# Rollover Control of AGV Combined with Differential Drive, Active Steering, and Centroid Adjustment under Slope Driving Condition

Authors: Wei Liu; Yue Yu; Yidong Wan
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

The centroid of the Automated Guided Vehicle (AGV) equipped with the load platform is high, and it is prone to rollover when driving sideways on a slope. To solve this problem, this study proposes an anti-rollover cooperative control strategy combining differential drive, active steering, and load platform. According to the characteristics of AGV, the dynamic coupling of each subsystem is analyzed. The expected yaw stability control torque is calculated based on optimal control. The active steering controller is designed based on the model predictive control. The expected roll control torque is solved by the proposed sliding mode variable structure control method to dynamically adjust the centroid. Evaluation of the overall control system is accomplished by simulations and experiments under different load conditions of the AGV in the lateral driving condition on a slope. Compared with the PI controller, the sideslip angle decreases by 14.62% and 18.24%, and the roll angle decreases by 12.38% and 14.93% under high and low load conditions, respectively. The error between the experimental and simulation results is within 7.8%. It shows that the proposed cooperative control strategy can improve the stability of AGV under different load conditions and reduce the rollover probability when the AGV is driving sideways on a slope. This research provides a theoretical and experimental basis for active safety control of AGVs.


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## Body

## 1. Introduction

Based on the characteristics that the driving wheel torque and speed can be controlled independently and accurately, the automated guided vehicle (AGV) has unique advantages in stability control, which meets the development needs of intelligent special vehicles in the future. When the AGV loaded with goods rolls over, it will cause the goods to fall or even damage the vehicle, so rollover is an extremely dangerous accident.Experts and scholars have conducted in-depth research on rollover control of traditional vehicles and electric vehicles and achieved relevant results. Ataei et al. [1] proposed a new rollover index (RI) and used the active front wheel steering system to prevent the vehicle from rollover. However, the active steering system will lose control when the tire lateral force reaches saturation. Zhao et al. [2] proposed a differential braking control strategy to improve lateral stability by applying braking pressure to the left and right wheels, but the active braking system may increase the braking distance during emergency braking and may interfere with the handling stability system. Qi et al. [3] proposed a comprehensive multi-factor evaluation index based on the energy method and realized anti-rollover control of distributed drive vehicles by differential drive. However, the uncertainty of the model is not considered. To suppress the influence of model uncertainty on the control system, Ji et al. [4] studied the lateral stability and anti-rollover control of vehicles based on adaptive radial function network and sliding mode control. Mehdizadeh et al. [5] proposed an improved sliding mode control system, which improved the vehicle handling performance through differential braking under the condition of road and model uncertainty. Bai et al. [6] and Wang et al. [7] proposed the method of adding yaw moment and improved the lateral stability of the vehicle by outputting the driving/braking torque of the target wheel in the lower controller. Active suspension can realize the optimal control of suspension performance under different driving conditions and is an ideal solution to enhance the dynamic performance of distributed drive vehicles. Jin et al. [8] analyzed the relationship between unsprung mass and rollover stability, and based on active suspension, improved the anti-rollover ability of vehicle when driving on uneven roads. Xia et al. [9, 10] designed a hydraulic support cylinder as the actuator and proposed an anti-rollover control strategy for counterbalance forklifts based on extension decision.To further improve the lateral stability of the vehicle, experts and scholars have adopted multi-system cooperative control for rollover control research. Yakub et al. [11] adopted the integrated control of active steering and differential braking. Mousavinejad et al. [12] integrated front-wheel active steering and direct yaw moment control to study the vehicle’s in-plane stability in the critical region and improve the vehicle’s transient response. However, body roll control is not considered. Zhang et al. [13] proposed the joint control of yaw and roll by a differential drive and active suspension system and decoupled control, which greatly improved the spatial stability of in-wheel motor-driven electric vehicles. Yim et al. [14, 15] proposed the synergy of differential braking and active suspension to control vehicle rollover, which effectively controlled the body roll angle and lateral acceleration. Chen et al. [16] proposed the control strategy of combining differential drive and differential braking and designed the coordinated control system of driving force distribution regulation and ESP differential braking regulation. In addition, the active anti-roll bar (AABB) can also control the body’s roll state. Chen et al. [17] further improved the anti-rollover ability of the vehicle using an active anti-roll bar combined with all-wheel differential braking. Considering the decoupling control effect of yaw and roll, Zhang et al. [18, 19] proposed a combined control method of integrated differential driving, active steering, and active suspension, which can effectively control the yaw and roll motion of 4-wheel-independently driven vehicles. Termous et al. [20] proposed the cooperative control strategy of active steering, differential braking, and active suspension to improve the stability, handling, and safety of distributed-drive vehicles. However, the damping adjustment range of the active suspension is not large and the response time is long, which needs to be improved in the future.At present, most studies focus on the rollover control of the vehicle under the condition of high-speed steering, and there are few studies that focus on the active adjustment of the rollover phenomenon caused by the change of its own centroid in the condition of slope lateral driving. Since the high centroid of the body when carrying goods, it is easy to cause rollover when the centroid changes. Due to the high centroid of AGV when loading goods, lateral instability occurs when the centroid position changes on the slope. At the same time, the change of the centroid position will lead to the nonlinear dynamic response of the entire vehicle system, and the dynamics in each direction are coupled with each other, which will further increase the rollover trend. The main contributions of this study are listed as follows:(1)
Considering the time-varying characteristics of the vehicle centroid when driving sideways on slopes, a control strategy of active adjustment of the centroid based on the load platform is proposed, differential drive and active steering are added for cooperative control.(2)
To improve the rollover stability of AGV, a cooperative controller is designed by using feedback optimal control, model predictive control, and sliding mode control, and verified by simulation and experiment under different load conditions.The rest of the study is organized as follows. The related models are presented in Section2. Section 3 is the coupling analysis between different subsystems. Section 4 gives a detailed description about the design of the cooperative controller. Its performance of control strategy is tested by MATLAB/Simulink-Carsim co-simulation and real test in Section 5. Section 6 gives the conclusions.


## 2. Vehicle Dynamic Modeling

### 2.1. Body Model

As shown in Figure1, the 7-DOF dynamics model includes longitudinal motion, lateral motion, roll motion, and four wheels’ rotations. Assumptions: ignore the changes in tire cornering characteristics of the left and right wheels due to load changes; ignore the effects of suspension roll stiffness and damping coefficient; and ignore the aerodynamic effects.Figure 1
Seven degrees of freedom vehicle dynamics model.


#### 2.1.1. Longitudinal Motion

(1)mV˙x−Vyγ=Fxfl+Fxfrcosδ−Fyfl+Fyfrsinδ+Fxrl+Fxrr,where m is the mass of AGV, Vx and Vy are the longitudinal and lateral velocities, γ is the yaw rate of the vehicle, Fxij and Fxij are longitudinal and lateral forces of 4 wheels, and the subscripts ij are fl, fr, rl, and rr, which represent the left front wheel, right front wheel, the left rear wheel, and the right rear wheel; δ is the steering angle of the front wheels; a and b represent the distance from the centroid to the front and rear axes respectively.


#### 2.1.2. Lateral Motion

(2)mVxβ˙+γ=Fxfl+Fxfrsinδ+Fyfl+Fyfrcosδ+Fyrl+Fyrr,where β is the sideslip angle.


#### 2.1.3. Yaw Motion

(3)Izγ˙=aFyflcosδ+Fyfrcosδ+Fxflsinδ+Fxfrsinδ−bFyrl+Fyrr+T2Fxfrcosδ−Fyfrsinδ+Fxrr−Fxflcosδ−Fyflsinδ+Fxrl,where Iz is the yaw moment of inertia of the AGV and T is the track width of the AGV.


#### 2.1.4. Rotational Motion of Each Wheel

(4)Jw⋅dωijdt=Ttij−Tbij−FxijR,where Jw is the wheel inertia moment about the rotary axis, dωij/dt is the wheel angular acceleration calculated by differencing the wheel angular speed ωij, Ttij and Tbij are the driving torque and braking torque on each wheel, and R is the radius of each wheel.The roll dynamics model, as shown in Figure2, gives the roll dynamics characteristic when the wheels do not lift off. Of and Or are roll centers of the front and rear suspension respectively.Figure 2
Roll dynamics model.


#### 2.1.5. Rolling Motion

(5)IxOφ¨=mayhOcosφ+mghOsinφ−Kφ−Cφ˙,where φ is the body roll angle of the vehicle, φ˙ is the roll rate of the vehicle; IxO is the roll inertial moment about roll axis Of and Or, ay is the lateral acceleration of the sprung mass, hO is the height from gravity center to the roll center, hg is the height of roll center, K is the suspension roll stiffness, and C is the suspension damping coefficient.The lateral accelerationay is composed of the derivative of the lateral acceleration and the product of the vehicle speed and the yaw rate. The lateral acceleration can be expressed as:(6)ay=V˙x+Vyγ.


