© Wang and Chen, Published by EDP Sciences 2025

1 Introduction

Under the influence of technological progress, robots have been more and more widely used in biological science [1], medical rehabilitation [2], industrial manufacturing [3], and other fields by virtue of their advantages of long working time, strong flexibility, and high efficiency [4]. An industrial manipulator is one of the most used robots in industrial scenes, which can replace humans to complete handling, assembly, welding, and other complex work [5]. The wide application of manipulators has promoted people's demand for more efficient and accurate manipulator movement trajectory control. More and more studies have been conducted on the optimization of manipulator control. Wang et al. [6] established a model based on the dynamic friction characteristics of the robot joint. Moreover, they optimized the square root cubature Kalman filter to estimate the external force of the robot manipulator, thereby improving the control accuracy of the model. Udomsap et al. [7] designed a 3-PRS parallel manipulator for platform stability. Through testing, it was found that the error between the input angle and the compensation angle was 0.4–3%, which was within the acceptable range. Hao et al. [8] designed a fully actuated air manipulator for industrial contact detection. It used a disturbance observer to estimate external disturbances and regarded them as feedforward compensations. The effectiveness of the proposed system in performing the task was verified. Yu et al. [9] designed a radial basis neural network algorithm based on acceleration feedback to control a 3-RRR parallel manipulator and demonstrated its stability through experiments. A 6R manipulator has the characteristics of a large control range and strong flexibility; therefore, it has extensive applications. As a higher degree of freedom of the manipulator will increase the complexity, the research on the control optimization algorithm of 6R manipulators has important application value in practice. Some current control methods for the motion trajectory of manipulators still have some deficiencies in terms of accuracy and efficiency, especially for multi-degree-of-freedom manipulators, and there is still the possibility of further optimization for their control. Therefore, in this paper, for 6R industrial manipulators, the parameter tuning method of the incremental proportional, integral, and differential (PID) algorithm was improved. Instead of using the traditional trial-and-error method, the intelligent optimization algorithm was introduced. An improved grey wolf optimization (IGWO) algorithm was developed, and the reliability of the proposed method was proved through simulation experiments. This paper provides a novel and reliable method for the movement trajectory control of industrial manipulators in practice and verifies the effectiveness of the intelligent optimization algorithm in parameter tuning. It also provides some theoretical support for the further optimization of the incremental PID algorithm.

2 The moving trajectory control method of industrial manipulators

2.1 Motor closed-loop control strategy

The drive component of the industrial manipulator is the motor. In order to meet the high precision movement requirements, each joint of the 6R manipulator uses a servo motor as the driver, and then the classical PID algorithm [10] is built into the control system to achieve closed-loop control and control the movement trajectory of each joint. The PID algorithm will feedback the output to the input. The deviation from the input is calculated, and the control quantity is obtained through the linear combination of each link and output to the target object, thus realizing the control of the manipulator. The structure of the PID algorithm is shown in Figure 1.

In Figure 1, r (t) is the input set value of the system, y (t) is the actual output value of the system, e(t) is the deviation between r (t) and y (t), e (t) = r (t) − y (t), and u (t) is the control quantity directly acting on the controlled object. Its control law can be written as: $$u\left(t\right)=K_p\left[e\left(t\right)+\frac{1}{T_I}{\displaystyle \underset{0}{\overset{t}{{\displaystyle \int }}}}e\left(t\right)dt+\frac{T_{D}de\left(t\right)}{\mathrm{d}t}\right],$$(1)

where K_p denotes the proportional coefficient, T_I is the integral time constant, and T_D is the differential time constant.

P, I, and D are the three correction links, as follows:

P: the proportional link used to reduce the system deviation;

I: the integral link used to reduce the system static difference;

D: the differential link used to advance correction of the system and speed up the system reaction speed [11].

