Multi-objective optimization of the suspension parameters in the in-wheel electric vehicle

Abstract

The hub motor significantly increases the unsprung mass of electric in-wheel vehicles, which deteriorates the ride comfort and safety of vehicles and which can be effectively improved by optimizing the main suspension parameters of vehicles reasonably, so a multi-objective optimization method of main suspension parameters based on adaptive particle swarm algorithm is proposed and the dynamic model of a half in-wheel electric vehicle is established. Taking the stiffness coefficient of the suspension damping spring and damping coefficient of the damper as independent variables, the vertical acceleration of the body, the pitch acceleration and the vertical impact force of the hub motor as optimization variables, and the dynamic deflection of the suspension and the dynamic load of the wheel as constraint variables, the multi-objective optimization function is constructed, and the parameters are simulated and optimized under the compound pavement. The simulation results show that the vertical acceleration and pitch acceleration are reduced by 20.2% and 18.4% respectively, the vertical impact force of the front hub motor is reduced by 3.7%, and the ride comfort and safety are significantly improved.

Keywords

In-wheel electric vehicle suspension multi-objective optimization adaptive particle swarm algorithm

1. Introduction

With the rapid increase of car ownership, environmental pollution, energy shortage and other issues are becoming serious increasingly. The development of electric vehicles has become an effective way to achieve energy conservation and reduce environmental pollution [1, 2, 3]. As a kind of pure electric vehicle, in-wheel electric vehicle is also called hub motor drive vehicle or distributed electric drive vehicle. Its main structural feature is that the hub motor is directly installed in wheel, and each electric wheel acts as an independent driving and braking unit [4, 5, 6]. Compared with the traditional fuel vehicle and the centralized drive electric vehicle, the in-wheel electric vehicle chassis structure is simplified greatly and realized the lightweight of the whole vehicle due to the removal of mechanical transmission components such as transmission and differential.

However, the wheel hub motor is connected with the wheel rim rigidly in the conventional electric wheels, which increases the vehicle’s unsprung mass significantly, and has a serious negative effect on the vertical vibration of the vehicle [7, 8, 9, 10, 11], which is mainly reflected in: (1) the vertical vibration is deteriorated, the ride comfort is reduced; (2) the dynamic load of the wheel is increased, which leading to the road companions poor, and reducing the driving safety of the whole vehicle; (3) the vertical impact load produced by uneven road surface on wheel directly acts on the hub motor, which reduces the working performance and service life of the motor. Therefore, hub motor reduces the ride comfort and driving safety of in-wheel electric vehicles directly.

The suspension, as the transmission connection between the body and the wheel, buffers the vertical excitation of the road. The vertical vibration transmitted from road to electric wheel or vehicle body can be effectively reduced by properly matching the damping spring stiffness and damping coefficient of suspension. At present, the mainstream optimization algorithms contain particle swarm optimization, genetic algorithm, pattern search optimization function in vehicle main suspension parameter optimization [12, 13, 14, 15, 16]. However, most of the studies only optimize one of index such as the vertical acceleration of the body and the dynamic load of the wheel and others, and lack the multi-objective optimization of the overall index. In addition, most of the parameters optimization process is only carried out under the specific road surface level by randomly, and the comprehensive evaluation of vehicle performance under different road surface level by randomly is lacked.

Therefore, a multi-objective parameter optimization method of main suspension parameters based on adaptive particle swarm algorithm is proposed and the dynamic model of a half in-wheel electric vehicle is established, which takes the stiffness coefficient of the damping spring and the damping coefficient of the damper as independent variables, the vertical acceleration, the pitch acceleration and the vertical impact force of the hub motor as optimization variables, and the dynamic deflection of the suspension and the dynamic load of the wheel as constraint variables, the parameters are simulated and optimized under the composite pavement composed by different level road randomly.