### 2.2. Tire Model

The magic formula fits the tire test data through a combination of trigonometric formulas and expresses the longitudinal force, lateral force, return moment, flip moment, and resistance moment of the tire in a complete way with a set of formulas of the same form. The tire model with the magic formula can be expressed as [21]:(7)yx=DsinCarctanBx−EBx−arctanBx,where y (x) is the lateral force or longitudinal force on the tire; the independent variable x can be expressed as the cornering angle or longitudinal slip rate of the tire; B, C, D, and E are the fitting coefficients, which are determined by the vertical load and camber angle of the tire in turn, where B is the stiffness factor; C is the curve shape factor; D is the peak factor; and E is the curve curvature factor.


### 2.3. Driving Motor Model

The driving motor used in the study is a permanent magnet synchronous motor, and its torque response can be simplified into a second-order delay system [22]. The transfer function is:(8)Gs=TmTm∗=12ξ2s2+2ξs+1,where Tm is the actual output torque of the motor, Tm∗ is the target output torque of the motor, and ξ is the structure parameter of the motor.


## 2.1. Body Model

As shown in Figure1, the 7-DOF dynamics model includes longitudinal motion, lateral motion, roll motion, and four wheels’ rotations. Assumptions: ignore the changes in tire cornering characteristics of the left and right wheels due to load changes; ignore the effects of suspension roll stiffness and damping coefficient; and ignore the aerodynamic effects.Figure 1
Seven degrees of freedom vehicle dynamics model.


### 2.1.1. Longitudinal Motion

(1)mV˙x−Vyγ=Fxfl+Fxfrcosδ−Fyfl+Fyfrsinδ+Fxrl+Fxrr,where m is the mass of AGV, Vx and Vy are the longitudinal and lateral velocities, γ is the yaw rate of the vehicle, Fxij and Fxij are longitudinal and lateral forces of 4 wheels, and the subscripts ij are fl, fr, rl, and rr, which represent the left front wheel, right front wheel, the left rear wheel, and the right rear wheel; δ is the steering angle of the front wheels; a and b represent the distance from the centroid to the front and rear axes respectively.


### 2.1.2. Lateral Motion

(2)mVxβ˙+γ=Fxfl+Fxfrsinδ+Fyfl+Fyfrcosδ+Fyrl+Fyrr,where β is the sideslip angle.


### 2.1.3. Yaw Motion

(3)Izγ˙=aFyflcosδ+Fyfrcosδ+Fxflsinδ+Fxfrsinδ−bFyrl+Fyrr+T2Fxfrcosδ−Fyfrsinδ+Fxrr−Fxflcosδ−Fyflsinδ+Fxrl,where Iz is the yaw moment of inertia of the AGV and T is the track width of the AGV.


### 2.1.4. Rotational Motion of Each Wheel

(4)Jw⋅dωijdt=Ttij−Tbij−FxijR,where Jw is the wheel inertia moment about the rotary axis, dωij/dt is the wheel angular acceleration calculated by differencing the wheel angular speed ωij, Ttij and Tbij are the driving torque and braking torque on each wheel, and R is the radius of each wheel.The roll dynamics model, as shown in Figure2, gives the roll dynamics characteristic when the wheels do not lift off. Of and Or are roll centers of the front and rear suspension respectively.Figure 2
Roll dynamics model.


### 2.1.5. Rolling Motion

(5)IxOφ¨=mayhOcosφ+mghOsinφ−Kφ−Cφ˙,where φ is the body roll angle of the vehicle, φ˙ is the roll rate of the vehicle; IxO is the roll inertial moment about roll axis Of and Or, ay is the lateral acceleration of the sprung mass, hO is the height from gravity center to the roll center, hg is the height of roll center, K is the suspension roll stiffness, and C is the suspension damping coefficient.The lateral accelerationay is composed of the derivative of the lateral acceleration and the product of the vehicle speed and the yaw rate. The lateral acceleration can be expressed as:(6)ay=V˙x+Vyγ.


## 2.1.1. Longitudinal Motion

(1)mV˙x−Vyγ=Fxfl+Fxfrcosδ−Fyfl+Fyfrsinδ+Fxrl+Fxrr,where m is the mass of AGV, Vx and Vy are the longitudinal and lateral velocities, γ is the yaw rate of the vehicle, Fxij and Fxij are longitudinal and lateral forces of 4 wheels, and the subscripts ij are fl, fr, rl, and rr, which represent the left front wheel, right front wheel, the left rear wheel, and the right rear wheel; δ is the steering angle of the front wheels; a and b represent the distance from the centroid to the front and rear axes respectively.


## 2.1.2. Lateral Motion

(2)mVxβ˙+γ=Fxfl+Fxfrsinδ+Fyfl+Fyfrcosδ+Fyrl+Fyrr,where β is the sideslip angle.


## 2.1.3. Yaw Motion

(3)Izγ˙=aFyflcosδ+Fyfrcosδ+Fxflsinδ+Fxfrsinδ−bFyrl+Fyrr+T2Fxfrcosδ−Fyfrsinδ+Fxrr−Fxflcosδ−Fyflsinδ+Fxrl,where Iz is the yaw moment of inertia of the AGV and T is the track width of the AGV.


## 2.1.4. Rotational Motion of Each Wheel

(4)Jw⋅dωijdt=Ttij−Tbij−FxijR,where Jw is the wheel inertia moment about the rotary axis, dωij/dt is the wheel angular acceleration calculated by differencing the wheel angular speed ωij, Ttij and Tbij are the driving torque and braking torque on each wheel, and R is the radius of each wheel.The roll dynamics model, as shown in Figure2, gives the roll dynamics characteristic when the wheels do not lift off. Of and Or are roll centers of the front and rear suspension respectively.Figure 2
Roll dynamics model.


## 2.1.5. Rolling Motion

(5)IxOφ¨=mayhOcosφ+mghOsinφ−Kφ−Cφ˙,where φ is the body roll angle of the vehicle, φ˙ is the roll rate of the vehicle; IxO is the roll inertial moment about roll axis Of and Or, ay is the lateral acceleration of the sprung mass, hO is the height from gravity center to the roll center, hg is the height of roll center, K is the suspension roll stiffness, and C is the suspension damping coefficient.The lateral accelerationay is composed of the derivative of the lateral acceleration and the product of the vehicle speed and the yaw rate. The lateral acceleration can be expressed as:(6)ay=V˙x+Vyγ.


## 2.2. Tire Model

The magic formula fits the tire test data through a combination of trigonometric formulas and expresses the longitudinal force, lateral force, return moment, flip moment, and resistance moment of the tire in a complete way with a set of formulas of the same form. The tire model with the magic formula can be expressed as [21]:(7)yx=DsinCarctanBx−EBx−arctanBx,where y (x) is the lateral force or longitudinal force on the tire; the independent variable x can be expressed as the cornering angle or longitudinal slip rate of the tire; B, C, D, and E are the fitting coefficients, which are determined by the vertical load and camber angle of the tire in turn, where B is the stiffness factor; C is the curve shape factor; D is the peak factor; and E is the curve curvature factor.


## 2.3. Driving Motor Model

The driving motor used in the study is a permanent magnet synchronous motor, and its torque response can be simplified into a second-order delay system [22]. The transfer function is:(8)Gs=TmTm∗=12ξ2s2+2ξs+1,where Tm is the actual output torque of the motor, Tm∗ is the target output torque of the motor, and ξ is the structure parameter of the motor.


