Applying machine learning for cars’ semi-active air suspension under soft and rigid roads

Xu Shaoyong Zhang Jianrun Nguyen Van Liem,

(1 School of Mechanical and Electrical Engineering, Hubei Polytechnic University, Huangshi 435003, China)(1 Hubei Key Laboratory of Intelligent Conveying Technology and Device, Hubei Polytechnic University, Huangshi 435003, China)(2 School of Mechanical Engineering, Southeast University, Nanjing 211189, China)

Abstract：To improve the ride quality and enhance the control efficiency of cars’ semi-active air suspensions (SASs) under various surfaces of soft and rigid roads, a machine learning (ML) method is proposed based on the optimized rules of the fuzzy control (FC) method and car dynamic model for application in SASs. The root-mean-square (RMS) acceleration of the driver’s seat and car’s pitch angle are chosen as the objective functions. The results indicate that a soft surface obviously influences a car’s ride quality, particularly when it is traveling at a high-velocity range of over 72 km/h. Using the ML method, the car’s ride quality is improved as compared to those of FC and without control under different simulation conditions. In particular, compared with those cars without control, the RMS acceleration of the driver’s seat and car’s pitch angle using the ML method are respectively reduced by 30.20% and 19.95% on the soft road and 34.36% and 21.66% on the rigid road. In addition, to optimize the ML efficiency, its learning data need to be updated under all various operating conditions of cars.

Key words：semi-active air suspension; ride quality; machine learning; fuzzy control; genetic algorithm

Existing studies show that the air suspension system using air springs improves cars’ ride quality better than the traditional suspension system using steel springs[1-2]. Semi-active air suspensions (SASs) use the fuzzy control (FC) and Hinfcontrol methods[3-4]to ameliorate cars’ ride quality. Investigations indicated that SASs controlled by the FC significantly ameliorated cars’ ride quality when compared to the passive suspension system. Moreover, the investigations showed that the efficiency of the FC is greatly affected by its control rules, which is also considered its disadvantage. To enhance the efficiency of the FC, its control rules were then optimized based on the genetic algorithm (GA)[3, 5-6]. The study results indicated that the efficiency of an FC used with optimized control rules is better than that of an FC without optimized control rules. However, the harmonic excitation or random excitation of rigid roads was mainly applied to evaluate the efficiency of the FC and cars’ ride comfort in the above studies.

The SAS efficiency controlled by an FC with its optimized control rules under random surfaces of the ISO levels A, B, C, D, and E of a rigid road[7]was investigated[3, 8]. The studies indicated that an FC with optimized control rules was only effective under each excitation of the ISO level A, B, C, D, or E of a rigid road surface. The control efficiency of an FC significantly decreased when the vehicle was moving along random surfaces of a rigid road changed in a large range. In addition, with the deformable surfaces of a soft road, its deformation was also changed in a large range when the vehicle was moving on the soft road[9-10]. However, the influence of soft roads on cars’ ride quality and SAS efficiency is less studied. Moreover, some studies on the elastic tire-soft road interaction indicated that vibration sources under an elastic tire are not only generated by a random surface but also by the deformable terrain of a soft road. Therefore, cars’ ride quality is strongly influenced by the vibration sources of soft roads[9, 11]. Hence, the FC efficiency and its control rule optimization can also be affected by soft road surfaces. However, this issue has not yet been concerned in existing studies.

A machine learning (ML) method is being investigated and used in adaptive controls. Based on the desired input data and output data of a machine system and an FC, a self-learning algorithm program[12-13]could be developed to control cars’ suspension systems under various simulation conditions. This topic is currently of particular interest to scholars. Thus, based on cars’ dynamic model and the control rules of an FC optimized by a GA on soft and rigid roads, an ML program was investigated and developed for optimizing SAS efficiency and cars’ ride quality. The root-mean-square (RMS) acceleration responses of the driver’s seat (awz1) and car’s pitch angle (awφ2) were chosen as the objective functions. The goal of the study is to enhance cars’ ride quality under soft and rigid roads.

