A Vibration Control Method Using MRASSA for 1/4 Semi-Active Suspension Systems

Abstract

The multi-subpopulation refracted adaptive salp swarm algorithm (MRASSA) was proposed for vibration control in 1/4 semi-active suspension systems. The MRASSA algorithm was applied to optimize suspension damping performance by addressing the local optimal and slow convergence speed challenge of the standard salp swarm algorithm for two-degrees-of-freedom 1/4 semi-active suspension systems. The developed MRASSA contains three key improvements: (1) partitioning multi-subpopulation; (2) applying refracted opposition-based learning; (3) adopting adaptive factors. In order to verify the performance of the MRASSA approach, a 1/4 suspension Simulink model was developed for simulation experiments. To further validate the results, a physical platform was built to test the applicability of the simulation model. The optimized suspension performance of MRASSA was also compared with three optimized models, namely, standard SSA, Single-Objective Firefly (SOFA) and Whale-optimized Fuzzy-fractional Order (WOAFFO). The experimental results showed that MRASSA outperformed the other models, achieving better suspension performance in complex environments such as a random road with a speed of 60 km/h. Compared to passive suspension, MRASSA led to a 41.15% reduction in sprung mass acceleration and a 15–25% reduction compared to other models. Additionally, MRASSA had a maximum 20% reduction in suspension dynamic deflection and dynamic load. MRASSA also demonstrated a faster convergence speed, finding the optimal solution faster than the other algorithms. These results indicate that MRASSA is superior to other models and has potential as a valuable tool for suspension performance optimization.

1. Introduction

A suitably designed suspension system improves vehicle smoothness and handling stability and prevents problems such as tire damage caused by excessive tire loads [1,2]. Most vehicles still use passive suspension. Due to the fixed stiffness and damping of passive suspension, the suspension cannot satisfy the requirement of smoothness under different vehicle speeds and road conditions. Intelligent suspension is a new type of suspension that is different from the traditional passive suspension. It can automatically adjust the system characteristics depending on the road conditions and take into account the smoothness and stability of the vehicle. Intelligent suspension is mainly divided into active suspension and semi-active suspension [3,4]. The semi-active suspension has the automatic adjustment capability of the active suspension and also has the advantage of the high reliability of the passive suspension.

The two-degrees-of-freedom quarter car model has been widely adopted by researchers because of its simple design and its ability to fully analyze the vertical motion of the vehicle [5]. Han [6] proposed a fuzzy PID control strategy based on road estimation to deal with road random disturbances and the uncertainty of 1/4 suspension parameters. The strategy used the flexible neural tree to estimate road disturbances in real time. Simulation results showed that the proposed control strategy can effectively adjust parameters based on changing road conditions. Sathishkumar [7] used a fuzzy control strategy to actively control the three-degrees-of-freedom air suspension seat model of the 1/4 vehicle. The vehicle body acceleration and velocity were used as inputs to the controller, and the acceleration of the seat and the output of the vehicle body were analyzed to evaluate the performance of the active suspension system. Vineet [8] proposed a self-tuning robust fractional-order fuzzy proportional-derivative (FO-FPD) controller for the nonlinear active suspension system of the 1/4 car. The control objective was to improve ride comfort by minimizing the root-mean-square acceleration of the vehicle body vertical vibration while maintaining hard constraint control. It exhibited excellent ride comfort in uncertain environments. Wang [9] proposed a new model-free fractional-order sliding mode control (MFFOSMC) based on the extended state observer (ESO) for the active suspension system of the 1/4 car, with the main goal of improving ride comfort while keeping the wheel dynamic load and suspension deflection within safe critical ranges. Devdutt [10] designed three different controllers to improve the passenger ride comfort and safety of the active 1/4 car model. Simulation results showed that the proposed HANFISPIDCR control scheme can successfully improve the required performance compared to passive and other control cases in the active quarter car model. Ding [11] proposed a multi-fuzzy PID suspension control system (MFRR) based on road recognition, which ensures the stability of the 1/4 suspension by identifying road grades and considering changes in road conditions. Experimental results showed that the spring mass acceleration (SMA) of MFRR is much smaller than that of the passive suspension system (PS) and FPID on single-bump and sine wave roads, and the control system exhibits good control performance. Youn [12] controlled the root mean square of acceleration of 1/4 suspension through road prediction simulation and optimal control theory. Time domain simulation shows that the preview control strategy can effectively reduce suspension vibration.

