# Dynamic Responses of 8-DoF Vehicle with Active Suspension: Fuzzy-PID Control

^{*}

## Abstract

**:**

## 1. Introduction

_{∞}control theory [27]. Sliding mode variable structure control can continuously change purposefully according to the current state of the system (such as deviation and its derivatives of each order) in the dynamic process, forcing the system to move according to the state track of the predetermined sliding mode, which includes single-mode sliding [28], multi-mode sliding [29], super-twisting sliding [30], and adaptive sliding [31]. Intelligent control methods include fuzzy control [32], neural network control [33], optimization control based on the genetic algorithm [34], expert control [35], and hierarchical intelligent control [36]. Fuzzy control is an intelligent control method that imitates human fuzzy reasoning and decision-making processes. Neural network control can perform neural network model identification on complex nonlinear objects that are difficult to accurately model. The optimization control based on a genetic algorithm simulates biological evolution processes such as natural selection, heredity, and mutation. Expert control combines the theory and technology of expert systems with control theory, methods, and technologies, imitating the experience of experts in unknown environments. Based on adaptive control and self-organizing control, hierarchical intelligent control is divided into the organizational level, coordination level, and execution level according to the level of intelligent control, and these three levels follow the principle of intelligent descending and increasing accuracy.

## 2. Mathematical Model of Suspension

_{si}of the suspension and the vertical displacement z

_{xy}of the seat system position are as follows:

- (1)
- Equation for the motion of the center of mass of the human-chair system:$${m}_{c}{\ddot{z}}_{c}=-{k}_{c}\left({z}_{c}-{z}_{xy}\right)-{c}_{c}\left({\dot{z}}_{c}-{\dot{z}}_{xy}\right),$$
- (2)
- Equation for the motion of the center of mass of the suspension body:$${m}_{b}{\ddot{z}}_{b}={\displaystyle \sum _{i=1}^{4}\left[-{k}_{si}\left({z}_{si}-{z}_{ui}\right)-{c}_{si}\left({\dot{z}}_{si}-{\dot{z}}_{ui}\right)-{u}_{i}\right]}+{k}_{c}\left({z}_{c}-{z}_{xy}\right)+{c}_{c}\left({\dot{z}}_{c}-{\dot{z}}_{xy}\right),$$
- (3)
- Differential equation for suspension pitch rotation:$$\begin{array}{ll}\hfill {J}_{\theta}\ddot{\theta}=& {\displaystyle \sum _{i=1,2}{l}_{1}\left[{k}_{si}\left({z}_{si}-{z}_{ui}\right)+{c}_{si}\left({\dot{z}}_{si}-{\dot{z}}_{ui}\right)+{u}_{i}\right]}-{\displaystyle \sum _{i=3,4}{l}_{2}\left[{k}_{si}\left({z}_{si}-{z}_{ui}\right)+{c}_{si}\left({\dot{z}}_{si}-{\dot{z}}_{ui}\right)+{u}_{i}\right]}-\hfill \\ & {r}_{x}\left[{k}_{c}\left({z}_{c}-{z}_{xy}\right)+{c}_{c}\left({\dot{z}}_{c}-{\dot{z}}_{xy}\right)\right],\hfill \end{array}$$
- (4)
- Differential equation for suspension roll rotation:$$\begin{array}{ll}\hfill {J}_{\phi}\ddot{\phi}=& {\displaystyle \sum _{i=1,3}-{l}_{3}\left[{k}_{si}\left({z}_{si}-{z}_{ui}\right)+{c}_{si}\left({\dot{z}}_{si}-{\dot{z}}_{ui}\right)+{u}_{i}\right]}-{\displaystyle \sum _{i=2,4}{l}_{4}\left[{k}_{si}\left({z}_{si}-{z}_{ui}\right)+{c}_{si}\left({\dot{z}}_{si}-{\dot{z}}_{ui}\right)+{u}_{i}\right]}-\hfill \\ & {r}_{y}\left[{k}_{c}\left({z}_{c}-{z}_{xy}\right)+{c}_{c}\left({\dot{z}}_{c}-{\dot{z}}_{xy}\right)\right],\hfill \end{array}$$
- (5)
- Equations for the movement of four unsprung masses:$${m}_{ui}{\ddot{z}}_{ui}={k}_{si}\left({z}_{si}-{z}_{ui}\right)+{c}_{si}\left({\dot{z}}_{si}-{\dot{z}}_{ui}\right)-{k}_{ui}\left({z}_{ui}-{z}_{ri}\right)+{u}_{i},i=1,\cdots ,4,$$
_{c}is the vertical displacement of the human, z_{b}represents the vertical displacement of the center of mass of the vehicle, z_{si}denotes the vertical displacement of the suspension, z_{ui}is the vertical displacement of the tire, θ means the pitch angle, φ is the roll angle, z_{xy}is the vertical displacement of the human-chair system, z_{ri}represents the vertical displacement excited by the road surface, and u_{i}is the suspension control input. The left-front wheel, right-front wheel, left-rear wheel, and right-rear wheel correspond to subscripts 1, 2, 3, and 4, respectively. Table 1 provides parameters for the inherent characteristics of the full vehicle.

