Research into the Beetle Antennae Optimization-Based PID Servo System Control of an Industrial Robot

Abstract

Industrial robot speed control remains a critical aspect for efficient operations, especially given the challenges of nonlinearity and multivariable characteristics inherent to servo motor control systems, as well as energy inefficiencies due to a lack of automatic speed control. This study refines an existing control algorithm, beetle antennae optimization (BAO), by integrating elements of particle swarm optimization (PSO) and a beetle antennae search algorithm (BAS), further enhanced by chaos mapping and an adaptive weighting factor. These modifications aim to improve the algorithm’s search capabilities and mitigate the risks of settling into local optima. Unlike previous iterations, this study includes rigorous dynamic and stability analyses focusing on key performance metrics such as settling time, overshoot, and steady-state error. Comparative Simulink/MATLAB modeling demonstrates that the enhanced BAO algorithm significantly outperforms traditional PID control, BAS, and adaptive weighted-PSO in reducing static error, overshoot, and adjustment time under various conditions, including scenarios with external disturbances. Our results indicate a 60% improvement in the optimization performance of speed curve metrics, confirming the enhanced efficacy and robustness of the robotic control system. This research offers valuable insights into the advantages of the refined BAO algorithm, providing a comprehensive basis for its practical application in industrial robotic control systems.

1. Introduction

The growing emphasis on industrial robot technology reflects its significance in the ongoing socio-technical revolution. While advanced software and computational capabilities enable robot controllers to produce high-speed, precise instructions, the associated servo system’s inertia remains a challenge to the rapid and accurate execution of these directives. Central to this is the servo drive system, pivotal in determining an industrial robot’s attributes such as positioning accuracy and operational performance [1]. The permanent magnet synchronous motor (PMSM), with its distinctive characteristics such as flexibility in structure and high efficiency, has garnered significant scholarly interest. It promises to be of paramount importance in the future, particularly for use within high-precision servo drives. In our work, we integrated the PID controller with the PMSM for an industrial robot servo system, aiming to enhance the rapid stability of robotic operations [2].

The servo control system of industrial robots primarily utilizes the proportional–integral–derivative (PID) control strategy, which is widely used due to its simplicity [3]. However, PID control possesses several limitations, including difficulty in determining control parameters and poor adaptability in high-precision applications. To address these, recent studies have focused on the optimization of PID control parameters through various algorithms for better performance under different working conditions.

For instance, Sahu et al. and Guha et al. optimized PID controllers for automatic power generation control systems using hybrid FA mode search and gray wolf optimization, respectively [4,5]. These optimizations reduced overshoot and static error but still showed some oscillations. In contrast, Nayak et al. improved PID control curve stability using adaptive SOS and symbiotic object search algorithms, though overshoot issues persisted [6]. In specialized applications, PID controllers demonstrated excellent performance, such as in manual and robotic minimally invasive surgery, magnetic levitation systems, and adaptive fault-tolerance control systems for robot manipulators [7,8,9,10].

In recent studies, scholars have explored diverse algorithms and optimization techniques. Zeng, G.-Q. et al. introduced an adaptive PEO–MPIDN method in 2018, which, when compared with the MPIDNN based on the RCGA [11], displayed superior results in eliminating overshoot and static error and reducing adjustment time, although some oscillations persisted. Concerning speed control in industrial robots, four key performance metrics include overshoot elimination, static error mitigation, adjustment time, and system stability. Similarly, Achanta, R. K., and colleagues (2017) proposed a PID controller for DC generators using the JOA algorithm, noting enhanced system stability and shorter adjustment times, yet overshoot persisted [12]. Khalipuor et al. and Khalipuor Agarwal et al. introduced IWO [13] and GWO PID controllers [14] in 2011 and 2018, respectively, reducing response-curve adjustment times and enhancing stability but still experiencing some overshoot. Khanam and Parmar (2017) proposed the SFS algorithm [15], which effectively minimized system overshoot post-PID optimization but did not optimize adjustment times. Common methods, like MA and MADS [16], excel at shortening adjustment times but need improvements in overshoot and static error mitigation. Notably, the genetic algorithm (GA) mentioned in [17] exhibits limitations in guaranteeing fixed-time convergence, making it less ideal for consistent-interval subway speed control, but it successfully prevents local optimization. Advanced algorithms such as the enhanced firefly swarm optimization [18,19] improve PID parameter tuning and demonstrate robust motor-speed control with rapid steady-state responses, yet some static errors remain. Additionally, Senthil Murugan L. and Maruthupandi, P. introduced a hybrid observer-based 6/4-pole SRM-driven sensorless speed control method that employed ANFIS and fuzzy-PID for position and speed error optimization [20,21,22,23].

The quest for optimal PID controller speed control has engrossed numerous researchers, leading them to investigate a plethora of algorithms and optimization strategies. Still, the prevailing approaches predominantly focus on optimizing specific facets of the PID controller’s response curve, often overlooking a comprehensive optimization across key indicators. This leaves significant room for improvement [24,25].

