
==== Front
Sci Rep
Sci Rep
Scientific Reports
2045-2322
Nature Publishing Group UK London

72156
10.1038/s41598-024-72156-x
Article
An improve nonlinear robust control approach for robotic manipulators with PSO-based global optimization strategy
Yue Peihao 13
Xu Bowen xbw_2014@163.com

2
Zhang Min 3
1 https://ror.org/05htk5m33 grid.67293.39 College of Electrical and Information Engineering, Hunan University, Changsha, 410082 China
2 https://ror.org/05d2yfz11 grid.412110.7 0000 0000 9548 2110 Hypersonic Technology Laboratory, National University of Defense Technology, Changsha, 410073 China
3 grid.216566.0 0000 0001 2104 9346 Hunan Academy of Forestry, Changsha, 410012 China
13 9 2024
13 9 2024
2024
14 2144730 3 2024
4 9 2024
© The Author(s) 2024
2024
https://creativecommons.org/licenses/by-nc-nd/4.0/ Open Access This article is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License, which permits any non-commercial use, sharing, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if you modified the licensed material. You do not have permission under this licence to share adapted material derived from this article or parts of it. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by-nc-nd/4.0/.
During the trajectory tracking of robotic manipulators, many factors including dead zones, saturation, and uncertain dynamics, greatly increase the modeling and control difficulty. Aiming for this issue, a nonlinear active disturbance rejection control (NADRC)-based control strategy is proposed for robotic manipulators. In this controller, an extended state observer is introduced on basis of the dynamic model, to observe the extend state of model uncertainties and external disturbances. Then, in combination with the nonlinear feedback control structure, the robust trajectory tracking of robotic manipulators is achieved. Furthermore, to optimize the key parameters of the controller, an improved particle swarm optimization algorithm (IPSO) is designed using chaos theory, which improves the tracking accuracy of the proposed NDRC strategy effectively. Finally, using comparative studies, the effectiveness of the proposed control strategy is demonstrated by comparing with several commonly used controllers.

Keywords

Robotic manipulator
Nonlinear dynamics
Active disturbance rejection controller
Nonlinear control
Particle swarm optimization
Subject terms

Mechanical engineering
Computational science
issue-copyright-statement© Springer Nature Limited 2024
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pmcIntroduction

With the continuous improvement of automation technology, robots have taken a more important role in industrial systems due to the flexible position movement pattern and strong adaptiveness to the work environment, and the phenomenon of “machines replacing people” has emerged in many fields, such as teleoperation, fault detection, health care, and other fields1–3. However, as a complex multi-input multi-output system, the robotic system is complex and highly nonlinear with various structural uncertainties, including dead zones, saturation, and uncertain dynamics, which can adversely affect the actuators4,5. In actual practice, these factors cannot be ignored, making it difficult to establish an accurate modeling and control for the robotic manipulators6. Therefore, it is necessary to research on the anti-disturbance control strategy of robotic mechanism, to suppress the influence of the uncertainty and disturbances on tracking accuracy of servo system7,8.

In the beginning, considering the advantages of PID algorithm not relying on dynamic models and simple principles, PID was applied for the trajectory tracking of robots9,10. When executing tasks, negative feedback control was adopted based on the deviation between the actual trajectory and the expected trajectory, and good control performance was achieved through simple parameter debugging. For instance, a robust controller was put forward in11, to compensate for the adverse effects of the unknown dead-zone input, which is ubiquitous in mobile robot motors. However, the studies assumed dead zones and saturation with predetermined parameters, while obtaining the parameter values of dead zones and saturation functions was often a challenging task. Besides, when there were multiple constraints and external uncertainties, these methods did not address the difficulties associated with controller design, which may lead to insufficient control performance. Therefore, many scholars have integrated other algorithms into the traditional controllers12–14. For instance, in14, an adaptive controller was developed on basis of linear extended state observer (LESO) for the industrial robot with time-varying load, uncertainties and disturbance. Besides, to improve the accuracy of robotic motion control, many model-based nonlinear control strategies have emerged15–17, such as sliding mode control (SMC)18, adaptive control19, neural network control (NN)20, model predictive control (MPC)21, fuzzy control22,23, and auto-disturbance rejection control (ADRC)24. For instance, in16, a Takagi–Sugeno (T–S) fuzzy model was employed to reconstruct the original nonlinear systems, and a hybrid-driven mechanism was applied for filter design to balance the systems' performance and the limited network resource consumption. In22, a fuzzy controller was designed to solve the control accuracy problem of a spherical robot under a complex underwater environment. In23, a self-organizing interval type-2 fuzzy wavelet neural network-based controller was designed for predicting and identifying nonlinear systems to improve the interaction performance. Among these controllers, as an effective robust control method, ADRC can effectively solve the control problems of nonlinear and uncertain systems25–27, since it was not only independent with the model, also had strong robustness and adaptability28. Besides, the internal disturbances, i.e., model, parameter perturbations and unmeasurable external disturbances, was attributed to the total disturbance, which can be observed by an extended state observer in real-time. In this way, the approximate linearization and deterministic linearization of nonlinear uncertain systems was achieved29. Therefore, the ADRC can effectively improve the dynamic tracking control performance and robustness of robots, and was widely used in the field of robotic motion control30–33. In34, the unknown uncertainties and external disturbances of robot manipulator was investigated by using a novel disturbance observer, which can actively compensate the total disturbance during the trajectory tracking process.