_{c}, z

_{b}, θ, φ, z

_{u}

_{1}, z

_{u}

_{2}, z

_{u}

_{3}, z

_{u}

_{4}]

^{T}, control variables U = [u

_{1}, u

_{2}, u

_{3}, u

_{4}]

^{T}, and excitation variables Z = [z

_{r}

_{1}, z

_{r}

_{2}, z

_{r}

_{3}, z

_{r}

_{4}]

^{T}. The mass matrix M, damping matrix C, stiffness matrix K, control matrix B, and excitation matrix W are denoted as follows:

## 3. Road Excitation Models

#### 3.1. Continuous Crossing Road Hump Model

_{f}and q

_{r}denote the longitudinal slope curves of the front and rear wheels, respectively, i

_{index}is the index of the speed bump, and z is the height of the speed bump z = 0.05 m.

#### 3.2. C-Level Road Model

_{q}(n) of the longitudinal slope curve q(I) is used as the fitted expression by the following equation:

_{q}(n

_{0}) denotes the coefficient for pavement unevenness at a spatial frequency of n

_{0}, n

_{0}= 0.1 m

^{−1}means the reference spatial frequency, and W = 2 is the frequency index.

_{q}(n) represented by n and the power spectral density G

_{q}(ω) represented by ω is as follows:

_{0}can be introduced into the power spectral density G

_{q}(ω). So, the power spectral density G

_{q}(ω) can be written as

_{0}= 2πn

_{00}u denotes the cutoff frequency, and n

_{00}= 0.01 m

^{−1}represents the cutoff spatial frequency under road roughness.

_{ω}= 1 through a linear system is equivalent to the spectral density of the road surface, as per the filtered white noise technique [55]. Additionally, the frequency domain response function of the linear system H

_{1}(jw) is assumed to be as follows:

_{00}, and b = 2πn

_{0}(G

_{q}(n

_{0})u)

^{1/2}. Then, the output power spectral density is as follows:

_{1}(jω) is as follows:

_{ω}. More information about the time-domain model of single-wheel road excitation based on filtered white noise can be found in [56].

#### 3.3. Time Domain for Front and Rear Wheels on the Same Track

_{1}+ l

_{2})/u. The first-order Pade approximate delay transfer function e

^{−Ts}is used to represent the time delay effect of the rear wheels on the front wheels [57], as shown below:

_{2}(s)|| = 1 is an all-pass system where the amplitude of the frequency component remains unchanged. Then G

_{qr}(ω) = |H

_{2}(jω)|

^{2}G

_{qf}(ω) = G

_{q}(ω), and G

_{qf}(ω) is the power spectral density function of the front wheel. The time-domain delay model of the rear wheel is obtained from this transfer function:

#### 3.4. Time Domain for Left and Right Wheels at the Same Axle

_{lr}(ω) of left and right wheels at the same axle is defined as [58]

_{lr}(ω) is the spectral density of the cross power between the left and right wheels. Assuming that the self spectra q

_{l}(t) and q

_{r}(t) of the left and right wheels are the same, and the phase spectrum between the two wheels is equal to zero, then the following relationship is established:

_{l}(t) as a reference, the above formula G

_{lr}(ω) can be considered as the output of G

_{q}(ω) through a linear system H

_{3}(jω), and the amplitude frequency characteristic of this frequency response function ||H

_{3}(jω)|| is equal to coh

_{lr}(ω). Assuming that the transfer function between the left and right wheels q

_{l}(t) and q

_{l}(t) is approximated by a second-order rational fraction, we obtain

_{0}= 3.1851, a

_{1}= 0.2063, a

_{2}= 0.0108, b

_{0}= 3.223, b

_{1}= 0.59, and b

_{2}= 0.0327 [57,59]. Transforming the above transfer function into the state equation between the left and right wheels and adding the intermediate state variable ξ(t) = [ξ

_{1}(t), ξ

_{2}(t)]

^{T}, we can get

## 4. Fuzzy-PID Control

#### 4.1. Design of Fuzzy Controller

_{p}, K

_{i}, and K

_{d}, respectively, as shown in Figure 3.