2. Dynamic model of one-half in-wheel electric vehicle

In order to facilitate the research, the following assumptions are made in modelling the in-wheel electric vehicles [17]: the vertical movement, wheel and wheel hub motor are only considered and the vibration is linear; the vehicle is symmetrical and the longitudinal axis is axis of symmetry, and the road roughness function of the left and right wheels is equal; the distribution mass of all suspension systems is equal; the vertical stiffness of all tires is equal; The body pitch angle changes in a small range. Then the simplified dynamic model of the half in-wheel electric vehicle is shown in Fig. 1.

Figure 1.

A half dynamic model of the in-wheel electric vehicle.

In Fig. 1, $m_{a}$ is 1/2 body mass, $m_{b}$ is 1/4 body mass, and $m_{a}=2m_{b}$ ; $m_{t}$ is the wheel mass, $m_{e}$ is the hub motor mass, $K_{S1}/K_{S2}$ is the stiffness coefficient of front/rear suspension damping spring, $C_{S1}/C_{S2}$ is the damping coefficient of front/rear suspension damper, $K_{t}$ is the vertical stiffness of the tyre, $x_{11}/x_{12}$ is the vertical displacement of the front/rear electric wheel, $x_{21}/x_{22}$ is the vertical displacement of the front/rear 1/4 body respectively, $x_{2}$ is the vertical displacement at the center of mass, $x_{q1}/x_{q2}$ is the vertical displacement excitation of the road surface on the front/rear electric wheels, $\varphi$ is the pitch angle of the body, $l$ is the wheelbase, $l_{f}/l_{r}$ is the distance between the vehicle centroid and the front/rear axle, and $l=l_{f}+l_{r}$ .

Assuming the moment inertia of vehicle pitching motion is ${\bm{I}}$ , the linear vibration equation of the model can be obtained from Newton’s second law as follows:

$\displaystyle m_{a}\ddot{x}_{2}+K_{s1}(x_{21}-x_{11})+K_{s2}(x_{22}-x_{12})+C_% {s1}(\dot{x}_{21}-\dot{x}_{11})+C_{s2}(\dot{x}_{22}-\dot{x}_{12})=0$ (1) $\displaystyle(m_{t}+m_{s})\ddot{x}_{11}+K_{s1}(x_{11}-x_{21})+C_{s1}(\dot{x}_{% 11}-\dot{x}_{21})+K_{t}(x_{11}-x_{q1})=0$ (2) $\displaystyle(m_{t}+m_{s})\ddot{x}_{12}+K_{s2}(x_{12}-x_{22})+C_{s2}(\dot{x}_{% 12}-\dot{x}_{22})+K_{t}(x_{12}-x_{q2})=0$ (3) $\displaystyle I\ddot{\varphi}+K_{s1}(x_{11}-x_{21})l_{f}+K_{s2}(x_{22}-x_{12})% l_{r}+C_{s1}(\dot{x}_{11}-\dot{x}_{21})l_{f}+C_{s2}(\dot{x}_{22}-\dot{x}_{12})% l_{r}=0$ (4) $\displaystyle x_{2}-l_{f}\varphi-x_{21}=0\qquad x_{2}+l_{r}\varphi-x_{22}=0$ (5)

3. Adaptive particle swarm optimization algorithm

Adaptive particle swarm optimization (APSO algorithm) is a kind of algorithm based on the social behavior of individuals and groups [18, 19, 20, 21], which has the advantages of simple programming, fast optimization speed and significant optimization effect. In this algorithm, the particle’s own state is represented by a set of specific position and velocity vectors, which respectively represent the feasible solution of the problem and its search direction in the search space. The update formula of speed and position for each particle is as follows:

$\displaystyle v_{ij}^{t+1}=w_{t}\cdot v_{ij}^{t}+r_{1}\cdot c_{1}\cdot(\textit% {pbest}_{ij}^{t}-x_{ij}^{t})+r_{2}\cdot c_{2}\cdot(\textit{gbest}_{j}^{t}-x_{% ij}^{t})$ (6) $\displaystyle x_{ij}^{t+1}=x_{ij}^{t}+v_{ij}^{t+1}$ (7)

Where, $i$ is the number of particles, $j$ is the number of optimization variables, and $v^{t}_{ij}$ and $x^{t}_{ij}$ represents the speed and position of the $j$ -th dimension optimization variables of particle $i$ in the $t$ -th iteration respectively $\textit{pbest}^{t}_{ij}$ , and $\textit{pbest}^{t}_{j}$ represent the optimal position of the $j$ -th optimization variable of the $i$ -th particle and the $j$ -th optimization variable of the whole population in the $t$ -th iteration. $w$ is inertia weight, $c_{1}$ and $c_{2}$ are cognitive learning factor and social learning factor, $r_{1}$ and $r_{2}$ are random number between [0,1].