However, the design of the ADRC is relatively complex, involving a large number of variable parameters, and different combinations of parameter values may cause significant variation of the control performance. Moreover, these parameter values have no rules to follow, which means that there is no complete and reliable reference standard form for parameter selection. In the past, most technical personnel relied on empirical testing for selection34–36. Besides, when adjusting the three parameters and six directions of the robotics, if there were many input variables, the combination of adjustment possibilities will be more. This work was often not competent for non-professionals36. In addition, it was easy to produce small amplitude oscillations near the extreme point. With the research of intelligent algorithms, a series of algorithms such as genetic algorithm (GA), ant colony algorithm (ACA), neural network algorithm (NN), particle swarm optimization algorithm (PSO), etc., have begun to be applied to the parameter design of controllers37–40. Gao40 proposed a glowworm swarm algorithm-based controller to improve the firefly algorithm using sine and cosine functions to accurately and quickly obtain parameters of the ADRC. Among these algorithms, the PSO is one of the most popular intelligent algorithms due to its simple structure, easy implementation and minor code. This algorithm was proposed by Kennedy and Eberhart in 1995, inspired by bird swarm motion41, and has been successfully applied to many control fields42–45. For instance, in42, an ADRC of an induction motor was proposed based on a particle swarm optimization (PSO) algorithm to realize the precise decoupling of the induction motor and the disturbance compensation. In43, the XK-I spherical robot PSO-ADRC motion controller with parameter tuning function was designed by combining the state equation with PSO algorithm. The PSO algorithm was improved by using the genetic algorithm theory to improve the optimization range and convergence rate45. However, the population diversity was losing during the optimization process, falling into local extremum, and low accuracy in actual practice, scholars have proposed various improvement measures, focusing on the selection of algorithm parameters, or the improvement of algorithm position and speed updating rules. Overall, the above algorithms have improved the PSO from different perspectives, which improved the optimization performance to some extent. However, the problem caused by premature convergence has not been fundamentally solved.

From the analysis, the robotic manipulator is characterized with complex dynamics, and the dynamic model of the robot manipulator is sensitive to the inaccuracies of the system parameters, which extremely increase the difficulties on control stability. Aiming for this problem, we developed an ADRC-based controller that considers both the model uncertainty and disturbance for robust tracking of robotic manipulator. When designing the controller, three main works are made, the main novelties are the following:A key point is that an extended state observer is introduced on basis of the dynamic model, to observe the extend state of model uncertainties and external disturbances.

In combination with the nonlinear control structure of the ADRC controller, the robust trajectory tracking of robotic manipulators is achieved.

To optimize the key parameters of the controller, an improved particle swarm optimization algorithm (IPSO) was designed using chaos theory, which improves the tracking accuracy of the proposed NDRC strategy effectively.

Finally, using the studies on a two-joint robotic arm, the performance for several commonly used controllers is detailly compared and analyzed under different types of disturbances, which demonstrates the satisfactory control effect of the proposed strategy during the operation processes.

Problem description

A robot can be defined as an electromechanical device that can operate autonomously or semi-autonomously to achieve a specific task. As shown in Fig. 1, manipulators consist of a series of links connected by either rotary or translational joints.Fig. 1 Structure of 6-degree of freedom robotic arm and its simplified schematic diagram.