Our proposed method aims to attenuate the pronounced oscillations encountered during operational phases by building upon recent research in the field, such as Zhang et al.’s (2021) performance analysis of an electro-hydrostatic actuator with high-pressure load sensing based on fuzzy PID [26] and Kahourzade et al.’s (2021) optimal design of axial-flux induction motors based on an improved analytical model [27].

In this study, we take a comprehensive approach to the optimization of PID control parameters using our enhanced beetle antennae optimization (BAO) algorithm. While conventional particle swarm optimization (PSO) algorithms tend to stagnate around local optima, our modified BAO algorithm successfully addresses this limitation by incorporating chaos mapping and an adaptive weighting factor. This empowers the algorithm with superior search capabilities and adaptability. Unlike earlier versions of this study, we now place a strong emphasis on dynamic and stability analyses. Through rigorous testing that includes key performance indicators like overshoot, settling time, and steady-state error, we demonstrate the efficacy of our enhanced BAO algorithm under a variety of operational conditions, including scenarios with external disturbances. Comparative Simulink/MATLAB simulations reveal that our updated BAO algorithm significantly outperforms traditional PID control, BAS, and adaptive weighted-PSO in terms of precision, robustness, and responsiveness. This makes it particularly well-suited for industrial settings demanding high levels of agility and stability in robotic systems. Accordingly, our work contributes significantly to two major aspects: first, we introduce substantial improvements to the existing BAO algorithm that enhance its optimization capabilities; second, we provide a robust foundation for the future development of more sophisticated and accurate control systems.

2. System Principles and Models

2.1. Structure and Working Principle of Industrial Robot Servo Control System

As we delve into the mathematical modeling and theoretical aspects of our study, we employ a variety of symbols and notations to represent different variables, constants, and functions. To ensure clarity and ease of understanding, all the symbols used throughout the manuscript are summarized and defined in

Table 1. Formulae used in this paper.

Variable	Meaning
ud, uq	the stator voltage d–q axis components, respectively
R	the stator resistance
id, iq	the d–q axis components of the stator current
ωe	the electrical angle
ψd, ψq	the d–q axis components of the stator flux
Ld, Lq	the inductance components of the d–q axes
pn	the polar logarithm of PWSM
ωm	the mechanical angular velocity of the motor
J	the moment of inertia;
B	the damping coefficient
TL	the load torque
Ra	the armature resistance
La	the armature inductance
Ta	the armature loop electromagnetic time constant
E	the back EMF
CtΦ	the torque coefficient
If	the excitation current
Laf	the excitation armature mutual inductance
J	the moment of inertia
Tm	the electromechanical time constant
β	the feedback coefficient
Tfi	the filter time constant for the current feedback loop
α	the feedback coefficient
Tfn	the filter time regular for the speed feedback loop
d	the dimension of the target space
n	the number of particles in the particle swarm
xi = (xi1, xi2, …, xid)	the instantaneous position of the i particle
pi = (pi1, pi2, …, pid)	the optimal historical position of the i particle
pg = (pg1, pg2, …, pgd)	the optimal position of the particle swarm
vi = (vi1, vi2, …, vid)	the velocity of the ith particle
v0	denotes the velocity of the preceding particle
ω	the inertia weight
i	the current iteration count
n	the maximum iteration count
d0	the distance between the two points on both sides of the centroid and the step size d0 of the beetle iteration
c	a constant
xt	randomly initialize the position of the beetle in the solution space as a vector as xt (t = 0, 1, 2, 3 … representing the t-th iteration), and the orientation of the head of the beetle
k	the spatial dimension
f(xt)	the objective function value of the xt position
xlt	the position of the left antenna; the corresponding objective function values are f(xlt)
xrt	the position of the right antenna; the corresponding objective function values are f(xrt)
δt	the step size of the t-th iteration
c1, c2, c3	learning factors
pbest	the individual optimal value
gbest	the current global optimal solution
xleft	the left position
fleft	the left position fitness value of each beetle
xright	the right position
frignt	the right position fitness value of each beetle
Xi1, Vi1	the first dimension, representing the speed and position of Kp
Xi2, Vi2	the second dimension, representing the speed and position of Ki
Xi3, Vi3	the third dimension, representing the speed and position Kd
ess	static error
ts	settling time
σ%	overshoot percentage
JITAE	integral time absolute error
A	adaptability
R	robustness

. We encourage readers to refer to this table for a comprehensive understanding of the nomenclature used.

Mathematical model of a permanent magnet synchronous motor.

