing the diameter of the motor rotor and shortening the magnetic circuit length, higher torque output per unit volume can be achieved. The torque can be directly controlled through current closed-loop, simplifying the control algorithm [22,23]. In practical applications, there are disturbance factors affecting the control accuracy of robotic arm joints. Therefore, the study considers all external disturbances and obtains the corresponding dynamic equation, as shown in equation (13).

$$\{\begin{array}{l}\tau =H(q){\displaystyle \ddot{q}}+B(q,{\displaystyle \dot{q}}){\displaystyle \dot{q}}+G(q)+{\tau }_d\hfill \\ y=q\hfill \end{array}.$$(13)

In equation (13), τ_d represents all external disturbances. The tracking error is shown in equation (14).

$$e_d=q_d(t)-q(t).$$(14)

In equation (14), q_d represents the expected angle. e_d represents the tracking error. q(t) → q_d(t) represents the optimal operating state. A filter is designed to estimate the load torque and suppress the noise impact during the motion of the robotic arm. The covariance matrix between noise and state variables is shown in equation (15).

$$\{\begin{array}{l}\{\begin{array}{l}Q=\mathrm{cov}(w)\hfill \\ R=\mathrm{cov}(v)\hfill \end{array}\hfill \\ P_k=\mathrm{cov}\left\{e_k{e}^T\right\}\hfill \end{array}.$$(15)

In equation (15), Q signifies the noise covariance matrix. R signifies the covariance matrix of the state variable. P signifies the covariance matrix of state estimation error. w represents the noise. v represents the adjustable noise. w_k represents the discrete state of noise. v_k represents the discrete state of adjustable noise. The control of industrial robotic arm based on dynamic model and nonlinear ADRC is presented in Figure 6.

In Figure 6, the study combines the dynamic model with ADRC and takes SMESO to estimate the total disturbance of the robotic arm in state observation, including Coriolis force, centrifugal force, external disturbances, etc. On disturbance compensation, the estimated total disturbance action is taken as a feedforward compensation term and added to the control input. Nonlinear feedback laws are designed based on state errors in nonlinear feedback to enhance the robustness.

Fig. 4 Joint-link modeling of robotic arm.

Fig. 5 Structure of DC torque motor.

Fig. 6 Industrial robotic arm control based on dynamic model and nonlinear ADRC.

3 Results

To verify the designed control method for industrial robotic arms based on nonlinear ADRC technology, experiments are conducted to validate it. The experimental data are analyzed, providing reference for the development of industrial robotic arms.

3.1 Experimental environment for verifying control methods of industrial robotic arms

The study selects an industrial robotic arm from a certain company and applies the proposed control method for industrial robotic arms based on nonlinear ADRC technology. A simulation model of an industrial robotic arm in Matlab environment is built. The experimental platform is equipped with 8 GB, 2 TB hard drives, Linux Ubuntu 18.04 system, and a 2.9 GHz Intel i5 processor. Table 1 displays the related parameters.

3.2 Analysis of the control effect of ADRC technology for industrial robotic arm

Figure 7 shows the new nf() function curve proposed in the study. The nf() function proposed in Figure 7a had good convergence properties. During system operation, the function could maintain good robustness and smoothness, which improved system stability. In Figure 7b, rounding was applied at the inflection point to address the jitter. This means that the function can provide a smoother response when dealing with dynamic changes in the system, thereby reducing system instability.

Figure 8 displays the simulation results of SMESO response curves in different signal waveforms. From Figure 8a, the step signal of the response variable showed a significant jump at 1 s. The response variables z1, z2, and z3 exhibited different dynamic response characteristics after the step signal jump. z1 and z2 quickly reached a stable value after the jump, while the response of z3 was relatively slow, showing a gradually rising and then stable trend. The sine signal in Figure 8b exhibited periodic changes within the time period of 0 to 5 s. The response variables z1, z2, and z3 of SMESO also exhibited periodic changes under the action of sinusoidal signals. The response of z1 and z2 was closer to the frequency and amplitude of a sine signal, while the response amplitude of z3 was smaller and the amplitude was slightly different. Overall, the SMESO proposed in the study was able to timely visualize nonlinear dynamics and external disturbances in the system. The state of the robot arm includes step signal parameter settings and joint motion trajectories. The step signal parameter is set to an amplitude of 0.5 rad, triggering a jump at 1 s; The joint motion trajectory adopts the typical trajectory of a 3-degree-of-freedom robotic arm (such as moving from the initial angle [0,0,0] rad to the target angle [0.8, 1.2, 0.6] rad). The simulation results show that after the step signal is triggered, the joint angle quickly tracks, verifying the response ability of SMESO to dynamic changes.

Figure 9 compares the error between Proportional-Derivative (PD) control and ADRC control before and after improvement. In Figure 9a, the error of PD control was relatively large and exhibited significant periodic fluctuations. The error of ADRC technology and the improved ADRC technology was relatively small, and the error of the improved ADRC technology was lower, showing better stability and accuracy. In Figure 9b, the error of PD control was still relatively large, and the fluctuation was more obvious, with an error of about 0.10–0.20. The error of ADRC technology and the improved ADRC technology was relatively small, within 0.18. The error of the improved ADRC technology was lower, showing better control effect. In both cases, the improved ADRC technology exhibits minimal error and optimal stability, indicating that it outperforms the other two control methods on control accuracy and stability.