_{Ci}as the inputs of fuzzy-PID. The force u

_{i}of the actuator is used as the controller output. The value of the ideal speed is always equal to 0. Figure 3 displays the three outputs of the fuzzy controller, namely K

_{p}, K

_{i}, and K

_{d}.

_{i}= [−0.1 m/s, 0.1 m/s], e

_{Ci}= [−1.5 m/s

^{2}, 1.5 m/s

^{2}], and the range of the force of the actuator is u

_{i}= [−200 N, 200 N].

_{i}and e

_{Ci}are transformed into integer theoretical domains {−3, −2, −1, 0, 1, 2, 3} with quantization factors of 30 and 2, respectively. The outputs K

_{p}, K

_{i}, and K

_{d}of the fuzzy controller are {−1, −2/3, −1/3, 0, 1/3, 2/3, 1}, and r = 1000, s = 6000, and t = 20 are set. To represent the fuzzy states of the input–output, a total of seven subsets that are fuzzy are used, namely {NB, NM, NS, ZE, PS, PM, PB}.

_{k}is the input values of the fuzzy system, and μ(y

_{k}) is the membership value of y

_{k}.

#### 4.2. Fuzzy Rules of Fuzzy Controller

_{p}, K

_{i}, and K

_{d}of fuzzy-PID control are determined by the fuzzy controller and presented in Table 2, Table 3 and Table 4 [60]. Lu et al. [60] showed that these fuzzy control rules can be applied to the pulsed MIG welding of aluminum alloys, which can serve as an important reference source for us to verify the stability of fuzzy control rules. Figure 5 displays the curve of the fuzzy controller’s relationship, which was designed using two input and three output parameters.

#### 4.3. Structure Analysis and Stability

_{k}: IF x

_{1}is $\mathcal{N}$

_{1}

^{k}AND x

_{2}is $\mathcal{N}$

_{2}

^{k}AND … x

_{m}is $\mathcal{N}$

_{m}

^{k}

THEN y

_{k}is $\mathcal{W}$

^{k}, k = 1, 2, …, n,

_{k}is the kth control rule, $\mathcal{W}$

^{k}and $\mathcal{N}$

_{j}

^{k}, j = 1, 2, …, m, denote fuzzy sets, and n is the total number of fuzzy rules. The Zadeh fuzzy logic AND operator (i.e., min()) is used to realize the AND operations in the above-mentioned rules. Thus, the membership value of y

_{k}can be written as

_{$\mathcal{N}$}is the membership function of variable x. Combining Equations (30) and (32), and replacing x

_{1}and x

_{2}with e

_{i}and e

_{Ci}, the output state of the fuzzy system is

_{p}, K

_{i}, and K

_{d}, respectively. By bringing Equations (34)–(36) into Equation (30), the control output u

_{i}(t) can be obtained.

_{1}, the activated fuzzy rules are as follows (seeing Table 2, Table 3 and Table 4):

- (1)
- For K
_{p},Rule R_{1}: IF e is NB AND e_{c}is NB, Then K_{p}is PB,Rule R_{2}: IF e is NB AND e_{c}is NM, Then K_{p}is PB,Rule R_{3}: IF e is NM AND e_{c}is NB, Then K_{p}is PB,Rule R_{4}: IF e is NM AND e_{c}is NM, Then K_{p}is PB, - (2)
- For K
_{i},Rule R_{1}: IF e is NB AND e_{c}is NB, Then K_{i}is NB,Rule R_{2}: IF e is NB AND e_{c}is NM, Then K_{i}is NB,Rule R_{3}: IF e is NM AND e_{c}is NB, Then K_{i}is NB,Rule R_{4}: IF e is NM AND e_{c}is NM, Then K_{i}is NB, - (3)
- For K
_{d},Rule R_{1}: IF e is NB AND e_{c}is NB, Then K_{d}is PS,Rule R_{2}: IF e is NB AND e_{c}is NM, Then K_{d}is PS,Rule R_{3}: IF e is NM AND e_{c}is NB, Then K_{d}is NS,Rule R_{4}: IF e is NM AND e_{c}is NM, Then K_{d}is NS.