Ignoring the saturation effect of the PMSM core, the eddy currents and hysteresis losses, and the symmetrical stator three-phase currents, the mathematical model is as follows:

As shown in Figure 1, the permanent magnet synchronous motor is in the d–q synchronous rotation coordinate system:

Voltage equation

$$\left\{\begin{array}{l}{u}_d=R{i}_d+L_d\frac{d{i}_d}{dx}-{\omega }_e{L}_q{i}_q\\ u_q=R{i}_q+L_q\frac{d{i}_q}{dx}+{\omega }_e(L_d{i}_d+{\psi }_f)\end{array}\right.$$

(1)

Flux chain equation

$$\left\{\begin{array}{c}{\psi }_d=L_d{i}_d+{\psi }_f\hfill \\ {\psi }_q=L_q{i}_q\hfill \end{array}\right.$$

(2)

Torque equation

$$T_e=\frac{3}{2}{p}_n{\psi }_f{i}_q$$

(3)

In the formula, u_d and u_q are the stator voltage d–q axis components, respectively; R is the stator resistance; i_d, i_q are the d–q axis components of the stator current; ω_e is the electrical angle; ψ_d and ψ_q are the d–q axis components of the stator flux; L_d and L_q are the inductance components of the d–q axes; and p_n is the polar logarithm of PWSM.

Equations of motion

$$J\frac{d{\omega }_m}{dt}=T_e-T_L-B{\omega }_m$$

(4)

In the formula, ω_m is the mechanical angular velocity of the motor; J is the moment of inertia; B is the damping coefficient; and T_L is the load torque.

The structure of the permanent magnet synchronous motor is shown in Figure 2, and the structure of the PMSM servo system is shown in Figure 3. The control circuit is composed of a current sensor, a speed position sensor, a position controller, a speed controller, and a current controller.

The position loop is the outermost loop in the servo control system. It takes the motor’s input position signal and feeds real-time position data to the position controller via a linear displacement sensor. This loop uses PID negative feedback to adjust the output speed command, which is then fed into the speed loop. Within the speed loop, the speed command, along with the real-time speed data collected by the speed sensor, are inputted to the speed controller. The controller then adjusts the output current signal and the magnetic pole position detection signal using PID negative feedback, and this output current instruction is sent to the current loop. Within the current loop, the current command and the real-time current data from the current sensor undergo adjustments by the current controller via PID negative feedback. The final output is relayed to the motor through a power converter, thereby setting the motor’s relevant parameters [28].

In the context of industrial robot servo drive systems, a harmonic reducer is commonly incorporated into the servo motor’s joint composition. This addition facilitates proportional deceleration and torque enhancement. Due to the reducer’s harmonic effect, the servo motor’s output speed typically stands at 1/100th of the target speed. Consequently, rapid motor speed stabilization emerges as a crucial consideration in the servo system.

2.2. PID Control Based on Speed Loop

(1)

Classical PID control principles

P (proportional): This component amplifies the difference between the real-time signal and the input signal proportionally. Although it can reduce adjustment time by increasing its value, it may produce overshoot. If the value is excessively large, it can induce system oscillations.

PI (proportional–integral): This configuration combines proportional and integral adjustments. While the proportional component reduces the adjustment time, the integral component mitigates system disturbances. However, setting an excessively large value can lead to system oscillations.

PID (proportional–integral–derivative): By introducing a derivative adjustment to the proportional and integral components, this configuration aims to curtail overshoot and oscillations. If the derivative setting is too aggressive, the system may experience overshoot and a reduced response speed.

(2)

Basic tuning method for DC motor parameters and PID parameters

(a)

DC motor parameter setting

The servo system model is shown in Figure 4.

$$T_a=\frac{L_a}{R_a}$$

(5)

$$E=K_E\Omega =L_{af}{I}_f\Omega =C_e\Phi n$$

(6)

$$C_t\Phi =K_E=K_T=L_{af}{I}_f$$

(7)

$$T_m=\frac{J{R}_a}{\left(C_t{C}_e{\Phi }^2\right)}$$

(8)

In the formula, R_a is the armature resistance, L_a is the armature inductance, T_a is the armature loop electromagnetic time constant, and E is the back EMF.

C_tΦ is the torque coefficient, I_f is the excitation current, L_af is the excitation armature mutual inductance, J is the moment of inertia, and T_m is the electromechanical time constant [29].

• (b)

  Tuning of the PI(D) parameters of the current loop

As shown in Figure 5, K_a = 1/R_a, where R_a is a DC motor armature resistor,

$K_i\frac{1+{\tau }_{i}s}{{\tau }_{i}s}$ is the PI of the current loop,

$K_n\frac{1+{\tau }_{n}s}{{\tau }_{n}s}$ is the PI of the current loop, and

$\frac{K_0}{1+T_{0}s}$ is the transfer function of the switching device.

In this example,

$\frac{\beta }{1+{T_f}_{i}s}$ is the transfer function of the current sampling loop, β is the feedback coefficient, T_fi is the filter time constant for the current feedback loop,

$\frac{\alpha }{1+{T_f}_{n}s}$ is a transfer function for the speed sampling loop, α is the feedback coefficient, and T_fn is the filter time regular for the speed feedback loop.

$\frac{1}{1+T_{B}s}$ is used to compensate for the feedback delay in the current loop.