In order to rectify and tune for the control parameters to improve the controller performance, intelligent controllers based on swarm intelligence algorithms are receiving more and more attention. Chao [13] improved the cuckoo search algorithm and designed the fuzzy adaptive cuckoo search algorithm, which can reduce the calculation time in the evolution process and improve the accuracy of multi-objective optimization problems. The improved cuckoo search algorithm was applied to fuzzy PID to effectively improve the robustness of suspension. Ghazally [14] proposed a model-free fuzzy controller based on particle swarm optimization, in which particle swarm optimization was used to select the sliding gain of FLC to optimize the linguistic space. Finally, it was verified that PSO-MFFLC has good controllability and adjustability for suspension. Gad [15] used a multi-objective genetic algorithm to optimize the fractional order PID (FOPID) controller to optimize the suspension smoothness. Dangor [16] compared the performance of particle swarm optimization (PSO), the genetic algorithm (GA) and the differential evolution algorithm (DA) for optimizing PID suspensions. The results showed that DA-optimized PID control produced the best suspension performance. Pang [17] combined neural network control with particle swarm optimization control to design a variable domain fuzzy controller. It was applied to a semi-active suspension system with magnetorheological dampers to improve the performance of the suspension system. Swethamarai [18] used the whale algorithm to optimize the fuzzy fractional-order PID by simulating a semi-vehicle driving seat model, which successfully reduced the suspension amplitude and made the driving seat more stable. Chen [19] used an improved fuzzy neural network to optimize the bat algorithm and design a new adaptive controller. Mahmoodabadi [20] used a Single-Objective Firefly Algorithm to optimize two- and five-degrees-of-freedom vehicle vibration models. The results demonstrate that the proposed method had good tuning capability. Talib [21] used an intelligent optimizer based on the advanced firefly algorithm (AFA) to control the magneto-rheological (MR) dampers in a semi-active system. Compared to other controllers, the proposed PID-AFA was able to significantly reduce the magnitudes of the sprung mass acceleration and the body acceleration responses. Zha [22] studied a new multi-objective genetic algorithm (MOGA) optimization method, which was applied to optimize the stiffness ratio of the seat suspension and the geometric parameters of the negative stiffness structure (NSS). The seat suspension with the optimized NSS had a good low-frequency vibration isolation effect under various excitation sources. The approaches in these papers were basically combining swarm algorithms with other more complex algorithms to improve the accuracy of computation, such as deep learning [23,24,25]. However, in this way, systems such as suspensions, which need to react quickly, become delayed because the complex algorithms require overly complex calculations.

As a new swarm intelligence optimization algorithm in recent years, the salp swarm algorithm (SSA) [26] has a better search accuracy and convergence performance than traditional swarm intelligence algorithms, such as the particle swarm algorithm, genetic algorithm and gray wolf optimization. The SSA search process is less prone to fall into a local optima solution compared to other algorithms. In addition, SSA has fewer control parameters, which greatly reduces the reliance of the algorithm on control parameters, and its convergence speed is thought to be fast for other swarm algorithms [27,28,29]. However, the standard SSA still suffers from the following shortcomings: it has low accuracy in finding the best, it easily falls into the local optimum and the convergence speed could be further improved to achieve the required tuning speed of the suspension system. In general, the main approaches for improving the performance of swarm algorithms are improving the algorithms themselves or combining them with other algorithms. Sayed [30] used chaotic mapping to initialize the position of the salp swarm population so that the individual positions could be more uniformly arranged during initialization. Ibrahim [31] combined SSA with PSO for feature selection and showed that the algorithm hybridized with slightly improved results. Both types of improvements did not address the characteristics of SSA itself. When the leader cannot obtain the global optimal solution, the followers cannot jump out of the local optimal solution either. The two kinds of populations division between leaders and followers in standard SSA also make it difficult for followers to jump out of the local optimal solution. The combination with other algorithms also cannot speed up the convergence rate.

In order to solve the problem of falling into local optimal solutions and slow convergence in standard SSA due to the over-dependence of followers on the leader’s position and only two population divisions, this paper designed the multi-subpopulation refracted adaptive salp swarm algorithm (MRASSA) and applied it to the 1/4 suspension system. First, the original two populations were both divided into leaders, followers and chain tailers. The refracted opposition-based learning mechanism was used by the followers to update the position. The adaptive factor was used by the chain tailor to update the position. The new algorithm is applied to adjust suspension performance based on specific evaluation indexes. In this paper, the ability of MRASSA to optimize the suspension performance was verified through Simulink simulation experiments and physical experiments on the 1/4 suspension platform. The superiority of MRASSA was determined by comparing the optimization results of the suspension with standard SSA, WOAFFO [18] and SOFA [20]. This paper focuses on changing the characteristics of SSA itself rather than combining it with other algorithms. By improving SSA population classification and population update functions, the algorithm can be more accurate and converge faster.

The structure of this article is as follows: In the second part, a mathematical model of 1/4 semi-active suspension as well as a road excitation model were constructed, and three suspension evaluation indexes were introduced. In the third part, the standard salp swarm algorithm and the improved algorithm were introduced. In the fourth part, the model in the second part was combined with the algorithm in the third part, and simulation experiments and physical platform experiments were conducted to verify the superiority of the algorithm. In the fifth part, a summary of the paper was given.

2. Suspension System Model

2.1. 1/4 Semi-Active Suspension Model

Currently, commonly used suspension system models include the seven-degrees-of-freedom model, the four-degrees-of-freedom model and the two-degrees-of-freedom model. Among them, the two-degrees-of-freedom model simplifies the system inputs and reduces performance parameters while not affecting the study of the suspension system. The two-degrees-of-freedom 1/4 semi-active suspension model was established [32], and its model schematic is shown in Figure 1.

In Figure 1, $m_s$ is the sprung mass; $m_u$ is the unsprung mass; $k_s$ is the suspension spring stiffness; $k_t$ is the tire stiffness; $c$ is the absorber damping coefficient; $u$ is the variable damping force; $x_s$ is the sprung mass displacement; $x_u$ is the unsprung mass displacement; $x_r$ is the tire vibration displacement.