^{2}(t) + (∫e(t)dt)

^{2})/2. The following expression can be obtained:

_{c}is N, K

_{p}, K

_{i}, and K

_{d}should be P, N, and P (or N with the condition $\dot{e}(t)$ > −∫e(t)dt). So, the stability of the subspace S

_{1}can be guaranteed. Similar operations can be extended to other subspaces, and the consequence of the fuzzy rule can be obtained systematically.

## 5. Results and Discussion

_{si}–z

_{ui}, and four displacements of left-rear tire z

_{ui}–z

_{ri}, a total of 12 evaluation indicators.

#### 5.1. Dynamic Response of Suspension under Continuous Crossing Road Hump

_{index}= 5) in the first 10 s and the flat road surface in the last 4 s. Figure 7 shows the variations of K

_{p}, K

_{i}, and K

_{d}for the fuzzy-PID control system under a continuous crossing road hump. Figure 8 shows the comparison of the suspension performance of the passive suspension system, PID active suspension, and fuzzy-PID active suspension under a continuous crossing road hump.

^{−2}~1.5 × 10

^{−2}m/s

^{2}, the vertical acceleration of the body ${\ddot{z}}_{b}$ is −1.5 × 10

^{−2}~1.5 × 10

^{−2}m/s

^{2}, the pitch angular acceleration of the body $\ddot{\phi}$ is −6.1 × 10

^{−3}~6.1 × 10

^{−3}rad/s

^{2}, the roll angular acceleration of the body $\ddot{\theta}$ is −1.6 × 10

^{−2}~1.6 × 10

^{−2}rad/s

^{2}, the deflections of suspension z

_{s}

_{i}–z

_{u}

_{i}are −2.0 × 10

^{−4}~2.0 × 10

^{−4}m, and the displacements of tire z

_{u}

_{i}–z

_{r}

_{i}are −3.0 × 10

^{−5}~3.0 × 10

^{−5}mm.

#### 5.2. Dynamic Response of Suspension under C-Level Road

_{p}, K

_{i}, and K

_{d}for the fuzzy-PID control on the C-level road are depicted in Figure 9. On the C-level road, suspension performances are compared between passive, PID control, and fuzzy-PID control suspensions, as depicted in Figure 10.

^{2}, the vertical acceleration of the body ${\ddot{z}}_{b}$ is −0.8~0.8 m/s

^{2}, the pitch angular acceleration of the body $\ddot{\phi}$ is −0.6~0.6 rad/s

^{2}, the roll angular acceleration of the body $\ddot{\theta}$ is −1.0~1.0 rad/s

^{2}, the deflections of suspension z

_{s}

_{i}–z

_{u}

_{i}are −1.0 × 10

^{−2}~1.0 × 10

^{−2}m, and the displacements of tire z

_{u}

_{i}–z

_{r}

_{i}are −4.0 × 10

^{−3}~4.0 × 10

^{−3}m.

#### 5.3. Comparison of Control Performance

^{λ}D

^{μ}controller based on a nine-degree-of-freedom air suspension system. Yang et al. [74] reported a fuzzy-PID controller with self-adjusting parameters for achieving the functions of PID parameter online amendment. Chen et al. [70] supported that the PID controller can be regulated by fuzzy parameters,

_{fuzzy}is a revision of the fuzzy controller.

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 4.**Design of fuzzy controller: (

**a**) fuzzy-PID control in Simulink, (

**b**) inputs and outputs of fuzzy-PID control, (

**c**) membership function of input quantities, and (

**d**) membership function of output quantities.

**Figure 5.**Surface graph of the relationship between K

_{p}, K

_{i}, K

_{d}, and e, e

_{C}: (

**a**) K

_{p}, (

**b**) K

_{i}, and (

**c**) K

_{d}.

**Figure 7.**Evolution diagrams of K

_{p}, K

_{i}, and K

_{d}under continuous crossing road hump: (

**a**) K

_{p}, (

**b**) K

_{i}, and (

**c**) K

_{d}.