In order to avoid the shortcomings of the traditional PSO algorithm that the iterative process is easy to fall into the local optimization, the APSO algorithm adopts the adaptive inertia weight updating method, and the updating formula is shown in Eq. (8). This method can not only enhance the global optimization ability, but also improve the local search ability of particles near the optimal solution, and accelerate the convergence speed [22].

$\displaystyle\left\{\begin{array}[]{ll}w_{t}=w_{\textit{min}}+\frac{(f_{i}-f_{% \textit{min}})}{(\bar{f}_{\textit{vag}}-f_{\textit{min}})}\cdot(w_{\textit{max% }}-w_{\textit{min}})&f_{i}\leqslant\bar{f}_{\textit{vag}}\\ w_{t}=w_{\textit{max}}&\textit{else}\end{array}\right.$ (8)

Where, $w_{t}$ is the inertia weight value at the $t$ -th iteration, $w_{\textit{min}}$ , $w_{\textit{max}}$ is the minimum value and maximum value of inertia weight respectively, $f_{i}$ is the objective function value of the particle number $i$ in the $t$ -th iteration, $f_{\textit{min}}$ , $f_{\textit{max}}$ , $\bar{f}_{\textit{avg}}$ are the minimum, maximum and average value of the objective function of all particles in the $t$ -th iteration.

Then, the suspension parameters of the in-wheel electric vehicle optimization flow chart based on the adaptive particle swarm algorithm is shown in Fig. 2.

Figure 2.

Block on suspension parameter optimization using APSO algorithm.

4. Multi-objective parameter optimization of the main suspension in-wheel electric vehicles

4.1 Objective function constructed

Taking the vertical acceleration, the acceleration of the pitch angle and the vertical impact force of the hub motor as the optimization variables, the dynamic deflection of the suspension and the dynamic load of the wheel as the constraint variables, and the stiffness coefficient of the damping spring and the damping coefficient of the damper in the main suspension as the parameters to be matched, the objective function is constructed as follows:

$\displaystyle F_{1}(K_{si},C_{si})=q_{1}\frac{\bar{a}_{2}}{\bar{a}_{1}}+q_{2}% \frac{\bar{p}_{2}}{\bar{p}_{1}}+q_{3}\sum\limits_{j=1}^{2}{\frac{\bar{F}_{e2j}% }{\bar{F}_{e1j}}}$ (9) $\displaystyle\min F(K_{si},C_{si})=\sum\limits_{k=1}^{3}{\min F_{1}(K_{si},C_{% si})}$ (10)

Where, $F_{1}(K_{si},C_{si})$ is the objective function value under single pavement condition, $F(K_{si},C_{si})$ is the objective function value under the condition of composite pavement, $i=$ 1, 2 represents the front/rear suspension, $j=$ 1, 2 represents the front/rear electric wheel, $\bar{a}_{2}$ , $\bar{p}_{2}$ , $F_{e2j}$ represents the RMS (Root mean square) values of the vertical acceleration, pitch acceleration and motor vertical impact force under the current matching parameters, $\bar{a}_{1}$ , $\bar{p}_{1}$ , $F_{e1j}$ represents the RMS(Root mean square) values of the vertical acceleration, pitch acceleration and motor vertical impact force under the reference matching parameter ( $\bar{K}_{si},\bar{C}_{si}$ ). In the objective function relationship, the quotient under each index between the two is used to achieve dimensional unification, $q_{1}$ , $q_{2}$ and $q_{3}$ are the weight coefficients of each optimization variable. $k$ represents the driving condition of the vehicle when the parameters are optimized. $k=$ 1, 2, 3 represent the parameter optimization under the random road surface of level B, level C and level D respectively.