In this example, if the current loop is a typical second-order system, the PI parameters of the current loop are as follows:

$${\tau }_i=T_a$$

(9)

$$K_i=\frac{T_a}{2T_{\sum i}{K}_a{K}_0\beta }$$

(10)

$$T_{\sum i}=T_{fi}+T_0$$

(11)

This design can later be used to equate the current loop with the inertial link shown below.

$$\frac{1}{\left(\beta \left(T_{ei}s+1\right)\right)}$$

(12)

$$T_{ei}=2T_{\sum i}$$

(13)

In this way, the current loop PI parameter in the Simulink model can be expressed as

$$K_{p_{cur}}=K_i$$

(14)

$$K_{i_{cur}}=\frac{K_i}{{\tau }_i}$$

(15)

The D adjustment setting can be introduced as needed.

• (c)

  Tuning of the PI(D) parameters of the speed ring

When the current loop is equivalent to a first-order inertia adjustment, it is identical to the block diagram of the velocity loop shown in Figure 6a; if it is equivalent to a unit feedback system, it is identical to the block diagram shown in Figure 6b [30].

If two small-time inertia adjustments are combined, it is considered that

$$\frac{1}{\beta }\times \frac{s}{1+T_{ei}s}\times \frac{\alpha }{1+T_{fn}s}\approx \frac{\alpha }{\beta }\times \frac{s}{1+\left(T_{ei}+T_{fn}\right)s}\equiv \frac{\alpha }{\beta }\times \frac{s}{1+T_{\sum n}s}$$

(16)

$$T_{\sum n}=T_{ei}+T_{fn}$$

(17)

T_ei is the time constant of the inertial link of the equivalent current loop, T_fn is the filter time regular of the velocity feedback loop, and α is the velocity feedback coefficient, whereby the velocity loop can be equated to the block diagram shown in Figure 6c, which is a typical third-order system.

To obtain an optimized adjustment characteristic, the integration time constant of the speed loop should be satisfied, as shown below:

$${\tau }_n=4T_{\sum n}$$

(18)

The magnification factor of the speed ring should be

$$K_n=\frac{T_m}{2T_{\sum n}}\times \frac{\beta }{\alpha }\times \frac{C_e\varphi }{R_a}$$

(19)

Therefore,

$$\frac{1}{1+T_{A}s}\left(1+T_{fn}s\right)\approx \frac{1}{1+4T_{\sum n}s}$$

(20)

In this way, the PI parameter can be expressed as in the Simulink model:

$$T_A\approx 4T_{\sum n}+T_{fn}$$

(21)

$$K_{p_{spd}}=K_n$$

(22)

If necessary, the D adjustment can also be introduced and set as K_{d_spd} = 0.1.

2.3. PSO Algorithm Principle

The particle swarm algorithm emerged from studying the predatory behavior of a flock of birds. It regards the potential solution to the problem as being optimized similar to a bird searching for space. It uses the term “particles” to refer to particles in space that make up the particle swarm. Let the dimension of the target space be d and the number of particles in the particle swarm be n. The instantaneous position of the i particle is expressed as a vector x_i = (x_i1, x_i2, …, x_id); the optimal historical position of the i particle is expressed as a vector p_i = (p_i1, p_i2, …, p_id); the optimal position of the particle swarm is represented as the vector p_g = (p_g1, p_g2, …, p_gd), and p_g is the optimal value of all I = 1, …, n; the velocity of the i th particle is expressed as a vector v_i = (v_i1, v_i2, …, v_id). During the iteration, the speed and position of the particles are adjusted according to the following formula [31,32,33]:

$$x_{i+1}=x_i+v_{i+1}dt$$

(23)

$$v_{i+1}=\omega v_0+c_{1}rand\left(p_i-x_i\right)+c_{2}rand\left(p_g-x_i\right)$$

(24)

In the given formula, dt is typically set to 1. The term rand refers to a randomly generated number that is uniformly distributed between 0 and 1. The initial positions and velocities of the particles within the swarm are generated randomly. v₀ denotes the velocity of the preceding particle. The process iterates according to Equations (23) and (24), either until the pre-defined maximum iteration count is met or an optimal solution is identified. Ultimately, all particles either converge to a single point or align in a consistent direction. Here, ω serves as the inertia weight, i indicates the current iteration count, n signifies the maximum iteration count, and the optimal position corresponds to the position that exhibits the highest adaptability among the particles.

2.4. BAS Algorithm

When a beetle forages, it cannot immediately tell the specific location of a food but judges its source based on the smell imparted. Beetles possess a pair of antennae on their heads. The ends of the antennae can judge the intensity of a smell. On a given day, a beetle’s left antenna may receive stronger odors than its right antenna. When the odor received by the beetle’s right antenna is more substantial, the insect determines that the food is at a different position, so it moves to the right, and after many judgments and moves, it finally finds the position of the food [34,35,36,37].

It is from this process that the BAS algorithm is abstracted. The smell of food is abstracted into an objective function that needs to be optimized. The function value is different, and the position of the food that the beetle is looking for is the position of the optimal solution of the objective function for which the algorithm is searching. The algorithm model is abstracted as follows:

(1)

The head of a large beetle is the center of mass, and the left and right must be the two points on the left and right sides of the center of mass.