## 3. Coupling Analysis of Different Subsystems

### 3.1. Coupling Analysis of Drive System and Suspension System

When the four-wheel drive is used for differential drive control, it is assumed that the drive wheel does not have a sliding phenomenon, and the additional yaw moment provided is:(9)ΔMz=Tfl+Tfrasinδr+Tfr−Tflcosδ+Trr−TrlT2r,where, Tfl, Tfr, Trl, and Trr are the corresponding driving torque of each wheel respectively.The analysis of the force of the reaction torque of the in-wheel motor on the vehicle body and the force characteristics of the suspension guiding rod system is shown in Figure3. In Figure 3(a), CG is the vehicle centroid position; ΔFf, ΔFr, ΔTf, and ΔTr are the ground driving force and reaction moment transferred from the front and rear wheels to the suspension, respectively; ΔP1, ΔP2, ΔP3 and ΔP4 are the force of the body on the front and rear suspensions, respectively; ΔP′1, ΔP′2, ΔP′3, and ΔP′4 are corresponding reaction forces; Cf and Cr are instant centers of motion; δf is the front wheel angle; Fzr and Fzf produce additional vertical force for wheels passing through suspension to the body; Z1, 2, 3, 4 are the corresponding vertical distances from the front and rear suspensions to the wheel center; θ1, 2, 3, 4 are the corresponding angles. In Figure 3(b), ΔFif, of and ΔTif, of are the ground driving force and reaction moment transmitted by the inner and outer driving wheels to the suspension, respectively (where i represents the inner wheel and o represents the outer wheel); ΔFy1 and ΔFy2 are the lateral forces generated by the left and right wheels passing through the suspension; ΔPi1, i2, o1, o2 and ΔP′i1, ′i2, ′o1, ′o2 are the force generated by the vehicle body on the suspension and the reaction force of the suspension on the vehicle body respectively; Qo and Qi are the instantaneous center of roll motion of inner and outer suspension respectively; Zi1, i2, i3, i4 are the vertical distances from the suspension to the center of the inner and outer wheels; δif and δof are the rotation angles of inner and outer front wheels; θi1, i2, o1, o2 are the corresponding angles; and h1, 2 is the corresponding vertical distance.Figure 3
Force analysis of AGV body and suspension. (a) Longitudinal plane force of body and suspension. (b) Transverse plane force of body and suspension.
(a)(b)The body roll control torque generated by the front and rear wheels can be expressed as:(10)ΔMxfl=−T2ΔP1l′sinθ1+ΔP2l′sinθ2,ΔMxfr=T2ΔP1r′sinθ1+ΔP2r′sinθ2,ΔMxrl=T2ΔP3l′sinθ3+ΔP4l′sinθ4,ΔMxrr=−T2ΔP3r′sinθ3+ΔP4r′sinθ4,An additional force of suspension on the body ΔP′1, ΔP′2, ΔP′3, and ΔP′4 can be obtained from the following formula:(11)ΔP1′=−ΔP1=z2ΔFfcosδfcosθ1−ΔTfcosδfcos2θ2cosθ1z1cos2θ1+z2cos2θ1,ΔP2′=−ΔP2=z1ΔFfcosδfcosθ2−Z2ΔTfcosδfcos2θ2cos2θ1z1cos2θ2+z2cos2θ1,ΔP3′=−ΔP3=z3ΔFrcos2θ3cosθ4−Z3ΔTrcosθ1z4cos2θ3+z3cos2θ1,ΔP4′=−ΔP4=z4ΔFrcosθ3−ΔTrcos2θ4cosθ3z4cos2θ3+z3cos2θ4.Taking the left and right connection points of the suspension and the wheel as the base point, it can be obtained according to the force balance:(12)ΔPo1′=zo2ΔFofsinδofcosθo1−ΔTofsinδofcos2θo2cosθo1zo1cos2θo2+zo2cos2θ1,ΔPo2′=−zo1ΔFofsinδfcosθo2−ΔTofsinδofcos2θo1cosθo2zo1cos2θo2+zo2cos2θo1,ΔPi1′=zi2ΔFifsinδifcosθi1−ΔTifsinδifcos2θi2cosθi1zo1cos2θo2+zo2cos2θ1,ΔPi2′=−zi1ΔFifsinδifcosθi2−ΔTifsinδifcos2θi1cosθi2zi1cos2θi2+zi2cos2θi1.The roll control torque generated by the suspension on the transverse plane of the vehicle body is:(13)Mxfo=ΔPo1′cosθo1h1+ΔP02′cosθo2h2,Mxfi=ΔPi1′cosθi1h1−ΔPi2′cosθi2h2.The total roll moment generated by the differential drive is:(14)ΔMxd=ΔMxf+ΔMxr+ΔMxfi+ΔMxfo,where ΔMxd is the total roll moment generated by the differential drive; ΔMxf is the roll control moment generated by the torque difference between the left and right front wheels; ΔMxr is the roll control moment generated by the torque difference between the left and right rear wheels; and ΔMxfi and ΔMxfo are the reaction forces of the front wheels acting on the body through the inner and outer suspensions, respectively.Based on the dynamic coupling analysis of the above drive system and suspension system, the yaw control torque and roll control torque can be generated by independently adjusting the driving torque of each drive wheel to control the yaw and roll motion of the body. In the normal driving process, it will be coupled with the suspension system and affect the roll characteristics of the vehicle. However, the force and torque directly transmitted by the suspension is relatively small, so the ability to control the roll of the vehicle using the differential drive alone is relatively weak, and other systems are required to assist and coordinate control.


### 3.2. Coupling Analysis of Drive System and Steering System

Since the output torque of each wheel of the in-wheel motor is accurate and controllable, the driving torque difference can be generated by controlling the driving force of each steering wheel. The torque around the kingpin generated by the driving torque difference and the offset of the main pin will make the steering wheel rotate around its kingpin, to realize the differential assist steering [23]. The force analysis of driving wheel steering is shown in Figure 4.Figure 4
Force analysis of driving wheel during steering.As shown in Figure4, the driving wheel and the body are connected by the double arm suspension; C and D are the upper and lower points of the main pin respectively. In order to facilitate the analysis, the driving wheel and steering knuckle are regarded as a whole. When the drive torque of the drive motor is Te, the drive torque is an internal torque relative to the overall system and has no effect on the differential steering torque. At this time, the forces on the system include the force Fx on the tire contact point, the force F′x1 on the upper point of the kingpin, and the force F′x2 on the lower point of the kingpin. The rotational moment generated by this force around the kingpin is denoted as Tst which can be expressed as:(15)Tst=Fxrσcosα,where Fx is the force of the ground on the wheel, rσ is the projection of the distance between the intersection of the kingpin extension line and the ground to the wheel grounding center on the y axis, and α is the angle between the kingpin and the z-axis.When the vehicle turns left, to reduce the roll angle to suppress the occurrence of lateral instability, the required roll control torque is negative. The absolute value of differential torque should be added to the left front wheel drive torque, and the absolute value of differential torque should be subtracted from the right front wheel drive torque. At this time, the steering torque is negative. If the front wheel angle is not regulated or the differential torque is limited at this time, insufficient steering will occur, resulting in the vehicle gradually deviating from the expected trajectory. The active steering adjusts the vehicle stability by adjusting the wheel angle, and the power steering effect caused by the differential drive will have a certain impact on the effect of the vehicle steering motor. Since the excitation force of the yaw control and roll control comes from the driving force of each wheel, the coupling between the drive system and the steering system will affect the yaw stability of the vehicle. The premise of roll stability control is to ensure yaw stability. Therefore, we choose the control mode of front wheel active steering and rear wheel differential driving to avoid the coupling effect between the steering system and the driving system affecting the yaw stability. The rear differential drive can reduce the influence on the active steering of the front wheel, and the roll torque generated by the rear differential drive can help adjust the vehicle roll attitude. To reduce the negative effect of the complex coupling between the two, only the front wheel active steering and the rear wheel differential drive are used in this study. Although the control effect of the differential drive is weakened, the yaw stability of the vehicle is ensured first, and a certain space margin is reserved for the control intervention of the load platform.