1 Mathematical Approaches

1.1 Car dynamic model

A car dynamic model with its suspension system used by an SAS was established to control it via ML, as shown in Fig. 1. In the figure,ziandmiare the vertical displacements and masses of the driver’s seat, car body, and axles, respectively;φ2is the angular displacement of the car body;c1,ct1,2, andkt1,2are the damping and stiffness values of the seat suspension and wheels;q1,2are the excitations at the front and rear wheels;ljis the distances of the car; andv0is the car’s moving velocity (i=1, 2, 3, 4;j=1, 2, 3).

Fig.1 Lumped model of the car

To simplify the computation process of the car’s motion equations, some assumptions are given as follows: 1) The car body is absolutely stiff, and its angular deformation is very small and hence ignored. 2) The displacements of the seat, car body, and axles around their equilibrium position are very small. 3) The vibration excitation is mainly in the vertical direction, and the longitudinal and horizontal excitations are very small and hence ignored. Based on the car’s dynamic model in Fig. 1 and Newton’s second law of motion, the general dynamic differential equation for the car is given by

(1)

whereFsis the vertical force of the driver’s seat suspension system, written as

(2)

F1andF2are the vertical forces of the front and rear SASs, respectively.Ft1andFt2are the vertical forces of the front and rear wheels, respectively.

1.2 Dynamic model of the SAS

To evaluate the ML efficiency in controlling the car’s SAS, the SAS used by an airbag spring and an active damper controlled by the FC were applied. The SAS simple structure includes a bag and reservoir connected by a surge pipe, as described in Fig. 2(a), whereVb,pb, andAeare the volume, pressure, and effective area of the airbag, respectively;As,ms, andlsare the cross-sectional area, air mass, and length of the pipe, respectively;Vrandprare the volume and pressure of the reservoir, respectively;zis the deformation of the airbag; andυsis the air displacement of the air in the pipe.

(a)

(b)

Some assumptions of the SAS model are given as follows: 1) the friction of the airbag’s material is very small; 2) the inertia force of the air mass in the SAS is also very small and neglected; and 3) the vibration excitation of the SAS model is mainly in the vertical direction.

The force balance betweenFand the restoring force of the airbag in Fig. 2(a) is expressed as

F=Ae(pb-pa)

(3)

wherepais the atmospheric pressure.

Based on the correlation among the reservoir, pipe, and airbag,pbis computed by the air mass flow rate into the airbag as follows:

(4)

The relationship ofρbandpbwas calculated based on the isentropic process in the airbag as follows[2]:

(5)

whereρ0bis the air density at the initial state and is determined byρ0b=p0/(RT0) andλis the polytropic constant,λ=1.4[3, 8].

(6)

The air mass flow rate into the reservoir is written as

(7)

(8)

(9)

The mass flow rate in the pipe is also expressed as

(10)

In addition, the mass flow rate of the air in the pipe was affected byAs,pr, andpbof the air suspension system[3]. Thus, the moving air mass in the pipe is written as[3]

(11)

By combining Eqs. (10) and (11), the mass flow rate in the pipe is expressed as

(12)

By combining Eqs. (6), (9), and (12),pbandFin Eq. (3) can be determined.

Based on the derivative ofF/z, the SAS nonlinear dynamic stiffness can be calculated as

(13)

(14)

By substituting Eq. (14) into Eq.(13),kis then rewritten as

(15)

The SAS dynamic model is described in Fig. 2(b).

From the car and SAS models given in Figs. 1 and 2(b), the vertical forces of the front/rear SAS are expressed as

k1(zi+1-z2+φ2li+1)i=1,2

(16)

wherecctrliis the damping values of the front/rear SAS.

1.3 Models of the wheel-road surface contact

1.3.1 Contact model of the wheel-soft road surface

When the car is traveling on the soft road surface, the elastic wheel interacts with the soft road. A dynamic model of the wheel-soft road interaction with the surface roughness of the soft roadqis built in Fig. 3(a). Under the effect of the dynamic and static loads of the wheel on the soft road surface, the road surface is sunk byzoa. Two deformable regions of the soft road surface-wheel (bob′ region) and soft road surface (b′aregion) appeared, as shown in Fig. 3(a).zx,z0, andzaare the sinkage of the soft road, static deformation of the soft road, and axle displacement, respectively, andr,Ft, andmtare the radius, dynamic force, and mass of the wheel, respectively.