For model-based controllers, a primary step is to develop a mathematical model of the system by identifying the relationship between the output and the input of the plant. Usually, the dynamic model of a robotic arm can be described as follows:1 M(q)q¨+C(q,q˙)q˙+G(q)=τ+w

wherein, τ is the input torque, it is a vector of n×1, n is the number of joints of the manipulator. q,q˙ and q¨ represent the position, velocity, and acceleration of the joints, respectively. M(q)∈Rn×n is the inertia matrix of the robot; C(q,q˙)∈Rn×n represents the Coriolis force and centrifugal matrix, which is usually bounded, namely, there is a known C¯(q) and C(q,q˙)≤C¯(q)q; the matrix D˙-2C is a skew symmetric matrix that satisfies xT(D˙-2C)x=0. G(q)∈Rn×n representing the gravity matrix, w is the generalized disturbance term of the robot, including model uncertainty and internal and external disturbances, satisfying w≤w¯ and w¯ is a known constant. However, during the operation process, dead zones are a common problem in robotic systems, including valves, gears, servo motors, and other industrial equipment. Except for this, saturation constraints can lead to actuator breakage, while friction can result in tracking errors and oscillations in a steady state. These factors can significantly impact the dynamic control performance of a robotic system. Therefore, main characteristics of the robotic models can be summarized as follows:The constraints related to the input dead zone, output saturation function and actuator dynamics in the dynamic model increases with the number of joints.

Each term in the equation contains terms such as sine and cosine, exhibiting a high degree of nonlinearity.

There are strong coupling factors, including coupling between joints, internal and external disturbances.

The model has strong nonlinear dynamics, for instance, the load force changes, the friction torque of the joint also changes over time.

These factors pose great challenges for the modeling and control of robotic manipulators. Then, the dynamics of the robot (1) can be expressed as follows:2 q¨=-[M(q)]-1C(q,q˙)q˙+G(q)-(τ+w)

Here, for the convenience of controller design, define x1=q,u=τ, then one has3 x˙1=x2x˙2=M-1(x1)(-C(x1,x2)x2-G(x1))+M-1(x1)(w+u)y=x1

From Eq. (3), it can be seen that the robotic manipulator is characterized with complex dynamics caused by internal parametrical perturbations and external disturbances (i.e., joint friction, actuator dynamics, vibration during operation, etc.). Besides, the dynamic model of the robot manipulator is sensitive to the inaccuracies of the system parameters such as, links’ lengths and masses, inertia and variable payload. Therefore, a significant deviation between the simulation and the experimentation may occur due these inaccuracies in dynamic model. Generally, the performance of controllers decreases when neglecting the effect of the dynamic model uncertainties, and there is a trade-off between performance and robustness in all the controllers designed for robotic systems. All these factors extremely increase the difficulties on actual model accuracy and control stability.

Therefore, an effective controller that considers both the model uncertainty and disturbance is required for robust tracking performance.

Controller design

Aiming for this problem, an ADRC-based controller considering both the model uncertainty and disturbance is developed for robust tracking of robotic manipulator. Besides, in order to obtain the optimal values of the core control parameters, an improved PSO is developed, which can search the solution space globally.

As shown in Fig. 2, the proposed control strategy mainly contains three parts:An extended state observer is introduced on basis of the dynamic model, to observe the extend state of model uncertainties and external disturbances. Then, in combination with the nonlinear feedback control structure, the robust trajectory tracking of robotic manipulators is achieved.

In order to optimize the key parameters of the controller, an improved particle swarm optimization algorithm (IPSO) was designed using chaos theory, which improves the tracking accuracy of the proposed NDRC strategy effectively.

The effectiveness of the proposed control strategy was verified through simulation experiments.

Fig. 2 Optimization framework of ADRC based on the improved PSO algorithm.

NADRC design for robotic manipulator

Here, to realize the robust control of robotic manipulators under noise and disturbances, the proposed NADRC strategy is designed and shown as Fig. 3. Based on the tracking trajectory of the robotic arm. In each control loop, the inverse kinematics model is used to obtain the angles of each joint, which serves as the control objective. An active disturbance rejection controller is designed for each joint, and a motor is used to control each joint to achieve the target angle, thereby achieving tracking control of the target position of each circuit.Fig. 3 Structure of the active disturbance rejection controller.

Integrating the nonlinear parts, dynamic time-varying characteristics, and disturbances present in the system (3) into a total disturbance, one has4 W=M-1(x1)(-C(x1,x2)x2-G(x1))+M-1(x1)(w)

Expanding W as the new system state x3, which represents the model uncertainty and external caused by joint friction, vibration, input dead zone and actuator dynamics during the operation, etc. the state equation is rewritten as follows:5 x˙1=x2x˙2=x2+b0ux˙3=W(t)y=x1

Then, the ADRC-based nonlinear control strategy for robotic manipulator is designed as Fig. 4. The controller mainly consists of three parts: the tracking differentiator (TD), the extended state observer (ESO) and nonlinear state error feedback (NLSEF) control law. v(k) is the control objective (given signal), v1(k) and v2(k) are the tracking signal and differential signal of v(k) respectively. u0 is the output of the nonlinear state error. z3(k) is the estimation of the internal and external disturbance.Fig. 4 Structure of the active disturbance rejection controller.