## 3.1. Coupling Analysis of Drive System and Suspension System

When the four-wheel drive is used for differential drive control, it is assumed that the drive wheel does not have a sliding phenomenon, and the additional yaw moment provided is:(9)ΔMz=Tfl+Tfrasinδr+Tfr−Tflcosδ+Trr−TrlT2r,where, Tfl, Tfr, Trl, and Trr are the corresponding driving torque of each wheel respectively.The analysis of the force of the reaction torque of the in-wheel motor on the vehicle body and the force characteristics of the suspension guiding rod system is shown in Figure3. In Figure 3(a), CG is the vehicle centroid position; ΔFf, ΔFr, ΔTf, and ΔTr are the ground driving force and reaction moment transferred from the front and rear wheels to the suspension, respectively; ΔP1, ΔP2, ΔP3 and ΔP4 are the force of the body on the front and rear suspensions, respectively; ΔP′1, ΔP′2, ΔP′3, and ΔP′4 are corresponding reaction forces; Cf and Cr are instant centers of motion; δf is the front wheel angle; Fzr and Fzf produce additional vertical force for wheels passing through suspension to the body; Z1, 2, 3, 4 are the corresponding vertical distances from the front and rear suspensions to the wheel center; θ1, 2, 3, 4 are the corresponding angles. In Figure 3(b), ΔFif, of and ΔTif, of are the ground driving force and reaction moment transmitted by the inner and outer driving wheels to the suspension, respectively (where i represents the inner wheel and o represents the outer wheel); ΔFy1 and ΔFy2 are the lateral forces generated by the left and right wheels passing through the suspension; ΔPi1, i2, o1, o2 and ΔP′i1, ′i2, ′o1, ′o2 are the force generated by the vehicle body on the suspension and the reaction force of the suspension on the vehicle body respectively; Qo and Qi are the instantaneous center of roll motion of inner and outer suspension respectively; Zi1, i2, i3, i4 are the vertical distances from the suspension to the center of the inner and outer wheels; δif and δof are the rotation angles of inner and outer front wheels; θi1, i2, o1, o2 are the corresponding angles; and h1, 2 is the corresponding vertical distance.Figure 3
Force analysis of AGV body and suspension. (a) Longitudinal plane force of body and suspension. (b) Transverse plane force of body and suspension.
(a)(b)The body roll control torque generated by the front and rear wheels can be expressed as:(10)ΔMxfl=−T2ΔP1l′sinθ1+ΔP2l′sinθ2,ΔMxfr=T2ΔP1r′sinθ1+ΔP2r′sinθ2,ΔMxrl=T2ΔP3l′sinθ3+ΔP4l′sinθ4,ΔMxrr=−T2ΔP3r′sinθ3+ΔP4r′sinθ4,An additional force of suspension on the body ΔP′1, ΔP′2, ΔP′3, and ΔP′4 can be obtained from the following formula:(11)ΔP1′=−ΔP1=z2ΔFfcosδfcosθ1−ΔTfcosδfcos2θ2cosθ1z1cos2θ1+z2cos2θ1,ΔP2′=−ΔP2=z1ΔFfcosδfcosθ2−Z2ΔTfcosδfcos2θ2cos2θ1z1cos2θ2+z2cos2θ1,ΔP3′=−ΔP3=z3ΔFrcos2θ3cosθ4−Z3ΔTrcosθ1z4cos2θ3+z3cos2θ1,ΔP4′=−ΔP4=z4ΔFrcosθ3−ΔTrcos2θ4cosθ3z4cos2θ3+z3cos2θ4.Taking the left and right connection points of the suspension and the wheel as the base point, it can be obtained according to the force balance:(12)ΔPo1′=zo2ΔFofsinδofcosθo1−ΔTofsinδofcos2θo2cosθo1zo1cos2θo2+zo2cos2θ1,ΔPo2′=−zo1ΔFofsinδfcosθo2−ΔTofsinδofcos2θo1cosθo2zo1cos2θo2+zo2cos2θo1,ΔPi1′=zi2ΔFifsinδifcosθi1−ΔTifsinδifcos2θi2cosθi1zo1cos2θo2+zo2cos2θ1,ΔPi2′=−zi1ΔFifsinδifcosθi2−ΔTifsinδifcos2θi1cosθi2zi1cos2θi2+zi2cos2θi1.The roll control torque generated by the suspension on the transverse plane of the vehicle body is:(13)Mxfo=ΔPo1′cosθo1h1+ΔP02′cosθo2h2,Mxfi=ΔPi1′cosθi1h1−ΔPi2′cosθi2h2.The total roll moment generated by the differential drive is:(14)ΔMxd=ΔMxf+ΔMxr+ΔMxfi+ΔMxfo,where ΔMxd is the total roll moment generated by the differential drive; ΔMxf is the roll control moment generated by the torque difference between the left and right front wheels; ΔMxr is the roll control moment generated by the torque difference between the left and right rear wheels; and ΔMxfi and ΔMxfo are the reaction forces of the front wheels acting on the body through the inner and outer suspensions, respectively.Based on the dynamic coupling analysis of the above drive system and suspension system, the yaw control torque and roll control torque can be generated by independently adjusting the driving torque of each drive wheel to control the yaw and roll motion of the body. In the normal driving process, it will be coupled with the suspension system and affect the roll characteristics of the vehicle. However, the force and torque directly transmitted by the suspension is relatively small, so the ability to control the roll of the vehicle using the differential drive alone is relatively weak, and other systems are required to assist and coordinate control.


## 3.2. Coupling Analysis of Drive System and Steering System

Since the output torque of each wheel of the in-wheel motor is accurate and controllable, the driving torque difference can be generated by controlling the driving force of each steering wheel. The torque around the kingpin generated by the driving torque difference and the offset of the main pin will make the steering wheel rotate around its kingpin, to realize the differential assist steering [23]. The force analysis of driving wheel steering is shown in Figure 4.Figure 4
Force analysis of driving wheel during steering.As shown in Figure4, the driving wheel and the body are connected by the double arm suspension; C and D are the upper and lower points of the main pin respectively. In order to facilitate the analysis, the driving wheel and steering knuckle are regarded as a whole. When the drive torque of the drive motor is Te, the drive torque is an internal torque relative to the overall system and has no effect on the differential steering torque. At this time, the forces on the system include the force Fx on the tire contact point, the force F′x1 on the upper point of the kingpin, and the force F′x2 on the lower point of the kingpin. The rotational moment generated by this force around the kingpin is denoted as Tst which can be expressed as:(15)Tst=Fxrσcosα,where Fx is the force of the ground on the wheel, rσ is the projection of the distance between the intersection of the kingpin extension line and the ground to the wheel grounding center on the y axis, and α is the angle between the kingpin and the z-axis.When the vehicle turns left, to reduce the roll angle to suppress the occurrence of lateral instability, the required roll control torque is negative. The absolute value of differential torque should be added to the left front wheel drive torque, and the absolute value of differential torque should be subtracted from the right front wheel drive torque. At this time, the steering torque is negative. If the front wheel angle is not regulated or the differential torque is limited at this time, insufficient steering will occur, resulting in the vehicle gradually deviating from the expected trajectory. The active steering adjusts the vehicle stability by adjusting the wheel angle, and the power steering effect caused by the differential drive will have a certain impact on the effect of the vehicle steering motor. Since the excitation force of the yaw control and roll control comes from the driving force of each wheel, the coupling between the drive system and the steering system will affect the yaw stability of the vehicle. The premise of roll stability control is to ensure yaw stability. Therefore, we choose the control mode of front wheel active steering and rear wheel differential driving to avoid the coupling effect between the steering system and the driving system affecting the yaw stability. The rear differential drive can reduce the influence on the active steering of the front wheel, and the roll torque generated by the rear differential drive can help adjust the vehicle roll attitude. To reduce the negative effect of the complex coupling between the two, only the front wheel active steering and the rear wheel differential drive are used in this study. Although the control effect of the differential drive is weakened, the yaw stability of the vehicle is ensured first, and a certain space margin is reserved for the control intervention of the load platform.


## 4. Rollover Prevention Overall Controller Architecture

To improve the rollover stability of the AGV, this study constructs an active anti-rollover control strategy that integrates differential drive, active steering, and load platform centroid active adjustment control. The architecture of the designed collaborative controller is shown in Figure5.Figure 5
Collaborative control block diagram.As shown in Figure5, βd is the ideal sideslip angle, γd is the ideal yaw rate, φd is the ideal body roll angle, Tdi∗ is the driving torque of normal output; ΔTdi∗ is the distribution value of rear wheel differential torque; Tdi∗∗ is the superposition value of the normal driving force output value and the rear wheel differential torque output value; Δδ∗ is the active front wheel steering angle calculated by the controller; Fxi∗ (i = 1, 2, 3, 4) is the output force of the push rod motor calculated by the controller. Since the differential drive consumes the smallest energy and responds rapidly [24], the control energy consumption of the load platform is the largest. Meanwhile, when the vehicle has no tendency to roll over, the primary task of the cooperative controller is to ensure the yaw stability of the vehicle. The differential drive can ensure the yaw stability of the vehicle by outputting the yaw control torque. From the perspective of energy consumption and safety, the intervention priority of each control system is load platform < active steering < differential drive. The main control process is as follows:(1)
The first step is to calculate the ideal value of each parameter of the vehicle according to the two-degree-of-freedom model and send it to the differential drive controller. The differential drive controller calculates the yaw control torque and controls the vehicle by the output torque difference of the driving wheel. The roll torque generated by the differential drive can assist in adjusting the vehicle roll motion while controlling the body yaw. The differential drive controller takes the yaw rate as the control value. When the yaw rate is less than the ideal value, it is considered that the vehicle is stable at this time, and the driving torqueTdi∗ is directly output to drive the vehicle. Conversely, the vehicle is considered unstable, that is, γ > γd, the torque difference distribution value ΔTdi∗ of the driving torque is calculated based on the feedback optimal control, and the calculation result is superimposed with ΔTdi∗ to generate Tdi∗∗ and sent to the vehicle.(2)
In the second step, the drive torqueTdi∗, the ideal yaw rate γd, and the ideal sideslip angle βd are sent to the active steering controller. When |Tdi∗| T < ΔMzm or β > βd, it is considered that the output yaw control torque is insufficient, and the sideslip angle is greater than the ideal value. At this time the active steering controller begins to intervene. Based on model predictive control, the front wheel active steering compensation angle Δδ∗ is calculated and sent to the vehicle.(3)
The third step is to send the lateral accelerationay and roll angle φ to the load platform controller. When φ > φd, it is considered that the vehicle is unstable at this time, and the load platform begins to intervene in the control. The roll control torque is calculated based on the sliding mode control. At this time, the roll torque ΔMxr generated by the differential drive needs to be considered to obtain the required additional roll control torque and then output the vertical force Fxi through the load platform motor.Each actuator sends the generated driving torqueTdi, active steering angle Δδ, load platform push rod motor driving force Fxi to the vehicle. The cooperative controller obtains the motion state of the vehicle and performs the next calculation to form a new control command, and repeats the command to complete the iterative control.