(a)

(b)

Assuming that the contact lengths ofbob′ andb′ain the horizontal direction arel1andl2, respectively, the pressurepgand shear stressτgof the soft road generated in the contact lengthsl1andl2are described by the reaction force of the soft road on the elastic wheel as follows[9]:

τg=(c+pgtanφ)(1-e-j/K)

(17)

It is assumed thatlis the average roughness line of the soft road surface, sozxcan be determined by

zx=zoa+q+zoa-Δ=za-z0+q+Ψ-r

(18)

The force responses at the front and rear axles of the elastic wheel are determined by

(19)

wheregis the gravitational acceleration andi=1, 2.

To determineqin Eq. (18), the power spectrum densityG(w0) of the soft road surface roughness was calculated as[3, 10]

(20)

wherew0= 0.1 m is the reference frequency.

Based on the white noise signalWandG(f),qis then determined by[14]

(21)

Mitschke[10]presented the spectral densities of soft road classifications, including good, medium, and poor soil surfaces, as provided in Tab. 1. Thus,qis generated using a parameter in these spectral densities.

Tab.1 Parameters of the soft road surface

1.3.2 Contact model of the wheel-rigid road surface

When a car is traveling on a rigid road surface, the surface greatly affects the car’s ride quality. Here, the point-contact model of the rigid road-wheel interaction was used to calculate the force response of the wheel[5, 14-15]. To calculate the force response of the wheel, the point-contact model of the rigid road surface-wheel interaction is also established in Fig. 3(b).

The vertical forces of the front and rear wheels (Fti) are written as

(22)

qof the rigid road surface is computed in Eq. (21) withG(w0) determined according to ISO-8068[7].

1.4 Evaluating index

To estimate the driver’s ride quality and the efficiency of isolation systems, the index of the RMS accelerations of the seat calculated according to ISO 2631-1[16]was used[6, 8, 14]. Based on the car’s dynamic model and ISO 2631-1, the RMS accelerations of the driver’s seat and car’s pitch angle are written as

(23)

whereχ={z1,φ2};aχ(t) is the acceleration inκ; andTis the simulation time.

To assess SAS efficiency controlled by the ML model on enhancing the car’s ride quality under various surfaces of soft and rigid roads, the decrease inawz1andawφ2was chosen as the evaluation index.

2 SAS Control Algorithm

The FC efficiency greatly depends on its control rules. The control rules of the FC are also optimized by the GA to enhance its efficiency[6]. However, the optimized control rules also strongly depend on the conditions of the road surface and car speed. The FC efficiency could be reduced when the high roughness of the road surface is changed in a large range. To overcome this shortcoming, based on the various surfaces of a soft road, including good, medium, and poor soil surfaces[10], and various surfaces of a rigid road, including level A, level B, and level C, according to the ISO 8068[7], each control rule of a road surface was optimized. Then, a data map of the FC control rules optimized under all the soft and rigid roads was applied for the ML model to controlcctrl.

2.1 Optimizing the FC control rules

Tab.2 LV input/output and value of the FC

…

To enhance the FC efficiency, the GA was then applied to optimize the “if-then” rule as follows.

To optimize the FC control rules and establish a data map for the ML model under soft and rigid roads, the values ofδandG(w0) of the soft road surfaces in Tab. 1,f<10 Hz, andv0=54 km/h were applied to build the various surfaces of the soft road. Moreover, the rigid road surfaces of levels A (good), B (medium), and C (poor) ofG(w0),δ=2,f<10 Hz, andv0=54 km according to ISO-8068[7]were applied to build the various surfaces of the rigid road. The simulation results of the soft and rigid roads with various surfaces are plotted in Figs. 4(a) and (b).

(a)

(b)

Based on the different vibration excitations ofqin Fig. 4, initial control rules of the FC in Tab. 2, and car’s dynamic parameters in Tab. 3, an algorithm program was then built to optimize the FC control rules, as plotted in Fig. 5 with “FC.” From the optimized control rules of the FC, three different types of data map, namely, 1) using the good, medium, and poor soil surfaces of the soft road; 2) using the surface of ISO levels A, B, and C of the rigid road; and 3) using the soft and rigid road surfaces, are plotted in Fig. 6.