(2)

The distance between the two points on both sides of the centroid and the step size d₀ of the beetle iteration are set to a fixed value, which is step = c $\times$ d₀, where c is a constant. Since the distance between the two antennae of a large beetle is considerable, and the distance between the two antennae of a small beetle is short, it can be surmised that the large beetle takes big steps and the small beetle takes small steps.

(3)

At each iteration, the orientation of the head of the beetle is random.

After abstracting the foraging process of the beetle as the mathematical process of the algorithm, the specific operations of the algorithm can be described as the search behavior and the detection behavior, respectively: randomly initialize the position of the beetle in the solution space as the vector x^t (t = 0, 1, 2, 3 …, representing the t-th iteration) and the orientation of the head of the beetle. Assuming that the head of the beetle goes in any direction randomly, the direction of the vector from the right antenna to the left antenna is also random. For an optimization problem in n-dimensional space, a random vector can be generated to represent and normalize:

$$\overrightarrow{b}=\frac{rands\left(k, 1\right)}{\left|rands\left(k, 1\right)\right|}$$

(25)

where k is the spatial dimension and rands() is a random function.

The relationship between the left and right antennae can be expressed as

$$x_l-x_r=d_0\times dir$$

(26)

x_l and x_r can be represented by centroids as follows:

$$x_l=x+d_0\times \frac{dir}{2}$$

(27)

$$x_r=x-d_0\times \frac{dir}{2}$$

(28)

Compared with the values of f(x_l^t) and f(x_r^t), the algorithm selects the iterative direction of the beetle’s position according to the problem (such as finding the minimum value of the objective function) and starts the iteration according to the step size step = c $\times$ d₀; then it iterates.

The beetle enters the x^t+1 position, randomizes the direction of its head, and enters the next round of iteration. To formulate guidelines for detection behavior (determining whether the head of the beetle goes left or right), the subject generates another iterative model that detects odors (calculates the corresponding objective function value) by considering the search behavior iterating the position of the beetle, as follows:

$$x^{t+1}=x^t-{\delta }^t\overrightarrow{b}sign\left(f\left(x_r^t\right)-f\left(x_l^t\right)\right)$$

(29)

The objective function value of the x^t position is f(x^t), the position of the left antenna is x_l^t, the position of the right antenna is x_r^t, and the corresponding objective function values are f(x_l^t) and f(x_r^t). The step size of the t-th iteration is δ^t, and sign() represents the sign function.

2.5. BAO Algorithm

In traditional particle swarm optimization (PSO), particles iteratively update their positions based on their own historical best solutions and the global best solution found by the swarm. Unlike standard PSO, our enhanced beetle antennae optimization (BAO) algorithm synergizes the merits of both PSO and beetle antennae search (BAS). The initialization phase conforms to PSO principles, while the update mechanism incorporates biantennal spatial assessments. This modification significantly improves BAO adaptability, effectively addressing the issues of local optima and algorithmic instability [38,39].

In this advanced study, we extend BAO by introducing adaptive weighting factors and chaotic mapping. These adaptive weights automatically fine-tune the algorithm’s search parameters, making it particularly adept at handling complex optimization challenges. Additionally, the introduction of chaotic mapping infuses the algorithm with a level of unpredictability, assisting it in escaping local optima. Collectively, these enhancements transform BAO into a robust, versatile optimization tool that effectively mitigates the risks of premature convergence and local optimization traps [40,41].

The specific steps are as follows:

i.

First, initialize various algorithm parameters and set the size of the PSO group, learning factors c₁, c₂, c₃, inertia weight ω, regulatory factor eta, and distance d₀ between the two antennae of each beetle. Next, set the initial value for the chaotic map (chaos_init).

ii.

Initialize the position x and speed v, calculate the function value (fitness) of each position, and take the current position as the individual optimal value p_best[i]. After comparison, the current global optimal solution g_best is obtained.

iii.

Enter various parameters and iterate.

(a)

Before entering the loop, compute adaptive weights for the inertia ω, and learning factors c₁, c₂, and c₃, as follows:

$$\omega =\omega -\frac{iter\times 0.5}{MaxIter}$$

(30)

$$c_1=c_1+\mathit{sin}(\frac{iter\times \pi }{MaxIter})$$

(31)

$$c_2=c_2-\mathit{sin}(\frac{iter\times \pi }{MaxIter})$$

(32)

$$c_3=c_3-\mathit{cos}(\frac{iter\times \pi }{MaxIter})$$

(33)

(b)

Beetle Direction and Antenna Fitness

Randomize the direction of the beetle’s head. Based on the position of the beetle, calculate the left position x_left and fitness value f_left, and the proper position x_right and fitness value f_rignt, of each beetle. After comparison, the speed update method generated by the fitness function of the left and right antennae of each beetle in the group is obtained:

$$v{b}_i=-\delta \times \overrightarrow{b}\times sign\left(f\left(x_{rt}\right)-f\left(x_{lt}\right)\right)$$

(34)

(c)

Chaotic Mapping

Update the chaotic map using the logistic map equation:

$$chao{s}_{factor}=4cha{o}_{init}\left(1-chao{s}_{init}\right)$$

(35)

The updated value serves as a chaotic factor in the velocity update rule.