According to Figure 1, the kinetic equations of the 1/4 suspension model can be developed as follows:

$$m_s{\ddot{x}}_s+k_s\left(x_s-x_u\right)+c\left({\dot{x}}_s-{\dot{x}}_u\right)-u=0$$

(1)

$$m_u{\ddot{x}}_u+k_t\left(x_u-x_r\right)+k_s\left(x_u-x_s\right)+c\left({\dot{x}}_u-{\dot{x}}_s\right)+u=0$$

(2)

Laplace transformation of Equations (1) and (2):

$${(m}_s{s}^2+cs+k_s)X_s\left(s\right)-\left(cs+k_s\right)X_u\left(s\right)=0$$

(3)

$${(m}_u{s}^2+cs+k_s+k_t)X_u\left(s\right)-\left(cs+k_s\right)X_s\left(s\right)=k_t{X}_r\left(s\right)$$

(4)

The solution is:

$$\left\{\begin{array}{c}{X}_s\left(s\right)=k_t(cs+k_s)\frac{X_r\left(s\right)}{∆}\\ X_u\left(s\right)=k_t(m_s{s}^2+cs+k_s)\frac{X_r\left(s\right)}{∆}\end{array}\right.$$

(5)

where $∆=m_s{m}_u{s}^4+\left(m_s+m_u\right)c{s}^3+\left(m_s{k}_s+m_s{k}_t+m_u{k}_s\right)s^2+k_{t}cs+k_s{k}_t$.

From this, the transfer functions of the sprung mass acceleration, suspension dynamic deflection and suspension dynamic load with respect to the road input can be obtained:

$$H_a\left(s\right)=\frac{\ddot{X_s}\left(s\right)}{X_r\left(s\right)}=\frac{k_t(cs+k_s)s^2}{∆}$$

(6)

$$H_d\left(s\right)=\frac{X_s\left(s\right)-X_u\left(s\right)}{X_r\left(s\right)}=\frac{{-k}_t{m_{s}s}^2}{∆}$$

(7)

$$H_f\left(s\right)=\frac{{k_t[X}_u\left(s\right)-X_r\left(s\right)]}{X_r\left(s\right)}=\frac{{-k_{t}m}_s{m}_u{s}^4-k_t\left(m_s+m_u\right){cs}^3-k_t{k}_s(m_s+m_u)s^2}{∆}$$

(8)

The kinetic equations are discretized, taking the state variables:

$$X=[x_s-x_u,{\dot{x}}_s,x_u-x_r,{\dot{x}}_u{]}^T$$

(9)

System output variables:

$$Y=[{\ddot{x}}_s,x_s-x_u,{k_t(x}_u-x_r),{\dot{x}}_u{]}^T$$

(10)

where ${\ddot{x}}_s$ is the acceleration of the reeded mass; $x_s-x_u$ is the dynamic deflection of the suspension; ${k_t(x}_u-x_r)$ is the suspension dynamic load; ${\dot{x}}_u$ is the unsprung mass velocity. Equations (1) and (2) are rewritten as the equation of state:

$$\left\{\begin{array}{c}\dot{X}=AX+BU\\ Y=CX+DU\end{array}\right.$$

(11)

where $A=\left[\begin{array}{cccc}0& 1& 0& -1\\ -\frac{k_s}{m_s}& -\frac{c}{m_s}& 0& \frac{c}{m_s}\\ 0& 0& 0& 1\\ \frac{k_s}{m_u}& \frac{c}{m_u}& -\frac{k_t}{m_u}& -\frac{c}{m_u}\end{array}\right], B=\left[\begin{array}{cc}0& 0\\ 0& \frac{1}{m_s}\\ -1& 0\\ 0& -\frac{1}{m_u}\end{array}\right], C=\left[\begin{array}{cccc}-\frac{k_s}{m_s}& -\frac{c}{m_s}& 0& \frac{c}{m_s}\\ 1& 0& 0& 0\\ 0& 0& k_t& 0\\ 0& 0& 0& 1\end{array}\right], D=\left[\begin{array}{cc}0& \frac{1}{m_s}\\ 0& 0\\ 0& 0\\ 0& 0\end{array}\right], U=\left[\begin{array}{c}\dot{x_r}\\ u\end{array}\right]$.

2.2. Road Profiles

In the study of vertical vehicle dynamics, a reasonable road excitation model is equally important as the vehicle model. Common road excitation models include the random road excitation model and the single-bump excitation model. The random road excitation model [33] represents the road roughness in the time domain as a random process, which is closest to the actual road roughness situation. The single-bump excitation [34] is used to simulate the bumps on the real road, such as speed bumps, manhole covers and stones. The road attributes studied in this paper include single-bump excitation and regular D-class road excitation, as shown in Figure 2.

The single bump is a transient high-intensity pavement excitation signal due to the bumpy undulations of the pavement. It is also often considered as a transient random signal:

$$q\left(t\right)=\left\{\begin{array}{c}\frac{h}{2}\left(1-\cos \left(\frac{2\pi v\left(t-t_0\right)}{l}\right)\right)t_0<t<\frac{l}{v}\\ 0 else\end{array}\right.$$

(12)

where $h$ is the bump height, which is 3 cm; $l$ is the length of the bump on the driving surface, which is 5 m; $v$ is the vehicle speed; $t_0$ is the time it takes for the suspension to enter the bump.