In the computer system, digital PID is usually used, so it is necessary to discretize the above formula. It is assumed that the sampling period is T and the sampling sequence is k. Let $$t=kT\left(k=\mathrm{0,1,2},\cdots \right),$$(2)

it is substituted to obtain a discrete PID: $$u\left(kT\right)=K_p\left[e\left(kT\right)+\frac{1}{T_I}{\displaystyle {\displaystyle \sum }_{j=0}^k}e\left(jT\right)+T_D\frac{e\left(kT\right)-e\left[\left(k-1\right)T\right]}{T}\right]\text{.}$$(3)

Let e (kT) = e_k. The above equation is simplified, then: $$u_k=k_p{e}_k+k_i{\displaystyle {\displaystyle \sum }_{j=0}^k}{e}_j+k_d\left(e_k-e_{k-1}\right),$$(4)

where k_p is the proportional link gain coefficient, k_i is the integral link gain coefficient, and k_d is the differential link gain coefficient.

In order to enable the servo motor to obtain better control effect, the incremental PID algorithm [12] is adopted. It does not need to accumulate deviation, and the output is only related to the first two samplings, so it can get better stability while reducing the calculation amount. At this moment, the increment of the output can be written as: $$\mathrm{\Delta }{u}_k=u_k-u_{k-1},$$(5) $$u_{k-1}=k_p{e}_{k-1}+k_i{\displaystyle {\displaystyle \sum }_{j=0}^{k-1}}{e}_j+k_d\left(e_{k-1}-e_{k-2}\right),$$(6) $$\mathrm{\Delta }{u}_k=k_p\left(e_k-e_{k-1}\right)+k_i{e}_k+k_d\left(e_k-2e_{k-1}+e_{k-2}\right),$$(7)

where e_k is the current deviation, e_k−1 is the previous deviation, and e_k−2 is the deviation before the previous deviation.

Fig. 1 The structure of the PID algorithm.

2.2 Parameter tuning based on the improved grey wolf optimization algorithm

The parameter setting of the incremental PID algorithm determines the control effect of the industrial manipulator motor. In practical application, the parameters are mostly determined by experience or by trial and error. In order to further optimize the control performance of the incremental PID algorithm, this paper introduces the grey wolf optimization (GWO) algorithm, improves it, and designs an IGWO algorithm to realize the parameter tuning of the incremental PID algorithm.

The GWO algorithm is an intelligent algorithm based on wolf hunting mode [13], which has good usability and feasibility [14] and has good performance in multi-objective solving [15] and parameter optimization [16]. The individuals in the wolf pack are divided into α, β, δ, and ω according to grades. α is the leader wolf, and ω is the lowest-ranking wolf. The wolf pack hunts according to the rank system, and its position update process is: $$X\left(t+1\right)=X_p\left(t\right)-A\cdot L,$$(8) $$L=\left|C\cdot X_p\left(t\right)-X\left(t\right)\right|,$$(9)

where X (t + 1) denotes the position vector of the grey wolf in the t + 1-th iteration, X_p (t) is the prey position vector at the t-th iteration, L is the distance between the grey wolf and prey, C is the coefficient vector, C = 2r₁, A is the coefficient vector, A = 2ar₂ − a (a is the convergence factor), r₁ and r₂ are random numbers in [0,1].When surrounding the prey, the location of each individual relative to the prey can be written as: $$\{\begin{array}{c}\hfill L_{\alpha }=\left|K_1\cdot X_{\alpha }\left(t\right)-X\left(t\right)\right|\hfill \\ \hfill L_{\beta }=\left|K_2\cdot X_{\beta }\left(t\right)-X\left(t\right)\right|\hfill \\ \hfill L_{\delta }=\left|K_3\cdot X_{\delta }\left(t\right)-X\left(t\right)\right|\hfill \end{array},$$(10)

where K₁, K₂, and K₃ are random disturbances. The updated location of each individual can be written as: $$\{\begin{array}{c}\hfill X_1=X_{\alpha }-A_1\cdot L_{\alpha }\hfill \\ \hfill X_2=X_{\beta }-A_2\cdot L_{\beta }\hfill \\ \hfill X_3=X_{\delta }-A_3\cdot L_{\delta }\hfill \end{array},$$(11) $$X\left(t+1\right)=\left(X_1+X_2+X_3\right)/3,$$(12)

where X₁, X₂ and X₃ are the step length and direction of wolf ω moving towards each individual, and X (t + 1) is the final location of wolf ω.