### 4.1. Determination of Ideal Values of Lateral Stability State Parameters

Assuming that the longitudinal speed and lateral speed of the vehicle are basically unchanged, ignoring the roll, pitch, and vertical motion, the linear 2-DOF reference model is selected as the dynamic prediction model of the MPC controller [25], and its state space equation is.(16)x˙=AX+BU,where,(17)X=βγT,U=δ,A=kf+krmvxakf−bkrmvx2−1−akf−bkrIz−akfIz,B=−kfmvx−akfIz.Considering the adhesion conditions of the road surface and the requirements of vehicle maneuverability and stability, the ideal values of sideslip angleβd and yaw rate γd can be calculated from the 2-DOF vehicle model:(18)βd=minb+mavx2/kra+ba+b1+Kvx2δ,μgbvx2+makra+b,π18sgnb+mavx2/kra+bL1+Kvx2,γd=minγref,μgvx⋅sgnδ,where μ is the road friction coefficient and γref is the nominal value of the yaw rate calculated by the 2-DOF reference model:(19)γd=minvxδa+b1+Kvx2,μgvx⋅sgnδ.The relationship between the ideal value of vehicle body roll angleφd control and the lateral acceleration ay can be expressed as [26]:(20)φd=max0.8ay+2.4,−4ay<−5,0−5≤ay≤5,min0.8ay−2.4,4ay≥5.Commonly used rollover evaluation indicators are load transfer ratio (LTR), critical roll angle, time to rollover (TTR), etc. LTR defines vehicle rollover as the left wheel or right wheel leaving the ground, which has good universality. LTR is defined as:(21)LTR=Fzfl+Fzrl−Fzfr−FzrrFzfl+Fzrl+Fzfr+Fzrr.Considering that the vertical tire forces are not easy to obtain directly, theLTR can also be expressed as [27]:(22)LTR=2Kφ+2Cφ˙mgT,where m is the mass of the AGV; φ is the roll angle of the AGV; φ˙ is the roll rate of the AGV; T is the track width of the AGV; K is the suspension roll stiffness; and C is the suspension damping coefficient.


### 4.2. Design of Differential Drive Controller

Since the stator of the in-wheel motor is rigidly connected to the suspension, the ground driving/braking force will be directly transmitted to the body through the suspension. By coordinating the distribution of the driving torque of the driving wheel, the additional roll control torque that controls the vehicle roll motion can be generated, thus forming a control effect similar to that of the active suspension [28]. When the rear wheel differential drive is adopted, the relationship between yaw moment and yaw motion can be expressed as.(23)Izγ˙=Mz.The space state equation is:(24)X˙1=A1X1+B1U1,Y1=C1X1,A1=1001;B1=1Ix0;C1=1001;X1=γ−γdϕ−ϕd;U1=ΔMzm.where U1 is the yaw control torque ∆Mzm and ϕ is the yaw angle.(25)ϕd=∫0TγddT.Based on the feedback optimal control, the performance indexJ is written in the matrix form.(26)J=limT⟶∞∫0TX1TQX1+U1TRU1+2X1TNU1dt.Bring into the Riccati equation to solve the feedback coefficient matrixK.(27)ΔMzm=−KX1t.The required yaw control torque is ∆Mzm.


### 4.3. Design of Active Steering Controller

Model predictive control has the characteristics of good control effect, strong robustness, and low requirements for model accuracy, and can effectively control complex processes [29]. As shown in Figure 6, the MPC algorithm is used to design the active steering controller. Taking the vehicle’s yaw rate and sideslip angle as the control variables, the lateral stability of the vehicle can be maintained by adjusting the front wheel angle.Figure 6
Flow chart of active steering controller.The system control inputs are yaw rate and sideslip angle, and the control output is the corrected value of the front wheel angle. Ignoring other interference, the system constraint output is:(28)y=C1X,where,(29)C1=1001,X=βγ.The continuous state equation of the system is transformed into the state space increment model of the discrete-time system.(30)Δβk+1Δγk+1=AoΔβkΔk+Bou⋅Δδ,yk=CΔβkΔγk,where,(31)Ao=expAts,Bou=∫0TsexpAτdτBu.To make the ideal controlled input close to the reference input, determine the objective function:(32)J=∑i=1p∑j=1ncΓyj,iβcj,ik+i|k00γcj,ik+i|k−rβ,jk+i|k00rγ,ik+i2+∑i=1mΓu,iΔδk+i−12,where Гu,i is the weighting matrix control input.(33)Γu,i=diagΓu,1,Γu,2,…Γu,pm×m,where rβ, j (k + i) and rγ, j (k + i) are the jth component of the reference output sequence respectively; Γu, i is the weighting matrix of the control input, and the prediction input of the system in the future step P can be expressed as:(34)Yp=k+1|k=SxΔxk+Iyck+SuΔUk,Sx=C1Ao∑i=12C1Ao⋮∑i=1pC1Aoip×1,I=Inc×ncInc×nc⋮Inc×ncp×1,where Inc × nc is the identity matrix and nc is the dimension of the control matrix.Considering the constraints of the control process, the control quantity, control increment, and output quantity of the system meet the following control constraints and output constraints:(35)δmink≤δk≤δmax∀k≥0,Δδmink≤Δδk≤Δδmax∀k≥0,βminkγmink≤βkγk≤βmaxkγmaxk∀k≥0.The constrained optimization problem is transformed into a quadratic programming problem through the numerical solution, and the objective function is at timek:(36)ΔΔUkminUKTHΔUk−Gk+1|kTΔUk,where,(37)H=SuTΓyTΓySu+ΓuTΓuGk+1|k=2SuTΓyTΓyEpk+1|k.Solve for the control input increment ΔU∗(k) at time k, and use the first step Δu∗(k) as the control input.(38)Δuk=Inu×nu0⋯0ΔU∗k,where, Δu ∗ (k) is the front wheel steering angle Δδ to be solved, and the predicted output acts on the system to form a closed-loop control.


### 4.4. Design of Load Platform Controller

The load platform adjusts the body through the left and right side push rod motors outputting vertical torque and thus forming a roll control torque, which works similarly to the active suspension. When the vehicle is in an unstable state after yaw stability control, start the load platform for the roll control. During the roll control of the load platform, the component of the sprung mass in the roll direction is reduced, the component of the whole body in the roll direction is reduced, and the difference between the vertical force borne by the left and right wheels is also reduced compared with before, so that the position of the centroid is adjusted. When the body rolls, the floating platform can produce a control torque opposite to the roll direction as follows:(39)ΔMc=T2∑i=14Fxi,where ΔMxc is the roll control torque of the load platform, which can first adjust the centroid of the platform and reduce the vertical load difference between the two wheels of the axle on the same side, so as to control the roll movement of the whole body and reduce the roll angle.The model tracking error of roll control can be expressed as:(40)e=φd−φ.Establishing sliding mode function.(41)s=ce+e˙,s˙=ce˙+e¨,=cφ˙d−cφ¨+φ¨d−ΔMx−ℑIxs,ℑ=moghosinφ+mohovy˙+Vxγ+Izxsγ˙+T2∑i=1.3Fpi−∑i=2,4Fpi,where Fpi is the reaction force of the push rod motor of the load platform to the vehicle body.Considering the control effect of rear wheel differential drive on a roll, the sliding mode controller is:(42)ΔMxc=Ixscφ˙d+φ¨d−cφ˙+ks+exp−λtεsgns−ΔMxr−ℑ.Take the Lyapunov function.(43)V2=12s2.Then,(44)V˙2=s−ks−exp−λtεsgns.It is not difficult to verify.(45)lims⟶0V˙2≤0.The load platform generates a force opposite to its motion direction while improving the roll of the vehicle body, which will change the motion state of the unsprung mass of the vehicle to a certain extent. If the active force on one side of the push rod motor is upward, the unsprung mass on this side will be subjected to a downward force. Simplify the calculation of the output torque of the linear motors on both sides of the load platform, whenFxi > 0,(46)Fx1Fx3=Fx2Fx4=ba.Then,(47)T2Fx1+Fx3−Fx2−Fx4=ΔMxc,where ΔMxc is the control torque output by each push rod motor of the load platform.