The standard random pavement time domain function is:

$${\dot{z}}_q\left(t\right)=-2\pi f_0{z}_q\left(t\right)+2\pi n_0\sqrt{G_q\left(n_0\right)\dot{x}}W\left(t\right)$$

(13)

where ${\dot{z}}_q\left(t\right)$ is the random pavement profile; $z_q\left(t\right)$ is the pavement excitation; $f_0={vn}_{00}$, where $v$ is the car driving speed and the lower cutoff frequency $n_{00}=0.11{\mathrm{m}}^{-1}$; $n_0=0.1{\mathrm{m}}^{-1}$ is the standard spatial frequency; $G_q\left(n_0\right)$ is the pavement unevenness coefficient, which takes the value $1024\times {10}^{-6}$ when it is a D-class pavement; $W\left(t\right)$ is the random Gaussian white noise.

2.3. Evaluation Indicators

The main function of a vehicle suspension is to ensure good ride comfort, driving stability and suspension dynamic deflection and to meet the requirements for vehicle posture during acceleration, braking and turning. These are also the control targets in the algorithm in this paper.

• Ride comfort: The evaluation of vehicle smoothness is divided into subjective evaluation and objective evaluation. Subjective evaluation may lead to large differences in evaluation results due to individual differences, making the evaluation results difficult to determine. Objective evaluation methods have been developed over many years of in-depth research, and there are now unified criteria. Currently, most of the literature uses the root mean square (RMS) value of the acceleration as the evaluation method, and this method is used in this paper as well. The RMS function is defined as follows:

$$\mathrm{R}\mathrm{M}\mathrm{S}\left(\ddot{{\mathrm{x}}_s}\right)=\sqrt{\frac{\sum _{i=1}^N|{\ddot{{\mathrm{x}}_s}|}^2}{N}}$$

(14)

In the equation, $N$ represents the number of data points to be solved in the time domain, and $\ddot{{\mathrm{x}}_s}$ represents the suspension sprung mass acceleration.

• Suspension dynamic deflection: suspension dynamic deflection describes the degree of suspension displacement changes relative to the static equilibrium position, and the suspension dynamic deflection should be limited within the allowable range. If the upper and lower travel limits are exceeded, the “bottoming out” phenomenon can occur, which seriously deteriorates the ride comfort. Therefore, the suspension dynamic deflection must be restricted to a certain range of displacement.

$$|x_s-x_u|\le X_{max}$$

(15)

In the equation, $X_{max}$ represents the upper limit of suspension dynamic deflection.

• Driving stability: driving stability refers to the ability of a vehicle to maintain contact between the tires and the road surface, which is also known as tire traction. When studying the driving dynamics of a vehicle, tire deformation or wheel load is often used as an evaluation index of driving stability. Only when the suspension dynamic load on the tire is less than the static load can the tire traction be guaranteed.

The above analysis shows that the goal of suspension system design is to improve the comfort of the vehicle while keeping the suspension dynamic deflection and suspension dynamic load within a reasonable range. Because there is a trade-off among the three factors, it is impossible to achieve the optimal result simultaneously. In this paper, the suspension sprung mass acceleration was taken as the optimization objective, and the suspension dynamic deflection and suspension dynamic load were taken as constraints. By selecting the control parameters appropriately, the vehicle can improve the comfort as much as possible while staying within the allowed suspension travel range.

2.4. System Framework

The main objective of this paper was to design an improved salp swarm algorithm by effectively combining a PID controller and dynamically adjusting the damper damping force based on the current suspension state. Different damping forces are adapted to different road conditions to achieve different performance requirements for driving comfort and safety. The system framework is shown in Figure 3.

The excitation generated by the road profile through the 1/4 suspension is the main source of suspension vibration. First, the 1/4 suspension sprung mass acceleration signals, which were collected by sensors, were used as the system evaluation indicators. The evaluation indicator was the controlled target. Then, the control target is input into MRASSA, and MRASSA performs parameter optimization on the control target. The control target, which was the sprung mass acceleration of the suspension, was as small as possible to ensure the smoothness of the suspension. MRASSA generates corresponding intermediate parameters and the proportional coefficients, integral coefficients and derivative coefficients of PID and then controls the damping size of the suspension damper.

3. Improved Salp Swarm Algorithm

Based on the above analysis, the suspension sprung mass acceleration was determined as the system control objectives. The smaller the value is, the smoother the suspension will be. In order to ensure that the value of the control objective was as small as possible, an algorithm with a high seeking accuracy that does not easily fall into the local optimal solution was needed for the seeking control. This paper chose the standard salp swarm algorithm as the benchmark algorithm and improved it.

3.1. SSA

The salp swarm is a type of marine colonial organism. During the foraging process, individuals in a population of salp swarm are connected to each other and form a circular chain. In the bionic salp swarm algorithm (SSA), the salp swarm chain is composed of two types of subpopulations: leaders and followers. The leader is located at the front of the salp swarm chain and is responsible for exploring the location of food, while the other individuals are followers and follow each other, one after the other.

Random initialization of the salp swarm population:

$$X_{N\times d}=rand\left(N,d\right)\times \left(ub-lb\right)+lb$$

(16)

where $N$ is the size of the salp swarm population, $d$ is the spatial dimension and the population vector consists of an $N\times d$ dimensional matrix.