Therefore, in response to Eq. (5), the ADRC is designed as follows:Design of the tracking differentiator.

The main purpose of arranging the transition process is to track the differentiator so that there is no jump in the input quantity and it is convenient for the real-time tracking of the actual system, namely,6 v1(k+1)=v1(k)+Tv2(k)v2(k+1)=v2(k)+Tfhanv1k-v0(k),v2(k),r0,h

where fhan(∙) is the time optimal control function. T is the sampling period, r0 is the speed factor, which is related to the tracking accuracy and transition time; h have the effect of noise filter, and is called filtering factor.(2) Design of the extended state observer.

Regarding the internal and external disturbance as an extended new system state, the ESO designs an extended observer to estimate this new state. Once the estimation of the new state is obtained, the estimated value can be subtracted directly in the control law, which is used to eliminate the uncertainty.7 e(k)=z1(k)-y(k)z1k+1=z1k+T[z2k-β01ek]z2k+1=z2k+T[z3k-β02falek,α1,δ+b0u(k)]z3(k+1)=z3(k)-Tβ03fale(k),α2,δ

where falx,α,δ is the continuous power function and β01,β02, β03 are the gain of the nonlinear error feedback.(3) Design of the nonlinear state error feedback.

According to the given signal and output feed obtained by TD and ESO, the NLSEF control law compensates the system disturbance and generates the reference input of the controlled object. In fact, the NLSEF is an advanced version of proportional-derivative (PD) control. While the control law of traditional PD controller is linear weighted directly, the NLSEF uses nonlinear function to deal with the error and its differential before the linear weighted procedure. The expression of NLSEF control law is:8 e1(k)=v1(k)-z1(k)e2(k)=v2(k)-z2(k)u0(k)=β1fale1(k),α1,δ+β2fale2(k),α2,δ

where β1, β2 represent the gain of the state error feedback control law.9 u(k)=u0(k)-z3(k)/b0

and b0 is the compensation coefficient. In the above rejection control law, z3(k)/b0 achieved compensation for total disturbance. In this way, due to its ability of estimation and compensation for total disturbances, strong robustness, high control accuracy can be realized.

The definition of those two nonlinear functions fal∙ and fhan∙ are:10 falx,α,δ=xδ1-α,x≤δ,xαsignx,x>δ,δ>0

11 d=rh2,a0=hx2,y=x1+a0a1=d(d+8y)a2=a0+sign(y)(a1-d)/2sy=(signy+d-signy-d)/2a=a0+y-a2sy+a2sa=(signa+d-signa-d)/2fhanx1,x2,r,h=-rad-signasa-rsign(a)

where β1, β2 represent the gain of the state error feedback control law.u=u0-z3/b0

Here, the h, r0, β01, β02, β03, β1, β2, b0, α1, α2 is parameters of the whole controller. Among those parameters,α1, α2 can take the value 0.5 and 0.25 according to experience; the speed factor r0 and filtering factor h guarantee the fast-tracking accuracy and noise reduction performance. Thus,r0,h,β01,β02,β03, β1, β2 and b0 have a closely relationship with the real-time output. For instance, the larger are the parameter values, the faster the convergence process will be, but bigger overshoot will be generated. Therefore, an effective strategy of parameter optimization is required to obtain satisfactory control effect.

Parameter identification

In PSO, the potential solution of every optimization problem is a particle in search space, which is called particle, and all particles have a fitness value determined by the optimized function. Among these particles, each particle has a speed that determines the direction and distance of their flight. Then the particles follow the current optimal particle to search in the solution space. However, the position of particles oscillates uniformly with constant amplitude in a certain range and does not converge to a stable value, which makes the system easy to fall into local optimum. Therefore, an improved PSO algorithm is designed to identify core parameters of the NADRC strategy.Oscillation Mechanism.

Suppose that in a d dimensional target search space, N particles form a community, and the position Xik and speed vi(k) of ith particle in kth iteration is represented as a d dimensional vector:Xik=xi1,xi2,xi3,⋯,xid,i=1,2,3,⋯,N

vik=vi1,vi2,vi3,⋯,vid,i=1,2,3,⋯,N

where xi1∈[xi1min,xi2max], vi1∈[xi1min,xi2max].k=1,2,3,⋯,M. M is the number of iterations.