Figure 10 displays the joint motion trajectory tracking error of different control methods. The research method is sliding mode control, and the traditional control method selects PD control and Neural Network (NN) for compensation. In Figure 10a, the errors of sliding mode control and NN compensation control were small, within 0.2 rad, and relatively stable. The PD control showed significant error peaks and fluctuations at approximately 10 s and 15 s, with the highest reaching 0.64 rad and the lowest reaching −0.3 rad. In Figure 10b, the errors of these three control methods were small, all less than 0.01 rad, and relatively stable. There were significant error peaks and fluctuations in PD control at approximately 10 and 15 s, with the highest reaching 0.05 rad and the lowest reaching 0.035 rad. Next was PD control, while the proposed nonlinear ADRC sliding mode control had the smallest fluctuation, which was the most stable.

Figure 11 shows the experimental results of the joint angle and angular velocity variation curves of an industrial robotic arm in practical applications. In Figure 11a, the joint angle under the research method and the comparative method tended to stabilize after about 2 s. In Figure 11b, the changes of joint angular velocity under the two methods were similar, but the joint angular velocity under the research method tended to stabilize after about 2 s, while the joint angular velocity under the comparative method tended to stabilize after about 2.5 s. In Figure 11c, the elastic displacement tended to stabilize after approximately 2.1 s, while the comparative method showed slight fluctuations after stabilizing. In Figure 11d, the control torque under the research method and the comparative method tended to stabilize after approximately 2 s. The joint angle, joint angular velocity, control torque, and change trend under the two methods were similar, but there were differences in reach a stable state and the final stable value. The research method has advantages in controlling the speed of robotic arm joints.

To verify the actual effectiveness of the algorithm, a certain model of 6-axis industrial robotic arm was selected for practical experiments. In the welding operation scenario, the tracking error data of the robotic arm end is as follows: when using traditional PD control, the average error is 0.042 rad, and the maximum error reaches 0.06 rad; When using the proposed nonlinear ADRC, the average error is reduced to 0.008 rad, the maximum error is only 0.012 rad, and there is no significant fluctuation, which is consistent with the simulation results and verifies the effectiveness of the algorithm in practical applications.

Fig. 7 Experimental verification of the proposed novel nf() function curve.

Fig. 8 Response curves of SMESO in different signal waveforms.

Fig. 9 Comparison of error simulation between PD control and ADRC.

Fig. 10 Tracking errors of joint motion trajectories using different control methods.

Fig. 11 Industrial robotic arm joint angle and angular velocity variation curve.

4 Discussion and conclusion

Nonlinear system control has always been an important research direction in robotics, automation systems, and many other engineering fields. Especially in robotic arms, it is particularly difficult to achieve stable and accurate control systems due to the highly nonlinear and unknown constraints of the system. Therefore, a control method for industrial robotic arms based on nonlinear ADRC technology was designed. Based on ESO to estimate and compensate for total interference, high-precision trajectory tracking was achieved using NLSEF control law. The errors of sliding mode control and NN compensation control were small, within 0.2 rad, and relatively stable. The PD control showed significant error peaks and fluctuations at approximately 10 s and 15 s, with the highest reaching 0.64 rad and the lowest reaching −0.3 rad. The errors of the three control methods were relatively small, all less than 0.01 rad, and they were relatively stable. PD control showed significant error peaks and fluctuations at approximately 10 s and 15 s, with the highest reaching 0.05 rad and the lowest reaching 0.035 rad, followed by PD control. The nonlinear ADRC sliding mode control had the smallest fluctuations and was the most stable. The method can effectively address the control challenges of industrial robotic arms under complex working conditions, significantly improve the dynamic performance and robustness, and has broad application prospects. However, the research on multi-modal switching of robotic arms in complex tasks such as precision assembly and compliant operation is still insufficient. In the future, the ADRC parameter tuning method is further optimized to improve the adaptive capability of control systems.

Funding

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

Conflicts of interest

The authors report there are no competing interests to declare.

Data availability statement

Data will be available on a reasonable request.

Author contribution statement

Zhiwei Geng and Teng Wan wrote the main manuscript text, Zhiwei Geng collected the data and prepared tables. Teng Wan did the data analysis. Zhiwei Geng designed and conducted the research. All authors reviewed the manuscript.

References

All Tables

All Figures

	Fig. 1 Structure diagram of ADRC.
In the text

	Fig. 2 Simulink simulation diagram of tracking differentiator.
In the text

	Fig. 3 Structure diagram of improved nonlinear ADRC.
In the text

	Fig. 4 Joint-link modeling of robotic arm.
In the text

	Fig. 5 Structure of DC torque motor.
In the text

	Fig. 6 Industrial robotic arm control based on dynamic model and nonlinear ADRC.
In the text

	Fig. 7 Experimental verification of the proposed novel nf() function curve.
In the text

	Fig. 8 Response curves of SMESO in different signal waveforms.
In the text

	Fig. 9 Comparison of error simulation between PD control and ADRC.
In the text

	Fig. 10 Tracking errors of joint motion trajectories using different control methods.
In the text

	Fig. 11 Industrial robotic arm joint angle and angular velocity variation curve.
In the text