**Figure 8.**Variations of dynamic responses under continuous crossing road hump: (

**a**) ${\ddot{z}}_{c}$, (

**b**) ${\ddot{z}}_{b}$, (

**c**) $\ddot{\phi}$, (

**d**) $\ddot{\theta}$, (

**e**) z

_{s}

_{1}–z

_{u}

_{1}, (

**f**) z

_{s}

_{2}–z

_{u}

_{2}, (

**g**) z

_{s}

_{3}–z

_{u}

_{3}, (

**h**) z

_{s}

_{4}–z

_{u}

_{4}, (

**i**) z

_{u}

_{1}–z

_{r}

_{1}, (

**j**) z

_{u}

_{2}–z

_{r}

_{2}, (

**k**) z

_{u}

_{3}–z

_{r}

_{3}, and (

**l**) z

_{u}

_{4}–z

_{r}

_{4}.

**Figure 9.**Evolution diagrams of K

_{p}, K

_{i}, and K

_{d}under continuous crossing road hump: (

**a**) K

_{p}, (

**b**) K

_{i}, and (

**c**) K

_{d}.

**Figure 10.**Variations of dynamic responses under C-level road: (

**a**) ${\ddot{z}}_{c}$, (

**b**) ${\ddot{z}}_{b}$, (

**c**) $\ddot{\phi}$, (

**d**) $\ddot{\theta}$, (

**e**) z

_{s}

_{1}–z

_{u}

_{1}, (

**f**) z

_{s}

_{2}–z

_{u}

_{2}, (

**g**) z

_{s}

_{3}–z

_{u}

_{3}, (

**h**) z

_{s}

_{4}–z

_{u}

_{4}, (

**i**) z

_{u}

_{1}–z

_{r}

_{1}, (

**j**) z

_{u}

_{2}–z

_{r}

_{2}, (

**k**) z

_{u}

_{3}–z

_{r}

_{3}, and (

**l**) z

_{u}

_{4}–z

_{r}

_{4}.

Property | Value | Property | Value |
---|---|---|---|

mass of human-chair system m_{c} (kg) | 80 | damping of human-chair system C_{c} (N⋅s/m) | 2200 |

sprung mass m_{b} (kg) | 1380 | suspension damping c_{si} (N⋅s/m) | 1000 |

unsprung mass m_{ui} (kg) | 40 | distances from body to front and rear axles l_{1}, l_{2} (m) | 1.3, 1.5 |

stiffness of human-chair system k_{c} (N/m) | 10,000 | distances from body to left and right axles l_{3}, l_{4} (m) | 0.74, 0.75 |

suspension stiffness k_{si} (N/m) | 17,000 | coordinates of human-chair system r_{x}, r_{y} (m) | 0.57, 0.33 |

wheel stiffness k_{ui} (N/m) | 200,000 |

e_{C} | e | ||||||
---|---|---|---|---|---|---|---|

NB | NM | NS | ZO | PS | PM | PB | |

NB | PB | PB | PM | PM | PS | ZO | ZO |

NM | PB | PB | PM | PS | PS | ZO | NS |

NS | PM | PM | PM | PS | ZO | NS | NS |

ZO | PM | PM | PS | ZO | NS | NM | NM |

PS | PS | PS | ZO | NS | NS | NM | NM |

PM | PS | ZO | NS | NM | NM | NM | NB |

PB | ZO | ZO | NM | NM | NM | NB | NB |

e_{C} | e | ||||||
---|---|---|---|---|---|---|---|

NB | NM | NS | ZO | PS | PM | PB | |

NB | NB | NB | NM | NM | NS | ZO | ZO |

NM | NB | NB | NM | NS | NS | ZO | ZO |

NS | NB | NM | NS | NS | ZO | PS | NS |

ZO | NM | NM | NS | ZO | PS | PM | PM |

PS | NM | NS | ZO | PS | PS | PM | PB |

PM | ZO | ZO | PS | PS | PM | PB | PB |

PB | ZO | ZO | PS | PM | PM | PB | PB |

e_{C} | e | ||||||
---|---|---|---|---|---|---|---|

NB | NM | NS | ZO | PS | PM | PB | |

NB | PS | NS | NB | NM | NB | NM | PS |

NM | PS | NS | NB | NM | NM | NS | ZO |

NS | ZO | NS | NM | NM | NS | NS | ZO |

ZO | ZO | NS | NS | NS | NS | NS | ZO |

PS | ZO | ZO | ZO | ZO | ZO | ZO | ZO |

PM | PB | NS | PS | PS | PS | PS | PB |

PB | PB | PM | PM | PM | PS | PS | PB |

**Table 5.**Comparison of root mean square of suspension performance under continuous crossing road hump.

Evaluating Indicator | Ratio of Root Mean Square | Optimization | |||
---|---|---|---|---|---|