In SSA, the position of the food source is the target position for all individuals of the snail chain, which is the global optimal solution during the exploration process. It affects the update of the leader’s position. The formula for updating the leader’s position is as follows:

$$x_j^i=\left\{\begin{array}{c}{F}_j+c_1\left(\left(u{b}_j-l{b}_j\right)c_2+l{b}_j\right)\hspace{1em}{c}_3\ge 0.5\\ F_j-c_1\left(\left(u{b}_j-l{b}_j\right)c_2+l{b}_j\right)\hspace{1em}{c}_3<0.5\end{array}\right.$$

(17)

where $x_j^i$ is the position of the i-th leader in the j-th-dimensional space; $F_j$ is the position of the food source in the j-th-dimensional space; $u{b}_j$ is the upper limit in the j-th-dimensional space; $l{b}_j$ is the lower limit in the j-th-dimensional space; $c_2$ and $c_3$ are the random numbers generated in the interval [0, 1], which determine the leader’s movement step and movement direction.

From Equation (17), it can be seen that the leader position update is mainly influenced by the food source position. The parameter $c_1$ is the distance control factor, which is defined as follows:

$$c_1=2e^{-{\left(\frac{4t}{T}\right)}^2}$$

(18)

where $t$ is the number of current iterations; $T$ is the maximum number of iterations. The parameter $c_1$ decreases nonlinearly to converge to 0 during the iterations. When the value of $c_1$ is larger in the early stage of the algorithm, it helps to enhance the exploration capability. When the value is smaller, it helps to develop local capabilities.

To update the positions of the followers, Newton’s motion theorem is used:

$$x_j^i=\frac{1}{2}\left(a{t}^2+v_0∆t\right)$$

(19)

where $i\ge 2$, and $x_j^i$ is the position of the i-th follower in the j-th-dimensional space; $∆t$ is time; $v_0$ is the initial velocity, where acceleration $a={(v}_{\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{a}\mathrm{l}}-v_0)/∆t$, $v_{\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{a}\mathrm{l}}=\left({x_j^{i-1}-x}_j^i\right)/∆t$ and $x_j^{i-1}$ is the position of the (i − 1)-th salp swarm in the j-th-dimensional space. Since the time is the difference in the number of iterations, $∆t=1$ and the initial velocity $v_0=0$. Equation (19) can be expressed as:

$$x_j^i=\frac{1}{2}\left(x_j^i+x_j^{i-1}\right)$$

(20)

The behavioral mechanism of the salp swarm chain can be simulated by Equations (17) and (20).

3.2. MRSSA

Similar to other swarm algorithms, the standard salp swarm algorithm also has the defects of poor search accuracy and a tendency to fall into local optimal solutions. Although the convergence speed of standard SSA is fast enough compared to other swarm intelligence algorithms, it still cannot reach the speed required by the suspension system. In order to improve the defects, this paper improves the algorithm on the basis of the standard salp swarm algorithm as follows:

• Multi-subpopulation population division

While maintaining the number of individuals in the original population, the population was evenly divided into three subpopulations according to their fitness values, from small to large. The original population is divided into three sub-populations: leaders, followers and chain-tailers. These three sub-populations execute different update strategies, focusing on balanced search, local search and global search.

• Refracted opposition-based learning mechanism to update followers

Opposition-Based Learning (OBL) is used to expand the search by computing the inverse of the current feasible solution to find a better candidate solution for a given problem. Assuming $x_1$ is a real number and $x_1\in [a,b]$, the inverse number of $x_1$ is formulated as:

$$x_2=a+b-x_1$$

(21)

Despite the fact that OBL can enhance the algorithm’s performance in finding the optimum in the early iterations, the algorithm is unable to jump out of the local optimum in the late iterations.

Refracted Opposition-Based Learning (ROBL) strategy [35] is based on OBL and combines the refraction law to find a better candidate solution. The basic principle of ROBL is shown in Figure 4.

In Figure 4, the search interval of the solution on the x-axis is $[a,b]$; the y-axis represents the normal; the lengths of the incident and refracted rays are $l$ and $l_2$; $\alpha$ and $\beta$ are the angles of incidence and refraction; the intersection point $O$ is the midpoint of the interval $[a,b]$. From the geometric relationship of the lines in the figure, the following can be obtained:

$$sin\alpha =\left(\left(a+b\right)/2-x_1\right)/l_1$$

(22)

$$sin\beta =\left(x_2-\left(a+b\right)/2\right)/l_2$$

(23)

From the definition of the refractive index, $n=sin\alpha /sin\beta$. Combining it with the above two equations yields the following:

$$n=\frac{\left(\left(a+b\right)/2-x_1\right)l_2}{\left(x_2-\left(a+b\right)/2\right)l_1}$$

(24)

Let $k=l/l_2$, and $k$ is called the scaling factor. Substituting the above equation and varying it yields the refraction direction learning solution:

$$x_2=\frac{a+b}{2}+\frac{a+b}{2kn}-\frac{x_1}{kn}$$

(25)

when $k=1$ and $n=1$, the above equation is equal to Equation (21). Obviously, OBL is a special case of ROBL. Compared with OBL, which can only obtain a fixed inverse solution, ROBL can obtain dynamic candidate solutions by adjusting the parameters, which will also improve the probability of the algorithm jumping out of the local optimum. Equation (25) is extended to the multidimensional space.

$$x_{j2}^i=\frac{a_j+b_j}{2}+\frac{a_j+b_j}{2kn}-\frac{x_{j1}^i}{kn}$$

(26)

where $x_{j1}^i$ denotes the position of the i-th individual in the current population on the j-th dimension; $x_{j2}^i$ is the refractive inverse solution; $a_j$ and $b_j$ are the upper and lower bounds of the j-th dimension on the search space. Obtaining two solutions in one calculation also speeds up the population search. The combination of Equations (20) and (26) is used to search for the optimal follower’s positions.