## 4.1. Determination of Ideal Values of Lateral Stability State Parameters

Assuming that the longitudinal speed and lateral speed of the vehicle are basically unchanged, ignoring the roll, pitch, and vertical motion, the linear 2-DOF reference model is selected as the dynamic prediction model of the MPC controller [25], and its state space equation is.(16)x˙=AX+BU,where,(17)X=βγT,U=δ,A=kf+krmvxakf−bkrmvx2−1−akf−bkrIz−akfIz,B=−kfmvx−akfIz.Considering the adhesion conditions of the road surface and the requirements of vehicle maneuverability and stability, the ideal values of sideslip angleβd and yaw rate γd can be calculated from the 2-DOF vehicle model:(18)βd=minb+mavx2/kra+ba+b1+Kvx2δ,μgbvx2+makra+b,π18sgnb+mavx2/kra+bL1+Kvx2,γd=minγref,μgvx⋅sgnδ,where μ is the road friction coefficient and γref is the nominal value of the yaw rate calculated by the 2-DOF reference model:(19)γd=minvxδa+b1+Kvx2,μgvx⋅sgnδ.The relationship between the ideal value of vehicle body roll angleφd control and the lateral acceleration ay can be expressed as [26]:(20)φd=max0.8ay+2.4,−4ay<−5,0−5≤ay≤5,min0.8ay−2.4,4ay≥5.Commonly used rollover evaluation indicators are load transfer ratio (LTR), critical roll angle, time to rollover (TTR), etc. LTR defines vehicle rollover as the left wheel or right wheel leaving the ground, which has good universality. LTR is defined as:(21)LTR=Fzfl+Fzrl−Fzfr−FzrrFzfl+Fzrl+Fzfr+Fzrr.Considering that the vertical tire forces are not easy to obtain directly, theLTR can also be expressed as [27]:(22)LTR=2Kφ+2Cφ˙mgT,where m is the mass of the AGV; φ is the roll angle of the AGV; φ˙ is the roll rate of the AGV; T is the track width of the AGV; K is the suspension roll stiffness; and C is the suspension damping coefficient.


## 4.2. Design of Differential Drive Controller

Since the stator of the in-wheel motor is rigidly connected to the suspension, the ground driving/braking force will be directly transmitted to the body through the suspension. By coordinating the distribution of the driving torque of the driving wheel, the additional roll control torque that controls the vehicle roll motion can be generated, thus forming a control effect similar to that of the active suspension [28]. When the rear wheel differential drive is adopted, the relationship between yaw moment and yaw motion can be expressed as.(23)Izγ˙=Mz.The space state equation is:(24)X˙1=A1X1+B1U1,Y1=C1X1,A1=1001;B1=1Ix0;C1=1001;X1=γ−γdϕ−ϕd;U1=ΔMzm.where U1 is the yaw control torque ∆Mzm and ϕ is the yaw angle.(25)ϕd=∫0TγddT.Based on the feedback optimal control, the performance indexJ is written in the matrix form.(26)J=limT⟶∞∫0TX1TQX1+U1TRU1+2X1TNU1dt.Bring into the Riccati equation to solve the feedback coefficient matrixK.(27)ΔMzm=−KX1t.The required yaw control torque is ∆Mzm.


## 4.3. Design of Active Steering Controller

Model predictive control has the characteristics of good control effect, strong robustness, and low requirements for model accuracy, and can effectively control complex processes [29]. As shown in Figure 6, the MPC algorithm is used to design the active steering controller. Taking the vehicle’s yaw rate and sideslip angle as the control variables, the lateral stability of the vehicle can be maintained by adjusting the front wheel angle.Figure 6
Flow chart of active steering controller.The system control inputs are yaw rate and sideslip angle, and the control output is the corrected value of the front wheel angle. Ignoring other interference, the system constraint output is:(28)y=C1X,where,(29)C1=1001,X=βγ.The continuous state equation of the system is transformed into the state space increment model of the discrete-time system.(30)Δβk+1Δγk+1=AoΔβkΔk+Bou⋅Δδ,yk=CΔβkΔγk,where,(31)Ao=expAts,Bou=∫0TsexpAτdτBu.To make the ideal controlled input close to the reference input, determine the objective function:(32)J=∑i=1p∑j=1ncΓyj,iβcj,ik+i|k00γcj,ik+i|k−rβ,jk+i|k00rγ,ik+i2+∑i=1mΓu,iΔδk+i−12,where Гu,i is the weighting matrix control input.(33)Γu,i=diagΓu,1,Γu,2,…Γu,pm×m,where rβ, j (k + i) and rγ, j (k + i) are the jth component of the reference output sequence respectively; Γu, i is the weighting matrix of the control input, and the prediction input of the system in the future step P can be expressed as:(34)Yp=k+1|k=SxΔxk+Iyck+SuΔUk,Sx=C1Ao∑i=12C1Ao⋮∑i=1pC1Aoip×1,I=Inc×ncInc×nc⋮Inc×ncp×1,where Inc × nc is the identity matrix and nc is the dimension of the control matrix.Considering the constraints of the control process, the control quantity, control increment, and output quantity of the system meet the following control constraints and output constraints:(35)δmink≤δk≤δmax∀k≥0,Δδmink≤Δδk≤Δδmax∀k≥0,βminkγmink≤βkγk≤βmaxkγmaxk∀k≥0.The constrained optimization problem is transformed into a quadratic programming problem through the numerical solution, and the objective function is at timek:(36)ΔΔUkminUKTHΔUk−Gk+1|kTΔUk,where,(37)H=SuTΓyTΓySu+ΓuTΓuGk+1|k=2SuTΓyTΓyEpk+1|k.Solve for the control input increment ΔU∗(k) at time k, and use the first step Δu∗(k) as the control input.(38)Δuk=Inu×nu0⋯0ΔU∗k,where, Δu ∗ (k) is the front wheel steering angle Δδ to be solved, and the predicted output acts on the system to form a closed-loop control.


## 4.4. Design of Load Platform Controller

The load platform adjusts the body through the left and right side push rod motors outputting vertical torque and thus forming a roll control torque, which works similarly to the active suspension. When the vehicle is in an unstable state after yaw stability control, start the load platform for the roll control. During the roll control of the load platform, the component of the sprung mass in the roll direction is reduced, the component of the whole body in the roll direction is reduced, and the difference between the vertical force borne by the left and right wheels is also reduced compared with before, so that the position of the centroid is adjusted. When the body rolls, the floating platform can produce a control torque opposite to the roll direction as follows:(39)ΔMc=T2∑i=14Fxi,where ΔMxc is the roll control torque of the load platform, which can first adjust the centroid of the platform and reduce the vertical load difference between the two wheels of the axle on the same side, so as to control the roll movement of the whole body and reduce the roll angle.The model tracking error of roll control can be expressed as:(40)e=φd−φ.Establishing sliding mode function.(41)s=ce+e˙,s˙=ce˙+e¨,=cφ˙d−cφ¨+φ¨d−ΔMx−ℑIxs,ℑ=moghosinφ+mohovy˙+Vxγ+Izxsγ˙+T2∑i=1.3Fpi−∑i=2,4Fpi,where Fpi is the reaction force of the push rod motor of the load platform to the vehicle body.Considering the control effect of rear wheel differential drive on a roll, the sliding mode controller is:(42)ΔMxc=Ixscφ˙d+φ¨d−cφ˙+ks+exp−λtεsgns−ΔMxr−ℑ.Take the Lyapunov function.(43)V2=12s2.Then,(44)V˙2=s−ks−exp−λtεsgns.It is not difficult to verify.(45)lims⟶0V˙2≤0.The load platform generates a force opposite to its motion direction while improving the roll of the vehicle body, which will change the motion state of the unsprung mass of the vehicle to a certain extent. If the active force on one side of the push rod motor is upward, the unsprung mass on this side will be subjected to a downward force. Simplify the calculation of the output torque of the linear motors on both sides of the load platform, whenFxi > 0,(46)Fx1Fx3=Fx2Fx4=ba.Then,(47)T2Fx1+Fx3−Fx2−Fx4=ΔMxc,where ΔMxc is the control torque output by each push rod motor of the load platform.