Using pbest(i) to record the optimal solution searched by individuals, and gbest to record the optimal solution searched by the whole group in one iteration.12 vik+1=w∗vik+c1∗r1∗pbestik-Xik+c2∗r2∗(gbest(k)-Xik)

13 Xik+1=Xik+vi(k+1)

where vi(k) is the speed of particle i; c1 and c2 represent the acceleration constants, which are used to constraint the speed of the particle learning factors. When c1=0, the particle has no cognitive ability and becomes a social-only model; when c2=0, there is no social information between particles and the model becomes a cognitive-only model; w is the inertia factor and indicates the ability to maintain its former speed. r1 and r2 are random constants bounded in [0,1], which reflect the particles' memory of their own historical experience and the group historical experience respectively. pbest(i) represents the currently best position of particle i, and gbest is the best position found by the whole population.

The optimization strategy is based on the fitness function, which is closely related to the research object. For particle i, the best position can be obtained by the following comparison process:pbestik+1=

14 pbestik,Fpbestik≥F(pbesti(k+1))pbestik,F(pbestik+1)<F(pbesti(k+1))

Similarly, the global optimal position of the whole population is:gbestk+1=

15 gbest(k),Fgbest(k+1)≥F(gbest(k))gbest(k+1),Fgbest(k+1)<F(gbest(k))

Here, by introducing a second-order oscillation link into the iterative equation of the standard PSO, the problem caused by oscillation non-convergence and local optimization in traditional particle swarm algorithms can be overcome. Here, a strict second-order oscillation mechanism is developed and represented as follows:s1=2φ1/φ1,s2=2φ2/φ2;k≤M/2s1=2/φ1,s2=2/φ2;t≥M/2

Then, the PSO can be improved as:16 vik+1=w∗vik+φ1∗pbestik-1+s1Xik+s1∗Xik-1+φ2∗gbest(k)-1+s2Xik+s2∗Xik-1

17 Xik+1=Xik+vi(k+1)

Similarly, we have18 Xi′′k=φ1∗pbestik-1+s1Xik+s1∗Xik-1+φ2∗gbestk-1+s2Xik+s2∗Xik-1

Then Eq. (18) can be simplified as:19 Xi′′k+φ1s1+φ2s2Xi′k+φ1+φ2Xik-φ1pbestik+φ2gbestk=0

By solving Eq. (19), we can get characteristic roots as follows:20 Xik1,2=-φ1s1+φ2s22±φ1s1+φ2s22-4φ1+φ22

According to the stability of the second order differential system, there are two convergence conditions. First, in order to make the particle position oscillatory convergence, the following situation must be satisfied:21 φ1s1+φ2s2>0φ1s1+φ2s22-4φ1+φ2<0

When s1, s2 satisfying s1=2φ1/φ1,s2=2φ2/φ2, we have φ1s1+φ2s2=2φ1+2φ2>0, due to φ1∈[0,1], φ2∈[0,1]. For the second condition, φ1s1+φ2s22=2φ1+2φ22=2φ1+φ2+4φ1φ2≤4φ1+φ2. Thus, φ1s1+φ2s22-4φ1+φ2<0 holds.

Then, to ensure the particle's asymptotic convergence, the following conditions also need to be satisfied:22 φ1s1+φ2s2>0φ1s1+φ2s22-4φ1+φ2>0

When s1, s2 satisfying s1=2/φ1,s2=2/φ2, the condition φ1s1+φ2s2>0 holds. Moreover, φ1s1+φ2s22=2φ1+2φ22=4φ1+φ2+8φ1φ2>4φ1+φ2, then φ1s1+φ2s22-4φ1+φ2>0.

Therefore, by introducing the oscillation convergence mechanism s1=2φ1/φ1,s2=2φ2/φ2 in the early stage of iterations, the algorithm has a strong global search ability.(2) Chaotic Disturbance Mechanism.

As described above, the former oscillation mechanism can ensure the optimization of the objective function in two stages, but there is a switch between these stages, which may influence the searching efficiency. It is known that Chaos is a common phenomenon in nonlinear system, which has the characteristics of ergodicity and internal randomness. It can traverse all states without repetition according to its own law in a certain range. As is a typical chaotic system, the mapping relationship of Logistic map is:23 ξi(k+1)=ρξi(k)1-ξi(k),ξi(k)∈0,1

where ρ is the control parameter, when ρ=4, the logistic map is completely in a chaotic state, and the generated chaotic variable ξk+1 has good performance of ergodicity in [0,1].