• Adaptive factor to update chain tailers

• From Equation (20), it is clear that the position of the i-th follower is only related to the position of itself and the adjacent i−1st individual. When the leader in the population falls into a local optimum, the followers are bound to fall into a local optimum with it. The adaptive factor will also accelerate the convergence of the population at a later stage. To enhance the flexibility of the follower position update mechanism, this paper applied the control factor $c_1$ in Equation (18) into Equation (20) and uses it for the chain-tailer update as follows:

  $$x_j^i=\frac{c_1}{2}\left(x_j^i+x_j^{i-1}\right)$$

  (27)

The process of MRASSA is shown in Figure 5:

4. Case Study

In order to verify the superiority of MRASSA, Simulink simulation experiments and 1/4 suspension platform experiments were conducted in this paper. MRASSA optimizes the suspension for damping according to the control objective. An MRASSA improvement effect was judged by comparing the optimization results of standard SSA, WOAFFO and SOFA, which were applied, respectively, to compare the respective optimized performance by corresponding evaluation indexes.

4.1. Experimental Environment

4.1.1. Simulink Simulation Experiment

Simulink suspension simulation experiments were conducted under single-bump and D-class random road excitation. The simulation evaluated the suspension performance at speeds of 30 km/h and 60 km/h using WOAFFO, SOFA, SSA and MRASSA, respectively, for testing. Sprung mass acceleration, suspension dynamic deflection and suspension dynamic load were used to evaluate the suspension control performance. The time required for the models to compute the PID parameters until convergence was also used as an evaluation index to assess the speed of convergence of each model. The time was timed by the MATLAB timer.

The suspension used in the physical experimental platform is a small 1/4 suspension. In order to ensure the realism of the Simulink simulation, the simulation parameters were the real parameters of the actual small suspension. The parameters can be seen in

Table 1. Parameter Table of the Suspension Model.

Model Param	Symbol	Value	Unit
Sprung mass	$m_s$	20	kg
Unsprung mass	$m_u$	5	kg
Absorber damping coefficient	$c$	2000	Ns/m
Spring stiffness	$k_s$	5	KN/m
Tire stiffness	$k_t$	200	KN/m

.

4.1.2. The 1/4 Suspension Platform Experiment

The small 1/4 suspension platform is shown in Figure 6.

The platform is mainly composed of two parts: the 1/4 suspension system and the control and data collection system.

Two vibration sensors were installed on the upper and lower parts of the shock absorber in the suspension system. The vibration sensors can output signals such as the velocity and displacement of the corresponding parts for the control system to perform corresponding control. After the control and data collection system collected the corresponding signals, the appropriate PID parameters were obtained through related algorithms for optimized control. The PID parameters were input to the PID program module of the PLC, and the PLC will output different voltages, which then output different currents through the current amplifier.

The magneto-rheological fluid absorber has the advantages of a simple structure, high reliability, a large damping force and continuous adjustability. Therefore, the absorber used in this paper was a magneto-rheological fluid absorber. The magneto-rheological fluid absorber received different currents to provide different damping forces, thereby feedback controlling the corresponding control targets. The relevant performance of the magneto-rheological fluid absorber used in the experiment is shown in Figure 7.

Where $f$ represents the damping force, $s$ represents the piston displacement, and $v$ represents the piston velocity. All values were obtained through testing.

The suspension system is connected to the vibration table, as shown in Figure 8. The vibration platform is placed at the bottom of the small 1/4 suspension tire to provide relevant excitation.

The relevant experimental equipment brands and models are shown in

Table 2. Experimental equipments brands and models.

Type	Brand	Model	Range
Vibration sensor	Wit-motion	WTVB01-485	0~50 mm/s
Magneto-rheological fluid absorber	Bohai	C30	−200~+200 N
PLC	Siemens	ST20	\
AD model	Siemens	AM06	0–10 V
Current amplifier	\	\	0–1 A
Laptop	HP	OMEN 15	\
Vibration platform	Econ	EDS	\

.

4.2. Experimental Results

4.2.1. Single-Bump Simulation Results

The selected single-bump height A = 3 cm, and the length L = 5 m. The tested speeds were 30 km/h and 60 km/h. Based on the results of passive suspension, the leading percentage of the other controller optimization results was calculated. The specific data of the optimized suspension performance are shown in

Table 3. Optimization Results for Simulation of the Single Bump.