According to the above equations, it can be found that GWO performance is closely related to convergence factor a. In the original GWO algorithm, a is linear, which is easy to fall into the local optimal. Based on the trigonometric function, this paper improves it to nonlinear: $$a=\sin \left(\frac{\pi t}{t_{max}}+\frac{\pi }{2}\right)+1\text{.}$$(13)

In addition, in the location update equation, variable proportional weights are added to achieve dynamic search of solutions. The proportional weights based on fitness values are: $$\{\begin{array}{c}\hfill W_{\alpha }=\frac{F_{\alpha }}{F_{\alpha }+F_{\beta }+F_{\delta }}\hfill \\ \hfill W_{\beta }=\frac{F_{\beta }}{F_{\alpha }+F_{\beta }+F_{\delta }}\hfill \\ \hfill W_{\delta }=\frac{F_{\delta }}{F_{\alpha }+F_{\beta }+F_{\delta }}\hfill \end{array},$$(14)

where F_α, F_β, and F_δ are the fitness values of α, β, and δ. The proportional weights of α, β, and δ leading wolf ω are: $$\{\begin{array}{c}\hfill W_1=\frac{\left|X_1\right|}{\left|X_1\right|+\left|X_2\right|+\left|X_3\right|}\hfill \\ \hfill W_2=\frac{\left|X_2\right|}{\left|X_1\right|+\left|X_2\right|+\left|X_3\right|}\hfill \\ \hfill W_3=\frac{\left|X_3\right|}{\left|X_1\right|+\left|X_2\right|+\left|X_3\right|}\hfill \end{array}\text{.}$$(15)

Variable proportional weights are obtained based on the above equations: $$\{\begin{array}{c}\hfill {\omega }_1=W_1\cdot W_{\alpha }\hfill \\ \hfill {\omega }_2=W_2\cdot W_{\beta }\hfill \\ \hfill {\omega }_3=W_3\cdot W_{\delta }\hfill \end{array}\text{.}$$(16)

The improved location equation of wolf ω is: $$X\left(t+1\right)=\left({\omega }_1{X}_1+{\omega }_2{X}_2+{\omega }_3{X}_3\right)/3\text{.}$$(17)

3 Experiment and analysis

The simulation experiment was carried out in MATLAB environment. A model was established by referring to the parameters of the manipulator (Tab. 1).

The parameters of the incremental PID algorithm were tuned by the trial-and-error method, GWO algorithm, and IGWO algorithm, respectively. Introductions to these methods are shown below.

(1) Trial-and-error method

Through repeated experiments and adjustments, the satisfactory corresponding curve was obtained, thereby determining the final parameters.

(2) GWO

K_p, T_I, and T_D were optimized using the aforementioned GWO method.

(3) IGWO

K_p, T_I, and T_D were optimized using the aforementioned IGWO method.

The results obtained by different methods are presented in Tables 2, 3 and 4.

In order to compare the effect of different algorithms on the movement trajectory control of the 6R manipulator, two points were randomly located in a Cartesian coordinate system: the starting point (263.12, 115.36, 362.17) and the target point (465.81, 269,55, 952.67). The experiment was repeated ten times to compare the deviations under the control of different algorithms. The results are shown in Table 5.

The comparison between the PID and incremental PID algorithms under the trial-and-error method in Table 5 showed that the 6R manipulator had a large movement deviation under the control of the PID algorithm, which proved the superiority of the incremental algorithm. Then, the comparison of the three incremental algorithms showed that under the trial-and-error method, the triaxial average deviation was 0.3–0.6 mm; under the GWO parameter tuning, the triaxial average deviation was 0.2–0.5 mm; under the IGWO parameter tuning, the triaxial average deviation was within 0.2 mm. In comparison, the incremental PID algorithm whose parameters were tuned using the IGWO algorithm achieved the best results in the control of the moving trajectory of the 6R manipulator, which proved the reliability of the proposed method.