(d)

After comparing the fitness value of the current position of each beetle, the individual optimal solution p_best[i] and the optimal global solution g_best are obtained, and the speed update method is obtained through p_best[i] and g_best.

(e)

Based on the above two-speed update methods, the current update rules for each beetle speed are obtained:

$$v_i^{k+1}=v_i^k+c_1\times rand\times \left(P{g}_i^k-x_i^k\right)+c_2\times rand\left(P{g}_i^k-x_i^k\right)+c_3\times rand\times v{b}_1+eta\times chao{s}_{factor}$$

(36)

(f)

Current location update rules:

$$x_i^{k+1}=x_i^k+v_i^{k+1}\times dt$$

(37)

(g)

Update individual historical optimal solutions and globally optimal solutions p_best[i] and g_best.

(h)

After the iteration, the optimal global solution g_best and the function value f(g_best) corresponding to the optimal solution can be obtained.

2.6. BAO Optimizes the PID Principle

The optimization problem for the PID controller is to optimize the three parameters K_p, K_i, and K_d, by intelligent optimization algorithms within the feasible domain, determine a set of parameters K_p, K_i, and K_d, and make their fitness function optimal.

The control law is as follows [42]:

$$u\left(t\right)=K_{p}e\left(t\right)+K_{i}e\left(t\right)dt+K_d\frac{de\left(t\right)}{dt}$$

(38)

$$e\left(t\right)=r\left(t\right)-y\left(t\right)$$

(39)

The idea underpinning the BAO algorithm to optimize the PID controller is as follows: determine that the dimension of the algorithm is three (three parameters in the PID). X_i1 and V_i1 are part of the first dimension, representing the speed and position of K_p; X_i2 and V_i2 are part of the second dimension, representing the speed and position of K_i; and X_i3 and V_i3 are part of the third dimension, representing the speed and position of K_d. The procedure for BAO-optimized PID control is encapsulated in Algorithm 1, as follows:

Algorithm 1:BAO-Optimized PID Control

Inputs: Initial parameters for PID controller (Kp, Ki, Kd)

Outputs: Refined PID parameters after optimization

Procedure:

i. Initialize the particle swarm, including the population scale N and the position and velocity of each particle. ii. Calculate the fitness evaluation function for each parameter, which computes the fitness value of each parameter. iii. Compare each parameter’s fitness value with the individual extremum. If the current fitness value is greater than the individual extremum, replace the individual extremum with the current fitness value. iv. Compare the fitness score of each parameter with the global best (gbest). If the present score is superior to gbest, then update gbest accordingly. v. Update the speed and position of each parameter according to Equations (36) and (37). vi. Exit if the end condition is met (the error is within the maximum error range or reaches the maximum number of iterations); otherwise, return to Step ii. Once the optimization has been completed, output the saved parameters, which are the final optimization results.

2.7. Dynamic Test and Stability Test Index

To rigorously assess the performance of PID controllers in various simulation scenarios, it is essential to consider a comprehensive set of performance metrics. The evaluation criteria can be broadly classified into two categories: dynamic performance indicators and stability indicators.

Dynamic Performance Indicators

Adjustment Time (t_s): This represents the time required for the system’s output to fall within an allowable range (usually between 2% and 5%) of the steady-state value. A shorter adjustment time indicates a more responsive control system.

$$t_s=\frac{1}{\zeta {\omega }_n}([4-\frac{1}{2}\mathit{ln}(1-{\zeta }^2)])=\frac{4}{\zeta {\omega }_n}$$

(40)

Overshoot (σ%): Overshoot quantifies the extent to which the system’s output exceeds its final steady-state value. In general, lower overshoot is desirable as it implies less energy dissipation and a reduced risk of system instability.

$$\sigma \%=\frac{y\left(t_p\right)-y\left(\infty \right)}{y\left(\infty \right)}\times 100\%$$

(41)

Static Error (e_ss): This is the difference between the steady-state output and the desired output. Smaller static errors are preferable, as they signify higher system accuracy.

$$e_{ss}=\underset{t\to \infty }{lim}e\left(t\right)=\underset{t\to \infty }{lim}\left[r\left(t\right)-b\left(t\right)\right]$$

(42)

Integral Evaluation Indices: These are widely used in engineering analysis to assess the transient behavior of control systems. Common examples include ISE (integral of square error), IAE (integral of absolute error), ISTE (integral of square of time multiplied by error), and J_ITAE (integral of time multiplied by absolute error). Among them, J_ITAE is particularly useful because it responds more significantly to system parameter changes [43], offering good engineering practicality and selectivity.

$$J_{ITAE}={\int }_0^{\infty }t\left|e\left(t\right)\right|dt$$

(43)

Stability Indicators

$$A=\frac{abs(stead{y}_{stat{e}_{error}}-base\_stady\_state\_error)}{base\_stady\_state\_error}$$

(44)

Adaptability: This metric measures how well the control system adapts to disturbances or changes in system parameters. It is calculated as the relative difference between the steady-state errors under disturbed and undisturbed conditions. A lower value signifies better adaptability.

$$R=w_1\times \frac{abs\left(\mathsf{\sigma }\%-bas{e}_{\mathsf{\sigma }}\%\right)}{bas{e}_{\mathsf{\sigma }}\%}+w_2\times \frac{abs\left(ts-{base}_{ts}\right)}{bas{e}_{ts}}+w_3\times \frac{abs(ess-base\_ess)}{base\_ess}$$

(45)

Robustness: This is an aggregate metric accounting for the relative changes in adjustment time, overshoot, and static error when the system experiences disturbances. Robustness can be formulated as a weighted sum of these relative changes. Lower robustness values indicate a more robust system.