Speed	Controller	Suspension Acceleration/(m/s2)	Lead/%	Suspension Dynamic Deflection/m	Lead/%	Suspension Dynamic Load/N	Lead/%	Time/s
30 km/h	Passive	0.0391	\	0.00011	\	1.3420	\	\
	SOFA	0.0307	21.48	0.00008	27.27	1.2815	4.51	1.753
	WOAFFO	0.0256	34.53	0.00007	36.36	1.1455	14.64	1.943
	SSA	0.0274	29.92	0.00007	36.36	1.2141	9.53	1.637
	MRASSA	0.0208	46.80	0.00006	45.45	1.0246	23.65	1.323
60 km/h	Passive	0.0945	\	0.00010	\	3.1818	\	\
	SOFA	0.0726	23.17	0.00008	20.00	3.0840	3.07	1.682
	WOAFFO	0.0473	49.95	0.00006	40.00	2.6712	16.05	2.135
	SSA	0.0518	45.19	0.00007	30.00	2.8078	11.75	1.746
	MRASSA	0.0342	63.81	0.00005	50.00	2.4065	24.37	1.342

.

The sprung mass acceleration curves, suspension dynamic deflection curves and suspension dynamic load curves at different speeds are shown in Figure 9.

4.2.2. Random Road Simulation Results

The test speeds were 30 km/h and 60 km/h under a D-class random road. The specific data of the suspension performance simulation after optimization by different PID controllers are shown in

Table 4. Optimization Results for Simulation of the Random Road.

Speed	Controller	Suspension Acceleration/(m/s2)	Lead/%	Suspension Dynamic Deflection/m	Lead/%	Suspension Dynamic Load/N	Lead/%	Time/s
30 km/h	Passive	0.6299	\	0.00099	\	22.6350	\	\
	SOFA	0.5302	15.83	0.00075	24.24	20.7701	8.24	2.893
	WOAFFO	0.4764	24.37	0.00068	31.31	18.6040	17.81	3.182
	SSA	0.4971	21.08	0.00070	29.29	19.2670	14.88	2.593
	MRASSA	0.3958	37.16	0.00059	40.40	16.2921	28.02	1.749
60 km/h	Passive	0.8682	\	0.00111	\	31.2323	\	\
	SOFA	0.7293	16.00	0.00090	18.92	28.5553	8.57	2.736
	WOAFFO	0.6391	26.39	0.00082	26.13	25.0919	19.66	3.045
	SSA	0.6677	23.09	0.00085	23.42	26.0329	16.65	2.642
	MRASSA	0.5109	41.15	0.00073	34.23	21.9281	29.79	1.636

.

The sprung mass acceleration curves, suspension dynamic deflection curves and suspension dynamic load curves at different speeds are shown in Figure 10.

4.2.3. Single-Bump Suspension Platform Experimental Results

The 1/4 suspension tires were placed on the ECON excitation platform to provide random excitation for performance testing. Due to the large acquisition interval of the sensor, the amount of data in similar simulation experiments cannot be collected. The displayed broken line was not smooth due to the insufficient data volume.

The specific data are shown in

Table 5. Optimization Results for Platform of the Single Bump.

Speed	Controller	Suspension Acceleration/(m/s2)	Lead/%	Suspension Dynamic Deflection/m	Lead/%	Time/s
30 km/h	Passive	0.0514	\	0.00013	\	\
	SOFA	0.0406	21.01	0.00010	23.08	1.544
	WOAFFO	0.0347	32.49	0.00008	38.46	1.764
	SSA	0.0382	25.68	0.00008	38.46	1.452
	MRASSA	0.0293	43.00	0.00008	38.46	1.166
60 km/h	Passive	0.1137	\	0.00013	\	\
	SOFA	0.0889	21.81	0.00011	15.38	1.503
	WOAFFO	0.0574	49.52	0.00008	38.46	1.905
	SSA	0.0659	42.04	0.00009	30.77	1.532
	MRASSA	0.0477	58.05	0.00007	46.15	1.135

.

The sprung mass acceleration curves and suspension dynamic deflection curves at different speeds are shown in Figure 11.

4.2.4. Random Road Suspension Platform Experimental Results

Due to the continuous vibration in the process of random road excitation vibration, the small suspension system produced unpredictable shaking. The shaking of the shelf led to some data deviation, but from the result, the overall optimization result was unchanged.

The specific data are shown in

Table 6. Optimization Results for Platform of the Random Road.

Speed	Controller	Suspension Acceleration/(m/s2)	Lead/%	Suspension Dynamic Deflection/m	Lead/%	Time/s
30 km/h	Passive	1.0831	\	0.00153	\	\
	SOFA	0.9689	10.54	0.00122	20.26	2.504
	WOAFFO	0.9098	16.00	0.00109	28.76	2.770
	SSA	0.9158	15.45	0.00115	24.84	2.173
	MRASSA	0.7264	32.93	0.00099	35.29	1.344
60 km/h	Passive	1.3531	\	0.00204	\	\
	SOFA	1.1864	12.32	0.00178	12.75	2.386
	WOAFFO	1.0894	19.49	0.00156	23.53	2.695
	SSA	1.1252	16.84	0.00162	20.59	2.301
	MRASSA	0.8951	33.85	0.00142	30.39	1.316

.

The sprung mass acceleration curves and suspension dynamic deflection curves at different speeds are shown in Figure 12.