Then, the manipulator's moving time was compared (Tab. 6).

In the process of moving from the start point to the target point, the traditional PID algorithm required the longest moving time (2–3 s), the incremental PID algorithm under the trial-and-error method required 1.8 s-2 s, and the incremental PID algorithm whose parameters were tuned by the GWO algorithm required 1.5 s and 1.7 s. Finally, the incremental PID algorithm whose parameters were tuned by the IGWO algorithm required a moving time between 1.1 s and 1.3 s. According to the results in Tables 5 and 6, it can be found that the incremental PID algorithm based on IGWO parameter tuning not only had a good control accuracy, but also had a higher moving efficiency in the movement trajectory control of the 6R manipulator. It could better meet the application requirements of the 6R manipulator in actual industrial scenarios.

The angle tracking of the 6R manipulator was analyzed. The following two classic signals were used:

• step signals with an amplitude of 1;

• sinusoidal signals with an amplitude of 1 and a frequency of 1 Hz.

Since signals similar to white noise rarely occur in the actual operation process of the manipulator, a sinusoidal signal with a step amplitude of 2 and a frequency of 4 Hz was taken as interference. Taking joint 6 as an example, the IGWO algorithm was used for parameter tuning, and the control effect of the PID and incremental PID algorithms were compared. The step response is shown in Figure 2.

It can be found that under ordinary PID control, there was an overshoot and oscillation in the step response, and the time required to enter the steady state was about 6 s. Under incremental PID control, the step response had no fluctuation and became steady in about 2.5 s. These results showed that under incremental PID control, the control of the moving trajectory by the 6R manipulator had better stability and could effectively inhibit external interference.

The sinusoidal response is displayed in Figure 3. It can be found that both control algorithms had good tracking performance for forward signals without too much obvious vibration. The comparison of the PID and incremental PID algorithms showed that the incremental PID algorithm had better tracking performance and smaller errors for sinusoidal signals, which further proved the stable control performance of the incremental PID algorithm for 6R manipulators. Therefore, in the context of actual industrial machinery, using incremental PID to achieve the control of 6R manipulators can ensure that the manipulators have higher accuracy, efficiency, and stability during the working process, thereby enabling their application in a wider range of industrial scenarios and providing reliable support for actual production.

Based on the above results, it can be concluded that the incremental PID based on IGWO parameter tuning showed smaller motion deviation and shorter motion time in controlling the motion trajectory of the 6R manipulator, with good stability and strong anti-interference ability, offering more advantages compared to the traditional PID algorithm.

Fig. 2 Step response.

Fig. 3 Sinusoidal response.

4 Conclusions

Based on the incremental PID algorithm, this paper studied the motion trajectory control of industrial manipulators and designed an IGWO algorithm to realize the parameter tuning of the incremental PID algorithm. From the results of simulation experiment, it can be found that the incremental PID algorithm based on IGWO parameter tuning had good performance in controlling the motion trajectory of the 6R manipulator. It exhibited high precision and efficiency and outperformed the traditional PID algorithm in step response and sinusoidal response. Therefore, it can be further promoted and applied in practical industrial robotic arms.

Funding

This research received no external funding.

Conflicts of interest

The authors have nothing to disclose.

Data availability statement

The datasets used and/or analyzed during the current study are available from the corresponding author on reasonable request.

Author contribution statement

Conceptualization, J.W. and F.C.; Methodology, J.W. and F.C.; Software, F.C.; Validation, J.W.; Formal Analysis, J.W.; Investigation, F.C.; Resources, F.C.; Data Curation, J.W.; Writing − Original Draft Preparation, J.W.; Writing − Review & Editing, F.C.; Visualization, J.W.; Supervision, J.W.; Project Administration, J.W.

References

All Tables

All Figures

	Fig. 1 The structure of the PID algorithm.
In the text

	Fig. 2 Step response.
In the text

	Fig. 3 Sinusoidal response.
In the text