4.2 Constraint established

In order to ensure the adhesion performance and handling stability, the optimized suspension parameters shall meet the requirements that the probability of the wheel leaving the road is less than 0.15%, and the probability of the suspension hitting the stopper is less than 0.3%. Therefore, the constraint conditions are established as follows [23]:

$\displaystyle\bar{F}_{w2j}\leqslant\frac{1}{3}\cdot(m_{b}+m_{t}+m_{e})\cdot g% \quad\bar{f}_{d2i}\leqslant\frac{1}{3}\cdot[f_{d}]$ (11)

Where, $F_{w2j}$ is the RMS value of tire dynamic load, $\bar{f}_{d2i}$ is the RMS value of suspension dynamic deflection, [ $f_{d}$ ] is the allowable value of dynamic deflection of suspension, taking 80 mm; $g$ is the acceleration of gravity, taking 9.8 m/s ${}^{2}$ .

At the same time, in order to ensure that the vehicle has ride comfort, the bias frequency and damping ratio of the suspension are properly constrained, and the limiting conditions are as follows [24]:

$\displaystyle\left\{{\begin{array}[]{l}\xi_{si}(K_{si},C_{si})=\frac{C_{si}}{2% \sqrt{m_{b}\cdot K_{si}}}\in[0.2,0.45]\\ \omega_{si}(K_{si})=\frac{1}{2\pi}\sqrt{\frac{K_{si}}{m_{b}}}\in[1,1.5]\\ \end{array}}\right.$ (12)

Where, $\xi_{si}$ and $\omega_{si}$ represent the relative damping coefficient and the natural frequency of the sprung mass respectively.

Table 1

Initial simulation parameters

Parameter	Value	Parameter	Value
$m_{a}$ (kg)	630	$I$ (kg $\cdot m^{2})$	1783
$m_{t}$ (kg)	35	$c_{1}/c_{2}$	1.5
$m_{e}$ (kg)	25	$w_{\textit{min}}$	0.4
$K_{t}$ (N/m)	217750	$w_{\textit{max}}$	0.9
$l_{f}$ (m)	1.035	Number of iterations $k$	100
$l_{r}$ (m)	1.655	Population size $m$	30
$K_{S1}/K_{S2}$ (N/m)	[15000, 25000]	$v$ (km/h)	60
$C_{S1}/C_{S2}$ (N $\cdot$ s/m)	[1000, 2500]	$G_{q}$ (n ${}_{0}$ ) – B level road	64 $\times$ 10 ${}^{-6}$
$\bar{K}_{S1}/\bar{K}_{S2}$ (N/m)	15000/25000	$G_{q}$ (n ${}_{0}$ ) – C level road	256 $\times$ 10 ${}^{-6}$
$\bar{C}_{S1}/\bar{C}_{S2}$ (N $\cdot$ s/m)	1000/2000	$G_{p}$ (n ${}_{0}$ ) – D level road	1024 $\times$ 10 ${}^{-6}$

Table 2

Parameter optimization results

Parameter	Before optimizations	After optimization	Optimization range
$K_{S1}$ (N/m)	15000	17935	–
$K_{S2}$ (N/m)	25000	17982	–
$C_{S1}$ (N $\cdot$ s/m)	1000	1093	–
$C_{S2}$ (N $\cdot$ s/m)	2000	1070	–
Vertical acceleration ( $m/s^{2}$ )	0.470	0.375	20.2%
Pitch acceleration (rad/s ${}^{2}$ )	0.266	0.217	18.4%
Vertical impact force of front hub motor (N)	208.84	201.14	3.7%

Figure 3.

Comparison of the body vertical acceleration in time domain.

Figure 4.

Comparison of body pitch acceleration in time domain.