Tab.3 Dynamic parameters of the car and SAS

Fig.5 SAS control model combined with the FC and ML

(a)

(b)

(c)

2.2 Control of ML from the FC control rules

The FC efficiency optimized by the GA obtains the maximum value when the car is moving on a type of road surface. However, under a type of the actual road surface, the good, medium, and poor surfaces of the soft road or surfaces of ISO levels A, B, and C of the rigid road can concurrently appear on each surface of the soft and rigid roads when the car is moving. Thus, the optimized FC efficiency is reduced. Thus, from the data map of the FC control rules optimized on all the surfaces of the soft and rigid roads in Fig.6, neuro-adaptive learning in an adaptive neuro-fuzzy inference system (ANFIS)[12-13]was used to optimize the control efficiency of the SAS under various conditions of the car. The model of neuro-adaptive learning is provided in Fig.5 and labeled with “ML.”

The input signalsnare defined by a vector ofX={a1,a2,…,an}T, and the output signalYof ML is computed by

(24)

whereγis the trigger function; ϑiis the weight ofai; andαis the neural activation threshold.

Based on the ANFIS tool in the MATLAB software, a neuro-fuzzy controller (NFC) was then designed to learn all the optimized control rules of the data map, as shown in Fig. 6. The data of the input/output values of “zand

(a)

(b)

(a)

(b)

Fig. 8 shows that the acceleration response of the driver’s seat using the ML method is similar to the training data of the FC when the car is moving on a medium surface of the soft road or a rigid road surface of ISO level B. The calculation results in Tab. 4 indicate that the value ofawz1is slightly decreased by 0.57% on the medium surface of the soft road and 0.22% on the ISO level B of the rigid road surface. There is a small error between the FC and ML, which is due to a small error in the learning process of the ML. Moreover, the value ofawz1with the SAS is smaller than that without control (WCtrl) by 17.86% on the medium surface of the soft road and 20.64% on the ISO level B of the rigid road surface. Hence, the car’s ride quality with the SAS is improved. To evaluate the efficiency of the ML for the SAS, various moving conditions of the car were simulated and assessed.

Tab.4 Seat’s RMS acceleration

3 Results and Analysis

3.1 Influence of soft road surfaces

A car mostly moves on rigid road surfaces, so its ride quality is mainly evaluated under the excitation of different rigid road surfaces[5-6, 8]. However, in the same cases, the car can also move on soft road surfaces of the soil or sand grounds. Thus, soft road surfaces also affect the car’s ride quality and control efficiency of the suspension systems. However, this issue has not been evaluated yet in existing studies. To clarify this issue, a soft road with lumped parameters ofn=1.01,kc=60 kN/mn+1,kφ=5 880 MN/mn+2,c=3.1 kPa, andφ=29.8o[17]of a medium surface was simulated and compared with the rigid road surface of ISO level B (medium level). The results of the acceleration responses and RMS values are provided in Fig. 9 and Tab. 5.

(a)

(b)

Tab.5 RMS accelerations under ISO level B of the rigid road and medium surface of the soft road

Fig. 9 indicates that under the same medium level of soft road and rigid road surfaces, the comparison results show that the accelerations of the vertical driver’s seat and car pitch angle on the soft road are higher than those on the rigid road. In particular,awz1andawφ2strongly increased by 17.60% and 74.60%, respectively. This outcome can be attributed to the effect of the deformable soil ground of the soft road under the impact of the static and dynamic loads of the wheels when the car is moving. Thus, the driver’s ride quality and car body’s shaking on the soft road were reduced in comparison with those on the rigid road. Hence, the soft road greatly reduces the driver’s ride quality and health in comparison with the rigid road.