Inspired by this, the chaotic disturbance mechanism is proposed to obtain a global optimization process, the detailed processes are summarized as: (1) Based on the iterative result generated by traditional PSO, the first 30% of particles with better fitness are retained and chaotic perturbations are applied. By utilizing the good ergodicity of chaotic variables, the position of this particle is further optimized to find the optimal solution around these particles. (2) Initializing the value of chaotic variable ξi(k), which meets the chaotic state of logistic map function, namely, ξi(k)=Xik-Ximin/Ximax-Ximin. 3) Generating chao components ξi(k+1), namely, ξi(k+1)=ρξi(k)1-ξi(k). 4) Generating new particle position according to chaotic variable iteratively, namely, Xik+1=Ximin+ξi(k+1)(Ximax-Ximin) , where Ximax and Ximin are represented as d dimensional vectors, Ximax=xi1max,⋯,xidmax,i=1,2,3,⋯,N. Ximin=xi1min,xi2min,xi3min,⋯,xidmin. Comparing the fitness value of Xik with Xik+1, and one has:24 Xi(k)=Xi(k),FXik+1≥F(Xi(k))Xik+1,F(Xik+1)<F(Xi(k))

(3) Boundary Contraction and Escape Strategy.

A typical problem faced by PSO is boundary control. If one can find an effective way to contract the boundary of parameters, the search efficiency and solution stability will be greatly improved. Thus, based on the global optimal solution of particles, a random shrinkage strategy is proposed for particle position boundary.25 Ximin=maxXimin,gbestk-rand∗(Ximax-Ximin)

26 Ximax=minXimax,gbestk+rand∗(Ximax-Ximin)

With the elaborated boundary contraction strategy, the upper and lower boundaries of particle positions are limited to a more reasonable range, namely,27 [gbestk-rand∗Ximax-Ximin,bestk+rand∗Ximax-Ximin]

For the multi-peak problem, it is easy for PSO to fall into the local optimum due to the decrease of population diversity, which is called premature phenomenon. Aiming for such problem, an effective escape strategy is developed.

In each iteration, once the global optimal solution for the population is obtained, the 30% outstanding particles with better fitness value are retained, and chaotic disturbance is applied to further optimize the local solution of these particles. For the rest part, the mutation operation is performed to make the particles spread out along the center of the local optimal solution and enter other areas of the solution space for searching. For the other 70% particles, the escape strategy is proposed as follows:28 Xik+1=Ximin+rand∗(Ximax-Ximin)

29 vik+1=rand(1,d)

Summary

Based on a large number of literatures consulting and simulation analysis, we found that five of those parameters, i.e., β01, β02, β03, β1, β2, which play a decisive role in the performance of convergence speed and robustness. Using the improve PSO method, values of these parameters can be obtained. In summary, the coding structure of each particle can be expressed as follows:30 Xik={β01,β02,β03,β1,β2}

In the PSO optimization process, the goal is to minimize the fitness function by adjusting the value of Xik. The detailed parameters optimization procedure of ADRC based on improved hybrid PSO can be described as Table 1. Table 1 the detailed optimization procedure of ADRC based on improved hybrid PSO.

Simulation and results

In this section, numerical simulations are conducted to validate the proposed control strategy and optimization method. First, to verify the effectiveness of the proposed PSO optimization algorithm, studies are conducted by several commonly used PSO algorithms. Then, using the two-joint robotic manipulator, the control simulation is carried out to verify the PSO-based control strategy.

Optimization simulation on standard functions

For Rastrigin function, there are many local minimum points in the search space, which is very suitable for verifying the global performance of the algorithm. The function can be expressed as:f3x=∑i=1n(xi2-10cos2πxi+10)

where xi∈[-10,10].

In this simulation, to improve the efficiency of SAPSO and CLSPSO algorithms, adaptive variable weights are adopted here. Set the size of the particle swarm to 50, taking the values of learning factors c1 and c2 as 0.5 and 0.5, respectively; wmin = 0.4, wmax = 0.9. The three test functions are run 15 times, and the termination condition of the algorithm is that the number of iterations is greater than 10,000, or the error is less than 1e-6. The algorithm remains stable as the complexity of the algorithm increases. The fitness values for each iteration are shown in Figs. 5 and 6.Fig. 5 Comparison results for Fitness values of the test function (10 dimensions).