5. Simulation analyses

The parameter optimization condition is that the vehicle drives at constant speed on the compound road composed of B-level, C-level and D-level roads by randomly, the road excitation model is calculated by filtering white noise, and the rear wheel road excitation is obtained by fixed time delay of the front wheel excited. The road excitation model can be established as follows:

$\displaystyle\left({{\begin{array}[]{l}{\dot{x}_{q1}(t)=-2\pi f_{0}\cdot x_{q1% }(t)+2\pi\sqrt{G_{q}(n_{o})\nu}\cdot w(t)}\\ {x_{q2}(t)=x_{q1}(t-\tau)}\\ \end{array}}}\right.$ (13)

Where, $x_{q1}(t)$ and $x_{q2}(t)$ is the road excitation of front and rear respectively; $\tau$ is the fixed time delay and satisfies $\tau=l/v$ , $v$ is the driving speed (km/h); $f_{0}$ is the lower cut-off frequency, taken as 0.1; $w(t)$ is white noise distributed evenly that the average is 0 and the intensity is 1; $G_{q}$ ( $n_{0}$ ) is the road roughness coefficient.

The weight coefficients of each index are $q_{1}=$ 5, $q_{2}=$ 5, $q_{3}=$ 1 by comprehensive considering that the vertical acceleration and pitching acceleration optimizing is primary, the vertical impact force of the hub motor optimizing is secondary. The above optimization method is used for multi-objective optimization of suspension parameters. The basic parameters of simulation model and the setting parameters of APSO algorithm are shown in Table 1.

The statistical results are shown in Table 2 after parameter optimization. The time-domain comparison curves of vertical acceleration, pitch acceleration and vertical impact force of front hub motor of the vehicle model before and after optimization are shown in Figs 3–5 respectively.

Figure 5.

Comparison of vertical impulse force of in-wheel motor in time domain.

It can be seen from the Fig. 3 to 5 that after the optimization of suspension parameters, the vertical acceleration and pitching angle acceleration of the in-wheel electric vehicle are optimized by 20.2% and 18.4% respectively, the vertical impact force of the front hub motor is optimized by 3.7%, and the comprehensive performance of the vehicle is significantly improved.

6. Conclusions

The multi-objective optimization function is constructed by taking the vertical acceleration, the pitch acceleration and the vertical impact force of the hub motor as the optimization variables, and the APSO algorithm is used to realize the reasonable optimization of the stiffness coefficient and the damping coefficient of the damping spring of the suspension of the in-wheel electric vehicle on the compound road. The simulation results show that the vertical acceleration and pitch acceleration are improved by 20.2% and 18.4% respectively, the vertical impact force of the front hub motor is improved by 3.7%, and the comprehensive performance of the vehicle is improved significantly, which proves that the multi-objective optimization of the main suspension parameters based on the APSO algorithm is effective.

Footnotes

Acknowledgments

The authors acknowledge the Henan Province Science and Technology Project (182102210508) for financial support.

References

Alimujiang

Jiang

Dong

H.J.

et al., Synergy and co-benefits of reducing CO2 and air pollutant emissions by promoting new energy vehicles: A case of Shanghai, Acta Scientiae Circumstantiae 40(5) (2020), 1873–1883.

Fan

J.Y.

and Xing

Y.Y.

, Influence of vehicle speed and acceleration different exhaust gas emission, Journal of Wuhan University of Technology (Transportation Science & Engineering) 43(5) (2019), 795–799.

Ingeborg

Q.W.

Warland

et al., Optimal planning of the Nordic transmission system with 100% electric vehicle penetration of passenger cars by 2050, Energy 107 (2016), 648–660.

Chu

W.B.

Luo

Y.G.

Han

Y.W.

et al., Rule-based traction system failure control of distributed electric drive vehicle, Journal of Mechanical Engineering 48(10) (2012), 90–95, 102.

Liu

L.J.

Shi

Yuan

X.F.

et al., Multiple model-based fault-tolerant control system for distributed drive electric vehicle, Journal of the Brazilian Society of Mechanical Sciences and Engineering 41(11) (2019), 531–540.

Z.P.

Feng

and Xiong

, Review on vehicle dynamics control of distributed drive electric vehicle, Journal of Mechanical Engineering 49(8) (2013), 105–114.