3.2 ML efficiency for the SAS

Based on the ML result learned via the optimized FC rules, the ML efficiency has not been clearly demonstrated yet under the medium surface of a soft road or ISO level B of a rigid road in Section 2.2. Thus, a random road surface built from a combination of the poor-good-medium surfaces of the soft road, and a random road surface built from a combination of the ISO level C-level A-level B of the rigid road were applied to evaluate the ML efficiency as follows:

(25)

(26)

The acceleration responses of the driver’s seat and car’s pitch angle under the soft and rigid roads are shown in Figs. 10 and 11, respectively. Under the excitation of the poor-good-medium soil surface of the soft road, Fig. 10 shows that the accelerations of the driver’s seat and car’s body used in the ML model greatly decreased as compared to those of the FC and WCtrl.awz1andawφ2using the FC in Tab. 6 obviously reduced by 18.02% and 12.31% in comparison with those using the WCtrl, respectively, whereasawz1andawφ2using the ML model significantly decreased by 30.20% and 19.95% compared to those using the WCtrl, respectively. Thus, the control efficiency of the ML model is better than that of the FC.

(a)

(b)

Tab.6 Calculation results of the RMS accelerations

Similarly, under excitation of the ISO level C-level A-level B of a rigid road surface, Fig. 11 also shows that the accelerations of the driver’s seat and car’s body using the ML model are lower than those of the FC and WCtrl. The calculation results ofawz1andawφ2with the ML model in Tab. 6 decreased by 34.36% and 21.66% in comparison with those using the WCtrl and by 14.56% and 9.62% compared with those using the FC, respectively. Therefore, the SAS controlled by the ML model can better improve the car’s ride quality in comparison with the FC under various excitations of the soft and rigid roads.

(b)

3.3 ML efficiency under different velocities

To fully assess the ML efficiency, a speed range from 2.5 to 20 m/s was also simulated in three cases: Case 1—the ML was only used by the data map of the soft road, Case 2—the ML was only used by the data map of the rigid road, and Case 3—the ML was used by the data map of the soft and rigid roads, as shown in Fig. 6. Theawz1results of the three cases are simulated and plotted in Fig.12.

(a)

(b)

(c)

(d)

(e)

(f)

Fig. 12(a) shows that under the influence of the soft road,awz1quickly augmented, whereas Fig. 12(b) indicates that under the influence of the rigid road,awz1insignificantly augmented, especially from 15 to 20 m/s. This result implies that the car’s traveling velocity needs to be limited when the car is traveling on a soft road. With the SAS controlled by the ML and FC, the result ofawz1is greatly reduced compared to that in the WCtrl under the car’s different velocities on both the soft and rigid road surfaces. Therefore, the car’s ride quality is obviously ameliorated by the SAS.

In Case 1, with the ML model only using the data map of the soft road, Fig. 12(a) reveals thatawz1with the ML model is also significantly reduced in comparison with the FC under all the different velocities of the car traveling on the soft road. However, Fig. 12(b) reveals thatawz1with the ML model is insignificantly changed in comparison with the FC on the rigid road under the car’s different velocities. This result can be due to the ML learning process only learning the optimized control rules of the deformable surfaces of the soft road. Therefore, the ML control efficiency has been limited on the rigid road.

In Case 2, similarly, with the ML model only using the data map of the rigid road, the result ofawz1in Fig. 12(c) is unchanged on the soft road, whereas the result ofawz1in Fig. 12(d) is significantly decreased on the rigid road as compared to the result ofawz1controlled by the FC.

In Case 3, with the ML using the data map of the soft and rigid roads, both Figs. 12(e) and (f) indicate that the results ofawz1are obviously reduced compared to the FC under the car’s different velocities. Consequently, the car’s ride quality with the SAS controlled by the ML model is better than that of the FC. Concurrently, the ML control efficiency also depends on the learning data.

4 Conclusions

1) The deformable surface of soft roads greatly influences cars’ ride quality as compared to rigid roads under the same car simulation conditions. Therefore, cars’ velocities on soft roads need to be limited to assess their ride quality.

2) The ML efficiency for the SAS to assess a car’s ride quality is better than the efficiency of the FC and WCtrl under all the simulation conditions of the car. In particular,awz1andawφ2of the ML model were greatly reduced by 30.20% and 19.95% on the deformable surfaces and 34.36% and 21.66% on the rigid surfaces in comparison with the WCtrl, respectively.

3) The ML efficiency significantly depends on learning data. Thus, to optimize its efficiency, the map of the learning data for the ML model should be further expanded under various conditions.

4) The ML has not only learned the optimized control rules of the FC from the data map to further enhance the SAS control efficiency but also improves the car’s ride quality more than the FC under the combined different road surfaces.