## 5. Simulation and Real Vehicle Test Results

### 5.1. Simulation and Experimental Setup

The vehicle accelerates from a flat road to a specified speed according to a predetermined path, drives up the slope with a 60° steering angle when approaching the slope, and then keeps driving straight until it exits the slope. The slope of the slope is 40%, the lateral width of the slope is 6 m, and the longitudinal length is 50 m. The road adhesion coefficient is set to 0.8. Considering that the working speed required by the AGV is not high, the maximum speed is set to 10 m/s. The parameters of the simulation model are set according to the parameters of the real vehicle. The main system parameters of the simulation vehicle are shown in Table1. In the simulation, the influence of road roughness and the interference of lateral wind are ignored.Table 1
Vehicle system parameters.
SymbolName and unitValuemFull load mass (kg)300TTrack width (m)1.0RWheel radius (m)0.23aDistance between centroid and front axis (m)0.76bDistance between centroid and rear axis (m)0.84LWheelbase1.6hgHeight of centroid (m)0.42IωWheel moment of inertia (kg·m2)0.579IzYaw moment of inertia (kg·m2)102IxRoll moment of inertia (kg·m2)39The test model is a small automated guided vehicle developed by the research group. The experimental platform is shown in Figure7. The test sample vehicle is independently developed by the research group and assembled by manual welding. The vehicle adopts a four-wheel independent drive, which is equipped with double wishbone independent suspension at the front and rear, and four-wheel independent steering. The vehicle controller is a DSP controller (Model: TMS320F28335), which is driven by a CAN signal. It also has a built-in inertial navigation system. The GPS is used to locate the vehicle and calculate the vehicle’s speed. The inertial navigation system can obtain information such as the three-axis acceleration, yaw angle, and position of the vehicle in the navigation coordinate system. The external parameters of the AGV are consistent with the simulation. In addition to the difference between the specific size and the simulation model, the system structure is the same as the simulation model.Figure 7
Experimental platform.The vehicle travels along the set trajectory according to the predetermined vehicle speed, and the slope has a buffer angle to avoid instability as soon as the vehicle enters the slope. The test slope condition is shown in Figure8. A dual-axis inclination sensor is used to measure the angular velocity and roll angle. The sensor is installed horizontally on the vehicle and has a built-in Kalman filter fusion algorithm, which can be directly read by software. Figure 8(b) shows the experimental data acquisition system. Through the dedicated software, online monitoring and real-time observation of experimental data can be realized. Since the slip angle parameter is difficult to obtain directly, we adopt the reference proposed method for estimation [30].Figure 8
Test condition and data collection. (a) Slope test condition. (b) Experimental data acquisition system.
(a)(b)In order to reduce the interference of external factors to clearly show the change of the results, the data is processed by filtering. Filters can be used to filter out the high-frequency fluctuations in the data simulation results to make them smooth, and data trends can be observed intuitively. There are external factors such as uneven road surface and lateral wind in the experiment, which are unavoidable factors. There will also be noise interference in the process of collecting data, and the superposition of noise and the original real signal will cause deviations in the collected data. We use high-pass filtering to process the collected data in MATLAB to attenuate signals below 50 Hz.


### 5.2. Simulation Results

The simulation results of vehicle response which include yaw rate, sideslip angle, roll angle, etc. (shown in Figures9–11), are compared to those without control, PI controller, and cooperative controller proposed in this study under different load conditions.Figure 9
Comparison of vehicle control response under different loads by PI controller and cooperative controller. (a) Yaw rate. (b) Sideslip angle. (c) Roll angle. (d) Comparison of high-load and low-load LTR under cooperative control.
(a)(b)(c)(d)Figure 10
Longitudinal velocity. (a) Comparison of PI controller and cooperative controller for longitudinal vehicle speed under 60 kg load condition. (b) Comparison of PI controller and cooperative controller for longitudinal vehicle speed under 120 kg load condition.
(a)(b)Figure 11
Reference differential torque and actual differential torque under different loads. (a) 60 kg load. (b) 120 kg load.
(a)(b)According to the simulation results, we can find that: without control, the yaw rate of the vehicle fluctuates obviously after 1 s, the sideslip angle and vehicle body roll angle will increase rapidly, the vehicle rollover occurs in 2.4 s, and the simulation stops. The results in Figure7 show that the PI controller and cooperative controller can control the yaw rate of the vehicle. The cooperative controller can control the vehicle roll, which can be successfully completed 5 s simulation. Compared with the PI controller, the sideslip angle decreases by 14.62% and the roll angle decreases by 12.38% under high load condition, the sideslip angle decreases by 18.24% and the roll angle decreases by 14.93% under low load condition. Figure 9(d) shows the comparison of LTR values of AGV under high load and low load conditions. The lowest LTR value reaches -0.76 under low load condition, and the lowest LTR value reaches -0.84 under high load condition, this indicates that no rollover occurred. Compared with the control effect of AGV under high load condition and low load condition, the response jitter of vehicle stability parameters under high load condition is larger, and the increase in load affects the stability of the vehicle. Under both high load and low load conditions, the vehicle stability parameters are controlled within a safe range, which proves that the proposed cooperative control strategy can ensure the safety and stability of the AGV when driving sideways on a slope. Compared with the low load condition, the vehicle response jitter is larger under the high load condition, indicating that different load conditions will affect the stability of the vehicle. The higher the load, the greater the probability of vehicle instability. The comparison between the reference output value of differential torque and the actual output value is shown in Figure 9. From the change of vehicle speed in Figure 10, it can be seen that the insufficient output torque of the differential drive causes the vehicle speed to drop, and the impact under high load is obvious. The differential torque reference value of the driving motor is larger than the actual allowable value of the motor. According to the results of other control parameters, it can be found that the vehicle does not lose stability, and the actuator does not fail, indicating that the actuator saturation has not yet led to the divergence of the control system. Meanwhile, when the vehicle is running sideways on a slope, the longitudinal force is large, and the lateral force that the tire can provide will be reduced, resulting in a sideslip on the slope. Therefore, it is necessary to further consider the impact of the vehicle sideslip, and the output torque of the motor must also be constrained during actual control.


### 5.3. Real Test Results

The load in the first experiment is 60 kg, and the load in the second experiment is 120 kg. The comparison between the results of the real vehicle test and simulation is shown in Figures12–15.Figure 12
Comparison of experimental results and simulation results of PI controller and cooperative controller for yaw rate under different loads. (a) 60 kg load PI controller. (b) 120 kg load PI controller. (c) 60 kg load cooperative controller. (d) 120 kg load cooperative controller.
(a)(b)(c)(d)Figure 13
Comparison of experimental results and simulation results of PI controller and cooperative controller for sideslip angle under different loads. (a) 60 kg load PI controller. (b) 120 kg load PI controller. (c) 60 kg load cooperative controller. (d) 120 kg load cooperative controller.
(a)(b)(c)(d)Figure 14
Comparison of experimental results and simulation results of PI controller and cooperative controller for roll angle under different loads. (a) 60 kg load PI controller. (b) 120 kg load PI controller. (c) 60 kg load cooperative controller. (d) 120 kg load cooperative controller.
(a)(b)(c)(d)Figure 15
Comparison of reference steering angle and control output under different loads. (a) 60 kg load. (b) 120 kg load.
(a)(b)As shown in Figures12–14, under high load and low load conditions, the cooperative controller and the PI controller can track the ideal sideslip angle and body roll angle, and achieve the desired control effect. The experimental results are consistent with the simulation results. At the same time, the effect of the cooperative controller is better than that of the PI controller. Figure 15 shows the comparison between the reference steering angle and the actual control output under different loads. When the vehicle runs uncontrolled, it deviates from the predetermined trajectory very quickly, and the auxiliary regulation of active steering in collaborative control enables the vehicle to better track the predetermined trajectory. Although there is an error between the reference control and the actual control output, the error is small, within 7.8%. Under cooperative control, the vehicle can maintain a normal driving state, and the vehicle does not lose stability. The real vehicle test results are consistent with the simulation results. The test results can prove that the combined control can control the vehicle’s yaw rate, sideslip angle, body roll angle, and other parameters, and reduce the risk of vehicle rollover.