Shao

X.X.

Naghdy

and Du

H.P.

, Reliable fuzzy H-infinity control for active suspension of in-wheel motor driven electric vehicles with dynamic damping, Mechanical Systems and Signal Processing (87) (2017), 365–383.

Tan

and Ren

C.B.

, Optimal matching between the suspension and the rubber bushing of the in-wheel motor system, Journal of Automobile Engineering 229(6) (2015), 758–769.

Dong

H.X.

Guo

J.G.

and Yan

K.K.

, Control strategy of regenerative braking for four-wheel-drive electric vehicles of in-wheel-motor, Mechanical Science and Technology for Aerospace Engineering 36(11) (2017), 1778–1784.

10.

X.G.

Zhang

and Chen

, Influences of driving system on the shimmy of four in-wheel motors independent drive electric vehicles, Noise and Vibration Control 39(6) (2019), 76–82.

11.

Zhou

Jia

Jing

et al., Coordinated longitudinal and lateral motion control for four wheel independent motor-drive electric vehicle, IEEE Transactions on Vehicular Technology 67(5) (2018), 3782–3790.

12.

Liu

M.C.

F.H.

and Zhang

Y.Z.

, Ride comfort optimization of in-wheel motor electric vehicles with in-wheel vibration absorbers, Energies 10(10) (2017), 1647.

13.

Liu

W.L.

Yang

Wang

P.P.

et al., Metamodel-based robust collaborative optimization for the suspension parameters of rail vehicles, Journal of the Chinese Institute of Engineers, Transactions of the Chinese Institute of Engineers, Series A 42(8) (2019), 643–652.

14.

Gao

Feng

J.Z.

and Zheng

S.L.

, Optimization design of the key parameters of McPherson suspension systems using generalized multi-dimension adaptive learning particle swarm optimization, Proceedings of the Institution of Mechanical Engineers, Part D: Journal of Automobile Engineering 233(13) (2019), 3403–3423.

15.

D.M.

Xiang

C.L.

Tao

et al., Kinematic analysis and optimization of key parameters of suspension for electric differential steering wheeled amphibious vehicle, Acta Armamentarii 40(8) (2019), 1580–1586.

16.

Chen

Y.J.

Zhang

L.M.

et al., Dynamic optimization design of the suspension parameters of car body-mounted equipment via analytical target cascading, Journal of Mechanical Science and Technology 34(5) (2020), 1957–1969.

17.

Cui

S.M.

, Vehicle system dynamics and simulation, Beijing: Peking University Press, 2014, pp. 28–32.

18.

Agarwallaa

and Mukhopadhyay

, Efficient player selection strategy based diversified particle swarm optimization algorithm for global optimization, Information Sciences 397 (2017), 69–90.

19.

Chen

Fan

Y.R.

and Deng

S.G.

, Adaptive particle swarm optimization algorithm with dynamic acceleration factor, Journal of China University of Petroleum (Edition of Natural Science) 34(6) (2010), 173–176, 184.

20.

Ali

N.A.

Han

and Ling

Q.H.

, An improved hybrid self-inertia weight adaptive particle swarm optimization algorithm with local search, Engineering Optimization 51(7) (2019), 1115–1132.

21.

Yin

Z.G.

Zhang

Y.P.

et al., Research on active disturbance rejection control with parameter autotune mechanism for induction motors based on adaptive particle swarm optimization algorithm with dynamic inertia weight, IEEE Transactions on Power Electronics 34(3) (2019), 2841–2855.

22.

Chinmaya

Biplab

K.P.

and Parhi Dayal

, An intelligent path planning approach for humanoid robots using adaptive particle swarm optimization, International Journal on Artificial Intelligence Tools 27(5) (2018), 1850015.

23.

Z.S.

, Automobile theory, Beijing: Beijing Machine Press, 2015, pp. 38–45.

24.

Liu

M.C.

F.H.

and Huang

J.H.

, Integration design and optimization control of a dynamic vibration absorber for electric wheels with in-wheel motor, Energies 10(12) (2017), 2069.