Fig. 6 Comparison results for Fitness values of the test function (20 dimensions).

According to the graphic results, it can be seen that SecVibratPSO and NCEPSO have fast speed and high optimization accuracy, while other algorithms are prone to falling into local optima. Due to the introduction of a second-order oscillation mechanism, which continuously adjusts the learning factor as iteration progresses, enabling the system to quickly skip local minimum points. Furthermore, the introduction of chaotic units improves the optimization accuracy of the algorithm near the local optimal value. In this way, the designed algorithm has achieved good results, and the global optimal solution is obtained within 10 generations at the earliest stage.

Simulation verification on two-joint robotic arm

As mentioned in Section “Controller design”, the parameter tuning of this controller involves 5 parameters, which seriously affects the popularity of the algorithm in the industrial field. In response to this situation, the IPSO is applied for adjusting parameters of the ADRC controller, and a simulation is conducted to verify the performance of the proposed ISO-ADRC strategy, in which the effectiveness of this algorithm is demonstrated.

Similar to Eq. (1), the dynamic equations of a dual joint rigid robotic arm can be expressed as:31 D11(q2)D12(q2)D21(q2)D22(q2)q¨1q¨2+-C12(q2)q˙2-C12(q2)(q˙1+q˙2)C12(q2)q˙10+g1(q1+q2)gg2(q1+q2)g+F(q,q˙,q¨)=τ1τ2

whereD11(q2)=(m1+m2)r12+m2r22+2m2r1r2cos(q2)

D12(q2)=D21(q2)=m2r22+m2r1r2cos(q2)

D22(q2)=m2r22,C12(q2)=m2r1r2sin(q2)

F(q,q˙,q¨)=Fr(q˙)+w

Setting the parameters as r1=1m, r1=0.8m, m1=1kg, m2=1.5kg. Here, the goal is to make the outputs of two joints q1 and q2 tracking along with the desired trajectory qd1=0.3sint and qd2=0.3sint. First, establishing the model of robotic arm, and designing the NECPSO according to Table 1, core parameters of the NADRC can be optimized. For NCEPSO, the first step is to encode several key parameters of the controller. The coding structure of each particle can be set as:Xik={β01,β02,β03,β1,β2}

Some parameters of NCEPSO are set as: the population size N=20; the number of iterations M=10; the acceleration constants c1=0.8 and c2=0.5; the inertia factor wmin=0.4, wmax=0.9; the initial velocity of the particle is v=zeros(10,5). The termination condition of the algorithm is that the number of iterations is greater than M. Other parameters are as follows: T=0.001, h=0.001, r=1e10, d=0.01, α1=0.5, α2=0.25.

Furthermore, to verify the anti-interference performance of the algorithm and the tracking ability of the reference signal, a mixture of the step signal and the sine signal are selected as the reference input. The fitness function is designed as:Fitness function for step signal.

The expectation of transition time and overshoot for fitness function FXik are set as ts0=5e-3, σ0=1e-2.32 FXik=lgtsts0+1+lg(σσ0+1)

(2) Fitness function for multi-frequency sine signal33 FXik=∑i=1n(yi-y^i)2

First, the simulation is carried out for step signal, to present the performance of this controller by using quantitate indicators, such as overshoot, transition time. Using the proposed controller, the optimization process, the optimal fitness value is 5.8239 and shown as Fig. 7.Fig. 7 Variation of fitness values with the increasing of iterations.

From Fig. 7, there are five parameters to be optimized for ADRC, while the system converges fast in 10 iterations. The optimal parameters are [β01,β02,β03,β1,β2] = [19.1169, 41.4084, 54.8496, 61.1518, 26.6085, 2.5204]. The tracking result is shown as Fig. 8.Fig. 8 tracking result on step reference signals.

Furthermore, to demonstrate the control effect, the multi-frequency sine signal is applied, and the trend of this signal is formatted as:34 rt=0.9∗sint+0.9∗sin(2t)

Besides, in order to demonstrate the tracking effect of the proposed controller, three commonly used controllers are applied for comparison, Namely, second-order sliding mode controller (SOSM), fuzzy-based SOSM controller (Fuzzy-SOSM), adaptive fuzzy approximation SOSM controller (Adapfuzzy-SOSM)46–48, the traditional PID controller and the proposed ISO-ADRC. Among these controllers, the switching function used for the robust law is the hyperbolic tangent function tanh(), which can effectively suppress the chattering of sliding mode controller, namely,35 s˙=-ε∗tanhs-K∗s

where s is the sliding surface, ε and K are the parameters of the approaching law.