4.3. Analysis

From Figure 9 and Table 3, it can be concluded that, for single-bump road surfaces under 30 km/h conditions, SOFA, WOAFFO, SSA and MRASSA all had significant improvements compared to passive suspensions. In particular, in the optimization of the control objective of sprung mass acceleration, all four models achieved an improvement of over 20%. Among the four models, MRASSA’s optimization effect was significantly better than that of the other three. Correspondingly, the magneto-rheological fluid absorber controlled by the swarm intelligence algorithm has a more intelligent damping force adjustment, so the tire can fit more closely to the ground when the vehicle passes over a bumpy road surface, avoiding the occurrence of floating and thereby reducing the suspension deflection and damage caused by excessive suspension dynamic load. The values of these two evaluation indicators also showed that MRASSA’s optimization effect was significantly better than that of the other models. When the speed increases, the road excitation will be more pronounced. Under more significant disturbances on high-speed roads, MRASSA’s optimization effect on sprung mass acceleration was more pronounced, with values leading by 46.80% to 63.81% compared with passive suspension. On the other hand, SOFA’s optimization effect did not improve much at high speeds and even showed a significant decline in the evaluation index of suspension deflection. According to the sprung mass acceleration, suspension dynamic deflection, suspension dynamic load and convergence speed results, the suspension optimization performance using the standard SSA without improvement already surpassed that of SOFA, and the results were close to the improved WOAFFO performance. This demonstrated that the standard SSA has excellent search accuracy as well as convergence speed compared to the general swarm algorithm. The comparison of the MRASSA results with those of the standard SSA showed that the improvements for the standard SSA’s own characteristics were also effective. MRASSA was 45.19% greater than the standard SSA in terms of the sprung mass acceleration at the 60 km/h speed. For the convergence speed, it can be seen from the convergence time that MRASSA took only a small number of iterations to find the exact solution. The WOAFFO with the addition of the fuzzy-fractional order took the longest time to find the solution.

Compared to single-bump excitation, random road excitation can provide a more complex excitation effect. As seen in Figure 10 and Table 4, the optimization effect of the four models under random road excitation was lower in sprung mass acceleration and suspension dynamic deflection than under single-bump excitation. However, the optimization effect of the suspension dynamic load was improved, which is due to the more complex road conditions and the more floating situations caused by passive suspensions, leading to excessive dynamic loads on the passive suspension. MRASSA still had a significant leading advantage in the three evaluation indicators. Although the convergence time has increased compared to the single bump, it was still the least time-consuming among the four models. Whether under low- or high-speed random road excitation, the optimized evaluation indicators of MRASSA were the best. Standard SSA also has good results for relevant performance indexes. Simulink simulations have proven the superiority of MRASSA over standard SSA, SOFA and WOAFFO.

In the experimental results of the 1/4 experimental platform, there was a deviation between the results and the simulation results due to the long collection frequency of the vibration sensor and the unpredictable shaking. Due to the short duration of vibration in the single-bump excitation, the results were basically the same as the simulation results. However, on the random road excitation, the experimental results differ from the simulation results, but the optimization ability of MRASSA can be seen to be better than the other three types of models. The trend of the real-time result is similar to that of the simulated time result. However, there are some errors in their values. This is mainly due to the fact that the amount of data collected by the sensor per unit time was not as much as that of the simulation. With less data, the algorithm converged faster. The results further confirm the authenticity of the Simulink simulation results, and the optimization ability of MRASSA for suspensions in real environments was also superior to the abovementioned three suspension optimization controllers.

From the experimental results, the improvement of MRASSA was effective and obvious compared to the standard SSA. MRASSA showed a higher leading effect on suspension optimization compared to SOFA and WOAFFO. This means that the optimized suspension of MRASSA can better control the contact between the tire and the road, reduce the impact and damage to the suspension and improve the comfort and stability of the vehicle. In addition, the performance of MRASSA suspension is better than that of other suspension types at all speeds. MRASSA is improved compared to the standard SSA’s own characteristics, so this convergence time of finding the best is not affected by other algorithms. MRASSA is more suitable for the field of suspension system optimization than other models which combine complex algorithms. However, there are two limitations to the experiments in this paper:

• In the 1/4 suspension platform experiment, the suspension dynamic load values of the suspension were not measured, so the suspension dynamic load values of Simulink simulation could not be verified;
• The 1/4 suspension platform used was a downsize version of the suspension system, not the 1/4 suspension system of a complete car. The tire material, spring performance and overall structure of the platform cannot be compared to those of a car suspension, which makes it difficult to better experimentally verify the performance results under real road conditions.

If future experiments can add the measurement of the suspension dynamic load and switch to using a car vibration testing platform, the results can better illustrate the superiority of MRASSA.

5. Conclusions

In this paper, a multi-subpopulation refracted adaptive salp swarm algorithm was proposed for the following problems: the standard salp swarm algorithm easily falls into the local optimal solution and it has a slow convergence speed. The MRASSA was combined with PID and applied to the control of 1/4 semi-active suspension. The experimental results show that MRASSA is more effective than other models in escaping the local optima and convergence speed, leading to better optimization of the suspension performance. Both simulation and physical experiments demonstrate that MRASSA outperforms standard SSA, the Whale Fuzzy Fractional Order Algorithm (WOAFFO) and the Single-Objective Firefly Algorithm (SOFA) in adapting to single-bump excitation and random road excitation. MRASSA effectively reduces sprung mass acceleration, suspension dynamic deflection and suspension dynamic load, resulting in a smoother ride and better ride quality. Notably, the MRASSA shows greater optimization capabilities at higher vehicle speeds. Faster convergence will also lead to a faster response of the suspension system. Future work could include adding measurements of the suspension dynamic load and converting the 1/4 suspension platform into a vehicle vibration test platform to further illustrate the superiority of the MRASSA.