Passive | PID | Fuzzy-PID | PID vs. Passive | Fuzzy-PID vs. PID | |

vertical acceleration of human ${\ddot{z}}_{c}$ (m/s^{2}) | 4.7 × 10^{−2} | 3.8 × 10^{−2} | 2.9 × 10^{−2} | 18.60% | 30.16% |

vertical acceleration of body ${\ddot{z}}_{b}$ (m/s^{2}) | 5.5 × 10^{−2} | 4.5 × 10^{−2} | 3.4 × 10^{−2} | 18.18% | 24.44% |

pitch angular acceleration of body $\ddot{\phi}$ (rad/s^{2}) | 2.1 × 10^{−2} | 1.5 × 10^{−2} | 1.0 × 10^{−2} | 28.57% | 33.33% |

roll angular acceleration of body $\ddot{\theta}$ (rad/s^{2}) | 3.7 × 10^{−2} | 2.6 × 10^{−2} | 1.8 × 10^{−2} | 29.72% | 30.76% |

deflection of left-front suspension z_{s}_{1}–z_{u}_{1} (m) | 5.65 × 10^{−5} | 5.09 × 10^{−5} | 4.43 × 10^{−5} | 10.54% | 12.96% |

deflection of right-front suspension z_{s}_{2}–z_{u}_{2} (m) | 5.07 × 10^{−5} | 4.27 × 10^{−5} | 4.17 × 10^{−5} | 15.77% | 23.41% |

deflection of left-rear suspension z_{s}_{3}–z_{u}_{3} (m) | 3.49 × 10^{−5} | 3.40 × 10^{−5} | 3.32 × 10^{−5} | 25.78% | 23.52% |

deflection of right-rear suspension z_{s}_{4}–z_{u}_{4} (m) | 4.64 × 10^{−5} | 4.56 × 10^{−5} | 4.56 × 10^{−5} | 17.24% | 17.76% |

displacement of left-front tire z_{u}_{1}–z_{r}_{1} (m) | 1.16 × 10^{−5} | 9.89 × 10^{−6} | 7.81 × 10^{−6} | 14.74% | 21.03% |

displacement of right-front tire z_{u}_{2}–z_{r}_{2} (m) | 1.06 × 10^{−5} | 8.82 × 10^{−6} | 7.04 × 10^{−6} | 16.79% | 20.18% |

displacement of left-rear tire z_{u}_{3}–z_{r}_{3} (m) | 7.59 × 10^{−6} | 6.26 × 10^{−6} | 4.81 × 10^{−6} | 17.52% | 23.16% |

displacement of right-rear tire z_{u}_{4}–z_{r}_{4} (m) | 9.53 × 10^{−6} | 7.91 × 10^{−6} | 6.19 × 10^{−6} | 16.99% | 21.74% |

Evaluating Indicator | Ratio of Root Mean Square | Optimization | |||
---|---|---|---|---|---|

Passive | PID | Fuzzy-PID | PID vs. Passive | Fuzzy-PID vs. PID | |

vertical acceleration of human ${\ddot{z}}_{c}$ (m/s^{2}) | 3.43 × 10^{−1} | 2.55 × 10^{−1} | 2.01 × 10^{−1} | 25.65% | 21.17% |

vertical acceleration of body ${\ddot{z}}_{b}$ (m/s^{2}) | 4.03 × 10^{−1} | 3.00 × 10^{−1} | 2.34 × 10^{−1} | 25.55% | 22.00% |

pitch angular acceleration of body $\ddot{\phi}$ (rad/s^{2}) | 2.25 × 10^{−1} | 1.45 × 10^{−1} | 1.14 × 10^{−1} | 35.56% | 21.37% |

roll angular acceleration of body $\ddot{\theta}$ (rad/s^{2}) | 4.50 × 10^{−1} | 3.02 × 10^{−1} | 2.29 × 10^{−1} | 32.89% | 24.17% |

deflection of left-front suspension z_{s}_{1}–z_{u}_{1} (m) | 3.70 × 10^{−3} | 3.29 × 10^{−3} | 2.72 × 10^{−3} | 11.08% | 17.32% |