For the Fuzzy-SOSM and Adap-Fuzzy-SOSM, the fuzzy model is the commonly used single valued fuzzier, and the parameters, fuzzy membership functions used for these two methods are based on reference48. During the simulation, two types of disturbances are introduced to simulate the force mutation (D1) and periodic disturbance (D2) in the operation process of the robots. The disturbances are generated as:36 D1t=rand0,0.2,D2t=sin(40π)

wherein, rand(0,0.2) is used to generate random numbers between 0 and 0.2. The duration of the simulation is 10s, and the occurrence of D1 and D2 are 7 – 8s and 3 – 4s, respectively. Values for the parameters in SOSM are ε=[1.50;01.5] and K=20∗[10;01]. The membership function used in the fuzzy model is the RBF function. The tracking result (trajectory tracking and tracking errors) on reference multi-frequency signals is illustrated as Figs.9 and 10. Besides, the control inputs of different controllers for two links are shown as Fig. 11.Fig. 9 Tracking result on reference signals.

Fig. 10 Tracking errors on reference signals.

Fig. 11 Control inputs of different controllers.

From these figures, all of these controllers can track the target trajectory with different accuracy. Compared with the other two controllers, the proposed control strategy can track the trajectory with good effect. However, compared with the Adapfuzzy-SOSM, due to the application of function approximation technique and the modeling performance of the fuzzy algorithm, the Adapfuzzy-SOSM has almost the same effect as the proposed ISO-ADRC, which also demonstrates the effectiveness of the designed control method from another perspective. Besides, under the action of different disturbances, the control input is more stable, indicating a positive impact on the components, which can be seen in Figs. 9 and 10. As shown in Fig. 10, except for the first 0.1 s, the control input over the control process is bounded in [-50, 50], and the curve is influent without chattering phenomena. It should be noted that the control signal applied to the system is significant at the initial time. Therefore, in actual practice, considering the safety of the servo system during the control process, the amplitude can be constrained in constant ranges. To some extent, this will reduce the tracking speed, but the control stability will be not influenced. Furtherly, to demonstrate the results clearly, quantitative analysis between the proposed method and the commonly used methods are carried out, and comparison results for mean absolute percentage error (MAPE) are shown in Table 2.Table 2 Comparison results of MAPE for different controllers.

Methods	MAPEs (Link1)	MAPEs (Link2)	
PID	0.2186	0.1953	
SOSM	0.1317	0.0914	
Fuzzy-SOSM	0.0493	0.0299	
ISO_ADRC	0.0284	0.0117	
Adapfuzzy-SOSM	0.0253	0.0109	

36 MAPE=1L∑k=1Lr(tk)-y(tk)r(tk)

From the table, the proposed ISO_ADRC has better control performance than the other three controllers (PID, SOSM, Fuzzy-SOSM), for the ISO_ADRC, MAPEs for Link 1 and Link 2 are [0.0284, 0.0117], which is smaller (almost 2%, twice better) than that of Fuzzy-SOSM. Besides, due to the application of function approximation technique, the tracking effect of the Adapfuzzy-SOSM is a litter better than that of the ISO_ADRC. For the Fuzzy-SOSM and SOSM, the former controller is far more effect than SOSM, that is because the application of fuzzy approximation during the tracking process, improved the adaptiveness of the controller for model uncertainty and disturbances.

Conclusion

In response to the affect of various structural uncertainties on tracking performance of the robots during the operation process, including dead zones, saturation, and uncertain dynamics, an anti-disturbance rejection control strategy (ISO-ADRC) is developed on basis of an improved particle swarm optimization algorithm, which ensures the trajectory tracking accuracy of the robot joint driver by applying control to its motion. The control strategy mainly contains two parts, on the one hand, this control method addresses internal and external disturbances of the system by introducing an improved extended state observer; on the other hand, an improved particle swarm optimization algorithm (IPSO) based on chaos theory is designed to optimize the key parameters in the controller. The effectiveness of this method was verified by comparing with several commonly used control methods. The results demonstrates that the proposed method has better anti-disturbance performance compared to most of these methods when facing sudden changes and periodic disturbances, and the tracking errors is bounded in 2%.

Author contributions

Peihao Yue: Conceptualization, Methodology, Original draft, Visualization, Investigation, Validation. Bowen Xu: Conceptualization, Methodology, Writing- Reviewing and Editing, Visualization, Supervision. Min Zhang: Data curation, Software, Validation, Writing- Reviewing and Editing.

Data availability

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

Competing interests

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Publisher's note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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