## 5.1. Simulation and Experimental Setup

The vehicle accelerates from a flat road to a specified speed according to a predetermined path, drives up the slope with a 60° steering angle when approaching the slope, and then keeps driving straight until it exits the slope. The slope of the slope is 40%, the lateral width of the slope is 6 m, and the longitudinal length is 50 m. The road adhesion coefficient is set to 0.8. Considering that the working speed required by the AGV is not high, the maximum speed is set to 10 m/s. The parameters of the simulation model are set according to the parameters of the real vehicle. The main system parameters of the simulation vehicle are shown in Table1. In the simulation, the influence of road roughness and the interference of lateral wind are ignored.Table 1
Vehicle system parameters.
SymbolName and unitValuemFull load mass (kg)300TTrack width (m)1.0RWheel radius (m)0.23aDistance between centroid and front axis (m)0.76bDistance between centroid and rear axis (m)0.84LWheelbase1.6hgHeight of centroid (m)0.42IωWheel moment of inertia (kg·m2)0.579IzYaw moment of inertia (kg·m2)102IxRoll moment of inertia (kg·m2)39The test model is a small automated guided vehicle developed by the research group. The experimental platform is shown in Figure7. The test sample vehicle is independently developed by the research group and assembled by manual welding. The vehicle adopts a four-wheel independent drive, which is equipped with double wishbone independent suspension at the front and rear, and four-wheel independent steering. The vehicle controller is a DSP controller (Model: TMS320F28335), which is driven by a CAN signal. It also has a built-in inertial navigation system. The GPS is used to locate the vehicle and calculate the vehicle’s speed. The inertial navigation system can obtain information such as the three-axis acceleration, yaw angle, and position of the vehicle in the navigation coordinate system. The external parameters of the AGV are consistent with the simulation. In addition to the difference between the specific size and the simulation model, the system structure is the same as the simulation model.Figure 7
Experimental platform.The vehicle travels along the set trajectory according to the predetermined vehicle speed, and the slope has a buffer angle to avoid instability as soon as the vehicle enters the slope. The test slope condition is shown in Figure8. A dual-axis inclination sensor is used to measure the angular velocity and roll angle. The sensor is installed horizontally on the vehicle and has a built-in Kalman filter fusion algorithm, which can be directly read by software. Figure 8(b) shows the experimental data acquisition system. Through the dedicated software, online monitoring and real-time observation of experimental data can be realized. Since the slip angle parameter is difficult to obtain directly, we adopt the reference proposed method for estimation [30].Figure 8
Test condition and data collection. (a) Slope test condition. (b) Experimental data acquisition system.
(a)(b)In order to reduce the interference of external factors to clearly show the change of the results, the data is processed by filtering. Filters can be used to filter out the high-frequency fluctuations in the data simulation results to make them smooth, and data trends can be observed intuitively. There are external factors such as uneven road surface and lateral wind in the experiment, which are unavoidable factors. There will also be noise interference in the process of collecting data, and the superposition of noise and the original real signal will cause deviations in the collected data. We use high-pass filtering to process the collected data in MATLAB to attenuate signals below 50 Hz.


## 5.2. Simulation Results

The simulation results of vehicle response which include yaw rate, sideslip angle, roll angle, etc. (shown in Figures9–11), are compared to those without control, PI controller, and cooperative controller proposed in this study under different load conditions.Figure 9
Comparison of vehicle control response under different loads by PI controller and cooperative controller. (a) Yaw rate. (b) Sideslip angle. (c) Roll angle. (d) Comparison of high-load and low-load LTR under cooperative control.
(a)(b)(c)(d)Figure 10
Longitudinal velocity. (a) Comparison of PI controller and cooperative controller for longitudinal vehicle speed under 60 kg load condition. (b) Comparison of PI controller and cooperative controller for longitudinal vehicle speed under 120 kg load condition.
(a)(b)Figure 11
Reference differential torque and actual differential torque under different loads. (a) 60 kg load. (b) 120 kg load.
(a)(b)According to the simulation results, we can find that: without control, the yaw rate of the vehicle fluctuates obviously after 1 s, the sideslip angle and vehicle body roll angle will increase rapidly, the vehicle rollover occurs in 2.4 s, and the simulation stops. The results in Figure7 show that the PI controller and cooperative controller can control the yaw rate of the vehicle. The cooperative controller can control the vehicle roll, which can be successfully completed 5 s simulation. Compared with the PI controller, the sideslip angle decreases by 14.62% and the roll angle decreases by 12.38% under high load condition, the sideslip angle decreases by 18.24% and the roll angle decreases by 14.93% under low load condition. Figure 9(d) shows the comparison of LTR values of AGV under high load and low load conditions. The lowest LTR value reaches -0.76 under low load condition, and the lowest LTR value reaches -0.84 under high load condition, this indicates that no rollover occurred. Compared with the control effect of AGV under high load condition and low load condition, the response jitter of vehicle stability parameters under high load condition is larger, and the increase in load affects the stability of the vehicle. Under both high load and low load conditions, the vehicle stability parameters are controlled within a safe range, which proves that the proposed cooperative control strategy can ensure the safety and stability of the AGV when driving sideways on a slope. Compared with the low load condition, the vehicle response jitter is larger under the high load condition, indicating that different load conditions will affect the stability of the vehicle. The higher the load, the greater the probability of vehicle instability. The comparison between the reference output value of differential torque and the actual output value is shown in Figure 9. From the change of vehicle speed in Figure 10, it can be seen that the insufficient output torque of the differential drive causes the vehicle speed to drop, and the impact under high load is obvious. The differential torque reference value of the driving motor is larger than the actual allowable value of the motor. According to the results of other control parameters, it can be found that the vehicle does not lose stability, and the actuator does not fail, indicating that the actuator saturation has not yet led to the divergence of the control system. Meanwhile, when the vehicle is running sideways on a slope, the longitudinal force is large, and the lateral force that the tire can provide will be reduced, resulting in a sideslip on the slope. Therefore, it is necessary to further consider the impact of the vehicle sideslip, and the output torque of the motor must also be constrained during actual control.


## 5.3. Real Test Results

The load in the first experiment is 60 kg, and the load in the second experiment is 120 kg. The comparison between the results of the real vehicle test and simulation is shown in Figures12–15.Figure 12
Comparison of experimental results and simulation results of PI controller and cooperative controller for yaw rate under different loads. (a) 60 kg load PI controller. (b) 120 kg load PI controller. (c) 60 kg load cooperative controller. (d) 120 kg load cooperative controller.
(a)(b)(c)(d)Figure 13
Comparison of experimental results and simulation results of PI controller and cooperative controller for sideslip angle under different loads. (a) 60 kg load PI controller. (b) 120 kg load PI controller. (c) 60 kg load cooperative controller. (d) 120 kg load cooperative controller.
(a)(b)(c)(d)Figure 14
Comparison of experimental results and simulation results of PI controller and cooperative controller for roll angle under different loads. (a) 60 kg load PI controller. (b) 120 kg load PI controller. (c) 60 kg load cooperative controller. (d) 120 kg load cooperative controller.
(a)(b)(c)(d)Figure 15
Comparison of reference steering angle and control output under different loads. (a) 60 kg load. (b) 120 kg load.
(a)(b)As shown in Figures12–14, under high load and low load conditions, the cooperative controller and the PI controller can track the ideal sideslip angle and body roll angle, and achieve the desired control effect. The experimental results are consistent with the simulation results. At the same time, the effect of the cooperative controller is better than that of the PI controller. Figure 15 shows the comparison between the reference steering angle and the actual control output under different loads. When the vehicle runs uncontrolled, it deviates from the predetermined trajectory very quickly, and the auxiliary regulation of active steering in collaborative control enables the vehicle to better track the predetermined trajectory. Although there is an error between the reference control and the actual control output, the error is small, within 7.8%. Under cooperative control, the vehicle can maintain a normal driving state, and the vehicle does not lose stability. The real vehicle test results are consistent with the simulation results. The test results can prove that the combined control can control the vehicle’s yaw rate, sideslip angle, body roll angle, and other parameters, and reduce the risk of vehicle rollover.


## 6. Conclusions

(1)
This study analyzes the rollover control effect of differential drive, active steering, and load platform, and conducts a co-simulation of the control effect. The simulation results show that the control method based on the coordination of the three can control the vehicle stability parameters in an ideal range and reduce the rollover instability probability. Compared with the low load condition, the vehicle response jitter is larger under the high load condition, indicating that different load conditions will affect the stability of the vehicle. Compared with the PI controller, the slip angle decreases by 14.62% and 18.24%, and the roll angle decreases by 12.38% and 14.93% under high and low load conditions, respectively. The error between the experimental results and the simulation results is within 7.8%, which verifies the effectiveness of the proposed control strategy.(2)
Compared with the single subsystem controller, the collaborative controller is complex, but the current calculation speed of the central controller can meet the calculation ability requirements. In order to meet the control requirements under fully critical instability conditions, the control accuracy and response speed of the system need to be further improved. At the same time, due to the influence of actuator saturation and vehicle sideslip, the differential torque needs to be further optimized. In addition, the intervention of the steering system during the stability control may lead to the deviation of vehicle trajectory, so the combination of rollover control and trajectory tracking control can be considered. Related research is the content of the next work.



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*Source: 9331756-2022-09-17.xml*