deflection of right-front suspension z_{s}_{2}–z_{u}_{2} (m) | 3.55 × 10^{−3} | 3.10 × 10^{−3} | 2.59 × 10^{−3} | 12.67% | 16.45% |

deflection of left-rear suspension z_{s}_{3}–z_{u}_{3} (m) | 2.53 × 10^{−3} | 2.24 × 10^{−3} | 1.99 × 10^{−3} | 11.46% | 11.16% |

deflection of right-rear suspension z_{s}_{4}–z_{u}_{4} (m) | 3.05 × 10^{−3} | 2.70 × 10^{−3} | 2.20 × 10^{−3} | 11.47% | 18.51% |

displacement of left-front tire z_{u}_{1}–z_{r}_{1} (m) | 9.07 × 10^{−4} | 6.87 × 10^{−4} | 5.86 × 10^{−4} | 24.03% | 14.70% |

displacement of right-front tire z_{u}_{2}–z_{r}_{2} (m) | 8.36 × 10^{−4} | 6.32 × 10^{−4} | 5.32 × 10^{−4} | 24.40% | 15.82% |

displacement of left-rear tire z_{u}_{3}–z_{r}_{3} (m) | 7.02 × 10^{−4} | 5.17 × 10^{−4} | 4.33 × 10^{−4} | 26.35% | 16.24% |

displacement of right-rear tire z_{u}_{4}–z_{r}_{4} (m) | 8.63 × 10^{−4} | 6.44 × 10^{−4} | 5.49 × 10^{−4} | 25.37% | 14.75% |

Indicator | Present | Jiang et al. [65] | Zhao et al. [66] | Peng et al. [67] | Wu et al. [68] | Nagarkar et al. [69] |
---|---|---|---|---|---|---|

Fuzzy-PID | Mixed Control | Particle Swarm | GA-PSO | LQR | GA-Optimized FLC | |

${\ddot{z}}_{c}$ | 41.4% | / | 40% | 15.19% | / | 45.6% |

${\ddot{z}}_{b}$ | 41.9% | 9.8% | / | 18.24% | 57.48% | 34% |

$\ddot{\phi}$ | 49.3% | / | / | / | 28.81% | / |

$\ddot{\theta}$ | 49.1% | / | / | / | 31.39% | / |

z_{s}_{i}–z_{u}_{i} | 26.5% | 18.9% | / | 21.95% | 34.78% | 8.7% |

z_{u}_{i}–z_{r}_{i} | 35.4% | 19.8% | / | 21.34% | 0.91% | 6.1% |

Indicator | Present | Chen et al. [70] | Ji et al. [71] | Chiou et al. [72] | Wang et al. [73] | Yang et al. [74] |
---|---|---|---|---|---|---|

Fuzzy-PID | Fuzzy-PID | EVUFP | Fuzzy-PSOPID | Fuzzy-PI^{λ}D^{μ} | Fuzzy-PID | |

${\ddot{z}}_{c}$ | 41.4% | / | / | 47.32% | / | |

${\ddot{z}}_{b}$ | 41.9% | 37.5% | 54.17% | 17.89% | 50.96% | 28.89% |

$\ddot{\phi}$ | 49.3% | 28.9% | / | 42.57% | / | |

$\ddot{\theta}$ | 49.1% | 26.2% | / | 48.84% | 20.30% | |

z_{s}_{i}–z_{u}_{i} | 26.5% | / | 50.00% | 78.31% | 21.49% | 15.33% |

z_{u}_{i}–z_{r}_{i} | 35.4% | / | 23.54% | 14.93% | 22.99% | 18.60% |

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## Share and Cite

**MDPI and ACS Style**

Yin, Z.; Su, R.; Ma, X.
Dynamic Responses of 8-DoF Vehicle with Active Suspension: Fuzzy-PID Control. *World Electr. Veh. J.* **2023**, *14*, 249.
https://doi.org/10.3390/wevj14090249

**AMA Style**

Yin Z, Su R, Ma X.
Dynamic Responses of 8-DoF Vehicle with Active Suspension: Fuzzy-PID Control. *World Electric Vehicle Journal*. 2023; 14(9):249.
https://doi.org/10.3390/wevj14090249

**Chicago/Turabian Style**

Yin, Zongjun, Rong Su, and Xuegang Ma.
2023. "Dynamic Responses of 8-DoF Vehicle with Active Suspension: Fuzzy-PID Control" *World Electric Vehicle Journal* 14, no. 9: 249.
https://doi.org/10.3390/wevj14090249