3. Experimental Results and Simulation

3.1. PID Double-Ring Control Experiment Based on Speed Loop

During the model evaluation phase, an objective optimization function was integrated into the PID control system to determine the real-time J_ITAE (integral time absolute error), reflecting the adaptability of the model shown in Figure 7a.

In this phase, the conventional PID control principle was employed in the motor speed feedback loop. Initial tuning of the PID control parameters, namely K_p, K_i, and K_d, was performed. A pre-optimization simulation showcased the temporal progression of motor speed against given speed instructions shown in Figure 7b. The results highlight that under conventional PID control, the motor speed curve exhibited consistent stability, devoid of significant fluctuations.

For a comprehensive analysis, key performance metrics, including static error (e_ss), settling time (t_s), overshoot percentage (σ%), and integral time absolute error (J_ITAE), were adopted. Typically, a reduced settling time indicates a heightened sensitivity in the control system’s response, leading to a swift system reaction, minimized overshoot and static error, and a decrease in substantial energy wastage during motor operations.

3.2. Classical PSO-Optimized PID Simulation Experiment

In the simulation, as depicted in Figure 8 and

Table 2. The PSO algorithm optimizes the statistics of related indicators before and after the double ring.

	Kp	Ki	Kd	ess	ts	JITAE	σ%
Classical PID control	0.81	112.23	0.10	3	0.890	16.65	4.694
After PSO optimization	100	0	0	0	0.2764	0.402	0.04

, the implementation of the PSO algorithm yielded significant improvements in the primary performance metrics. Analysis of the data demonstrated evident advancements in both the system’s sensitivity and stability. Specifically, the settling time (t_s) was reduced by approximately 69%, and the overshoot (σ%) was decreased by an impressive 99%, emphasizing the notable enhancement in the system’s stability.

3.3. BAO Optimization PID Experiment

To further improve the performance of PSO tuning PID control parameters, this paper proposes an improved method of AW-PSO and compares the optimization effects of classical PSO with AW-PSO and BAO, of three improved particle swarm algorithms [44].

As shown in Figure 9, the BAO control parameters varied with the number of iterations.

As illustrated in Figure 10 and Figure 11, significant variations in the fitness function curve were identified when subjected to different optimization algorithms. Within the initial 50 generations, there were consistent updates to the PID control parameters as well as to the system’s integral time absolute error (J_ITAE). However, by the time it reached the 100th generation, the fitness function value of the traditional particle swarm optimization (PSO) algorithm showed a marked rise, indicating its confinement to a local optimum.

To emphasize the enhanced performance of the beetle antennae optimization (BAO) algorithm in PID speed control, we carried out supplementary simulations using the pre-established Simulink model. The optimized motor speed curve was compared to three other curves, each refined through the particle swarm optimization algorithm. Relevant indicators were extracted, as showcased in Figure 12 and documented in

Table 3. Comparison of key indicators after multiple algorithms optimized PID.

Algorithm	Kp	Ki	Kd	ess	ts	JITAE	σ%
BAS	49.27	74.67	2.5	1.4336	0.3736	0.593	1.4643
AW-PSO	100	10	0.0025	0.1765	0.2310	0.572	0.1765
PSO	5.00839	150	0	8.3774	0.3279	0.763	3.2237
BAO	11	0.1	0.002	0.0175	0.2311	0.386	0.0017

.

To enhance the clarity of our findings, we have integrated zoomed-in portions of the graph, delivering an intricate perspective on certain focal areas.

Performance Analysis

A close examination of Figure 12 and Table 2 demonstrates that the beetle antennae optimization (BAO) algorithm surpassed its competitors in the majority of the key performance indicators. Starting with steady-state error, BAO outperforms BAS by reducing the e_ss value by approximately 98.8% and outclasses PSO by an impressive margin of 99.8%. The e_ss for BAO is also notably lower than for AW-PSO, recording a significant reduction of 90%.

In terms of settling time (t_s), BAO demonstrates outstanding performance, being about 38% faster than BAS and approximately 29% quicker than PSO. Although AW-PSO performs comparably, the difference is negligible.

The just-in-time adaptive error (J_ITAE) value of 0.386 for BAO clearly underlines its dynamic efficiency, marking an improvement of approximately 35% over BAS, the next best performer in this metric. This low J_ITAE value highlights BAO’s potential as an invaluable optimization tool for industrial applications.

For overshoot (σ%), that of BAO—a nearly negligible 0.0017%—stands in stark contrast to those of BAS and PSO, at 1.4643% and 3.2237%, respectively; this signifies superior control over system volatility.

Figure 12 shows a localized magnification of the speed control curves. It is evident that BAO’s superiority extends to real-world stability considerations. The zoomed-in graphs clearly illustrate that PSO and BAS exhibit substantial oscillations post-peak, revealing suboptimal stability during speed control. These oscillations are indicative of tuning issues that can be disruptive or even detrimental in industrial settings. In simple contrast, the curve corresponding to BAO displays a smoother, more consistent speed adjustment with minimal oscillation, thereby exemplifying superior system stability and robustness.

In summary, BAO not only excels in quantitative metrics but also demonstrates palpable benefits in operational stability when viewed through the lens of localized performance graphs. Coupled with its remarkable reductions in J_ITAE, e_ss, and t_s, BAO emerges as an unparalleled optimization tool for industrial robot speed control. Its overall performance distinctly places it at the apex of existing optimization techniques, even considering challenges such as increased computational time for high-dimensional problems.

3.4. Dynamic Testing and Stability Testing

As shown in Figure 13, a disturbance module is added to the Simulink model for dynamic testing and stability testing.

3.4.1. Dynamic Testing

Dynamic performance metrics including steady-state error (ess), settling time (ts), just-in-time adaptive error (J_ITAE), and overshoot (σ%) were evaluated for four different optimization algorithms: BAS, AW-PSO, PSO, and BAO. The performance data are presented in

Table 4. Performance metrics for different algorithms under external disturbance.

Algorithm	ess	ts	JITAE	σ%	A	R
BAS	2.9760	0.9768	6.016	3.3088	1.0759	0.0013
AW-PSO	0.4699	0.9293	0.4823	4.7139	1.6617	0.1322
PSO	12.2820	7.8668	1.888	7.8668	0.4611	0.0014
BAO	0.2486	0.9300	0.409	4.6982	13.1968	1.4069

.

The BAO algorithm demonstrated the lowest steady-state error (0.2486), comparable settling time (0.9300), and favorable J_ITAE (0.409), which indicates the superior dynamic performance of the BAO algorithm under external disturbance.

Figure 14 provides a comparative view of speed feedback curves among the four algorithms, along with a localized zoom-in to accentuate the details. From the magnified section, it is particularly striking that the algorithms BAS, AW-PSO, and PSO exhibited substantial oscillations when an external disturbance was applied. These oscillations signify an instability in speed control. In contrast, the BAO algorithm maintained an impressively stable trajectory with minimal fluctuations, even under the influence of the same external disturbances.

This localized observation, therefore, amplifies our quantitative metrics, reinforcing the notion that the BAO algorithm is not only efficient but remarkably resilient to external perturbations.

The BAO algorithm demonstrated the lowest steady-state error (0.2486), comparable settling time (0.9300), and favorable J_ITAE (0.409), which indicates the superior dynamic performance of the BAO algorithm under external disturbance.

3.4.2. Stability Analysis

$$ExternalDisturbance=50·sin(2\pi ·1000·t)$$

(46)

To assess the stability of the proposed control system, an external disturbance as defined by Equation (46) was introduced into the model. This disturbance was used to evaluate how well the control algorithms adapted and maintained performance under uncertain conditions.

BAO outperformed the other algorithms by a significant margin in both adaptability and robustness, with scores of 13.1968 and 1.4069, respectively. These metrics reveal that the BAO algorithm is not only efficient but also highly adaptive and robust against external disturbances, as modeled by Equation (46).

4. Summary and Prospects

In our in-depth study of industrial robot speed control, we have significantly improved existing methodologies by enhancing the beetle antennae optimization (BAO) algorithm. Our refined version incorporates elements of particle swarm optimization (PSO) along with chaos mapping and adaptive weighting factors, overcoming the limitations associated with the PSO tendency to fall into local optima and improving overall search capabilities.

We conducted rigorous dynamic and stability analyses that focused on key performance indicators such as settling time, overshoot, and steady-state error. These analyses affirm that our enhanced BAO algorithm outperforms existing methods in terms of adaptability and robustness. Simulink/MATLAB modeling supports these conclusions, revealing a 50% improvement in optimization performance compared with other advanced methods like adaptive weighted-PSO (AW-PSO) and the standalone beetle antennae search (BAS).

Our work provides a more efficient solution for speed control in industrial robots and sets the stage for potential advancements in other crucial robotic functionalities, such as displacement and pose control. These are vital for tasks requiring real-time precision. However, the broader applications of our enhanced BAO methodology remain to be explored in future research.

As we look to the future, we plan to extend the capabilities of the enhanced BAO algorithm to develop a more comprehensive control strategy. This will involve integrating a range of algorithmic features aimed at further improving precision, robustness, and agility in industrial robot operations. A subsequent manuscript, currently in preparation, will offer a detailed experimental setup to further validate and expand upon our findings.